In other words, the average of the Schoenfeld residuals for coefficient \(p\) at time \(k\) estimates the change in the coefficient at time \(k\). 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. model lenfol*fstat(0) = gender|age bmi hr;
Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. To specify a Cox model with start and stop times for each interval, due to the usage of time-varying covariates, we need to specify the start and top time in the model statement: If the data come prepared with one row of data per subject each time a covariate changes value, then the researcher does not need to expand the data any further. class gender;
In the code below, we show how to obtain a table and graph of the Kaplan-Meier estimator of the survival function from proc lifetest: Above we see the table of Kaplan-Meier estimates of the survival function produced by proc lifetest. This confidence band is calculated for the entire survival function, and at any given interval must be wider than the pointwise confidence interval (the confidence interval around a single interval) to ensure that 95% of all pointwise confidence intervals are contained within this band. run;
Instead, the survival function will remain at the survival probability estimated at the previous interval. As we know, each subject in the WHAS500 dataset is represented by one row of data, so the dataset is not ready for modeling time-varying covariates. 147-60. class gender;
Click here to report an error on this page or leave a comment, Your Email (must be a valid email for us to receive the report! run; proc phreg data = whas500(where=(id^=112 and id^=89));
Notice there is one row per subject, with one variable coding the time to event, lenfol: A second way to structure the data that only proc phreg accepts is the “counting process” style of input that allows multiple rows of data per subject. run; proc phreg data = whas500;
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. Examples of response variables include the failure time of a machine part in engineering, the customer lifetime in customer churn analysis, the time to default in credit scoring, and so on. For example, if the event of interest is cancer, then the survival time can be the time in years until a person develops cancer. 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. Thus, we can expect the coefficient for bmi to be more severe or more negative if we exclude these observations from the model. Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. output out=residuals resmart=martingale;
Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. The mean time to event (or loss to followup) is 882.4 days, not a particularly useful quantity. 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). Survival analysis is a set of methods for analyzing data in which the outcome variable is the time until an event of interest occurs. At this stage we might be interested in expanding the model with more predictor effects. 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. scatter x = age y=dfage / markerchar=id;
run; lenfol: length of followup, terminated either by death or censoring. One caveat is that this method for determining functional form is less reliable when covariates are correlated. Now let’s look at the model with just both linear and quadratic effects for bmi. survival analysis is used to refer to a statistical analysis of the time at which the event of interest occurs (Kalbfleisch and Prentice, 2002 and Allison, 1995). None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). run;
The graphical presentation of survival analysis is a significant tool to facilitate a clear understanding of the underlying events. Springer: New York. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). 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)). 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. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). run;
Researchers who want to analyze survival data with SAS will find just what they need with this fully updated new edition that incorporates the many enhancements in SAS procedures for survival analysis in SAS 9. Widening the bandwidth smooths the function by averaging more differences together. Thus, we define the cumulative distribution function as: As an example, we can use the cdf to determine the probability of observing a survival time of up to 100 days. Another great feature is that it also tests a linear hypothesis about the regression coefficients. Notice the survival probability does not change when we encounter a censored observation. \[F(t) = 1 – exp(-H(t))\] Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. In the code below we demonstrate the steps to take to explore the functional form of a covariate: In the left panel above, “Fits with Specified Smooths for martingale”, we see our 4 scatter plot smooths. This text is suitable for researchers and statisticians working in the medical and other life sciences as well as statisticians in academia who teach introductory and second-level courses on survival analysis. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time \(t\). run; proc phreg data = whas500;
We see that the uncoditional probability of surviving beyond 382 days is .7220, since \(\hat S(382)=0.7220=p(surviving~ up~ to~ 382~ days)\times0.9971831\), we can solve for \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). A simple transformation of the cumulative distribution function produces the survival function, \(S(t)\): The survivor function, \(S(t)\), describes the probability of surviving past time \(t\), or \(Pr(Time > t)\). run; proc phreg data = whas500;
model lenfol*fstat(0) = ;
We could test for different age effects with an interaction term between gender and age. model (start, stop)*status(0) = in_hosp ;
