March 28 Analyses of binary outcomes 2 x 2 tables Relative Risks and Relative Odds (Odds Ratio) Lee 6.1 through 6.5 C & S Chapter 3 (G, I, K, L, M, O)

Estimating a Single Proportion p = proportion in population with characteristic Take random sample of size n x = number in sample with characteristic p = x/n estimate of p SE(p) = Assumptions: n is large enough for CLT Then p is normally distributed 95% CI for p: p ± 1.96 SE(p) p – p0 Z = SQRT(p(1-p) (1/n))

Example p = proportion favoring a certain candidate n = 625 x = 300 favor the candidate p = 300/625 = 0.48 is estimate of p SE(p) = = 0.020 95% CI for p: 0.48 ± 1.96 (0.02) 0.48 ± 0.04 Note: Samplers use n=625 frequently

1-Sample Z-Test: Matched Pair Data Control Positive Control Negative a b c d Pos Case Neg Analyses is done on discordant pairs b and c Called McNemar’s chi-square Ho: p = 0.5 where n=b+c and x=b Z = (b/(b+c) – 0.5)/sqrt(.5*.5/(b+c)) Z = (b-c)/sqrt(b+c) c2 = (b-c)2/(b+c)

Example – Vitamin Use/Disease (440 Pairs) Control Vitamin + Control Vitamin - 100 50 90 200 Vit + Case Vit - Ho: p = 0.5 where n=140 and b = 50 c2 = (50-90)2/(50+90) = 11.43 (p=.0007)

Comparing Two Proportions Ho: p1 = p2 Ha: p1 ≠ p2 p1 = x1/ n1 p2 = x2 / n2 p = (x1+x2)/(n1+n2) This is the pooled proportion p2 – p1 Z = p(1-p) (1/n1 + 1/n2) Compare to standard normal distribution Assume n1 and n2 are large enough to use normal approximation

CI: Difference in Proportions 95% CI for difference in proportions:

Example – Asthma and SES Ho: p1 = p2 Ha: p1 ≠ p2 p1 = 30/ 160 = 0.188 p2 = 40 / 140 = 0.286 p = 70/300 = 0.233 This is the pooled proportion 0.286 – 0.188 Z = 0.233(.767) (1/160 + 1/140) c2 = Z2 = 4.03 X1 is number with asthma in high SES group X2 is number with asthma in low SES group = 0.098/0.049 = 2.01

2 by 2 Table a b c d c2 = ( a + b + c + d ) (ad – bc)2 Factor Present Factor Absent a b c d Sample 1 n1 = a + b Sample 2 n2 = c + d a + c b+ d c2 = ( a + b + c + d ) (ad – bc)2 ( a + c) (b + d) (a + b) (c + d)

2 by 2 Table Have Asthma No Asthma 30 130 40 100 High SES n1 =160 Low SES n2 =140 70 230 c2 = ( 30 + 130 + 40 + 100 ) (3000 – 5200)2 ( 70) (230) (160) (140) = 4.02

Relative Risks and Relative Odds Factor Present Factor Absent a b c d Sample 1 n1 = a + b Sample 2 n2 = c + d a + c b + d RR = a/(a+b) c/(c+d) RO = a/b c/d = ad/bc If a+b is approximately equal to b and If c+d is approximately equal to d then RR will be close to RO

Calculation RR and RO 30 130 40 100 160 140 70 140 RR = 30/160 40/140 Have Asthma No Asthma 30 130 40 100 High SES 160 Low SES 140 70 140 RR Asthma (High v Low SES) RO Asthma (High v Low SES) RR = 30/160 40/140 = 0.66 RO = 30/130 40/100 = 0.58 34% Lower risk of asthma in high SES compared to low SES

Confidence Interval for Relative Risk This CI looks a little different from usual form It is calculated on log scale Distribution of RR possible values is skewed Can’t be less than zero Can be extremely large positive values Usually transformed back when presented Calculated automatically by SAS

Confidence Interval for Odds Ratio Similar to CI for relative risk Can calculate by hand easily; SAS calculates automatically

Notes About CI for RR and RO Confidence intervals are not symmetric around the point estimate Cannot use RR ± SE notation Point estimate: 0.66 95% CI (0.43 – .99) 0.23 below 0.33 above

Odds Ratio Property 30 130 40 100 160 140 70 140 RO = 30/130 40/100 Have Asthma No Asthma 30 130 40 100 High SES 160 Low SES 140 70 140 RO Asthma (High v Low SES) RO High SES (Asthma v No Asthma) RO = 30/130 40/100 = 0.58 RO = 30/40 130/100 = 0.58 Same Answer – Not true for RR

Cohort Versus Case Control Study Cohort (Prospective) Find a population of low SES persons and a population of high SES persons For each person determine if he/she has asthma Case-Control (Retrospective) Find a group of persons with asthma and a group of persons without asthma. Determine if person is of low or high SES

What Can Be Estimated Cohort (Prospective) Can estimate probability of asthma for both low and high SES groups Can compute relative risk of asthma (high versus low SES) Case-Control (Retrospective) Can not estimate probability of asthma Can not estimate relative risk of asthma (H versus L SES) Can estimate relative odds (H versus L SES) If disease is fairly rare the RO can estimate RR

Cohort Versus Case-Control Cohort (Prospective) May not be possible to do Case-Control (Retrospective) May be only way to assess risk factors for disease

INPUT ses asthma count; DATALINES; 1 1 30 1 2 130 2 1 40 2 2 100 ; USING SAS DATA asthma; INFILE DATALINES; INPUT ses asthma count; DATALINES; 1 1 30 1 2 130 2 1 40 2 2 100 ; PROC FREQ DATA=asthma; TABLES ses*asthma/CHISQ RELRISK; WEIGHT count; RUN; Insert 2 x 2 table. The variable count contains the number in each cell of the table Get c2 value Get RR and RO Very important statement !

