# BreslowDayTest: Breslow-Day Test for Homogeneity of the Odds Ratios In DescTools: Tools for Descriptive Statistics

## Description

Calculates the Breslow-Day test of homogeneity for a 2 x 2 x k table, in order to investigate if all k strata have the same OR. If OR is not given, the Mantel-Haenszel estimate is used.

## Usage

 `1` ```BreslowDayTest(x, OR = NA, correct = FALSE) ```

## Arguments

 `x` a 2 x 2 x k table. `OR` the odds ratio to be tested against. If left undefined (default) the Mantel-Haenszel estimate will be used. `correct` If TRUE, the Breslow-Day test with Tarone's adjustment is computed, which subtracts an adjustment factor to make the resulting statistic asymptotically chi-square.

## Details

For the Breslow-Day test to be valid, the sample size should be relatively large in each stratum, and at least 80% of the expected cell counts should be greater than 5. Note that this is a stricter sample size requirement than the requirement for the Cochran-Mantel-Haenszel test for tables, in that each stratum sample size (not just the overall sample size) must be relatively large. Even when the Breslow-Day test is valid, it might not be very powerful against certain alternatives, as discussed in Breslow and Day (1980).

Alternatively, it might be better to cast the entire inference problem into the setting of a logistic regression model. Here, the underlying question of the Breslow-Day test can be answered by investigating whether an interaction term with the strata variable is necessary (e.g. using a likelihood ratio test using the `anova` function).

## Author(s)

Michael Hoehle <[email protected]>

## References

source: Bug fixed version of https://onlinecourses.science.psu.edu/stat504/sites/onlinecourses.science.psu.edu.stat504/files/lesson04/breslowday.test_.R

Breslow, N. E., N. E. Day (1980) The Analysis of Case-Control Studies Statistical Methods in Cancer Research: Vol. 1. Lyon, France, IARC Scientific Publications.

Tarone, R.E. (1985) On heterogeneity tests based on efficient scores, Biometrika, 72, pp. 91-95.

Jones, M. P., O'Gorman, T. W., Lemka, J. H., and Woolson, R. F. (1989) A Monte Carlo Investigation of Homogeneity Tests of the Odds Ratio Under Various Sample Size Configurations Biometrics, 45, 171-181

Breslow, N. E. (1996) Statistics in Epidemiology: The Case-Control Study Journal of the American Statistical Association, 91, 14-26.

`WoolfTest`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```migraine <- xtabs(freq ~ ., cbind(expand.grid(treatment=c("active", "placebo"), response =c("better", "same"), gender =c("female", "male")), freq=c(16, 5, 11, 20, 12, 7, 16, 19)) ) # get rid of gender tab <- xtabs(Freq ~ treatment + response, migraine) Desc(tab) # only the women female <- migraine[,, 1] Desc(female) # .. and the men male <- migraine[,, 2] Desc(male) BreslowDayTest(migraine) BreslowDayTest(migraine, correct = TRUE) salary <- array( c(38, 12, 102, 141, 12, 9, 136, 383), dim=c(2, 2, 2), dimnames=list(exposure=c("exposed", "not"), disease =c("case", "control"), salary =c("<1000", ">=1000")) ) # common odds ratio = 4.028269 BreslowDayTest(salary, OR = 4.02) ```

### Example output

```------------------------------------------------------------------------------
tab (xtabs, table)

Summary:
n: 106, rows: 2, columns: 2

Pearson's Chi-squared test (cont. adj):
X-squared = 7.3178, df = 1, p-value = 0.006827
Fisher's exact test p-value = 0.00491
McNemar's chi-squared = 5.0256, df = 1, p-value = 0.02497

estimate lwr.ci upr.ci'

odds ratio             3.370  1.462  7.772
rel. risk (col1)       2.164  1.237  3.783
rel. risk (col2)       0.642  0.471  0.875

Phi-Coefficient        0.282
Contingency Coeff.     0.272
Cramer's V             0.282

response
better   same    Sum
treatment

active    freq       28     27     55
perc    26.4%  25.5%  51.9%
p.row   50.9%  49.1%      .
p.col   70.0%  40.9%      .

placebo   freq       12     39     51
perc    11.3%  36.8%  48.1%
p.row   23.5%  76.5%      .
p.col   30.0%  59.1%      .

Sum       freq       40     66    106
perc    37.7%  62.3% 100.0%
p.row       .      .      .
p.col       .      .      .

----------
' 95% conf. level

------------------------------------------------------------------------------
female (table)

Summary:
n: 52, rows: 2, columns: 2

Pearson's Chi-squared test (cont. adj):
X-squared = 6.7595, df = 1, p-value = 0.009325
Fisher's exact test p-value = 0.005189
McNemar's chi-squared = 1.5625, df = 1, p-value = 0.2113

estimate lwr.ci upr.ci'

odds ratio             5.818  1.676 20.203
rel. risk (col1)       2.963  1.274  6.891
rel. risk (col2)       0.509  0.310  0.836

Phi-Coefficient        0.400
Contingency Coeff.     0.371
Cramer's V             0.400

response
better   same    Sum
treatment

active    freq       16     11     27
perc    30.8%  21.2%  51.9%
p.row   59.3%  40.7%      .
p.col   76.2%  35.5%      .

placebo   freq        5     20     25
perc     9.6%  38.5%  48.1%
p.row   20.0%  80.0%      .
p.col   23.8%  64.5%      .

Sum       freq       21     31     52
perc    40.4%  59.6% 100.0%
p.row       .      .      .
p.col       .      .      .

----------
' 95% conf. level

------------------------------------------------------------------------------
male (table)

Summary:
n: 54, rows: 2, columns: 2

Pearson's Chi-squared test (cont. adj):
X-squared = 0.88353, df = 1, p-value = 0.3472
Fisher's exact test p-value = 0.2635
McNemar's chi-squared = 2.7826, df = 1, p-value = 0.09529

estimate lwr.ci upr.ci'

odds ratio             2.036  0.648  6.398
rel. risk (col1)       1.592  0.741  3.418
rel. risk (col2)       0.782  0.526  1.163

Phi-Coefficient        0.167
Contingency Coeff.     0.164
Cramer's V             0.167

response
better   same    Sum
treatment

active    freq       12     16     28
perc    22.2%  29.6%  51.9%
p.row   42.9%  57.1%      .
p.col   63.2%  45.7%      .

placebo   freq        7     19     26
perc    13.0%  35.2%  48.1%
p.row   26.9%  73.1%      .
p.col   36.8%  54.3%      .

Sum       freq       19     35     54
perc    35.2%  64.8% 100.0%
p.row       .      .      .
p.col       .      .      .

----------
' 95% conf. level

Breslow-Day test on Homogeneity of Odds Ratios

data:  migraine
X-squared = 1.4929, df = 1, p-value = 0.2218

Breslow-Day Test on Homogeneity of Odds Ratios (with Tarone
correction)

data:  migraine
X-squared = 1.4905, df = 1, p-value = 0.2221

Breslow-Day test on Homogeneity of Odds Ratios

data:  salary
X-squared = 0.080143, df = 1, p-value = 0.7771
```

DescTools documentation built on Dec. 11, 2017, 5:10 p.m.