Kasun Meetiyagoda, the issue of using tests of normality prior to estimating some kind of OLS model has been discussed many times on RG. See the following thread, for example, and pay particular attention to the Feb 21 post by Jochen Wilhelm. (And see the image attached to my Feb 23 post.)

https://www.researchgate.net/post/Can_the_use_of_a_parametric_test_ANOVA_after_a_failed_normality_test_be_justified

I agree with Bruce Weaver .

Just as a quick summary:

• One issue is that of testing normality as a precondition to an analysis. This is not helpful. And apparently makes Hulk yell.
• A different issue is looking at the distribution of the whole of the dependent variable. This doesn't make any sense, as you mention. (Edit: See comments by Jochen Wilhelm in this thread.)
• It is fine to look at the distribution of the residuals from the analysis. These don't need to be tested for normality. Plots are helpful (q-q or histogram).
• In addition, there are other plots of residuals that are helpful. This includes looking at the homoscedasticity by plotting residuals vs. fitted values.

Dear Sal Mangiafico ,

your second point uses kind of misleading wording, to my opinion. You certainly mean the (empirical) distribution of the whole of the sample data, not the distribution (model) of the variable.

I think it is important (and irgnored to much in education) to make a distinction between the empirical distribution of data (y1, y2, ... yn) and a distribution model or random variable (Y) used in or as a statistical model. This model is typically represented by a parametric formula describing the density or the distribution function of the random variable. A concrete distribution of this model has concrete moments (expectation, variance, skew, kurtosis, ...), and concrete values of parameters of the distribution function.

Only the distribution model is relevant for the analysis. The empirical distribution of the data does not matter. Huh? Yes :) Each and every individual observed value is assumed to be a realization from a random variable defined by that distribution model, not neccesarily sharing the same values of the parameters. If I have a single observation of, say, the germination success of a seedling, and I encode this numerically as 1 for success and 0 for failure, then this single individual value is reasonably be assumed a realization of random variable with a binomial distribution. This distribution model has one parameter, and this single value provides extremely little information about the value of this parameter.

A larger sample of data has an empirical distribution, and with increasing sample size this empirical distribution should be an increasingly good approximation of the distribution model (and allow more precise estimations of the parameter values). I could plot the empirical distribution against the corresponding quantiles of the distribution model with the estimated or hypothesized parameter value(s) to check that (not working well for the binomial distribution as this has onlytwo distinct outcomes).

In an experiment I gather more data, typically under different conditions, assuming that the value of the parameter depends on the condition. In this case it makes no sense to compare the empirical distribution against one concrete distribution. I need to consider and separate all different conditions. In a regression setting, each individual observation is assumed a realization of a random variable with different parameter value(s), bringing us back to my introductory statement. The observed sample of data will not "fit" of any one concrete distribution, but still is each value a realization of a random variable with the same distribution model.

The normal distribution provides a special case here. The model has two parameters which independently determine the expectation and the variance. The shape of the distribution in independent from the expectation. This means, when (only) the expectation depends on the conditions, then the difference of the observed value and the corresponding expectation can be modelled as a realization from a random variable with zero expectation. This is the case for the difference of each and every observed value, so that the sample of such differences will have an empirical distribution that approximates the normal distribution model with expectation 0 (and the constant variance, whatever its value might be). We typically don't know the expected values that correspond to the observed values, but these are estimated from the statistical model. The difference between the observed values and the estimated values are the residuals you mentioned. Their empirical distribution will still approximate the normal distribution with expectation zero (but with a slightly larger variance).

When this is not the case, than the functional relationship between the expectation and the conditions and/or the assumptions about the distribution model are wrong.

Coming back to your point: I think that looking at the qq-plot of the residuals is looking at the distribution of the variable (comparing the distribution of the data to an assumed distribution model of the random variable). This works only in the special case of a normal distribution with constant variance, where the shape is independent of the expectation.

Jochen Wilhelm adresses this on, and to the point. Where "Y" is the random variable and "y" is assumed a realization of Y. We assume Y=y (which is not adressed in the "statistics" classes I had). Y=y never really holds exactly, so testing for it, is sometimes, if not always, without clear reasons a difficult topic. Mostly, if the observables y"are not normal "we" switch to ranked statistics, which adress a completely different question. Even when suggested against, this as not be the most reasonable choice and explain why it is suggested too complex (e.g., GLM logit link and beta distribution). This dichotomy normal vs. non-normal is also rather sad considering as it does not explain "why" and additionally, we have +d distributions to choose from. I have the idea (belive) "we" are not intersted anymore in assumptions and learning from our data by making an assumption on Y connecting it to something meaningful. We just want some flow chart with rules to follow. Note that this is an personal epistemic view.

