Normality test in a case-control study

I have conducted a research testing the associations between vitamin D, CRP, C3, and insulin resistance. The design was a case-control design. the sample size was 134, of which 67 are cases (obese) and 67 are controls (normal weight). of note, each group contains 35 females and 32 males. The Journal which i intended to publish in asked for the normality test. However, I assumed that the sample are normality distributed as the sample size is greater than 30. now the questions is: 1) do i need to conduct the normality test in the first place or not? 2) If the first question is ye, do I have to conduct the normality test for the whole variables including, age and BMI? 3) which normality test is better in mmy case? 4) do I have to conduct the normality b group and gender or for the whole sample at once? 5) what If the SD is greater or almost equal the mean?

You don't identify which is your dependent variable(s) and which are independent variables.

I assumed that the sample are normality distributed as the sample size is greater than 30.

Changing the sample size doesn't change the distribution of the population.

do I need to conduct the normality test in the first place or not?

Statistically, a formal test of normality is pretty much irrelevant, since it answers the wrong question (failure to reject doesn't imply that the distribution the data are drawn from is sufficiently close for your inferences to have near enough to the desired properties, rejection doesn't imply that they don't).

(That's not to say that an assessment of the assumptions is irrelevant, just that a hypothesis test isn't a good way to do that)

My own advice would be to look at a Q-Q plot to see the kind and degree of deviation from normality. With a reasonable sample size like this, the impact of moderate deviations form normality on type I error rate may be fairly small, but power can be more heavily affected.

[However, check the other assumptions relating to mean specification and variance about the model first -- if those are wrong any consideration of normality is useless]

I think it's more a matter of strategy -- if they won't be convinced to let you do anything but a formal test, you will need to do a formal test.

do I have to conduct the normality test for the whole variables including, age and BMI

None of the original variables are assumed to be normal. Residuals would presumably be the only thing you should look at.

which normality test is better in my case?

If it's just a matter of ticking a box to get a publication, "better" means more likely not to reject (if they're going to make you do something so pointless I have no compunction about the fact that this is bad practice -- lumping bad practice on top of the terrible practice of making you jump through useless hoops isn't necessarily making things worse). You will want the lowest power test you would get away with.

The lowest power at-all-commonly-used test available would be the chi-square test but they might balk at that. Lilliefors test (Kolmogorov-Smirnov with estimated parameters, some packages just call this Kolmogorov-Smirnov too) is sort of middling and that might be accepted.

A better approach might be not assuming normality in the first place

4) do I have to conduct the normality b group and gender or for the whole sample at once?

"Have to" is again a matter of what they'll accept. If I was doing a QQ plot and the other residual diagnostics were fine I'd lump them all together. If someone is forcing a test on you, you might lean toward testing by subset.

what If the SD is greater or almost equal the mean?

This would only be relevant if your variable was restricted to be non-negative (which you haven't mentioned).

If that's the case then you likely will have a skewed distribution for the conditional response but you will likely have an even bigger issue with non-constant variance. Either working on the log-scale (for positive variables) or modelling with GLMs (/GLMMs) would be a better choice than assuming normality.

Either way, dealing with that issue of unequal variance and skewness of conditional response would be important and may avoid the entire issue.

• I assume the outcome variables are vitamin D, CRP, C3, and insulin resistance. If so, then in my experience some of those variables are usually better modelled by a log-normal distribution. I have the most experience with CRP, for which that would have been my up-front choice, because it can be pretty non-normal depending ont he kind of disease we are talking about. I am less sure about the others. Perhaps for point (4) it should be clarified that it's not for the raw data, but rather for the residuals from the model. – Björn Apr 19 '17 at 6:42
• @Björn with respect to point 4 (given I had already explicitly stated that you check the residuals rather than the observations for normality earlier) I'd have thought the phrase "other residual diagnostics" (implying the normality check was also a residual diagnostic) was sufficient reiteration of the point there. – Glen_b Apr 19 '17 at 6:50