Package {metaConvert}

metaConvert: An R Package Dedicated to Automated Effect Size Calculations

Description

The metaConvert package automatically estimates 14 effect size measures from a well-formatted dataframe. Various other functions can help, for example, removing dependency between several effect sizes, or identifying differences between two dataframes. This package is mainly designed to assist in conducting a systematic review with a meta-analysis, but it can be useful to any researcher interested in estimating an effect size.

Overview of the package

To visualize all the types of input data that can be used to estimate the 14 effect size measures available in metaConvert, you can use the see_input_data() function.

Estimate effect sizes

To automatically estimate effect sizes directly from a dataset, you can use the convert_df() function.

Aggregate dependent effect sizes

To automatically aggregate dependent effect sizes using Borenstein's formulas, you can use the aggregate_df() function. This function can handle dependent effect sizes from multiple subgroups, or dependent effect sizes from the same participants.

Flag differences between two datasets

If pairs of data extractors have generated similar datasets that should be compared, you can use the compare_df() function.

Prepare a dataset extraction sheet

If you have not started data extraction yet, you can use the data_extraction_sheet() function to obtain a perfectly formatted data extraction sheet.

Well-formatted dataset

One of the specificities of the metaConvert package is that its core function (convert_df) does not have arguments to specify the names of the variables contained in the dataset. While this allow using a convenient automatic process in the calculations, this requires that the datasets passed to this function respect a very precise formatting (which we will refer to as well-formatted dataset).

Rather than a long description of all column names, we built several tools that help you find required information.

1. You can use the data_extraction_sheet() function that generates an excel/csv/txt file containing all the column names available, as well as a description of the information it should contain.

2. You can use the see_input_data() function that generates a list of all available types of input data as well as their estimated/converted effect size measures. This function also points out to the corresponding helper tables available in https://metaconvert.org

Effect size measures available

Eleven effect size measures are accepted:

• "d": standardized mean difference (i.e., Cohen's d)

• "g": Hedges' g

• "md": mean difference

• "r": Correlation coefficient

• "z": Fisher's r-to-z correlation

• "or" or "logor": odds ratio or its logarithm

• "rr" or "logrr": risk ratio or its logarithm

• "irr" or "logirr": incidence rate ratio or its logarithm

• "nnt": number needed to treat

• "logcvr": log coefficient of variation

• "logvr": log variability ratio

Output

All the functions of the metaConvert package that are dedicated to effect size calculations (i.e., all the functions named es_from_*) return a dataframe that contain, depending on the function - some of the following columns:

Author(s)

Maintainer: Corentin J. Gosling corentin.gosling@parisnanterre.fr

Authors:

• Corentin J. Gosling corentin.gosling@parisnanterre.fr

• Samuele Cortese samuele.cortese@gmail.com

• Marco Solmi marco.solmi83@gmail.com

• Belen Haza hazabelen@gmail.com

• Eduard Vieta EVIETA@clinic.cat

• Richard Delorme richard.delorme.rdb@gmail.com

• Paolo Fusar-Poli Paolo.fusar-poli@kcl.ac.uk

• Joaquim Radua radua@clinic.cat

Aggregate a dataframe containing dependent effect sizes

Description

Aggregate a dataframe containing dependent effect sizes

Usage

aggregate_df(
  x,
  dependence = "outcomes",
  cor_unit = 0.8,
  agg_fact,
  es = "es",
  se = "se",
  col_mean = NA,
  col_weighted_mean = NA,
  weights = NA,
  col_sum = NA,
  col_min = NA,
  col_max = NA,
  col_fact = NA,
  na.rm = TRUE
)

Arguments

Details

1. In the dependence argument, you should indicate "outcomes" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced from the same participants at the same time-point (e.g., multiple outcome measures)

2. In the dependence argument, you should indicate "times" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced from the same participants at the different time-points (e.g., an RCT with several follow-up waves). This option requires a time_agg column (the time-point of each effect size) and a cor_unit column giving the within-cluster correlation (constant within each cluster; falls back to the cor_unit argument if the column is absent).

3. In the dependence argument, you should indicate "subgroups" if the dependence within the same clustering unit (e.g., study) is due to the presence of multiple effect sizes produced by independent subgroups (e.g., one effect size for boys, and one for girls).

If you are working with ratio measures, make sure that the information on the effect size estimates (i.e., the column passed to the es argument of the function) is presented on the log scale.

Value

The object returned by the aggregate_df contains, is a dataframe containing at the very least, the aggregating factor, and the aggregated effect size values and standard errors. All columns indicated in the col_* arguments will also be included in this dataframe.

Examples

res <- summary(convert_df(df.haza, measure = "d"))
aggregate_df(res, dependence = "outcomes", cor_unit = 0.8,
             agg_fact = "study_id", es = "es_crude", se = "se_crude",
             col_fact = c("outcome", "type_publication"))

Convert a “metaConvert” object to a dataframe

Description

Convert a “metaConvert” object to a dataframe

Usage

## S3 method for class 'metaConvert'
as.data.frame(x, ...)

Arguments

Details

Convenience wrapper around summary.metaConvert(x, ...).

Value

A dataframe containing the effect size results, diagnostics, and all input data columns.

See Also

summary.metaConvert

Examples

### get the full output as a dataframe
as.data.frame(convert_df(df.haza, measure = "g"))

Flag the differences between two dataframes.

Description

Flag the differences between two dataframes.

Usage

compare_df(
  df_extractor_1,
  df_extractor_2,
  ordering_columns = NULL,
  tolerance = 0,
  tolerance_type = "ratio",
  output = "html",
  file_name = "comparison.xlsx"
)

Arguments

Details

This function aims to facilitate the comparison of two datasets created by blind data extractors during a systematic review. It is a wrapper of several functions from the 'compareDF' package.

Value

This function returns a dataframe composed of the rows that include a difference (all identical rows are removed). Several outputs can be requested :

1. setting output="xlsx" returns an excel file. A message indicates the location of the generated file on your computer.

2. setting output="html" returns an html file

3. setting output="html2" returns an html file (only useful when the "html" command did not make the html pane appear in R studio).

4. setting output="wide" a wide dataframe

5. setting output="long" a long dataframe

References

Alex Joseph (2022). compareDF: Do a Git Style Diff of the Rows Between Two Dataframes with Similar Structure. R package version 2.3.3. https://CRAN.R-project.org/package=compareDF

Examples

df.compare1 = df.compare1[order(df.compare1$author), ]
df.compare2 = df.compare2[order(df.compare2$year), ]

compare_df(
  df_extractor_1 = df.compare1,
  df_extractor_2 = df.compare2,
  ordering_columns = c("author", "year")
)

Compute Smallest Detectable Change (SDC) from SEM

Description

Compute Smallest Detectable Change (SDC) from SEM

Usage

compute_sdc(sem, sem_se)

Arguments

Details

Computes the smallest detectable change (SDC) from the standard error of measurement:

SDC = 1.96 \times \sqrt{2} \times SEM

The SDC represents the smallest change in scores that can be detected beyond measurement error with 95% confidence. If sem_se is provided, the uncertainty in SDC is propagated:

SDC\_se = 1.96 \times \sqrt{2} \times SEM\_se

Value

A data.frame with the SDC, its standard error and 95% CI.

References

de Vet, H. C. W., Terwee, C. B., Mokkink, L. B., & Knol, D. L. (2011). Measurement in Medicine: A Practical Guide. Cambridge University Press.

Examples

sem_res <- compute_sem(sd = 10, icc = 0.85, n_sample = 100)
compute_sdc(sem = sem_res$sem, sem_se = sem_res$sem_se)

Compute Standard Error of Measurement (SEM) from reliability and SD

Description

Compute Standard Error of Measurement (SEM) from reliability and SD

Usage

compute_sem(sd, icc, n_sample, icc_se, n_measurements = 2)

Arguments

Details

Computes the standard error of measurement (SEM) from a reliability coefficient and the standard deviation of scores:

SEM = SD \times \sqrt{1 - ICC}

The sampling variance of SEM is computed in one of two ways:

1. When icc_se is supplied, the ICC is treated as an external estimate, independent of the within-sample SD, and the bivariate delta method (with zero covariance) is used:

   Var(SEM) = \frac{SD^2}{4(1 - ICC)} \times Var(ICC) + (1 - ICC) \times Var(SD)

   with Var(ICC) = icc\_se^2 and Var(SD) \approx SD^2 / (2(n-1)).

2. When icc_se is not supplied, the ICC and SD are assumed to come from the same reliability sample, where they are strongly positively correlated. Because SEM = SD\sqrt{1 - ICC} equals the within-subject residual root-mean-square \sqrt{MSE}, and MSE / \sigma_e^2 follows a scaled chi-square with (n-1)(k-1) degrees of freedom, the exact sampling variance is

   Var(SEM) = \frac{SEM^2}{2(n-1)(k-1)}

   where k is the number of measurement occasions/raters (n_measurements, default 2 for test-retest). Treating the same-sample SD and ICC as independent (the delta method of case 1) would overestimate this variance by roughly 2-6x, so the exact form is used instead.

The 95% CI is likewise regime-specific. In the same-sample regime (case 2) the degrees of freedom df = (n - 1)(k - 1) are known, so the exact chi-square interval

[SEM \sqrt{df / \chi^2_{0.975, df}},\; SEM \sqrt{df / \chi^2_{0.025, df}}]

is returned; the symmetric Wald interval undercovers at small df (89.7% instead of 95% at n = 10, k = 2). When icc_se is supplied (case 1) the df behind the external estimate is unknown, so the Wald interval SEM \pm 1.96 \times SE, truncated at 0, is kept.

Degenerate inputs: n_measurements = 1 makes the same-sample df zero, so the SE and CI are returned as NA with a warning; an ICC > 1 is impossible and yields NA for the SEM, its SE and CI (with a warning).

Value

A data.frame with the SEM, its standard error and 95% CI.

References

Weir, J. P. (2005). Quantifying test-retest reliability using the intraclass correlation coefficient and the SEM. Journal of Strength and Conditioning Research, 19(1), 231-240.

Examples

compute_sem(sd = 10, icc = 0.85, n_sample = 100)

Automatically compute effect sizes from a well formatted dataset

Description

Automatically compute effect sizes from a well formatted dataset

Usage

convert_df(
  x,
  measure = c("d", "g", "md", "logor", "logrr", "logirr", "nnt", "rd", "r", "z", "rp",
    "zp", "logvr", "logcvr"),
  main_es = TRUE,
  es_selected = c("auto", "hierarchy", "minimum", "maximum"),
  selection_auto = c("crude", "paired", "adjusted"),
  split_adjusted = TRUE,
  format_adjusted = c("wide", "long"),
  verbose = TRUE,
  max_asymmetry = 50,
  hierarchy = "means_sd > means_se > means_ci",
  table_2x2_to_cor = "tetrachoric",
  rr_to_or = "metaumbrella",
  or_to_rr = "metaumbrella_cases",
  or_to_cor = "bonett",
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  smd_denom = "pooled",
  pre_post_to_smd = "bonett",
  r_pre_post = 0.8,
  cor_to_smd = "viechtbauer",
  unit_type = "raw_scale",
  yates_chisq = FALSE,
  pool_sd = FALSE,
  prop_to_es = "raw",
  alpha_to_es = "bonett",
  icc_to_es = "bonett",
  correct_inputs = TRUE,
  flag_options = list()
)

Arguments

Details

This function automatically computes or converts between 14 effect sizes measures from any relevant type of input data stored in the dataset you pass to this function.

Input validation

Before the calculations, the input data are checked (negative sample sizes/SD/SE/p-values, inverted CI bounds, values outside their CI, asymmetric CI). By default invalid values are set to NA; issues appear in the 'es_flags' column of summary().

Effect size measures

Possible effect size measures are:

1. Cohen's d ("d")

2. Hedges' g ("g")

3. mean difference ("md")

4. (log) odds ratio ("or" and "logor")

5. (log) risk ratio ("rr" and "logrr")

6. (log) incidence rate ratio ("irr" and "logirr")

7. correlation coefficient ("r")

8. transformed r-to-z correlation coefficient ("z")

9. partial correlation coefficient ("rp")

10. Fisher's z of partial correlation ("zp")

11. log variability ratio ("logvr")

12. log coefficient of variation ("logcvr")

13. number needed to treat ("nnt")

14. risk difference ("rd")

Computation of a main effect size

If you enter multiple types of input data (e.g., means/sd of two groups and a student t-test value) for the same comparison i.e., for the same row of the dataset, the convert_df() function can have two behaviours. If you set:

• main_es = FALSE the function will estimate all possible effect sizes from all types of input data (which implies that if a comparison has several types of input data, it will result in multiple rows in the dataframe returned by the function)

• main_es = TRUE the function will select one effect size per comparison (which implies that if a comparison has several types of input data, it will result in a unique row in the dataframe returned by the function)

Selection of input data for the computation of the main effect size

If you choose to estimate one main effect size (i.e., by setting main_es = TRUE), you have several options to select this main effect size. If you set:

• es_selected = "auto": the main effect size will be automatically selected, by prioritizing specific types of input data over other (see next section "Hierarchy").

• es_selected = "hierarchy": the main effect size will be selected, by prioritizing specific types of input data over other (see next section "Hierarchy").

• es_selected = "minimum": the main effect size will be selected, by selecting the lowest effect size available.

• es_selected = "maximum": the main effect size will be selected, by selecting the highest effect size available.

Hierarchy

More than 70 different combinations of input data can be used to estimate an effect size. You can retrieve the effect size measures estimated by each combination of input data in the see_input_data() function and online https://metaconvert.org/input.html.

You have two options to use a hierarchy in the types of input data.

• an automatic way (es_selected = "auto")

• an manual way (es_selected = "hierarchy")

Automatic

If you select an automatic hierarchy, here are the types of input data that will be prioritized.

