# GLMM to test change between two periods

I have made an analysis to test whether the weight of a mice population has changed between two periods. Data have been collected in the period 1978-81 and 2005-07. Many mice were captured through the whole set of years (all months) in the first period and a much smaller number was captured in the second period. The unbalanced data are due to a different capture effort in the field and to a strong reduction of mice population size.

Weight may vary according to: (i) sex, (ii) age class, (iii) month of capture, (iv) total body length, and (iv) type of capture (e.g. individuals captured by live- and death-traps, both used, may tend to provide data on individuals of different average conditions or age). Moreover, a random deviation in weight may be result of environmental stochasticity with some years "better" and other "worse". For this reason, to test whether weight has changed BETWEEN the two periods I have chosen to use a GLMM for each sex by including all the predictors I mentioned before (using notation of package lme4 in R):

weight ~ age + month + TBL + TypeCap + year + (1|yearF) where all the predictors are categorical except TBL (total body length, continuous variable) and year (integer variable, i.e. 1978,1979,1980,1981,2005,2006,2007 - yearF is categorical). I have checked my model visually as usually and looked also for influential point (individuals or years). The model seems really OK, it converges well, it has good fit to data, and does not seem overfitted.

I find a strong and significant decrease over time (year effect highly significant). According to the first and last year (1978 and 2007) point estimates and net of the above mentioned controlling variables, it would correspond to a decrease of about 30% (similar for both females and males).

So, what is my question about? I feel fine with this approach and the interpretation I am doing of the results but, until now, reviewers (this forms part of a scientific manuscript) do not seem to agree. There have been already three reviewers who claim against using the GLMM because there are not data in the middle and suggest some type of ANOVA test. I do not agree and would like to convince them that there is nothing wrong in using a GLMM to test whether there has been a change between the beginning and final of period study even if we do not have data in the middle. What actually matters is that we cannot say nothing about what happens in the middle (and in fact we do not) but we can do it about the change in weight between 1978 and 2007. We never talk about any linear trend or anything similar. By using the GLMM we can control for several potential confounding variables (e.g age, month, capture type, etc.) and also for random effect as that of year. However, I don't see an immediate way of doing that by using some kind of t-test.

I would appreciate any thought, commentary or suggestion (even some reference I could cite) on this.

You have no reason to add random effect in your model. Random effect is used to model the correlation between error terms.

The general linear model is:

$Y=X\beta + \epsilon$

We assume that $\epsilon$ ~ $N(0, I\sigma^2)$.

If you find the assumption above is not good and $\epsilon$ ~ $N(0, \Sigma)$, and $\Sigma$ has non-zero off-diagonal elements, it is the time to add random effect to incorporate these non-zero off-diagonal elements.

But from your description, I could not find any evidence of existence of non-zero off-diagonal elements in $\Sigma$.

So reviewers' suggestions are correct, just use GLM, instead of GLMM.

• Thanks for your input. However, the reviewers suggestion is not about using a GLM instead of a GLMM, it is about using an ANOVA or t-test, i.e. a test to compare difference between groups (in this case difference in weight between period I and period II). I do not understand your point. The reason why I am using a GLMM with a random intercept is because I think that the yearly average weight may vary randomly among years. – simone May 8 '17 at 18:02
• t-test and ANOVA are special cases of GLM. Random effect is used to deal with correlation among $\epsilon$. But it seems there is no this kind of correlation in you data. So no random effect is necessary for your analysis. – user158565 May 8 '17 at 20:03
• I think that my skills in math are too limited to follow your reasoning about the non-zero off diagonal elements. However, I understand the conceptual meaning of adding a random intercept as that I have included in my model. Actually, I have compared by Likelihood Ratio Test and also by AIC the fit of one model with and another without the random intercept and the former is better than the latter. Anyway, my question was a bit different. Is there anything wron in using a GLMM (or a GLM) for my case of study? if, as I think, it is OK, how may I convince a reviewer? – simone May 8 '17 at 20:44
• Suppose you have two mice with exact age, month, TBL, TypeCap and year. You did not see them. Someone else measured weight for one mouse and tell you result. Does this weight provide additional information (apart from age, month, TBL, TypeCap and year) for you to guess the weight of another mouse? If yes, it means there is correlation and random effect should be added. If no, no random effect is necessary. – user158565 May 9 '17 at 2:52
• Right, this is a clear explanation of random intercept. My question is: is it wrong using a GLM or GLMM instead of an ANOVA or t-test for my case study? Note that I have unbalanced data and lot of variables related to my response variable. how may I convince a reviewer that it is OK? – simone May 9 '17 at 8:58