Can non normal data be used for factor analysis and multiple regression? If so what is the procedure to justify it?

Question

While I was writing up the analysis in my thesis, I just came across when rechecking my test for normality, that the p-value for most continuous variables was .000, which is less than .05, and it rejects the null hypothesis which means to my understanding my data is not normally distributed.

I have already completed the factor analysis with Extraction Method: Principal Component Analysis & Rotation Method: Oblimin with Kaiser Normalization. All 88 continuous variables were reduced to 17 factors. Reliability was checked and all items were alpha .8 and above, and then performed multiple regression (standard) to analyze the overall effectiveness against the 17 predictor variables (factors). Then I ran a t-test to see how two groups of professionals responded to the overall effectiveness and the p-value was more than .05 in this test. All of a sudden now I am panicking whether I have done the it right in terms of the continuous variables since all had a p-value of .000? Is my final analysis valid?

Answer 1

I just came across when rechecking my test for normality, that the p value for most continuous variables have been .000.

Certainly in the case of multiple regression there is no such assumption.

Let me be completely explicit. Neither the IVs nor the DV in multiple regression are assumed normal, not even for inference. It is not an assumption.

When conducting inference, you assume something to be normal, but not any of those.

Formal hypothesis testing of normality isn't a particularly useful idea in any case. It largely answers a question we already know the answer to and gives little information about what actually matters. (for a little more discussion, see here)

For what I understand of factor analysis it doesn't seem to be an assumption either (*). It would perhaps impact some methods of assessing the number of factors.

(*) e.g. Here's Wikipedia on factor analysis

Also we will impose the following assumptions on F.

$F$ and $\varepsilon$ are independent.
$\mathrm{E}(F) = 0$
$\mathrm{Cov}(F)=I$ (to make sure that the factors are uncorrelated)

... no mention of normality there.

Some later tests (say a chi-square test if you do one) would be affected by non-normality, but my understanding is that there are alternatives (e.g. this paper mentions the existence of alternatives in that case).

Edit: Your t-test will rely on the normality of whatever you were testing; note that principal components (or factors for that matter) are weighted sums - if there are many components they may end up looking reasonably normal even if the components aren't. You can always check how off the distribution looks for that with say a normal scores plot (/Q-Q plot).