Solutions to the practice problems. - UCLA Biostatistics

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as the probability that the event happens in a tiny interval around time t. The hazard function is like the density function except that it is conditional on having survived up to time t. It gives the relative likelihood of the event happening at time t given that the event has not happened up to time t. You can think of it as the probability that the event will happen in a tiny interval after time t assuming it hasn’t happened prior to t. The hazard function is given by h(t) = f(t)/S(t). Note that all these quantities are inter-related. If you know one of them for all values of t then you can calculate all the others. (c) Decribe the two major categories of approaches to estimating the survival quantities above. Solution: There are both parametric and non-parametric techniques for estimating the key survival functions. In the parametric approach you assume that the survival time, T, has a particular distribution (e.g. an exponential distribution) or equivalently that the survival curve or hazard function has a particular shape, and you then use maximum likelihood ideas to estimate the parameters of that distribution (e.g. mean, variance, etc.). In the non-parametric framework you use the empirical distribution given by the observed event times and censoring times in your sample. The Kaplan-Meier product limit estimator of the survival curve is the classic non-parametric technique. There is also a famous estimator of the cumulative hazard function, H(t) which “adds up” the total hazard a subject has been exposed to up to time t, called the Nelson-Aalen estimator. Both require only that you know the event and censoring times. (d) Explain briefly the ideas behind the accelerated failure time model and the Cox proportional hazards model for incorporating covariates into a survival model. Solution: The accelerated failure time (AFT) model is basically a generalized linear model with a log link and the usual systematic component consisting of a linear combination of the predictor variables or X’s. It can be fit in conjunction with a number of distribution functions and is a parametric model. The trick is that the censoring times have to be built into the likelihood function that you are trying to maximize. The interpretation of the coefficients proceeds in the usual way as it would for a Poisson or other model that has a log link. The Cox Proportonal Hazards model is what is called a “semi-parametric” model. You make some assumptions about the form of the distribution/hazard function but stop short of estimating the whole thing. Specifically the proportional hazards model assumes that the hazard function can be written as h(t) = h 0 (t)c(Xβ) where h 0 (t) is called the baseline hazard function and c is a known function (frequently the exponential so that c(Xβ) = e Xβ ) applied to our usual linear combination of the X’s. This is a multuiplicative model. If c is the exponential function then taking logs puts us back on our favorite additive scale. Note that while we assume the form of the function c is known we make no assumptions about the form of the baseline hazard function (other than that it is non-negative and usually continuous). This is why the model is called semi-parametric. In fact to understand the impact of the X’s we do not need to estimate the shape of the baseline hazard because if we take the hazard ratio for two people with different covariate values the baseline hazard cancels out, leaving us a piece that depends only on c. (2) Weighted Analysis Basics: (a) Give four examples of situations in which you might want to perform a weighted analysis. Solution: There are many situations in which you might want to perform a weighted analysis. Examples include fitting an OLS regression model in which the constant variance assumption is violated; fitting a regression model in which the observed Y’s are measured with different levels of accuracy (this occurs, for example, if the Y’s are actually averages of several replications and the number of replicates varies from observation to observation); fitting a model where the observational units have differing sizes or perceived importance (e.g. measurements on countries); observational studies where some types of subjects were harder 2
to reach than others so are not represented equally in the population; studies in which sampling was deliberately stratified to allow for obtaining adequate estimates for rare groups–in this case the rare group needs to be down-weighted when the estimates from the strata are combined; studies where the sampling was done in a hierarchical or clustered manner–e.g. by neighborhood or school–so that certain areas are over-represented (and their measurements are correlated); and many, many others. Basically you need weighting when your sample is not a nice independent and identically distributed draw from the population of interest so that it is in some way un-representative of that population. (b) Describe briefly some of the ways in which you could incorporate weights into an estimation procedure. Solution: For many estimation and modeling situations the weights (once you have them!) can be built right into the likelihood function you need to maximize. For instance, in OLS regression the maximum likelihood solution is the same as the least squares solution which minimizes the sum of squared differences between the observed Y’s and the values predicted by the regression line. In the weighted regression setting you simply minimize the sum of squared errors for each point multiplied by their weights. A similar strategy applies to all sorts of other regression models. When you are doing something like stratified sampling, rather than writing down the likelihood, it may be obvious how to apply the weights. For instance, if you have stratified your sample into men and women and calculated the mean value of your variable of interest separately in both groups then you could create the weighted mean simply by multiplying the group means by the relative proportions of men and women in the sample. Sometimes you do this even if the sample was not deliberately obtained by stratification. For instance, in an observational sample where you were concerned about representativeness, you might group the observations into “census” bins (could be age ranges, neighborhoods, or any other grouping of interest) and then use the census proportions for those bins to weight the estimates calculated separately for the bins. Sometimes the calculation of the appropriate weights can get very complex depending on the sampling scheme and the sources of bias. (c) Define as carefully as you can what a propensity score is and give two examples of when and how you might want to use it. Solution: A propensity score is simply the probability of having or not having a characteristic of interest. The most common use of propensity scores is in observational studies where you wish to know the causal impact of a treatment or other binary characterstic on an outcome of interest but the subjects weren’t randomized to the treatment and so there may be confounding factors. In this case, the propensity score is the probability that a subject receives the treatment given their observed values of the potential confounders. Propensity scores provide an alternative to matching or covariate adjustment. It may be very difficult to match subjects adequately if the number of possible confounders is large and the number of available cases is small. One could add the confounders as predictors, along with the treatment, into the model for the outcome. However this requires knowing the “right” model and if there are large numbers of confounders this may be complicated by multicollinearity or overfitting issues. Instead, one creates a logistic regression model with the treatment (yes or no) as the outcome variables and the potential confounders as predictors. The resulting predicted probabilities of treatment are the propensity scores. These scores can be used in a number of ways. First, one can match subjects on their propensity score. This is much easier than matching on many covariates, some values of which may be rare. Second, one my stratify the analysis by creating bins of people with similar propensity scores. One then examines the difference between the treatment and non-treatment groups within each bin and combines them, A common choice is to create 5 bins based on the quintiles of the propensity score distribution but you need to check that within these bins the original confounders are well enough balanced between the treatment and control group. If they are not you may need to use finer binning. Thirdly, you can simply use the propensity score as a covariate, along with the treatment variable, in the analysis of the outcome. It can be shown under certain not to rigid requriements that matching. stratifying or adjusting by the propensity score will correctly account for the confounders, giving you an improve estimate of the “true” treatment effect. 3
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Solutions to the practice problems. - UCLA Biostatistics

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