Crude SMD or MD (measure=c("d", "g", "md") and selection_auto="crude")

1. User's input effect size value

2. SMD value

3. Means at post-test

4. ANOVA/Student's t-test/point biserial correlation statistics

5. Linear regression estimates

6. Mean difference values

7. Quartiles/median/maximum values

8. Post-test means extracted from a plot

9. Pre-test+post-test means or mean change

10. Paired ANOVA/t-test statistics

11. Odds ratio value

12. Contingency table

13. Correlation coefficients

14. Phi/chi-square value

Paired SMD or MD (measure=c("d", "g", "md") and selection_auto="paired")

1. User's input effect size value

2. Paired SMD value

3. Pre-test+post-test means or mean change

4. Paired ANOVA/t-test statistics

5. Means at post-test

6. ANOVA/Student's t-test/point biserial correlation

7. Linear regression estimates

8. Mean difference values

9. Quartiles/median/maximum values

10. Odds ratio value

11. Contingency table

12. Correlation coefficients

13. Phi/chi-square value

Adjusted SMD or MD (measure=c("d", "g", "md") and selection_auto="adjusted")

1. User's input adjusted effect size value

2. Adjusted SMD value

3. Estimated marginal means from ANCOVA

4. F- or t-test value from ANCOVA

5. Adjusted mean difference from ANCOVA

6. Estimated marginal means from ANCOVA extracted from a plot

Odds Ratio (measure=c("or"))

1. User's input effect size value

2. Odds ratio value

3. Contingency table

4. Risk ratio values

5. Phi/chi-square value

6. Correlation coefficients

7. (Then hierarchy as for "d" or "g" option crude)

Risk Ratio (measure=c("rr"))

1. User's input effect size value

2. Risk ratio values

3. Contingency table

4. Odds ratio values

5. Phi/chi-square value

Incidence rate ratio (measure=c("irr"))

1. User's input effect size value

2. Number of cases and time of disease free observation time

Correlation (measure=c("r", "z"))

1. User's input effect size value

2. Correlation coefficients

3. Contingency table

4. Odds ratio value

5. Phi/chi-square value

6. SMD value

7. Means at post-test

8. ANOVA/Student's t-test/point biserial correlation

9. Linear regression estimates

10. Mean difference values 11 Quartiles/median/maximum values

11. Post-test means extracted from a plot

12. Pre-test+post-test means or mean change

13. Paired ANOVA/t-test

Variability ratios (measure=c("vr", "cvr"))

1. User's input effect size value

2. means/variability indices at post-test

3. means/variability indices at post-test extracted from a plot

Number needed to treat (measure=c("nnt"))

1. User's input effect size value

2. Contingency table

3. Odds ratio values

4. Risk ratio values

5. Incidence rate ratio (person-time NNT, requires baseline_rate)

6. Phi/chi-square value

NNT values should not be pooled directly (the NNT confidence interval is disjoint when the risk difference crosses zero). You should pool RD/OR/RR values and convert the pooled estimate to NNT (Deeks, 2002; Cochrane Handbook, Chapter 15).

Risk difference (measure=c("rd"))

1. User's input effect size value

2. Contingency table

3. Odds ratio values

4. Risk ratio values

5. Incidence rate ratio (requires baseline_rate)

6. Phi/chi-square value

Manual

If you select a manual hierarchy, you can specify the order in which you want to use each type of input data. You can prioritize some types of input data by placing them at the begining of the hierarchy argument, and you must separate all input data with a ">" separator. For example, if you set:

• hierarchy = "means_sd > means_se > student_t", the convert_df function will prioritize the means + SD, then the means + SE, then the Student's t-test to estimate the main effect size.

• hierarchy = "2x2 > or_se > phi", the convert_df function will prioritize the contingency table, then the odds ratio value + SE, then the phi coefficient to estimate the main effect size.

Importantly, if none of the types of input data indicated in the hierarchy argument can be used to estimate the target effect size measure, the convert_df() function will automatically try to use other types of input data to estimate an effect size.

Adjusted effect sizes

Some datasets will be composed of crude (i.e., non-adjusted) types of input data (such as standard means + SD, Student's t-test, etc.) and adjusted types of input data (such as means + SE from an ANCOVA model, a t-test from an ANCOVA, etc.).

In these situations, you can decide to:

• treat crude and adjusted input data the same way split_adjusted = FALSE

• split calculations for crude and adjusted types of input data split_adjusted = TRUE

If you want to split the calculations, you can decide to present the final dataset:

• in a long format (i.e., crude and adjusted effect sizes presented in separate rows format_adjusted = "long")

• in a wide format (i.e., crude and adjusted effect sizes presented in separate columns format_adjusted = "wide")

Value

The convert_df() function returns a list of more than 70 dataframes (one for each function automatically applied to the dataset). These dataframes systematically contain the columns described in metaConvert-package. The list of dataframes can be easily converted to a single, calculations-ready dataframe using the summary function (see summary.metaConvert).

Examples

res <- convert_df(df.haza,
  measure = "g",
  split_adjusted = TRUE,
  es_selected = "minimum",
  format_adjusted = "long"
)
summary(res)

Data extraction sheet generator

Description

Data extraction sheet generator

Usage

data_extraction_sheet(
  measure = c("d", "g", "md", "dw", "gw", "mdw", "or", "rr", "nnt", "rd", "r", "z",
    "logvr", "logcvr", "irr", "prop"),
  type_of_measure = c("natural", "natural+converted"),
  name = "mcv_data_extraction",
  extension = c("data.frame", ".txt", ".csv", ".xlsx"),
  verbose = TRUE
)

Arguments

Details

This function generates, on your computer, a data extraction sheet that contains the name of columns that can be used by our tools to estimate various effect size measures.

If you select a specific measure (e.g., measure = "g"), you will be presented only with most common information allowing to estimate this measure (e.g., you will not be provided with columns for contingency tables if you request a data extraction sheet for measure = "g").

Measure

You can specify a specific effect size measures (among those available in the metaConvert-package). Doing this, the data extraction sheet will contain only the columns of the input data allowing a natural estimation of the effect size measure. For example, if you request measure="d" the data extraction sheet will not contain the columns for the contingency table since, although the convert_df function allows you to convert a contingency table into a "d", this requires to convert the "OR" that is naturally estimated from the contingency table into a "d".

This table is designed to be used in combination with tables showing the combination of input data leading to estimate each of the effect size measures (https://metaconvert.org/html/input.html)

Extension

You can export a file in various formats outside R (by indicating, for example, ".txt", ".xlsx", or ".csv") in the extension argument. You can also visualise this dataset directly in R by setting extension = "data.frame".

Value

This function returns a data extraction sheet that contains all the information necessary to estimate any effect size using the metaConvert tools.

Examples

data_extraction_sheet(measure = "md", extension = "data.frame")

Fictitious dataset 1

Description

First fictitious dataset aiming to understand how the compare_df function works. Slightly different from df.compare1

Usage

df.compare1

Format

An object of class data.frame with 5 rows and 7 columns.

Fictitious dataset 2

Description

First fictitious dataset aiming to understand how the compare_df function works. Slightly different from df.compare2

Usage

df.compare2

Format

An object of class data.frame with 6 rows and 7 columns.

Meta-analytic dataset inspired from Haza and colleagues (2024)

Description

Dataset of a meta-analysis exploring the specificity of social functioning of children with ADHD (compared to healthy controls) in case-control studies. This dataset contains: 1. several information coming from the same participants (due to the completion of multiple outcomes). 1. several information coming from the same study (due to the presence of multiple subgroups). 1. overlapping information for the same comparison 1. several information types from which a standardized mean difference can be estimated/converted

Usage

df.haza

Format

An object of class data.frame with 170 rows and 106 columns.

Source

Haza B, Gosling CJ, Conty L & Pinabiaux C (2024). Social Functioning in Children and Adolescents with ADHD: A Meta-analysis. Journal of Child Psychology and Psychiatry and Allied Disciplines.

Simulated dataset for a COSMIN-based systematic review of PROMs

Description

Simulated dataset of a systematic review evaluating measurement properties of a fictional patient-reported outcome measure (the Mental Health Wellbeing Scale, MHWS) following the COSMIN framework. Designed to demonstrate the psychometric functions of the metaConvert package.

Usage

df.psychom

Format

An object of class data.frame with 30 rows and 25 columns.

Details

The dataset contains studies reporting:

• Internal consistency (Cronbach's alpha)

• Test-retest reliability (ICC)

• Criterion and construct validity (Pearson and Spearman correlations)

• Responsiveness (change-score correlations)

• Floor/ceiling effects (proportions)

Additional columns support standalone psychometric utility functions (SEM, SDC, disattenuation, change-score reliability).

Short version of the df.haza dataset

Description

This dataset is a shoter version of the df.haza dataset.

Usage

df.short

Format

An object of class grouped_df (inherits from tbl_df, tbl, data.frame) with 37 rows and 109 columns.

Source

Haza B, Gosling CJ, Conty L & Pinabiaux C (2024). Social Functioning in Children and Adolescents with ADHD: A Meta-analysis. Journal of Child Psychology and Psychiatry and Allied Disciplines.

Disattenuate (correct for unreliability) a correlation coefficient

Description

Disattenuate (correct for unreliability) a correlation coefficient

Usage

es_disattenuate(r, r_se, reliability_x, reliability_y, n_sample)

Arguments

Details

Corrects an observed correlation for attenuation due to measurement error in both measures, using the classical disattenuation formula (Hunter & Schmidt, 2004; Spearman, 1904):

r_c = \frac{r_{obs}}{\sqrt{rel_x \times rel_y}}

where rel_x and rel_y are reliability coefficients for the two measures (e.g., Cronbach's alpha, test-retest ICC).

The standard error of the corrected correlation is approximated as:

r_c\_se = \frac{r\_se}{\sqrt{rel_x \times rel_y}}

This approximation treats the reliabilities as known constants, so the SE and CI are a lower bound when the reliabilities are themselves estimated (Hunter & Schmidt, 2004, Ch. 3).

When r_se is not supplied, it is first obtained from the sample size as the large-sample Pearson correlation SE \sqrt{(1 - r^2)^2 / (n - 1)} (Cooper et al., 2019).

The function also provides Fisher's z transformation of the corrected correlation for use in meta-analysis, with SE(z_c) = SE(r_c) / (1 - r_c^2) (delta method). The corrected-r confidence interval is obtained by back-transforming the Fisher-z interval (\tanh), so it always lies within (-1, 1).

When the corrected correlation is extreme (|r_c| > 0.999, including the mathematically impossible |r_c| \ge 1 that inconsistent inputs produce), no meaningful Fisher's z or CI exists: the corrected-r CI and all Fisher's z outputs (z_corrected, z_corrected_se and its CI bounds) are set to NA and a warning is emitted. The corrected point estimate and its first-order standard error are always returned as computed, so the offending inputs can be inspected.

This function is typically applied to the results of summary(convert_df(..., measure = "r")):

res <- summary(convert_df(my_data, measure = "r"))
corrected <- es_disattenuate(
  r = res$es, r_se = res$se,
  reliability_x = my_data$rel_target,
  reliability_y = my_data$rel_comparator,
  n_sample = my_data$n_sample
)

Value

A data.frame with the corrected correlation and its Fisher's z transformation (with their standard errors and 95% CIs), and the attenuation factor.

References

Hunter, J. E., & Schmidt, F. L. (2004). Methods of Meta-Analysis: Correcting Error and Bias in Research Findings (2nd ed.). Sage Publications.

Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72-101.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_disattenuate(r = 0.50, r_se = 0.05,
                reliability_x = 0.85, reliability_y = 0.80,
                n_sample = 100)

# Only correct for one measure's unreliability (set other to 1)
es_disattenuate(r = 0.50, r_se = 0.05,
                reliability_x = 0.85, reliability_y = 1.0,
                n_sample = 100)

Convert a 2x2 table into several effect size measures

Description

Convert a 2x2 table into several effect size measures

Usage

es_from_2x2(
  n_cases_exp,
  n_cases_nexp,
  n_controls_exp,
  n_controls_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_2x2
)

Arguments

Details

This function first computes (log) odds ratio (OR), (log) risk ratio (RR) and number needed to treat (NNT) from the 2x2 table. Note that if a cell is equal to 0, we applied the typical adjustment (add 0.5) to all cells. This adjustment is used for the OR/RR only; the RD and NNT are obtained from the raw cell counts. Cohen's d (D), Hedges' g (G) and correlation coefficients (R/Z) are then estimated from the OR.

To estimate an OR, the formulas used (Box 6.4.a in the Cochrane Handbook) are:

logor = log(\frac{n\_cases\_exp / n\_cases\_nexp}{n\_controls\_exp / n\_controls\_nexp})

logor\_se = \sqrt{\frac{1}{n\_cases\_exp} + \frac{1}{n\_cases\_nexp} + \frac{1}{n\_controls\_exp} + \frac{1}{n\_controls\_nexp}}

To estimate an RR, the formulas used (Box 6.4.a in the Cochrane Handbook) are:

logrr = log(\frac{n\_cases\_exp / n\_exp}{n\_cases\_nexp / n\_nexp})

logrr\_se = \sqrt{\frac{1}{n\_cases\_exp} - \frac{1}{n\_exp} + \frac{1}{n\_cases\_nexp} - \frac{1}{n\_nexp}}

To estimate a risk difference (RD) and NNT, the formulas used are (Wen et al., 2005; Altman, 1998):

pt = \frac{n\_cases\_exp}{n\_cases\_exp + n\_controls\_exp}

pc = \frac{n\_cases\_nexp}{n\_cases\_nexp + n\_controls\_nexp}

rd = pc - pt

rd\_se = \sqrt{\frac{pt(1-pt)}{n\_exp} + \frac{pc(1-pc)}{n\_nexp}}

nnt = \frac{1}{rd}

nnt\_se = \frac{rd\_se}{rd^2}

Note that NNT confidence intervals are set to NA when the RD confidence interval crosses zero (discontinuous CI; Altman, 1998).

