# T tests on proportions - Wrong, but how wrong?

Background: In psychology, and probably a number of other disciplines, it's common practice to test between-groups effects on a binary variable, such as accuracy, by aggregating data within participants, and then running a t test on the aggregates.

aggregate.data = data %>%
group_by(subject_nr, condition) %>%
summarise(accuracy=mean(accuracy))

t.test(aggregate.data[aggregate.data$condition==0,]$accuracy,
aggregate.data[aggregate.data$condition==1,]$accuracy,
paired=T)


We all know at this stage that it's wrong to analyse proportion data like this using t tests/ANOVA. Researcher should at least by applying the arcsine transform to normalize the data (which I've never seen in a psychology journal), but ideally should use multilevel logistic regression

glmer(accuracy ~ condition + (1|subject_nr), data=data, family=binomial)


By way of an example, I was just reading this study, with 61 participants, which reports,

A large decrease in the proportion of base-rate responses was evident for incongruent relative to congruent items, t(60) = 11.66, SE = .04, p < .001, d = 1.49.

Question: We all know this is bad practice, but how bad of a practice is it?

It's hard to know if the matter is a minor statistical squabble, a problem for t tests which are only just significant (say p > .01), or something which casts doubt on the results of thousands of studies.

More practically, analyzing my own data, while I know that the logistic mixed model is the right tool for the job, I see untransformed t tests used in all of the top journals. Am I actually hurting my chances of publication by using the less well-known analysis?

• I think the problem might be many medical journals or psychology journals do not invite statisticians to review their papers. Therefore, many papers with inappropriate statistical methods can be published. – Deep North Jun 30 '15 at 13:56
• Part of the answer is available in a thread comparing the t-test to logistic regression. It sheds some light on when the "ideal" approach might be preferred and when it should not (because it may be a less powerful way to achieve the study objectives). Even use of the arcsine transform is questionable when the groups have different sizes. This leads me to ask, to what are you referring by "bad practice"? The t-test? The arcsine? The logistic regression? All three? Or maybe a policy of using one procedure regardless of the nature of the data? – whuber Jun 30 '15 at 14:06
• From my limited understanding (i.e. this paper), untransformed t test < arcine < multilevel logistic model. – Eoin Jun 30 '15 at 14:44
• In the linked question, the analysis is a little different, in that they're not talking about analysing proportions with a t test, but using the binary outcome as predictors in one. See my edit to the question. – Eoin Jun 30 '15 at 14:46
• There are some old simulation studies from the 1950's showing ANOVA on binary data works OK. Of course, it probably depends on the proportions and sample size. – David Lane Mar 19 '17 at 18:23