1.1 Outline of NORTA

The goal of the NORTA procedure is to produce identically independently distributed (iid) samples from random variables with a given correlation structure (Pearson correlation matrix) and given marginal distributions, thereby e.g. approximating existing datasets.

Following Ghosh and Henderson (2003), we want to sample iid replicates of the random vector \(X = (X_1, X_2, \ldots, X_k)\). Denote by \(F_i(s) = P(X_i \leq s)\) the distribution functions (i.e. the marginal distributions) of the components of \(X\), and by \(\Sigma_X\) the \(k \times k\) correlation matrix of \(X\). Then NORTA proceeds as follows:

• Generate multivariate standard normal random vectors (i.e mean 0, variance 1) \(Z = (Z_1, Z_2, \ldots, Z_k)\) with a correlation matrix \(\Sigma_Z\).
• Compute the random vector \(X\) via \(X_i := F_i^{-1}(\Phi(Z_i))\), where \(\Phi\) denotes the distribution function of the standard normal distribution, and \(F_i^{-1}(t) := \inf\{x: F_i(x) \geq t\}\) is the quantile function of \(X_i\).

The resulting vector \(X\) then has the desired marginal distribution. To obtain the target correlation structure \(\Sigma_X\), the correlation matrix \(\Sigma_Z\) for the first step has to be chosen appropriately. This can be achieved via solving univariable optimisation problems for each pair of variables \(X_i\) and \(X_j\) in \(X\) and is part of the simdata package.

1.2 Caveats of NORTA

The NORTA procedure has some known limitations, which may lead to discrepancies between the target correlation structure and the correlation structure obtained from the sampling process. These are, however, partly alleviated when using existing datasets as templates, or by special techniques within simdata.

• Not all combinations of given marginal distributions and target correlation are feasible by the nature of the variables. This is not an issue when an existing dataset is used as template, since that demonstrates that the combination exists.
• The optimisation procedure to obtain \(\Sigma_Z\) may lead to a matrix which is not positive definite (since the optimisation is only done for pairs of variables), and therefore not a proper correlation matrix. To alleviate this, the simdata package ensures positive definiteness by using the closest positive definite matrix instead. This may lead to discrepancies between the target correlation and the achieved correlation structure.
• NORTA cannot reproduce non-linear relationships between variables. This may lead to issues for continuous variables and categorical variables with more than two categories when the goal is to faithfully re-create a real dataset that features non-linear relations.
• The optimisation procedure to obtain \(\Sigma_Z\) may take a while to compute when the number of variables increases. This is alleviated through the fact that this computation has to be done only a single time, during definition of the simulation design. All further simulation iterations only use the optimisation result and are therefore not subject to this issue.
• When applied to an existing dataset, NORTA relies on the estimation of the target correlation matrix and marginal distributions. More complex data (e.g. special marginal distributions, complex correlation structure) therefore requires more observations for an accurate representation.

1.3 Comparison to other methods

NORTA is well suited to re-create existing datasets through an explicit parametric approximation. Similar methods exist, that achieve this through other means. A particularly interesting alternative is the generation of synthetic datasets using an approach closely related to multiple imputation, and is implemented in e.g. the synthpop R package (Nowok, Raab, and Dibben (2016)). Its’ primary aim is to achieve confidentiality by re-creating copies to be shared for existing, sensitive datasets.

In comparison, synthpop potentially offers more flexible data generation than NORTA, thereby leading to a better approximation of an original dataset. However, synthpop is also more opaque than the explicit, user defined specification of correlation and marginal distributions of NORTA. This also entails that synthpop can be generally used more like a black-box approach, which requires little user input, but is also less transparent than the manual curation of the simulation setup in NORTA. Furthermore, NORTA allows easy changes to the design to obtain a wide variety of study designs from a single template dataset, whereas synthpop is more targeted at re-creating the original dataset. Both methods therefore have their distinct usecases and complement each other.

2.1 Quantile functions for some common distributions

The required marginal distributions are given as quantile functions. R provides implementations of many standard distributions which can be directly used, see the help on distributions. The quantile functions use the prefix “q”, as in e.g. qnorm or qbinom. Further implementations can be found in the packages extraDistr, actuar and many others (see https://CRAN.R-project.org/view=Distributions).

