# Difference between t-test on betas from individual regressions vs linear mixed modeling

I often come across the following practice in my field; for example, people want to predict participants responses on a Dependent variable (e.g 0 or 1) based on a few independent variables - continuous, ordinal and categorical (that vary at a trial level. E.g. DV ~ IV1 +Iv2 + IV3

Typically one would fit a regression on an individual participant level, that is a regression for each participant and then get the betas for each participant (that is n betas where n is the number of participants) and submit it against a single sample t-test. I can see that this ensures that some individual variability is accounted for at the level of the participant.

But what is the benefit of doing this compared to fitting a linear mixed model (in this case a generalised binomial mixed model) where participant id and trial id can be specified as random factors?

But what is the benefit of doing this compared to fitting a linear mixed model

I don't think there is much, if any, benefit. Is it really the "typical" approach in your field ? Whatb field is this, out of interest ? There are a few things to note here:

• by fitting a mixed model you are making much more use of the data, than fitting individual models. That is, you lose a lot of statistical power with individual regressions due to the small sample sizes.

• the mixed model is more flexible in that you can also fit random slopes in addition to random intercepts, if the underlying theory and data support this. Doing so allows you to specify that one of more fixed effects also vary accross participants and/or trials.

• one possible downside to the mixed model approach is that the random effects are assumed to follow a normal distribution, and this may not be appropriate in some situations. On the other hand, in my experience mixed models are quite robust to even quite severe departures.

• if you have very few participants or trials, then a mixed model may not be a good idea - however you could still fit a global model with participants / trials as fixed effects.

• the fixed effects estimates for generalised linear mixed models are conditional on the random effects. This may not always be what is required. On the other hand, some mixed model packages are able to compute marginal as well as conditional estimates.

• Thank you @RobertLong, this supports some of what I had in mind my field is Experimental Psychology, it's not necessarily a common practice of the entire field but rather specific research areas. Following on the number of trials, I have another question, and I know from experience trying to simulate data that there is not one fit all answer for when it comes sample size, but I wonder of there is a minimum required (for soundness) sample of participants and trials for mixed models, say a general rule akin to e.g correlations are not stable with < ~ 25 observations? Thank you Oct 7, 2020 at 13:45
• You're welcome. You are quite right that there is no simple rule of thumb or consensus. With 25 I think most would agree that this is sufficient. It's when you get into single figures that some people object. Others seem happy with a minimum of 6. It will also depend on the number of measurements per participant / trial. Are these crossed or nested factors ? Oct 7, 2020 at 13:55
• Thanks again, participants and trials are crossed, the fixed effects predictors are also crossed Oct 7, 2020 at 14:54
• No probs. I thought that would be the case. Oct 7, 2020 at 15:17