Table of ses by asthma ses asthma Frequency| Percent | Row Pct | Col Pct | 1| 2| Total ---------+--------+--------+ 1 | 30 | 130 | 160 | 10.00 | 43.33 | 53.33 | 18.75 | 81.25 | | 42.86 | 56.52 | 2 | 40 | 100 | 140 | 13.33 | 33.33 | 46.67 | 28.57 | 71.43 | | 57.14 | 43.48 | Total 70 230 300 23.33 76.67 100.00 Ho: p1 = p2 p1 = 30/160 = 0.1875 p2 = 40/140 = 0.2857 Statistic DF Value Prob ------------------------------------------------------ Chi-Square 1 4.0262 0.0448 Likelihood Ratio Chi-Square 1 4.0234 0.0449 Continuity Adj. Chi-Square 1 3.4959 0.0615 Mantel-Haenszel Chi-Square 1 4.0128 0.0452

Table of ses by asthma ses asthma Frequency| Percent | Row Pct | Col Pct | 1| 2| Total ---------+--------+--------+ 1 | 30 | 130 | 160 | 10.00 | 43.33 | 53.33 | 18.75 | 81.25 | | 42.86 | 56.52 | 2 | 40 | 100 | 140 | 13.33 | 33.33 | 46.67 | 28.57 | 71.43 | | 57.14 | 43.48 | Total 70 230 300 23.33 76.67 100.00 Ho: RR =1 or RO = 1 RR = 0.1875/0.2857 = 0.6563 RO = (30/130) / (40/100) = 0.5769 Row1 = High SES Row2 = Low SES Col1 = Have asthma Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ----------------------------------------------------------------- Case-Control (Odds Ratio) 0.5769 0.3361 0.9904 Cohort (Col1 Risk) 0.6563 0.4331 0.9943 Cohort (Col2 Risk) 1.1375 1.0003 1.2935

Adjusting for Other Factors Other factors must be categorical Estimated RR and RO are a pooled estimate across all combinations of adjustment variables Analyses called Mentel-Haenszel c2 PROC FREQ DATA=asthma; WEIGHT count; TABLES gender*ses*asthma/CHISQ CMH; RUN; Dependent variable Risk factor of interest Adjustment Variable (s)

Adjusting for Other Factors Perhaps some or all of the SES/ASTHMA relationship is due to sex/gender PROC FREQ DATA=asthma; WEIGHT count; TABLES gender*ses*asthma/CHISQ CMH; RUN; Dependent variable Risk factor of interest Adjustment Variable (s)

INPUT gender ses asthma count; DATALINES; 1 1 1 10 1 1 2 70 1 2 1 10 USING SAS DATA asthma; INFILE DATALINES; INPUT gender ses asthma count; DATALINES; 1 1 1 10 1 1 2 70 1 2 1 10 1 2 2 30 2 1 1 20 2 1 2 60 2 2 1 30 2 2 2 70 ; PROC FREQ DATA=asthma; TABLES gender*ses*asthma/CHISQ CMH; WEIGHT count; 2 x 2 table for men 2 x 2 table for women

Analysis for men ses asthma Frequency| Percent | Row Pct | Table of ses by asthma ses asthma Frequency| Percent | Row Pct | Col Pct | 1| 2| Total ---------+--------+--------+ 1 | 10 | 70 | 80 | 8.33 | 58.33 | 66.67 | 12.50 | 87.50 | | 50.00 | 70.00 | 2 | 10 | 30 | 40 | 8.33 | 25.00 | 33.33 | 25.00 | 75.00 | | 50.00 | 30.00 | Total 20 100 120 16.67 83.33 100.00 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ----------------------------------------------------------------- Case-Control (Odds Ratio) 0.4286 0.1616 1.1366 Cohort (Col1 Risk) 0.5000 0.2269 1.1018 Analysis for men

Analysis for women Table of ses by asthma ses asthma Frequency| Percent | Row Pct | Col Pct | 1| 2| Total ---------+--------+--------+ 1 | 20 | 60 | 80 | 11.11 | 33.33 | 44.44 | 25.00 | 75.00 | | 40.00 | 46.15 | 2 | 30 | 70 | 100 | 16.67 | 38.89 | 55.56 | 30.00 | 70.00 | | 60.00 | 53.85 | Total 50 130 180 27.78 72.22 100.00 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ----------------------------------------------------------------- Case-Control (Odds Ratio) 0.7778 0.4010 1.5087 Cohort (Col1 Risk) 0.8333 0.5139 1.3513 Analysis for women

Tests if Odds Ratio is same for men and women POOLED ANALYSES Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits ------------------------------------------------------------------------- Case-Control Mantel-Haenszel 0.6491 0.3749 1.1241 (Odds Ratio) Logit 0.6443 0.3725 1.1147 Cohort Mantel-Haenszel 0.7222 0.4794 1.0879 (Col1 Risk) Logit 0.7251 0.4801 1.0951 Breslow-Day Test for Homogeneity of the Odds Ratios ------------------------------ Chi-Square 0.9880 DF 1 Pr > ChiSq 0.3202 Pooled Analyses Tests if Odds Ratio is same for men and women

Class Exercise Among the 668 patients in TOMHS randomized to active treatment 74 experienced a CVD event during the study. Among the 234 patients randomized to placebo 38 had a CVD event. Compute the RR of CVD, for active versus placebo Compute the RO of CVD, for active versus placebo Use SAS to create the 2 x 2 table Using SAS compute the c2 statistic Using SAS compute the 95% for the RR and RO above