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2	Probability Models I define parametric, semiparametric, and nonparametric models in the two sample setting My definition of semiparametric models is a little stronger than some statisticians The distinction is to isolate models with assumptions that I think too strong Notation for two sample probability model
3	Probability Models Parametric models F, G are known up to some finite dimensional parameter vectors
4	Probability Models Parametric models: Examples
5	Probability Models Semiparametric models Forms of F, G are unknown, but related to each other by some finite dimensional parameter vector G can be determined from F and a finite dimensional parameter (Most often: under the null hypothesis, F = G)
6	Probability Models Semiparametric models Forms of F, G are unknown, but related to each other by some finite dimensional parameter vector G can be determined from F and a finite dimensional parameter
7	Probability Models Semiparametric models: Examples
8	Probability Models Nonparametric models Forms of F, G are completely arbitrary and unknown An infinite dimensional parameter is needed to derive the form of G from F (I demand that the above hold under all hypotheses, unless the test is consistent when F ¹ G) Examples of truly nonparametric analyses: Kolmogorov-Smirnov test t-test with unequal variances (large samples)
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11	The Problem In the development of statistical models, and even moreso in the teaching of statistics, parametric probability models have received undue emphasis Examples: t test is typically presented in the context of the normal probability model theory of linear models stresses small sample properties random effects specified parametrically Bayesian (and especially hierarchical Bayes) models are replete with parametric distributions
12	The Problem ASSERTION: Such emphasis is not typically in keeping with the state of knowledge as an experiment is being conducted The parametric assumptions are more detailed than the hypothesis being tested, e.g.,: Question: How does the intervention affect the first moment of the probability distribution? Assumption: We know how the intervention affects the 2nd, 3rd, …, ∞ central moments of the probability distribution.
13	The Problem Conditions under which an intervention might be expected to affect many aspects of a probability distribution Example 1: Cell proliferation in cancer prevention Within subject distribution of outcome is skewed (cancer is a focal disease) Such skewed measurements are only observed in a subset of the subjects The intervention affects only hyperproliferation (our ideal)
14	The Problem Conditions under which an intervention might be expected to affect many aspects of a probability distribution (cont.) Example 2: Treatment of hypertension Hypertension has multiple causes Any given intervention might treat only subgroups of subjects (and subgroup membership is a latent variable) The treated population has a mixture distribution (and note that we might expect greater variance in the group with the lower mean)
15	The Problem Conditions under which an intervention might be expected to affect many aspects of a probability distribution (cont.) Example 3: Effects on rates The intervention affects rates The outcome measures a cumulative state Arbitrarily complex mean-variance relationships can result
16	The Problem These and other mechanisms would seem to make it likely that the problems in which a fully parametric model or even a semiparametric model is correct constitute a set of measure zero Exception: independent binary data must be binomially distributed in the population from which they were sampled randomly (exchangeably?)
17	The Problem Impact on what we teach about optimality of statistical models Clearly, parametric theory may be irrelevant in an exact sense (though as guidelines it is still useful) Much of what we teach about the optimality of nonparametric tests is based on semiparametric models e.g., Lehmann, 1975: location-shift models
18	The Problem Example: the Wilcoxon rank sum test Common teaching: Not too bad against normal data Better than t test when data have heavy tails More accurate guidelines: Above holds when a shift model holds for some monotonic transformation of the data If propensity to outliers (mixture distributions) is different between groups, the t test may be better even in presence of heavy tails In the general case, the t test and the Wilcoxon are not testing the same summary measure