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Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. One can request that SAS estimate the survival function by exponentiating the negative of the Nelson-Aalen estimator, also known as the Breslow estimator, rather than by the Kaplan-Meier estimator through the method=breslow option on the proc lifetest statement. Easy to read and comprehensive, Survival Analysis Using SAS: A Practical Guide, Second Edition, by Paul D. Allison, is an accessible, data-based introduction to methods of survival analysis. Here we see the estimated pdf of survival times in the whas500 set, from which all censored observations were removed to aid presentation and explanation. Both proc lifetest and proc phreg will accept data structured this way. hrtime = hr*lenfol; Non-parametric methods are appealing because no assumption of the shape of the survivor function nor of the hazard function need be made. For exponential regression analysis of the nursing home data the syntax is as follows: data nurshome; infile 'nurshome.dat'; input los age rx gender married health fail; label los='Length of stay' rx='Treatment' married='Marriage status' However, widening will also mask changes in the hazard function as local changes in the hazard function are drowned out by the larger number of values that are being averaged together. 147-60. A complete description of the hazard rate’s relationship with time would require that the functional form of this relationship be parameterized somehow (for example, one could assume that the hazard rate has an exponential relationship with time). Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. Thus, if the average is 0 across time, then that suggests the coefficient $$p$$ does not vary over time and that the proportional hazards assumption holds for covariate $$p$$. model lenfol*fstat(0) = gender|age bmi|bmi hr; $df\beta_j \approx \hat{\beta} – \hat{\beta_j}$. Covariates are permitted to change value between intervals. The regression analyses of relative survival can be conducted easily using mainstream statistical software packages (SAS and STATA), thereby removing the reliance on special-purpose software. class gender; For each subject, the entirety of follow up time is partitioned into intervals, each defined by a “start” and “stop” time. where $$n_i$$ is the number of subjects at risk and $$d_i$$ is the number of subjects who fail, both at time $$t_i$$. None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). Alternatively, the data can be expanded in a data step, but this can be tedious and prone to errors (although instructive, on the other hand). Using the assess statement to check functional form is very simple: First let’s look at the model with just a linear effect for bmi. $f(t) = h(t)exp(-H(t))$. The other covariates, including the additional graph for the quadratic effect for bmi all look reasonable. download 1 file . A solid line that falls significantly outside the boundaries set up collectively by the dotted lines suggest that our model residuals do not conform to the expected residuals under our model. assess var=(age bmi bmi*bmi hr) / resample; The Kaplan_Meier survival function estimator is calculated as: $\hat S(t)=\prod_{t_i\leq t}\frac{n_i – d_i}{n_i},$. (1993). Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. proc sgplot data = dfbeta; Springer: New York. Only as many residuals are output as names are supplied on the, We should check for non-linear relationships with time, so we include a, As before with checking functional forms, we list all the variables for which we would like to assess the proportional hazards assumption after the. We will thus let $$r(x,\beta_x) = exp(x\beta_x)$$, and the hazard function will be given by: This parameterization forms the Cox proportional hazards model. Once outliers are identified, we then decide whether to keep the observation or throw it out, because perhaps the data may have been entered in error or the observation is not particularly representative of the population of interest. This relationship would imply that moving from 1 to 2 on the covariate would cause the same percent change in the hazard rate as moving from 50 to 100. (1994). The hazard function is also generally higher for the two lowest BMI categories. Notice that the baseline hazard rate, $$h_0(t)$$ is cancelled out, and that the hazard rate does not depend on time $$t$$: The hazard rate $$HR$$ will thus stay constant over time with fixed covariates. Thus, it appears, that when bmi=0, as bmi increases, the hazard rate decreases, but that this negative slope flattens and becomes more positive as bmi increases. var lenfol; If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. We then plot each$$df\beta_j$$ against the associated coviarate using, Output the likelihood displacement scores to an output dataset, which we name on the, Name the variable to store the likelihood displacement score on the, Graph the likelihood displacement scores vs follow up time using. Thus, both genders accumulate the risk for death with age, but females accumulate risk more slowly. Here are the steps we use to assess the influence of each observation on our regression coefficients: The dfbetas for age and hr look small compared to regression coefficients themselves ($$\hat{\beta}_{age}=0.07086$$ and $$\hat{\beta}_{hr}=0.01277$$) for the most part, but id=89 has a rather large, negative dfbeta for hr. The outcome in this study. This indicates that omitting bmi from the model causes those with low bmi values to modeled with too low a hazard rate (as the number of observed events is in excess of the expected number of events). class gender; For example, if males have twice the hazard rate of