# Measurement level of changes in test score

I have a question. In a study that I am acquainted with, the goal is to test for the effectiveness of a new psychotherapy method. The dependent variable is stress as assessed per PSQ (Perceived stress questionnaire).

If I want to look for differences between various groups (e.g. sex, education, experience with psychotherapy etc.) after the psychotherapy, does it make sense to study CHANGES between two time points (t2-t1) or do I have to inspect every data point on its own.

So can I for example calculate an independent t-test between males and females with CHANGES OF STRESS as a dependent variable, so not the stress score per se but the change of it between the two time points. Do you know what I mean, does my question make sense? And is the measurement level of stress score CHANGE also at least interval scale just as the Perceived stress score itself?

I hope my question makes sense. :)

Kind regards, Helena

I think you are asking whether a paired t test is the same thing as a one-sample t test on the paired differences. If so, the answer is Yes.

Simulated data. I illustrate using fake paired data simulated in R as follows.

set.seed(519)  # for reproducibility
before = round(rnorm(100, 200, 25))
increase = round(rnorm(100, 6, 6))
after = before + increase
diff = after - before


In the case of actual experimental data, you would see before and after as you look at test scores, and increase would be available only on upon taking differences. [Please ignore the variable increase used only for simulation; it is the same as diff/]

Datapoints for the first six of 100 simulated subjects are as follows:

head(cbind(before, after, diff))

before after diff
[1,]    192   195     3
[2,]    230   236     6
[3,]    202   197    -5
[4,]    153   151    -2
[5,]    239   249    10
[6,]    178   179     1


We show a stripchart (dotplot) of diff. Over 3/4, but not all, of the subjects show positive differences. [For our purposes, individual graphs of before and after are not useful; they would not show pairing and so might be misleading.]

stripchart(diff, meth="stack", pch=20, ylim=c(1,1.1),
main="Differences [After - Before]]")
abline(v=0, col="green2")
mean(diff > 0)
[1] 0.77


Correlation. On expects before and after scores to be correlated because of the pairing: the first 'before' score is from the same subject as the first after score. Thus, there will typically be positive association between before and after, even if not typically as strong as in my simulated data.

The correlation $$r$$ between 'before' and 'after' is nearly $$1$$ and a scatterplot shows a nearly linear pattern for pairs. Points below the 45-degree line correspond to subjects that showed no improvement.

cor(before, after)
[1] 0.9696317

plot(before, after, pch=20)
abline(a=0, b=1, col="green3")


Tests. Now we compare results from two t tests in R: (a) a paired test for before and after and (b) a one-sample t test for diff. Notice that results are the same: the t statistic 8.3978, estimated improvement 5.53, and (very small) P-value 3.347e-13 are identical between the two tests.

t.test(after, before, pair=T)  # paired test

Paired t-test

data:  after and before
t = 8.3978, df = 99, p-value = 3.347e-13
alternative hypothesis:
true difference in means is not equal to 0
95 percent confidence interval:
4.223386 6.836614
sample estimates:
mean of the differences
5.53

t.test(diff)   # one-sample test of 'diff'

One Sample t-test

data:  diff
t = 8.3978, df = 99, p-value = 3.347e-13
alternative hypothesis:
true mean is not equal to 0
95 percent confidence interval:
4.223386 6.836614
sample estimates:
mean of x
5.53


Incorrect two-sample t tests. However, if you were incorrectly to do a two-sample t test, then the before and after are treated as two independent samples ('pairing' is ignored), and results of the test are substantially different---both numerically and in format.

The 'Welch' t test, which does not assume equal variances, is shown in full; only the P-value of the 'pooled' t test is shown. Neither of these incorrect tests shows a significant difference between 'before' and 'after'.

t.test(before, after)  # 'before' & 'after'
# treated as if independent

Welch Two Sample t-test

data:  before and after
t = -1.4757, df = 197.79, p-value = 0.1416
alternative hypothesis:
true difference in means is not equal to 0
95 percent confidence interval:
-12.919855   1.859855
sample estimates:
mean of x mean of y
201.31    206.84

t.test(before, after, var.eq=T)\$p.val  # pooled
[1] 0.1416086


Note on nonparametric tests: If data are not normal, you may choose to use a Wicoxon signed-rank test to judge whether differences are mostly positive. Equivalently, you might do a paired test on After and Before.

wilcox.test(diff)

Wilcoxon signed rank test
with continuity correction

data:  diff
V = 4069, p-value = 3.052e-11
alternative hypothesis: true location is not equal to 0

wilcox.test(after, before, pair=T)

Wilcoxon signed rank test
with continuity correction

data:  after and before
V = 4069, p-value = 3.052e-11
alternative hypothesis:
true location shift is not equal to 0


However, it would be incorrect to do a two-sample Mann-Whitney-Wilcoxon rank sum test as if Before and After were two independent samples.

wilcox.test(after, before)

Wilcoxon rank sum test
with continuity correction

data:  after and before
W = 5602, p-value = 0.1416
alternative hypothesis:
true location shift is not equal to 0