Mixed models for repeated measures

Question

I have measures of weight for three countries for 3 time points. I want to find whether there is a difference in change of weight over time, based on gender accounting for groups (country). I also have a bit different age ranges for countries, so I was thinking to use an age variable only, which would account for differences in groups. The final model would look something like this:

weight ~ gender + (age_at_weight_measure | participant_id)

1. Do I think in the right direction or should I anyway add a country group variable somewhere?

2. If I want to find the influence of variable X on the weight change (let's say, income (categorical)), would the model change into this:

weight ~ gender*income + (age_at_weight_measure | participant_id)

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Data look somewhere like this:

participant_id	gender	age_at_weight_measure	weight	income	country
1	F	17	56	1	Italy
2	M	35	74	4	US
3	F	15	65	4	Italy
4	M	27	67	3	US
5	F	28	58	2	US

Answer 1

Regarding your first question, I agree with @user2974951 . Country should be included. Whether it should be fixed or random depends on what you are interested in. If you are actually interested in the differences, then I think it should be a fixed effect; if not, then it should be random. You write "based on gender accounting for groups (country)" and so far, you have not accounted for country.

Another way of deciding is to ask whether, if you did the study again, you would use the same countries. If "yes" then fixed, if "no" then random.

For your second question, you have added the interaction of gender and income. That would look at whether the relationship between gender and weight is different at different levels of income and also whether the relationship between income and weight is different for the two (or more) genders. (As an aside, don't confuse gender and sex).

If you want to just add the main effect of income, then the * in your second equation should be a +.

Also, the relationship between weight and age might not be linear. In your sample data you have already got a range from 15-35. Weight is increasing much faster at 15 than 35, especially for boys. Depending on the full range of age in your data, you might need a spline of age and you might need the interaction of age and sex (because girls go through puberty sooner than boys do).

You will need quite a lot of data to estimate all these parameters.