Key

(1) GENERIC Data Sets

GENERIC1.REC. This data set shows clear evidence of statistic interaction. OR₁= 60.0, OR₂= 2.0 and c²_int(1, N = 285) = 17.93, p = .000023
GENERIC2.REC. This data set has strata-specific odds ratios of OR₁= 6.0 and OR₂= 4.2 (c²_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 OR₁=1.2 and OR₂= 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 (H₀: OR₁ = OR₂ = . . . = OR₁₀ vs. H₁: at least on age-specific odds ratio differs). The c²_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 (p₁) = 312 / 844 = 37% Helmet use rate in Contra Costa county (p₂) = 335 / 807 = 42%.
In testing H₀: p₁ = p₂, 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
H₀: RR₁ = RR₂ = RR₃ = RR₄ = RR₅ vs. H₁: 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 H₀. 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) OR₁ = 2.7; OR₂ = 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:
H₀: OR₁ = OR₂ vs. H₁: OR₁ not = OR₂
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.