Tests for Two Proportions in a Stratified Design (Cochran/Mantel-Haenszel Test)

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1 Chater 225 Tests for Two Proortions in a Stratified Design (Cochran/Mantel-Haenszel Test) Introduction In a stratified design, the subects are selected from two or more strata which are formed from imortant covariates such as gender, income level, or marital status. The number of subects in each of the two grous in each strata is set (fixed) by the design. A searate 2-by-2 table is formed for each stratum. Although resonse rates may vary among strata, hyotheses about the overall odds ratio can be tested the Cochran-Mantel-Haenszel test. This module allows you to determine ower and samle size for such a study. Technical Details This rocedure is based on the results of Woolson, Bean, and Roas (1986) which were extended to include a continuity correction by Nam (1992). For more details, consult those articles or chater 4 in Lachin (2). We will now briefly summarize these results. Suose you are interested in comaring the disease resonse rates of two grous (treatment and control). Further suose that resonse rate is known to be related to another covariate (such as age, race, or gender). It is often desirable to remove the covariate s imact from the comarison of the two roortions. This is accomlished by stratifying on the covariate and forming hyotheses about a common odds ratio across all strata. Data from such a stratified design may be analyzed by the Cochran-Mantel-Haenszel test. There are two versions of the Cochran-Mantel-Haenszel test: one that is continuity corrected and one that is not. The continuity-corrected test is more commonly used. The comutation of the test statistic is as follows. Suose there are strata. The result of each 2-by-2 table may be summarized as follows. Grous Grou 1 Grou 2 Resonse Treatment Control Total Yes x 1 x 2 x. No n1 x1 n2 x2 N x. Total n 1 n 2 N 225-1

2 where 1, 2,, and N 1 N. The arameters of interest are the success roortions 1 and 2. These arameters are estimated by 1 x1 and n 1 2 x n The odds of resonse in each of the two grous in each strata is given by o 1 1 and o 1 The strata odds ratio ψ is calculated using the equation ψ o o In the sequel, it is assumed that the strata odds ratios are all equal. That is, it is assumed that ψ ψ ψ ψ. Solving this relationshi for 1 in terms of ψ and 2 gives ψ ψ 2 2 If values for the odds ratio under the null hyothesis( ψ ), under the alternative hyothesis( ), and 2 are secified, values for as follows 1 under the null hyothesis ( 1 ) and the alternative hyothesis ( ) ψ 2, 1 + ψ 1, 2,, 2 2 ψ12, 1 + ψ 1, 2,, ψ can be calculated Assuming a common odds ratio across all strata of ψ (that is, assumingψ1 ψ 2 ψ ψ ), hyotheses of the form H :ψ ψ versus H 1 :ψ > ψ may be tested using Cochran s U statistic (Woolson et al. 1986, age 928) {( 1 2 ) ( 1 2 )} U w w n n G, where N

3 Note that when ψ 1, U G reduces to ( ) U w The value U is commonly used to form the Cochran-Mantel-Haenszel statistic. U G is an extension of this statistic which allows ψ 1. The calculation of the asymtotically normal test statistic, z c, may or may not include a continuity correction factor deending on whether the arameter cc is set to 1/2 or. The formula for z CMH is where ( ) v U G x N. 1 z CMH ( 1 ) 2 ( 1 2 ) w n1 n 2 1 w ( ) U G cc ( ) v U G if ψ 1 1 if ψ 1 The name Cochran-Mantel-Haenszel test actually refers to two tests: the Cochran test and the the Mantel-Haenszel test. The difference is between these test is that Cochran s test uses v ( U G ) to estimate the unconditional variance assuming that the grou samle sizes are fixed, while the Mantel-Haenszel test relaces v ( U G ) with an estimate of the conditional variance of U assuming that both row and column marginals are fixed. Asymtotically the two variances are equivalent, so the test is often called the Cochran-Mantel-Haenszel statistic. Power Calculations The asymtotic ower of z CMH for testing a one-sided hyothesis of the form H :ψ ψ versus H 1 :ψ > ψ is where ( G ) ( ) ( ) ( G ) ( G ) V ( U ) z V U E U + cc 1 α Power 1 Φ 1 G { } E U w ( ) V U G 1 w n 1 ( 1 ) 2 ( 1 2 ) w + ( ) n 2 if ψ 1 1 if ψ

