# Paired t-test to test trends in regression coefficients between groups

I have data where I estimate regression coefficients/slopes for two groups (e.g. sex-specific regression coefficients) for a number of single effect regression models. I have used Bayesian methods which allow me to test, individually for each model, whether the regression coefficients of each group are significantly different (and they are generally not significant).

However, there seems to be a trend, that one group has consistently lower coefficients (nearer to zero) than the other.

Could I then use (/would it be appropriate to use) a paired t-test to see if the (absolute) value of the regression coefficients for one group is consistently lower than the other to test whether one group is more sensitive to explanatory variables?

E.g. (in R)

> a = data.frame("Coefficient" = c(1:10), "GroupA" = abs(rnorm(10,1,1)), "GroupB" = abs(rnorm(10,4,1)))
> a
Coefficient    GroupA   GroupB
1            1 1.2359143 4.528682
2            2 0.1260180 5.703323
3            3 1.1529601 5.998172
4            4 2.3689296 4.343029
5            5 1.8734228 4.245404
6            6 1.1287780 4.699337
7            7 1.8684325 3.829195
8            8 0.2723389 4.646488
9            9 3.1309934 4.158523
10          10 3.3550409 5.042786
> t.test(a[,2],a[,3],paired=TRUE)

Paired t-test

data:  a[, 2] and a[, 3]
t = -6.4382, df = 9, p-value = 0.0001198
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-4.146265 -1.990157
sample estimates:
mean of the differences
-3.068211

• Are these coefficients from different models run on the same dataset, different subsets of the same dataset or different datasets entirely ? – Robert Long Jul 6 '16 at 8:50
• all subsets of the same data (for the response variable and groups), different explanatory variables used in each model. It's data on a trait and regression models test sex-specific responses to various environmental variables. I want to test whether one sex is less sensitive to environment generally (visual suggests that one is consistently less affected than the other, and for some environmental variables this is significant) – rg255 Jul 6 '16 at 8:57
• So you are running different models each on a different subset of a larger dataset. The models have one predictor in common that you want to compare across the different models ? – Robert Long Jul 6 '16 at 11:11
• All models have the same response variable, and a common fixed effect of sex (male or female), but vary in the other fixed effect – rg255 Jul 6 '16 at 11:50
• IIUC, in the data you have posted, each row refers to a model (run on a different subset of the data), and Group A refers to one sex and Group B to the other sex ? – Robert Long Jul 7 '16 at 7:43