Direction convention. The risk difference is defined as rd = pc - pt (control risk minus exposed risk), so a POSITIVE RD means the control group has the higher risk. This is the OPPOSITE direction to the OR and RR produced from the same 2x2 table, which are exposed-over-non-exposed (an OR/RR > 1 means the exposed group has the higher risk). Consequently, for the same table, a protective exposure yields OR < 1, RR < 1 but RD > 0; keep this in mind when pooling RD alongside OR/RR, and use reverse_2x2 if you need to align the directions.

To convert the 2x2 table into a SMD, the function estimates an OR value from the 2x2 table (formula above) that is then converted to a SMD (see formula in es_from_or_se()).

To convert the 2x2 table into a correlation coefficient, For now, only the tetrachoric correlation is currently proposed

• table_2x2_to_cor = "tetrachoric". Given the heavy calculations required for this effect size measure, we relied on the implementation of the formulas of the 'metafor' package. More information can be retrieved here (https://wviechtb.github.io/metafor/reference/escalc.html#-b-measures-for-two-dichotomous-variables).

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Available from www.training.cochrane.org/handbook.

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Sedgwick, P. (2013). What is number needed to treat (NNT)? Bmj, 347.

Altman, D. G. (1998). Confidence intervals for the number needed to treat. BMJ, 317(7168), 1309-1312.

Wen, S., Zhang, L., & Yang, B. (2005). Two approaches to incorporate clinical data uncertainty into number needed to treat. Journal of Clinical Pharmacy and Therapeutics, 30(2), 105-109.

Examples

es_from_2x2(n_cases_exp = 467, n_cases_nexp = 22087, n_controls_exp = 261, n_controls_nexp = 8761)

Convert the proportion of occurrence of a binary event in two independent groups into several effect size measures

Description

Convert the proportion of occurrence of a binary event in two independent groups into several effect size measures

Usage

es_from_2x2_prop(
  prop_cases_exp,
  prop_cases_nexp,
  n_exp,
  n_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_prop
)

Arguments

Details

This function uses the proportions and sample size to recreate the 2x2 table, and then relies on the calculations of the es_from_2x2_sum() function.

The formulas used is to obtain the 2x2 table are

n\_cases\_exp = prop\_cases\_exp * n\_exp

n\_cases\_nexp = prop\_cases\_nexp * n\_nexp

n\_controls\_exp = (1 - prop\_cases\_exp) * n\_exp

n\_controls\_nexp = (1 - prop\_cases\_nexp) * n\_nexp

Value

This function estimates and converts between several effect size measures.

Examples

es_from_2x2_prop(prop_cases_exp = 0.80, prop_cases_nexp = 0.60, n_exp = 10, n_nexp = 20)

Convert a table with the number of cases and row marginal sums into several effect size measures

Description

Convert a table with the number of cases and row marginal sums into several effect size measures

Usage

es_from_2x2_sum(
  n_cases_exp,
  n_exp,
  n_cases_nexp,
  n_nexp,
  table_2x2_to_cor = "tetrachoric",
  reverse_2x2
)

Arguments

Details

This function uses the number of cases in both the exposed and non-exposed groups and the total number of participants exposed and non-exposed to recreate a 2x2 table. Then relies on the calculations of the es_from_2x2 function.

n\_controls\_exp = n\_exp - n\_cases\_exp

n\_controls\_nexp = n\_nexp - n\_cases\_nexp

Value

This function estimates and converts between several effect size measures.

Examples

es_from_2x2_sum(n_cases_exp = 10, n_exp = 40, n_cases_nexp = 25, n_nexp = 47)

Convert a F-statistic obtained from an ANCOVA model into several effect size measures.

Description

Convert a F-statistic obtained from an ANCOVA model into several effect size measures.

Usage

es_from_ancova_f(
  ancova_f,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_f
)

Arguments

Details

This function first computes an "adjusted" Cohen's d (D), and Hedges' g (G) from the F-value of an ANCOVA (binary predictor). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a Cohen's d the formula used is (table 12.3 in Cooper):

cohen\_d = \sqrt{ancova\_f * \frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp}} * \sqrt{1 - cov\_out\_cor^2}

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_f(ancova_f = 4, cov_outcome_r = 0.2, n_cov_ancova = 3, n_exp = 20, n_nexp = 20)

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Description

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Usage

es_from_ancova_f_pval(
  ancova_f_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_f_pval
)

Arguments

Details

This function converts the p-value of an ANCOVA (binary predictor) into a t value, and then relies on the calculations of the es_from_ancova_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.3 in Cooper):

df = n\_exp + n\_nexp - 2 - n\_cov\_ancova

t = | qt(ancova\_f\_pval/2, df = df) |

Then, calculations of the es_from_ancova_t() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_f_pval(
  ancova_f_pval = 0.05, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_ci(
  ancova_md,
  ancova_md_ci_lo,
  ancova_md_ci_up,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_md
)

Arguments

Details

This function converts the mean difference (MD) 95% CI into a standard error, and then relies on the calculations of the es_from_ancova_md_se function.

To convert the 95% CI into a standard error, the following formula is used (table 12.3 in Cooper):

md\_se = \frac{ancova\_md\_ci\_up - ancova\_md\_ci\_lo}{(2 * qt(0.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova))}

Calculations of the es_from_ancova_md_se() are then applied.

Value

This function estimates and converts between several effect size measures.

Examples

es_from_ancova_md_ci(
  ancova_md = 4, ancova_md_ci_lo = 2,
  ancova_md_ci_up = 6,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_pval(
  ancova_md,
  ancova_md_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_md
)

Arguments

Details

This function converts the mean difference (MD) p-value into a standard error, and then relies on the calculations of the es_from_ancova_md_se() function.

To convert the p-value into a standard error, the following formula is used (table 12.3 in Cooper):

t = qt(p = \frac{ancova\_md\_pval}{2}, df = n\_exp + n\_nexp - 2 - n\_cov\_ancova)

ancova\_md\_se = | \frac{ancova\_md}{t} |

Calculations of the es_from_ancova_md_se() are then applied.

Value

This function estimates and converts between several effect size measures.

Examples

es_from_ancova_md_pval(
  ancova_md = 4, ancova_md_pval = 0.05,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and adjusted standard deviation between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_sd(
  ancova_md,
  ancova_md_sd,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_md
)

Arguments

Details

This function first computes an "adjusted" Cohen's d (D), Hedges' g (G) from the adjusted mean difference (MD). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the SE of the adjusted MD:

md\_se = ancova\_md\_sd * \sqrt{\frac{1}{n\_exp} + \frac{1}{n\_nexp}}

md\_lo = md - md\_se * qt(.975, n\_exp + n\_nexp-2-n\_cov\_ancova)

md\_up = md + md\_se * qt(.975, n\_exp + n\_nexp-2-n\_cov\_ancova)

To recover the unadjusted pooled SD (for Cohen's d, table 12.3 in Cooper):

md\_sd = \frac{ancova\_md\_sd}{\sqrt{1 - cor\_outcome\_r^2}}

To estimate the Cohen's d (table 12.3 in Cooper):

d = \frac{ancova\_md}{md\_sd}

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

Note

The standardized effect size variance assumes the covariate is balanced across groups (Cooper eq. 12.26) and treats cov_outcome_r as known; it omits the covariate-imbalance ("leverage") term of the exact ANCOVA variance, which cannot be recovered from summary statistics. In observational or otherwise covariate-imbalanced designs the standard error is therefore a lower bound (anti-conservative), while the point estimate remains unbiased. See Lai and Kelley (2012).

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65(2), 350-370.

Examples

es_from_ancova_md_sd(
  ancova_md = 4, ancova_md_sd = 2,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

Convert an adjusted mean difference and standard error between two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert an adjusted mean difference and standard error between two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_md_se(
  ancova_md,
  ancova_md_se,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_md
)

Arguments

Details

This function converts the mean difference (MD) standard error into a standard deviation, and then relies on the calculations of the es_from_ancova_md_sd function.

To convert the standard error into a standard deviation, the following formula is used.

ancova\_md\_sd = \frac{ancova\_md\_se}{\sqrt{1 / n_exp + 1 / n_nexp}}

Calculations of the es_from_ancova_md_sd() are then applied.

Value

This function estimates and converts between several effect size measures.

Examples

es_from_ancova_md_se(
  ancova_md = 4, ancova_md_se = 2,
  cov_outcome_r = 0.5, n_cov_ancova = 5,
  n_exp = 20, n_nexp = 22
)

Convert means and 95% CIs of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and 95% CIs of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_ci(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_ci_lo_exp,
  ancova_mean_ci_up_exp,
  ancova_mean_nexp,
  ancova_mean_ci_lo_nexp,
  ancova_mean_ci_up_nexp,
  cov_outcome_r,
  n_cov_ancova,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_means
)

Arguments

Details

This function converts the adjusted means 95% CI of two independent groups into a standard error, and then relies on the calculations of the es_from_ancova_means_se() function.

To convert the 95% CIs into standard errors, the following formula is used (table 12.3 in Cooper):

ancova\_mean\_se\_exp = \frac{ancova\_mean\_ci\_up\_exp - ancova\_mean\_ci\_lo\_exp}{2 * qt(0.975, df = n\_exp + n\_nexp - 2 - n\_cov\_ancova)}

ancova\_mean\_se\_nexp = \frac{ancova\_mean\_ci\_up\_nexp - ancova\_mean\_ci\_lo\_nexp}{2 * qt(0.975, df = n\_exp + n\_nexp - 2 - n\_cov\_ancova)}

Calculations of the es_from_ancova_means_se() are then applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_ci(
  n_exp = 55, n_nexp = 55, cov_outcome_r = 0.5, n_cov_ancova = 4,
  ancova_mean_exp = 25, ancova_mean_ci_lo_exp = 15, ancova_mean_ci_up_exp = 35,
  ancova_mean_nexp = 18, ancova_mean_ci_lo_nexp = 12, ancova_mean_ci_up_nexp = 24
)

Convert means and standard deviations of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and standard deviations of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_sd(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_sd_exp,
  ancova_mean_sd_nexp,
  cov_outcome_r,
  n_cov_ancova,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_means
)

Arguments

Details

This function first computes an "adjusted" mean difference (MD), Cohen's d (D) and Hedges' g (G) from the adjusted means and standard deviations. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

This function start by estimating the non-adjusted standard deviation of the two groups (formula 12.24 in Cooper);

mean\_sd\_exp = \frac{ancova\_mean\_sd\_exp}{\sqrt{1 - cov\_outcome\_r^2}}

mean\_sd\_nexp = \frac{ancova\_mean\_sd\_nexp}{\sqrt{1 - cov\_outcome\_r^2}}

To obtain the mean difference, the following formulas are used (authors calculations):

md = ancova\_mean\_exp - ancova\_mean\_nexp

md\_se = \sqrt{\frac{ancova\_mean\_sd\_exp^2}{n\_exp} + \frac{ancova\_mean\_sd\_nexp^2}{n\_nexp}}

md\_ci\_lo = md - md\_se * qt(.975, n\_exp+n\_nexp-2-n\_cov\_ancova)

md\_ci\_up = md + md\_se * qt(.975, n\_exp+n\_nexp-2-n\_cov\_ancova)

To obtain the Cohen's d, the following formulas are used (table 12.3 in Cooper):

mean\_sd\_pooled = \sqrt{\frac{(n\_exp - 1) * ancova\_mean\_sd\_exp^2 + (n\_nexp - 1) * ancova\_mean\_sd\_nexp^2}{n\_exp+n\_nexp-2}}

cohen\_d = \frac{ancova\_mean\_exp - ancova\_mean\_nexp}{mean\_sd\_pooled}

cohen\_d\_se = \sqrt{\frac{(n\_exp+n\_nexp)*(1-cov\_outcome\_r^2)}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2(n\_exp+n\_nexp)}}

cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova)

cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, n\_exp + n\_nexp - 2 - n\_cov\_ancova)

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

Note

The sampling variance follows Cooper's eq. 12.26 and assumes the covariate is balanced across groups. It omits the covariate-imbalance ("leverage") term of the exact ANCOVA variance, \sigma^2_{res}\,(1/n\_exp + 1/n\_nexp + (\bar{x}\_exp-\bar{x}\_nexp)^2 / SS_x), which is not recoverable from summary statistics, and treats cov_outcome_r as known. In balanced/randomised designs the omission is negligible; in observational or otherwise covariate-imbalanced designs the standard error is a lower bound (anti-conservative), while the point estimate remains unbiased. See Lai and Kelley (2012).

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65(2), 350-370.

Examples

es_from_ancova_means_sd(
  n_exp = 55, n_nexp = 55,
  ancova_mean_exp = 2.3, ancova_mean_sd_exp = 1.2,
  ancova_mean_nexp = 1.9, ancova_mean_sd_nexp = 0.9,
  cov_outcome_r = 0.2, n_cov_ancova = 3
)

Convert means and adjusted pooled standard deviation of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and adjusted pooled standard deviation of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_sd_pooled_adj(
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_sd_pooled,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_means
)

Arguments

Details

This function converts the adjusted pooled standard deviations of two independent groups into a crude pooled standard deviation. and then relies on the calculations of the es_from_ancova_means_sd_pooled_crude() function.

