Transcript Testing hypotheses with tests of significance

Significance testing
Ioannis Karagiannis
(based on previous EPIET material)
18th EPIET/EUPHEM Introductory course
28.09.2012
The idea of statistical inference
Generalisation to the population
Conclusions based
on the sample
Population
Hypotheses
Sample
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Inferential statistics
• Uses patterns in the sample data to draw
inferences about the population represented,
accounting for randomness
• Two basic approaches:
– Hypothesis testing
– Estimation
• Common goal: conclude on the effect of an
independent variable on a dependent variable
3
The aim of a statistical test
To reach a deterministic decision (“yes” or “no”)
about observed data on a probabilistic basis.
4
Why significance testing?
Norovirus outbreak on a Greek island:
“The risk of illness was higher among people
who ate raw seafood (RR=21.5).”
Is the association due to chance?
5
The two hypotheses
There is NO difference between Null Hypothesis (H0)
the two groups
(=no effect)
(RR=1)
There is a difference between
the two groups
(=there is an effect)
Alternative Hypothesis
(H1)
(e.g.: RR=21.5)
When you perform a test of statistical significance,
you reject or do not reject the Null Hypothesis (H0)
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Norovirus on a Greek island
• Null hypothesis (H0): “There is no association
between consumption of raw seafood and
illness.”
• Alternative hypothesis (H1): “There is an
association between consumption of raw
seafood and illness.”
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Hypothesis testing
• Tests of statistical significance
• Data not consistent with H0 :
– H0 can be rejected in favour of some alternative
hypothesis H1 (the objective of our study).
• Data are consistent with the H0 :
– H0 cannot be rejected
You cannot say that the H0 is true.
You can only decide to reject it or not reject it.
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p value
p value = probability that our result (e.g. a
difference between proportions or a RR) or
more extreme values could be observed under
the null hypothesis
H0 rejected using reported p value
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p values – practicalities
Low p values = low degree of compatibility between H0
and the observed data:
association unlikely to be by chance
you reject H0, the test is significant
High p values = high degree of compatibility between H0
and the observed data:
association likely to be by chance
you don’t reject H0, the test is not
significant
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Levels of significance – practicalities
We need of a cut-off !
1%
5%
10%
p value > 0.05 = H0 not rejected (non significant)
p value ≤ 0.05 = H0 rejected (significant)
BUT:
Give always the exact p-value rather than „significant“
vs. „non-significant“.
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Examples from the literature
• ”The limit for statistical significance was set at p=0.05.”
• ”There was a strong relationship (p<0.001).”
• ”…, but it did not reach statistical significance (ns).”
• „ The relationship was statistically significant (p=0.0361)”
p=0.05 
Agreed convention
Not an absolute truth
”Surely, God loves the 0.06 nearly as much
as the 0.05” (Rosnow and Rosenthal, 1991)
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p = 0.05 and its errors
• Level of significance, usually p = 0.05
• p value used for decision making
But still 2 possible errors:
H0 should not be rejected, but it was rejected :
Type I or alpha error
H0 should be rejected, but it was not rejected :
Type II or beta error
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Types of errors
Truth
No diff
H0
Decision
based
on the
p value
H0 not rejected
No diff
H0 rejected (H1)
Diff
to be not rejected
Right decision
1-

Type I error
Diff
H0 to be rejected (H1)

Type II error
Right decision
1-
• H0 is “true” but rejected: Type I or  error
• H0 is “false” but not rejected: Type II or  error
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More on errors
• Probability of Type I error:
– Value of α is determined in advance of the test
– The significance level is the level of α error that we
would accept (usually 0.05)
• Probability of Type II error:
– Value of β depends on the size of effect (e.g. RR, OR)
and sample size
– 1- β: Statistical power of a study to detect an effect on
a specified size (e.g. 0.80)
– Fix β in advance: choose an appropriate sample size
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Quantifying the association
•
•
•
•
•
Test of association of exposure and outcome
E.g. chi2 test or Fisher’s exact test
Comparison of proportions
Chi2 value quantifies the association
The larger the chi2 value, the smaller the
p value
– the more the observed data deviate from the
assumption of independence (no effect).
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Chi-square value
 
2

(observed
num.  expected
expected
num.)
2
num.
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Norovirus on a Greek island
2x2 table
Ill
Raw seafood
Expected number
of ill and not ill
for each cell :
Non ill
29
38
9
31
6
x19% ill
x 81% non-ill
No raw seafood
5
Expected proportion
of ill and not ill :
27
136
34
145
19 %
81%
114
141
x 19% ill
x 81% non-ill
179
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Chi-square calculation
χ2= 125
Ill
Raw seafood
Non ill
p < 0.001
(29-6)2/6
(9-31)2/31
38
(5-27)2/27
(136-114)2/
114
141
34
145
179
No raw seafood
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Norovirus on a Greek island
“The attack rate of illness among consumers of
raw seafood was 21.5 times higher than
among non consumers of these food items
(p<0.001).”
The p value is smaller than the chosen
significance level of α = 5%.
→ The null hypothesis is rejected.
There is a < 0.001 probability (<1/1000) that the observed association could have
occured by chance, if there were no true association between
eating imported raw seafood and illness.
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C2012 vs facilitators
The ultimate (eye) test.
H0: the proportion of facilitators wearing glasses
during the Tuesday morning sessions was equal
to the proportion of fellows wearing glasses.
H1: the above proportions were different.
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C2012 vs facilitators
Glasses
Fellow
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Expected proportion
of ill and not ill :
6
38
27
25
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Facilitator
Expected number
of ill and not ill
for each cell :
No glasses
4.6
8
17
35
33%
67%
9.4
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x33% +ve
x67% -ve
x33% +ve
x67% -ve
52
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Chi-square calculation
χ2= 1.11
Fellow
Facilitator
Glasses
No glasses
(11-13)2/13
(27-25)2/25
(6-4.6)2/4.6
(8-9.4)2/9.4
p = 0.343
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t-test
• Used to compare means of a continuous
variable in two different groups
• Assumes normal distribution
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t-test
• H0: fellows with glasses do not tend to sit
further in the back of the room compared to
fellows without glasses
• H1: fellows with glasses tend to sit further in
the back of the room compared to fellows
without glasses
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t-test
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Epidemiology and statistics
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Criticism on significance testing
“Epidemiological application need more than a
decision as to whether chance alone could have
produced association.”
(Rothman et al. 2008)
Estimation of an effect measure (e.g. RR, OR)
rather than significance testing.
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Suggested reading
• KJ Rothman, S Greenland, TL Lash, Modern Epidemiology,
Lippincott Williams & Wilkins, Philadelphia, PA, 2008
• SN Goodman, R Royall, Evidence and Scientific Research,
AJPH 78, 1568, 1988
• SN Goodman, Toward Evidence-Based Medical Statistics.
1: The P Value Fallacy, Ann Intern Med. 130, 995, 1999
• C Poole, Low P-Values or Narrow Confidence Intervals:
Which are more Durable? Epidemiology 12, 291, 2001
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Previous lecturers
•
•
•
•
Alain Moren
Paolo D’Ancona
Lisa King
Ágnes Hajdu
• Preben Aavitsland
• Doris Radun
• Manuel Dehnert
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