Description Usage Arguments Details Author(s) References See Also Examples

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.

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

`x` |
a |

`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. |

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).

Michael Hoehle <[email protected]>

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.

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)
``` |

```
------------------------------------------------------------------------------
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
```

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