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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.