4 1 1 n1 N + 2 n 2 N V ( U ) w 1 G 1 n 1 ( 1 ) 2 ( 1 2 ) n 2 Note that Woolson et al. (1986) and Nam (1992) give results for the usual case when ψ 1. The above results are our extension to the imortant case when ψ 1. We could not find ublished results for this case, so we have made this extension. When ublished results become available, we will adot those results. If you have ψ 1, you must use U G, rather than U, in the calculation of the test statistic. Similar calculations may also be made for testing the other one-sided hyothesis H :ψ ψ versus H 1 :ψ < ψ and the two-sided hyothesis H :ψ ψ versus H 1 :ψ ψ. Procedure Otions This section describes the otions that are secific to this rocedure. These are located on the Design 1 and Design 2 tabs. For more information about the otions of other tabs, go to the Procedure Window chater. Design 1 and Design 2 Tabs The Design tabs contain most of the arameters and otions of interest for this rocedure. Solve For Solve For This otion secifies the arameter to be solved for using the other arameters. The arameters that may be selected are OR1, Alha, Power, or Samle Size. In most cases, you will select either Power or Samle Size. Select Samle Size when you want to calculate the samle size needed to achieve a given ower and alha level. Select Power when you want to calculate the ower. Test H1 (Alternative Hyothesis) This otion secifies whether a one-sided or two-sided hyothesis is analyzed. One-Sided (H1: OR1 < OR) refers to a one-sided test in which the alternative hyothesis is of the form H1: OR1 < OR. One-Sided (H1: OR1 > OR) refers to a one-sided test in which the alternative hyothesis is of the form H1: OR1 > OR. Two-Sided refers to a two-sided test in which the alternative hyothesis is of the tye H1: OR1 OR. Here means is not equal to or is less than or greater than. Note that the alternative hyothesis enters into ower calculations by secifying the reection region of the hyothesis test. Its accuracy is critical

5 Continuity Correction Secify whether to use the Continuity Correction. When selected, a continuity correction is made that is recommend by Fleiss et al. (23) to make the alha and beta values achieved by the test more accurate. Power and Alha Power This otion secifies one or more values for ower. Power is the robability of reecting a false null hyothesis, and is equal to one minus Beta. Beta is the robability of a tye-ii error, which occurs when a false null hyothesis is not reected. In this rocedure, a tye-ii error occurs when you fail to reect the null hyothesis of equal roortions when in fact they are different. Values must be between zero and one. Historically, the value of.8 (Beta.2) was used for ower. Now,.9 (Beta.1) is also commonly used. A single value may be entered here or a range of values such as.8 to.95 by.5 may be entered. Alha This otion secifies one or more values for the robability of a tye-i error. A tye-i error occurs when a true null hyothesis is reected. For this rocedure, a tye-i error occurs when you reect the null hyothesis of equal roortions when in fact they are equal. Values must be between zero and one. Historically, the value of.5 has been used for alha. This means that about one test in twenty will falsely reect the null hyothesis. You should ick a value for alha that reresents the risk of a tye-i error you are willing to take in your exerimental situation. You may enter a range of values such as or.1 to.1 by.1. Samle Size M (Samle Size Multilier) M and the values of R1 and R2 are used to calculate the grou samle sizes within each strata using the formulas N1 M x R1 and N2 M x R2. The total samle size, N, is found by summing N1 and N2 across all strata. Note that fractional values for N, N1, and N2 will usually result. In ractice these values are rounded u to the next integer value. One or more values, searated by blanks or commas, may be entered. A searate analysis is erformed for each value. Using M as the Grou Size To use M as the samle size in each grou, the values of R1 and R2 must each be set to one. Using M as the Strata Size To use M as the samle size in each strata, the values of R1 and R2 must sum to one within each strata. For examle, suose M 3 and R1 R2.5. The values of N1 and N2, the grou samle sizes within a stratum, will be.5 x Thus, the total samle size within the strata is Using M as Total Samle Size To use M as the total samle size across all strata, the values of R1 and R2 must sum to one across all values. Note that the resulting value of N may not exactly equal M because of rounding. For examle, suose there are three strata with R1.1,.2, and.2 and R2.1,.3, and.1. (Note that these values sum to one.) If M were 1, then the values of N1 would be 1, 2, and 2 and the values of N2 would be 1, 3, and 1. These sum to 1, the value of M

6 Effect Size OR1 (Odds Ratio H1) This otion secifies the odds ratio of the two roortions P1 and P2 at which the ower is to be comuted. This odds ratio is used to secify the size of the difference between the two roortions at which the ower is calculated. You may enter a range of values such as or 1.25 to 2. by.25. Odds ratios must greater than zero. OR (Odds Ratio H) Secify the odds ratio under the null hyothesis, H. For each strata, this value is used with the value of Pr(Success) to calculate the robability of obtaining a success in grou one (the treatment grou) assuming the null hyothesis. In the standard Cochran-Mantel-Haenszel test, this value is assumed to be (and should be entered as) one. If you enter a value other than one, your data analysis should use the more general test statistic. Note that OR must be greater than zero and cannot be equal to OR1. Strata Information Strata This otion secifies the number of strata secified on this line. Usually, you will enter a 1 to secify a single stratum, or you will enter a to ignore this line. However, this otion lets you secify several strata at once. The total number of strata is equal to the sum of these values. R1 N1 / M, R2 N2 / M R1 and R2 are used to obtain the samle sizes in grous 1 (treatment) and 2 (control) within a strata using the formulas N1 R1 x M and N2 R2 x M. The only limitation on R1 and R2 is that they are ositive (non-zero) values. See the comments under M for more information. Note that only a single value may be entered for this arameter you cannot enter several values. Pr(Success) This is the baseline robability of a successful resonse. This value is used with OR1 to calculate the robability of a success in grou 1 (the treatment or numerator grou). Since this value is a robability, it must be between zero and one. Note that only one value may be entered here

