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Evaluation of Logistic and Cox Regression Models Using Simulated Survival Data

and Clinical Practice Research DatalinkChenyi Pan1, Jessica Kim, Ph.D.2, Clara Kim, Ph.D.2, and Esther Zhou, M.D., Ph.D.3

1. ORISE Fellow, University of Virginia; 2. Division of Biometrics VII/Office of Biostatistics; 3. Division of Epidemiology II/Office of Surveillance and Epidemiology

Background

Simulation Specifications

Simulation Results

Simulation Summary

Summary

• Safety analysis

– Acute outcome

– Short exposure/follow-up time

• Model selection

– Logistic versus Cox

• Compare logistic and Cox model

– Relative Bias (R1): 100* 𝜷−𝜷

𝜷

– Relative Ratio (R2): 100*𝑯𝑹−𝑶𝑹

𝑶𝑹

• Event time: Cox-Weibull

distribution

𝑻 = (−𝒍𝒐𝒈(𝑼)

𝝀𝒆𝒙𝒑(𝜷𝑿))𝟏/𝜸,

where

U : Random variable~Uniform[0,1]

𝜷 : Coefficients

𝛌 : Scale parameter 𝛌 > 𝟎

𝜸 : Shape parameter 𝜸 > 𝟎

• Treatment indicator:

𝑿~𝒃𝒆𝒓𝒏𝒐𝒖𝒍𝒍𝒊(0.5)

• Censoring times: 𝑪~Weibull(𝝀𝒄, 𝛄)

• Follow-up time: t = min(T, C)

• If T > C

– Cox model: Subject censored

– Logistic model: Event not occurred

• Simulation sample size: 1000

• Angiotensin Receptor Blocker

(ARB)

– Indication: Hypertension

• Cardiovascular risks in diabetics

• Cohort study using UK Clinical

Practice Research Datalink

• Sample Size

– Total cohort: n = 58,617

– High-dose subgroup: n = 7,460

• Outcome: Major Adverse Cardiac

Events

– Composite of cardiovascular

death, non-fatal myocardial

infarction, and stroke

• Olmesartan vs. Other ARBs

• Compared logistic vs. Cox

– Various lengths of follow-up

time

– Total cohort

– High-dose subgroup

AnalysisCox Logistic

R2

Hazard Ratio

(95% CI)

Odds Ratio

(95% CI)

Total

Cohort

0.77

(0.60, 1.00)

0.72

(0.56, 0.92) 8.06

High-Dose

Subgroup

1.72

(0.80, 3.68)

1.28

(0.59, 2.76) 34.64

Follow-

up Time

(Months)

Cox Logistic

R2Hazard Ratio

(95%CI)

Odds Ratio

(95%CI)

0–20.6

(0.4, 1.0)

0.6

(0.4, 1.1)

-3

2–61.2

(0.8, 2.0)

1.2

(0.7, 1.9)

4

6–170.6

(0.3, 1.1)

0.6

(0.3, 1.0)

10

17–1010.7

(0.5, 1.2)

0.7

(0.4, 1.1)

6

Case Study Results

By Follow-Up Time

Case Study — Olmesartan

• Effect of treatment effect (𝜷 )

– R1: Logistic < Cox

– R2

• Effect of mean survival time (𝝀𝒕 )

– R1: Logistic ≈ Cox

– R2

• Effect of follow-up time ( )

– R1: Logistic > Cox

– R2

• Effect of sample size ( )

– R1: Logistic > Cox

– R2

Total Cohort and High-Dose Subgroup

Simulation Estimators

• 𝜷: Mean of 1000 simulated 𝜷

• Relative Bias

• Relative Ratio

Effects of Interest

• Treatment effect (𝜷)

• Mean survival time (𝝀𝒕)

• Follow-up time

• Sample size

• Follow-up time : R2

• Sample size : R2

• Model with smaller relative bias

depends on

– Stronger treatment effect: Logistic

– Longer follow-up time: Cox

– Smaller sample size: Cox

• Models performed similarly

regarding survival time

Treatment Effect (β)

Mean Survival Time (𝝀𝒕)

Follow-up Time

Sample Size

References

Bender R, Augustin T, Blettner M. Generating

survival times to simulate Cox proportional hazards

models. Statistics in Medicine 2005; 24:1713–1723

D’Agostino RB, Lee ML, Belanger AJ. Relation of

pooled logistic regression to time dependent Cox

regression analysis: The Framingham heart study.

Statistics in Medicine 1990; 9: 1501–1515