Transcript of "Testing point null hypothesis, a discussion by Amira Mziou"
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Testing a point null hypothesis: IPE Testing a point null hypothesis: The Irreconcilability of P-values and Evidence Authors: James O.Berger & Thomas Sellke Source: Journal of the American Statistical Association 1987 Presented by: MZIOU Amira Reading Seminar in Statistical Classics: C.P Robert Univ Paris Dauphine February
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 2
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 2
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 2
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Testing a point null hypothesis: IPEOutline 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion
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Testing a point null hypothesis: IPE Introduction 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 3
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test 4
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test Let X be a characteristic of the population whose distribution depends on an unknown parameter θ. We want to make a decision about the value of this parameter θ from a sample. 4
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test Let X be a characteristic of the population whose distribution depends on an unknown parameter θ. We want to make a decision about the value of this parameter θ from a sample. Question :How to decide on a population from the examination of a sample from this population ? 4
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test Let X be a characteristic of the population whose distribution depends on an unknown parameter θ. We want to make a decision about the value of this parameter θ from a sample. Question :How to decide on a population from the examination of a sample from this population ? Deﬁnition A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study. 4
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 3 Specify signiﬁcance level 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 3 Specify signiﬁcance level 4 State decision rule 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 3 Specify signiﬁcance level 4 State decision rule 5 Collect data and perform calculations 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 3 Specify signiﬁcance level 4 State decision rule 5 Collect data and perform calculations 6 Make statistical decision : reject or accept H0 5
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Testing a point null hypothesis: IPE Introduction Statistical Hypothesis TestStatistical Hypothesis Test There are six steps to do a Statistical Hypothesis Test : 1 State hypothesis 2 Identify test statistic 3 Specify signiﬁcance level 4 State decision rule 5 Collect data and perform calculations 6 Make statistical decision : reject or accept H0 5
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in Statistics 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 6
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in StatisticsP-Value and Evidence in statistics 7
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in StatisticsP-Value and Evidence in statistics P-value Statistical hypothesis tests answer the question : Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed ? 7
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in StatisticsP-Value and Evidence in statistics P-value Statistical hypothesis tests answer the question : Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed ? That probability is known as the P-value 7
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in StatisticsP-Value and Evidence in statistics P-value Statistical hypothesis tests answer the question : Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed ? That probability is known as the P-value Statistical evidence A set or collection of numbers that prove a theory or story to be true.
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Testing a point null hypothesis: IPE Introduction P-value and Evidence in StatisticsExample Let x = (x1 , x2,... , xn ) a sample of X = (X1 , X2,... , Xn ) where the Xi are iid N (θ, σ2 ) It’s desired to test the null hypothesis : H0 : θ = θ0 VS H1 : θ = θ0 A classical test is based on consideration of test statistic T (X ) where large values of T (X ) cause doubt on H0 . The P-value of observed data x is then p = Prθ=θ0 (T (X ) ≥ t = T (x )) (1) √ T (X ) = n|X − θ0 |/σ (2) 8
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Testing a point null hypothesis: IPE Problematic 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 9
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? Irreconcilability By a Bayesian analysis with a ﬁxed prior and for values of t chosen to yield a given ﬁxed p, the posterior probability of H0 is greater than p. 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? Irreconcilability By a Bayesian analysis with a ﬁxed prior and for values of t chosen to yield a given ﬁxed p, the posterior probability of H0 is greater than p. =⇒ P-values can be highly misleading measures of the evidence provided by the data against the null hypothesis 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? Irreconcilability By a Bayesian analysis with a ﬁxed prior and for values of t chosen to yield a given ﬁxed p, the posterior probability of H0 is greater than p. =⇒ P-values can be highly misleading measures of the evidence provided by the data against the null hypothesis Relationship between the P-value and conditional and Bayesian measures of evidence against the null hypothesis. 10
