I'm currently working on a comparative study of a species where I compare certain immune function metrics across captive and wild populations. Studies of these measures for this species typically report the mean and SD (or SE) of the population, which makes comparing between the various studies (and my own dataset of these measures in several sampled populations) difficult.
On the one hand, I could run summary statistic-based t tests (tsum.test in BSDA in R) and then correct for multiple comparisons between populations. But this is limited in that is confines us to one statistical approach, assumes a normal distribution (which likely is rarely the case for this immune variable, which is given in a proportion), and restricts us in terms of data visualization. I've been more keen to use a simulation approach, where we can convert the mean/SD into parameters of a more justified distribution, simulate the sample given those summary stats and the sample size, and then run the appropriate statistical models for our question.
However, I've had quite the time trying to find a decent citation that uses this kind of approach. If anyone is aware of a similar sort of approach or study, I'd be much grateful. Thanks in advance
Daniel -
This is way outside of my expertise, though I'd like to know more, and find your question interesting.
Are you trying to compare populations for mean values? A distribution can be any shape, but normality becomes of interest for means because of the Central Limit Theorem, and in the distribution of 'errors.' If you are comparing two means, then instead of a t-test, you could estimate confidence intervals about the difference between two means.
You mentioned proportions, and that is handled somewhat differently than continuous data. You might want to consult a book such as
Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons.
Regarding hypothesis tests, you might consider this:
https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS
A couple of other books that might be of interest:
Blair, E. and Blair, J(2015), Applied Survey Sampling, Sage Publications.
Lohr, S.L(2010), Sampling: Design and Analysis, 2nd ed., Brooks/Cole.
I like simulation for many things, but if you are going to "...convert the mean/SD into parameters of a more justified distribution..." then you may have a hard time choosing between justified distributions, as there are other factors such as skewness and kurtosis, though I am not clear as to whether that would matter to your problem.
It may be good to use more than one approach to see how your results might differ.
If you are looking at distributions, graphics may be very helpful. I did see something on ResearchGate about a medical statistics problem with histograms being misleading, but that might be alleviated, I think, by repeating, trying several different bin sizes. At any rate, graphics are generally, I think, more likely to give you a clearer picture. If you compare x,y values, scatterplots are very helpful. My point is that you are likely to find some helpful graphics, whatever the need.
Best wishes - Jim
Article Practical Interpretation of Hypothesis Tests - letter to the...
James,
Thanks for your feedback. The data in question are differential leukocyte counts -- so percent neutrophils, percent lymphocytes, percent eosinophils, etc. I know from my own field data that the distribution of most of these cells is often not normal -- sometimes N and L approach normality, but M/E/B typically do not.
Because the data are bound between 0 and 1, I've been considering using the beta distribution given the shape flexibility and the proportional nature of the data. From my own field data, the beta distribution fits better than the normal distribution (based on AIC), so this feels moderately justified. And graphically, as you mention, they don't really appear normal.
But the main justification for the simulation approach for the other populations (versus the t test one) is to be able to use beta regression models to compare each leukocyte % between the populations (which fits substantially better than a linear regression). I'll certainly consult with the texts you've mentioned though. And using the simulation approach but with several different distributional forms would appear to be the best kind of sensitivity analysis I could pull here.
Thanks for the feedback
Daniel -
Ah. I recall working with the gamma and maybe beta distributions many years ago. Sounds interesting. So will you use the standard error of the beta distribution to estimate confidence intervals? That's a bit more involved than the ones you usually see for normality, but sounds promising.
If you have to resort to nonparametric ("distribution-free") statistics, you can still use the Chebyshev Inequality. That might still be a reasonable check on results.
Your simulation idea sounds promising. Thanks for the information. Cool.
Cheers - Jim
PS - Maybe you could simulate confidence intervals about differences in your means by a "resampling" method???
Please pardon me if you already know all of this, but resampling, to be clear, does not require additional data. When someone mentions resampling, they probably mean a bootstrap or jackknife approach. I think Brad Efron was a pioneer in this area, and there is a great deal of current work going on, I think.
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