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Lately, I have had several data sets at hand that I want to use to study the effect of a treatment. The data is in the form of:

\begin{array} {|c|c|c|c|} \hline \text{Participant} \quad & \text{value before} \quad & \text{value after} \quad & \text{difference} \quad \\ \hline 1 & 64 & 62 &2 \\ 2 & 58 & 53 & 5 \\ \vdots & & & \\ \hline \end{array}

(best attempt to display a table)

I assume the paired t-test to be a good test to decide if the difference in values before and after the treatment are significant. However, I'm unsure about the pre-conditions of this test.
Half of the sources I find state that I need to check for normality of the distribution of values before and after the test (this is given in my sample).
The other half states that I need to check the normality of the difference of the values (this is not given in my sample).

Which (if not both?) of the checks is truly necessary and why?

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  • $\begingroup$ I think you can use MathJax t create arrays which serve the same function as your table. This is explained in one of the answers to the post you link to. $\endgroup$ – mdewey Dec 21 '18 at 15:19
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    $\begingroup$ It is the differences which are relevant (the test is often referred to as the one-sample t-test which gives you a clue). Having said that moderate departures from normality ar eunlikely to affect much. $\endgroup$ – mdewey Dec 21 '18 at 15:20
  • $\begingroup$ thanks @mdewey, didn't scroll down long enough... The long headers where a bit tricky, but it looks nice now. Also thanks for the answer. $\endgroup$ – nirolo Dec 21 '18 at 16:46
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If you are doing a paired t-test, you should look for normality in the differences. A paired t-test uses the differences to compute the test statistic, so the differences are all that matter, not the original data. See Bland's "Introduction to Medical Statistics" for more.

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  • $\begingroup$ Thanks @demetri-pananos, that book helped. I think I understand now why the differences are of interest and why there is the assumption that the difference is normally distributed. One last thing: I can test for this normality assumption by testing my sample differences because if the difference follows a normal distribution in the population, then I would also assume that my sample should follow a normal distribution, correct? $\endgroup$ – nirolo Dec 21 '18 at 18:15
  • $\begingroup$ @nirolo Yea, sure $\endgroup$ – Demetri Pananos Dec 21 '18 at 19:22
  • $\begingroup$ In reality the Gauss-Markov theorem specifies that the model errors are normally distributed, but this would imply normality of the differences. $\endgroup$ – StatsStudent Dec 22 '18 at 3:48

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