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Testing a point null hypothesis: IPE ProblematicProblematic Most statisticians prefer use of P-values feeling it to be important to indicate how strong the evidense against H0 Given the observed value, is it likely that H0 is true i.e how high is P (H0x ) : the posterior probability of H0 giving x ? Irreconcilability By a Bayesian analysis with a ﬁxed prior and for values of t chosen to yield a given ﬁxed p, the posterior probability of H0 is greater than p. =⇒ P-values can be highly misleading measures of the evidence provided by the data against the null hypothesis Relationship between the P-value and conditional and Bayesian measures of evidence against the null hypothesis. 10
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDS 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 11
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ π1 :Prior probability of H1 (θ = θ0 ) : π1 = 1 − π0 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ π1 :Prior probability of H1 (θ = θ0 ) : π1 = 1 − π0 We suppose that the mass on H1 is spread out according to a density g (θ). 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ π1 :Prior probability of H1 (θ = θ0 ) : π1 = 1 − π0 We suppose that the mass on H1 is spread out according to a density g (θ). The marginal density of X is : m(x ) = f (x /θ0 )π0 + (1 − π0 )mg (x ) (3) where mg (x ) = f (x /θ)g (θ)d θ 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior probability X has density f (x θ) , θ ∈ Θ π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ π1 :Prior probability of H1 (θ = θ0 ) : π1 = 1 − π0 We suppose that the mass on H1 is spread out according to a density g (θ). The marginal density of X is : m(x ) = f (x /θ0 )π0 + (1 − π0 )mg (x ) (3) where mg (x ) = f (x /θ)g (θ)d θ Assuming that f (x /θ) > 0, the posterior probability of H0 is given by : π0 1 − π0 mg (x ) −1 Pr (H0 x ) = f (x θ0 ) × = [1 + × ] (4) m (x ) π0 f (x /θ0 ) 12
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior ODDS Ratio Deﬁntion The Odds Ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression. 13
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior ODDS Ratio Deﬁntion The Odds Ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression. The posterior ODDS ratio of H0 to H1 is : 13
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSPosterior ODDS Ratio Deﬁntion The Odds Ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression. The posterior ODDS ratio of H0 to H1 is : Pr (H0 /x ) π0 f (x /θ0 ) PosteriorODDSRatio = = × (5) 1 − Pr (H0 /x ) 1 − π0 mg ( x ) Prior ODDS Ratio Bayes Factor Bg (x ) 13
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Testing a point null hypothesis: IPE Posterior Probability, Bayes Factor and Posterior ODDSRemark f (x /θ0 ) • The Bayes factor Bg (x ) = does not involve the prior mg (x ) probabilities of the hypotheses. It’s the evidence reported by the data alone.It can be interpreted as the likelihood ratio where the likelihood of H1 is calculated with respect to g (θ)
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x) 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 15
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Astronomer’s Example 16
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Astronomer’s Example An astronomer how learned that many statistical users rejected null normal hypothesis at the 5% level when t = 1, 96 was observed. He decides to do an experiment to verify the validity of rejecting H0 when t = 1, 96 by testing normal hypothesis. He supposes that about half of the point nulls are false and half are true. When he concentrates attention on the subset in which t is between 1, 96 and 2 , he discovers that 22% of null hypotheses are true. 16
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Example(continued) 2 Again X ∼ N (θ, σ2 ), X ∼ N (θ, σ ) n
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Example(continued) 2 Again X ∼ N (θ, σ2 ), X ∼ N (θ, σ ) n Suppose that π0 is arbitrary and g is N (θ0 , σ2 ) , We have that mg (x ) ∼ N (θ0 , σ2 (1 + n−1 )) Thus f (x /θ0 ) 1 Bg (x ) = mg (x ) = (1 + n) 2 exp{− 1 t 2 /(1 + n−1 )} 2 and 1 P (H0 x ) = [1 + 1−π0 ]−1 =[1 + 1−π0 (1 + n)− 2 × exp[ 1 t 2 /(1 + n−1 )]]−1 π B π 2 0 g 0 17
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Jeffrey’s Bayesian Analysis 1 Let π0 = 2 and g is N (θ0 , σ2 ) 1 =⇒ Pr (H0 x ) = (1 + (1 + n)− 2 × exp[ 1 t 2 (1 + n−1 )])−1 2 18
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Jeffrey’s Bayesian Analysis 1 Let π0 = 2 and g is N (θ0 , σ2 ) 1 =⇒ Pr (H0 x ) = (1 + (1 + n)− 2 × exp[ 1 t 2 (1 + n−1 )])−1 2 F IGURE : Pr(H0 x ) for Jeffreys Type prior If n = 50 and t = 1, 96 : classically we reject H0 at signiﬁcance level p = 0, 05 , but Pr(H0 ) = 0, 52 which indicates the evidence favors H0 . 18
