Biostat Didactic Seminar SeriesBiostat Didactic Seminar Series

Analyzing Binary Outcomes:  Analyzing Binary Outcomes: 

An Introduction to Logistic An Introduction to Logistic RegressionRegression

Robert Boudreau, PhDRobert Boudreau, PhD

Co-Director of Methodology CoreCo-Director of Methodology Core

PITT-Multidisciplinary Clinical Research Center PITT-Multidisciplinary Clinical Research Center

for Rheumatic and Musculoskeletal Diseasesfor Rheumatic and Musculoskeletal Diseases

Core Director for BiostatisticsCore Director for Biostatistics

Center for Aging and Population Health Center for Aging and Population Health

Dept. of Epidemiology, GSPH Dept. of Epidemiology, GSPH 10/8/201010/8/2010

Flow chart for group Flow chart for group comparisonscomparisons

Measurements to be compared

continuous

Distribution approx normal or N ≥ 20?

No Yes

Non-parametrics T-tests

discrete

( binary, nominal, ordinal with few values)

Chi-squareFisher’s Exact

Flow chart for regression Flow chart for regression modelsmodels

(includes adjusted group comparisons)(includes adjusted group comparisons)Outcome variable continuous or dichotomous?

Dichotomous (binary)continuous

Time-to-event available (or relevant)?

No Yes

Multiple logistic regression

Cox proportionalhazards regression

Predictor variable categorical?

No Yes(e.g. groups)

Multiple linear regression

ANCOVA(Multiple linear regression -using dummyvariable(s) forcategorical var(s)

Analysis From Last Analysis From Last Didactic …Didactic …

In Health, Aging and Body Composition Knee-OA Substudy:In Health, Aging and Body Composition Knee-OA Substudy:

Examine Association between SxRxKOA (knee OA) and CRP Examine Association between SxRxKOA (knee OA) and CRP adjusted for BMI.adjusted for BMI.

Motivation:Motivation: Sowers M, Hochberg M et. al. C-reactive protein as a biomarker

of emergent osteoarthritis. Osteoarthritis and CartilageVolume 10, Issue 8, August 2002, Pages 595-601

Conclusion: “CRP is highly associated with Knee OA; however, its high correlation with obesity limits its utility as an exclusive marker for knee OA”

Logistic RegressionOutline for today Definition and interpretation of odds-ratio for binary

outcome Essential equivalence of odds-ratio ↔ testing for

group differences in rates (or percentages) when evaluated using 2 x 2 table, chi-square and p-values

Logistic regression as “binary outcome” version of multiple linear regression: group (and covariate adjustment) effects are interpreted as odds-ratios affecting the binary outcome

Detailed example: relating obesity to odds of knee OA

- adjusted for race and gender

HABC: Obese x KneeOA

Obese:BMI > 30

Chi-squareP < 0.0001

Obese=1: Odds of kneeOA = p/(1-p)=0.2444/0.7556 = 0.32345Obese=0: Odds of kneeOA = p/(1-p)=0.0911/0.9089 = 0.10023Obesity odds-ratio for kneeOA OR = 0.32345/0.10023=3.225

HABC: Obese x KneeOA

proc logistic data=worst_knee_vs_noOA;

model kneeOA(event="1")=obese;

run;

Note OR and C.I.

ConfidenceInterval (C.I.)(2.56,4.04)

doesn’t cover 1.0 => stat signif.

HABC: Obese x KneeOA

Prob[kneeOA│obese=0]= exp(-2.3)/(1+exp(-2.3) = 0.0911

Prob[kneeOA│obese=0]= exp(-2.3+1.17)/(1+exp(-2.3+1.17) = 0.2444

HABC: Obese x KneeOA

Obese:BMI > 30

Chi-squareP < 0.0001

Prob[kneeOA│obese=0]= exp(-2.3)/(1+exp(-2.3) = 0.0911

Prob[kneeOA│obese=0]= exp(-2.3+1.17)/(1+exp(-2.3+1.17) = 0.2444General logistic regression form:Prob[kneeOA│obese] = exp(int+obese)/(1+exp(int+obese)

Gender x PAD

Gender x PAD(referent=female)

proc logistic data=pad;

model y1ppad(event=“1”)=male;

run;

Gender x PAD(ref=male)

proc logistic data=pad;

model y1ppad(event=“1”)=female;

run;

Gender x PAD(compare models: ref=female vs

ref=male)

(vs females)Male OR= 1.891

(vs males)Female OR= 0.529 = 1/1.891

CHD x KneeOACHD x KneeOACHD Knee OAassociation notstatisticallysignificant

C.I.=(0.79,1.34)

Self-reported rheumatoid Self-reported rheumatoid arthritis as binary outcome arthritis as binary outcome (or covariate) for analyses ?(or covariate) for analyses ?

(NOT ?#!)(NOT ?#!)

White Females: Obesity x KneeOA

White vs Black FemalesObesity x KneeOA: Similar

OR’s

WhiteFemales

BlackFemales

Black females have about two times higher rates of

kneeOA than white women

proc logistic data=worst_knee_vs_noOA; model kneeOA(event="1")= black ; where female;run;

Obesity odds-ratio is same for white and black women

(interaction term is NS)proc logistic data=worst_knee_vs_noOA;

model kneeOA(event="1")=obese black

obese_x_black;

where female;

run;

Non-obese black women have OR=1.53 higher rates of knee OA, but obesity is associated with increased OR=3.61 for knee OA that applies within each race

Obesity explains some, but not all of the difference in rates of knee OA between black

and white females

(Note the “black race” OR attenuation from 2.08 to 1.53

after “adjusting” for obesity)

model kneeOA= black

model kneeOA= black obese

White Females: White Females: Continuous CRP

Difference in average logCRP: 0.76 – 0.43 = 0.33

Knee OA

P-value

No (n=752) Yes (n=92)

Mean (SD) Mean (SD)

Equal vars Unequal

logCRP 0.43 (0.83) 0.76 (0.58) 0.0002 < 0.0001

logCRP SD’s were signif diff (p<0.0001) => Use Satterthwaite unequal variance test

All White Females in HABC (N=844)[includes SxRxKOA (n=93); also rest of parent study cohort]

N=5N=5 had CRP > 30 (max=63.2)

log CRP

White Females Continuous CRP as predictor

of kneeOA

Standardized var: mean-centered, divided by SD

logCRP_perSD= (logCRP-0.4728)/0.8625

Units of standardized logCRP is SD’s

White Females: Per SD higher logCRP,

rates of knee OA increase by OR=1.5

proc logistic data=worst_knee_vs_noOA3;

model kneeOA(event="1")=logCRP_perSD ;

where female and white;

run;

Thank youThank you

Questions, comments, suggestions or insights?Questions, comments, suggestions or insights?

Remaining time: Open consultation …Remaining time: Open consultation …