# Is there any statistical test that is parametric and non-parametric?

Is there any statistical test that is parametric and non-parametric? This question was asked by an interview panel. Is it valid question?

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Studying the wikipedia entry for nonparametric statistics might be enough to prepare you for an interviewer. You could answer the question with a question, as in "what do you mean by non-parametric? Distribution-free models or rank-order statististics?" –  jrhorn424 Nov 16 '11 at 4:10
As a point of departure, it might help you, as well as your respondents, to consult an authority (not the Internet!) concerning definitions. "The parametric cases ... are all those in which the class of all [states of nature] can be represented in terms of a vector $\theta$ consisting of a finite number of real components in a natural way. (...the distribution and loss function depend on $\theta$ in a reasonably smooth fashion.) All other problems are called nonparametric. --J. C. Kiefer, Introduction to Statistical Inference, p. 23. –  whuber Nov 16 '11 at 14:38
One of the Professor told me that 'Chi-Square test' has both behaviors (i.e., parametric and nonparametric as well). I did not understand at all, why 'chi square test' has both behaviors. –  Biostat Nov 16 '11 at 14:56
It's not the test that's parametric, it's the model that is. Chi-square distributions arise in both situations (in a natural way in the general linear model with Normal distributional assumptions, and as an approximation for a difference of log likelihoods--both of them parametric applications--and also as an approximation for the multinomial distributions that arise in many nonparametric applications), so there are many different tests sharing the name "chi-squared." This is probably what suggested your professor's comment. –  whuber Nov 16 '11 at 19:28
@whuber: Does your last comment mean that chi-square test for goodness-of-fit is nonparametric? –  Tim May 28 '13 at 12:12

It is fundamentally difficult to tell exactly what is meant by a "parametric test" and a "non-parametric test", though there are many concrete examples where most will agree on whether a test is parametric or non-parametric (but never both). A quick search gave this table, which I imagine represents a common practical distinction in some areas between parametric and non-parametric tests.

Just above the table referred to there is a remark:

"... parametric data has an underlying normal distribution .... Anything else is non-parametric."

It may be a accepted criterion in some areas that either we assume normality and use ANOVA, and this is parametric, or we don't assume normality and use non-parametric alternatives.

It's perhaps not a very good definition, and it's not really correct in my opinion, but it may be a practical rule of thumb. Mostly because the end goal in the social sciences, say, is to analyze data, and what good is it to be able to formulate a parametric model based on a non-normal distribution and then not be able to analyze the data?

An alternative definition, is to define "non-parametric tests" as tests that do not rely on distributional assumptions and parametric tests as anything else.

The former as well as the latter definition presented defines one class of tests and then defines the other class as the complement (anything else). By definition, this rules out that a test can be parametric as well as non-parametric.

The truth is that also the latter definition is problematic. What if there are certain natural "non-parametric" assumptions, such as symmetry, that can be imposed? Will that turn a test statistic that does otherwise not rely on any distributional assumptions into a parametric test? Most would say no!

Hence there are tests in the class of non-parametric tests that are allowed to make some distributional assumptions $-$ as long as they are not "too parametric". The borderline between the "parametric" and the "non-parametric" tests has become blurred, but I believe that most will uphold that either a test is parametric or it is non-parametric, perhaps it can be neither but saying that it is both makes little sense.

Taking a different point of view, many parametric tests are (equivalent to) likelihood ratio tests. This makes a general theory possible, and we have a unified understanding of the distributional properties of likelihood ratio tests under suitable regularity conditions. Non-parametric tests are, on the contrary, not equivalent to likelihood ratio tests per se $-$ there is no likelihood $-$ and without the unifying methodology based on the likelihood we have to derive distributional results on a case-by-case basis. The theory of empirical likelihood developed mainly by Art Owen at Stanford is, however, a very interesting compromise. It offers a likelihood based approach to statistics (an important point to me, as I regard the likelihood as a more important object than a $p$-value, say) without the need of typical parametric distributional assumptions. The fundamental idea is a clever use of the multinomial distribution on the empirical data, the methods are very "parametric" yet valid without restricting parametric assumptions.

Tests based on empirical likelihood have, IMHO, the virtues of parametric tests and the generality of non-parametric tests, hence among the tests I can think of, they come closest to qualify for being parametric as well as non-parametric, though I would not use this terminology.

