Gender
Female
Reli
gious
ident
ity
Male
Abrahamic
16
29
Agnostic
9
20
Atheist
32
194
Other
18
40
Asd
Asd
Ad
H0: Pr(row i and column j)=Pr(row i)*Pr(column j)
H1: Anything goes as long as the
probabilities sum to one
Example of zero-hypothesis:
Pr(female and atheist)=
Pr(female)Pr(atheist)
Pr(data | H1)
Bayes factor: B= --------------------Pr(data | H0)
(Where the parameters in H0 and H1 have
been summed (integrated) over, using their
prior distribution.)
Bayes formula:
1
Pr(H0 | D)= --------------------------1 + Pr(H1)/Pr(H0) B
1
= ------ if Pr(H0)=Pr(H1)=50%
1+B
Me:
J
I
x
x
x..
∫
B x∗∗∣all margins H1 vs H0=1/
p ∏i=1 pi, . ∏i, j dpi , j
∏
j=1 ., j
x∗∗ pij ≥0, ∑ pij=1
x..
x..
where
=
and a '.' stand for sum over the index.
x∗∗ ∏i, j x i, j
 
., j
i,.
 
Gunel & Dickey (1974) – Bayes factors for independence in
contingency tables, Biometrika, Vol 61, No 3, pp 545-557
'
  
B x∗∗∣all margins  H1 vs H0=B IJ a∗ ∗ ∑ g  x∗∗ '∣x.. , a∗∗ /{B IJ  x∗∗a∗∗ x.. x.. }
x ∗. x.∗
where * is the set of possible indexes,
a∗∗ is the prior Dirichlet-parameters, with ai , j =1 being a flat prior,
K
∏k=1   y k 
B K  y∗ ≡
K
 ∑ k=1 y k 
 
and
g  x∗∗∣x.. , a= x.. B IJ  x∗ ∗a∗ ∗/B IJ a ∗∗
x∗∗
is the Dirichlet-multinomial distribution function
Frequentist test: calculate the maximal probability for
getting something as extreme as you did, given that H0
is true. H1 defines what extreme means.
Reject when this goes below a given significance level.
The lower the level, the more seldom do you want to
reject a correct zero-hypothesis. Usual; 5%, 1% or 0.1%.
If H0 is rejected with a significance level of 5%, it is
rejected with a confidence of 95%.
Frequentist test: calculate the maximal probability for
getting something as extreme as you did, given that H0
is true. H1 defines what extreme means.
Reject when this goes below a given significance level.
The lower the level, the more seldom do you want to
reject a correct zero-hypothesis. Usual; 5%, 1% or 0.1%.
If H0 is rejected with a significance level of 5%, it is
rejected with a confidence of 95%.
Bonferroni: If a significance level of p % is wanted for
a string of k tests, we put the individual significance
level to p/k %.
Two frequentist tests:
1) Fisher's exact method. Distribution under H0 of data
conditioned on the row and column sums, is hyper-geometric.
Use that to calculate the p-value, identifying extremes through
the alternative (dependency) hypothesis.
2) Pearson's Chi-square method. Use central limit theorem to
justify that the sum over cells of (observed-estimate)2/estimate
is approximately chi-square distributed with (I-1)(J-1)-1 degrees
of freedom. Calculate the probability to get a result higher than
you did, using that assumption. Only gives reliable results when
the expected count is larger than 5 (and preferably larger than
10, too).
Joining of groups in order to make the calculations
manageable:
Joining of groups in order to make the calculations
manageable:
Countries divided into USA and the rest
Joining of groups in order to make the calculations
manageable:
Countries divided into USA and the rest
Religious identity divided into: Abrahamic, agnostic, atheist
and 'other'. Same for religious upbringing.
Joining of groups in order to make the calculations
manageable:
Countries divided into USA and the rest
Religious identity divided into: Abrahamic, agnostic, atheist
and 'other'. Same for religious upbringing.
Belief in creationism, evolution, big bang etc. divided into
yes/no for each separate belief.
Joining of groups in order to make the calculations
manageable:
Countries divided into USA and the rest
Religious identity divided into: Abrahamic, agnostic, atheist
and 'other'. Same for religious upbringing.
Belief in creationism, evolution, big bang etc. divided into
yes/no for each separate belief.
Should also join groups in the feminism issue, but unsure
how to precede. (Alternatives: yes, no, male supporter of
equal rights, non-answer)
Gender
Female
Reli
gious
ident
ity
Male
Abrahamic
16
29
Agnostic
9
20
Atheist
32
194
Other
18
40
Asd
Asd
Ad
Gender
Female
Reli
gious
ident
ity
Male
Abrahamic
16 (9,4)
[2.14]
Agnostic
9 ( 6.07)
[1.18]
20 (22.9)
[-0.62]
Atheist
32 (47.3)
[-2.23]
194 (178.5)
[1.14]
Other
18 (12.1)
[1.68]
40 (45.8)
[-0.86]
Asd
Asd
Ad
29 (35.5)
[-1.10]
Chi-square, pval=0.05%
Fisher, pval=0.05%
Bayesian: B=26.35
Pr(H0|D)=3.65%
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups p-value
p-value factor (Pr(H0)=0.5)
Feminism
4
(18%)
Country
2
64.2%
64.6%
Age group
5
(58%)
-------
Sexual
orientation
4
(0.096%)
0.2%
*
Switch religion?
