Apr 21, 2023

Chapter 19Chapter 19Stratified 2-by-2 TablesStratified 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 – ExampleGender bias? Are male applicants more likely to get accepted into a particular graduate school? Data reveal:

Accepted Rejected TotalMale 198 162 360Female 88 112 200Total 286 274 560

Male incidence of acceptance = 198/360 = 0.55

RR = 0.55 / 0.44 = 1.25 (males 25% more likely to be accepted)

Female incidence of acceptance = 88/200 =0.44

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:

Stratified Data – Example

Business School ApplicantsSuccess Failure Total

Male 18 102 120Female 24 96 120

Total 42 198 240

p^male = 18 / 120 = 0.15

p^female = 24 / 120 = 0.20

All ApplicantsAccepted Rejected Total

Male 198 162 360Female 88 112 200Total 286 274 560

Art School ApplicantsSuccess Failure Total

Male 180 60 240Female 64 16 80

Total 244 76 320

p^male = 180 / 240 = 0.75

p^female = 64 / 80 = 0.80

Stratify

• 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 Data, cont.

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

k

kk

k

kk

n

nan

na

RR12

21

H-Mˆ

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

19.3 Mantel-Haenszel Methods

M-H Summary RR - ExampleBusiness School (Stratum 1)

Success Failure TotalMale 18 102 120

Female 24 96 120Total 42 198 240

RR^1 = (18 / 120) / (24 / 120) = 0.75

Art School (Stratum 2)Success Failure Total

Male 180 60 240Female 64 16 80

Total 244 76 320

RR^2= (180 / 240) / (64 / 80) = 0.94

90.0

320

24064

240

12024320

80180

240

12018

ˆ12

21

H-M

k

kk

k

kk

n

nan

na

RR

This RR suggests that men were 10% less likely than women to be accepted to the Grad school.

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

(95% CI 0.78 - 1.04)

M-H RR = 0.90

X2stat = 1.84, df = 1, P = 0.175

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 InteractionA. Hypotheses.

H0: Strata-specific measures in population are homogeneous (no interaction) vs.Ha: Strata-specific measures are heterogeneous (interaction)

B. 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)

C. P-value. Convert the chi-square statistic to a P-value; interpret.

Test for Interaction – Example

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.

Strata-specific RR estimates from the illustrative example are submitted to a test of interaction

Interaction Statistic – Hand Calculation

Ad hoc interaction statistic presented in the text:

strata) of no. ( 1

ˆlnˆln

2ˆln

22int

KKdf

SE

RRRR

kRR

MHk

Example of Interaction Asbestos, Lung Cancer, Smoking

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”).

Case-control data

Test for Interaction – Asbestos Example

A. H0:OR1 = OR2 versus Ha:OR1 ≠ OR2 B. Chi-square interaction = 21.38, 1 df Output

from WinPepi > Compare2.exe > Program B:

C. P = 3.8 × 10−6 Conclude “significant interaction.”

When interaction is present, avoid the summary adjustments because this would obscure the interaction.