Duration Models: Parametric Models - Login

1 Bradford S. Jones, UC-Davis, Dept. of Political Science Duration Models : Parametric Models Brad Jones1. 1 Department of Political Science University of California, Davis January 28, 2011. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric survival Models Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Some Motivation for Parametrics I Consider the hazard rate: dh(t). > 0, dt Hazard increasing wrt time. dh(t). < 0, dt Hazard decreasing wrt time. dh(t). = 0, dt Hazard flat wrt time. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Parametric Models give structure (shape) to the hazard function.

2 I : the structure is a function of the , not necessarily of the real world.. I .. though some do a good job of approximating some failure-time processes. I Any with positive support on the real number line will work. I Lots of choices: exponential, Weibull, gamma, Gompertz, log-normal, log-logistic .. etc. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I For parametrics, we work with standard likelihood methods. I Specify a distribution function and write out the log-likelihood for the data. I The question is, which distribution function?

3 I In all software programs/computing environments, youre given a menu. I Stata:streg, R:survreg, eha Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Advantages of Parametric Models ? I If S(t) is known to follow, or closely approximate a known distribution, then estimates will be consistent the the theoretical survivor function. I Unlike K-M or Cox (discussed later), the hazard may be used for forecasting (under KM or Cox, the hazard is only defined up until the last observed failure). I Will return smooth functions of h(t) or S(t). Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I As noted, there are a wide variety of choices.

4 I I sometimes refer to these choices as plug and play . estimators. I Why? Consider the survivor function: Z Z t S(t) = Pr(T > t) = f (u)d(u) = 1 f (u)d(u) = 1 F (t). t 0. (1). I If we know this function follows some distribution, then we write a likelihood function in terms of this distribution .. I If it follows a different distribution, just replace the previous likelihood with another pdf. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Most texts, including ours, typically begin with the exponential distribution. I The reason is easy: it's an easy distribution to work with and visualize. I It also may be unrealistic in many settings.

5 I The basic feature: the hazard rate is flat wrt time. I That is: h(t) = (2). Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Recall from the first week: S(t) = exp{ H(t)} (3). where Z t H(t) = h(u)du 0. I Substituting into (3) , Z t S(t) = exp{ du}. 0. and so S(t) = exp( t). I This is the survivor function for the exponential distribution. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Since we know f (t) = h(t)S(t) then f (t) = exp( t). I This is the pdf of a random variable T that is exponentially distributed. I Note how the unconditional probability of failure, f (t), handles censored cases.

6 I Consider the hazard function: Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I What is ? I Or put differently, where are the predictor variables? I Typically will be parameterized in terms of regression coefficients and covariates, X . I A model : h(t) = = exp( 0 + 1 T ). I Suppose T is a treatment indicator and we're interested in the hazard of failure for the treated and untreated. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Two hazards: h(tT =1 ) = exp( 0 + 1 ).

7 H(tT =0 ) = exp( 0 ). I If we plotted the hazards, we would have two parallel lines separated by exp( 1 ). I Or analogously, if we want to compare hazards: h(tT =1 ) exp( 0 + 1 ). =. h(tT =0 ) exp( 0 ). item 4- This expression must simplify to exp( 1 ). Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I In words (sort of)..the ratio of the treated to the untreated simplifies to exp( 1 ). I So all we need to know to know the differences in the hazards is the coefficient for the treated. I This is an important result because it shows the hazards are proportional hazards. I Some simulated data. I h(t) = + .96(Z ).

8 I Let Z denote whether or not a subject was exposed to some condition. Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Since 1 is positive, this implies exposure increases the risk. I The hazard is higher for the exposed than for the unexposed. I Treatment estimate is .96 implies difference in hazard is exp(.96) I Risk for exposed is about times greater than for the unexposed. I Consider the hazards: Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I PH property is important to understand.

9 I By way of analogy, think about what odds ratios are in a logit-type setting or recall the ordered logit model : the OR are invariant to the scale scores. I The proportional difference in hazards is invariant to time. I So under the exponential we are making two assumptions: 1. The hazards are flat wrt time. 2. The difference in hazards across levels of a covariate is a fixed proportion. I Which is the stronger assumption? Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models I Note that even with the PH assumption, we are not saying (in general) the hazards are invariant to time (though in the exponential case, we are).

10 I The hazards may change but the proportional difference between (say) two groups, does not change. I That's the basic result of proportionality. I Suppose it does not hold. Then what? I Consider another model that relaxes the assumption of flat hazards (but not the PH assumption). Jones POL 290G. Bradford S. Jones, UC-Davis, Dept. of Political Science Parametric survival Models Parametric Models : Weibull I A more flexible distribution function is given by the Weibull. I Named for Waloddi Weibull, who derived it (1939, 1951). I Why more general than the exponential? I It is a two-parameter distribution: h(t) = pt p 1 (4). where is a positive scale parameter and p is a shape parameter.