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### Reading Testing a point-null hypothesis, by Jiahuan Li, Feb. 25, 2013

1. 1. Testing a Point Null Hypothesisi: The Irreconcilability of P Values and Evidence JAMES O.BERGER and THOMAS SELLKE 25.02.2013JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
2. 2. Content 1 INTRODUCTION JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
3. 3. Content 1 INTRODUCTION 2 POSTERIOR PROBABILITIES AND ODDS JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
4. 4. Content 1 INTRODUCTION 2 POSTERIOR PROBABILITIES AND ODDS 3 LOWER BOUNDS ON POSTERIOR PROBABILI- TIES Introduction Lower Bounds for GA ={All Distributions} Lower Bounds for GS ={Symmetric Distributions} Lower Bounds for GUS ={Unimodal,Symmetric Distributions} Lower Bounds for GNOR ={Normal Distributions} JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
5. 5. Content 1 INTRODUCTION 2 POSTERIOR PROBABILITIES AND ODDS 3 LOWER BOUNDS ON POSTERIOR PROBABILI- TIES Introduction Lower Bounds for GA ={All Distributions} Lower Bounds for GS ={Symmetric Distributions} Lower Bounds for GUS ={Unimodal,Symmetric Distributions} Lower Bounds for GNOR ={Normal Distributions} 4 MORE GENERAL HYPOTHESES AND CONDITIONAL CALCULATOINS General Formulation More General Hypotheses JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
6. 6. Content 1 INTRODUCTION 2 POSTERIOR PROBABILITIES AND ODDS 3 LOWER BOUNDS ON POSTERIOR PROBABILI- TIES Introduction Lower Bounds for GA ={All Distributions} Lower Bounds for GS ={Symmetric Distributions} Lower Bounds for GUS ={Unimodal,Symmetric Distributions} Lower Bounds for GNOR ={Normal Distributions} 4 MORE GENERAL HYPOTHESES AND CONDITIONAL CALCULATOINS General Formulation More General Hypotheses 5 CONCLUSIONS JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
7. 7. 1. Introduction The paper studies the problem of testing a point null hypothesis, of interest is the relationship between the P value and conditional and Bayesian measures of evidence against the null hypothesis ∗ The overall conclusion is that P value can be highly misleading measures of the evidence provided by le data against the null hypothesis JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
8. 8. 1. Introduction Consider the simple situation of observing a random quantity X having density f (x | θ) , θ ⊂ R 1 , it is desired to test the null hypothesis H0 : θ = θ0 versus the alternative hypothesis H1 : θ = θ0 . p = Prθ=θ0 (T (X ) ≥ T (x)) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
9. 9. 1. lntroduction Example Suppose that X = (X1 , ......., Xn ) where the Xi are iid N(θ, σ 2 ) Then the usual test statistic is √ ¯ T (X ) = n | X − θ0 | /σ ¯ where X is the sample mean, and p = 2(1 − Φ(t)) where Φ is the standard normal cdf and √ t = T (x) = n | x − θ0 | /σ ¯ JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
10. 10. 1. Introduction We presume that the classical approcach is the report of p, rather than the report of a Neyman-Perason error probability. This is because Most statistician prefer use of P values, feeling it to be impor- tant to indicate how strong the evidence against H0 is . The alternative measures of evidence we consider are based on knowledge of x. There are several well-known criticisms of testing a point null hypothesis. One is the issue of ’statistical’ versus ’practical’ signiﬁcance, that one can get a very small p even when | θ − θ0 | is so small as to make θ equivalent to θ0 for practical purposes. Another well known is ’Jeﬀrey’s paradox’ JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
