The notes for the topics on this page can be found in the lecture 19 folder on Canvas.
To conduct a Brewlosw-Day test in R you can use BreslowDayTest() from the DescTools package. BreslowDayTest() requires a 2 x 2 x k table as its main argument; you can also specify an odds ratio for the null hypothesis (if not specified, the Mantel-Haenszal OR will be used). To create 2 x 2 x k tables from a data-set you can use xtabs().
library(DescTools)
teeth <- tibble(
exposure = rep(c("tetra", "tetra", "notetra", "notetra"), 2),
discolor = rep(c("yes", "no"), 4),
age = c(rep("child", 4), rep("adult", 4)),
count = c(30, 25, 12, 17, 4, 6, 68, 94)
)
teeth_tabs <- xtabs(count ~ exposure + discolor + age, data = teeth)
teeth_tabs## , , age = adult
##
## discolor
## exposure no yes
## notetra 94 68
## tetra 6 4
##
## , , age = child
##
## discolor
## exposure no yes
## notetra 17 12
## tetra 25 30BreslowDayTest(teeth_tabs)##
## Breslow-Day test on Homogeneity of Odds Ratios
##
## data: teeth_tabs
## X-squared = 0.57493, df = 1, p-value = 0.4483cervical <- tibble(
smoke = rep(c("yes", "yes", "no", "no"), 2),
cancer = rep(c("yes", "no", "yes", "no"), 2),
partners = c(rep("zeroone", 4), rep("twomore", 4)),
count = c(12, 21, 25, 118, 96, 148, 92, 150)
)
cervical_tabs <- xtabs(count ~ smoke + cancer + partners, data = cervical)
cervical_tabs## , , partners = twomore
##
## cancer
## smoke no yes
## no 150 92
## yes 148 96
##
## , , partners = zeroone
##
## cancer
## smoke no yes
## no 118 25
## yes 21 12BreslowDayTest(cervical_tabs)##
## Breslow-Day test on Homogeneity of Odds Ratios
##
## data: cervical_tabs
## X-squared = 4.2069, df = 1, p-value = 0.04026To conduct a Breslow-Day test in SAS you use the frequency procedure with the cmh argument specified in the options of the table statement. The 2 x 2 x k tables are built as usual, but with the following adjustment: table k*exposure*disease.
title 'Teeth Discoloration Example';
data teeth;
input exposure $ discolor $ age $ count;
cards;
tetra yes child 30
tetra no child 25
notetra yes child 12
notetra no child 17
tetra yes adult 4
tetra no adult 6
notetra yes adult 68
notetra no adult 94
;
run;
proc freq data = teeth order = data;
table age*exposure*discolor / chisq relrisk cmh expected;
weight count;
run; ## Teeth Discoloration Example
##
## The FREQ Procedure
##
## Table 1 of exposure by discolor
## Controlling for age=child
##
## exposure discolor
##
## Frequency|
## Expected |
## Percent |
## Row Pct |
## Col Pct |yes |no | Total
## ---------+--------+--------+
## tetra | 30 | 25 | 55
## | 27.5 | 27.5 |
## | 35.71 | 29.76 | 65.48
## | 54.55 | 45.45 |
## | 71.43 | 59.52 |
## ---------+--------+--------+
## notetra | 12 | 17 | 29
## | 14.5 | 14.5 |
## | 14.29 | 20.24 | 34.52
## | 41.38 | 58.62 |
## | 28.57 | 40.48 |
## ---------+--------+--------+
## Total 42 42 84
## 50.00 50.00 100.00
##
##
## Statistics for Table 1 of exposure by discolor
## Controlling for age=child
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 1.3166 0.2512
## Likelihood Ratio Chi-Square 1 1.3216 0.2503
## Continuity Adj. Chi-Square 1 0.8426 0.3586
## Mantel-Haenszel Chi-Square 1 1.3009 0.2540
## Phi Coefficient 0.1252
## Contingency Coefficient 0.1242
## Cramer's V 0.1252
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 30
## Left-sided Pr <= F 0.9160
## Right-sided Pr >= F 0.1794
##
## Table Probability (P) 0.0954
## Two-sided Pr <= P 0.3588
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 1.7000 0.6844 4.2226
