"Many statistical textbooks state that parametric significance tests require a normal distribution of the samples' data points."

That's either wrong or a wrong interpretation.

The parametric tests are about parameters in a model. The models are fitted to data based on assumptions about properties of the "errors" (differences between the "best" model predictions and the observed data). These assumptions may imply a normal distribution of the "errors", but they may also imply any other distribution. The key is that the whole maths behind model fitting and assessment of statistical significance becomes very simple when the distribution of "errors" is normal. In ancient days, where no computers were available, people were restricted to such models because everything else was next to impossible to calculate, and R.A. Fisher developed nice and simple-to-use "shortcuts" to get the result without applying any complicated math (like the calculation fo the t-value as mean-difference divided by the standard error of the mean difference; the corresponding p-value could then be obtained by a provided look-up table; everything could be done with pencil and paper).

So there are two misconceptions:

- "parametric" is related to a model and not to the normal distribution

- the often required normal distribution is about the "errors" and not about the data.

Now to your citation of Norman and Steiner:

It is half-way correct.

Given the "errors" are exactly normal distributed, then statistical models based on this assumptions are exact. When the error-distribution is not exactly normal, then these tests are not exact. In reality, nothing has a perfect, exact normal distribution. It is like with a "circle": a circle is a mathematical concept of a set of points that have all the identical distance to the same (center)point. Although this concept is very useful in practice, there is nothing in nature that is really exactly circular. So in reality, there is no exact test based on such assumptions anyway. The key question is if the model is good enough for the purpose.

All parametric models internally work on the likelihood function of the data. For instance, the t-test is nothing but a likelihood-ratio test that compares the likelihoods of the data for two values of the parameter (the parameter is the mean difference between groups; one value is the null-hypothesis value, the other one is the value under which the likelihood of the data is maximal). The shape of this likelihood function approximates the shape of the normal distribution, and the approximation gets better for increasing sample sizes (this follows from the central limit theorem). Therefore, a test that is assuming that the likelihood function has the shape of a normal distribution will give quite reasonable or useful results even when the distribution of the errors is clearly not normal, as long as the sample size is large enough, so that the shape of the likelihood function is a reasonable approximation of the shape of a normal distribution.

Thank you for this detailed clarification, Jochen.

So far, I have never read evidences that errors have normal distribution in models found to fit datapoints -even if these ones are asumed apriori to have a "normal" distribution, or not asumed in that way-. I see no good reasons that may answer and explain the question posted by Peter Klaren. The measure of errors come from contrasting the dataset points with the estimated model values for those points and only requires to know the dataset and the model values for those points, it does not need to declare any condition on errors distribution model. I refuse to accept arguments that can not be explained with simple words and examples. Perhaps I am expecting too much from statistics, but a simple visual test of graphs of dataset versus model gives better hints about fitting quality. Is it time to question normal distributions and forget them forever as a dangerous prejudiced concept and source of doubtful statistical practices? emilio  

Emilio, I recommend you to read

E.T.Janes: Probability Theory: The Logic of Science

http://bayes.wustl.edu/etj/prob/book.pdf

Therein you will find a lot of concepts, deeper explanations and links enlightning some of the concepts behing using probability theory as a tool to quantify what and how much we can learn from data. The references to the derivation of the normal distribution will surely be helpful to understand why this is a *model*, and how this is relalated to assumptionsm adm eventually, why it is often (not always!) useful in statistical analyses.

Jochen. I could download only first 1/4 of 600 pages book, the missing part contains the subject of interest related to normal model. May you recommend me a shorter text  completely available at the net? Meanwhile I will follow the question. Or please, may you give me a 100 points univariate dataset that illustrate your point, so I may work it, and see if it contains some particular model that fits data better than some parametric one? There is here an english version of a short article (12 pages) of mine to be published this term, called "Laplace Criterion, central premise in statistical induction"; it shows a simpler method to model non linear, non-parametric datasets with good fittings.  Thanks, emilio

Emilio, I am sure that you will find different sources where to download "The logic of science", either the whole book or as individual chapters.

https://www.google.de/search?q=The+logic+of+science

OK Jochen, I could read (Chapter 7: The Central, Gaussian Or Normal Distribution) searching it with Google. It defends this parametric premise. It deals with many physics topics I hardly understood, and it contains more subjective remarks than factual explanations about Normal models. I think it resembles somewhat to the handling contained in the university textbook of Morris H. DeGroot (spanish version of Probability and Statistics, 2d. edition, 1988) when introducing Normal distributions. Thanks, emilio

Don't forget to mention that when a continuous variable has normal distribution, then, the distribution of the means, is also normal. In addition, the mean remains the same and the variance will be root(Variance/n)

Then, (from your question) "Although it is true that parametric statistics hang on the idea of a normal distribution, all we need is a normal distribution of the means, not of the original data." is wrong, both have normal (gaussian) distribution. This is "basic statistics" so you can find references on any basic Statistic book.

Jochen, amd José Jacobo. Recently I read this from a paper by Hagel about "Gauss versus Pareto":

"As McKelvey and Andriani observe, “the lesson that we can draw . . . is that extreme events, which in a Gaussian world could be safely ignored, are not only more common than expected but also of vastly larger magnitude and far more consequential.”