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+1 Very interesting comments. As far as the borderline becoming "blurred," I take that as a correct statement about perception, but there is no blurriness in the definitions themselves: the distinction between parametric and non-parametric is as clear and sharp as that between, say, finite and infinite. –  whuber Nov 16 '11 at 14:45
@whuber, regarding what is "blurred", I was specifically referring to the fact that there can be distributional assumptions for non-parametric tests too, thus my second definition does not work either. If I should attempt a sharp definition, a parametric test is based on a model that can be parametrized by a subset of a finite dimensional Euclidean space. What I think is most "blurred" is that it is unclear, to me, how far from "no distributional assumptions" you can go before non-parametric assumptions become as much an issue as parametric assumptions. –  NRH Nov 16 '11 at 16:05
@whuber, I now read your comment to the question with reference to Kiefer, and yes it is definitely a good idea to consult an authority for a formal definition! I was actually more concerned with what people generally mean when they say "non-parametric", and I guess that few have a Kiefer-definition on their mind. –  NRH Nov 16 '11 at 16:21
See my quotation from Kiefer in a comment to the original question. In particular, "non-parametric" does not mean "no distributional assumptions." On the contrary, the most well-known non-parametric tests all make distributional assumptions. I think I do understand your sense of "blurred": I chose the finite/infinite analogy out of respect for that, because in practice a very large (but finite) number of parameters might just as well be considered infinite. –  whuber Nov 16 '11 at 16:24

Parametric is used in (at least) two meanings: A- To declare you are assuming the family of the noise distribution up to it's parameters. B- To declare you are assuming the specific functional relationship between the explanatory variables and the outcome.

Some examples:

• A quantile regression with a linear link would qualify as B-parametric and A-non-parametric.
• Spline smoothing of a time series with Gaussian noise can quality as A-non-parametric and B-parametric.

The term "semi-parametric" usually refers to case B and means you are not assuming the whole functional relation, but rather you have milder assumptions such as "additive in some smooth transformation of the predictors".

You could also have milder assumptions on the distribution of the noise- such as "all moments are finite", without specifically specifying the shape of the distribution. To the best of my knowledge, there is no term for this type of assumption.

Note that the answer relates to the underlying assumptions behind the data generating process. When saying "a-parametric test", one usually refers to non-parametric in sense A. In this is what you meant, then I would answer "no". It would be impossible to be parametric and non-parametric in the same sense at the same time.

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The two meanings in the first paragraph frequently have a unified treatment in the literature: that is, there appears to be no fundamental or important distinction between them. BTW, the "all moments are finite" case is definitely a non-parametric problem. –  whuber Nov 16 '11 at 14:41
@whuber: the definition in Keifer seems to cover both cases (I admit- I never read it and I am still looking for exceptions). On the other hand, terms do change their meanings. "Empirical-Bayes" no longer means what Robbins used it for in 1955. You cannot ignore the fact there is more than one interpretation circulating. –  JohnRos Nov 16 '11 at 19:09
OK, but we should be a little choosy: it's obvious many interpretations and attempted definitions of "parametric" and "non-parametric" are expressions of ignorance, not of understanding. Can you cite an alternative definition that is at once clear, rigorous, and authoritative (to be precise, authoritative in the sense that it would be accepted without question by a credible peer-reviewed journal)? –  whuber Nov 16 '11 at 19:23
@whuber: I accept the challenge! :-) Although note, since all researchers start their lookups in Wikipedia, it is a matter of time until credible peer-reviewed journals align with the Wiki definition. ("if you can't beat them...") –  JohnRos Nov 16 '11 at 19:35
The Wikipedia article quotes Wolfowitz from the 1940's, who not only is the first to use "non-parametric," but is also one of Kiefer's direct intellectual ancestors. I don't think we'll find any real difference there. (Kiefer only adds a technical requirement about the loss function.) However, I suspect that very few (if any) genuine researchers take Wikipedia as a point of departure, especially not in fields with mathematical foundations! –  whuber Nov 16 '11 at 19:58

I suppose that depends on what they mean by "parametric and non-parametric"? At the same time exactly both, or a blend of the two?