11. 11. 1. Introduction Example Consider a Bayesian who chooses the prior distribution on θ, which gives probability 0,5 to H0 and H1 and spreads mass out on H1 according to an N(θ, σ 2 ) density. It will be seen in Section 2 that the posterior probability, Pr(H0 | x), of H0 is given by Pr (H0 | x) = (1 + (1 + n)−1/2 exp{t 2 /[2[(1 + 1/n)]})−1 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
12. 12. 1. Introduction Table 1 : Pr(H0 | x) for Jeﬀreys-Type Prior) n p t 1 5 10 20 50 100 1,000 .10 1.645 .42 .44 .47 .56 .65 .72 .89 .05 .1.960 .35 .33 .37 .42 .52 .60 .82 .01 .2.576 .21 .13 .14 .16 .22 .27 .53 .001 3.291 .086 .026 .024 .026 .034 .045 .124 The conﬂict between p and Pr(H0 | x) is apparent. JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
13. 13. 1. Introduction Example Again consider a Bayesian who gives each hypothesis prior probabil- ity 0.5, but now suppose that he decides to spread out the mass on H1 in the symmetric fashion that is as favorable to H1 as possible. The corresponding values of Pr (H0 | x) are determined in Section 3 and are given in Table 2 for certain values of t. Table 2 : Pr(H0 | x) for a Prior Biased Towar H1 P value(p) t Pr (H0 | x) .10 1.645 .340 .05 .1.960 .227 .01 .2.576 .068 .001 3.291 .0088 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
14. 14. 1. Introduction Example (A Likelihood Analysis) It is common to perceive the comparative evidence provided by x for two pssible parameter values, θ1 andθ2 , as being measured by the likelihood ratio lx (θ1 : θ2 ) = f (x | θ1 )/f (x | θ2 ) A lower bound on the comparative evidence would be f (x | θ0 ) lx = inf lx (θ0 : θ) = = exp{−t 2 /2} θ supθ f (x | θ) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
15. 15. 1. Introduction Values of lx for various t are given in Table 3 Table 3 : Bounds on the Comparative Likelihood Likelihood ratio P value(p) t lower bound (lx ) .10 1.645 .340 .05 .1.960 .227 .01 .2.576 .068 .001 3.291 .0088 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
16. 16. 2. Posterior probabilities and odds let 0< π0 < 1 denote the prior probability of H0 , and let π1 = 1−π0 denote the prior probability of H1 , suppose that the mass on H1 is spread out according to the density g (θ). Realistic hypothesis: H0 :| θ − θ0 |≤ b Prior probability π0 would be assigned to {θ :| θ − θ0 |≤ b} (To a Bayesian, a point null test is typically reasonable only when the prior distribution is of this form) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
17. 17. 2. Posterior probabilities and odds Noting that the marginal density of X is m(x) = f (x | θ0 )π0 + (1 − π0 )mg (x) Where mg (x) = f (x | θ)g (θ)dθ The posterior probability of H0 is given by Pr (H0 | x) = f (x | θ0 ) × π0 /m(x) (1 − π0 ) mg (x) −1 = [1 + × ] π0 f (x | θ0 ) Also of interest is the posterior odds ratio of H0 to H1 which is Pr (H0 | x) π0 f (x | θ0 ) = × 1 − Pr (H0 | x) (1 − π0 ) mg (x) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
18. 18. 2. Posterior probabilities and odds The posterior odds ratio of H0 to H1 Pr (H0 | x) π0 f (x | θ0 ) = × 1 − Pr (H0 | x) (1 − π0 ) mg (x) Post odds Prior odds Bayes factor Bg (x) Interest in the Bayes factor centers around the fact that it does not involve the prior probabilities of the hypotheses and hence is sometimes interpreted as the actual odds of the hypotheses implied by the data alone. JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
