Chapter 19 Stratified 2-by-2 Tables

Chapter 19Stratified 2-by-2 Tables

In Chapter 19: • 19.1 Preventing Confounding • 19.2 Simpson’s Paradox • 19.3 Mantel-Haenszel Methods • 19.4 Interaction

§19.1 Confounding • Confounding is a systematic distortion in a measure of association due to the influence of “lurking” variables • Confounding occurs when the effects of an extraneous lurking factor get mixed with the effects of the explanatory variable (The word confounding means “to mix together” in Latin.) • When groups are unbalanced with respect to determinants of the outcome, comparisons will tend to be confounded.

Techniques that Mitigate Confounding • Randomization – see Ch 2; randomization of an exposure balances group with respect to potential confounders (especially effective in large samples) • Restriction – imposes uniformity in the study base; participants are made homogenous with respect to the potential confounder

Mitigating Confounding, cont. • Matching – balances confounders; require matched analyses techniques (e.g., §18.6) • Regression models – mathematically adjusts for confounding variables • Stratification – subdivides data into homogenous groups before pooling results

§19.2 Simpson’s Paradox Simpson’s paradox is a severe form of confounding in which there is a reversal in the direction of an association caused by the confounding variable

Simpson’s Paradox – Example Gender bias? Are male applicants more likely to get accepted into a particular graduate school? Data reveal: Male incidence of acceptance = 198/360 = 0.55 Female incidence of acceptance = 88/200 =0.44 RR = 0.55 / 0.44 = 1.25 (males 25% more likely to be accepted)

Simpson’s Paradox – Example • Consider the lurking variable "major applied to” • Business School (240 applicants) • Art School (320 applicants) • Perhaps males were more likely to apply to the major with the higher acceptance rate? • To evaluate this hypothesis, stratify the data according to the lurking variable as follows:

Stratify Stratified Data – Example

Stratified Data, cont. • Overall, men had the higher acceptance rate • Within each school, women had the higher acceptance rate • How do we reconcile this paradox? • The answer lies in the fact that men were more likely to apply to the art school, and the art school had much higher acceptance rate. • The lurking variable MAJOR confounded the observed relation between GENDER and ACCEPT

Stratified Analysis, cont. • By stratifying the data, we achieved like-to-like comparisons and mitigated confounding • We can then combine the strata-specific estimates to derive an summary measure of effect that shows the true relation between GENDER and ACCEPT

19.3 Mantel-Haenszel Methods The Mantel-Haenszel estimate is a summary measure of effect adjusted for confounding

This RR suggests that men were 10% less likely than women to be accepted to the Grad school. M-H Summary RR - Example

M-H RR = 0.90 (95% CI 0.78 - 1.04) X2stat = 1.84, df = 1, P = 0.175 Mantel-Haenszel Inference • CIs for M-H estimates are calculated by computer • Results are tested for significance with chi-square test statistic (H0: RR = 1) • See text for formulas

Other Mantel-Haenszel Statistics Mantel-Haenszel methods are available for other measures of effect, such as odds ratio, rate ratios, and risk difference. Mantel-Haenszel methods for ORs are described on pp. 471–3.

19.4 Interaction • Statistical interaction occurs when a statistical model does not adequately predict the joint effects of two or more explanatory factors • Statistical interaction = heterogeneity of the effect measures • Our example had strata-specific RRs of 0.75 and 0.94. Do these effect measures reflect the same underlying relationship, or is there heterogeneity? • We can test this question with a chi-square interaction statistic.

Test for Interaction • Hypotheses. H0: Strata-specific measures in population are homogeneous (no interaction) vs.Ha: Strata-specific measures are heterogeneous (interaction) • Test statistic. A chi-square interaction statistic is calculated by the computer program. (Several such statistics are used. WinPepi cites Rothman, 1986, Formula 12-59 and Fleiss, 1981, Formula 10.35) • P-value. Convert the chi-square statistic to a P-value; interpret.

Test for Interaction – Example Strata-specific RR estimates from the illustrative example are submitted to a test of interaction A. H0: RR1 = RR2 (no interaction) vs. H0: RR1≠RR2 (interaction) B. Hand calculation (next slide) shows chi-sq = 0.78 with 1 df. [WinPepi calculated 0.585 using a slightly different formula.] C. P = 0.38. The evidence against H0 is not significant. Retain H0 and assume no interaction.

Interaction Statistic – Hand Calculation Ad hoc interaction statistic presented in the text:

Example of Interaction Asbestos, Lung Cancer, Smoking Case-control data Smokers had an OR of lung cancer for asbestos of 60. Non-smokers had an OR of 2. Apparent heterogeneity in the effect measure (“interaction”).

Test for Interaction – Asbestos Example • H0:OR1= OR2versus Ha:OR1≠OR2 • Chi-square interaction = 21.38, 1 df Output from WinPepi > Compare2.exe > Program B: • P= 3.8 × 10−6Conclude “significant interaction.” When interaction is present, avoid the summary adjustments because this would obscure the interaction.

Chapter 19 Stratified 2-by-2 Tables