**(1) GENERIC Data Sets**

**GENERIC1.REC**. This data set shows clear evidence of statistic interaction.*OR*_{1 }= 60.0,*OR*_{2 }= 2.0 and c^{2}_{int}(1,*N*= 285) = 17.93,*p*= .000023**GENERIC2.REC**. This data set has strata-specific odds ratios of*OR*_{1 }= 6.0 and*OR*_{2 }= 4.2 (c^{2}_{int}(1,*N*= 200) = 0.07,*p*= .78. I would therefore conclude "no significant interaction." The M-H summary*aOR*= 5.5 and*cOR*= 4.8. Since the crude odds ratio and adjusted odds ratio are similar (both suggest a strong positive association between E and D -- an approximate 5-fold elevation in risk when exposed), the potential for confounding is small and I would report the crude odds ratio.**GENERIC3.REC**. Here, the crude odds ratio = 4.9, and strata-specific odds ratios are*OR*_{1 }=1.2 and*OR*_{2 }= 1.5. The chi-square interaction statistic derives*p*= .84. The M-H summary odds ratio (*aOR*) = 1.3. The adjusted odds ratio suggests little or no association between E and D, contradicting the crude analysis. This suggests a serious potential for confounding.

**(2) BD2.REC**. There are 10 single-year age-strata (ages 0 through 9). The question is whether odds ratios are heterogeneous (*H*_{0}: *OR*_{1}
= *OR*_{2} = . . . = *OR*_{10} vs. *H*_{1}: at least on age-specific odds ratio differs). The c^{2}_{int} statistic has 9 *df* and derives *p* = .25. We therefore
proceed under the assumption of no interaction. The *aOR* = *cOR* = 1.65. Therefore, the crude odds ratio seems to provide the "best"
estimate given available information: *cOR* = 1.65 (95% confidence interval: 1.48, 1.85). This suggests is between a 48% and a 85%
increase in leukemia and lymphoma risk with *in utero *X-rays exposure.

**(3) BI-HELM1.REC**. Incidence of bicycle helmet use.

- (A) Crude analysis

Helmet use rate in Santa Clara county (*p*_{1}) = 312 / 844 = 37%

Helmet use rate in Contra Costa county (*p*_{2}) = 335 / 807 = 42%. - In testing
*H*_{0}:*p*_{1}=*p*_{2},*p*= .059 by the uncorrected chi-square test

(B) Stratified by School Area

Statum | MATCHVAR: S. C. / C. C. |
Santa Clara Use Rate | Contra Costa Use Rate | Incidence Ratio | p value (uncorrected
chi-square) |

1 | 3: Miner / Fair Oaks | 21% | 22% | 0.97 | .91 (NS) |

2 | 4: Sedg / Strandwood | 55% | 36% | 1.54 | .0014 |

3 | 5: Sakamoto / WalAcres | 33% | 58% | 0.58 | .00000065 |

4 | 6: Toyon / Disco Bay | 32% | 22% | 1.46 | .048 |

5 | 7: Lietz / Belshaw | 38% | 42% | 0.91 | .46 (NS) |

(C) Test for Interaction

*H*_{0}: *RR*_{1} = *RR*_{2} = *RR*_{3} = *RR*_{4} = *RR*_{5} vs. *H*_{1}: At least one stratum-specific relative risk parameter differs ("interaction")

Let alpha = .05

Chi-square / interaction (4, *N *= 1651) = 32.69, *p* = .0000014

Conclusion: Reject *H*_{0}. Significant interaction is confirmed.

(D) Summary

The crude (unstratified) comparison was confounded. Interaction was also present. There was no significant difference in two of the
five strata (strata 3 and strata 7). In two of the strata, Santa Clara schools showed better helmet-use rates. In one stratum, the Contra
Costa district fared better.

(4) **CERVICAL**

(A) *cOR* ~= 1.5

(B) *OR*_{1} = 2.7; *OR*_{2} = 1.1

(C) Data suggest an interaction between smoking and number of sexual partners. In women with no or one partner, smoking is
positively associated with cervical cancer. In women with multiple sexual partners, this association seems small or non-existent. We
might also view the crude odds ratio as confounded, since it fails to reflect these complex relationships. Recommendation: report
strata-specific odds ratios. (Comment: This interpretation differs from that in the text by Pagano & Gauvreau.)

**(5) ASBESTOS.REC**

(A) Smoking & lung cancer: *OR* = 4.8 (Cornfield 95% confidence limits for *OR*: 2.5, 9.5). There is a significant association between
smoking and lung cancer, with smokers having approximately 5-times the risk as non-smokers.

(B) Asbestos & lung cancer: *OR* = 21.3 (Cornfield 95% confidence limits for *OR*: 10.5, 43.9). There is an exceedingly strong
association between asbestos and lung cancer, with exposed individuals having approximatley 21-times the risk as non-exposed individuals.

(C) Asbestos & lung cancer controlling for smoking (stratified analysis):

Strata-specific odds ratios:

*OR*_{smokers} = 60.0 (95% confidence limits, mid-P exact method: 21.9, 181.1)

*OR*_{non-smokers} = 2.0 (95% confidence limits, mid-P exact method: 0.6, 6.6)

Test for interaction:

*H*_{0}: *OR*_{1} = *OR*_{2} vs. *H*_{1}: *OR*_{1} not = *OR*_{2}

Let alpha = .05

Chi-square, interaction(1, *N* = 285) = 17.93, *p* = .000023

Reject the null hypothesis

Conclude: significant interaction

Recommendation: report strata-specific odds ratios.

Confounder analysis:

*cOR* = 21.3, *aOR* = 16.2; also see strata-specific results, above. Confounding is present, since the crude odds ratio fails to accurately
reflect the relationship between asbestos exposure and lung cancer risk.

Discussion of results:

- Strata-specific results should be reported and interpreted.

- It should be made clear that there is a very high, significant risk between asbestos exposure and lung cancer in smokers (60-times
the risk in asbestos-exposed smokers).

- It should be made clear that there is an insignificant, weak, positive association between asbestos exposure and lung cancer in non-smokers.

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