Risk Ratio and Odds Ratio

Transcription

1 Risk Ratio and Odds Ratio

2 Risk and Odds Risk is a probability as calculated from one outcome probability = ALL possible outcomes Odds is opposed to probability, and is calculated from one outcome Odds = all OTHER outcomes Both are measures of likelihood but differ in Denominator 0 Risk (or probability) 1 whereas 0 Odds

3 Calculation Examples The probability and the odds of flipping a coin and getting a head Outcome: H Other outcome: T All possible outcome: H, T probability = Odds = one outcome ALL possible outcomes = 1 2 = 0.5 one outcome all OTHER outcomes = 1 1 = 1: 1 The probability and the odds of flipping a coin twice and getting two heads Outcome: HH Other outcome: HT, TH, TT All possible outcome: HH, HT, TH, TT one outcome probability = = 1 = 0.25 ALL possible outcomes 4 one outcome Odds = = 1 = 1: 3 all OTHER outcomes 3

4 Another Example The probability and the odds of having 2 short-hair kittens in a litter with 5 kittens Outcome: 2 short-hair kittens (S S) Other outcome: 3 long-hair kittens (L L L) All possible outcome: S S L L L probability = Odds = one outcome ALL possible outcomes = 2 5 = 0.4 one outcome all OTHER outcomes = 2 3 = 2: The probability and the odds of having 4 short-hair kittens in a litter with 5 kittens Outcome: 4 short-hair kittens (S S S S) Other outcome: 1 long-hair kittens (L) All possible outcome: S S S S L one outcome probability = = 4 = 0.8 ALL possible outcomes 5 one outcome Odds = = 4 = 4: 1 = 4 all OTHER outcomes 1

5 Graphical Representations of Risk and Odds Risk = Odds =

6 Epidemiology Wiki definition: Epidemiology is the study and analysis about the distribution and determinants of health and diseases in defined population Many types of epidemiological studies Randomized control study Cohort study Case-control study

7 Epidemiological studies: Some assessments Experimental study or Observational study If a researcher assigns mixture of participants to groups (i.e. randomization), it s the experimental study If the researcher does not assign participants to any groups, but let participants characteristics determined which group they should fall in, it s an observational study Directionality Forward study the exposure is known, then follow up to see what outcomes occurred Backward study the outcomes are occurred, then exposure is determined Timing Prospective the study starts before the outcome occurred Retrospective the study starts after the outcomes occurred

8 Epidemiology Study Types: Cohort study An observational study; forward directionality; prospective timing, or can be a retrospective timing Conceptually, Start with a population of disease-free individuals Identify individuals that are exposed to a risk factor(s) and those that are NOT exposed to the same risk factor(s) then follow up both groups over time to find out the risk of specific outcomes (e.g. diseases) occurring in each individual Relative Risk is used to determined association between the exposure and the outcomes Hypotheses (tentative!!!) Ho: Proportions of the outcome in exposed and unexposed groups are equal H1: Proportions of the outcome in exposed and unexposed groups are not equal

9 Cohort study time Exposure e.g. smoking outcome e.g. lung cancer smoke not smoke Lung cancer No Lung cancer versus

10 Breast cancer and Hormone replacement therapy in the million-women study Outcome Exposure Breast cancer No breast cancer Used HRT ,936 Never used HRT ,863 Question: what is the risk of using HRT on breast cancer occurrence in women? Outcome Exposure Breast cancer No breast cancer Used HRT a b Never used HRT c d The Lancet 362:

11 Relative Risk (RR) and confident intervals Outcome Exposure Breast cancer No breast cancer Used HRT a b Relative Risk = Never used HRT c d a a + b c c + d limits = ln(rr) ±z b a a + b + d c c + d CI = e lower limit upper limit, e

12 Interpretation of RR If RR = 1 or CI includes 1, there is no risk for the outcome to the exposed group nor the unexposed group If RR is more than 1 and CI does not includes 1, the relative risk of the outcome in the exposed group was increased by 1 RR 100% relative to the unexposed group Or the risk of the outcome has RR times more likely to occur in the exposed group than in the unexposed group If RR is less than 1 and CI does not includes 1 in its range, the relation risk of the outcome in the exposed group was reduced by 1 RR 100% relative to the unexposed group

13 Breast cancer and Hormone replacement therapy in the million-women study Risk HRT = Risk No HRT = = = Outcome Exposure Breast cancer No breast cancer Used HRT ,936 Never used HRT ,863 Reletive Risk = Risk HRT = Risk No HRT = limit a = limit a = ln limit b = ln limit b = CI = e.5573, e.6120 CI = 1.746,1.958

14 Relative Risk (RR) and confident intervals Exposure Outcome Breast cancer No breast cancer Used HRT ,936 Never used HRT ,863 RR=1.824 with CI=(1.746,1.958) RR is significantly different from 1, reject Ho then accept H1 stating that proportions of breast cancer in both groups are not equal Interpretation: Relative risk of breast cancer in women who used HRT is increased by %=82.4% relative to women who did not use HRT. Or the risk of breast cancer is 1.8 times more likely to occur in women who used HRT than in the women who did not use HRT.