3.1 Load dataset

First we will load the dataset and extract several variables of interest, namely gender (‘Gender’), age (‘Age’), race (‘Race’), weight (‘Weight’), bmi (‘BMI’), systolic (‘BPsys’) and diastolic blood pressure (‘BPdia’). These variabes demonstrate several different kinds of distributions. For a detailed description of the data, please see the documentation at https://www.cdc.gov/nchs/nhanes.htm and for the NHANES R package on CRAN. Here we are not concerned with the exact codings of the variables, so we will remove labels to factor variables and work with numeric codes. Further, we will only use data from a single survey round to keep the dataset small.

if (requireNamespace("NHANES", quietly = TRUE)) {
    df = NHANES::NHANES %>% 
        filter(SurveyYr == "2011_12") %>%
        select(Gender, Age, Race = Race1, Weight, 
               BMI, BPsys = BPSys1, BPdia = BPDia1) %>% 
        filter(complete.cases(.)) %>% 
        filter(Age > 18) %>% 
        mutate(Gender = if_else(Gender == "male", 1, 2), 
               Race = as.numeric(Race))
    
    print(head(df))
} else {
    message("Package 'NHANES' not available.")
}

## # A tibble: 6 x 7
##   Gender   Age  Race Weight   BMI BPsys BPdia
##    <dbl> <int> <dbl>  <dbl> <dbl> <int> <int>
## 1      2    43     1   98.6  33.3   100    70
## 2      1    80     4   85.9  28.5   124    72
## 3      1    35     4   89    27.9   108    62
## 4      1    57     4   96.9  28     112    70
## 5      1    57     4   96.9  28     112    70
## 6      1    57     4   96.9  28     112    70

3.2 Estimate target correlation

Using this dataset, we first define the target correlation cor_target and plot it.

cor_target = cor(df)
ggcorrplot::ggcorrplot(cor_target, lab = TRUE)

3.3 Define marginal distributions

Further, we define a list of marginal distributions dist representing the individual variables. Each entry of the list must be a function in one argument, defining the quantile function of the variable. The order of the entries must correspond to the order in the target correlation cor_target.

dist_auto = quantile_functions_from_data(df, n_small = 15) 

dist = list()

# gender
dist[["Gender"]] = function(x) qbinom(x, size = 1, prob = 0.5)

# age
dens = stats::density(df$Age, cut = 1) # cut defines how to deal with boundaries
# integrate
int_dens = cbind(Age = dens$x, cdf = cumsum(dens$y))
# normalize to obtain cumulative distribution function
int_dens[, "cdf"] = int_dens[, "cdf"] / max(int_dens[, "cdf"])
# derive quantile function
# outside the defined domain retun minimum and maximum age, respectively
dist[["Age"]] = stats::approxfun(int_dens[, "cdf"], int_dens[, "Age"], 
                          yleft = min(int_dens[, "Age"]), 
                          yright = max(int_dens[, "Age"]))

# race
dist[["Race"]] = function(x) 
    cut(x, breaks = c(0, 0.112, 0.177, 0.253, 0.919, 1), 
        labels = 1:5)

# weight
fit = fitdistrplus::fitdist(as.numeric(df$Weight), "gamma")
summary(fit)
dist[["Weight"]] = function(x) qgamma(x, shape = 16.5031110, rate = 0.2015375)

# bmi
fit = fitdistrplus::fitdist(as.numeric(df$BMI), "lnorm")
summary(fit)
dist[["BMI"]] = function(x) qlnorm(x, meanlog = 3.3283118, sdlog = 0.2153347)

# systolic blood pressure
fit = fitdistrplus::fitdist(as.numeric(df$BPsys), "lnorm")
summary(fit)
dist[["BPsys"]] = function(x) qlnorm(x, meanlog = 4.796213, sdlog = 0.135271)

# diastolic blood pressure
fit = fitdistrplus::fitdist(as.numeric(df %>% 
                                         filter(BPdia > 0) %>% 
                                         pull(BPdia)), "norm")
summary(fit)
dist[["BPdia"]] = function(x) qnorm(x, mean = 71.75758, sd = 11.36352)

3.4 Simulate data

Now we can use simdata::simdesign_norta to obtain designs using both the manual and automated marginal specifications. After that, we simulate datasets of the same size as the original data set using simdata::simulate_data, and compare the resulting summary statistics and correlation structures.

