Hi, Dennis. I'm not sure this answers precisely the question you asked, but FWIW:
I learned to distinguish between the "rejection region" and the "acceptance region" associated with a null hypothesis; although, now that I think about it, possibly "retention" is a better label than "acceptance"; although the decision associated with that region was not so much to "accept" or "retain" the null hypothesis as to "fail to reject" it. In any case, the two regions are complementary, jointly covering the entire real line. If the data lead to a test value beyond the critical value (that is, farther from the value specified by the null hypothesis), one decides to reject the null, and the region of possible values of the test statistic that satisfies this condition is the "rejection region"; for a two-sided test, it is a bifurcated region, rather like Pakistan when it was first created. The complementary region (between the critical values in a two-sided test) is the "acceptance" or "retention" region, where one cannot reject the null hypothesis but must logically consider it consistent with the data (at the significance level chosen). Now these regions appear to me to control _decisions_ we choose to make; I'm not sure that decisions are necessarily "actions", as you write, but I've no particular desire to quibble about that point. Does this speak adequately to your question? On Mon, 28 Oct 2002, dennis roberts wrote: > the terms retention and rejection refer to actions we take ... based > on comparing some sample data to some stated null hypothesis > > clearly, there are a number of ways one could go about making > retention and/or rejection decisions ... one could perform the > standard "test" and see if the value (t for example) calculated from > your sample data was = or >, or < than some critical value > > one could also construct a confidence interval and see if the null > value was inside or outside of that CI Which yields exactly the same decision as the hypothesis test, if the CI is the (1-alpha) interval corresponding to the significance level at which one chose to test the null hypothesis. > the question i have is: do the terms "retention REGION" and > "rejection REGION" have variable meanings? I don't really see how they should. > for example, most books will show some theoretical null distribution > with CVs at either end (if a 2 tailed test) and label the middle > around the null value as the RETENTION area or region and the > extremes as the REJECTION area or region ... BUT, could not the > range between the boundaries of the CI also be called the RETENTION > area or region and the area outside be called the REJECTION region > or area? The distinction to be made is between, on the one hand, values of the test statistic (that is, of the data) that are incompatible (or not) with the null hypothesis (which is a statement about the value of an unobserved and probably unobservable parameter of the underlying population), where "incompatible" is understood in terms of the chosen level of significance, alpha; and, on the other, values of the parameter of the population (that is, values that might have been specified in the null hypothesis) that are incompatible with the data (in that they lie outside the CI). The _decision_ is the same from either perspective; provided the CI is the (1-alpha) CI and is two-sided if the hypothesis test is two-sided. > is it THE areas that give forth to specific decisions that is key > here or, areas ON some specific distribution that are key? that is > ... are the terms retention and rejection regions ONLY appropriate > when considering two overlapping null and true distributions for the > purposes of establishing power, etc? I don't understand what you mean by "two overlapping null and true distributions"; unless you're making the logical error of plotting values of the parameter and values of the statistic on the same axis. I remember reading an article some years ago, in the days when we were both at OISE, whose author advocated (at least for pedagogical purposes and for clarity of distinction) plotting parameter values on one axis and statistic values on the orthogonal axis. One could then clearly see the distinction between a CI and an "acceptance region": for the decision rule was represented by a pair of parallel 45-degree lines (if one used the same scale on both axes, as was easy to do on graph paper and is much less easy in one's standard software package!), and the one interval was a vertical line segment, and the other region was a horizontal line segment. One could see clearly the separate effects of (i) choosing a different parameter value for H0 and (ii) obtaining different observed values from one's sample. Ciao! -- Don. ----------------------------------------------------------------------- Donald F. 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