Setup

Functions for Reanalysis

stdCoef.merMod <- function(object) {
  sdy <- sd(getME(object,"y"))
  sdx <- apply(getME(object,"X"), 2, sd)
  sc <- fixef(object)*sdx/sdy
  se.fixef <- coef(summary(object))[,"Std. Error"]
  se <- se.fixef*sdx/sdy
  return(data.frame(stdcoef=sc, stdse=se))}

ZREG <- function(B1, B2, SEB1, SEB2) {
  Z = (B1-B2)/sqrt((SEB1)^2 + (SEB2)^2)
  return(Z)}

Rationale

Richardson (2021) implied he found a sex difference in the effects of sex ratio on wellbeing. He could not legitimately test for an interaction by sex due to singularity but also did not test for a difference in the relationship by sex, instead preferring to conclude that there was a difference based on significance levels in the groups separated by sex. The validity of the conclusion has to be tested, as that method alone cannot, in principle, provide any support for it. Note that there were no corrections for multiple comparisons included in the study either. I do not deal with that issue here.

Reanalysis

I compare the male model 1 ASR \(\beta\) with the female models 1 through 3 \(\beta\) values, followed by comparing the male model 2 to the same models.

m1 <- stdCoef.merMod(men1)
m2 <- stdCoef.merMod(men2)

w1 <- stdCoef.merMod(women1)
w2 <- stdCoef.merMod(women2)
w3 <- stdCoef.merMod(women3)
#Model 1 versus Models 1 - 3

ZREG(m1$stdcoef[2], w1$stdcoef[2], m1$stdse[2], w1$stdse[2])
## [1] -0.7703652
ZREG(m1$stdcoef[2], w2$stdcoef[2], m1$stdse[2], w2$stdse[2])
## [1] -0.2278899
ZREG(m1$stdcoef[2], w3$stdcoef[2], m1$stdse[2], w3$stdse[2])
## [1] -0.4897749
#Model 2 versus Models 1 - 3

ZREG(m2$stdcoef[2], w1$stdcoef[2], m2$stdse[2], w1$stdse[2])
## [1] -1.649519
ZREG(m2$stdcoef[2], w2$stdcoef[2], m2$stdse[2], w2$stdse[2])
## [1] -1.146719
ZREG(m2$stdcoef[2], w3$stdcoef[2], m2$stdse[2], w3$stdse[2])
## [1] -1.424246

None of these are conventionally significant. One of the differences is actually at a Z of 1.6495, which is very close to p = 0.05 for a one-sided test. With the relatively large samples used here, thereโ€™s definitely no room to conclude there were male-female differences (especially since the p value should be scaled to maintain comparability with the standard 0.05 level anyway). The number of observations in the male model 2 is 14034 with 133 regions and 11 countries; in the female model 3, it is 14772 with 133 regions and 11 countries; scaling the p values, they would need to be 5.28e-06 and 4.96e-06 for the male and female groups, respectively. The fixed effects help to compound the issue by reducing the variance (but this is not really a bad thing). Overall, the study showcased no support for an interaction - though it could not even be formally tested - and, as such, it was unjustified to conclude one appeared (i.e., concluding there was differential prediction by group, a finding that resulted from capitalizing on chance and did not warrant substantial interpretation).

References

Richardson, T. (2021). The adult sex ratio of European regions predicts female, but not male, subjective well-being. Current Psychology. https://doi.org/10.1007/s12144-021-01619-5