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BMS 617

BMS 617. Lecture 15: Proportional Hazards Regression. Survival Analysis Recap. Last time, we looked at “survival analysis” Analysis of data where the outcome is the time to a non-recurring event Event may not occur during the timeframe of the study

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BMS 617

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  1. BMS 617 Lecture 15: Proportional Hazards Regression

  2. Survival Analysis Recap • Last time, we looked at “survival analysis” • Analysis of data where the outcome is the time to a non-recurring event • Event may not occur during the timeframe of the study • In this case we only know that the event did not occur for some specific time • We call such data “censored” • We saw how to plot such data • Kaplan-Meier curve • And how to compare survival data from two different groups • LogRank or Cox-Mantel test

  3. Survival Data and Hazard • For survival data, the hazard function h(t) is the risk of the event occurring at time t • Formally, it is the probability of the event occurring in some small time interval [t, t+Δt], given that the subject has survived to time t. • For the logrank test, we assume that the hazard is proportional; i.e. the ratio of the hazards for the two groups is the same at all times

  4. Formalization of hazard ratio • Formally, suppose the hazard ratio for one group is h(t) • Then by assumption, the hazard ratio for the other group is r×h(t) • The logrank test is a test for the null hypothesis that r=1 • We call r here the hazard ratio • Plays a similar role to relative risk

  5. Multiple Regression Revisited In previous lectures, we studied models where the outcome was estimated by some function of many different variables For example, in logistic regression, we usedYi = β0 + β1X1,i + β2X2,i + β3X3,i+… Here Yi is the log(odds) that subject i is in one of the two outcome groups Xi are the independent variables, and β are the corresponding parameters We can do the same thing for survival data (or hazard functions)

  6. Proportional Hazards Regression In proportional hazards regression, we assume that the hazard function for any given subject is the product of factors depending on the values of any number of independent variables:hX(t)=h0(t) × exp(β1X1+β2X2+β3X3+…) Each parameter βi represents the increase in the log of the hazard function for each unit increase in the corresponding independent variable Note we are assuming the contribution of each variable is independent of time

  7. LogRank test is a special case of proportional hazards regression • We can think about the case of two groups as an example of proportional hazards regression • Example from last time: Patients with chronic active hepatitis, treated with either prednisolone or with a placebo • We have one variable, with just two values: • Let x=0 for patients in the placebo group and x=1 for patients in the prednisolone group

  8. LogRank test as proportional hazards regression The proportional hazards model for this (very) special case looks like:hx(t)=h0(t)×eβx For the placebo group, x=0, so we havehx(t)=h0(t)×e0 =h0(t)×1=h0(t) For the prednisolone group, x=1, sohx(t)=h0(t)×eβ=r×h0(t)where β=ln(r), i.e. β is natural log of the hazard ratio

  9. Proportional Hazards Regression Example • Rosman et al. studied the effect of diazepam on febrile seizures in children • Febrile seizures are seizures/convulsions associated with high body temperature, but not associated with, e.g. epilepsy • Recruited 400 children who had experienced at least one febrile seizure • Parents instructed to give either diazepam or a placebo to the children whenever they had high fever • Measured time to first seizure

  10. Rosman et al. study First analysis was a simple logrank (Mantel-Cox) test between the diazepam and placebo groups Did not produce a statistically significant result (p=0.064) Instead, fitted a proportional hazards model with age, number of previous seizures, time between last seizure and entry into study, existence of developmental problems, and diazepam/placebo as independent variables

  11. Rosman et al. Results • For the proportional hazards model in the Rosman et al. study, the parameter for diazepam/placebo had an estimate of -0.494, with a 95% confidence interval of [-0.942,-0.0619] and a p-value of 0.027. • This means that we estimate the the natural log of the hazard function for the diazepam group is 0.494 lower than the natural log of the hazard function for the placebo group • Alternatively, the hazard for the diazepam group is e-0.494=0.61 times the hazard for the placebo group. • We can think of this as a relative risk; it’s the risk of seizures for the diazepam group relative to the control group, assuming all other variables are held constant • The 95% confidence interval for the relative risk is [e-0.942,e-0.0619]=[0.39,0.94].

  12. Summary • Proportional Hazards Regression allows us to model survival data as a function of multiple independent variables • Proportional Hazards Regression is also called Cox Regression (after David Cox, who pioneered most of the techniques in survival analysis) • Means we can correct for confounding factors in survival analysis • Or compare the effects of different factors

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