Post on 24-Dec-2015
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 …