Significance testingand confidence intervals

Ágnes HajduEPIET Introductory course

3.10.2011

The idea of statistical inference

Sample

PopulationConclusions basedon the sample

Generalisation to the population

Hypotheses

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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 (exposure) on a dependent variable (outcome).

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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.

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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?

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The two hypothesis!

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)

There is NO difference between the two groups

(=no effect)

Null Hypothesis (H0)

(e.g.: RR=1)

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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.”

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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.

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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

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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

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Levels of significance – practicalities

We need of a cut-off !

0.01 0.05 0.10

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“.

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• ”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)”

Examples from the literature

p=0.05 Agreed conventionNot 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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• H0 is “true” but rejected: Type I or error• H0 is “false” but not rejected: Type II or error

Types of errors

H0 to be not rejected H0 to be rejected (H1)

H0 not rejected Right decision

1-

Type II error

H0 rejected (H1)

Type I error

Right decision

1-

Decision based on the p value

Truth

No diff

No diff

Diff

Diff

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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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H0 is true H1 is true

Test statistics T

1 1

ok

1 -

error2. kind

error 1. kind

ok

1 -

H0 Reality H1

H0

Decisionaccording to p value

H1

1- Power

1- Significance

error

Even more on errors

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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!

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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).18

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

num. expected

num.) expectednum. (observed 22

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Botulism outbreak in Italy2x2 table

9 43

4 79

Olives

Noolives

Ill Non ill

13 122

52

83

1352010 % 90 %

Expected proportion of ill and not ill :

x10% ill

x 90% non-ill

x10% ill

x 90% non-ill

Expected number of ill and not ill for each cell :

5

8 75

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5.01

5.01)-(9 2

46.99

46.99)-(43 2

7.99

7.99)-(4 2

75.01

75.01)-(79 2

Chi-square value

Botulism outbreak in Italy

Olives

Noolives

Ill Non ill

2 = 5.73p = 0.016

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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 0.016 probability (16/1000) that the observed association could have occured by chance, if there were no true association between

eating olives and illness.22

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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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?

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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

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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 .”

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Confidence interval (CI)

Indicates the amount of random error in the estimateCan be calculated for any „test statistic“, e.g.: means, proportions, ORs, RRs

e.g. CI for means95% CI = x – 1.96 SE up to x + 1.96 SE

1 - αα/2 α/2

Lower limit upper limitof 95% CI of 95% CI

= 5%

s

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CI terminology

RR = 1.45 (0.99 – 2.1)

Confidence intervalPoint estimate

Lower confidence limit

Upper confidence limit

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• 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%)

Width of confidence interval depends on …

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

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Study A, large sample, precise results, narrow CI – SIGNIFICANTStudy B, small size, large CI - NON SIGNIFICANT

Looking the CI

Study A, effect close to NO EFFECTStudy B, no information about absence of large effect

RR = 1

A

B

Large RR

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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.

1RR

20 studies with different results...

Need to look at the point estimation and its CI

But also consider its clinical or biological significance

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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%.

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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).”

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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.

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Example: Chlordiazopoxide use and congenital heart disease

C use No C use

Cases 4 386

Controls 4 1250

OR = (4 x 1250) / (4 x 386) = 3.2

p=0.08 ; 95% CI=0.81–13

From Rothman K

Odds ratio

3.2

p=0.08

0.81 - 1337

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=1.05-1.2From Rothman K

Precision and strength of association

Strength

Precision39

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 !

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2 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 associationif > 1 risk factor if < 1 protective factor(independently from sample size)

CI Magnitude and precision of effect

What we have to evaluate the study

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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”.

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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.

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Cases Non cases Total 2 = 1.3E 9 51 60 p = 0.13NE 5 55 60 RR = 1.8Total 14 106 120 95% CI [ 0.6 - 4.9 ]

Cases Non cases Total 2 = 12E 90 510 600 p = 0.0002NE 50 550 600 RR = 1.8Total 140 1060 1200 95% CI [ 1.3-2.5 ]

2 and Relative Risk

« Too large a difference and you are doomed to statistical significance » 44

Exposure cases non cases AR%Yes 15 20 42.8%No 50 200 20.0%

Total 65 220

Common source outbreak suspected

REMEMBER: These values do not provide any information on the possibility that the observed association is due to a bias or confounding.

2 = 9.1 p = 0.002RR = 2.195%CI = 1.4-3.4

23%

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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.

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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

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