# Today s lecture. Lecture 6: Dichotomous Variables & Chi-square tests. Proportions and 2 2 tables. How do we compare these proportions

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1 Today s lecture Lecture 6: Dichotomous Variables & Chi-square tests Sandy Eckel Dichotomous Variables Comparing two proportions using 2 2 tables Study Designs Relative risks, odds ratios and their confidence intervals Chi-square tests 29 April / 36 2 / 36 Proportions and 2 2 tables How do we compare these proportions Population Success Failure Total Population 1 x 1 n 1 x 1 n 1 Population 2 x 2 n 2 x 2 n 2 Total x 1 + x 2 n (x 1 + x 2 ) n Row 1 shows results of a binomial experiment with n 1 trials Row 2 shows results of a binomial experiment with n 2 trials Often, we want to compare p 1, the probability of success in population 1, to p 2, the probability of success in population 2 Usually: Success = Disease Population 1 = Treatment 1 Population 2 = Treatment 2 (maybe placebo) How do we compare these proportions? We ve talked about comparing proportions by looking at their difference But sometimes we want to look at one proportion relative to the other This approach depends on the type of study the data came from 3 / 36 4 / 36

2 Observational epidemiologic study designs Cross-sectional study design Cross-sectional Cohort Case-control A snapshot of a sample of the population Commonly a survey/questionnaire or one time visit to assess information at a single point in time Measures existing disease and exposure levels Difficult to argue for a causal relation between exposure and disease Example: A group of Finnish residents between the ages of 25 and 64 are mailed a questionnaire asking about their daily coffee consumption habits and their systolic BP from their last visit to the doctor. 5 / 36 6 / 36 Cohort study design Case-control study design Find a group of individuals without the disease and separate into those with the exposure and without the exposure Follow over time and measure the disease rates in both groups Compare the disease rate in the exposed and unexposed If the exposure is harmful and associated with the disease, we would expect to see higher rates of disease in the exposed group versus the unexposed group Allows us to estimate the incidence of the disease (rate at which new disease cases occur) Example: A group of Finnish residents between the ages of 25 and 64 and currently without hypertension are asked about their current levels of daily coffee consumption. These individuals are followed for 5 years at which point they are asked if they have been diagnosed with hypertension. 7 / 36 Identify individuals with the disease of interest (case) and those without the disease (control) and then look retrospectively (at prior records) to find what the exposure levels were in these two groups The goal is to compare the exposure levels in the case and control groups If the exposure is harmful and associated with the disease, we would expect to see higher levels of exposure in the cases than in the controls Very useful when the disease is rare Example: A group of Finnish residents with hypertension and a group of Finnish residents without hypertension are identified. These individuals are contacted and asked about their level of daily coffee consumption in the two years prior to diagnosis with hypertension. 8 / 36

3 (Almost) Cohort Study Example Trial Outcome: 12 month mortality Aceh Vitamin A Trial Exposure levels (Vitamin A) assigned at baseline and then the children are followed to determine survival in the two groups. 25,939 pre-school children in 450 Indonesian villages in northern Sumatra 200,000 IU vitamin A given 1-3 months after the baseline census, and again at 6-8 months Consider 23,682 out of 25,939 who were visited on a pre-designed schedule References: 1 Sommer A, Djunaedi E, Loeden A et al, Lancet Sommer A, Zeger S, Statistics in Medicine Alive at 12 months? Vit A No Yes Total Yes 46 12,048 12,094 No 74 11,514 11,588 Total ,562 23,682 Does Vitamin A reduce mortality? Calculate risk ratio or relative risk Relative Risk abbreviated as RR Could also compare difference in proportions: called attributable risk 9 / / 36 Relative Risk Calculation Confidence interval for RR Relative Risk = = ˆp 1 ˆp 2 Rate with Vitamin A Rate without Vitamin A = 46/12,094 74/11, 588 = = 0.59 The death rate with vitamin A is 0.60 times (or 60% of) the death rate without vitamin A. Equivalent interpretation: Vitamin A group had 40% lower mortality than without vitamin A group! 11 / 36 Step 1: Find the estimate of the log RR log(ˆp 1 ˆp 2 ) Step 2: Estimate the variance of the log(rr) as: 1 p 1 n 1 p p 2 n 2 p 2 Step 3: Find the 95% CI for log(rr): log(rr) ± 1.96 SD(log RR) = (lower, upper) Step 4: Exponentiate to get 95% CI for RR; e (lower, upper) 12 / 36

