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Significance testing and confidence intervals Ágnes Hajdu EPIET Introductory course 3.10.2011

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

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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). 3

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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. 4

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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? 5

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The two hypothesis! There is a difference between the two groups (=there is an effect) Alternative Hypothesis (H 1 ) (eg: RR=3.6) When you perform a test of statistical significance you usually reject or do not reject the Null Hypothesis (H 0 ) There is NO difference between the two groups (=no effect) Null Hypothesis (H 0 ) (e.g.: RR=1) 6

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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. 7

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Hypothesis, testing and null hypothesis Tests of statistical significance Data not consistent with H 0 : – H 0 can be rejected in favour of some alternative hypothesis H 1 (the objective of our study). Data are consistent with the H 0 : – H 0 cannot be rejected You cannot say that the H 0 is true. You can only decide to reject it or not reject it. 8

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How to decide when to reject the null hypothesis? H 0 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 9

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p values – practicalities Small p values = low degree of compatibility between H 0 and the observed data: you reject H 0, the test is significant Large p values = high degree of compatibility between H 0 and the observed data: you dont reject H 0, the test is not significant We can never reduce to zero the probability that our result was not observed by chance alone 10

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Levels of significance – practicalities We need of a cut-off ! 0.01 0.05 0.10 p value > 0.05 = H 0 non rejected (non significant) p value 0.05 = H 0 rejected (significant) BUT: Give always the exact p-value rather than significant vs. non-significant. 11

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

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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: H 0 should not be rejected, but it was rejected : Type I or alpha error H 0 should be rejected, but it was not rejected : Type II or beta error 13

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H 0 is true but rejected: Type I or error H 0 is false but not rejected: Type II or error Types of errors Decision based on the p value Truth No diff Diff 14

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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 15

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H 0 is true H 1 is true Test statistics T Even more on errors 16

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Principles of significance testing Formulate the H 0 Test your sample data against H 0 The p value tells you whether your data are consistent with H 0 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 H 0, or fail to reject it! 17

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Quantifying the association Test of association of exposure and outcome E.g. Chi 2 test or Fishers exact test Comparison of proportions Chi 2 -value quantifies the association The larger the Chi 2 -value, the smaller the p value – the more the observed data deviate from the assumption of independence (no effect). 18

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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 19

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Botulism outbreak in Italy 2x2 table 943 479 Olives No olives IllNon ill 13122 52 83 135 20 10 %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 : 5 875 47

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

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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

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Epidemiology and statistics 23

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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. 24

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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? 25

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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 26

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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. 27

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

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

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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 30

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

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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. 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 32

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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%. 33

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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). 34

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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. 35

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Example: Chlordiazopoxide use and congenital heart disease C useNo C use Cases4386 Controls41250 OR = (4 x 1250) / (4 x 386) = 3.2 p=0.08 ; 95% CI=0.81–13 From Rothman K

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Odds ratio 3.2 p=0.08 0.81 - 13 37

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Example: Chlordiazopoxide use and congenital heart disease – large study C useNo C use Cases109014 910 Controls100015 000 OR = (1090 x 15000) / (1000 x 14910) = 1.1 p=0.04 ; 95% CI=1.05-1.2 From Rothman K

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Precision and strength of association Strength Precision 39

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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 ! 40

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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 association if > 1risk factor if < 1protective factor (independently from sample size) CI Magnitude and precision of effect What we have to evaluate the study 41

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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. 42

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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. 43

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Cases Non casesTotal 2 = 1.3 E 9 51 60p = 0.13 NE 5 55 60RR = 1.8 Total 1410612095% CI [ 0.6 - 4.9 ] Cases Non casesTotal 2 = 12 E 90 510 600p = 0.0002 NE 50 550 600RR = 1.8 Total 1401060120095% CI [ 1.3-2.5 ] 2 and Relative Risk « Too large a difference and you are doomed to statistical significance » 44

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Exposurecases non casesAR% Yes152042.8% No5020020.0% Total65220 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 = 0.002 RR= 2.1 95%CI = 1.4-3.4 23% 45

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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. 46

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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 47

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Previous lecturers Alain Moren Paolo DAncona Lisa King Preben Aavitsland Doris Radun Manuel Dehnert 48

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