# use automated specification
dsgn_auto = simdata::simdesign_norta(cor_target_final = cor_target, 
                                     dist = dist_auto, 
                                     transform_initial = data.frame,
                                     names_final = names(dist), 
                                     seed_initial = 1)

simdf_auto = simdata::simulate_data(dsgn_auto, nrow(df), seed = 2)

# use manual specification
dsgn = simdata::simdesign_norta(cor_target_final = cor_target, 
                                dist = dist, 
                                transform_initial = data.frame,
                                names_final = names(dist), 
                                seed_initial = 1)

simdf = simdata::simulate_data(dsgn, nrow(df), seed = 2)

3.5 Results

Summary statistics of the original and simulated datasets.

summary(df)

##      Gender           Age             Race           Weight      
##  Min.   :1.000   Min.   :19.00   Min.   :1.000   Min.   : 39.30  
##  1st Qu.:1.000   1st Qu.:32.00   1st Qu.:3.000   1st Qu.: 66.58  
##  Median :1.000   Median :46.00   Median :4.000   Median : 79.40  
##  Mean   :1.494   Mean   :46.97   Mean   :3.539   Mean   : 81.88  
##  3rd Qu.:2.000   3rd Qu.:60.00   3rd Qu.:4.000   3rd Qu.: 93.20  
##  Max.   :2.000   Max.   :80.00   Max.   :5.000   Max.   :198.70  
##       BMI            BPsys           BPdia       
##  Min.   :15.70   Min.   : 74.0   Min.   :  0.00  
##  1st Qu.:23.80   1st Qu.:110.0   1st Qu.: 64.00  
##  Median :27.60   Median :120.0   Median : 72.00  
##  Mean   :28.57   Mean   :122.2   Mean   : 71.63  
##  3rd Qu.:31.90   3rd Qu.:132.0   3rd Qu.: 78.00  
##  Max.   :80.60   Max.   :224.0   Max.   :110.00

summary(simdf_auto)

##      Gender           Age             Race           Weight      
##  Min.   :1.000   Min.   :19.00   Min.   :1.000   Min.   : 39.30  
##  1st Qu.:1.000   1st Qu.:32.00   1st Qu.:3.000   1st Qu.: 65.50  
##  Median :1.000   Median :46.00   Median :4.000   Median : 78.90  
##  Mean   :1.499   Mean   :46.72   Mean   :3.522   Mean   : 81.04  
##  3rd Qu.:2.000   3rd Qu.:60.00   3rd Qu.:4.000   3rd Qu.: 92.49  
##  Max.   :2.000   Max.   :80.00   Max.   :5.000   Max.   :143.60  
##       BMI            BPsys           BPdia       
##  Min.   :15.70   Min.   : 74.0   Min.   :  0.00  
##  1st Qu.:23.60   1st Qu.:112.0   1st Qu.: 64.00  
##  Median :27.30   Median :120.0   Median : 72.00  
##  Mean   :28.34   Mean   :122.0   Mean   : 71.31  
##  3rd Qu.:31.90   3rd Qu.:132.0   3rd Qu.: 78.00  
##  Max.   :48.23   Max.   :174.6   Max.   :102.00

summary(simdf)

##      Gender            Age             Race           Weight      
##  Min.   :0.0000   Min.   :16.01   Min.   :1.000   Min.   : 27.56  
##  1st Qu.:0.0000   1st Qu.:32.10   1st Qu.:4.000   1st Qu.: 66.90  
##  Median :1.0000   Median :46.05   Median :4.000   Median : 79.85  
##  Mean   :0.5098   Mean   :46.85   Mean   :3.542   Mean   : 81.75  
##  3rd Qu.:1.0000   3rd Qu.:60.21   3rd Qu.:4.000   3rd Qu.: 94.44  
##  Max.   :1.0000   Max.   :82.91   Max.   :5.000   Max.   :175.01  
##       BMI            BPsys            BPdia       
##  Min.   :12.57   Min.   : 67.03   Min.   : 34.98  
##  1st Qu.:23.90   1st Qu.:111.23   1st Qu.: 64.42  
##  Median :27.67   Median :121.13   Median : 71.99  
##  Mean   :28.53   Mean   :122.61   Mean   : 72.00  
##  3rd Qu.:32.44   3rd Qu.:132.88   3rd Qu.: 79.73  
##  Max.   :56.66   Max.   :189.50   Max.   :115.30

Correlation structures of the original and simulated datasets.

We may also inspect the continuous variables regarding their univariate and bivariate distributions. The original data is shown in black, the simulated data is shown in red. (Note that we only use the first 1000 observations to speed up the plotting.)

From this we can observe, that the agreement between the original data and the simulated data is generally quite good. Both, automated and manual specification work equally well for this dataset. Note, however, that e.g. the slightly non-linear relationship between age and diastolic blood pressure cannot be fully captured by the approach, as expected. Furthermore, the original data shows some outliers, which are also not reproducible due to the parametric nature of the NORTA procedure.