Many consider the Cox proportional hazards model to be semi-parametric, as it doesn't parametrically estimate the baseline hazard.

Or you might choose to view many non-parametric statistics as actually massively parametric.

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This seems to be a dodge. The question is probing whether one appreciates the distinction between "parametric" and "non-parametric," whether or not it is clear-cut. A good answer will illuminate that distinction, not confuse it. –  whuber Nov 16 '11 at 3:52
@whuber Which "the question"? The panel, or the OP? Because in my mind, the OP isn't probing at the distinction of anything. Which then means it depends on where people draw the line. I don't think providing both a common and philosophical example for "Well, it depends" is a dodge. I think it's an answer. Like whether or not one wants to consider a "parametric" to be fully parametric, or merely having parameters. –  Fomite Nov 17 '11 at 4:11
The point about "which question" is good. I think where I begin to have some trouble with your reply is that it makes distinctions that according to my resources make no sense (a "blend" is nonsensical, as well as the idea that a "statistic" can be parametric), which suggests you are using a different definition of "parametric" and "non-parametric" than I am. Although you make the excellent point that an answer has to depend on what these terms mean, you don't actually offer a definition to make your subsequent comments clear or understandable. –  whuber Nov 17 '11 at 7:25
@whuber Fair enough. I found the original question to be somewhat nonsensical, so was doing what I could. The question now has better answers that make some assumptions about what the OP means. –  Fomite Nov 17 '11 at 17:51

Bradley, in his classic Distribution-Free Statistical Tests (1968, p. 15–16 - see this question for a quote) clarifies the difference between distribution-free and nonparametric tests, which he says are often conflated with each other, and gives an example of a parametric distribution-free test as the Sign test for the median. This test makes no assumption about the underlying distribution of the sampled population of variate values, so it is distribution-free. However, if the selected median is correct, values above and below it should be selected at equal probability, testing random samples from the sampled variates as to whether they are above or below the median estimate should be binomial with $p=0.5$ so it is simultaneously parametric.

Update

Based on the discussion in the comments (thank you, whuber), it seems as if Bradley is in the minority, and what Bradley calls distribution-free, most others call parametric. And while nothing can really be $(A \cap \neg A)$ simultaneously, the answer to the question may just well depend on how you define the term, whether you make Bradley's distinction or call both elements of Bradley "parametric".

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I like the start of this answer because it makes an interesting distinction and supports it with a good reference. However, it seems to me that the rest of the answer confuses assumptions about the data with properties of the test statistic. The assumptions of the sign test are indeed "distribution free." However, the fact that the sampling distribution of the test statistic is binomial is a completely separate issue and does not make the procedure parametric! –  whuber Jun 20 '14 at 18:00
Well, Bradley himself calls the Sign test distribution-free yet parametric on page 15. The comment box is too small to bring the two key sentences in their entirety. Please read the other answer, specifically the sentences that start "Roughly speaking…" and "In order to be entirely clear…". Thank you. –  Avraham Jun 20 '14 at 18:15
If that's the case with Bradley, then either the meanings of these terms have changed since then or (I hate to say it) you misinterpret what he wrote. (I haven't access to a copy I can check.) It is definitely not the case now--nor has it been for at least the last 30 years--that "parametric" has referred to the distribution of a test statistic. See the Wolfowitz quotation in the Wikipedia article. –  whuber Jun 20 '14 at 19:13
Thank you for the offer, but this isn't a big deal so you needn't bother. My favorite source for clear information on basic concepts is Kiefer, Introduction to Statistical Inference (Springer 1987). "The parametric cases of statistical problems are all those in which the class of all df's $F$ in $\Omega$ can be represented in terms of a vector $\theta$ consisting of a finite number of real components, in a natural way. ... All other problems are called nonparametric" [at p. 23, emphases in the original]. –  whuber Jun 20 '14 at 20:56
For what it is worth, I looked at two other statistical texts, DeGroot's Probability and Statistics (2nd ed, pp 520-521) and Larson's Introduction to Probability Theory and Statistical Inference (3rd edition, pp.508-509) and both use the term parametric to mean what Bradly calls distribution-free, which is like Kiefer, I reckon. So, to answer the OP, it depends on how you define "parametric". –  Avraham Jun 24 '14 at 13:56