19. 19. 2. Posterior probabilities and odds Example (Jeﬀreys-Lindley paradox) Suppose that π0 is arbitrary and g is again N(θ0 , σ 2 ).Since a suﬃ- cent statistic for θ is X ¯ N(θ0 , σ 2 /n) ,we have that mg (¯) is an x N(θ0 , σ 2 (1 + n−1 )) distribution. Thus Bg (x) = f (x | θ0 )/mg (¯) x [2πσ 2 /n]−2 exp{− n (¯ − θ0 )2 /σ 2 } 2 x = 2 (1 + n−1 )]−1/2 exp{− 1 (¯ − θ 2 )/[σ 2 (1 + n−1 ]} [2πσ 2 x 0 1 = (1 + n)1/2 exp{− t 2 /(1 + n−1 )} 2 and Pr (H0 | x) = [1 + (1 − π0 )/(π0 Bg )]−1 (1 − π0 ) 1 = [1 + (1 + n)−1/2 × exp{ t 2 /(1 + n−1 )}]−1 π0 2 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
20. 20. 3. Lower bounds on posterior probabilites3.1 Introduction This section will examine some lower bounds on Pr (H0 | x) when g (θ), the distribution of θ given that H1 is true is allowed to vary within some class of distribitions G GA ={all distributions} GS ={all distributions symmetric about θ0 } GUS ={all unimodal distribution symmetric about θ0 } GNOR ={all N(θ0 , τ 2 )distributions, 0≤ τ 2 < ∞} Even though these G’s are supposed to consist only of distribution on {θ | θ = θ0 }, il will be convenient to allow them to include distributions with mass at θ0 , so the lower bounds we compute are always attained. JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
21. 21. 3. Lower bounds on posterior probabilites3.1 Introduction Letting Pr (H0 | x, G ) = inf Pr (H0 | x) g ∈G and B(xmG ) = inf Bg (x) g ∈G we see immediately form formulas before that B(x, G ) = f (x | θ0 )/ sup mg (x) g ∈G and (1 − π0 ) 1 Pr (H0 | xmG ) = [1 + × ]−1 π0 B(x, G ) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
22. 22. 3. Lower bounds on posterior probabilites3.2 Lower bounds for GA ={All distributions} Theorem Suppose that a maximum likelihood estimate of θ0 , exists for the observed x. Then ˆ B(x, GA ) = f (x | θ0 )/f (x | θ(x)) and ˆ (1 − π0 ) f (x | θ(x)) −1 Pr (H0 | x, GA ) = [1 + × ] π0 f (x | θ0 ) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
23. 23. 3. Lower bounds on posterior probabilites3.2 Lower bounds for GA ={All distributions} Example In this situation,we have 2 B(x, GA ) = e −t /2 ¯ and (1 − π0 ) t 2 /2 −1 Pr (H0 | x, GA ) = [1 + e ] π0 For servral choices of t, Table 4 gives the corresponding two-sided P values,p, and the values of Pr (H0 | x, GA ),with π0 = 0.5. JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
24. 24. 3. Lower bounds on posterior probabilites3.2 Lower bounds for GA ={All distributions} For servral choices of t, Table 4 gives the corresponding two-sided P values,p, and the values of Pr (H0 | x, GA ),with π0 = 0.5. Table 4 : Comparison of P values and Pr (H0 | x, GA ) when π0 = 0.5 P value(p) t Pr (H0 | x, GA ) Pr (H0 | x, GA )/(pt) .10 1.645 .205 1.25 .05 .1.960 .128 1.30 .01 .2.576 .035 1.36 .001 3.291 .0044 1.35 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
25. 25. 3. Lower bounds on posterior probabilites3.2 Lower bounds for GA ={All distributions} Theorem For t > 1.68 and π0 = 0.5 in Example 1, Pr (H0 | x, GA )/pt > π/2 1.253 Furthermore lim Pr (H0 | x, GA )/pt = π/2 t→∞ JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
26. 26. 3. Lower bounds on posterior probabilites3.3 Lower bounds for GS ={Symmetric distributions} There is a large gap between Pr (H0 | x, GA )andPr (H0 | x) for the Jeﬀreys-type single prior analysis.This reinforces the suspicion that using GA unduly biases the conclusion against H0 and suggests use of more reasonable classes of priors. Theorem sup mg (x) = sup mg (x), g ∈G2ps g ∈GS so B(x, G2PS ) = B(x, GS ) and Pr (H0 | x, G2ps ) = Pr (H0 | x, GS ) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