To convert the adjusted pooled SD into a crude pooled SD (table 12.3 in Cooper):

mean\_sd\_pooled = \frac{ancova\_mean\_sd\_pooled}{\sqrt{1 - cov\_outcome\_r^2}}

Calculations of the es_from_ancova_means_sd_pooled_crude() are then applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_sd_pooled_adj(
  ancova_mean_exp = 98, ancova_mean_nexp = 87,
  ancova_mean_sd_pooled = 17, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

Convert adjusted means obtained from an ANCOVA model and crude pooled standard deviation of two independent groups into several effect size measures

Description

Convert adjusted means obtained from an ANCOVA model and crude pooled standard deviation of two independent groups into several effect size measures

Usage

es_from_ancova_means_sd_pooled_crude(
  ancova_mean_exp,
  ancova_mean_nexp,
  mean_sd_pooled,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_means
)

Arguments

Details

This function first computes an "adjusted" mean difference (MD) and Cohen's d (D) from the adjusted means and crude pooled standard deviation of two independent groups. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the Cohen's d:

d = \frac{ancova\_mean\_exp - ancova\_mean\_nexp\_adj}{mean\_sd\_pooled}

To estimate the mean difference:

md = ancova\_mean\_exp - ancova\_mean\_nexp\_adj

md\_se = \sqrt{\frac{n\_exp + n\_nexp}{n\_exp * n\_nexp} * (1 - cov\_outcome\_r^2) * mean\_sd\_pooled^2}

Then, calculations of the es_from_ancova_means_sd() and es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_means_sd_pooled_crude(
  ancova_mean_exp = 29, ancova_mean_nexp = 34,
  mean_sd_pooled = 7, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

Convert means and standard errors of two independent groups obtained from an ANCOVA model into several effect size measures

Description

Convert means and standard errors of two independent groups obtained from an ANCOVA model into several effect size measures

Usage

es_from_ancova_means_se(
  n_exp,
  n_nexp,
  ancova_mean_exp,
  ancova_mean_nexp,
  ancova_mean_se_exp,
  ancova_mean_se_nexp,
  cov_outcome_r,
  n_cov_ancova,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_means
)

Arguments

Details

This function converts the adjusted means standard errors of two independent groups into standard deviations, and then relies on the calculations of the es_from_ancova_means_sd function.

To convert the standard errors into standard deviations, the following formula is used.

ancova\_mean\_sd\_exp = ancova\_mean\_se\_exp * \sqrt{n\_exp}

ancova\_mean\_sd\_nexp = ancova\_mean\_se\_nexp * \sqrt{n\_nexp}

Calculations of the es_from_ancova_means_sd() are then applied.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_ancova_means_se(
  n_exp = 55, n_nexp = 55,
  ancova_mean_exp = 2.3, ancova_mean_se_exp = 1.2,
  ancova_mean_nexp = 1.9, ancova_mean_se_nexp = 0.9,
  cov_outcome_r = 0.2, n_cov_ancova = 3
)

Convert a t-statistic obtained from an ANCOVA model into several effect size measures.

Description

Convert a t-statistic obtained from an ANCOVA model into several effect size measures.

Usage

es_from_ancova_t(
  ancova_t,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_t
)

Arguments

Details

This function first computes an "adjusted" Cohen's d (D), and Hedges' g (G) from the t-value of an ANCOVA (binary predictor). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a Cohen's d the formula used is (table 12.3 in Cooper):

cohen\_d = ancova\_t* \sqrt{\frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp}}\sqrt{1 - cov\_out\_cor^2}

To estimate other effect size measures, Calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

Note

The Cohen's d point estimate is recovered from the reported ANCOVA t (which already embeds the covariate adjustment), but its sampling variance is rebuilt from Cooper's eq. 12.26, which assumes a covariate balanced across groups and treats cov_outcome_r as known. In observational or otherwise covariate-imbalanced designs the standard error is a lower bound (anti-conservative), while the point estimate remains unbiased. See Lai and Kelley (2012).

References

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65(2), 350-370.

Examples

es_from_ancova_t(ancova_t = 2, cov_outcome_r = 0.2, n_cov_ancova = 3, n_exp = 20, n_nexp = 20)

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Description

Convert a two-tailed p-value of an ANCOVA t-test into several effect size measures.

Usage

es_from_ancova_t_pval(
  ancova_t_pval,
  cov_outcome_r,
  n_cov_ancova,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_ancova_t_pval
)

Arguments

Details

This function converts the p-value of an ANCOVA (binary predictor) into a t value, and then relies on the calculations of the es_from_ancova_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.3 in Cooper):

df = n\_exp + n\_nexp - 2 - n\_cov\_ancova

t = | qt(ancova\_t\_pval/2, df = df) |

Then, calculations of the es_from_ancova_t() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_ancova_t_pval(
  ancova_t_pval = 0.05, cov_outcome_r = 0.2,
  n_cov_ancova = 3, n_exp = 20, n_nexp = 20
)

Convert a one-way independent ANOVA F-value to several effect size measures

Description

Convert a one-way independent ANOVA F-value to several effect size measures

Usage

es_from_anova_f(
  anova_f,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_anova_f
)

Arguments

Details

This function converts the F-value (one-way, binary predictor) into a t-value, and then relies on the calculations of the es_from_student_t() function.

To convert the F-value into a t-value, the following formula is used (table 12.1 in Cooper):

student\_t = \sqrt{anova\_f}

Then, calculations of the es_from_student_t() are applied.

Important - single numerator degree of freedom only. The identity \sqrt{F} = |t| holds only when the F-test has a single numerator degree of freedom, i.e. a one-way ANOVA comparing exactly two groups (a binary predictor). Supplying an omnibus F from a factor with three or more levels (numerator df > 1) produces a meaningless effect size and is not detected by the function. In addition, \sqrt{F} discards the sign of the effect, so the generated effect sizes are always non-negative; use reverse_anova_f to encode direction.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_anova_f(anova_f = 2.01, n_exp = 20, n_nexp = 22)

Convert a p-value from a one-way independent ANOVA to several effect size measures

Description

Convert a p-value from a one-way independent ANOVA to several effect size measures

Usage

es_from_anova_pval(
  anova_f_pval,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_anova_f_pval
)

Arguments

Details

This function converts the p-value from the F-value of an ANOVA (one-way, binary predictor) into a t-value, and then relies on the calculations of the es_from_student_t() function.

To convert the p-value into a t-value, the following formula is used (table 12.1 in Cooper):

student\_t = qt(\frac{anova\_f\_pval}{2}, df = n\_exp + n\_nexp - 2)

Then, calculations of the es_from_student_t() are applied.

As for es_from_anova_f, this conversion is valid only for an F-test with a single numerator degree of freedom (a two-group comparison): the p-value of a multi-level (numerator df > 1) omnibus F is inverted here as if it were a two-sided two-group t p-value, which is incorrect. The two-sided p-value also carries no direction, so the generated effect sizes are always non-negative; use reverse_anova_f_pval to encode the correct sign.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_anova_pval(anova_f_pval = 0.0012, n_exp = 20, n_nexp = 22)

Convert a standardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Description

Convert a standardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Usage

es_from_beta_std(
  beta_std,
  sd_dv,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_beta_std
)

Arguments

Details

This function converts a standardized linear regression coefficient (coming from a model with only one binary predictor), into an unstandardized linear regression coefficient.

sd\_dummy = \sqrt{\frac{n_exp - (n_exp^2 / (n_exp + n_nexp))}{(n_exp + n_nexp - 1)}}

unstd\_beta = beta\_std * \frac{sd\_dv}{sd\_dummy}

Calculations of the es_from_beta_unstd functions are then used.

Value

This function estimates and converts between several effect size measures.

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_beta_std(beta_std = 0.35, sd_dv = 0.98, n_exp = 20, n_nexp = 22)

Convert an unstandardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Description

Convert an unstandardized regression coefficient and the standard deviation of the dependent variable into several effect size measures

Usage

es_from_beta_unstd(
  beta_unstd,
  sd_dv,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_beta_unstd
)

Arguments

Details

This function estimates a Cohen's d (D) and Hedges' g (G) from an unstandardized linear regression coefficient (coming from a model with only one binary predictor), and the standard deviation of the dependent variable. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

The formula used to obtain the Cohen's d is:

N = n\_exp + n\_nexp

sd\_pooled = \sqrt{\frac{sd\_dv^2 * (N - 1) - unstd\_beta^2 * \frac{n\_exp * n\_nexp}{N}}{N - 2}}

cohen\_d = \frac{unstd\_beta}{sd\_pooled}

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_beta_unstd(beta_unstd = 0.7, sd_dv = 0.98, n_exp = 20, n_nexp = 22)

Convert the number of cases and the person-time of disease-free observation in two independent groups into an incidence rate ratio (IRR)

Description

Convert the number of cases and the person-time of disease-free observation in two independent groups into an incidence rate ratio (IRR)

Usage

es_from_cases_time(
  n_cases_exp,
  n_cases_nexp,
  time_exp,
  time_nexp,
  baseline_rate,
  reverse_irr
)

Arguments

Details

This function estimates the incidence rate ratio from the number of cases and the person-time of disease-free observation in two independent groups.

The formula used to obtain the IRR and its standard error are (Cochrane Handbook (section 6.7.1):

logirr = log(\frac{n\_cases\_exp / time\_exp}{n\_cases\_nexp / time\_nexp)}

logirr\_se = \sqrt{\frac{1}{n\_cases\_exp} + \frac{1}{n\_cases\_nexp}}

To estimate a person-time NNT (Mayne et al., 2006), the following formulas are used:

ird = baseline\_rate \times (1 - irr)

nnt = \frac{1}{ird}

where ird is the incidence rate difference and baseline_rate is the incidence rate in the control group.

To estimate the standard error of the IRD, two formulas are used. When baseline_rate is missing:

ird\_se = \sqrt{\frac{n\_cases\_exp}{time\_exp^2} + \frac{n\_cases\_nexp}{time\_nexp^2}}

When baseline_rate is entered by users, the delta method is used:

ird\_se = baseline\_rate \times IRR \times logirr\_se

Value

This function estimates IRR and, when baseline rate information is available, NNT.

References

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Mayne, T. J., Whalen, E., & Rost, K. (2006). Annualized was found better than absolute risk reduction in the calculation of number needed to treat in chronic conditions. Journal of clinical epidemiology, 59(3), 217-223.

Examples

es_from_cases_time(
  n_cases_exp = 241, n_cases_nexp = 554,
  time_exp = 12.764, time_nexp = 19.743
)

Convert a chi-square value to several effect size measures

Description

Convert a chi-square value to several effect size measures

Usage

es_from_chisq(
  chisq,
  n_sample,
  n_cases,
  n_exp,
  yates_chisq = FALSE,
  reverse_chisq
)

Arguments

Details

This function converts a chi-square value (with one degree of freedom) into a phi coefficient (Lipsey et al. 2001):

phi = \sqrt{\frac{chisq}{n\_sample}}

and then converts it to other effect size measures exactly as in es_from_phi() (including the correlation-based R/Z/D/G fallback with standard sampling variances when the 2x2 table cannot be reconstructed).

Note that if yates_chisq = TRUE, the chi-square value is interpreted as Yates-corrected when back-transforming to a 2x2 contingency table; this is propagated row by row when a vector is supplied.

Then, the phi coefficient is converted to other effect size measures (see es_from_phi).

Value

This function estimates and converts between several effect size measures.

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_chisq(chisq = 4.21, n_sample = 78, n_cases = 51, n_exp = 50)

Convert a p-value of a chi-square to several effect size measures

Description

Convert a p-value of a chi-square to several effect size measures

Usage

es_from_chisq_pval(
  chisq_pval,
  n_sample,
  n_cases,
  n_exp,
  yates_chisq = FALSE,
  reverse_chisq_pval
)

Arguments

Details

This function converts a chi-square value (with one degree of freedom) into a chi-square coefficient (Section 3.12 in Lipsey et al., 2001):

chisq = qchisq(chisq\_pval, df = 1, lower.tail = FALSE)

Note that if yates_chisq = TRUE, the chi-square value is interpreted as Yates-corrected when back-transforming to a 2x2 contingency table; this is propagated row by row when a vector is supplied.

Then, the chisq coefficient is converted to other effect size measures (see es_from_chisq).

Value

This function estimates and converts between several effect size measures.

References

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage Publications, Inc.

Examples

es_from_chisq_pval(chisq_pval = 0.2, n_sample = 42, n_exp = 25, n_cases = 13)

Convert a Cohen's d value to several effect size measures

Description

Convert a Cohen's d value to several effect size measures

Usage

es_from_cohen_d(
  cohen_d,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_d
)

Arguments

Details

This function estimates the standard error of a Cohen's d value and computes a Hedges' g (G). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate the standard error of Cohen's d, the following formula is used (formula 12.13 in Cooper):

cohen\_d\_se = \sqrt{\frac{n\_exp+n\_nexp}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2*(n\_exp+n\_nexp)}}

cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, df = n\_exp+n\_nexp-2)

cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, df = n\_exp+n\_nexp-2)

To estimate the Hedges' g and its standard error, the following formulas are used (Hedges, 1981):

df = n\_exp + n\_nexp - 2

J = exp(\log_{gamma}(\frac{df}{2}) - 0.5 * \log(\frac{df}{2}) - \log_{gamma}(\frac{df - 1}{2}))

hedges\_g = cohen\_d * J

hedges\_g\_se = \sqrt{cohen\_d\_se^2 * J^2}

hedges\_g\_ci\_lo = hedges\_g - hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)

hedges\_g\_ci\_up = hedges\_g + hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)

To estimate the log odds ratio and its standard error, the following formulas are used (formulas 12.34-12.35 in Cooper):

logor = \frac{cohen\_d * \pi}{\sqrt{3}}

logor\_se = \sqrt{\frac{cohen\_d\_se^2 * \pi^2}{3}}

logor\_lo = logor - logor\_se * qnorm(.975)

logor\_up = logor + logor\_se * qnorm(.975)

Note that this conversion assumes that responses within the two groups follow logistic distributions.

To estimate the correlation coefficient and its standard error, various formulas can be used.

