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I have a dataset consisting of 12 subsets, each subset contains between 10 and 15 temperature observations from one person.

I would like to determine whether or not it is reasonable to assume that, within each subset (person), the temperature observations are normally distributed. That is, I would like to be able to say "the data supports the assumption that temperature observations for an individual person follow a normal distribution".

Mean temperature and variance varies considerably from subset to subset (person to person). Therefore, my proposed analysis is to qualitatively test for normality using a QQ plot as follows:

  1. For each subset, standardize data (for each observation within a subset, subtract the mean of that subset, and divide by the standard deviation of that subset)
  2. Pool this standardized data for all subsets
  3. Plot pooled data against theoretical quantiles for N(0,1)

Is this an appropriate analysis?

I'm concerned that pooling in this way may not be valid since subsets are unequal sample size (more weight is given to larger subsets). Would interpolating the larger subsets make sense here?

Is there a better way to test the assumption of normality?

My reason for wanting to assume normality of temperature observations: Firstly, it would be helpful to make predictions about future temperature observations. For example, it would be useful to be able to say "based on the average standard deviation found in this study, a 95% prediction interval can be calculated for future temperature observations, in other words, based on an average person in this study, 95% of the time we can reasonably expect someone's temperature to be within ±X° of their mean temperature". My understanding is that I can only make such a prediction if the assumption of normality is reasonable.
Secondly, it would be interesting to know if temperature observations are (or at least appear to be) normally distributed (given the conditions of my study), from a 'basic science' understanding of body temperature.

Edit: Researching this issue further, I have come to learn what I am proposing might be described as the equivalent of generating a QQ plot of the studentized residuals. From my non-technical viewpoint, this suggests it is a valid approach to checking the assumption of normality:

ANOVA:How to detect non-normality with a QQPlot in the presence of non-homogeneous variance

http://data.princeton.edu/wws509/notes/c2s9.html

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  • $\begingroup$ If you standardise each to N(0,1) then surely they will be normal? Can you clarify what exactly you are doing? You might also expand on why you think testing for normality is important for your scientific question. $\endgroup$
    – mdewey
    Commented Oct 13, 2017 at 12:57
  • $\begingroup$ Pooling is the right thing to do,but I would only subtract means, not divide by standard deviation. That corresponds to plotting the residuals from a common-variance oneway anova. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Oct 13, 2017 at 19:24
  • $\begingroup$ Thanks @kjetil, I initially thought to do exactly what you described (subtract the means only before pooling). But then I realized that the standard deviation varies quite a lot between each subset (i.e. inhomogeneous variance). By dividing by the SD for each subgroup, I'm scaling them to be the same width. If I pool subsets before scaling, low variance subsets (tall and thin) would combine with high variance subsets (wide and flat) and even if each was normal individually, the result would be very different to normal. $\endgroup$
    – Benjamin Straker
    Commented Oct 14, 2017 at 14:18
  • $\begingroup$ Ok, that means you need a version of anova not assuming constant variance $\endgroup$
    – kjetil b halvorsen ♦
    Commented Oct 14, 2017 at 16:27

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I don't yet have enough reputation to respond directly to the proposal of @Klaus, but statistical tests are not a rigorous way of answering the question. They check that the samples are not normal but not the opposite. Graphical methods are therefore preferable. See this question for example: Normality Test: Normality Test: Accept hypothesis null with uncertainty

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There are a lot of normalitytests. For instance the Shapiro-Wilk–test. I would test for each subsets, because this is what you want to know.

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  • $\begingroup$ Thank you for the suggestion, I will research the Shapiro-Wilk test. If I understand you correctly, this would mean conducting the test 12 times, once for each subset. How would I interpret the results if the null hypothesis is rejected for, say, one subset, and not rejected for the other 11? Should I adjust for multiplicity? $\endgroup$
    – Benjamin Straker
    Commented Oct 13, 2017 at 15:44
  • $\begingroup$ If you assume that each subset have a different mean and variance then you should delete the subsets that are not normaldistributet. $\endgroup$
    – Klaus
    Commented Oct 13, 2017 at 17:44

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