Statistik for MPH: september Odds, odds ratio, analyse af case control studier (Silva:

Transcription

1 Statistik for MPH: september Odds, odds ratio, analyse af case control studier (Silva: ) Per Kragh Andersen 1

2 Fra den 3. uges statistikundervisning: skulle jeg gerne 1. forstå, at en rate er et alternativt frekvensmål, som i højere grad end risiko-parameteren tager hensyn til varierende follow-up tider 2. kunne beregne raten og dens tilhørende sikkerhedsgrænser 3. kunne sammenligne to rater ved at beregne rate ratio (med sikkerhedsgrænser) og tilsvarende χ 2 -test 4. forstå, at der er en sammenhæng mellem rate og risiko 5. forstå formålet med og principperne i direkte og indirekte standardisering 2

3 Fra den 3. uges statistikundervisning: behøver jeg derimod ikke nødvendigvis 1. at have forstået, hvordan formlen for SD(ln RR) fremkommer 2. at have forstået, hvorfor sammenhængen mellem rate og risiko er, som den er 3. at have forstået, hvordan formlen for standard afvigelsen i Mantel-Haenszel testet er fremkommet 4. at have forstået begrænsningerne ved standardiseringsteknikkerne, og hvorfor stratificeret analyse og regressionsanalyse er bedre 3

4 The odds parameter Alternative measure than prevalence and risk for disease frequency: odds which expresses the ratio of disease risk (P ) and one minus disease risk (1 P ), i.e., odds = P 1 P. This implies the relation P = odds 1 + odds. 4

5 Population Classification of a population by risk factor status and disease development in an incidence study (i.e., A, B, C, D are large numbers) Risk factor Disease incidence Total at classification + risk + (present) A B A + B (absent) C D C + D Total A + C B + D T P + = A/(A + B), 1 P + = B/(A + B), Odds + = A/(A+B) B/(A+B) = A B P = C/(C + D), 1 P = D/(C + D), Odds = C/(C+D) D/(C+D) = C D 5

6 Odds ratio Corresponding measure of association Estimated by: Odds ratio = A/B C/D = AD BC OR = a d b c, where a, b, c, d are the values observed in the sample. 6

7 Example Died Survived Standard a = 65 b = New c = 40 d = OR std. vs. new = = 2.46 Note: OR = 2.46 > 1.60 = R > 1 (R=relative risk or risk ratio) This is no coincidence! OR is always further away from 1 than R. Some times, but not in this example: OR R When the disease is rare, i.e., when the disease risk is small then OR and R will be close. In such cases, OR is a good surrogate for R, but in the present example with higher disease risks this is not the case. 7

8 Example: The other way around Died Survived New a = 40 b = Standard c = 65 d = OR new vs. std. = Note: OR = 0.41 < 0.63 = R < = 0.41 = This is still no coincidence! OR is always further away from 1 than R. Note that a two-by-two table always gives rise to calculating two (reciprocal) ORs. 8

9 Why on earth be interested in odds and OR? horse racing etc. (DR s TTV p. 266 ff, e.g. OR is a good surrogate for R for rare diseases OR can be estimated from case-control studies OR is the focus of analysis in logistic regression 9

10 Confidence limits for odds ratio Simple, but usually too crude method: OR ± 1.96 SD(OR) for some suitable SD(OR). Better (but more complicated) method: 1. Calculate LOR = ln(or) 1 Calculate L 2 = LOR a b + 1 c + 1 d 1 L 1 = LOR 1.96 a + 1 b + 1 c + 1 d 3. The desired 95% confidence limits are: from exp(l 1 ) to exp(l 2 ) This is illustrated in the figure on the next slide where OR = 1.5 (or ln(or) = 0.405). 10

11 11

12 Example OR = ln(or) = SD(LOR) = = L 2 = = L 1 = = The desired 95% confidence limits are from exp(0.353) = 1.42 to exp(1.443) = 4.23 The confidence limits the other way around are from 1/4.23=0.236 to 1/1.42=