27. 27. 3. Lower bounds on posterior probabilites3.3 Lower bounds for GS ={Symmetric distributions} Example If t ≤ 1, a calculus argument show that the symmetric two point distribution that strictly maximizes mg (x) is the degenerate ”two- point”distribution putting all mass at θ0 . Thus B(x, GS ) = 1 and Pr (H0 | x, GS ) = π0 for t ≤ 1. If t ≥ 1 , then mg (x) is maximized by a nondegenerate element of G2ps . For moderately large t, the maximum value of mg (x) for g∈ G2ps is very well approximated by taking g to be the two-point ˆ ˆ distribution putting equal mass at θ(x) and at 2θ − θ(x). so ϕ(t) B(x, GS ) 2 exp {−0.5t 2 } 0.5ϕ(0) + 0.5ϕ(2t) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
28. 28. 3. Lower bounds on posterior probabilites3.3 Lower bounds for GS ={Symmetric distributions} Example For t ≤ 1.645, the ﬁrst approximation is accurate to within 1 in the fourth signiﬁcant digit and the second approximation to within 2 in the third signiﬁcant digit. Table 5 gives the value of Pr (H0 | x, Gs ) of several choices of t. Table 5 : Comparison of P values and Pr (H0 | x, GS ) when π0 = 0.5 P value(p) t Pr (H0 | x, GS ) Pr (H0 | x, GS )/(pt) .10 1.645 .340 2.07 .05 .1.960 .227 2.31 .01 .2.576 .068 2.62 .001 3.291 .0088 2.68 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
29. 29. 3. Lower bounds on posterior probabilites3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions} Minimizing Pr (H0 | x) over all symmetric priors still involves considerable bias against H0 . A further ’objective’ restriction, which would seem reasonable to many, is to require the prior to be unimodal, or non-increasing in | θ − θ0 | . Theorem sup mg (x) = sup mg (x), g ∈Gus g ∈US with US ={all symmetric uniform distributions} so B(x, GUS ) = B(x, US ) and Pr (H0 | x, GUS ) = Pr (H0 | x, US ) JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
30. 30. 3. Lower bounds on posterior probabilites3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions} Theorem If t ≤ 1 in example 1, then B(x, GUS ) = 1 and Pr (H0 | x.GUS ) = π0 . Ift > 1 then 2ϕ(t) B(x.GUS ) = ϕ(K + t) + ϕ(K − t) and (1 − π0 ) ϕ(K + t) + ϕ(K = t) −1 Pr (H0 | x, GUS ) = [1 + × ] π0 2ϕ(t) where K > 0 Figures 1 and 2 give values of K and B for various val- ues of t in this problem JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
31. 31. 3. Lower bounds on posterior probabilites3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions} JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
32. 32. 3. Lower bounds on posterior probabilites3.4 Lower bounds for GUS ={Unimodal, Symmetric distributions} Table 6 gives Pr (H0 | x, GUS ) for some speciﬁc important values of t and π0 = 0.5 Table 6 : Comparison of P values and Pr (H0 | x, GUS ) when π0 = 0.5 P value(p) t Pr (H0 | x, GUS ) Pr (H0 | x, GUS )/(pt) .10 1.645 .390 1.44 .05 .1.960 .290 1.51 .01 .2.576 .109 1.64 .001 3.291 .018 1.66 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
33. 33. 3. Lower bounds on posterior probabilites3.5 Lower bounds for GNOR ={Normal distributions} We have seene that minimizing Pr (H | x)overg ∈ GUS is the same as minimizing over g ∈ US . Althought using US is much more reasonable than using GA ,there is still some residual bias against H0 involved in using US . Theorem If t ≤ 1in Example 1, then B(x, GNOR ) = 1 and Pr (H0 | x, GNOR ) = π0 . If t > 1, then √ 2 B(x, GNOR ) = ete −t /2 and (1 − π0 ) exp{t 2 /2} −1 Pr (H0 | x, GNOR ) = [1 + × √ ] π0 et JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
