# Random effects vs. fixed effects in twin studies

I am conducting a study on identical twins, such that the data can be considered as paired observations. I want to run a co-twin control analysis on the data to study the relationship between the independent variable Entrepreneurship (binary variable, 1 being entrepreneur) and the dependent variable Stress (ordinal variable ranging from 1 to 3, with 1 being least stress).

I have followed the work of Hu et al. (1998) (see paper here) who apply the random-effects ordinal logistic model.

• Hu et al. (1998). Modelling ordinal responses from co-twin control studies, Statistics in Medicine, 17, 957Ð970.

My question is, why do they use a random-effects model? Why not fixed effects? What are the reasons for using a random-effects model instead of a fixed-effects model? I'm more familiar with fixed effects models, and can't seem to understand their justification of using random effects.

• I notice you are studying entrepreneurship, & your question contrasts "random effects models" vs "fixed effects models". OTOH, the paper you cite is from Statistics in Medicine. Part of the confusion might be that these names are used differently in econometrics than biostatistics. It might help you to read: What is a difference between random effects-, fixed effects- and marginal model? – gung Jan 19 '18 at 20:07

Their specific reasons, from the linked paper, include:

Use of the random-effects approach in modelling co-twin control data is intuitively appealing. We can think of the responses from each member of an MZ twin pair as repeated measures for one pair. The random-effects model assumes that the logit varies from one pair to the next by $\nu_i$. This assumption is reasonable because each pair of twins has its own unique shared genetic and environmental background. Thus, this variability reflects natural heterogeneity due to unmeasured genetic and environmental factors among all the twin pairs. This heterogeneity is represented by a Gausssian probability distribution, that is, the random pair effect, $\nu_i$, which is assumed normally distributed in the population.

and

One important feature of the random-effects approach is that it can readily accommodate missing data in the response variables.

and

In random-effects models, the effects of the individual-level confounding variables also involve both within-pair and between-pair comparisons. Thus, they may differ from the effects estimated through the conditional likelihood approach. In addition, we can estimate the effects of pair-level variables (for example, age) with use of the random-effects approach, but not with use of the conditional likelihood approach, though we can estimate the interactions between pair-level variables and the exposure using both approaches.

Do you have particular disagreements with these or are they unclear? Or are you looking for general principles that are not particularly tied to this study at all?

In terms of generally why you'd use a random effect, that's pretty complex and unfortunately different fields define "fixed effect" and "random effect" differently. There are also different names given to mixed-effects modeling, including "hierarchical" modeling. (Not to mention a lot of mixed-effects modeling now-a-days is Bayesian rather than Frequentist.)

In this particular study, they were looking at intra- and inter-twin-pair differences, so mixed-effects models naturally come to mind. As I understand it, a fixed-effect model will either pool completely across all observations (if you leave out a group variable), or will not pool at all across observations (if you include a group variable). A mixed-effects model allows for limited pooling, driven by the data. ("Borrowing strength" is the term.) Not sure how much borrowing strength applies in a paired study like this, though.

Last, since they are treating twins as repeated measurements, their errors will be correlated, which violates the independence assumption of a fixed-effects model.

• Thank you for your detailed answer! All the points you have made are really helpful! My question was related to Monozygotic (identical) twins specifically since these twins can be regarded as repeated measures of the same subject. I'm not quite sure I understand the last point you made however. While I understand the idea of when to use fixed vs. random effects, I still don't quite grasp why a fixed effects model wouldn't work here, where each dummy variable would represent the individuals in a twin pair. Can you please elaborate more on this point? – roland Jul 25 '16 at 16:08
• @roland: I may be wrong, but the errors of the twins will be correlated and that is not allowed in fixed-effects regressions. I don't do such models so can't give you details, but if you search for "fixed effect model strict exogeneity" you'll find lectures like: schmidheiny.name/teaching/panel2up.pdf which hopefully can clarify whether this is an issue for you or not. – Wayne Jul 25 '16 at 17:30
• @Wayne, re: "the errors of the twins will be correlated and that is not allowed in fixed-effects regressions", see my comment to the OP above. – gung Jan 19 '18 at 20:09