13 Significance tests The null hypothesis: is the same as H 0 : Risk ratio = 1. H 0 : Odds ratio = 1 Therefore, the chi-squared test derived for the risk ratio applies equally well for the odds ratio. In the example the value is 10.57, and P is between and 0.01 (in fact, P = ). 13

14 Exposed B A Population C Non-Exposed D The population consists of A + B exposed and C + D non-exposed individuals (i.e., A, B, C, D are large numbers). After some time, (say, 2 years) A out of the exposed and C out of the non-exposed have developed the disease. That is, Relative risk = A/(A + B) C/(C + D) Odds ratio = A/B C/D = A D B C 14

15 Exposed B A C b a c d Population: cohort sample Non-Exposed D Relative risk = A/(A + B) C/(C + D) Odds ratio = A/B C/D = A D B C Exposed are sampled at baseline with probability k 1 and non-exposed with probability k 2. 15

16 Cohort sample: Exposed: k 1 (A + B) = k 1 A + k 1 B a b Non-exposed: k 2 (C + D) = k 2 C + k 2 D Then: a a + b c c + d c d k 1 A k 1 A + k 1 B = A A + B k 2 C k 2 C + k 2 D = C C + D We can estimate relative risk and odds ratio, since a d b c k 1A k 2 D k 1 B k 2 C = A D B C 16

17 Exposed B A Population: case-control sample Non-Exposed D b d a C c Relative risk = A/(A + B) C/(C + D) Odds ratio = A/B C/D = A D B C Cases are sampled at end of follow-up with probability k 3 and controls with probability k 4. 17

18 Case-control sample: Diseased: k 3 (A + C) = k 3 A + k 3 C (cases) a c Non-diseased: k 4 (B + D) = k 4 B + k 4 D (controls) b d Then: a a + b k 3 A k 3 A + k 4 B, c c + d k 3 C k 3 C + k 4 D We cannot estimate relative risk but odds ratio, since a d b c k 3A k 4 D k 4 B k 3 C = A D B C 18

19 Conclusion From the data collected in the case-control study we may: estimate odds ratio by OR = a d b c and its confidence limits test for no association between exposure and outcome using the chi-square test provided that there is no bias, that is, when selecting cases and controls, no consideration of exposure must be taken ( no selection bias ) and cases and controls should report on exposure equally reliably ( no informatin bias ). 19

20 Example from Silva, p.205: Cervical cancer Cases Controls No schooling Some schooling OR no school vs. some = 95% confidence interval from 1.24 to M-H chi-square test 11.04, P < = Note that the risk calculations 119/187 or 317/636 make no sense since their results will depend on how many controls were selected. 20

21 Exposure odds ratio cases controls Exposed a b Non-exposed c d Odds for being exposed among cases = a/c Odds for being exposed among controls = b/d exposure odds ratio = a/c the disease OR. b/d = ad bc It is the latter that we are interested in., i.e. the exposure OR estimates 21

22 Matching Some times, controls are not sampled randomly but, rather, they are matched to cases based on some factors like: age, sex, family membership, neighbourhood etc. Then the odds ratio cannot be estimated simply as ad/bc. Two kinds of matching: Individual matching where, to each case, one or more private controls are matched. Here, in the analysis, cases should be compared to their matched control(s) within pairs (or matched sets). Frequency matching where the distribution of, e.g. sex and age, are made identical among cases and controls, but each control does not correspond to any particular case. 22

23 Exercise Calculate OR and its associated 95% confidence limits for the data from Abbott et al. and compare with the results for the relative risk. Stroke Smoker Yes No Total Yes No Total R = 171/ /4437 =

24 Solution OR = = 1.93 (> R = 1.89), ln(or) = LOR = SD(ln OR) = a + 1 b + 1 c + 1 d = L 1 = = L 2 = = exp(l 1 ) = 1.52, exp(l 2 ) = 2.46 (For R : ) In this example results for OR and R are close. 24