When Fm in ANOVA is significant, Schefft?s procedure provides no limit to the number of contrasts which can be declared significant. The new procedure described here provides a limit (r) to the number of significant contrasts which can be asserted. If assertions are made for u1 contrasts, which include r declarations of significance, the expectation of r/ul will be a linearly decreasing function of ul if traditional critical F values are used. A table is provided of new critical F values which set this expectation at a constant (0.05 or 0.01). The new critical F values are recommended for use whenever contrasts are sought, post hoc, for significance. RODGERThe new procedure is not limited to simple ANOVA. A description of its value in analysing data from experiments designed factorially, particularly as it affects the power of tests for interactions, is given in Rodger (1974). One of the many possible non-parametric applications can be found in Rodger (1969).
A method is given for choosing (post hoc) a set of v1 mutually orthogonal null contrasts such that the maximum number (r) rejectable by a new criterion are rejected and v1 – r retained. Type I error rate is defined as the expected proportion of the v1 decisions which will be errors when all nulls are true. Because the new criterion uses F, the expected proportion of rejections can be calculated for the case when all nulls are not true, and this is taken as the definition of power. Theoretical means are deducible from decisions for v1 mutually orthogonal contrasts and, if rejected null contrasts are given suitable non‐zero values, there is no ambiguity about the theoretical means. The traditional values of FΓ; v1, v2 ‘load the dice’ against finding true interactions ‘significant’ in factorial analyses. The new values (table given) do not have this property and their use is illustrated using a one‐way anova on data from a factorially designed experiment. This procedure permits the analysis of unequal replications in factorially designed experiments.
The psychometric properties of the Autism Behavior Checklist (ABC; Krug, Arick, & Almond, 1980a, 1980b), a 57-item screening checklist for autism was investigated. Professional Informants completed the ABC on 67 autistic and 56 mentally retarded and learning-disabled children. The autistic children were the total population of autistic children aged 6-15 in two circumscribed suburban and rural regions. Using the total score, the ABC accurately discriminated 91% of the children, with 87% of the autistic and 96% of the nonautistic group correctly classified. Moreover, the accuracy of classification was virtually identical when only the more heavily weighted checklist items were used. A 3-factor model accounted for 32% of the total variance in the checklist. Seventeen items loaded .4 or more on Factor 1, 12 items loaded on Factor 2, and 10 items loaded on Factor 3. The present results fail to provide empirical support for a single unidimensional scale for autism. Also, there is little support for subdividing the checklist into five subscales based on symptom areas.
Rodger (1 967) proposed a procedure for selecting contrasts for decision-making post hoc. Using new critical F values (Rodger, 1975), the method keeps the expected proportion of null contrasts rejected (when all nulls are true) at a fixed level Ea (e.g. 0.05) by rejecting a calculated number r and retaining v1 -Y nulls. The present paper reports parameters A[Ep] ; vl, uZ from the non-central variance-ratio distribution which has the overall, non-central parameter Am = vl A[EN ; vl, v2 = vI Nga, This Am is appropriate when vl mutually orthogonal contrasts, across the true means p,, each have the value 6 = f go 42 c,*. T h e tabled parameters allow the user to calculate the sample size N required to set the expected null contrast rejection rate at EP (e.g. 0-95) when the above A,,, is true.t Requests for reprints should be sent to D r R. S. Rodger, REJECTION RATE FOR CONTRASTS 215(3) with some gain in generality. These features are rarely used, but see Rodger, 1967, pp. 200-202 for an example.When (1) is true, then all contrasts of form (3) are true. Furthermore, if any H = J-1 linearly independent contrasts of form (3) are all true, then (1) is true. I n such cases, F, in (2) is distributed in central variance-ratio form with degrees of freedom v1 and v2. When contrasts constitute the questions of primary scientific interest, as is usually true in the fixed-effects context, then Rodger (1967, 1974) has advocated making decisions for H = J-1 of them by rejecting those which satisfyThe H chosen should preferably be mutually orthogonal, but certainly linearly independent of one another. If the H contrasts are selected independently of the m,, preferably before these are collected, then v1 = 1 and this is the method of planned tests. If the H contrasts are selected in the light of, or as a function of, how the mj have turned out, this is a method of post-hoc tests, v1 = J-1 is used and no more thannulls should be rejected. I n (4) and (5) the F[Eor] ; vl, v2 are special critical values given by Rodger (1975) and he showed there that rule (5) sets the expected rejection rate E(r/vl) at Ea (cf. (13) below). The emphasis in this paper will be on contrasts selected for decision making post hoc. WHEN SOME NULLS ARE FALSEWhen the overall null hypothesis (1) is false then A, = N C (pip.)2/d > 0 (6) is true and F,, in (2) does not follow the central variance-ratio distribution but a non-central form (Fisher, 1928, distribution C). The latter distribution has degrees of freedom v1,v2 and overall, non-central parameter 118; v1,v2. I n the non-central F distribution which has A, = AB; v1,v2, the area to the right of Fa; vl, v2 is p, which is the power of test (2). One can therefore ensure a power at least / 3 by choosing N to satisfy N 2 u2 AB ; v1, V 2 l X (Pjp.)2, (7) but / 3 drops dramatically, for fixed A, , as v1 increases (see Rodger, 1974, Table 5).The hth non-zero contrast across the pi may be written l K h = C h l t L l + C h 2 E 1 2 + . . . + C h J l l J = 6,. (8) T h e ch5 are defined for eqn. (8) as they were for (3)...
Methods are given for testing linear statistical hypotheses about population means of a variate which is normally distributed in each population, with common variance. The view is adopted that error‐rates (or the expected proportions of decision errors) be prespecified for each decision to be made. This was the view that led to the invention of the R technique and methods for setting type II error‐rates for this technique are given here. Experiment‐based error‐rates control the probability of making one or more decision errors. Although not an essential part of the doctrine, the decision‐based error‐rate view leads naturally to a consideration of the probability distributions of r decision errors and such distributions are given in this paper.
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