# accounting for age effects: two age groups and within-group age differences

I am analyzing data from a study with two groups of about 25 healthy participants each, one young (mean age ~25 [~18-~30] years) and one older (mean age 59 [45-75] years). The study was set up mainly to test for differences in several dependent variables between the two age groups.

I am currently using 'glm' in R, because the two most relevant variables are not actually normally distributed, and because 'glmer' quickly fails on more complex models due to lack of repeats (in the most complex model there are only two entries per participant in each cell of the fixed effects structure). However, due to conventions in the field I may have to fall back on conventional AN(C)OVA at some point.

Now to the actual question: I realized that it is also possible that the rate of change (if any) itself changes, and that if this is indeed the case, it would be nice to quantify this aspect. For these data, one would expect the scores of older participants to deteriorate at an increasing rate. In theory, the deterioration should be monotonic, i.e., for a particular dependent variable scores should always change in the same direction (though deterioration might manifest as either higher or lower scores for different dependent variables).

If this is the case, then neither a factor Group (Young/Older) or a linear fit on age would completely describe the data. I see two basic alternatives: 1) a non-linear (e.g., quadratic) fit, or 2) a two-level factor Group (Young/Older) plus a linear fit of a continuous variable (let's call it VarAge) representing the difference between individual age and the mean age of the group (the increase in rate would manifest as an interaction between the Group and VarAge).

The main problem I see with option 1 is that data from the middle of the age distribution (~30-45) are lacking. The problems I see with the second option is that Group and VarAge are both measuring linear effects of age, and that it is not clear whether the (linear) interaction would be a reasonable approximation, in particular for the older group.

Is there a better option? If not, which of these approaches should I choose? And does the answer depend strongly on the precise statistical approach (regression, ANCOVA, glm, mixed models)?

Regarding the effect of subtracting mean age per group: this does not change the overall fit of the model, but it does change the coefficient estimates (and hence the p-value etc) for the main effect of group:

Without subtraction

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)     0.660284   0.009415  70.133  < 2e-16 ***
groupyoung      0.389899   0.128509   3.034  0.00243 **
age            -0.007157   0.001104  -6.485 9.92e-11 ***
groupyoung:age  0.016773   0.003708   4.524 6.24e-06 ***


With subtraction

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)      0.660284   0.009415  70.133  < 2e-16 ***
groupyoung       0.043393   0.013003   3.337 0.000854 ***
sage            -0.007157   0.001104  -6.485 9.92e-11 ***
groupyoung:sage  0.016773   0.003708   4.524 6.24e-06 ***


## 1 Answer

You second option seems appropriate to me. I would fit a model with Group, a factor with two levels, and Age plus the interaction between Age and Group. I do not think you would gain anything by subtracting out the mean age for their group as the age effect is just per year (that is to say assuming that effect is the same for each group) and so is the interaction. As you state it does affect the group effect as one might expect.

I cannot see any reason to suppose that the model will affect things: regression, generalised linear modelling and mixed effect models should act similarly.

• Where I wrote ANOVA I meant ANCOVA (which yes, still has it's problems for this application). – Ishisht Feb 6 '19 at 16:25
• One qualification to accepting this answer: subtracting the mean age per group does not change the overall fit of the model, but it does change the coefficient estimates for the group effect (and hence the significance of that effect) - see my edited question for details. – Ishisht Feb 6 '19 at 16:29
• I have edited the question accordingly – Ishisht Feb 6 '19 at 16:43