A. To estimate the 'biserial' correlation (smd_to_cor="viechtbauer"), the following formulas are used (formulas 5, 8, 13, 17, 18, 19 in Viechtbauer):

df = n\_exp + n\_nexp - 2 - n\_cov\_ancova

h = \frac{df}{n\_exp} + \frac{df}{n\_nexp}

r.pb = \frac{cohen\_d}{\sqrt{cohen\_d^2 + h}}

p = \frac{n\_exp}{n\_exp + n\_nexp}

q = 1 - p

f = dnorm(qnorm(1-p))

R = \frac{\sqrt{p*q}}{f} * r.pb

R\_var = \frac{1}{n\_exp + n\_nexp - 1} * (\frac{p*q}{f^2} - (\frac{3}{2} + (1 - \frac{p*qnorm(1-p)}{f})(1 + \frac{q*qnorm(1-p)}{f})) R^2 + R^4)

R\_se = \sqrt{R\_var}

a = \frac{\sqrt{f}}{(p*q)^\frac{1}{4}}

Z = \frac{a}{2} * \log(\frac{1+a*R}{1-a*R})

Z\_var = \frac{1}{n - 1}

Z\_se = \sqrt{Z\_var}

Z\_ci\_lo = Z - qnorm(.975) * Z\_se

Z\_ci\_up = Z + qnorm(.975) * Z\_se

R\_ci\_lo = \frac{1}{a} * tanh(\frac{Z\_lo}{a})

R\_ci\_up = \frac{1}{a} * tanh(\frac{Z\_up}{a})

Note: this Z is Jacobs & Viechtbauer's (2017) variance-stabilizing transform of the biserial correlation (with variance 1/(n-1)), which is analogous to – but NOT the same as – Fisher's atanh(R) z-transform for a Pearson correlation (the two coincide only as the effect approaches 0). It is intended for constructing the confidence interval of a single coefficient. For a meta-analysis that mixes standardized-mean-difference and genuine-correlation studies, pool on the correlation (R) scale, where biserial and product-moment correlations are directly comparable (Jacobs & Viechtbauer, 2017), rather than on this Z scale (which summary() labels "Fisher's z" for all correlation inputs).

B. To estimate the correlation coefficient according to Cooper et al. (2019) (formulas 12.40-42) and Borenstein et al. (2009) (formulas 54-56), the following formulas are used (smd_to_cor="lipsey_cooper"):

p = \frac{n\_exp}{n\_exp + n\_nexp}

R = \frac{cohen\_d}{\sqrt{cohen\_d^2 + 1 / (p * (1 - p))}}

a = \frac{(n\_exp + n\_nexp)^2}{(n\_exp*n\_nexp)}

var\_R = \frac{a^2 * cohen\_d\_se^2}{(cohen\_d^2 + a)^3}

R\_se = \sqrt{R\_var}

R\_ci\_lo = R - qt(.975, n\_exp+n\_nexp- 2) * R\_se

R\_ci\_up = R + qt(.975, n\_exp+n\_nexp- 2) * R\_se

Z = atanh(R)

Z\_var = \frac{cohen\_d\_se^2}{cohen\_d^2 + (1 / p*(1-p))}

Z\_se = \sqrt{Z\_var}

Z\_ci\_lo = Z - qnorm(.975) * Z\_se

Z\_ci\_up = Z + qnorm(.975) * Z\_se

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Borenstein, M., Hedges, L. V., Higgins, J. P., & Rothstein, H. R. (2021). Introduction to meta-analysis. John Wiley & Sons.

Hedges LV (1981): Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational and Behavioral Statistics, 6, 107-28

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research synthesis methods, 8(2), 161-180.

Examples

es_from_cohen_d(cohen_d = 1, n_exp = 20, n_nexp = 20)

Convert an adjusted Cohen's d value to several effect size measures

Description

Convert an adjusted Cohen's d value to several effect size measures

Usage

es_from_cohen_d_adj(
  cohen_d_adj,
  n_cov_ancova,
  cov_outcome_r,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_d
)

Arguments

Details

This function estimates the standard error of an adjusted Cohen's d value and Hedges' g (G), and converts an odds ratio (OR) and correlation coefficients (R/Z).

To estimate the standard error of Cohen's d, the following formula is used (table 12.3 in Cooper):

d\_se = \sqrt{\frac{n\_exp+n\_nexp}{n\_exp*n\_nexp} * (1 - cov\_outcome\_r^2) + \frac{cohen\_d\_adj^2}{2*(n\_exp+n\_nexp)}}

To estimate other effect size measures, calculations of the es_from_cohen_d() function are used (with the exception of the degree of freedom that is estimated as df = n_exp + n_nexp - 2 - n_cov_ancova).

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_cohen_d_adj(cohen_d_adj = 1, n_cov_ancova = 4, cov_outcome_r = .30, n_exp = 20, n_nexp = 20)

Convert Cronbach's alpha into an effect size measure

Description

Convert Cronbach's alpha into an effect size measure

Usage

es_from_cronbach_alpha(
  cronbach_alpha,
  n_sample,
  n_items,
  alpha_to_es = "bonett"
)

Arguments

Details

This function computes an effect size from a Cronbach's alpha.

1. When alpha_to_es = "bonett" (default), the Bonett (2002) transformation is applied:

   T(\alpha) = \ln(1 - \alpha)

   T\_se = \sqrt{\frac{2k}{(k - 1)(n - 2)}}

2. When alpha_to_es = "raw", the raw alpha is used. Its standard error is the delta-method back-transform of the Bonett (2002) transformed variance (equivalently the van Zyl, Neudecker & Nel, 2000, asymptotic variance; this is the form implemented by metafor's measure = "ARAW"):

   \alpha\_se = (1 - \alpha) \sqrt{\frac{2k}{(k - 1)(n - 2)}}

   Note that Feldt et al.'s (1987) classical asymptotic variance instead uses an (n - 1) denominator; the (n - 2) form above follows Bonett (2002) for consistency with the Bonett transformation used in method 1.

The Bonett transformation stabilizes the variance and is recommended for meta-analysis.

Value

This function estimates the standard error of the Cronbach's alpha.

References

Bonett, D. G. (2002). Sample size requirements for testing and estimating coefficient alpha. Journal of Educational and Behavioral Statistics, 27(4), 335-340.

Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11(1), 93-103.

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65(3), 271-280.

Examples

es_from_cronbach_alpha(
  cronbach_alpha = 0.85, n_sample = 200, n_items = 10
)

Convert an eta-squared value to various effect size measures

Description

Convert an eta-squared value to various effect size measures

Usage

es_from_etasq(
  etasq,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_etasq
)

Arguments

Details

This function first computes a Cohen's d (D) and Hedges' g (G) from the eta squared of a binary predictor (ANOVA model). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a Cohen's d the following formula is used (Cohen, 1988):

d = 2 * \sqrt{\frac{etasq}{1 - etasq}}

Note that this closed form is the large-sample, equal-groups (n\_exp = n\_nexp) limit of the exact F-based conversion. For markedly unequal group sizes it is approximate; supplying the ANOVA F (es_from_anova_f()), which carries the per-group sample sizes, is more accurate.

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Routledge.

Examples

es_from_etasq(etasq = 0.28, n_exp = 20, n_nexp = 22)

Convert an adjusted eta-squared value (i.e., from an ANCOVA) to various effect size measures

Description

Convert an adjusted eta-squared value (i.e., from an ANCOVA) to various effect size measures

Usage

es_from_etasq_adj(
  etasq_adj,
  n_exp,
  n_nexp,
  n_cov_ancova,
  cov_outcome_r,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_etasq
)

Arguments

Details

This function converts the adjusted (partial) eta-squared of a binary predictor (ANCOVA model) into the ANCOVA F-statistic it implies, and then relies on the calculations of the es_from_ancova_f() function. The returned Cohen's d (D) and Hedges' g (G) are therefore expressed on the marginal (unadjusted) SD scale — consistent with the other es_from_ancova_* functions — not on the residual (covariate-adjusted) SD scale on which a partial eta-squared is natively defined. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To convert the adjusted eta-squared into an ANCOVA F-statistic, the partial-eta-squared identity \eta_p^2 = F / (F + df) is inverted (this is why a partial eta-squared, not a classical one, is required):

df = n\_exp + n\_nexp - 2 - n\_cov\_ancova

ancova\_f = \frac{etasq\_adj * df}{1 - etasq\_adj}

To estimate a Cohen's d the formula used is (table 12.3 in Cooper):

cohen\_d = \sqrt{ancova\_f * \frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp}} * \sqrt{1 - cov\_out\_cor^2}

Note that the back-transformation to the marginal SD scale requires cov_outcome_r; when it is missing, the effect size estimates are returned as NA (the marginal scale is not identified from the adjusted eta-squared alone).

To estimate other effect size measures, calculations of the es_from_cohen_d_adj() are applied.

Value

This function estimates and converts between several effect size measures.

Note

Two cautions apply. (1) etasq_adj must be the partial eta-squared (see the parameter description); a classical eta-squared biases the effect size toward zero. (2) The sampling variance is rebuilt from Cooper's eq. 12.26, which assumes a covariate balanced across groups and treats cov_outcome_r as known; in covariate-imbalanced designs the standard error is a lower bound (anti-conservative), while the point estimate remains unbiased. See Lai and Kelley (2012).

References

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65(2), 350-370.

Examples

es_from_etasq_adj(etasq_adj = 0.28, n_cov_ancova = 3, cov_outcome_r = 0.2, n_exp = 20, n_nexp = 22)

Convert a Fisher's z (r-to-z transformation) to several effect size measures

Description

Convert a Fisher's z (r-to-z transformation) to several effect size measures

Usage

es_from_fisher_z(
  fisher_z,
  n_sample,
  unit_type = "raw_scale",
  n_exp,
  n_nexp,
  cor_to_smd = "viechtbauer",
  sd_iv,
  unit_increase_iv,
  reverse_fisher_z
)

Arguments

Details

This function converts estimates the standard error of the Fisher's z and performs the z-to-r Fisher's transformation.

Last, it converts this r value into a Cohen's d and OR (see details in es_from_pearson_r()).

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Mathur, M. B., & VanderWeele, T. J. (2020). A Simple, Interpretable Conversion from Pearson's Correlation to Cohen's for d Continuous Exposures. Epidemiology (Cambridge, Mass.), 31(2), e16-e18. https://doi.org/10.1097/EDE.0000000000001105

Viechtbauer W (2010). "Conducting meta-analyses in R with the metafor package." Journal of Statistical Software, 36(3), 1-48. doi:10.18637/jss.v036.i03.

Examples

es_from_fisher_z(
  fisher_z = .21, n_sample = 44,
)

Convert a Hedges' g value to other effect size measures (G, OR, COR)

Description

Convert a Hedges' g value to other effect size measures (G, OR, COR)

Usage

es_from_hedges_g(
  hedges_g,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_g
)

Arguments

Details

This function estimates the standard error of the Hedges' g and the Cohen's d (D). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate standard error of Hedges'g, the following formula is used (Hedges, 1981):

df = n\_exp + n\_nexp - 2

hedges\_g\_se = \sqrt{cohen\_d\_se^2 * J^2}

hedges\_g\_ci\_lo = hedges\_g - hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)

hedges\_g\_ci\_up = hedges\_g + hedges\_g\_se * qt(.975, df = n\_exp+n\_nexp-2)

To estimate the Cohen's d value, the following formula is used (Hedges, 1981):

J = exp(\log_{gamma}(\frac{df}{2}) - 0.5 * \log(\frac{df}{2}) - \log_{gamma}(\frac{df - 1}{2}))

cohen\_d = \frac{hedges\_g}{J}

cohen\_d\_se = \sqrt{(\frac{n\_exp+n\_nexp}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2*(n\_exp+n\_nexp)})}

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Hedges LV (1981): Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational and Behavioral Statistics, 6, 107-28

Examples

es_from_hedges_g(hedges_g = 0.243, n_exp = 20, n_nexp = 20)

Convert an intraclass correlation coefficient (ICC) into an effect size measure

Description

Convert an intraclass correlation coefficient (ICC) into an effect size measure

Usage

es_from_icc(
  icc,
  n_sample,
  n_measurements,
  icc_type = "agreement",
  icc_to_es = "bonett"
)

Arguments

Details

This function computes an effect size from an ICC.

1. When icc_to_es = "bonett" (default), the Bonett (2002) transformation is applied:

   T(ICC) = \ln(1 - ICC)

   with the one-way random-model / two-way-consistency leading-order sampling variance (Bonett, 2002; Donner & Eliasziw, 1987):

   T\_se = \sqrt{\frac{2 (1 + (k-1) ICC)^2}{k (k - 1)(n - 1)}}

2. When icc_to_es = "raw", the raw ICC is used and its standard error is obtained by the delta method ((1 - ICC) times the transformed-scale SE).

Scope of the SE formula. For the two-way consistency ICC(3,1) (icc_type = "consistency") this SE is exact at leading order: deriving it from F_0 = MSR/MSE (with degrees of freedom n-1 and (n-1)(k-1)) reduces to the same expression, and it still depends on the ICC value (it is not \rho-free). For the two-way absolute-agreement ICC(2,1) (icc_type = "agreement", the default) the same formula is only a one-way approximation that assumes negligible between-rater variance: when raters differ systematically (\sigma^2_{rater} > 0), the ICC(2,1) estimator depends on the between-rater mean square, which has only k - 1 degrees of freedom, so its true sampling variance does not shrink at the 1/n rate this formula assumes and the reported SE/CI can be markedly anti-conservative (simulation: 95\ n grows). The exact ICC(2,1) variance requires the rater-variance component, which summary data do not report; a per-row informational flag (V31) marks agreement-type rows for this reason. If the raters are known to be exchangeable (negligible rater variance), the approximation is accurate.

Scale note. Under the default icc_to_es = "bonett" the returned icc_se column is on the \ln(1 - ICC) scale. If you feed it to compute_sem (whose icc_se argument expects the RAW-scale SE), convert it first: raw_se = icc_se * (1 - icc) - or call es_from_icc with icc_to_es = "raw".

Value

This function estimates the standard error of the ICC.

References

Bonett, D. G. (2002). Sample size requirements for estimating intraclass correlations with desired precision. Statistics in Medicine, 21(9), 1331-1335.

Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: uses in assessing rater reliability. Psychological Bulletin, 86(2), 420-428.

