# How do I interpret regression coefficients with PCA scores as dependent variable?

I am performing a PCA on four variables all measuring performance on a test in some way. Now I want to use the first principal component (test1_pca) as dependent variable in a linear regression, such that I can regress the newly found principal component scores on several continuous (e.g. age) and categorical (e.g. sex) variables without having to run four separate linear regressions with the four variables separately as dependent variables. In my opinion it makes sense to combine them into one score, since they all measure performance and that is how I would interpret the principal component accordingly.

However, I am not sure how to interpret the regression coefficients that follow in terms of performance. To be more specific, all four original variables could be interpreted as: lower score means better performance. The variables were scaled (and centered) using prcomp(), which makes that I don't know what the resulting coefficients indicate of the regression. Usually, you would say something like "with every unit increase in X, Y increases ...", but how does that work with a PCA dependent variable?

lm_mod <- lm(test1_pca ~ Group + Age + Ed, data = data)
summary(lm_mod)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)             2.56987    1.17483   2.367 0.004378 **
Group2                 -1.23648    0.57839  -3.893 0.000120 ***
Age                    -0.32894    0.03478  -2.938 0.026473 *
EdNo                   -0.23405    0.34589  -0.537 0.538949
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


with Ed as factor (yes/no), Age continuous, and Group factor (Group1/Group2)

pca_res <- prcomp(data, scale=TRUE)
test1_pca <- pca_res\$x[,1]

PC1        PC2        PC3           PC4
perf1       0.5578949 -0.3908578 -0.1957844 -6.738934e-01
perf2       0.5578949 -0.3908578 -0.1957844  6.738934e-01
perf3       0.4578934  0.2455783  0.7957830 -3.683985e-17
perf4       0.3689048  0.8346758 -0.4783748  0.000000e+00


EDIT:

The PCA I am doing is only on continuous variables perf1, perf2, perf3, and perf4. In the regression, I use both categorical and continuous predictors and the PCA scores created with the perf (=performance) variables serve as the dependent variable, which is where I get stuck on the interpretation of the coefficients.

• From what I understand, you are making the conscious decision to regard PCA1 as performance. Hence it should be reasonable to interpret it in the same manner as if it were observable. All test coefficients are quite equal for PC1, and more importantly they are all positive. So I think your assumptions are sound. Commented Nov 28, 2021 at 15:54
• You can't have it both ways. Either using PC1 makes sense because the original variables are so highly correlated that they can be replaced by one, or else you're losing interpretability that you care about and should reconsider your strategy. Commented Jul 18 at 11:25