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x)Jeffrey’s Bayesian Analysis 1 Let π0 = 2 and g is N (θ0 , σ2 ) 1 =⇒ Pr (H0 x ) = (1 + (1 + n)− 2 × exp[ 1 t 2 (1 + n−1 )])−1 2 F IGURE : Pr(H0 x ) for Jeffreys Type prior If n = 50 and t = 1, 96 : classically we reject H0 at signiﬁcance level p = 0, 05 , but Pr(H0 ) = 0, 52 which indicates the evidence favors H0 . =⇒ the conﬂict between p and Pr (H0 x ). 18
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x) Bad choice of priors 19
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Testing a point null hypothesis: IPE Irreconcilability and Conﬂit between P-value and Pr(H0 x) Bad choice of priors To prevent having a non-Bayesian reality, working with lowers bounds on Pr (H0 x ) (translate into lower bounds on Bg ), and raisonable densities’ classes can be considered to be objective. 19
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Testing a point null hypothesis: IPE Various lower bounds on Posterior Probabilities 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 20
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesLower bounds 21
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesLower bounds This section examine some lower bounds on Pr (H0 x ) when g (θ) the distribution of θ given that H1 is true is alowed to vary within some class of distributions G. 1 GA : all distributions 2 GS : all distributions symmetric about θ0 3 GUS : all unimodal distributions symmetric about θ0 4 GNOR :all N (θ0 , τ2 ) Letting Pr (H0 x , G)= infg ∈G Pr (H0 x ) B (x , G) = infg ∈G Bg (x ) =⇒ B (x , G) = f (x θ0 )/supg ∈G mg (x ) Pr (H0 x , G) = [1 + 1−π0 × B (x ,G) ]−1 π 1 0 21
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesGA : all distribution 1 F IGURE : Comparison of P-values and Pr (H0 x , GA ) when π0 = 2 22
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesGS : all distributions symmetric about θ0 1 F IGURE : Comparison of P-values and Pr (H0 x , GS ) when π0 = 2 23
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesGUS : all unimodal distributions symmetric aboutθ0 1 F IGURE : Comparison of P-values and Pr (H0 x , GUS ) when π0 = 2 24
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesGNOR :all N (θ0 , τ2 ) 1 F IGURE : Comparison of P-values and Pr (H0 x , GNOR ) when π0 = 2 25
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Testing a point null hypothesis: IPE Various lower bounds on Posterior ProbabilitiesComparaison of the lower bounds B(x,G) for thefour G’s considered F IGURE : Values of B (x , G) in the normal example for different choices of G. 26
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Testing a point null hypothesis: IPE Solution 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 27
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Testing a point null hypothesis: IPE SolutionSolution Again B (x , G) can be considered to be a reasonable lower bound on the comparative likelihood measure of evidence against H0 (under an unimodal symmetric prior on H1 ). 28
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Testing a point null hypothesis: IPE SolutionSolution Again B (x , G) can be considered to be a reasonable lower bound on the comparative likelihood measure of evidence against H0 (under an unimodal symmetric prior on H1 ). Replacing t by (t − 1)+ , the lower bound on Bayes factor is quite similar to the p-value obtained by (t − 1)+ . F IGURE : Comparison of P-values and B (x , GUS ) 28
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Testing a point null hypothesis: IPE Conclusion 1 Introduction Statistical Hypothesis Test P-value and Evidence in Statistics 2 Problematic 3 Posterior Probability, Bayes Factor and Posterior ODDS 4 Irreconcilability and Conﬂit between P-value and Pr(H0 x) 5 Various lower bounds on Posterior Probabilities 6 Solution 7 Conclusion 29
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Testing a point null hypothesis: IPE Conclusion P-values can be dangerous to quantify evidence against a point null hypothesis because they lead to talk about posterior distribution without going through Bayesian Analysis. What should a statistician desiring a point null hypothesis do ? Lower bound of Pr (H0 x ) can be argued to be useful to test evidence : if the lower bound is large we know not to reject H0 but if the lower bound is small we still do not know if H0 can be rejected. For normal distribution, replacing t by (t − 1)+ , p-value= 0, 05 can conﬁrms the rejection of H0 30
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Testing a point null hypothesis: IPE ConclusionREFERENCES JAMES O.BERGER, THOMAS SELKE, The Irreconcilability of P-value and Evidence, JOURNAL OF AMERICAN STATISTICAL ASSOCIATION (March 1987). MATAN GAVISH, A Note on Berger and Selke LINDLEY, D.V, The Use of Prior Probability Distributions in Statistical Inference and Decision, UNIVERSITY OF CALIFORNIA (1961) LINDLEY, D.V, A Statistical Paradox, BIOMETRIKA(1957) CHRISTIAN P.ROBERT,Faut-il Accepter ou rejeter les p-values ? , PRIX DU STATISTICIEN DE LANGUE FRANCAISE WIKIPEDIA 31
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Testing a point null hypothesis: IPE Conclusion THANK YOU FOR YOUR ATTENTION 32
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