Examples

es_from_icc(
  icc = 0.80, n_sample = 50, n_measurements = 2, icc_type = "agreement"
)

Convert an unstandardized regression coefficient and its confidence interval into several effect size measures

Description

Convert an unstandardized regression coefficient and its confidence interval into several effect size measures

Usage

es_from_linreg_b_ci(
  linreg_b,
  linreg_b_ci_lo,
  linreg_b_ci_up,
  n_sample,
  n_covariates,
  sd_iv,
  unit_increase_iv,
  unit_type = "raw_scale",
  n_exp,
  n_nexp,
  cor_to_smd = "viechtbauer",
  reverse_linreg_b
)

Arguments

Details

This function derives the standard error from the 95% confidence interval using a t-distribution with df = n\_sample - n\_covariates - 2 degrees of freedom:

SE(b) = \frac{ci\_up - ci\_lo}{2 \times qt(.975, df)}

Then, calculations of the es_from_linreg_b_se function are applied. For binary predictors without covariates, use es_from_beta_unstd instead.

Value

This function estimates and converts between several effect size measures.

References

Aloe, A. M., & Thompson, C. G. (2013). The synthesis of partial effect sizes. Journal of the Society for Social Work and Research, 4(4), 390–405.

Examples

es_from_linreg_b_ci(
  linreg_b = 1.5, linreg_b_ci_lo = 0.3, linreg_b_ci_up = 2.7,
  n_sample = 100, n_covariates = 2
)

Convert an unstandardized regression coefficient and its p-value into several effect size measures

Description

Convert an unstandardized regression coefficient and its p-value into several effect size measures

Usage

es_from_linreg_b_pval(
  linreg_b,
  linreg_b_pval,
  n_sample,
  n_covariates,
  sd_iv,
  unit_increase_iv,
  unit_type = "raw_scale",
  n_exp,
  n_nexp,
  cor_to_smd = "viechtbauer",
  reverse_linreg_b_pval
)

Arguments

Details

This function recovers the t-statistic from the two-sided p-value:

t = qt(1 - pval/2, df) \times sign(b)

where df = n\_sample - n\_covariates - 2.

The sign of the regression coefficient gives the direction, which the two-tailed p-value does not carry. Then, calculations of the es_from_linreg_t function are applied. For binary predictors without covariates, use es_from_beta_unstd instead.

Value

This function estimates and converts between several effect size measures.

References

Aloe, A. M., & Thompson, C. G. (2013). The synthesis of partial effect sizes. Journal of the Society for Social Work and Research, 4(4), 390–405.

Examples

es_from_linreg_b_pval(
  linreg_b = 1.5, linreg_b_pval = 0.01,
  n_sample = 100, n_covariates = 2
)

Convert an unstandardized regression coefficient and its standard error into several effect size measures

Description

Convert an unstandardized regression coefficient and its standard error into several effect size measures

Usage

es_from_linreg_b_se(
  linreg_b,
  linreg_b_se,
  n_sample,
  n_covariates,
  sd_iv,
  unit_increase_iv,
  unit_type = "raw_scale",
  n_exp,
  n_nexp,
  cor_to_smd = "viechtbauer",
  reverse_linreg_b
)

Arguments

Details

This function derives a t-statistic from a regression coefficient and its standard error, then converts it into a partial correlation and other effect size measures.

The Wald t-statistic is computed as:

t = \frac{b}{SE(b)}

Once the t-statistic is obtained, all subsequent conversions follow the same formulas as in es_from_linreg_t.

For binary predictors without covariates, use es_from_beta_unstd instead, which converts directly to Cohen's d.

Value

This function estimates and converts between several effect size measures.

References

Aloe, A. M., & Thompson, C. G. (2013). The synthesis of partial effect sizes. Journal of the Society for Social Work and Research, 4(4), 390–405.

van Aert, R. C. M., & Goos, C. (2023). A critical reflection on computing the sampling variance of the partial correlation coefficient. Research Synthesis Methods, 14(3), 520–525.

Examples

es_from_linreg_b_se(linreg_b = 1.5, linreg_b_se = 0.6, n_sample = 100, n_covariates = 2)

Convert a t-statistic from a linear regression model to several effect size measures

Description

Convert a t-statistic from a linear regression model to several effect size measures

Usage

es_from_linreg_t(
  linreg_t,
  n_sample,
  n_covariates,
  sd_iv,
  unit_increase_iv,
  unit_type = "raw_scale",
  n_exp,
  n_nexp,
  cor_to_smd = "viechtbauer",
  reverse_linreg_t
)

Arguments

Details

This function converts a t-statistic from a linear regression model into a partial correlation coefficient (Rp) and its Fisher's z transformation (Zp).

The partial correlation is obtained as (Aloe & Thompson, 2013, Eq. 2; Gustafson, 1961):

r_p = \frac{t}{\sqrt{t^2 + df}}

where df = n\_sample - n\_covariates - 2 is the residual degrees of freedom of the regression model (i.e. n minus the intercept, the focal predictor, and the n_covariates covariates).

Its sampling variance is estimated as recommended by van Aert & Goos (2023, Eq. 5):

var(r_p) = \frac{(1 - r_p^2)^2}{df}

The Fisher's z transformation and its variance are:

z_p = atanh(r_p)

var(z_p) = \frac{1}{n - n\_covariates - 3}

Cohen's d, Hedges' g and odds ratio are then converted from the partial correlation using the calculations of the es_from_pearson_r function. A partial correlation controls for covariates and thus targets a different estimand than a two-group comparison or a bivariate correlation (Aloe & Thompson, 2013); the converted d/g/OR values should not be pooled with such effect sizes. For meta-analyses of regression results, use measure = "rp" or measure = "zp" in convert_df.

Value

This function estimates and converts between several effect size measures.

References

Aloe, A. M., & Thompson, C. G. (2013). The synthesis of partial effect sizes. Journal of the Society for Social Work and Research, 4(4), 390–405.

Gustafson, R. L. (1961). Partial correlations in regression computations. Journal of the American Statistical Association, 56, 363–367.

Mathur, M. B., & VanderWeele, T. J. (2020). A simple, interpretable conversion from Pearson's correlation to Cohen's d for continuous exposures. Epidemiology, 31(2), e16–e18.

van Aert, R. C. M., & Goos, C. (2023). A critical reflection on computing the sampling variance of the partial correlation coefficient. Research Synthesis Methods, 14(3), 520–525.

Examples

es_from_linreg_t(linreg_t = 2.5, n_sample = 100, n_covariates = 2)

# Reproduce Aloe & Thompson (2013) Table 1, Cole et al. (2004)
es_from_linreg_t(linreg_t = 6.19, n_sample = 232, n_covariates = 6)

Convert an odds ratio or risk ratio and a Wald t-statistic from a regression model into several effect size measures

Description

Convert an odds ratio or risk ratio and a Wald t-statistic from a regression model into several effect size measures

Usage

es_from_logreg_t(
  or,
  logor,
  rr,
  logrr,
  logreg_t,
  baseline_risk,
  small_margin_prop,
  n_exp,
  n_nexp,
  n_cases,
  n_controls,
  n_sample,
  or_to_rr = "metaumbrella_cases",
  or_to_cor = "bonett",
  rr_to_or = "metaumbrella",
  reverse_logreg_t
)

Arguments

Details

This function derives the standard error of the log odds ratio (or log risk ratio) from a Wald t-statistic reported in a regression model.

To estimate the standard error of the log OR (or log RR), the formulas used are:

t = \frac{\beta}{SE(\beta)}

SE(\beta) = \frac{|\beta|}{|t|}

where \beta is \log(OR) or \log(RR) depending on the model.

Then, if an OR (or logOR) is entered, calculations of es_from_or_se() are applied. If a RR (or logRR) is entered, calculations of es_from_rr_se() are applied.

Note that the standardized-mean-difference and correlation conversions (D, G, R, Z) are produced for OR inputs only. RR inputs are treated as a ratio measure and yield RR + OR + NNT + RD: RR is not converted to a standardized mean difference or correlation, because that would require going through the OR and the baseline risk (see es_from_rr_se). To obtain a SMD or correlation from an RR, convert it to an OR first (supplying the baseline risk) and then use the OR path.

Value

This function estimates and converts between several effect size measures.

References

Sanchez-Meca, J., Marin-Martinez, F., & Chacon-Moscoso, S. (2003). Effect-size indices for dichotomized outcomes in meta-analysis. Psychological Methods, 8(4), 448–467.

Examples

es_or <- es_from_logreg_t(
  or = 2.12, logreg_t = 3.21,
  n_cases = 50, n_controls = 150
)

es_rr <- es_from_logreg_t(
  rr = 1.5, logreg_t = 2.8,
  n_exp = 100, n_nexp = 100
)

Convert a mean difference between two independent groups and 95% CI into several effect size measures

Description

Convert a mean difference between two independent groups and 95% CI into several effect size measures

Usage

es_from_md_ci(
  md,
  md_ci_lo,
  md_ci_up,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  max_asymmetry = 10,
  reverse_md
)

Arguments

Details

This function converts 95% CI of a mean difference into a standard error (Cochrane Handbook section 6.5.2.3):

md\_se = \frac{md\_ci\_up - md\_ci\_lo}{2 * qt(0.975, df = n\_exp + n\_nexp - 2)}

Calculations of the es_from_md_se() function are then used to estimate the Cohen's d and other effect size measures.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_md_ci(md = 4, md_ci_lo = 2, md_ci_up = 6, n_exp = 20, n_nexp = 22)

Convert a mean difference between two independent groups and its p-value into several effect size measures

Description

Convert a mean difference between two independent groups and its p-value into several effect size measures

Usage

es_from_md_pval(
  md,
  md_pval,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_md
)

Arguments

Details

This function converts the p-value of a mean difference into a standard error (Cochrane Handbook section 6.5.2.3):

t = qt(\frac{md\_pval}{2}, df = n\_exp + n\_nexp - 2)

md\_se = |\frac{md}{t}|

Calculations of the es_from_md_se function are then used to estimate the Cohen's d and other effect size measures.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_md_pval(md = 4, md_pval = 0.024, n_exp = 20, n_nexp = 22)

Convert a mean difference between two independent groups and standard deviation into several effect size measures

Description

Convert a mean difference between two independent groups and standard deviation into several effect size measures

Usage

es_from_md_sd(
  md,
  md_sd,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_md
)

Arguments

Details

This function converts the mean difference and 95% CI into a Cohen's d (D) and Hedges' g (G). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

The formula used to obtain the Cohen's d is:

d = \frac{md}{md\_sd}

Note that this formula is perfectly accurate only if the md_sd has been estimated by assuming that the variance of the two groups is equal.

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

Examples

es_from_md_sd(md = 4, md_sd = 2, n_exp = 20, n_nexp = 22)

Convert a mean difference between two independent groups and its standard error into several effect size measures

Description

Convert a mean difference between two independent groups and its standard error into several effect size measures

Usage

es_from_md_se(
  md,
  md_se,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_md
)

Arguments

Details

This function the standard error of a mean difference into a standard deviation:

inv\_n = \frac{1}{n\_exp} + \frac{1}{n\_nexp}

md\_sd = \frac{md\_se}{\sqrt{inv\_n}}

Calculations of the es_from_md_sd function are then used to estimate the Cohen's d and other effect size measures.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_md_se(md = 4, md_se = 2, n_exp = 20, n_nexp = 22)

Convert mean changes and 95% CI of two independent groups into standard effect size measures

Description

Convert mean changes and 95% CI of two independent groups into standard effect size measures

Usage

es_from_mean_change_ci(
  mean_change_exp,
  mean_change_ci_lo_exp,
  mean_change_ci_up_exp,
  mean_change_nexp,
  mean_change_ci_lo_nexp,
  mean_change_ci_up_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  n_exp,
  n_nexp,
  max_asymmetry = 10,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  pool_sd = FALSE,
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and 95% CI of two independent groups into a Cohen's d. The Cohen's d is then converted to other effect size measures.

This function simply internally calls the es_from_means_ci_pre_post function but setting:

mean\_pre\_exp = 0

mean\_pre\_ci\_lo\_exp = 0

mean\_pre\_ci\_up\_exp = 0

mean\_exp = mean\_change\_exp

mean\_ci\_lo\_exp = mean\_change\_ci\_lo\_exp

mean\_ci\_up\_exp = mean\_change\_ci\_up\_exp

mean\_pre\_nexp = 0

mean\_pre\_ci\_lo\_nexp = 0

mean\_pre\_ci\_up\_nexp = 0

mean\_nexp = mean\_change\_nexp

mean\_ci\_lo\_nexp = mean\_change\_ci\_lo\_nexp

mean\_ci\_up\_nexp = mean\_change\_ci\_up\_nexp

To know more about the calculations, see es_from_means_sd_pre_post function.

Value

This function estimates and converts between several effect size measures.

References

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_mean_change_ci(
  n_exp = 36, n_nexp = 35,
  mean_change_exp = 8.4,
  mean_change_ci_lo_exp = 6.4, mean_change_ci_up_exp = 10.4,
  mean_change_nexp = 2.43,
  mean_change_ci_lo_nexp = 1.43, mean_change_ci_up_nexp = 3.43,
  r_pre_post_exp = 0.2, r_pre_post_nexp = 0.2
)

Convert mean change and 95% CI of a single group into standard effect size measures

Description

Convert mean change and 95% CI of a single group into standard effect size measures

Usage

es_from_mean_change_ci_single_group(
  mean_change_exp,
  mean_change_ci_lo_exp,
  mean_change_ci_up_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  max_asymmetry = 10,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and 95% CI of a single group into within-group Cohen's d, Hedges' g, and mean difference. The effect sizes are then converted to other measures.

The effect size (d_rm) depends linearly on the pre-post correlation (0.8 assumed when r_pre_post_exp is missing); see es_from_mean_change_sd_single_group.