34. 34. 3. Lower bounds on posterior probabilites3.5 Lower bounds for GNOR ={Normal distributions} Table 7 gives Pr (H0 | x, GNOR ) for servral values of t Table 7 : Comparison of P values and Pr (H0 | x, GNOR ) when π0 = 0.5 P value(p) t Pr (H0 | x, GNOR ) Pr (H0 | x, GNOR )/(pt) .10 1.645 .412 1.52 .05 .1.960 .321 1.67 .01 .2.576 .133 2.01 .001 3.291 .0235 2.18 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
35. 35. 4. More general hypotheses and conditional calculations4.1 General formulation Consider the Bayesian calculation of Pr (H0 | A) , where H0 is of the form H0 : θ ∈ Θ0 and A is the set in which x is known to reside. Then letting π0 and π1 denote the prior probabilities of H0 and H1 and g1 and g2 as the densities on Θ0 and Θ1 . Then we have : 1 − π0 mg 1 (A) −1 Pr (H0 | A) = [1 + × ] π+0 mg 0 (A) where mg 1 (A) = Prθ (A)gi (θ)dθ Θ0 JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
36. 36. 4. More general hypotheses and conditional calculations4.1 General formulation For the general formulation, one can determine lower bounds on Pr (H0 | A) by choosing sets G0 and G1 of g0 and g1 , respectively, calculating B(A, G0 , G1 ) = inf mg0 (A)/ sup mg 1 (A) g0 ∈G0 g1 ∈G1 (1−π0 ) 1 −1 and deﬁning Pr (H0 | A, G0 , G1 ) = [1 + π0 × B(A,G0 ,G1 ) ] JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
37. 37. 4. More general hypotheses and conditional calculations4.2 More general hypotheses Assume in this section that A= {x}. The lower bounds can be applied to a variety of generalizations of point null hypotheses. If Θ0 is a small set about θ0 , the negeral lower bounds turn out to be essentially equivalent to the point null lower bounds. In example 1, suppose that the hypotheses were H0 : θ ∈ (θ0 − √ b, θ0 + b) and H1 : θ ∈ (θ0 − b, θ0 + b) If | t − nb/σ |≥ 1 / and G0 = G1 = GS , then B(x, G0 , G1 ) and Pr (H0 | x, G0 , G1 ) are exactly the same as B and P for testing the point null. JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
38. 38. 5. Conclusion B(x, GUS ) and the P value calculated at (t − 1)+ in instead of t, this last will be called the P value of (t − 1)+ Figure shows that this comparative likelihood is close to the P value that would be obtained if we replaced t by (t − 1)+ . The implication is that : t = 1 means only mild evidence against H0 , t = 2 means signiﬁcant evidence against H0 , and t = 3 means highly evidence against H0 , and t = 4 means over whelming evidence against H0 , should at least replaced by the rule of thumb, that is : t = 1 means no evidence against H0 , t = 2 means only mild evidence against H0 , and t = 3 means signiﬁcant evidence against H0 , and t = 4 means highly signiﬁcant evidence against H0 . JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
39. 39. 5. Conclusion t = 1 means no evidence against H0 , t = 2 means only mild evidence against H0 , and t = 3 means signiﬁcant evidence against H0 , and t = 4 means highly signiﬁcant evidence against H0 . JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val
40. 40. References EDWARDS, W, LINDMAN, H, and SAVAGE, L.J (1963). Bayesian statistical inference for psychological research. Psycho- logical Review 70, 193-242 Jayanta Ghosha, Sumitra Purkayastha and Tapas Samanta Role of P-values and other Measures of Evidence in Bayesian Anal- ysis. Handbook of Statistics, Vol. 25 James O. Berger Statistical Decision Theory and Bayesian Analy- sis. Springer Series in Statistics JAMES O.BERGER and THOMAS SELLKE Testing a Point Null Hypothesisi: The Irreconcilability of P Val