I have performed an in vivo study evaluating splenic white:red pulp ratio as determined by AI on histopathologic slides (neural network trained and validated per published guidelines). The questions being posed are what impacts do radiation source and/or dose have on splenic lymphodepletion as measured by white:red pulp ratio. The experimental design is four sources of radiation, each with four different doses. Each source/dose combination has 3-5 biological replicates (individual mice). When assessing for normality, should these biological replicates be averaged, then assessed for normality (4 values for each source or dose) or should all values of each subgroup (source and dose) be tested for normality as independent data points (resulting in 14-20 values for each source or dose)? Ultimately I’m trying to decide if parametric testing (two way ANOVA with Tukey multiple comparisons) should be used or no parametric testing (Krystal-Wallis with Dunn’s multiple comparisons).

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    $\begingroup$ Please tell us what you are assessing for Normality and why. $\endgroup$
    – whuber
    Commented Feb 29 at 19:12
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    $\begingroup$ Please also say more about the data: How do you measure the "splenic white:red pulp ratio"? What hypothesis are you trying to test? My guess is that there will be a better way to accomplish your goals than with the ANOVA or linear regression that you seem to have in mind. Please provided this information by editing the question itself, as comments are easy to overlook and can be deleted. $\endgroup$
    – EdM
    Commented Feb 29 at 19:44
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    $\begingroup$ What method requires normal data? Some (like OLS) require normal errors for some purposes. $\endgroup$
    – Peter Flom
    Commented Feb 29 at 20:01
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    $\begingroup$ Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. $\endgroup$ Commented Mar 1 at 0:56

2 Answers 2


Choosing whether to do a parametric or non-parametric test based on an initial assessment of normality is not good practice. See this page for extensive discussion. Even if you were to evaluate normality, that is best done on the residuals between observed and predicted values after building a model, not on the original values before modeling.

It's probably risky to do a standard parametric ANOVA when the outcomes are ratios, as you have here. It might be OK, depending on the details of the data, but if there are extremes of lymphodepletion at some dose/source combinations then it probably won't work well. As Frank Harrell points out, a proportional odds ordinal regression model is a useful semi-parametric generalization of Kruskal-Wallis and similar non-parametric tests. In your case it could quite easily handle your 4 source by 4 dose experimental design.


"Replicates" in bioassays are usually used as a method for reducing measurement noise, and sometimes as a way to decide on whether a run of the assay was a technical success. The variation between technical replicates is of little interest and so the technical replicates are typically averaged to give a single datum that is used in statistical analyses.

The variability of interest will be the variability between mice. Use the averages of your technical replicates and you will have one measurement per mouse. Using technical replicates as individual data points is often called 'pseudo-replication' and it is a very bad thing in many circumstances.

Having said that, a sample of 3 to 5 mice is far, far too small for any meaningful result from a test for normality. Many would argue that it is far too small for a meaningful result from any experiment, but I will allow that some biological effects are so large that a tiny sample like that can occasionally suffice.

  • $\begingroup$ Averaging technical replicates simplifies the analysis for the reasons you describe. However, what if there are different (& known) number of replicates per experimental unit (mice)? Should this be taken into account because the average measurements have different std. error? $\endgroup$
    – dipetkov
    Commented Mar 2 at 11:07
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    $\begingroup$ @dipetkov I would guess that it would probably not matter much, but would advise researchers to nail down their protocols for experiments in preliminary runs. The preliminary data can be analysed, but only to give indicative results that are used to design the 'proper' experiments. Good experimental design is far better than band-aid statistical approaches. $\endgroup$ Commented Mar 2 at 20:17

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