females 1 day after followup, the Cox model assumes that males have twice the hazard rate at 1000 days after follow up as well. format gender gender. The output for the discrete time mixed effects survival model fit using SAS and Stata is reported in Statistical software output C7 and Statistical software output C8, respectively, in Appendix C in the Supporting Information. Some examples of time-dependent outcomes are as follows: As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. ; Modeling Survival Data: Extending the Cox Model. In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. As an example, imagine subject 1 in the table above, who died at 2,178 days, was in a treatment group of interest for the first 100 days after hospital admission. class gender; The hazard rate can also be interpreted as the rate at which failures occur at that point in time, or the rate at which risk is accumulated, an interpretation that coincides with the fact that the hazard rate is the derivative of the cumulative hazard function, $$H(t)$$. Above, we discussed that expressing the hazard rate’s dependence on its covariates as an exponential function conveniently allows the regression coefficients to take on any value while still constraining the hazard rate to be positive. It appears that for males the log hazard rate increases with each year of age by 0.07086, and this AGE effect is significant, AGE*GENDER term is negative, which means for females, the change in the log hazard rate per year of age is 0.07086-0.02925=0.04161. run; proc phreg data = whas500; Please login to your account first; Need help? 68 Analysis of Clinical Trials Using SAS: A Practical Guide, Second Edition A detailed description of model-based approaches can be found in the beginning of Chapter 1. hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5; In other words, the average of the Schoenfeld residuals for coefficient $$p$$ at time $$k$$ estimates the change in the coefficient at time $$k$$. Suppose that you suspect that the survival function is not the same among some of the groups in your study (some groups tend to fail more quickly than others). Once again, the empirical score process under the null hypothesis of no model misspecification can be approximated by zero mean Gaussian processes, and the observed score process can be compared to the simulated processes to asses departure from proportional hazards. ISBN 13: 9781629605210. 51. From these equations we can see that the cumulative hazard function $$H(t)$$ and the survival function $$S(t)$$ have a simple monotonic relationship, such that when the Survival function is at its maximum at the beginning of analysis time, the cumulative hazard function is at its minimum. It is not at all necessary that the hazard function stay constant for the above interpretation of the cumulative hazard function to hold, but for illustrative purposes it is easier to calculate the expected number of failures since integration is not needed. Therneau, TM, Grambsch PM, Fleming TR (1990). Several covariates can be evaluated simultaneously. We will model a time-varying covariate later in the seminar. The SAS Enterprise Miner Survival node is located on the Applications tab of the SAS Enterprise Miner tool bar. There are $$df\beta_j$$ values associated with each coefficient in the model, and they are output to the output dataset in the order that they appear in the parameter table “Analysis of Maximum Likelihood Estimates” (see above). Here are the typical set of steps to obtain survival plots by group: Let’s get survival curves (cumulative hazard curves are also available) for males and female at the mean age of 69.845947 in the manner we just described. Click here to download the dataset used in this seminar. A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. run; proc phreg data = whas500; The WHAS500 data are stuctured this way. In this seminar we will be analyzing the data of 500 subjects of the Worcester Heart Attack Study (referred to henceforth as WHAS500, distributed with Hosmer & Lemeshow(2008)). We see that beyond beyond 1,671 days, 50% of the population is expected to have failed. Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject. Using the equations, $$h(t)=\frac{f(t)}{S(t)}$$ and $$f(t)=-\frac{dS}{dt}$$, we can derive the following relationships between the cumulative hazard function and the other survival functions: $S(t) = exp(-H(t))$ If proportional hazards holds, the graphs of the survival function should look “parallel”, in the sense that they should have basically the same shape, should not cross, and should start close and then diverge slowly through follow up time. For example, if there were three subjects still at risk at time $$t_j$$, the probability of observing subject 2 fail at time $$t_j$$ would be: $Pr(subject=2|failure=t_j)=\frac{h(t_j|x_2)}{h(t_j|x_1)+h(t_j|x_2)+h(t_j|x_3)}$. The mean time to event (or loss to followup) is 882.4 days, not a particularly useful quantity. However, we can still get an idea of the hazard rate using a graph of the kernel-smoothed estimate. 1 Notes on survival analysis using SAS These notes describe how some of the methods described in the course can be implemented in SAS. We compare 2 models, one with just a linear effect of bmi and one with both a linear and quadratic effect of bmi (in addition to our other covariates). However, despite our knowledge that bmi is correlated with age, this method provides good insight into bmi’s functional form. Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. SAS Survival Handbook. The assess statement with the ph option provides an easy method to assess the proportional hazards assumption both graphically and numerically for many covariates at once. download 1 file . These techniques were developed by Lin, Wei and Zing (1993). Easy to read and comprehensive, Survival Analysis Using SAS: A Practical Guide, Second Edition, by Paul D. Allison, is an accessible, data-based introduction to methods of survival analysis. In this model, this reference curve is for males at age 69.845947 Usually, we are interested in comparing survival functions between groups, so we will need to provide SAS with some additional instructions to get these graphs. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time $$t$$. For example, the time interval represented by the first row is from 0 days to just before 1 day. run; proc phreg data = whas500; Looking at the table of “Product-Limit Survival Estimates” below, for the first interval, from 1 day to just before 2 days, $$n_i$$ = 500, $$d_i$$ = 8, so $$\hat S(1) = \frac{500 – 8}{500} = 0.984$$. The survival curves for females is slightly higher than the curve for males, suggesting that the survival experience is possibly slightly better (if significant) for females, after controlling for age. For observation $$j$$, $$df\beta_j$$ approximates the change in a coefficient when that observation is deleted. Because this likelihood ignores any assumptions made about the baseline hazard function, it is actually a partial likelihood, not a full likelihood, but the resulting $$\beta$$ have the same distributional properties as those derived from the full likelihood. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. If the observed pattern differs significantly from the simulated patterns, we reject the null hypothesis that the model is correctly specified, and conclude that the model should be modified. Indeed, exclusion of these two outliers causes an almost doubling of $$\hat{\beta}_{bmi}$$, from -0.23323 to -0.39619. So what is the probability of observing subject $$i$$ fail at time $$t_j$$? Nevertheless, in both we can see that in these data, shorter survival times are more probable, indicating that the risk of heart attack is strong initially and tapers off as time passes. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. Confidence intervals that do not include the value 1 imply that hazard ratio is significantly different from 1 (and that the log hazard rate change is significanlty different from 0). run; proc phreg data = whas500; The log-rank and Wilcoxon tests in the output table differ in the weights $$w_j$$ used. Expressing the above relationship as $$\frac{d}{dt}H(t) = h(t)$$, we see that the hazard function describes the rate at which hazards are accumulated over time. For more detail, see Stokes, Davis, and Koch (2012) Categorical Data Analysis Using SAS, 3rd ed. During the interval [382,385) 1 out of 355 subjects at-risk died, yielding a conditional probability of survival (the probability of survival in the given interval, given that the subject has survived up to the begininng of the interval) in this interval of $$\frac{355-1}{355}=0.9972$$. Biometrika. Instead, we need only assume that whatever the baseline hazard function is, covariate effects multiplicatively shift the hazard function and these multiplicative shifts are constant over time. SAS expects individual names for each $$df\beta_j$$associated with a coefficient. Checking the Cox model with cumulative sums of martingale-based residuals. This includes, for example, logistic regression models used in the analysis of binary endpoints and the Cox proportional hazards model in settings with time-to-event endpoints. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. The Kaplan-Meier curve, also called the Product Limit Estimator is a popular Survival Analysis method that estimates the probability of survival to a given time using proportion of patients who have survived to that time. In the graph above we see the correspondence between pdfs and histograms. In the Cox proportional hazards model, additive changes in the covariates are assumed to have constant multiplicative effects on the hazard rate (expressed as the hazard ratio ($$HR$$)): In other words, each unit change in the covariate, no matter at what level of the covariate, is associated with the same percent change in the hazard rate, or a constant hazard ratio. The Wilcoxon test uses $$w_j = n_j$$, so that differences are weighted by the number at risk at time $$t_j$$, thus giving more weight to differences that occur earlier in followup time. Nevertheless, the bmi graph at the top right above does not look particularly random, as again we have large positive residuals at low bmi values and smaller negative residuals at higher bmi values. Send-to-Kindle or Email . The interpretation of this estimate is that we expect 0.0385 failures (per person) by the end of 3 days. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that proportional hazards holds for all of our covariates. class gender; We previously saw that the gender effect was modest, and it appears that for ages 40 and up, which are the ages of patients in our dataset, the hazard rates do not differ by gender. In the relation above, $$s^\star_{kp}$$ is the scaled Schoenfeld residual for covariate $$p$$ at time $$k$$, $$\beta_p$$ is the time-invariant coefficient, and $$\beta_j(t_k)$$ is the time-variant coefficient. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. We could thus evaluate model specification by comparing the observed distribution of cumulative sums of martingale residuals to the expected distribution of the residuals under the null hypothesis that the model is correctly specified. We can similarly calculate the joint probability of observing each of the $$n$$ subject’s failure times, or the likelihood of the failure times, as a function of the regression parameters, $$\beta$$, given the subject’s covariates values $$x_j$$: $L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{iin R_j}exp(x_i\beta)}\Bigg\rbrace$. Easy to read and comprehensive, Survival Analysis Using SAS: A Practical Guide, Second Edition, by Paul D. Allison, is an accessible, data-based introduction to methods of survival analysis. output out=residuals resmart=martingale; Easy to read and comprehensive, Survival Analysis Using SAS: A Practical Guide, Second Edition, by Paul D. Allison, is an accessible, data-based introduction to methods of survival analysis. Survival Analysis Using SAS®: A … Most of the time we will not know a priori the distribution generating our observed survival times, but we can get and idea of what it looks like using nonparametric methods in SAS with proc univariate. The Schoenfeld residual for observation $$j$$ and covariate $$p$$ is defined as the difference between covariate $$p$$ for observation $$j$$ and the weighted average of the covariate values for all subjects still at risk when observation $$j$$ experiences the event. During the next interval, spanning from 1 day to just before 2 days, 8 people died, indicated by 8 rows of “LENFOL”=1.00 and by “Observed Events”=8 in the last row where “LENFOL”=1.00. The probability P(a < T < b) is the area under the curve . Additionally, a few heavily influential points may be causing nonproportional hazards to be detected, so it is important to use graphical methods to ensure this is not the case. Note: A number of sub-sections are titled Background. 81. However, if that is not the case, then it may be possible to use programming statement within proc phreg to create variables that reflect the changing the status of a covariate. For example, if $$\beta_x$$ is 0.5, each unit increase in $$x$$ will cause a ~65% increase in the hazard rate, whether X is increasing from 0 to 1 or from 99 to 100, as $$HR = exp(0.5(1)) = 1.6487$$. From these equations we can also see that we would expect the pdf, $$f(t)$$, to be high when $$h(t)$$ the hazard rate is high (the beginning, in this study) and when the cumulative hazard $$H(t)$$ is low (the beginning, for all studies). model martingale = bmi / smooth=0.2 0.4 0.6 0.8; Let’s take a look at later survival times in the table: From “LENFOL”=368 to 376, we see that there are several records where it appears no events occurred. Because the observation with the longest follow-up is censored, the survival function will not reach 0. As we see above, one of the great advantages of the Cox model is that estimating predictor effects does not depend on making assumptions about the form of the baseline hazard function, $$h_0(t)$$, which can be left unspecified. The sudden upticks at the end of follow-up time are not to be trusted, as they are likely due to the few number of subjects at risk at the end. Applied Survival Analysis. In the 15 years since the first edition of the book was published, statistical methods for survival analysis and the SAS system have both evolved. Below, we show how to use the hazardratio statement to request that SAS estimate 3 hazard ratios at specific levels of our covariates. Pages: 426. between time a and time b. (Book), View 2 excerpts, cites background and methods, View 5 excerpts, cites methods and background, View 3 excerpts, cites background and methods, View 4 excerpts, cites background and methods, View 15 excerpts, cites methods and background, By clicking accept or continuing to use the site, you agree to the terms outlined in our. Survival Analysis Approaches and New Developments using SAS, continued . download 1 file . output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; In large datasets, very small departures from proportional hazards can be detected. The estimate of survival beyond 3 days based off this Nelson-Aalen estimate of the cumulative hazard would then be $$\hat S(3) = exp(-0.0385) = 0.9623$$. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). The LIFEREG procedure produces parametric regression models with censored survival data using maximum likelihood estimation. Our goal is to transform the data from its original state: to an expanded state that can accommodate time-varying covariates, like this (notice the new variable in_hosp): Notice the creation of start and stop variables, which denote the beginning and end intervals defined by hospitalization and death (or censoring). Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. We can see this reflected in the survival function estimate for “LENFOL”=382. Let us further suppose, for illustrative purposes, that the hazard rate stays constant at $$\frac{x}{t}$$ ($$x$$ number of failures per unit time $$t$$) over the interval $$[0,t]$$. The survival function drops most steeply at the beginning of study, suggesting that the hazard rate is highest immediately after hospitalization during the first 200 days. First, each of the effects, including both interactions, are significant. ... View the article PDF and any associated supplements and figures for a period of 48 hours. Similarly, because we included a BMI*BMI interaction term in our model, the BMI term is interpreted as the effect of bmi when bmi is 0. Below we demonstrate use of the assess statement to the functional form of the covariates. Survival Analysis Using SAS: A Practical Guide, Second Edition. The hazard function for a particular time interval gives the probability that the subject will fail in that interval, given that the subject has not failed up to that point in time. model (start, stop)*status(0) = in_hosp ; run; proc phreg data = whas500; SAS computes differences in the Nelson-Aalen estimate of $$H(t)$$. Plots of covariates vs dfbetas can help to identify influential outliers.