An personal ontological point would be platonistonism from an agnostic-realism perspective. The perfect circle "C", can be considered the random variable in a universe independent to human perception are the realizations "c" (circles we observe are car tires, bottle caps, flower pots) are an imperfect observed reflection of it. Whether we believe C exists as transcedental entity or simply reproducable abstract is ontological. C is in this case a universal, the perfect form. The epistemic point comes in what we want to learn and do with the association between c more-or-less being C. The normality assumption is not exact and the assumption that a domain -Inf till Inf is never really "true". However, normality it is an extremely simple and elegant distribution. I believe "C" does not exist, so normality, gamma, beta, exponential distributions etc. do not exist as physicals forms. This does not matter too much though, but most of us never got tought the elegance of these distributions (and others) as what the idea/use/applications/violations are.

From an pragmatic epistemic point, given the residuals are "non-normal" one can perfectly assume normallity under particulair conditions. Ephasizing the particulair condtions. Generating a 1000 realized values from an exponential(1/0.5) and beta(3, 1), clearly both distributions are "non-normal" and neither are the residuals. This can be adressed looking at thehistogram or qqplot. Yet, whether we bootstrap, manually calculate a z-test (with n-1) and t.test (in R) the point and interval estimates are similair. The underlying assumption on Y (or C) as being unreasonable matter little when n is huge, even though both distributions are skwed in opposite directions. Assumptions matter when the sample size is small and even for small samples, some kinds of violations have no relevant impact on the analyses.* Yet, we never got thought these things. So Jochen Wilhelm is correct (please correct me when my comment is incorrect then I adjust it; edit=*).

Wim Kaijser , thank you for elaborating on my point and for giving the nice analogy with the abstract idea of a "circle" (as the mathematical set of all points in a two-dimensional Euklidean space with the same Euklidean distance to a given point) as compared to a concrete realization of some object with some shape property that we approximate as being "circle-like".

I slightly disagree with your statement that "The underlying assumption on Y (or C) as normal is reasonable in this case". Maybe just a wording thing. I'd rather say that the asusmption - although being unreasonable - does not matter much (when the sample size is huge). Assumptions matter when the sample size is small. And even for small samples, some kinds of violations have no relevant impact on the analyses, whereas other kinds of violations may have a severe impact.

For instance, violations regarding moments that do not impact the skewness are of little relevance for tests assuming a normal distribution even for small samples. As long as the distribution (from which the observations are considered realizations) is symmetric, t-tests and alikes work really well. Violations of the symmetry are way more problematic. They typically strongly compromise the power of the tests - something that is again not a problem in huge samples. Unfortunately, the empirical distributions of real data often have a strong skew, and here it matters that the assumption (that these data are realizations of a random variable with a normal dirtibution) is not reasonable, possibly even for moderately large samples (depending on the nature and strength of the skew).

Jochen Wilhelm I agree, bad wording (not my strongest "quality"), I made some minor edit ;). Your statement is more catious in this regared. Mine only hinges on n being huge.

Dear colleagues

trying to understand the context of the pronouncement that opens the debate, I see that the answers given and very well argued do not necessarily attend all the implications of what was initially exposed.

It is evident that there are areas of science in which 1000 measurements are extremely easy to obtain, but in others 4 or 5 have been the maximum, W. Cochran mentions that in small samples the sensitivities in the differences increase.

In several cases and when studying much better the experimental design process, some behaviors of the variables and residuals are related to the way of obtaining the measurements, the number of groups, the form of the groups and the origin of the groups, being this last one the one that most affects in declaring an experiment as pure or quasi, which has implications on how the assumption of normality is analyzed (here what you have mentioned).

but what happens between "parametric and non-parametric"?

I have had some cases in which researchers brought me results, in which one result rejected Ho and the other accepted it. obviously my question was why do both tests (typical T-test and Mann-Whitney U), and the answer was that so far it always confirmed because in both it tended to give me the same results.

just to exemplify and I hope I have explained myself well

Case 1 we tested the impact of 3 doses of a product on a type of soil, we analyzed the response in mineral recomposition, the difference is the dose of a product that being the same product for all treatments.

Case 2: three treatments (common product, organic compound, and algae compound) were tested on one type of soil, the response in mineral recomposition is analyzed. in this case the products are very different from the base, with different responses on the same type of soil.

Due to experimental costs, some of the treatments could not be applied more than 4 times.

one of these is a pure experiment, and it is worked as a single residue.

the other is a quasi-experiment and the residues should be treated per treatment.