# How do I interpret the regression coefficients when making comparisons between the change of slope between control and disease?

I have 15 genes and two groups (58 control and 51 disease). I have relative abundances of the genes at two time points (time point 1 and a few years later time point 2, the average age difference is 4.9 years (max 11, min is 1).

I have basic line plots to visualise the change in the average expression at time point 1 and timepoint 2 for each group.

I would like to compare the change in slope between the second and first time point comparing the two groups. I have subtracted the relative abundances at the first time point from the second for each group, for each gene (TP2-TP1). To make the comparison between the control and disease, I have then used a linear mixed effect model using the TP2-TP1 difference values and taking into consideration certain fixed and random effects.

Below is my code;

  output4 <- apply(control_disease[, 24:38], 2, function(i){
fit <- lmer (i ~ disease + BMI + Age + Gender + Age.diff + (1|Fam), data = control_disease, REML=F,
na.action=na.omit)
results <-summary(fit)\$coef[2, c(1, 2, 5)]  ( this gets effect size, st.error and p value).


Control is marked as 0, disease is 4.

An effect size for gene 1 was -0.027 which was significant. Am I interpreting this correctly if the change in slope between the second and first decreased by this amount in iT2D relative to control? In other words, the negative value suggests that the difference between TP2-TP1 is less in iT2D. ( I calculated the difference with the average TP2-TP1 relative abundance manually for this gene for both groups and this is the case).

The age difference coefficient for this is -0.0118. This suggests that for every unit increase in age difference you would expect this gene to decrease. A quick eye ball of the graph for this gene and in each group, the average gene expression of the line graph increases at the second time point, which confuses me.

In addition, as I am making the comparisons between the two groups, from this value I don’t believe I can determine the change in direction from TP2 to TP1, for instance, whether there is an increase or a decrease at TP2 compared to TP1 when just considering the disease group, without looking at the line graph?

Lastly, in my complete data set I have three groups. Control, Disease 1 and Disease 2. Would a Repeated Measures ANOVA be useful to make the comparison between the change in slope between each group and also between the first and second-time point for each group? I am reading up about this now.

I would appreciate any help please.

It's quite hard to interpret your results purely from words and some bits of code which we cannot reproduce. Below I try to comment based on my understanding. Suggest you update your post with more information:

An effect size for gene 1 was -0.027 which was significant. Am I interpreting this correctly if the change in slope between the second and first decreased by this amount in iT2D relative to control?

I guess you are referring to the effect size of disease. I am not so sure it is the slope, because slope will be $$\Delta(expression)/\Delta(age)$$ and as far as I understood, you only have $$\Delta(expression)$$. So the coefficient is the change in gene expression from control to disease.

However, as I am making the comparisons between the two groups, from this value I don’t believe I can determine the change in direction from TP2 to TP1, for instance, whether there is an increase or a decrease at TP2 compared to TP1 when just considering the disease group, without looking at the line graph?

This is more or less correct, as you have noticed, this is the effect of disease relative to control. So in R models, 0 (control) is taken as the reference level, and you estimate the effect of other groups relative to this reference. To know the change in direction (overall change of gene expression), you need to add the intercept. So if intercept is say 0.2, then overall change for disease = 0.2 + -0.027 = 0.19.. which makes it not much.

Would a Repeated Measures ANOVA be useful to make the comparison between the change in slope between each group and also between the first and second-time point for each group?

Not very sure how you would set that up given the times are different.

• Thank you for your reply. As I have calculated the TP2-TP1 relative abundance for each gene, I thought the effect size would be reflective of whether the difference between these timepoints is greater or less in control. With regards to change in expression/age, if keeping with TP2-TP1, is this the age difference between these points? Also please could you let me know whether you would approach trying to identify the changes in slope between the two timepoints comparing both groups in a different way? Commented Apr 23, 2020 at 23:01
• Also some genes decrease in expression at TP2 in the disease group and with control the expression has increased. So the direction of the lines are different with the line graph. I don't know whether doing TP2-TP1 is logical, as there is a greater change between TP1 and TP2 ( but in the negative direction for disease group). i.e Gene 1 Control (TP2-TP1; 0.745904-0.725227= 0.020677- line from TP1-TP2 goes up). In the disease group (TP2-TP1 0.608-0.692= -0.084 -line from TP1-TP2 goes down). I guess, ignoring the signs, there is a bigger difference in disease. Commented Apr 23, 2020 at 23:09
• so i think what you are doing is ok, i would explain the coefficient as overall increase in gene expression, after controlling for difference in time period.. The slight problem is this, you are expecting the age coefficient to be the same for both groups Commented Apr 23, 2020 at 23:10
• Thank you for your reply. I do have the difference in age TP2-TP1 as a fixed effect, please could you let me know whether you would still advise having the change in TP2-TP1/age diff for each gene as the response variable? Commented Apr 23, 2020 at 23:14
• ok so if you want to model it as a difference in slope, the model will be a bit more complicated. What you have done so far should be ok, it's a matter of explaining the coefficients. So in the sense of your data, you are modeling it like this, the change in expression with time is explained by the age coefficient, i.e x units per year, regarding of disease or not, let's say it's like a biological process for any human Commented Apr 23, 2020 at 23:19