15 Exposure Cardiovascular diseases among users of estrogen with progestin as compared to nonusers Outcome Major coronary disease No disease Estrogen with progestin 8 27, Not Used , RR = Τ = with CI=(0.103,0.419) RR is significantly different from 1, reject Ho ten accept H1 stating that proportions of breast cancer in both groups are not equal Interpretation: Relative risk of major coronary disease is reduced by % = 79.2% in users of estrogen with progestin relative to users who has not used any hormone Or the risk of major coronary disease is 0.2 times less likely to occur in user who of estrogen with progestin than in users who do not use any hormone. Risk = = Adapted from NEJM 1996:

16 Epidemiology Study Types: Case-Control study An observational study; backward directionality; retrospective timing Conceptually, Start with the case, i.e. a group of individuals having the outcomes (e.g. disease), and the control, i.e. a group of individuals not having the outcomes, then look back in time in both groups to find out what exposure(s) in both case and control that lead to specific outcomes or diseases Odds ratio is used to determined association Hypotheses (tentative!!!) Ho: Proportions of the exposure in case and control are equal H1: Proportions of the exposure in case and control are not equal

17 Case-Control study time Exposure e.g. smoking outcome e.g. lung cancer Smoking Not smoking Lung cancer No Lung cancer versus

18 Hay fever and eczema in 11 years old children Eczema Hay fever Yes (case) No (control) Yes No Case is children with hay fever and control is children without hay fever. Exposure is the children has experienced eczema or not. What is the odds of children having hay fever will develop eczema compared to children without hay fever? Eczema Hay fever Yes (case) No (control) Yes a b BMJ May 27; 320(7247): No c d

19 Odds ratio (OR) and confident intervals Eczema Hey fever Yes (case) No (control) Yes a b No c d Odds Ratio = aτ c b d limits = ln(or) ±z 1 a + 1 b + 1 c + 1 d CI = e lower limit upper limit, e

20 Interpretation of OR If OR = 1 or CI includes 1, the odds are equal for the case group and the control group to experience the exposures If OR is more than 1 and CI does not includes 1, the odds of the exposure in the case group was higher relative to the control group If OR is less than 1 and CI does not includes 1, the odd of the exposure in the case group was lower relative to the control group Normally, there should be switched the case and the control (NOT shuffling data!) so that OR is greater than 1

21 Odds ratio (OR) and confident intervals Odds Ratio = = Hey fever Eczema Yes (case) No (control) Yes No limit a = ln(4.893) 1.96 limit b = ln(4.893) limit a = limit b = CI = e 1.386, e CI = 3.998,5.998

22 Hay fever and eczema in 11 years old children Eczema Hay fever Yes (case) No (control) Yes No OR = with CI=(3.998,5.998) OR is significantly different from 1, reject Ho then accept H1 stating that proportions of eczema developed in the case and the control are not equal Interpretation: children having hay fever has the odds of times to develop eczema compared to children without hay fever BMJ May 27; 320(7247): 1468.

23 Leukemia and parental smoking in pregnancy Leukemia Yes (case) No (control) Smoking Yes No Case is patients with leukemia and control is patients without leukemia. Exposure is whether or not there is parental smoking during pregnancy. What is the odd of patients with leukemia (the case) to have been exposed to parental smoking in pregnancy?

24 Leukemia and parental smoking in pregnancy Smoking Leukemia Yes (case) No (control) Yes No Odds = = OR = 0.433Τ = with CI=(1.096,2.042) OR is significantly different from 1, reject Ho then accept H1 stating that proportions of exposing to parental smoking in pregnancy in case and control are not equal Interpretation: In patients with leukemia (case group), the odds is times to have been exposed to parental smoking in pregnancy

25 Choosing a test [after a thought!] If you want to know whether or not the observation deviates from the theory choose the test for goodness of fit If you have only 2 outcomes, use binomial test; else, use 2 test If observations is less than 1000, you may find exact probability [you are using a computer, aren t you?]; else, asymptotic probability will suffice it However, you want to find association between 2 nominal variables, a) You may choose 2 test or Fisher s exact test for a test of independence if what you really want to know is whether one or more categories in variable A affect one or more categories in variable B; or b) You may choose 2 test for a test of homogeneity if you just want to know whether proportions of one category in variable A are equal in 2 or more groups (=variable B); or c) You may choose to find relative risk if you want to know causality of the outcome (=one of two categories in variable A) in 2 different exposed groups from the cohort study; or d) You may choose to find odds ratio if you want to know odds of the exposure (=one of two categories in variable A) in the case and control groups from the case-control study

26 Strength of association by crosstab 2 test For 2x2 tables, i.e. 2 binary nominal variables Phi that is defined as φ = χ2 Example: if 2 = and n=150, then φ = = 0.25 For table larger than 2x2 tables n Cramer s V that is defined as V = n min(r 1,c 1) Example: if 2 = , n=566, r=4 rows and c=3 columns, so r-1=4-1=3 and c-1=3-1=2, then V = = = χ 2

27 Interpretation of Phi and Cramer s V Reminder: Phi and Cramer s V are the measures of association between two nominal variables, i.e. how strong the association is Both Phi and Cramer s V do not identify the pattern nor direction To assess the pattern of association, interpret the column percentages in the bivariate table Here the guideline Measure of association Between 0.00 and 0.10 Between 0.11 and 0.30 Greater than 0.30 Strength of association Weak Moderate Strong