3
6*10-6
***
4.3*10-6 ≈2330 ≈0.004% Strong indications of
***
dependency. More abrahamic
followers answering 'never'
than expected. Agnostics
switching more than exp.
Religious
upbringing
4
(5.3%)
20% ≈0.0083 ≈99.2%
Conclusion
0.012
98.8%
≈4*10-5 ≈100%
≈2
≈33%
3.7% 0.018
98.2%
*
unstable
No dependency detected
No dependency detected.
Bayesian evidence for
independence.
No dependency detected.
Too many groups.
Indications of dependency
but in the end,
inconclusive.
Inconclusive. Bayesian test
suggests evidence for
independence, but it's
very unrealistic to say
Pr(H0)=0.5 before data, here.
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups
p-value p-value factor (Pr(H0)=0.5)
0.13%
10.72
8.5%
Conclusion
Parents know?
2
(0.90%)
Dependency probably
detected. Less abrahamic
answering 'no' than exp.
Speak with
Parents?
5
(52%)
--------- ≈5.6*10-5 ≈100%
Friends gender
5
(35%)
-------- ≈0.00038 ≈99.96% Too many groups!
But no dependency seen.
Friends religion
5
(34%)
---------- ≈9*10-5 99.999% No indication of
dependency. Too many
groups?
Supernatural
experiences?
2
3.6*10-10
***
Tell about
supernatural
experience?
2
(12%)
7.2%
1.03
49%
Own videos?
3
15.4%
13.4%
0.016
96.6%
Too many groups!
But no dependency seen.
2.8*10-10 39933527 2.5*10-8 Clear case of dependency
***
More 'other' and less atheists
answering 'yes' than exp.
No clear evidence of
dependency. People
answering 'no' to the
previous removed.
Evidence for independence.
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups
p-value p-value factor (Pr(H0)=0.5)
Videos with
religious topic?
2.1%
*
3
(2.2%)
Discuss religion? 5
(34%)
-------- ≈0.0055 ≈99.4% Too many groups! No
dependency found.
Judge concerns? 2
0.05%
**
0.06%
**
27.9
Fear judgement?
4
(3.6%)
--------
≈0.016
≈98.4% Too many groups, but
indicates independence
Running (against
clock or people)
2
76.7%
75.7%
0.010
99.0%
Indicates independence
Contest attitude
2
(79.5%)
71.0%
0.138
98.6%
Indicates independence
≈0.64
61%
Conclusion
3.46%
Faint indications of
dependence but nowhere
near conclusive.
Probably dependency. More
abrahamic answering 'yes'
than expected.
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups p-value p-value factor (Pr(H0)=0.5)
Conclusion
Creationism
(yes/no)
2
(<2.2*10-16) <2.2*10-16 4.6*1025 2.17*10-27 Clear case of dependence.
***
More abrahamic answering
yes than exp.
Evolution
(yes/no)
2
(<2.2*10-16) 8.5*10-15 2.2*1012 4.5*10-13 Clear case of dependence.
***
More abrahamic answering
no than exp.
Big bang
2
(2.6*10-13)
2.0*10-12 6.0*109 1.7*10-10 Clear case of dependence.
***
More abrahamic answering
no than exp.
Abiogenesis
2
4.5*10-15
***
<2.2*10-16 2.8*1014 3.6*10-15 Clear case of dependence.
***
Less abrahamic answering
yes than exp.
Life after death
2
<2.2*10-16 <2.2*10-16 2.3*1035 4.3*10-36 Clear case of dependence.
***
***
More abrahamic answering
yes than exp.
(Curiously enough: 8 'no's)
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups
p-value p-value factor (Pr(H0)=0.5)
Reincarnation
2
(<2.2*10-16)<2.2*10-16 1.4*1014 7.1*10-15
***
Age of Earth
5
(<2.2*10-16) 5.3*10-14 high
***
but
Low?
unstable
Attitude towards
stemcell
research
3
Education
9
Education of
parents
(<2.2*10-16) 1.4*10-14 ≈1.46*1012 ≈6.84*1013
***
(8.7%)
--------
---------
------------
Conclusion
Clear case of dependence.
More 'other' answering
yes than exp.
Clear case of dependence.
More abrahamic answering
'25000-10mill' and '500010000' than expected.
Clear case of dependence.
More abrahamic opposed
than expected.
Way too many groups, but
chi-square does not indicate
dependency.
Same results
Religious identity #gro Chi-square Fisher's Bayes- Pr(H0|D)
versus
ups
p-value p-value factor (Pr(H0)=0.5)
Stance towards
mastrubation
5
Kids?