## Relative Risk (Column 1) 1.3182 0.8028 2.1643
## Relative Risk (Column 2) 0.7754 0.5089 1.1814
##
## Sample Size = 84
##
##
## Table 2 of exposure by discolor
## Controlling for age=adult
##
## exposure discolor
##
## Frequency|
## Expected |
## Percent |
## Row Pct |
## Col Pct |yes |no | Total
## ---------+--------+--------+
## tetra | 4 | 6 | 10
## | 4.186 | 5.814 |
## | 2.33 | 3.49 | 5.81
## | 40.00 | 60.00 |
## | 5.56 | 6.00 |
## ---------+--------+--------+
## notetra | 68 | 94 | 162
## | 67.814 | 94.186 |
## | 39.53 | 54.65 | 94.19
## | 41.98 | 58.02 |
## | 94.44 | 94.00 |
## ---------+--------+--------+
## Total 72 100 172
## 41.86 58.14 100.00
##
##
## Statistics for Table 2 of exposure by discolor
## Controlling for age=adult
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 0.0151 0.9022
## Likelihood Ratio Chi-Square 1 0.0152 0.9020
## Continuity Adj. Chi-Square 1 0.0000 1.0000
## Mantel-Haenszel Chi-Square 1 0.0150 0.9025
## Phi Coefficient -0.0094
## Contingency Coefficient 0.0094
## Cramer's V -0.0094
##
## WARNING: 25% of the cells have expected counts less
## than 5. Chi-Square may not be a valid test.
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 4
## Left-sided Pr <= F 0.5876
## Right-sided Pr >= F 0.6687
##
## Table Probability (P) 0.2564
## Two-sided Pr <= P 1.0000
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 0.9216 0.2504 3.3919
## Relative Risk (Column 1) 0.9529 0.4367 2.0796
## Relative Risk (Column 2) 1.0340 0.6131 1.7441
##
## Sample Size = 172
##
##
##
## Teeth Discoloration Example
##
## The FREQ Procedure
##
## Summary Statistics for exposure by discolor
## Controlling for age
##
## Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
##
## Statistic Alternative Hypothesis DF Value Prob
## ---------------------------------------------------------------
## 1 Nonzero Correlation 1 0.7531 0.3855
## 2 Row Mean Scores Differ 1 0.7531 0.3855
## 3 General Association 1 0.7531 0.3855
##
##
## Common Odds Ratio and Relative Risks
##
## Statistic Method Value 95% Confidence Limits
## -------------------------------------------------------------------------------------
## Odds Ratio Mantel-Haenszel 1.3893 0.6643 2.9056
## Logit 1.3909 0.6597 2.9327
##
## Relative Risk (Column 1) Mantel-Haenszel 1.1959 0.7877 1.8156
## Logit 1.2007 0.7901 1.8248
##
## Relative Risk (Column 2) Mantel-Haenszel 0.8606 0.6200 1.1944
## Logit 0.8684 0.6256 1.2054
##
##
## Breslow-Day Test for
## Homogeneity of the Odds Ratios
## ------------------------------
## Chi-Square 0.5749
## DF 1
## Pr > ChiSq 0.4483
##
##
## Total Sample Size = 256title 'Cervical Cancer Example';
data cervical;
input smoke $ cancer $ partners $ count;
cards;
yes yes zeorone 12
yes no zeorone 21
no yes zeorone 25
no no zeorone 118
yes yes twomore 96
yes no twomore 148
no yes twomore 92
no no twomore 150
;
run;
proc freq data = cervical order = data;
table partners*smoke*cancer / chisq relrisk cmh;
weight count;
run; ## Cervical Cancer Example
##
## The FREQ Procedure
##
## Table 1 of smoke by cancer
## Controlling for partners=zeorone
##
## smoke cancer
##
## Frequency|
## Percent |
## Row Pct |
## Col Pct |yes |no | Total
## ---------+--------+--------+
## yes | 12 | 21 | 33
## | 6.82 | 11.93 | 18.75
## | 36.36 | 63.64 |
## | 32.43 | 15.11 |
## ---------+--------+--------+
## no | 25 | 118 | 143
## | 14.20 | 67.05 | 81.25
## | 17.48 | 82.52 |
## | 67.57 | 84.89 |
## ---------+--------+--------+
## Total 37 139 176
## 21.02 78.98 100.00
##
##
## Statistics for Table 1 of smoke by cancer
## Controlling for partners=zeorone
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 5.7571 0.0164
