Paired t-test, non-normality, and checking for equal variances

[Vanessa Ojeda]

Hello,
I need some direction and hope someone could help me.

I have a research question that wants to know if there is a mean difference between a pre-test (pre) and a post-test, and post-test (post) and a separate post test (post2). Same sample (and group of people). n=74

I would assume I would use a paired-sample t-test. However, some of the data is normally distributed while other data is not. I used a shapiro will test for normality and also looked at histograms and skewness and kurtosis.

Questions:
1) I had read that even though I may have some non-normal distributions, it would be okay to move forward with a paired sample t-test because it is robust. True?
2) If I have some data that are normally distributed, while others are not, do I need to run a paired samples ttest on those data and a Wilcoxin-Rank Sum on the others?
3) I understand that for unpaired or independent samples, that checking for equal variances is important, but I have been told for paired (dependent) data, I do not. Is this because the data have the same sample size?

Thank you in advance!

[Clyde Schechter]

1. Yes, the normality "requirement" for a paired t-test is that the difference between the two variables have a normal distribution. But even that is only important in small samples. With n = 74 and the kind of skewness and kurtosis you are showing, the central limit theorem will assure that the estimate of the mean difference has a close to normal enough distribution that you get correct results.

2. I wouldn't.

3. The paired t-test cares only about the distribution of the difference between the two variables. The variances of the two variables separately are completely irrelevant. Otherwise said, the paired t-test is actually a one-sample calculation. There aren't two different variances to be equal or unequal.

[Rich Goldstein]

comment on #3 - my comment in your earlier post on this was meant to show that you should NOT test for equal variances for unpaired t-tests because it introduces bias; if you are worried about unequal variances just use the "unequal" option on the t-test command

see #'s 6 and 8 in https://www.statalist.org/forums/for...sdtest-results

also, it may help to think about the t-test as performed via linear regression:

if you want to do a paired t-test via regression, first generate a variable that is the difference of interest and regress that with only a constant

if you want to do an unpaired t-test via regression, use the indicator variable as the only predictor (other than the constant); e.g., try this:

sysuse auto

ttest weight, by(foreign)
regress weight i.foreign

and compare the results

[Vanessa Ojeda]

Clyde Schechter and Rich Goldstein . I really appreciate all your help! Thank you so much.