1. Significance testing and confidence intervals
Ágnes Hajdu
EPIET Introductory course

2. The idea of statistical inference
Generalisation to the population
Conclusions based
on the sample
Population
Hypotheses
Sample

3. 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 (exposure) on a dependent variable (outcome).

4. The aim of a statistical test
To reach a scientific decision (“yes” or “no”) on a difference (or effect), on a probabilistic basis, on observed data.

5. Why significance testing?
Botulism outbreak in Italy:
“The risk of illness was higher among diners who ate home preserved green olives (RR=3.6).”
Is the association due to chance?

6. The two hypothesis! There is NO difference between the two groups
(=no effect)
Null Hypothesis (H0)
(e.g.: RR=1)
There is a difference between the two groups
(=there is an effect)
Alternative Hypothesis (H1)
(eg: RR=3.6)
When you perform a test of statistical significance you usually reject or do not reject the Null Hypothesis (H0)

7. Botulism outbreak in Italy
Null hypothesis (H0): “There is no association between consumption of green olives and Botulism.”
Alternative hypothesis (H1): “There is an association between consumption of green olives and Botulism.”

8. Hypothesis, testing and null hypothesis
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.

9. How to decide when to reject the null hypothesis?
H0 rejected using reported 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

10. p values – practicalities
Small p values = low degree of compatibility between H0 and the observed data:
you reject H0, the test is significant
Large p values = high degree of compatibility between H0 and the observed data:
you don’t reject H0, the test is not significant
We can never reduce to zero the probability
that our result was not observed by chance alone

11. Levels of significance – practicalities
We need of a cut-off !
p value > 0.05 = H0 non rejected (non significant)
p value ≤ 0.05 = H0 rejected (significant)
BUT:
Give always the exact p-value rather than „significant“
vs. „non-significant“.

12. 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=  Agreed convention
Not an absolute truth
”Surely, God loves the 0.06 nearly as much
as the 0.05” (Rosnow and Rosenthal, 1991)

13. 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

14. Types of errors Truth Decision based on the p value
No diff
Diff
Decision based on the p value
No diff
Diff
H0 is “true” but rejected: Type I or  error
H0 is “false” but not rejected: Type II or  error

15. 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

16. Even more on errors
H1 is true
H0 is true b a
Test statistics T

17. Principles of significance testing
Formulate the H0
Test your sample data against H0
The p value tells you whether your data are
consistent with H0
i.e, whether your sample data are consistent with a chance finding (large p value), or whether there is reason to believe that there is a true difference (association) between the groups you tested
You can only reject H0, or fail to reject it!

18. 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).

19. Chi-square value
= sum of all cells: for each cell, subtract the expected number from the observed number, square the difference, and divide by the expected number

20. Botulism outbreak in Italy 2x2 table
Expected number
of ill and not ill
for each cell :
Ill
Non ill 9 43 4 79
Olives 52
x10% ill
x 90% non-ill 5 47 No
olives 83
x10% ill 8 75
x 90% non-ill 13
122
135
Expected proportion
of ill and not ill :
10 %
90 %

21. Chi-square value = 5.73 Botulism outbreak in Italy p = 0.016 Ill
Non ill
Olives No
olives

22. Botulism outbreak in Italy
“The relative risk (RR) of illness among diners who ate home preserved green olives was 3.6 (p=0.016).”
The p-value is smaller than the chosen significance level of a = 5%.
→ Null hypothesis can be rejected.
There is a probability (16/1000) that the observed association could have occured by chance, if there were no true association between
eating olives and illness.

23. Epidemiology and statistics

24. 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.

25. Why estimation? Botulism outbreak in Italy:
“The risk of illness was higher among diners who ate home preserved green olives (RR=3.6).”
How confident can we be in the result?
What is the precision of our point estimate?

26. The epidemiologist needs measurements rather than probabilities
2 is a test of association
OR, RR are measures of association on a continuous scale
infinite number of possible values
The best estimate = point estimate
Range of values allowing for random variability:
Confidence interval  precision of the point estimate

27. Confidence interval (CI)
Range of values, on the basis of the sample data, in which the population value (or true value) may lie.
Frequently used formulation:
„If the data collection and analysis could be replicated many times, the CI should include the true value of the measure 95% of the time .”

