16 CaseControl Odds Ratios Casecontrol studies get around

16: Case-Control Odds Ratios Case-control studies get around several limitations of cohort studies 2/25/2021 Case-Control Odds Ratios 1

Relative risks from cohort studies n Use incidences to assess risk n n Exposed group incidence 1 Non-exposed group incidence 2 RR = ratio of incidence relative measure of effect Hindrances in Cohort Studies n n n 2/25/2021 Long induction period between exposure & disease Study of rare diseases require large sample sizes When studying many people information is limited in scope & accuracy Case-Control Odds Ratios 2

n Case-control sample Study all cases (Proportion exposed reflects exposure proportion of cases in the population) n Select random sample of non-cases (Proportion exposed reflects exposure proportion of noncases in the population) n This design forfeits the ability to estimate incidence, but maintains ability to estimate relative risk via the odds ratio (Cornfield, 1951; Disease + - Total Exposed + a b n 1 Exposed - c d n 2 Total m 1 m 2 N Miettinen 1976) 2/25/2021 Case-Control Odds Ratios 3

Illustrative Example (Breslow & Day, 1980) n Dataset = bd 1. sav n n Exposure variable (alc 2) = Alcohol use dichotomized Disease variable (case) = Esophageal cancer Alcohol 80 g/day < 80 g/day Total 2/25/2021 Case 96 104 200 Control 109 666 775 Case-Control Odds Ratios Total 205 770 975 4

Interpretation of Odds Ratio n Odds ratios are relative risk estimates n Risk multiplier n n e. g. , odds ratio of 5. 64 suggests 5. 64× risk with exposure Percent increase or decrease in risk (in relative terms) = (odds ratio – 1) × 100% n n 2/25/2021 e. g. , odds ratio of 5. 64 Percent relative risk difference = (5. 64 – 1) × 100% = 464% Case-Control Odds Ratios 5

95% Confidence Interval n Calculation n n Illustrative example n n n Convert OR estimate to ln scale SEln. OR = sqrt(a-1 + b-1 + c-1 + d-1) 95% CI for ln. OR = (ln OR^) ± (1. 96)(SE) Take anti-logs of limits ln(OR^) = ln(5. 640) = 1. 730 SEln. OR = sqrt(96 -1 + 104 -1 + 109 -1 + 666 -1) = 0. 1752 95% CI for ln. OR= 1. 730 ± (1. 96)(0. 1752) = (1. 387, 2. 073) 95% CI for OR = e(1. 387, 2. 073) = (4. 00, 7. 95) Interpretation of confidence interval (discuss) 2/25/2021 Case-Control Odds Ratios 6

SPSS Output Odds ratio point estimate and confidence limits Ignore “For cohort” information when data derived by case-control sample 2/25/2021 Case-Control Odds Ratios 7

Testing H 0: OR = 1 with the CI n n 95% CI corresponds to a = 0. 05 If 95% CI for odds ratio excludes 1 odds ratio is significant n n n e. g. , (95% CI: 4. 00, 7. 95) is a significant positive association e. g. , (95% CI: 0. 25, 0. 65) is a significant negative association If 95% CI includes 1 odds ratio NOT significant e. g. , (95% CI: 0. 80, 1. 15) is not significant (i. e. , cannot rule out odds ratio parameter of 1 with 95% confidence Also use a chi-square test or Fisher’s test as needed n 2/25/2021 Case-Control Odds Ratios 8

Chi-Square, Pearson (Do not review) OBSER VED D+ D- Total EXPEC TED D+ D- Total E+ 96 109 205 E+ 42. 051 162. 949 205 E- 104 666 770 E- 157. 949 612. 051 770 Total 200 775 975 c 2 Pearson's = (96 - 42. 051)2 / 42. 051 + (109 – 162. 949)2 / 162. 949 + (104 - 157. 949)2 / 157. 949 + (666 – 612. 051)2 / 612. 051 = 69. 213 + 17. 861 + 18. 427 + 4. 755 = 110. 256 c = sqrt(110. 256) = 10. 50 off chart (way into tail) P <. 0001 2/25/2021 Case-Control Odds Ratios 9

Chi-Square, Yates (Do not review) OBSER VED D+ D- Total EXPEC TED D+ D- Total E+ 96 109 205 E+ 42. 051 162. 949 205 E- 104 666 770 E- 157. 949 612. 051 770 Total 200 775 975 c 2 Pearson's = (|96 - 42. 051| - ½)2 / 42. 051 + (|109 – 162. 949| - ½)2 / 162. 949 + (|104 - 157. 949| - ½)2 / 157. 949 + (|666 – 612. 051| - ½)2 / 612. 051 = 67. 935 + 17. 532 + 18. 087 + 4. 668 = 108. 221 c = sqrt(108. 22) = 10. 40 P <. 0001 2/25/2021 Case-Control Odds Ratios 10

SPSS Output Pearson = uncorrected Yates = continuity corrected Fisher’s unnecessary here Linear-by-linear not covered 2/25/2021 Case-Control Odds Ratios 11

Matched-Pairs Matching can be employed to help control for confounding (e. g. , matching on age and sex), with each pair representing an observation. Case E+ Case E− 2/25/2021 Control E+ Control E− Doesn’t matter n. A n. B Doesn’t matter Calculate 95% CI for ln OR with (ln estimate) ± (1. 96)(SE) and then take anti-logs of limits Case-Control Odds Ratios 12

Example (Matched Pairs) 2/25/2021 Control E+ Control E− Case E+ 5 30 Case E− 10 5 Case-Control Odds Ratios 13

Validity Conditions! No info bias (data accurate) n No selection bias (cases and controls random reflection of population analogues) n No confounding (association not explained by lurking factors!) n Validity conditions are nearly always the limiting factor in practice. 2/25/2021 Case-Control Odds Ratios 14

Mc. Nemar’s Test for Matched Pairs (not covered, use CI instead) Use Mc. Nemar’s chi-square to test H 0: OR = 1 (“no association”) for binary matched paired P for current example = 0. 0016 2/25/2021 Case-Control Odds Ratios 15