This function simply internally calls the es_from_means_ci_pre_post_single_group function but setting:

mean\_pre\_exp = 0

mean\_pre\_ci\_lo\_exp = 0

mean\_pre\_ci\_up\_exp = 0

mean\_exp = mean\_change\_exp

mean\_ci\_lo\_exp = mean\_change\_ci\_lo\_exp

mean\_ci\_up\_exp = mean\_change\_ci\_up\_exp

To know more about the calculations, see es_from_means_sd_pre_post_single_group function.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_mean_change_ci_single_group(
  n_exp = 36,
  mean_change_exp = 8.4,
  mean_change_ci_lo_exp = 6.4, mean_change_ci_up_exp = 10.4,
  r_pre_post_exp = 0.2
)

Convert mean changes and p-values of two independent groups into standard effect size measures

Description

Convert mean changes and p-values of two independent groups into standard effect size measures

Usage

es_from_mean_change_pval(
  mean_change_exp,
  mean_change_pval_exp,
  mean_change_nexp,
  mean_change_pval_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  pool_sd = FALSE,
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and associated p-values of two independent groups into a Cohen's d. The Cohen's d is then converted to other effect size measures.

To start, this function estimates the mean change standard errors from the p-values:

t\_exp <- qt(p = mean\_change\_pval\_exp / 2, df = n\_exp - 1, lower.tail = FALSE)

t\_nexp <- qt(p = mean\_change\_pval\_nexp / 2, df = n\_nexp - 1, lower.tail = FALSE)

mean\_change\_se\_exp <- |\frac{mean\_change\_exp}{t\_exp}|

mean\_change\_se\_nexp <- |\frac{mean\_change\_nexp}{t\_nexp}|

Then, this function simply internally calls the es_from_means_se_pre_post function but setting:

mean\_pre\_exp = 0

mean\_pre\_se\_exp = 0

mean\_exp = mean\_change\_exp

mean\_se\_exp = mean\_change\_se\_exp

mean\_pre\_nexp = 0

mean\_pre\_se\_nexp = 0

mean\_nexp = mean\_change\_nexp

mean\_se\_nexp = mean\_change\_se\_nexp

To know more about other calculations, see es_from_means_sd_pre_post function.

Value

This function estimates and converts between several effect size measures.

References

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_mean_change_pval(
  n_exp = 36, n_nexp = 35,
  mean_change_exp = 8.4, mean_change_pval_exp = 0.13,
  mean_change_nexp = 2.43, mean_change_pval_nexp = 0.61,
  r_pre_post_exp = 0.8, r_pre_post_nexp = 0.8
)

Convert mean change and p-value of a single group into standard effect size measures

Description

Convert mean change and p-value of a single group into standard effect size measures

Usage

es_from_mean_change_pval_single_group(
  mean_change_exp,
  mean_change_pval_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and associated p-value of a single group into within-group Cohen's d, Hedges' g, and mean difference. The effect sizes are then converted to other measures.

The effect size (d_rm) depends linearly on the pre-post correlation (0.8 assumed when r_pre_post_exp is missing); see es_from_mean_change_sd_single_group.

To start, this function estimates the mean change standard error from the p-value:

t <- qt(p = mean\_change\_pval\_exp / 2, df = n\_exp - 1, lower.tail = FALSE)

mean\_change\_se\_exp <- |\frac{mean\_change\_exp}{t}|

Then, this function simply internally calls the es_from_means_se_pre_post_single_group function but setting:

mean\_pre\_exp = 0

mean\_pre\_se\_exp = 0

mean\_exp = mean\_change\_exp

mean\_se\_exp = mean\_change\_se\_exp

To know more about other calculations, see es_from_means_sd_pre_post_single_group function.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_mean_change_pval_single_group(
  n_exp = 36,
  mean_change_exp = 8.4, mean_change_pval_exp = 0.13,
  r_pre_post_exp = 0.8
)

Convert mean changes and standard deviations of two independent groups into standard effect size measures

Description

Convert mean changes and standard deviations of two independent groups into standard effect size measures

Usage

es_from_mean_change_sd(
  mean_change_exp,
  mean_change_sd_exp,
  mean_change_nexp,
  mean_change_sd_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  pool_sd = FALSE,
  reverse_mean_change
)

Arguments

Details

This function computes a Cohen's d (D) and Hedges' g (G) from the mean change and standard deviation of change scores of two independent groups. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from Cohen's d.

Two formulas can be used to obtain the SMD (Morris & DeShon, 2002):

d_{z} = \frac{mean\_change}{sd\_change}

d_{rm} = d_{z} \times \sqrt{2 \times (1 - r\_pre\_post)}

The 'morris_drm' formula (default, alias 'cooper') requires the pre-post correlation while 'morris_dz' does not. Note that d_rm and d_z are not expressed on the same scale and should not be combined in a same meta-analysis (Morris & DeShon, 2002). Under 'morris_drm', the resulting effect size directly depends on the r_pre_post value entered; if r is unknown, the 'morris_dz' formula can be used instead. The 'bonett' and 'morris_dav' formulas require separate pre/post SDs and are thus not available for mean change data.

This function simply internally calls the es_from_means_sd_pre_post function but setting:

mean\_pre\_exp = 0

mean\_pre\_sd\_exp = 0

mean\_exp = mean\_change\_exp

mean\_sd\_exp = mean\_change\_sd\_exp

mean\_pre\_nexp = 0

mean\_pre\_sd\_nexp = 0

mean\_nexp = mean\_change\_nexp

mean\_sd\_nexp = mean\_change\_sd\_nexp

To know more about the calculations, see es_from_means_sd_pre_post function.

Value

This function estimates and converts between several effect size measures.

References

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_mean_change_sd(
  n_exp = 36, n_nexp = 35,
  mean_change_exp = 8.4, mean_change_sd_exp = 9.13,
  mean_change_nexp = 2.43, mean_change_sd_nexp = 6.61,
  r_pre_post_exp = 0.2, r_pre_post_nexp = 0.2
)

Convert mean change and standard deviation of a single group into standard effect size measures

Description

Convert mean change and standard deviation of a single group into standard effect size measures

Usage

es_from_mean_change_sd_single_group(
  mean_change_exp,
  mean_change_sd_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  reverse_mean_change
)

Arguments

Details

This function computes within-group Cohen's d (dw), Hedges' g (gw), and mean difference (mdw) from the mean change (MC) and standard deviation of a single group. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from Cohen's d.

The effect size (d_rm) depends linearly on the pre-post correlation:

d\_rm = (mean\_change\_exp / sd\_change) \times \sqrt{2 \times (1 - r\_pre\_post\_exp)}

If r_pre_post_exp is not provided, a value of 0.8 is assumed. A misspecified correlation biases the point estimate, not only its precision; when the correlation is not reported, a sensitivity analysis using several plausible values (e.g., 0.3, 0.5, 0.7) is advisable.

This function simply internally calls the es_from_means_sd_pre_post_single_group function but setting:

mean\_pre\_exp = 0

mean\_pre\_sd\_exp = 0

mean\_exp = mean\_change\_exp

mean\_sd\_exp = mean\_change\_sd\_exp

To know more about the calculations, see es_from_means_sd_pre_post_single_group function.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_mean_change_sd_single_group(
  n_exp = 36,
  mean_change_exp = 8.4, mean_change_sd_exp = 9.13,
  r_pre_post_exp = 0.2
)

Convert mean changes and standard errors of two independent groups into standard effect size measures

Description

Convert mean changes and standard errors of two independent groups into standard effect size measures

Usage

es_from_mean_change_se(
  mean_change_exp,
  mean_change_se_exp,
  mean_change_nexp,
  mean_change_se_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  pool_sd = FALSE,
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and standard errors of two independent groups into a Cohen's d. The Cohen's d is then converted to other effect size measures.

This function simply internally calls the es_from_means_se_pre_post function but setting:

mean\_pre\_exp = 0

mean\_pre\_se\_exp = 0

mean\_exp = mean\_change\_exp

mean\_se\_exp = mean\_change\_se\_exp

mean\_pre\_nexp = 0

mean\_pre\_se\_nexp = 0

mean\_nexp = mean\_change\_nexp

mean\_se\_nexp = mean\_change\_se\_nexp

To know more about the calculations, see es_from_means_se_pre_post function.

Value

This function estimates and converts between several effect size measures.

References

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_mean_change_se(
  n_exp = 36, n_nexp = 35,
  mean_change_exp = 8.4, mean_change_se_exp = 9.13,
  mean_change_nexp = 2.43, mean_change_se_nexp = 6.61,
  r_pre_post_exp = 0.2, r_pre_post_nexp = 0.2
)

Convert mean change and standard error of a single group into standard effect size measures

Description

Convert mean change and standard error of a single group into standard effect size measures

Usage

es_from_mean_change_se_single_group(
  mean_change_exp,
  mean_change_se_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "cooper",
  reverse_mean_change
)

Arguments

Details

This function converts the mean change and standard error of a single group into within-group Cohen's d, Hedges' g, and mean difference. The effect sizes are then converted to other measures.

The effect size (d_rm) depends linearly on the pre-post correlation (0.8 assumed when r_pre_post_exp is missing); see es_from_mean_change_sd_single_group.

This function simply internally calls the es_from_means_se_pre_post_single_group function but setting:

mean\_pre\_exp = 0

mean\_pre\_se\_exp = 0

mean\_exp = mean\_change\_exp

mean\_se\_exp = mean\_change\_se\_exp

To know more about the calculations, see es_from_means_se_pre_post_single_group function.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_mean_change_se_single_group(
  n_exp = 36,
  mean_change_exp = 8.4, mean_change_se_exp = 1.52,
  r_pre_post_exp = 0.2
)

Convert means and 95% CI of two independent groups several effect size measures

Description

Convert means and 95% CI of two independent groups several effect size measures

Usage

es_from_means_ci(
  mean_exp,
  mean_ci_lo_exp,
  mean_ci_up_exp,
  mean_nexp,
  mean_ci_lo_nexp,
  mean_ci_up_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  smd_denom = "pooled",
  max_asymmetry = 10,
  reverse_means
)

Arguments

Details

This function converts the 95% CI of two independent groups into a standard error, and then relies on the calculations of the es_from_means_se() function.

To convert the 95% CIs into standard errors, the following formula is used (table 12.3 in Cooper):

mean\_se\_exp = \frac{mean\_ci\_up\_exp - mean\_ci\_lo\_exp}{2 * qt{(0.975, df = n\_exp - 1)}}

mean\_se\_nexp = \frac{mean\_ci\_up\_nexp - mean\_ci\_lo\_nexp}{2 * qt{(0.975, df = n\_nexp - 1)}}

Calculations of the es_from_means_se() are then applied.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_means_ci(
  n_exp = 55, n_nexp = 55,
  mean_exp = 25, mean_ci_lo_exp = 15, mean_ci_up_exp = 35,
  mean_nexp = 18, mean_ci_lo_nexp = 12, mean_ci_up_nexp = 24
)

Convert pre-post means of two independent groups into various effect size measures

Description

Convert pre-post means of two independent groups into various effect size measures

Usage

es_from_means_ci_pre_post(
  mean_pre_exp,
  mean_exp,
  mean_pre_ci_lo_exp,
  mean_pre_ci_up_exp,
  mean_ci_lo_exp,
  mean_ci_up_exp,
  mean_pre_nexp,
  mean_nexp,
  mean_pre_ci_lo_nexp,
  mean_pre_ci_up_nexp,
  mean_ci_lo_nexp,
  mean_ci_up_nexp,
  n_exp,
  n_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "bonett",
  max_asymmetry = 10,
  pool_sd = FALSE,
  reverse_means_pre_post
)

Arguments

Details

This function converts the bounds of the 95% CI of the pre/post means of two independent groups into standard errors (Section 6.3.1 in the Cochrane Handbook).

mean\_pre\_se\_exp = \frac{mean\_pre\_ci\_up\_exp - mean\_pre\_ci\_lo\_exp}{2 * qt{(0.975, df = n\_exp - 1)}}

mean\_pre\_se\_nexp = \frac{mean\_pre\_ci\_up\_nexp - mean\_pre\_ci\_lo\_nexp}{2 * qt{(0.975, df = n\_nexp - 1)}}

mean\_se\_exp = \frac{mean\_ci\_up\_exp - mean\_ci\_lo\_exp}{2 * qt{(0.975, df = n\_exp - 1)}}

mean\_se\_nexp = \frac{mean\_ci\_up\_nexp - mean\_ci\_lo\_nexp}{2 * qt{(0.975, df = n\_nexp - 1)}}

Then, calculations of the es_from_means_se_pre_post are applied.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_means_ci_pre_post(
  n_exp = 36, n_nexp = 35,
  mean_pre_exp = 98,
  mean_pre_ci_lo_exp = 88,
  mean_pre_ci_up_exp = 108,
  mean_exp = 102,
  mean_ci_lo_exp = 92,
  mean_ci_up_exp = 112,
  mean_pre_nexp = 96,
  mean_pre_ci_lo_nexp = 86,
  mean_pre_ci_up_nexp = 106,
  mean_nexp = 102,
  mean_ci_lo_nexp = 92,
  mean_ci_up_nexp = 112,
  r_pre_post_exp = 0.8, r_pre_post_nexp = 0.8
)

Convert pre-post means and 95% CI of a single group into standard effect size measures

Description

Convert pre-post means and 95% CI of a single group into standard effect size measures

Usage

es_from_means_ci_pre_post_single_group(
  mean_pre_exp,
  mean_exp,
  mean_pre_ci_lo_exp,
  mean_pre_ci_up_exp,
  mean_ci_lo_exp,
  mean_ci_up_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  pre_post_to_smd = "bonett",
  smd_to_cor = "viechtbauer",
  max_asymmetry = 10,
  reverse_means_pre_post
)

Arguments

Details

This function converts the bounds of the 95% CI of the pre/post means of a single group into standard errors (Section 6.3.1 in the Cochrane Handbook).

mean\_pre\_se\_exp = \frac{mean\_pre\_ci\_up\_exp - mean\_pre\_ci\_lo\_exp}{2 * qt{(0.975, df = n\_exp - 1)}}

mean\_post\_se\_exp = \frac{mean\_post\_ci\_up\_exp - mean\_post\_ci\_lo\_exp}{2 * qt{(0.975, df = n\_exp - 1)}}

Then, calculations of the es_from_means_se_pre_post_single_group() are applied.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022.

