What is the difference between z.ratio and t.ratio in the pairwise comparison output using emmeans function?

Question

I don't understand why the output of pairwise comparison using emmeans function is z.ratio when analysing response time data. What is the difference between z.ratio and t.ratio? And is this reasonable that the estimate is too large? Here is my code.

# model    
e1_model1 <- lmer(data = mix1, formula = RT ~ soc_s*persp_n*consis_c + (1 + soc_s*persp_n*consis_c|Subject), REML=TRUE, control = lmerControl(optimizer = "bobyqa",optCtrl=list(maxfun=(20000))))
summary(e1_model1)

anova(e1_model1)

# pairwise comparison
e1 <- emmeans(e1_model1, pairwise~soc_s|persp_n + consis_c, adjust = "mvt")
e1$contrasts %>%
     summary(infer = TRUE)

And this is pairwise comparison output

How can I command to get t.ratio?

Answer 1

The test statistic has the same form in either case, a difference divided by its standard error, but the comparison is either against a t distribution with the specified number of degrees of freedom or against the limiting normal distribution (an infinite number of degrees of freedom) in a z test.

There is some disagreement about the best way to evaluate the number of degrees of freedom in a linear mixed model like yours. See this answer, or several other pages on this site, for an introduction to the statistical issues.

A vignette for emmeans notes the methods for estimating degrees of freedom that are available in that package. If you do not have the necessary packages installed for the "kenward-roger" or "satterthwaite" estimates of the number of degrees of freedom, the software will use the "asymptotic" method based on a z test. That's probably what happened in your case.

Install the necessary packages and specify the way that you want the degrees of freedom to be estimated. If you have a very large number of observations, there won't be much of a difference between the t- and z-test results, and with a large enough sample even a small absolute difference can have a very small p value.