Breslow-Day Test for Homogeneity of Odds Ratios

Tests for homogeneity of odds ratios across strata in $2 \times 2 \times k$ tables (i.e., whether a common odds ratio fits all strata). Generalized to handle tables of any dimensionality beyond 3. For 4-dimensional tables, optionally provides a two-way decomposition of the homogeneity test into row effects, column effects, and residual, analogous to woolf_test() with decompose = TRUE.

Usage

breslow_day_test(x, OR = NA, correct = FALSE, decompose = FALSE)

# S3 method for class 'breslow_day_test'
print(x, digits = 4, ...)

Arguments

x: An object of class "breslow_day_test"
OR: The common odds ratio to test against. If NA (the default), the Mantel-Haenszel estimate is used.
correct: Logical. If TRUE, the Tarone (1985) correction is applied. Defaults to FALSE.
decompose: Logical. If TRUE and x is 4-dimensional (a $2 \times 2 \times R \times C$ table), the test is decomposed into row effects, column effects, and residual. Defaults to FALSE. Ignored for non-4-dimensional tables.
digits: Number of significant digits for the common OR. Default 4.
...: Additional arguments (currently unused).

Value

A list of class "breslow_day_test" (also inheriting from "htest") containing:

statistic: the chi-squared test statistic.
parameter: degrees of freedom.
p.value: $p$-value.
method: character string describing the test.
data.name: character string giving the name of the data.
or_vars: names of the first two dimensions (the 2x2 table variables).
strata_vars: names of the stratifying variables (dimensions 3 and beyond).
OR: the common odds ratio used (MH estimate if OR = NA was supplied).
correct: logical indicating whether Tarone correction was applied.
observed: observed $a_j$ counts (cell $[1,1]$ of each stratum).
expected: expected $\tilde{a}_j$ counts under the common OR.
decomposed: logical indicating if decomposition was performed.

When decompose = TRUE (only for 4-dimensional tables), additional components:

rows: list with statistic, df, p.value for row effects.
cols: list with statistic, df, p.value for column effects.
residual: list with statistic, df, p.value for residual (interaction).

Details

The Breslow-Day test (Breslow & Day, 1980) tests the hypothesis that a common odds ratio $\psi$ fits all $k$ strata. Given a common OR (by default the Mantel-Haenszel estimate), the expected cell count $\tilde{a}_j$ in cell $[1,1]$ of stratum $j$ is found by solving the quadratic:

$$(\psi - 1)\tilde{a}_j^2 - [n_{2j} - m_{1j} + \psi(n_{1j} + m_{1j})]\tilde{a}_j + \psi \, m_{1j} n_{1j} = 0$$

where $m_{1j}$ and $m_{2j}$ are the row margins and $n_{1j}$ and $n_{2j}$ are the column margins of stratum $j$. The test statistic is:

$$\chi^2_{BD} = \sum_{j=1}^{k} \frac{(a_j - \tilde{a}_j)^2}{\widehat{\text{Var}}(a_j)}$$

where $\widehat{\text{Var}}(a_j) = (1/\tilde{a}_j + 1/\tilde{b}_j + 1/\tilde{c}_j + 1/\tilde{d}_j)^{-1}$ and $\tilde{b}_j, \tilde{c}_j, \tilde{d}_j$ are the remaining expected cell counts. Under the null hypothesis, $\chi^2_{BD}$ follows a chi-squared distribution with $k - 1$ degrees of freedom.

The Tarone (1985) correction subtracts a term to account for estimation of the common OR:

$$\chi^2_{BD,T} = \chi^2_{BD} - \frac{(\sum_j a_j - \sum_j \tilde{a}_j)^2}{\sum_j \widehat{\text{Var}}(a_j)}$$

Comparison with the Woolf test: The Woolf test uses log odds ratios and tests deviation from their weighted mean, whereas the Breslow-Day test works on the cell-count scale against a specified common OR. For large samples they agree closely; Breslow-Day is generally preferred when the Mantel-Haenszel common OR is the quantity of interest.

Decomposition for 4-way tables: For a $2 \times 2 \times R \times C$ table, when decompose = TRUE, the overall test is decomposed as:

$$\chi^2_{\text{Total}} = \chi^2_{\text{Rows}} + \chi^2_{\text{Cols}} + \chi^2_{\text{Residual}}$$

where the row and column components use the row- and column-marginal (pooled) tables, all tested against the same common OR as the overall test. The residual is defined by subtraction and has $(R-1)(C-1)$ degrees of freedom.

References

Breslow, N. E. & Day, N. E. (1980). Statistical Methods in Cancer Research. Vol. 1: The Analysis of Case-Control Studies. IARC Scientific Publications No. 32. Lyon: International Agency for Research on Cancer.

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

Lachin, J. M. (2000). Biostatistical Methods: The Assessment of Relative Risks. Wiley, p. 124-125.

Author

Andri Signorell, Michael Friendly (enhancements)

Examples

# 3-way table
data(CoalMiners, package = "vcd")
breslow_day_test(CoalMiners)
#> 
#> Breslow-Day Test on Homogeneity of Odds Ratios 
#> 
#> Data:          CoalMiners 
#> OR variables:  Breathlessness, Wheeze 
#> Strata:        Age 
#> Common OR:     16.33 
#> 
#> X-squared = 26.9457, df = 8, p-value = 0.0007224
breslow_day_test(CoalMiners, correct = TRUE)   # Tarone correction
#> 
#> Breslow-Day Test on Homogeneity of Odds Ratios (with Tarone correction) 
#> 
#> Data:          CoalMiners 
#> OR variables:  Breathlessness, Wheeze 
#> Strata:        Age 
#> Common OR:     16.33 
#> 
#> X-squared = 26.3787, df = 8, p-value = 0.0009045

# Compare with Woolf test
woolf_test(CoalMiners)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 3-way association) 
#> 
#> Data:          CoalMiners 
#> OR variables:  Breathlessness, Wheeze 
#> Strata:        Age 
#> 
#> X-squared = 26.2034, df = 8, p-value = 0.0009694

data(Heart, package = "vcdExtra")
breslow_day_test(Heart)
#> 
#> Breslow-Day Test on Homogeneity of Odds Ratios 
#> 
#> Data:          Heart 
#> OR variables:  Disease, Gender 
#> Strata:        Occup 
#> Common OR:     4.622 
#> 
#> X-squared = 98.4871, df = 2, p-value = 0

# 4-way table without decomposition
data(Fungicide, package = "vcdExtra")
breslow_day_test(Fungicide)
#> 
#> Breslow-Day Test on Homogeneity of Odds Ratios 
#> 
#> Data:          Fungicide 
#> OR variables:  group, outcome 
#> Strata:        sex, strain 
#> Common OR:     0.3248 
#> 
#> X-squared = 0.8659, df = 3, p-value = 0.8337

# 4-way table with decomposition
breslow_day_test(Fungicide, decompose = TRUE)
#> 
#> Breslow-Day Test on Homogeneity of Odds Ratios 
#> 
#> Data:          Fungicide 
#> OR variables:  group, outcome 
#> Strata:        sex, strain 
#> Common OR:     0.3248 
#> 
#> Overall homogeneity test:
#>   X-squared = 0.8659, df = 3, p-value = 0.8337
#> 
#> Decomposition:
#>   Rows (sex):    X-squared = 0.0086, df = 1, p-value = 0.9261
#>   Cols (strain): X-squared = 0.8336, df = 1, p-value = 0.3612
#>   Residual:      X-squared = 0.0236, df = 1, p-value = 0.8779
#> 
#> Note: Overall = Rows + Columns + Residual