7 Examle 1 Finding Power Nam (1992) discusses a case-control study investigating the ossible association between chlorinated water and colon cancer among males in Iowa. Since age is known to affect colon cancer rates, the oulation is stratified into four age grous with weights of 1%, 4%, 35%, and 15%. An equal number of cases and controls will be selected in each age-grou. Prior studies had shown the robability of chlorinated water exosure among noncancer subects was.75,.7,.65, and.6, resectively, among the four age grous. The significance level is set to.5. The investigators want to consider various total samle sizes from 5 to 5. They also want to consider odds ratios of 2 and 3. Setu This section resents the values of each of the arameters needed to run this examle. First, from the PASS Home window, load the rocedure window by exanding Proortions, then Two Indeendent Proortions, then clicking on Stratified, and then clicking on. You may then make the aroriate entries as listed below, or oen Examle 1 by going to the File menu and choosing Oen Examle Temlate. Otion Value Design 1 Tab Solve For... Power H1 (Alternative Hyothesis)... One-Sided (H1:OR1>OR) Continuity Correction... Checked Alha....5 M (Samle Size Multilier)... 5 to 5 by 5 OR (Odds Ratio H)... 1 OR1 (Odds Ratio H1) Strata(1)... 1 R1(1)....5 (half of 1%) R2(1)... R1 Pr(Success)(1) Strata(2)... 1 R1(2)....2 (half of 4%) R2(2)... R1 Pr(Success)(2)....7 Strata(3)... 1 R1(3) (half of 35%) R2(3)... R1 Pr(Success)(3) Design 2 Tab Strata(4)... 1 R1(4) (half of 15%) R2(4)... R1 Pr(Success)(4)

8 Annotated Outut Click the Calculate button to erform the calculations and generate the following outut. Numeric Results Numeric Results of Cochran-Mantel-Haenszel Test of an Odds Ratio H: OR1 OR. H1: OR1 > OR. Test: Continuity-Corrected Z-Test. Total Samle Samle Samle H Actual Samle Size Size of Size of Odds Odds Signif. Size Multilier Grou 1 Grou 2 Ratio Ratio Level Power (N) (M) (N1) (N2) (OR) (OR1) Alha Beta Reort Definitions 'Power' is the robability of reecting a false null hyothesis. It should be close to one. 'N' is the total samle size summed across all grous and strata. 'M' is the factor by which the values of R1 and R2 are multilied. 'N1 and N2' are the samle sizes from grous 1 and 2 summed across all strata. 'OR' is the odds ratio [P1/(1-P1)] / [P2/(1-P2)] assuming the null hyothesis (H). 'OR1' is the value of the odds ratio at which the ower is comuted. 'Alha' is the robability of reecting a true null hyothesis. 'Beta' is the robability of acceting a false null hyothesis. In a treatment vs. control design, the treatment grou is 1 and the control grou is 2. Summary Statements A stratified design, which divides the samle among 4 strata, is analyzed using the one-sided, Cochran-Mantel-Haenszel test. Samle sizes, summed across all strata, of 25 in grou 1 (treatment grou) and 25 in grou 2 (control grou) achieve 18% ower to reect the odds ratio set by the null hyothesis of 1. when the odds ratio is actually 2.. The significance level of the test was set at.5. Samle Sizes: N, N1, and N2 The value of N is the sum of N1 and N2. The values of N1 and N2 are found by summing the individual stratagrou samle sizes. These are found by multilying R1 and R2 by M. Note that this multilication will usually result in fractional samle sizes across the strata. As a ractical matter, we recommend rounding each fractional value u to the next integer when imlementing a given design

9 Strata-Detail Reort Strata-Detail Reort Proortion Proortion Proortion Strata Number of Total of this of this Grou 1 Grou 2 Probability of Samle in Strata in Strata in Multilier Multilier of Strata each Strata Grou 1 Grou 2 (R1) (R2) Success This reort shows the values of the individual, strata-level arameters that were used. These arameters are the same for all rows of the Numerical Results Reort (shown above), so they are only dislayed once. Plots Section The values from the Numerical Results reort are dislayed in these lots. These charts rovide a quick view of the ower that is achieved for various samle sizes

