Can you run another t-test on the differences after your 1st t-test results?

Question

Let's say we are measuring the amount of anger (numerically) among males and females before and after watching a short video.

Males	Anger (before video)	Anger (after video)
Aaron	71	75
Bob	68	81
Carl	70	77
...	...	...

Imagine we do the same for females (see the table below).

Females	Anger (before video)	Anger (after video)
Alice	73	75
Bailey	66	78
Carol	72	76
...	...	...

1st t-test (males): I can use a t-test on the 1st table (males) to determine whether the two columns are significantly different, i.e. did the video make them angry.

1st t-test (females): Exactly the same except using the 2nd table (females).

Conclusions: Let's pretend that both males & females had significantly higher levels of anger after watching the short video.

Question: The short video gets males and females angry. More specifically, the males got more angry by an amount (respectively) of 4,13, and 7. The females have similarly calculated differences. My question is whether Males got more angry than females from watching the short video. So, can I perform a t-test on the two columns below (the difference columns)?

	Difference (male)	Difference (female)
Subject 1	4	2
Subject 2	13	12
Subject 3	7	4
...	...	...

A significant result would mean that one of the sexes was affected more by the video. A non-significant result would mean that they both were affected about equal. (I know that I am simplifying significance a little bit here)

Answer 1

Speculation with fictitious data: If your real data are sufficiently similar in key respects to my fictitious data, then maybe you have your answer. If not, please use what is below to edit your Question, saying what possible difficulties you think might arise with your real data.

Suppose you nave male differences w1 and female differences w2 as summarized below:

summary(w1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.00    5.75    7.00    6.91    8.00   12.00 
length(w1); sd(w1)
[1] 100
[1] 2.165361
summary(w2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   5.000   6.000   6.007   7.000  10.000 
length(w2); sd(w2)
[1] 150
[1] 2.031618


par(mfrow=c(1,2))
 qqnorm(w1); qqline(w1, col="blue")
 qqnorm(w2); qqline(w2, col="blue")
par(mfrow=c(1,1))

Normal probability plots (quantile-quantile plots) are roughly linear, so it seems OK to compare differences for men and women with a two-sample t test. I will use a Welch version of the test because I can't be sure the variances of the two populations are equal. For my fictitious data, the difference in sample means $6.9$ and $6.0$ is highly significant with P-value near $0.$

t.test(w1,w2)

        Welch Two Sample t-test

data:  w1 and w2
t = 3.3117, df = 202.87, p-value = 0.001098
alternative hypothesis: 
 true difference in means is not equal to 0
95 percent confidence interval:
 0.3655023 1.4411643
sample estimates:
mean of x mean of y 
 6.910000  6.006667 

The 'flat sequences' in the normal probability plots indicate numerous ties. But with sample sizes over 100 the implementation of the Wilcoxon rank sum test in R, handles the ties without giving a warning message.

So it would also be OK to use a Wilcoxon rank sum test to see if there is a difference in location of the two populations.

wilcox.test(w1,x2)

        Wilcoxon rank sum test 
        with continuity correction

data:  w1 and x2
W = 0, p-value < 2.2e-16
alternative hypothesis: 
 true location shift is not equal to 0

The shapes of the two samples is sufficiently similar to say that the medians of the two populations differ. Also the 'notches' in the sides of the boxes do not overlap, which is a further indication of different medians.

boxplot(w1,w2, horizontal=T, 
        col="skyblue2", notch=T)

Note: The fictitious data for this demonstration were sampled in R as below. The rounding gives rise to lots ot ties.

set.seed(12)
w1 = round(rnorm(100, 7, 2.5))
w2 = round(rnorm(150, 6, 2))