4
(<2.2*10-16) ---------- High but Low?
unstable
(10%)
--------- ≈0.0032 ≈99.7%
Conclusion
Way too many groups.
Approximate method
suggests dependency. Exact
methods used only on
atheists vs abrahamic
followers clearly indicates
dependency.
Too many groups. No
suggestion of dependency.
Religious upbringing
Abrahamic Agnostic Atheist
Reli
gious
ident
ity
Other
Abrahamic
38 (33.1)
[0.85]
1 (2.7)
[-1.06]
0 (2.1)
[-1.46]
6 (7.0)
[-0.39]
Agnostic
28 (21.3)
[1.45]
1 (1.8)
[-0.58]
0 (1.4)
[-1.17]
0 (4.5)
[-2.13]
Asd
Asd
Ad
Atheist
158 (166) 17(13.8) 12(10.7) 39 (35.4)
[-0.62]
[0.83] [0.39]
[0.61]
Other
39 (42.6)
[-0.55]
3 (3.6) 5 (2.8)
[-0.30] [1.35]
11 (9.1)
[0.64]
Chi-square, pval=5.2%
Fisher, pval=3.7%
Bayesian: B≈0.018
Pr(H0|D)≈81.6%
Evolution
Reli
gious
ident
ity
No
Yes
Abrahamic
20 (4,0)
[7.96]
25 (41.0)
[-2.50]
Agnostic
1 ( 2.6)
[-0.99]
28 (26.4)
[0.31]
Atheist
3 (20.2)
[-3.83]
223 (205.8)
[1.20]
Other
8 (5.2)
[1.24]
50 (52.8)
[-0.38]
Asd
Asd
Ad
Chi-square, pval<2.2*10-16
Fisher, pval=8.5*10-15
Bayesian: B=2.2*1012
Pr(H0|D)=4.55*10-13
Feminism
Male supporter No
Reli
gious
ident
ity
Yes
Abrahamic
25 (27.4)
[-0.45]
6 (2.4) 11 (12.2)
[2.32] [-0.35]
Agnostic
16 (18.2)
[-0.52]
2 (1.6)
[0.31]
Atheist
153 (146)
[0.59]
9 (12.8) 62 (65.3)
[-1.06] [-0.41]
Other
34 (36.5)
[-0.41]
3 (3.2)
[-0.11]
Asd
Asd
Ad
Chi-square, pval=18.7%
Bayesian: B≈0.083
10 (8.2)
[0.64]
19 (16.3)
[0.66]
Fisher, pval=20.0%
Pr(H0|D)≈99.2%
Country
not USA
Reli
gious
ident
ity
USA
Abrahamic
15 (18.7)
[-0.85]
30 (26.3)
[0.72]
Agnostic
12 (12.0)
[-0.01]
17 (17.0)
[0.01]
Atheist
98 (93.4)
[0.47]
127 (131.6)
[-0.40]
Other
22 (22.8)
[-0.18]
33 (32.2)
[-0.15]
Asd
Asd
Ad
Chi-square, pval=64.2%
Bayesian: B=0.012
Fisher, pval=64.7%
Pr(H0|D)=98.8%
Switch religion?
Multiple times
Reli
gious
ident
ity
Abrahamic
2 (10.1)
[-2.54]
Never
Once
31 (17.5) 12 (17.5)
[3.24] [-1.31]
Agnostic
13 (6.5)
[2.56]
4 (11.3)
[-2.16]
12 (11.3)
[0.22]
Atheist
47 (50.5)
[-0.49]
88(87.7) 91 (87.7)
[0.02] [0.34]
Other
18 (13.0)
[1.40]
16 (22.5) 24 (22.5)
[-1.37]
[0.31]
Chi-square, pval=0.00061%
Fisher, pval=0.00043%
Bayesian: B≈2330
Pr(H0|D)≈0.043%
Concerns for judgement
No
Reli
gious
ident
ity
Yes
Abrahamic
17 (27.1)
[-1.95]
28 (17.8)
[2.40]
Agnostic
12 (17.5)
[-1.31]
17 (11.5)
[1.62]
Atheist
147 (135)
[1.02]
77 (88.8)
[-1.26]
Other
37 (33.2)
[0.66]
18 (21.8)
[-0.82]
Asd
Asd
Ad
Chi-square, pval=0.052%
Fisher, pval=0.059%
Bayesian: B=27.92
Pr(H0|D)=3.46%
Concerns for judgement
No
Stan
ce
to
wards
mast
urb
ation
Fun for others
not for me
Yes
6 (4.9)
[0.52]
2 (3.14)
[-0.65]
40 (37.6)
[0.39]
22 (24.4)
[-0.48]
Bad behaviour
6 (11.5)
[-1.63]
13 (7.5)
[2.03]
Fun & healthy
151 (150.5)
[0.04]
97 (97.5)
[-0.05]
It's immature
4 (2.43)
[1.01]
0 (1.57)
[-1.25]
Indifferent
Chi-square, pval=3.4%
Fisher, pval=3.7%
Bayesian: B=0.82
Pr(H0|D)=55%