4 Confidence interval for RR from Vitamin A Trial What if the data were from a case-control study? 95% CI for log relative risk is: 95% CI for relative risk log(rr) ± 1.96 SD(log RR) = log(0.59) ± = 0.53 ± 0.37 = ( 0.90, 0.16) (e 0.90,e 0.16 ) = (0.41,0.85) Does this confidence interval contain 1? 13 / 36 Recall: in case-control studies, individuals are selected by outcome status Disease (mortality) status defines the population, and exposure status defines the success p 1 and p 2 have a difference interpretation in a case-control study than in a cohort study Cohort: p 1 = P(Disease Exposure) p 2 = P(Disease No Exposure) Case-Control: p 1 = P(Exposure Disease) p 2 = P(Exposure No Disease) This is why we cannot estimate the relative risk from case-control data! 14 / 36 The Odds Ratio Which p 1 and p 2 do we use? The odds ratio measures association in Case-Control studies P(event occurs) Odds = P(event does not occur) = p 1 p Remember the odds ratio is simply a ratio of odds! odds in group 1 OR = odds in group 2 OR = ˆp 1/(1 ˆp 1 ) ˆp 2 /(1 ˆp 2 ) We can actually calculate OR using either case-control or cohort set up Using case-control p 1 and p 2 where we condition on disease or no disease OR = (46/120)/(74/120) (12048/23562)/(11514/23562) = 46/ /11514 = 0.59 Using cohort p 1 and p 2 where we condition on exposure or no exposure OR = (46/12094)/(12048/12094) (74/11588)/(11514/11588) = 46/ /11514 = 0.59 We get the same answer either way! 15 / / 36

5 Bottom Line Confidence interval for OR Step 1: Find the estimate of the log OR The relative risk cannot be estimated from a case-control study The odds ratio can be estimated from a case-control study OR estimates the RR when the disease is rare in both groups The OR is invariant to cohort or case-control designs, the RR is not We are introducing the OR now because it is an essential idea in logistic regression log(ˆp 1/(1 ˆp 1 ) ˆp 2 /(1 ˆp 2 ) ) Step 2: Estimate the variance of the log(or) as: 1 n 1 p n 1 q n 2 p n 2 q 2 Step 3: Find the 95% CI for log(or): log(or) ± 1.96 SD(log OR) = (lower, upper) Step 4: Exponentiate to get 95% CI for OR; e (lower, upper) 18 / / 36 All kinds of χ 2 tests The χ 2 distribution Test of Goodness of fit Test of independence Test of homogeneity or (no) association All of these test statistics have a χ 2 distribution under the null Derived from the normal distribution χ 2 1 = ( y µ σ )2 = Z 2 χ 2 k = Z1 2 + Z Zk 2 where Z 1,...,Z k are all standard normal random variables k denotes the degrees of freedom A χ 2 k random variable has mean = k variance = 2k Since a normal random variable can take on values in the interval (, ), a chi-square random variable can take on values in the interval (0, ) 19 / / 36

6 χ2 Family of Distributions χ2 Critical Values We generally use only a one-sided test for the χ2 distribution Area under the curve to the right of the cutoff for each curve is 0.05 Increasing critical value with increasing number of degrees of freedom df=1 df=3 df= Chi square value 21 / 36 χ2 Table 22 / 36 But...we ll use R For χ2 random variables with degrees of freedom = df we ll use / 36 pchisq(a, df, lower.tail=f) to find P(χdf a) =? P(ChiSq > a ) =? P(ChiSq >? ) = b a? qchisq(b, df, lower.tail=f) to find P(χdf?) = b 24 / 36