10 Examle 2 Validation using Nam To validate the rocedure, we will comare PASS s results to those on age 392 of Nam (1992). Most of the settings in this examle are the same as those of Examle 1, excet that the ower is 9% and the odds ratio is 3. Nam (1992) found the necessary samle sizes to be 192 for the corrected test and 171 for the uncorrected test. Setu This section resents the values of each of the arameters needed to run this examle. First, from the PASS Home window, load the rocedure window by exanding Proortions, then Two Indeendent Proortions, then clicking on Stratified, and then clicking on. You may then make the aroriate entries as listed below, or oen Examle 2a or Examle 2b by going to the File menu and choosing Oen Examle Temlate. Otion Value Design 1 Tab Solve For... Samle Size H1 (Alternative Hyothesis)... One-Sided (H1:OR1>OR) Continuity Correction... Checked/Unchecked Power....9 Alha....5 OR (Odds Ratio H)... 1 OR1 (Odds Ratio H1)... 3 Strata(1)... 1 R1(1)....5 (half of 1%) R2(1)... R1 Pr(Success)(1) Strata(2)... 1 R1(2)....2 (half of 4%) R2(2)... R1 Pr(Success)(2)....7 Strata(3)... 1 R1(3) (half of 35%) R2(3)... R1 Pr(Success)(3) Design 2 Tab Strata(4)... 1 R1(4) (half of 15%) R2(4)... R1 Pr(Success)(4)

11 Outut Click the Calculate button to erform the calculations and generate the following outut. Numeric Results Numeric Results of Cochran-Mantel-Haenszel Test of an Odds Ratio H: OR1 OR. H1: OR1 > OR. Test: Continuity-Corrected Z-Test. Total Samle Samle Samle H Actual Samle Size Size of Size of Odds Odds Signif. Size Multilier Grou 1 Grou 2 Ratio Ratio Level Power (N) (M) (N1) (N2) (OR) (OR1) Alha Beta The value of 192 agrees exactly with that of Nam (1992). If you uncheck the Continuity Correction otion and rerun the analysis, you will get the following results. Numeric Results No Continuity Correction Numeric Results of Cochran-Mantel-Haenszel Test of an Odds Ratio H: OR1 OR. H1: OR1 > OR. Test: Uncorrected Z-Test. Total Samle Samle Samle H Actual Samle Size Size of Size of Odds Odds Signif. Size Multilier Grou 1 Grou 2 Ratio Ratio Level Power (N) (M) (N1) (N2) (OR) (OR1) Alha Beta The value of 171 agrees exactly with that of Nam (1992)

12 Examle 3 Finding Power of a Comleted Exeriment Suose you want to find the ower for a comleted exeriment in which the individual strata samle sizes are known. In this examle there are three strata with success robabilities.72,.66, and.69. The samle sizes for the treatment grou in each stratum are 12, 113, and 97. The samle sizes for the control grou in each stratum are 98, 11, and 114. The exeriment was designed to detect an odds ratio of at least 1.5 with alha equal to.5 for a one-sided test. To calculate the ower in this situation, we set M to 1 and enter the samle sizes directly into R1 and R2. Setu This section resents the values of each of the arameters needed to run this examle. First, from the PASS Home window, load the rocedure window by exanding Proortions, then Two Indeendent Proortions, then clicking on Stratified, and then clicking on. You may then make the aroriate entries as listed below, or oen Examle 3 by going to the File menu and choosing Oen Examle Temlate. Otion Value Design 1 Tab Solve For... Power H1 (Alternative Hyothesis)... One-Sided (H1:OR1>OR) Continuity Correction... Checked Alha....5 M (Samle Size Multilier)... 1 OR (Odds Ratio H)... 1 OR1 (Odds Ratio H1) Strata(1)... 1 R1(1) R2(1) Pr(Success)(1) Strata(2)... 1 R1(2) R2(2) Pr(Success)(2) Strata(3)... 1 R1(3) R2(3) Pr(Success)(3)

13 Outut Click the Calculate button to erform the calculations and generate the following outut. Numeric Results Numeric Results of Cochran-Mantel-Haenszel Test of an Odds Ratio H: OR1 OR. H1: OR1 > OR. Test: Continuity-Corrected Z-Test. Total Samle Samle Samle H Actual Samle Size Size of Size of Odds Odds Signif. Size Multilier Grou 1 Grou 2 Ratio Ratio Level Power (N) (M) (N1) (N2) (OR) (OR1) Alha Beta The ower to detect an odds ratio of 1.5 is only.698 in this exeriment

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