The example calculation of the Chi-square test of homogeneity statistic in the Aschengrau and Seage textbook on page 356-357 has a typo. The equation for the pooled log odds ratio uses an incorrect weight of 62.15 instead of 78.9 for the never-breastfed strata, though the actual calculated log odds ratio of 0.32 is correct. Below, the calculations are corrected, with more detail on the equations used in each step of the calculation.
Here is the 2x2 table, unstratified by womenโs breastfeeding history:
| DDE levels | Breast cancer cases | Breast cancer controls |
|---|---|---|
| High | 500 | 600 |
| Low | 2000 | 4000 |
| Total | 2500 | 4600 |
Naive odds ratio:
\[ \frac{ad}{bc} = \frac{500 \times 4000}{600 \times 2000} = 1.7 \]
Breastfed
| DDE levels | Breast cancer cases | Breast cancer controls |
|---|---|---|
| High | 140 | 300 |
| Low | 550 | 2600 |
| Total | 690 | 2900 |
Stratum-specific odds ratio:
\[ \frac{ad}{bc} = \frac{140 \times 2600}{300 \times 550} = 2.2 \]
Strata specific weight:
\[ w_i = [\frac{1}{a_i}+\frac{1}{b_i}+\frac{1}{c_i}+\frac{1}{d_i}]^{-1} \]
\[ w_1 = [\frac{1}{140}+\frac{1}{300}+\frac{1}{550}+\frac{1}{2600}]^{-1} = 78.9 \]
Not breastfed
| DDE levels | Breast cancer cases | Breast cancer controls |
|---|---|---|
| High | 360 | 300 |
| Low | 950 | 800 |
| Total | 1310 | 1100 |
Stratum-specific odds ratio:
\[ \frac{ad}{bc} = \frac{360 \times 800}{300 \times 950} = 1.0 \]
Strata specific weight:
\[ w_2 = [\frac{1}{360}+\frac{1}{800}+\frac{1}{300}+\frac{1}{950}]^{-1} = 118.9 \]
\[ lnOR_{weighted} = \frac{\sum^{i=1,...,k}(W_i \times lnOR_i)}{\sum^{i=1,...,k}W_i} \]
\[ lnOR_{weighted} = \frac{w_1 \times lnOR_1 + w_2 \times lnOR_2}{w_1 + w_2} \]
\[ lnOR_{weighted} = \frac{78.9 \times ln(2.2) + 118.9 \times ln(1)}{78.9 + 118.9} = \frac{78.9 \times 0.788 + 118.9 \times 0}{78.9 + 118.9} = 0.32 \]
\[ \chi^2_{HOM}=\sum^{i=1,...,k}W_i(lnOR_i-lnOR_{weighted})^2 \]
\[ \chi^2_{HOM}=78.9(0.791-0.32)^2 + 118.9(0.01-0.32)^2 = 28.94 \] Referring to a chi-square table, one can see that the corresponding P-value is less than 0.001, so we can reject the null hypothesis that the odds ratios are the same across strata of breastfeeding. Mothers with high DDE exposure who never breastfed had 2.2 times the odds of developing breast cancer compared to mothers with low DDE exposure who never breastfed, but there was no difference in the odds of breast cancer among those who breastfed, regardless of DDE exposure (OR=1).