The notes for the topics on this page can be found in the lecture 20 folder on Canvas.
To conduct a Cochran-Mantel-Haenszel test in R you can use mantelhaen.test() from the stats package. In this example, we first create our dataset and then 2 x 2 x 2 table as shown elsewhere. The 2 x 2 x 2 table is then piped into mantelhaen.test(). Notice that you must set correct = FALSE as the default is to conduct a test with a continuity correction.
library(tidyverse)
library(magrittr)
health_policy <- tibble(
stress = rep(c("low", "low", "high", "high"), 2),
opinion = rep(c("favorable", "unfavorable"), 4),
region = c(rep("rural", 4), rep("urban", 4)),
count = c(55, 135, 7, 53, 48, 12, 96, 94)
)
health_policy %<>%
mutate(stress = fct_relevel(stress, c("low", "high")))
xtabs(count ~ stress + opinion + region, data = health_policy) %>%
mantelhaen.test(correct = FALSE)##
## Mantel-Haenszel chi-squared test without continuity correction
##
## data: .
## Mantel-Haenszel X-squared = 23.05, df = 1, p-value = 1.578e-06
## alternative hypothesis: true common odds ratio is not equal to 1
## 95 percent confidence interval:
## 2.066744 6.069367
## sample estimates:
## common odds ratio
## 3.541726To conduct a Cochran-Mantel-Haenszel test in SAS you use the frequency procedure with the cmh argument specified in the options of the table statement.
data healthpolicy;
input stress $ opinion $ region $ count;
cards;
low favorable rural 55
low unfavorable rural 135
high favorable rural 7
high unfavorable rural 53
low favorable urban 48
low unfavorable urban 12
high favorable urban 96
high unfavorable urban 94
;
run;
proc freq data = healthpolicy order = data;
table region*stress*opinion / chisq relrisk cmh;
weight count;
run;## The FREQ Procedure
##
## Table 1 of stress by opinion
## Controlling for region=rural
##
## stress opinion
##
## Frequency|
## Percent |
## Row Pct |
## Col Pct |favorabl|unfavora| Total
## ---------+--------+--------+
## low | 55 | 135 | 190
## | 22.00 | 54.00 | 76.00
## | 28.95 | 71.05 |
## | 88.71 | 71.81 |
## ---------+--------+--------+
## high | 7 | 53 | 60
## | 2.80 | 21.20 | 24.00
## | 11.67 | 88.33 |
## | 11.29 | 28.19 |
## ---------+--------+--------+
## Total 62 188 250
## 24.80 75.20 100.00
##
##
## Statistics for Table 1 of stress by opinion
## Controlling for region=rural
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 7.3016 0.0069
## Likelihood Ratio Chi-Square 1 8.1976 0.0042
## Continuity Adj. Chi-Square 1 6.4044 0.0114
## Mantel-Haenszel Chi-Square 1 7.2724 0.0070
## Phi Coefficient 0.1709
## Contingency Coefficient 0.1685
## Cramer's V 0.1709
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 55
## Left-sided Pr <= F 0.9988
## Right-sided Pr >= F 0.0041
##
## Table Probability (P) 0.0029
## Two-sided Pr <= P 0.0061
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 3.0847 1.3207 7.2045
## Relative Risk (Column 1) 2.4812 1.1945 5.1539
## Relative Risk (Column 2) 0.8044 0.7069 0.9153
##
## Sample Size = 250
##
##
## Table 2 of stress by opinion
## Controlling for region=urban
##
## stress opinion
##
## Frequency|
## Percent |
## Row Pct |
## Col Pct |favorabl|unfavora| Total
## ---------+--------+--------+
## low | 48 | 12 | 60
## | 19.20 | 4.80 | 24.00
## | 80.00 | 20.00 |
## | 33.33 | 11.32 |
## ---------+--------+--------+
## high | 96 | 94 | 190
## | 38.40 | 37.60 | 76.00
## | 50.53 | 49.47 |
## | 66.67 | 88.68 |
## ---------+--------+--------+
## Total 144 106 250
## 57.60 42.40 100.00
##
##
## Statistics for Table 2 of stress by opinion
## Controlling for region=urban
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 16.2198 <.0001
## Likelihood Ratio Chi-Square 1 17.3520 <.0001
## Continuity Adj. Chi-Square 1 15.0354 0.0001
## Mantel-Haenszel Chi-Square 1 16.1549 <.0001
## Phi Coefficient 0.2547
## Contingency Coefficient 0.2468
## Cramer's V 0.2547
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 48
## Left-sided Pr <= F 1.0000
## Right-sided Pr >= F <.0001
##
## Table Probability (P) <.0001
## Two-sided Pr <= P <.0001
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 3.9167 1.9575 7.8366
## Relative Risk (Column 1) 1.5833 1.3104 1.9131
## Relative Risk (Column 2) 0.4043 0.2389 0.6841
##
## Sample Size = 250
##
##
##
## The FREQ Procedure
##
## Summary Statistics for stress by opinion
## Controlling for region
##
## Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
##
## Statistic Alternative Hypothesis DF Value Prob
## ---------------------------------------------------------------
## 1 Nonzero Correlation 1 23.0502 <.0001
## 2 Row Mean Scores Differ 1 23.0502 <.0001
## 3 General Association 1 23.0502 <.0001
##
##
## Common Odds Ratio and Relative Risks
##
## Statistic Method Value 95% Confidence Limits
## -------------------------------------------------------------------------------------
## Odds Ratio Mantel-Haenszel 3.5417 2.0667 6.0694
## Logit 3.5593 2.0805 6.0891
##
## Relative Risk (Column 1) Mantel-Haenszel 1.7518 1.4071 2.1808
## Logit 1.6286 1.3560 1.9560
##
## Relative Risk (Column 2) Mantel-Haenszel 0.6607 0.5554 0.7861
## Logit 0.7735 0.6823 0.8769
##
##
## Breslow-Day Test for
## Homogeneity of the Odds Ratios
## ------------------------------
## Chi-Square 0.1830
## DF 1
## Pr > ChiSq 0.6688
##
##
## Total Sample Size = 500