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386.

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_means_ci_pre_post_single_group(
  n_exp = 36,
  mean_pre_exp = 98,
  mean_pre_ci_lo_exp = 88, mean_pre_ci_up_exp = 108,
  mean_exp = 102,
  mean_ci_lo_exp = 92, mean_ci_up_exp = 112,
  r_pre_post_exp = 0.8
)

Convert means and standard deviations of two independent groups into several effect size measures

Description

Convert means and standard deviations of two independent groups into several effect size measures

Usage

es_from_means_sd(
  mean_exp,
  mean_sd_exp,
  mean_nexp,
  mean_sd_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  smd_denom = "pooled",
  reverse_means
)

Arguments

Details

This function first computes a Cohen's d (D), Hedges' g (G) and mean difference (MD) from the means and standard deviations of two independent groups. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a mean difference (formulas 12.1-12.6 in Cooper):

md = mean\_exp - mean\_nexp

md\_se = \sqrt{\frac{mean\_sd\_exp^2}{n\_exp} + \frac{mean\_sd\_nexp^2}{n\_nexp}}

The confidence interval uses the unequal-variance (Welch-Satterthwaite) degrees of freedom that match this standard error (as in stats::t.test(var.equal = FALSE)):

df = \frac{(s_e^2/n\_exp + s_n^2/n\_nexp)^2}{(s_e^2/n\_exp)^2/(n\_exp - 1) + (s_n^2/n\_nexp)^2/(n\_nexp - 1)}

md\_ci\_lo = md - md\_se * qt(.975, df)

md\_ci\_up = md + md\_se * qt(.975, df)

To estimate a Cohen's d the following formulas are used (formulas 12.10-12.18 in Cooper):

mean\_sd\_pooled = \sqrt{\frac{(n\_exp - 1) * sd\_exp^2 + (n\_nexp - 1) * sd\_nexp^2}{n\_exp+n\_nexp-2}}

cohen\_d = \frac{mean\_exp - mean\_nexp}{mean\_sd\_pooled}

cohen\_d\_se = \sqrt{\frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2(n\_exp+n\_nexp)}}

cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, df = n\_exp + n\_nexp - 2)

cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, df = n\_exp + n\_nexp - 2)

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_means_sd(
  n_exp = 55, n_nexp = 55,
  mean_exp = 2.3, mean_sd_exp = 1.2,
  mean_nexp = 1.9, mean_sd_nexp = 0.9
)

Convert means of two groups and the pooled standard deviation into several effect size measures

Description

Convert means of two groups and the pooled standard deviation into several effect size measures

Usage

es_from_means_sd_pooled(
  mean_exp,
  mean_nexp,
  mean_sd_pooled,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  reverse_means
)

Arguments

Details

This function first computes a Cohen's d (D), Hedges' g (G) and mean difference (MD) from the means of two independent groups and the pooled standard deviation across the groups. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

To estimate a mean difference (formulas 12.1-12.6 in Cooper):

md = mean\_exp - mean\_nexp

md\_se = \sqrt{\frac{n_exp+n_nexp}{n_exp*n_nexp} * mean_sd_pooled^2}

md\_ci\_lo = md - md\_se * qt(.975, df = n\_exp + n\_nexp - 2)

md\_ci\_up = md + md\_se * qt(.975, df = n\_exp + n\_nexp - 2)

To estimate a Cohen's d the following formulas are used (formulas 12.10-12.18 in Cooper):

cohen\_d = \frac{mean\_exp - mean\_nexp}{means\_sd\_pooled}

cohen\_d\_se = \sqrt{\frac{(n\_exp+n\_nexp)}{n\_exp*n\_nexp} + \frac{cohen\_d^2}{2(n\_exp+n\_nexp)}}

cohen\_d\_ci\_lo = cohen\_d - cohen\_d\_se * qt(.975, df = n\_exp + n\_nexp - 2)

cohen\_d\_ci\_up = cohen\_d + cohen\_d\_se * qt(.975, df = n\_exp + n\_nexp - 2)

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Examples

es_from_means_sd_pooled(
  n_exp = 55, n_nexp = 55,
  mean_exp = 2.3, mean_nexp = 1.9,
  mean_sd_pooled = 0.9
)

Convert pre-post means of two independent groups into various effect size measures

Description

Convert pre-post means of two independent groups into various effect size measures

Usage

es_from_means_sd_pre_post(
  mean_pre_exp,
  mean_exp,
  mean_pre_sd_exp,
  mean_sd_exp,
  mean_pre_nexp,
  mean_nexp,
  mean_pre_sd_nexp,
  mean_sd_nexp,
  n_exp,
  n_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "bonett",
  pool_sd = FALSE,
  reverse_means_pre_post
)

Arguments

Details

This function converts pre-post means of two independent groups into a Cohen's d (D) and Hedges' g (G). Odds ratio (OR) and correlation coefficients (R/Z) are then converted from the Cohen's d.

Four approaches can be used to compute the Cohen's d.

In these approaches, the standard deviation of the difference within each group first needs to be obtained:

adj\_exp = 2*r\_pre\_post\_exp*mean\_pre\_sd\_exp*mean\_sd\_exp

sd\_change\_exp = \sqrt{mean\_pre\_sd\_exp^2 + mean\_sd\_exp^2 - adj\_exp}

adj\_nexp = 2*r\_pre\_post\_nexp*mean\_pre\_sd\_nexp*mean\_sd\_nexp

sd\_change\_nexp = \sqrt{mean\_pre\_sd\_nexp^2 + mean\_sd\_nexp^2 - adj\_nexp}

1. In the approach described by Bonett (pre_post_to_smd = "bonett"), one Cohen's d per group is obtained by standardizing the pre-post mean difference by the standard deviation at baseline (Bonett, 2008):

   cohen\_d\_exp = \frac{mean\_exp - mean\_pre\_exp}{mean\_pre\_sd\_exp}

   cohen\_d\_nexp = \frac{mean\_nexp - mean\_pre\_nexp}{mean\_pre\_sd\_nexp}

2. In the approach described by Cooper (pre_post_to_smd = "cooper" or "morris_drm"), the following formulas are used (Cooper et al., 2019; Morris & DeShon, 2002):

   cohen\_d\_exp = \frac{mean\_exp - mean\_pre\_exp}{sd\_change\_exp} * \sqrt{2 * (1 - r\_pre\_post\_exp)}

   cohen\_d\_nexp = \frac{mean\_nexp - mean\_pre\_nexp}{sd\_change\_nexp} * \sqrt{2 * (1 - r\_pre\_post\_nexp)}

3. In the approach described by Morris & DeShon (pre_post_to_smd = "morris_dz"), the pre-post mean difference is standardized by the standard deviation of the change (Morris & DeShon, 2002):

   cohen\_d\_exp = \frac{mean\_exp - mean\_pre\_exp}{sd\_change\_exp}

   cohen\_d\_nexp = \frac{mean\_nexp - mean\_pre\_nexp}{sd\_change\_nexp}

4. In the approach described by Morris (pre_post_to_smd = "morris_dav"), the pre-post mean difference is standardized by the average of the baseline and follow up SDs (Morris, 2008). Note that Morris (2008, p.384) recommended against this standardizer because its sampling variance was unknown to him; the variance used here is the heteroscedasticity-robust form of Bonett (2008, eq. 10/19):

   cohen\_d\_exp = \frac{mean\_exp - mean\_pre\_exp}{\sqrt{(mean\_pre\_sd\_exp^2 + mean\_sd\_exp^2)/2}}

   cohen\_d\_nexp = \frac{mean\_nexp - mean\_pre\_nexp}{\sqrt{(mean\_pre\_sd\_nexp^2 + mean\_sd\_nexp^2)/2}}

Last, the Cohen's d reflecting the within-group change from baseline to follow-up are combined into one Cohen's d:

cohen\_d = d\_exp - d\_nexp

cohen\_d\_se = \sqrt{cohen\_d\_se\_exp^2 + cohen\_d\_se\_nexp^2}

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_means_sd_pre_post(
  n_exp = 36, n_nexp = 35,
  mean_pre_exp = 98, mean_exp = 102,
  mean_pre_sd_exp = 16, mean_sd_exp = 17,
  mean_pre_nexp = 96, mean_nexp = 102,
  mean_pre_sd_nexp = 14, mean_sd_nexp = 15,
  r_pre_post_exp = 0.8, r_pre_post_nexp = 0.8
)

Convert pre-post means of a single group into standard effect size measures

Description

Convert pre-post means of a single group into standard effect size measures

Usage

es_from_means_sd_pre_post_single_group(
  mean_pre_exp,
  mean_exp,
  mean_pre_sd_exp,
  mean_sd_exp,
  n_exp,
  r_pre_post_exp = 0.8,
  pre_post_to_smd = "bonett",
  smd_to_cor = "viechtbauer",
  reverse_means_pre_post
)

Arguments

Details

This function converts pre-post means of a single group into within-group Cohen's d (dw), Hedges' g (gw), and mean difference (mdw). These within-group effect sizes quantify the standardized change from baseline to follow-up within one group. Odds ratio (OR) and correlation coefficients (R/Z) are then converted from Cohen's d.

Four formulas can be used to convert the pre/post means into a SMD (pre_post_to_smd argument):

1. "bonett": baseline SD standardizer (Bonett, 2008; equivalent to the SMCRH measure in metafor)

   d\_w = \frac{mean\_post\_exp - mean\_pre\_exp}{mean\_pre\_sd\_exp}

   var(g\_w) = \frac{sd\_change^2}{sd\_pre^2(n-1)} + \frac{g\_w^2}{2(n-1)}

   where

   sd\_change = \sqrt{sd\_pre^2 + sd\_post^2 - 2r\,sd\_pre\,sd\_post}

2. "cooper" (alias: "morris_drm"): raw score standardizer (Cooper 2019, Morris & DeShon 2002)

   d\_rm = \frac{mean\_post\_exp - mean\_pre\_exp}{sd\_change} * \sqrt{2 * (1 - r\_pre\_post\_exp)}

   var(d\_rm) = \frac{2 * (1 - r\_pre\_post\_exp)}{n\_exp} + \frac{d\_rm^2}{2 * n\_exp}

3. "morris_dz": change score standardizer (Morris & DeShon, 2002; equivalent to the SMCC measure in metafor)

   d\_z = \frac{mean\_post\_exp - mean\_pre\_exp}{sd\_change}

   var(g\_z) = \frac{1}{n} + \frac{g\_z^2}{2n}

4. "morris_dav": average SD standardizer (Bonett, 2008; equivalent to the SMCRP measure in metafor)

   d\_av = \frac{mean\_post\_exp - mean\_pre\_exp}{\sqrt{(mean\_pre\_sd\_exp^2 + mean\_post\_sd\_exp^2)/2}}

   g\_av = d\_av \times J(mi), \quad mi = \frac{2(n\_exp - 1)}{1 + r\_pre\_post\_exp^2}

   var(g\_av) = J^2 \left[\frac{2(1 - r)}{n} + \frac{d^2(1 + r^2)}{4n}\right]

The within-group Hedges' g is obtained by applying a bias correction factor J to d:

g\_w = d\_w * J(n\_exp-1)

The within-group mean difference is simply:

md\_w = mean\_post\_exp - mean\_pre\_exp

var(md\_w) = \frac{sd\_change^2}{n\_exp}

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied, treating the single group as a matched design.

Value

This function estimates and converts between several effect size measures.

References

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99-109.

Morris, S. B. (2008). Estimating effect sizes from pretest-posttest-control group designs. Organizational Research Methods, 11(2), 364-386. https://doi.org/10.1177/1094428106291059

Cooper, H., Hedges, L.V., & Valentine, J.C. (Eds.). (2019). The handbook of research synthesis and meta-analysis. Russell Sage Foundation.

Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7(1), 105-125. https://doi.org/10.1037/1082-989X.7.1.105

Examples

es_from_means_sd_pre_post_single_group(
  n_exp = 36,
  mean_pre_exp = 98, mean_exp = 102,
  mean_pre_sd_exp = 16, mean_sd_exp = 17,
  r_pre_post_exp = 0.8
)

Convert means and standard errors of two independent groups several effect size measures

Description

Convert means and standard errors of two independent groups several effect size measures

Usage

es_from_means_se(
  mean_exp,
  mean_se_exp,
  mean_nexp,
  mean_se_nexp,
  n_exp,
  n_nexp,
  smd_to_cor = "viechtbauer",
  smd_var = "borenstein",
  smd_denom = "pooled",
  reverse_means
)

Arguments

Details

This function converts the standard errors of two independent groups into standard deviations, and then relies on the calculations of the es_from_means_sd() function.

To convert the standard errors into standard deviations, the following formula is used.

mean\_sd\_exp = mean\_se\_exp * \sqrt{n\_exp}

mean\_sd\_nexp = mean\_se\_nexp * \sqrt{n\_nexp}

Then, calculations of the es_from_means_sd() are applied.

To estimate other effect size measures, calculations of the es_from_cohen_d() are applied.

Value

This function estimates and converts between several effect size measures.

References

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect size measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.3 (updated February 2022). Cochrane, 2022. Available from www.training.cochrane.org/handbook.

Examples

es_from_means_se(
  mean_exp = 42, mean_se_exp = 11,
  mean_nexp = 42, mean_se_nexp = 15,
  n_exp = 43, n_nexp = 34
)

Convert pre-post means of two independent groups into various effect size measures

Description

Convert pre-post means of two independent groups into various effect size measures

Usage

es_from_means_se_pre_post(
  mean_pre_exp,
  mean_exp,
  mean_pre_se_exp,
  mean_se_exp,
  mean_pre_nexp,
  mean_nexp,
  mean_pre_se_nexp,
  mean_se_nexp,
  n_exp,
  n_nexp,
  r_pre_post_exp,
  r_pre_post_nexp,
  smd_to_cor = "viechtbauer",
  pre_post_to_smd = "bonett",
  pool_sd = FALSE,
  reverse_means_pre_post
)

Arguments