# Effect size in GLMM

In the output of a GLMM, using a binary variable as response variable and continuous variables as explanatory variables [family = binomial(link="logit")], I obtain, for each variable, an estimate value, standard error, a z-value and a Pr(>|z|).

1) Is the z-value simmilar to the effect size?

2) If not, how can I obtain the effect size for each variable?

• 1: No, it's a test statistic to test the null hypothesis that the estimate is zero. 2: don't you mean "how can I obtain the effect size" ? Since you don't say what kind of glmm you are running, this is very hard to say, – Robert Long Jul 31 '16 at 14:09
• @RobertLong Yes, I meant effect size, the question is updated! – mtao Jul 31 '16 at 14:14

1) Is the z-value similar to the effect size?

No, it is a Wald statistic to test the null hypothesis that the estimate is zero.

2) If not, how can I obtain the effect size for each variable?

Since this is a generalized linear mixed model, you can't calculate effect sizes such as cohen's d, but since it is a logistic model with a logit link you can report odds ratios as effect sizes. The raw coefficients are on the log-odds scale, so to calculate the odds ratios, these are just exponentiated.

• Thank you for your answer! Just didn't understand very well how to obtain the odds ratios; do I need to exponentiate each estimate value? – mtao Jul 31 '16 at 14:33
• @Teresa yes that is correct. – Robert Long Jul 31 '16 at 15:02
• Ok, thanks!! Do you know of any book/paper with that information that I can cite? And which one (effect size/ Wald statistics) do you suggest to report relative differences between variabes? – mtao Jul 31 '16 at 15:09
• @Teresa try Hosmer Jr, David W., and Stanley Lemeshow. Applied logistic regression. John Wiley & Sons, 2004. – Robert Long Jul 31 '16 at 15:19
• One other thing: I have my variables transformed (z-score),so does that change the interpretation of the odds ratio? Which, I assume, is: higher odds ratio, higher "importance" of a given variable. – mtao Jul 31 '16 at 16:13

I´ll be trying to help you with your second question. Recently was added a function in the package emmeans that calculate the Effect Size, the Cohen´s d. To use it, you will need the GLMM adjusted, the Sigma, and the df. But, to calculate the Sigma from a Mixed Model had some trick. Let me copy here a part of the information you can find at https://rdrr.io/cran/emmeans/man/eff_size.html

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Oats.lme <- lme(yield ~ Variety + factor(nitro), random = ~ 1 | Block / Variety, data = Oats)

VarCorr(Oats.lme)

# Combine variance estimates

totSD <- sqrt(214.4724 + 109.6931 + 162.5590); emmV <- emmeans(Oats.lme, ~ Variety); print(eff_size(emmV, sigma = totSD, edf = 5)); print(eff_size(emmV, sigma = totSD, edf = 51)

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