Testing point null hypothesis, a discussion by Amira Mziou
1. 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
2. Testing a point null hypothesis: IPE
Outline
1 Introduction
Statistical Hypothesis Test
P-value and Evidence in Statistics
3. Testing a point null hypothesis: IPE
Outline
1 Introduction
Statistical Hypothesis Test
P-value and Evidence in Statistics
2 Problematic
4. Testing a point null hypothesis: IPE
Outline
1 Introduction
Statistical Hypothesis Test
P-value and Evidence in Statistics
2 Problematic
3 Posterior Probability, Bayes Factor and Posterior ODDS
5. Testing a point null hypothesis: IPE
Outline
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
6. Testing a point null hypothesis: IPE
Outline
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
7. Testing a point null hypothesis: IPE
Outline
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
8. Testing a point null hypothesis: IPE
Outline
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. 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
10. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical Hypothesis Test
4
11. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
12. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
13. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
14. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical Hypothesis Test
There are six steps to do a Statistical Hypothesis Test :
5
15. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical Hypothesis Test
There are six steps to do a Statistical Hypothesis Test :
1 State hypothesis
5
16. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical Hypothesis Test
There are six steps to do a Statistical Hypothesis Test :
1 State hypothesis
2 Identify test statistic
5
17. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical Hypothesis Test
There are six steps to do a Statistical Hypothesis Test :
1 State hypothesis
2 Identify test statistic
3 Specify significance level
5
18. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
19. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
20. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
21. Testing a point null hypothesis: IPE
Introduction
Statistical Hypothesis Test
Statistical 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
22. 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
23. Testing a point null hypothesis: IPE
Introduction
P-value and Evidence in Statistics
P-Value and Evidence in statistics
7
24. Testing a point null hypothesis: IPE
Introduction
P-value and Evidence in Statistics
P-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
25. Testing a point null hypothesis: IPE
Introduction
P-value and Evidence in Statistics
P-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
26. Testing a point null hypothesis: IPE
Introduction
P-value and Evidence in Statistics
P-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.
27. Testing a point null hypothesis: IPE
Introduction
P-value and Evidence in Statistics
Example
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
28. 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
29. Testing a point null hypothesis: IPE
Problematic
Problematic
Most statisticians prefer use of P-values feeling it to be important to
indicate how strong the evidense against H0
10
30. Testing a point null hypothesis: IPE
Problematic
Problematic
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
31. Testing a point null hypothesis: IPE
Problematic
Problematic
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
32. Testing a point null hypothesis: IPE
Problematic
Problematic
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
33. Testing a point null hypothesis: IPE
Problematic
Problematic
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
34. Testing a point null hypothesis: IPE
Problematic
Problematic
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
35. Testing a point null hypothesis: IPE
Problematic
Problematic
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
36. Testing a point null hypothesis: IPE
Problematic
Problematic
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
37. 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
38. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior probability
X has density f (x θ) , θ ∈ Θ
12
39. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior probability
X has density f (x θ) , θ ∈ Θ
π0 :Prior probability of H0 (θ = θ0 ) θ0 ∈ Θ
12
40. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
41. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
42. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
43. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
44. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
45. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
46. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Posterior 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
47. Testing a point null hypothesis: IPE
Posterior Probability, Bayes Factor and Posterior ODDS
Remark
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)
48. 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
49. Testing a point null hypothesis: IPE
Irreconcilability and Conflit between P-value and Pr(H0 x)
Astronomer’s Example
16
50. 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
51. 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
52. 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
53. 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
54. 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
55. 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
56. Testing a point null hypothesis: IPE
Irreconcilability and Conflit between P-value and Pr(H0 x)
Bad choice of priors
19
57. 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
58. 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
59. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
Lower bounds
21
60. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
Lower 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
61. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
GA : all distribution
1
F IGURE : Comparison of P-values and Pr (H0 x , GA ) when π0 = 2
22
62. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
GS : all distributions symmetric about θ0
1
F IGURE : Comparison of P-values and Pr (H0 x , GS ) when π0 = 2
23
63. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
GUS : all unimodal distributions symmetric about
θ0
1
F IGURE : Comparison of P-values and Pr (H0 x , GUS ) when π0 = 2
24
64. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
GNOR :all N (θ0 , τ2 )
1
F IGURE : Comparison of P-values and Pr (H0 x , GNOR ) when π0 = 2
25
65. Testing a point null hypothesis: IPE
Various lower bounds on Posterior Probabilities
Comparaison of the lower bounds B(x,G) for the
four G’s considered
F IGURE : Values of B (x , G) in the normal example for different choices
of G.
26
66. 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
67. Testing a point null hypothesis: IPE
Solution
Solution
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
68. Testing a point null hypothesis: IPE
Solution
Solution
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
69. 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
70. 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
71. Testing a point null hypothesis: IPE
Conclusion
REFERENCES
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
72. Testing a point null hypothesis: IPE
Conclusion
THANK YOU FOR YOUR ATTENTION
32