Sal Mangiafico Thank you for your comment. I am asking this because my concern is, would a binary DV violate the assumptions of mixed model ANOVA, which state that the DV must be continuous and approximately normally distributed?

Oh... I misunderstood the question... With a binary DV, you should use logistic regression. You know this variable can't be conditionally normal or even continuous. I don't know about SPSS, but software can conduct mixed effects (for repeated measures) logistic regression.

Sal Mangiafico I am still relatively new to data analysis, so sorry if I am asking a stupid question, but cuz from my understanding, logistic regression is used when you want to know whether the IV can predict the binary DV. However, I am more interested in knowing whether the two (or more) groups in the between-subject IV differ significantly in their responses of the binary DV at this stage. In this case, may I know whether using repeated measure logistic regression is still relevant?

In SPSS, the GENLINMIXED command can be used to estimate a multilevel (or mixed effects) logistic regression model.

• https://www.ibm.com/docs/en/spss-statistics/28.0.0?topic=reference-genlinmixed

Chapter 4 in this book should be helpful, if you can get your hands on it:

• https://books.google.ca/books/about/Multilevel_Modeling_of_Categorical_Outco.html?id=i9D2Hu2OFW0C&redir_esc=y

Logistic regression can have multiple IV's of different types (nominal, continuous). Software can build a mixed effects model to deal with the repeated measures aspect.

I am concerned with the formulation "...IV differ significantly in their responses...". Typically, you should be interested in how much the groups differ and and 1) if this is relevant from a practical point of you (practical relevance) and in a 2) if this difference is also statistically different and you have evidence for the sign/direction of your effect. People tend to start with 2) and in case of significance they totally forget about the relevance, e.g. the effect might be significant, but too small to be of practical relevance. Furthermore, only looking at the significance may be misleading if you do not know what is siginificant.

I point this out, because in logistic regression (as suggested by Bruce Weaver and Sal Mangiafico ) you do not estimate differences between the groups anymore, but log-odds, which can be "backtransformed" into odds ratios or probabilites. Therefore, you should be very careful, since a significant results in that case will not tell you anything about the difference between the groups as such.

Hi Rainer Duesing. Did you mean to say that the difference in log-odds is "backtransformed" (i.e., exponentiated) to yield an odds ratio?

**EDIT/CORRECTION** The statement about differences in log-odds by me, is not correct! In logistic regression the regression coefficients will give the difference in log-odds.

No, that is not what I wanted to say, sorry if this was unclear.

Quite the contrary, I wanted to express that you cannot directly infer the differences in probabilities, nor the difference in log-odds (whatever that may mean), from the logistic regression coefficients. Unlike OLS regression for example with dummy coding, the β1​ regression coefficient in logistic regression may not be interpreted as a difference at all.

What you can do is to exponentiate the regression coefficient β1​ and get the odds ratio. But this is not a difference and does not have an additive interpretation, but a multiplicative.

Researchers may be more interested in the probabilites for each group (e.g. control vs. experimental group) or their differences. You can get this via the inverse logit function, plogis, but you need also the β0 coefficient.

logit−1(β0​+β1​)−logit−1(β1​)

which can be calculated by hand (or with marginaleffects package for example) in R like this, if you use the glm function:

plogis(coef(mod.1)[1] + coef(mod.1)[2]) - plogis(coef(mod.1)[1])

With a dichotomous predictor (control vs. experimental) this will give you the difference in probabilities.

So, all I wanted to say that it is not sufficient to know that your coefficient is significant, if you do not know WHAT exactly is significant. In logistic regression, it is not the difference at all.

Imbens, G. W., & Rubin, D. B. (2015). Causal inference in statistics, social, and biomedical sciences: An Introduction. Cambridge University Press.

Hi Rainer Duesing. I must be misunderstanding your point. I say that because when I estimate a logistic regression model on a 2x2 table, the z-test that is shown for the OR is the same z-test that is shown for the difference in log-odds. I can also see that the OR = Odds1/Odds2 = Exp(LogOdds1 - LogOdds2). Therefore, I find it baffling when you say that one cannot infer anything about the "difference in log-odds (whatever that may mean), from the logistic regression coefficients".

I've attached a text file showing an example using Stata. I think the calculations should be fairly easy to follow even for folks who do not use Stata. I hope it helps you work out how/why I am failing to grasp your point. ;-)

Cheers,

Bruce

Thx Bruce for pointing that out! Maybe I misunderstood something. I will have a look at your file asap. Many thanks!

It's better to check www.Stats4Edu.com

Dear Bruce Weaver , sorry for the long delay to answer. You are absolutely right, my answer was not correct about the differences in the log-odds. Somehow, I was thinking in a totally wrong direction, maybe because I find the log-odds very unintuitive to grasp. That's why I would prefer the transformations proposed in my former post.

But nonetheless, my answer was not correct concerning the differences!

I will correct my former post.

For binary dependent, there are specific models that allow independent continuious variables such as logistic regression, probit ..or multinomial for more than 2 outcomes. For binary IV dummies can be created and fioolw dummy variable regression.