"Our institutions (not just businesses, but also educational and governmental) are largely designed for a Gaussian world where averages and forecasts are meaningful. As a result, we have evolved a sophisticated set of push programs that have delivered significant efficiency. In a world of sudden, severe and difficult to anticipate shifts, push programs become much less viable and we need to become a lot more creative in terms of designing pull platforms – something that JSB and I have written extensively about in the past. Bottom line: our institutional architectures, not to mention our technology architectures, will need to be redesigned to cope with a Paretian world."

While Gaussians always use the arithmetic mean of data to estimate media, and then use it to estimate deviation, they forget that they build it completely from data and assume that each one has same frequence; what for? Just to postulate a posterior normal distribution that denies the distribution media obtained from assuming f=1/N for each data value -which is assumed also as the corresponding media of each individual 1/N frequence-.

Thanks a lot to the person that downloaded my article. It is typical attitude of people without arguments neither courage to show own face and own name -as a signal of respect for debate and participants-. Best wishes for the ethical recovery of that human being.

OK.

Dear Emilio,

thank you for your comment.

I like to make three notes/comments:

1) Not everyone who calculates an arithmetic mean is a "Gaussian". The arithmetic mean is the maximum likelihood estimator of parameters of other probability models, for instance for the binomial and the Poisson model.

2) The arithmetic mean is a general estimate for the expectation (E(X)), absolutely independent of the underlying distributional model. The only required assumption is that the smplaing probability of each possible value in the population is equal (what many people call "random sampling").

3) Nothing in nature or real life is identical to what any of our models state. Models are simplified caricatures of particular aspects of reality. They are never "correct", but they can be useful. There is nothing really like a "circle" in nature, but the mathematical model "circle" is very helpful to describe and understand many things and processes that we observe. Simiar applies to the Gaussian distribution. It is a model, and it is in fact helpful. You should note that many practical things really work, although their development was based on conclusions derived under the assumption of some Gaussian distributions. Obviousely, the Gaussian model is not always appropriate and helpful, and it is one of the larger problems in applied statistics today that people seem to have stoped thinking about this and considering alternatives, well, even being aware of the existence of alternatives.

"If the only tool you know is a hammer, then everything looks like a nail" (does anyone know who said this?) - this is surely a common habit and it is to blame. But to substitute this by its inverse is also not ok: "When not everything is a nail, you should abandon the hammer"...

HI Jochen, your post requires answer and time to think.

I think that it is not rational to use premise A (U=arithmetic mean of data values) that conduces to model M1 (Build Lorenz Curve from A, then Cummulative distribution function is the derivate of Lorenz Curve with respect to cummulative frequence), and to postulate at the same time a model M2 (a parametric Probability Density Function, or [PDF(X)= dP/dX vs. X] that uses A at the start plus some “parameters” taken from a magic hat,  but contradicts A soon, to “develop” a theory around model M2, forgetting that you have denied premise A. It is an unacceptable logic trick. The two PDF obtained from both methods result different, so they can not be true at the same time, for the same estimated initial U mean.

It is absurd to explain PDF as the most important curve of statistics, just to say that probability is measured from the integral of that PDF, but sorry students, there is not mathematical solution for that integral, but I may give it to you in Tables at the end of book. Later people of statistics created the concept of “estimator” which is just another trick to hide the lack of high quality mathematics.

About 2) “the expectation (E(X)), absolutely independent of the underlying distributional model. The only required assumption is that the sampling probability of each possible value in the population is equal (what many people call "random sampling").

If you assume that each possible value has equal probability, then you loose your freedom to declare that E(x) is absolutely independent of distribution model, because you postulated the value of frequences for your data values in such a way that they add to unity, and are equal.

For example, if you have an unbiassed 4-faces die it means each side has ¼ probability. Supose experiment A is made with a die with faces 0, 1, 2, 3, and experiment B with faces 0,1,4,9.

In case A the expected value is Ua=6/4=1.5, in case B it is Ub=14/4= 3.5.

In both cases the U value is dependent from frequences, the same happends with the distributive structure. However, both experiments are random ones because they guarantee equal probabilities to each one of the possible events. In first case there is a linear relationship among outcomes and frequences; in second case there is a cuadratic relationship among them.

My thesis is that if we measure a natural event many times, it tends to show its tendences both in frequences and outcomes, so we may estimate proxy values of expected values, even considering the avatars brought by randomness that distort the estimated U value from arithmetic mean. If we do not accept it, then we should resign any effort to make statistical inferences.

Using a priori strange and imaginative models like normal curves that are presented as posterior distributions is amnesic with respect to the arithmetic mean premises, and thousand parametric ones only helps to obtain arbitrary results that only eventually will fit data if submitted to impartial tests. The only way to prove it is by working similar datasets to model them with the two methods with mathematics and graphing: 1) Lorenz curve, 2) Cummulative distribution curve and finally if you want, 3) The Function of probability density –the only one that was handled in the 19th century statistics theory-, a non informative curve that may be easily derived from CDF with calculus.

If you do not want to accept it, I understand it is hard to abandon previous wrong theories, so you may take the whole time you need. Meanwhile science will advance with other different inductive methods of data interpretation. My wish is that youth deserves to be educated demanding strict logic mathematic reasons and models and not accepting unclear methods that break initial premises during analysis.

If a person misuses its hammer and thinks everything takes the shape of a nail, that person requires psychic aid. I think that of parametric statistics. Many tools worked before Gauss bells existed as an analytical method to handle data. And I can obtain better inferences without talking about standard deviations, nor Gauss bells, nor freedom degrees, nor parametric models, nor estimators.

Thanks for representing one side of this debate, emilio