7 χ 2 Goodness-of-Fit Test The χ 2 test statistic Determine whether or not a sample of observed values of some random variable is compatible with the hypothesis that the sample was drawn from a population with a specified distributional form, i.e. Normal Binomial Poisson etc... Here, the expected cell counts would be derived from the distributional assumption under the null hypothesis χ 2 = k i=1 [(O i ) 2 ] where O i = i th observed frequency = i th expected frequency in the i th cell of a table Degrees of freedom = (# categories 1) Note: This test is based on frequencies (cell counts) in a table, not proportions 25 / / 36 Example: Handgun survey I Example: Handgun survey II Survey 200 adults regarding handgun bill: Statement: I agree with a ban on handguns Four categories: Strongly agree, agree, disagree, strongly disagree Can one conclude that opinions are equally distributed over four responses? Response (count) Strongly Strongly agree disagree agree disagree Responding (O i ) Expected ( ) χ 2 = k i=1 [ (O i ) 2 ] (102 50)2 = + 50 = (30 50) (60 50) (8 50)2 50 df = 4 1 = 3 27 / / 36

8 Example: Handgun survey III χ 2 Test of Independence I Critical value: χ 2 4 1,0.05 = χ2 3,0.05 = 7.81 Since > 7.81, we conclude that our observation was unlikely by chance alone (p < 0.05) Based on these data, opinions do not appear to be equally distributed among the four responses Test the null hypothesis that two criteria of classification are independent r c contingency table Criterion 2 Criterion c Total 1 n 11 n 12 n 13 n 1c n 1 2 n 21 n 22 n 23 n 2c n 2 3 n 31 n 32 n 33 n 3c n r n r1 n r2 n r3 n rc n r Total n 1 n 2 n 3 n c n 29 / / 36 χ 2 Test of Independence II χ 2 Test of Homogeneity (No association) Test statistic: χ 2 = k i=1 [ (O i ) 2 ] Degrees of freedom = (r 1)(c 1) where r is the number of rows and c is number of columns Assume the marginal totals are fixed Test the null hypothesis that the samples are drawn from populations that are homogenous with respect to some factor i.e. no association between group and factor Same test statistic as χ 2 test of independence Test statistic: χ 2 = k i=1 [ (O i ) 2 ] Degrees of freedom = (r 1)(c 1) where r is the number of rows and c is number of columns 31 / / 36

9 Example: Treatment response I Example: Treatment response II χ 2 Test of Homogeneity (No association) Observed Numbers Response to Treatment Treatment Yes No Total A B Total Test H 0 that there is no association between the treatment and response Calculate what numbers of Yes and No would be expected assuming the probability of Yes was the same in both treatment groups Condition on total the number of Yes and No responses Expected proportion with Yes response = = 0.45 Expected proportion with No response = = 0.55 Response to Treatment Treatment Yes No Total Observed A 37 (22.5) 13 (27.5) 50 (Expected) B 17 (31.5) 53 (38.5) 70 Total Get expected number of Yes responses on treatment A: = = Using a similar approach you get the other expected numbers 33 / / 36 Example: Treatment response III Lecture 6 summary Test statistic: χ 2 = k i=1 [ (O i ) 2 ] ( )2 ( )2 = ( )2 ( ) = 29.1 Degrees of freedom = (r-1)(c-1) = (2-1)(2-1) = 1 Critical value for α = is so we see p<0.001 Reject the null hypothesis, and conclude that the treatment groups are not homogenous (similar) with respect to response Response appears to be associated with treatment 35 / 36 Dichotomous variables cohort studies - relative risk case-control - odds ratios Chi-square tests Next time, we ll talk about analysis of variance (ANOVA) 36 / 36

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