28. Indicates the amount of random error in the estimate
Confidence interval (CI)
e.g. CI for means
95% CI =
x – 1.96 SE up to x SE
a = 5%
1 - α
α/2
α/2 s
Lower limit upper limit
of 95% CI of 95% CI
Indicates the amount of random error in the estimate
Can be calculated for any „test statistic“, e.g.: means, proportions, ORs, RRs

29. CI terminology RR = 1.45 (0.99 – 2.1) Point estimate
Confidence interval
RR = 1.45 (0.99 – 2.1)
Lower confidence limit
Upper confidence limit

30. Width of confidence interval depends on …
The amount of variability in the data
The size of the sample
The arbitrary level of confidence you desire for your study (usually 90%, 95%, 99%)
A common way to use CI regarding OR/RR is :
If 1.0 is included in CI  non significant
If 1.0 is not included in CI  significant

31. Looking the CI
RR = 1 A B
Large RR
Study A, large sample, precise results, narrow CI – SIGNIFICANT
Study B, small size, large CI - NON SIGNIFICANT
Study A, effect close to NO EFFECT
Study B, no information about absence of large effect

32. More studies are better or worse?
Decision making based on results from a collection of studies is not facilitated when each study is classified as a YES or NO decision. 1 RR 
20 studies with different results...
Need to look at the point estimation and its CI
But also consider its clinical or biological significance

33. Botulism outbreak in Italy
How confident can we be in the result?
Relative risk = 3.6 (point estimate)
95% CI for the relative risk:
(1.17 ; 11.07)
The probability that the CI from 1.17 to 11.07
includes the true relative risk is 95%.

34. Botulism outbreak in Italy
“The risk of illness was higher among diners who ate home preserved green olives (RR=3.6, 95% CI 1.17 to 11.07).”

35. The p-value (or CI) function
A graph showing the p value for all possible values of the estimate (e.g. OR or RR).
Quantitative overview of the statistical relation between exposure and disease for the set of data.
All confidence intervals can be read from the curve.
The function can be constructed from the confidence limits in Episheet.

36. Example: Chlordiazopoxide use and congenital heart disease
C use
No C use
Cases 4
386
Controls
1250
OR = (4 x 1250) / (4 x 386) = 3.2
p=0.08 ; 95% CI=0.81–13
From Rothman K

37. 3.2
p=0.08
Odds ratio

38. Example: Chlordiazopoxide use and congenital heart disease – large study
C use
No C use
Cases
1090
14 910
Controls
1000
15 000
OR = (1090 x 15000) / (1000 x 14910) = 1.1
p=0.04 ; 95% CI=
From Rothman K

39. Precision and strength of association

40. Confidence interval provides more information than p value
Magnitude of the effect (strength of association)
Direction of the effect (RR > or < 1)
Precision of the point estimate of the effect (variability)
p value can not provide them !

41. What we have to evaluate the study
 A test of association. It depends on sample size.
p value Probability that equal (or more extreme) results can be observed by chance alone
OR, RR Direction & strength of association if > 1 risk factor if < 1 protective factor (independently from sample size)
CI Magnitude and precision of effect
Significance testing evaluates only the role of chance as alternative explanation of observed difference or effect.
Confidence intervals are more informative than p values.

42. Comments on p-values and CIs
Presence of significance does not prove clinical or biological relevance of an effect.
A lack of significance is not necessarily a lack of an effect: “Absence of evidence is not evidence of absence”.

43. Comments on p values and CIs
A huge effect in a small sample or a small effect in a large sample can result in identical p values.
A statistical test will always give a significant result if the sample is big enough.
p values and CIs do not provide any information on the possibility that the observed association is due to bias or confounding.

44. 2 and Relative Risk
Cases Non cases Total 2 = 1.3
E p = 0.13
NE RR = 1.8
Total % CI [ ]
Cases Non cases Total 2 = 12
E p =
NE RR = 1.8
Total % CI [ ]
« Too large a difference and you are doomed to statistical significance »

45. Common source outbreak suspected
Exposure cases non cases AR%
Yes %
No %
Total
2 = 9.1
p = 0.002
RR = 2.1
95%CI =
23%
REMEMBER: These values do not provide any information on the possibility that the observed association is due to a bias or confounding.

46. Recommendations
Always look at the raw data (2x2-table). How many cases can be explained by the exposure?
Interpret with caution associations that achieve statistical significance.
Double caution if this statistical significance is not expected.
Use confidence intervals to describe your results.
Report p values precisely.

47. 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

48. Previous lecturers Alain Moren Preben Aavitsland Paolo D’Ancona
Doris Radun
Lisa King
Manuel Dehnert
Previous lecturers