## Likelihood Ratio Chi-Square 1 5.2087 0.0225
## Continuity Adj. Chi-Square 1 4.6760 0.0306
## Mantel-Haenszel Chi-Square 1 5.7244 0.0167
## Phi Coefficient 0.1809
## Contingency Coefficient 0.1780
## Cramer's V 0.1809
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 12
## Left-sided Pr <= F 0.9941
## Right-sided Pr >= F 0.0184
##
## Table Probability (P) 0.0126
## Two-sided Pr <= P 0.0304
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 2.6971 1.1758 6.1868
## Relative Risk (Column 1) 2.0800 1.1705 3.6961
## Relative Risk (Column 2) 0.7712 0.5895 1.0089
##
## Sample Size = 176
##
##
## Table 2 of smoke by cancer
## Controlling for partners=twomore
##
## smoke cancer
##
## Frequency|
## Percent |
## Row Pct |
## Col Pct |yes |no | Total
## ---------+--------+--------+
## yes | 96 | 148 | 244
## | 19.75 | 30.45 | 50.21
## | 39.34 | 60.66 |
## | 51.06 | 49.66 |
## ---------+--------+--------+
## no | 92 | 150 | 242
## | 18.93 | 30.86 | 49.79
## | 38.02 | 61.98 |
## | 48.94 | 50.34 |
## ---------+--------+--------+
## Total 188 298 486
## 38.68 61.32 100.00
##
##
## Statistics for Table 2 of smoke by cancer
## Controlling for partners=twomore
##
## Statistic DF Value Prob
## ------------------------------------------------------
## Chi-Square 1 0.0903 0.7638
## Likelihood Ratio Chi-Square 1 0.0903 0.7638
## Continuity Adj. Chi-Square 1 0.0430 0.8357
## Mantel-Haenszel Chi-Square 1 0.0901 0.7640
## Phi Coefficient 0.0136
## Contingency Coefficient 0.0136
## Cramer's V 0.0136
##
##
## Fisher's Exact Test
## ----------------------------------
## Cell (1,1) Frequency (F) 96
## Left-sided Pr <= F 0.6530
## Right-sided Pr >= F 0.4179
##
## Table Probability (P) 0.0709
## Two-sided Pr <= P 0.7805
##
##
## Odds Ratio and Relative Risks
##
## Statistic Value 95% Confidence Limits
## ------------------------------------------------------------------
## Odds Ratio 1.0576 0.7341 1.5237
## Relative Risk (Column 1) 1.0349 0.8273 1.2947
## Relative Risk (Column 2) 0.9786 0.8497 1.1270
##
## Sample Size = 486
##
##
##
## Cervical Cancer Example
##
## The FREQ Procedure
##
## Summary Statistics for smoke by cancer
## Controlling for partners
##
## Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
##
## Statistic Alternative Hypothesis DF Value Prob
## ---------------------------------------------------------------
## 1 Nonzero Correlation 1 1.3361 0.2477
## 2 Row Mean Scores Differ 1 1.3361 0.2477
## 3 General Association 1 1.3361 0.2477
##
##
## Common Odds Ratio and Relative Risks
##
## Statistic Method Value 95% Confidence Limits
## -------------------------------------------------------------------------------------
## Odds Ratio Mantel-Haenszel 1.2153 0.8697 1.6985
## Logit 1.2309 0.8812 1.7194
##
## Relative Risk (Column 1) Mantel-Haenszel 1.1312 0.9194 1.3918
## Logit 1.1346 0.9209 1.3979
##
## Relative Risk (Column 2) Mantel-Haenszel 0.9315 0.8221 1.0554
## Logit 0.9294 0.8202 1.0532
##
##
## Breslow-Day Test for
## Homogeneity of the Odds Ratios
## ------------------------------
## Chi-Square 4.2069
## DF 1
## Pr > ChiSq 0.0403
##
##
## Total Sample Size = 662