A cross-level interaction occurs when a level-2 variable (e.g., school climate) moderates the relationship between a level-1 predictor (e.g., student SES) and the outcome (e.g., math achievement).
Formally, the two-level model is:
Level 1: \[math_{ij} = \beta_{0j} + \beta_{1j} \cdot SES_{ij} + e_{ij}\]
Level 2: \[\beta_{0j} = \gamma_{00} + \gamma_{01} \cdot Climate_j + u_{0j}\] \[\beta_{1j} = \gamma_{10} + \gamma_{11} \cdot Climate_j + u_{1j}\]
Substituting gives the composite model: \[math_{ij} = \gamma_{00} + \gamma_{10} SES_{ij} + \gamma_{01} Climate_j + \gamma_{11} SES_{ij} \times Climate_j + u_{0j} + u_{1j} SES_{ij} + e_{ij}\]
The term \(\gamma_{11}\) is the cross-level interaction: it captures how much the SES slope varies as a function of school climate.
For cross-level interactions, the recommended centering strategy is:
This separates within-school from between-school variance in SES, and makes the intercept interpretable as the school-average outcome at average climate.
dat <- mlm_center(school_data,
vars = "ses",
cluster = "school",
type = "group") # within-school SES
dat <- mlm_center(dat,
vars = "climate",
type = "grand") # grand-mean-centred climate
# Check
cat("Within-school SES mean (should be ~0):", round(mean(dat$ses_c), 4), "\n")
#> Within-school SES mean (should be ~0): 0
cat("Grand-mean climate (should be ~0):", round(mean(dat$climate_c), 4), "\n")
#> Grand-mean climate (should be ~0): 0Because we hypothesise that the SES slope varies across schools (hence the cross-level interaction), we include a random slope for SES.
mod <- lmer(
math ~ ses_c * climate_c + gender + (1 + ses_c | school),
data = dat,
REML = TRUE
)
summary(mod)
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: math ~ ses_c * climate_c + gender + (1 + ses_c | school)
#> Data: dat
#>
#> REML criterion at convergence: 18414.9
#>
#> Scaled residuals:
#> Min 1Q Median 3Q Max
#> -3.7607 -0.6591 -0.0070 0.6580 3.1325
#>
#> Random effects:
#> Groups Name Variance Std.Dev. Corr
#> school (Intercept) 7.841 2.8002
#> ses_c 0.168 0.4099 -0.28
#> Residual 24.916 4.9916
#> Number of obs: 3000, groups: school, 100
#>
#> Fixed effects:
#> Estimate Std. Error t value
#> (Intercept) 49.84009 0.30871 161.446
#> ses_c 1.42623 0.10133 14.075
#> climate_c 1.01234 0.28421 3.562
#> gendermale 0.07486 0.18608 0.402
#> ses_c:climate_c 0.46554 0.09932 4.687
#>
#> Correlation of Fixed Effects:
#> (Intr) ses_c clmt_c gndrml
#> ses_c -0.105
#> climate_c -0.002 0.000
#> gendermale -0.300 0.012 0.007
#> ss_c:clmt_c 0.001 -0.009 -0.105 -0.005The ses_c:climate_c interaction is the cross-level
interaction of interest.
probe <- mlm_probe(mod,
pred = "ses_c",
modx = "climate_c",
modx.values = "mean-sd",
conf.level = 0.95)
probe
#>
#> --- Simple Slopes: mlm_probe ---
#> Focal predictor : ses_c
#> Moderator : climate_c
#> Confidence level: 0.95
#>
#> climate_c slope se t df p ci_lower ci_upper
#> -1.036 0.944 0.145 6.504 2995 < .001 0.659 1.228
#> 0.000 1.426 0.101 14.075 2995 < .001 1.228 1.625
#> 1.036 1.909 0.144 13.277 2995 < .001 1.627 2.191Interpretation: At schools with above-average climate (+1 SD), the positive effect of SES on math is amplified. At schools with below-average climate (−1 SD), the SES effect is weaker (and may not be significant).
The JN interval pinpoints the exact climate values where the SES effect transitions from non-significant to significant.
jn <- mlm_jn(mod, pred = "ses_c", modx = "climate_c")
jn
#>
#> --- Johnson-Neyman Interval: mlm_jn ---
#> Focal predictor : ses_c
#> Moderator : climate_c
#> Alpha : 0.05
#> Moderator range : -3.026 to 2.254
#>
#> Johnson-Neyman boundary/boundaries ( climate_c ):
#> -2.088
#>
#> Interpretation: The simple slope of 'ses_c' is statistically
#> significant (p < 0.05) for values of 'climate_c'
#> between -2.07 and 2.254.mlm_plot(
mod,
pred = "ses_c",
modx = "climate_c",
modx.values = "mean-sd",
interval = TRUE,
x_label = "Student SES (group-mean centred)",
y_label = "Mathematics Achievement",
legend_title = "School Climate"
)
mlm_summary(mod,
pred = "ses_c",
modx = "climate_c",
modx.values = "mean-sd",
jn = TRUE)
#>
#> ========================================
#> Multilevel Moderation Summary
#> ========================================
#> Focal predictor : ses_c
#> Moderator : climate_c
#> Confidence level: 0.95
#>
#> --- Interaction Term ---
#> ses_c:climate_c b = 0.466 SE = 0.099 t(2995) = 4.687 p = < .001 [0.271, 0.66]
#>
#> --- Simple Slopes ---
#> climate_c slope se t df p 95% CI lo 95% CI hi
#> ------------------------------------------------------------------------------------
#> -1.036 0.944 0.145 6.504 2995.0 < .001 0.659 1.228
#> 0.000 1.426 0.101 14.075 2995.0 < .001 1.228 1.625
#> 1.036 1.909 0.144 13.277 2995.0 < .001 1.627 2.191
#>
#> --- Johnson-Neyman Region ---
#> JN boundary/boundaries for 'climate_c': -2.088
#>
#> ========================================When reporting a cross-level interaction, include:
Example write-up:
There was a significant cross-level interaction between student SES and school climate, \(\hat{\gamma}_{11}\) = [value], SE = [value], t([df]) = [value], p = [value]. Simple slope analysis indicated that the positive effect of SES on math achievement was significantly stronger at schools with high climate (+1 SD: \(b\) = [value], 95% CI [lo, hi]) than at schools with low climate (−1 SD: \(b\) = [value], 95% CI [lo, hi]). Johnson–Neyman analysis revealed that the SES slope was statistically significant for climate values above [JN boundary].