# Should I use the mean difference or mean ratio for a t-test?

I have a quantitative variable (medication dose) that is paired: dose of medication before surgery and after surgery for 76 patients.

I was under the impression that I should calculate the mean of the differences of the before and after doses and then calculate the standard error of the mean difference, CI and t-test. In SPSS a paired-sample t-test would do this.

I was told instead to do the percentage of the second dose (after surgery, i.e., 600 mg) as a function of the first dose (before surgery, i.e., 1200 mg) for every patient and then calculate the mean, standard error, CI and t-test of the percentages.

Is this a valid analysis?

• By "do the percentage...", do you mean compute the quotient 2nd/1st? Jun 1, 2015 at 22:34
• that is correct Jun 1, 2015 at 22:41
• sorry for getting the terms wrong, obviously math is not my forte. but just out of curiosity, if all of the quotients are multiplied by 100 to get a percentage, prior to calculating all other measures, does that change the meaning of my question? Jun 1, 2015 at 23:30
• any thoughts on this gung? Jun 2, 2015 at 12:47
• my research question would be if the surgery decreases the dose of medication necessary. So in my results I would like to express in milligrams and by percentage what that reduction was. For example, "surgery reduces the dose of medication by 600mg" or "surgery reduces the dose of medication by 50%" Jun 2, 2015 at 12:53

A $t$-test assumes the relevant data (differences or ratios) are drawn from a normally distributed population. Thus, at most, one of those possibilities affords a valid $t$-test. In addition, a $t$-test of the mean ratio against $0$ makes no sense whatsoever, so you would need to specify some value that makes intrinsic sense given your data and the theoretical context (most commonly $1$). Moreover, they ask subtly different questions—by asking about constant ratios, you wonder if those with larger starting values have a proportionally greater increase.
I am a firm believer that the statistical analysis should match the substantive question that motivated the study. Thus, your first job is to get as clear as possible on what you want to know, then pick the analysis that matches that. Subsequently, you will need do determine that the assumptions of the analysis obtain. You can check the normality of the differences or ratios with a qq-plot. Ratios are not typically normally distributed. A common strategy is to take the logarithm of every value, and compute the differences of the logs. Those differences can be tested against $0$ with a one-sample $t$-test.