Here, we will learn what are the procedures used in SAS survival analysis: PROC ICLIFETEST, PROC ICPHREG, PROC LIFETEST, PROC SURVEYPHREG, PROC LIFEREG, and PROC PHREG with syntax and example. 1 … The calculation of the statistic for the nonparametric “Log-Rank” and “Wilcoxon” tests is given by : \[Q = \frac{\bigg[\sum\limits_{i=1}^m w_j(d_{ij}-\hat e_{ij})\bigg]^2}{\sum\limits_{i=1}^m w_j^2\hat v_{ij}},\]. Several covariates can be evaluated simultaneously. It appears the probability of surviving beyond 1000 days is a little less than 0.2, which is confirmed by the cdf above, where we see that the probability of surviving 1000 days or fewer is a little more than 0.8. 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. The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. However, often we are interested in modeling the effects of a covariate whose values may change during the course of follow up time. Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. if lenfol > los then in_hosp = 0;
Below we demonstrate use of the assess statement to the functional form of the covariates. Not only are we interested in how influential observations affect coefficients, we are interested in how they affect the model as a whole. 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. Applied Survival Analysis. categories. Imagine we have a random variable, \(Time\), which records survival times. We see that beyond beyond 1,671 days, 50% of the population is expected to have failed. Let’s know about Multivariate Analysis Procedure – SAS/STAT. The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. Things become more complicated when dealing with survival analysis data sets, specifically because of the hazard rate. 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 LIFETEST procedure in SAS/STAT is a non-parametric procedure for analyzing survival data. hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5;
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. Grambsch, PM, Therneau, TM, Fleming TR. As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. run;
The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. 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. In the second table, we see that the hazard ratio between genders, \(\frac{HR(gender=1)}{HR(gender=0)}\), decreases with age, significantly different from 1 at age = 0 and age = 20, but becoming non-signicant by 40. Because the observation with the longest follow-up is censored, the survival function will not reach 0. These are indeed censored observations, further indicated by the “*” appearing in the unlabeled second column. 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. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). model lenfol*fstat(0) = gender|age bmi|bmi hr in_hosp ;
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. Below is an example of obtaining a kernel-smoothed estimate of the hazard function across BMI strata with a bandwidth of 200 days: The lines in the graph are labeled by the midpoint bmi in each group. \[df\beta_j \approx \hat{\beta} – \hat{\beta_j}\]. Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. 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). model lenfol*fstat(0) = gender|age bmi|bmi hr ;
1. class gender;
Data that measure lifetime or the length of time until the occurrence of an event are called lifetime, failure time, or survival data. The survival probability at time t is equal to the product of the percentage chance of surviving at time t and each prior time. Here, we cannot use linear regression methods because survival times are typically positive numbers and also ordinary linear regression may not be the best choice unless these times are first transformed in some way so that this restriction is removed. Cox models are typically fitted by maximum likelihood methods, which estimate the regression parameters that maximize the probability of observing the given set of survival times. Maximum likelihood methods attempt to find the \(\beta\) values that maximize this likelihood, that is, the regression parameters that yield the maximum joint probability of observing the set of failure times with the associated set of covariate values. Event history data can be categorized into broad categories: 1. longitudinal It performs other tasks such as computing variances of the regression parameters and producing observation level output statistics. Censored observations are represented by vertical ticks on the graph. 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. SAS computes differences in the Nelson-Aalen estimate of \(H(t)\). Survival Analysis (Life Tables, Kaplan-Meier) using PROC LIFETEST in SAS Survival data consist of a response (time to event, failure time, or survival time) variable that measures the duration of time until a specified event occurs and possibly a set of independent variables thought to be associated with the failure time variable. where \(R_j\) is the set of subjects still at risk at time \(t_j\). In all of the plots, the martingale residuals tend to be larger and more positive at low bmi values, and smaller and more negative at high bmi values. To do so: It appears that being in the hospital increases the hazard rate, but this is probably due to the fact that all patients were in the hospital immediately after heart attack, when they presumbly are most vulnerable. model lenfol*fstat(0) = gender|age bmi|bmi hr ;
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. The dfbeta measure, \(df\beta\), quantifies how much an observation influences the regression coefficients in the model. Survival analysis is a branch of statistics for analyzing the expected duration of time until one or more events happen, such as death in biological organisms and failure in mechanical systems. Second, all three fit statistics, -2 LOG L, AIC and SBC, are each 20-30 points lower in the larger model, suggesting the including the extra parameters improve the fit of the model substantially. 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. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. Before we dive into survival analysis, we will create and apply a format to the gender variable that will be used later in the seminar. 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. The second edition of Survival Analysis Using SAS: A Practical Guide is a terrific entry-level book that provides information on analyzing time-to-event data using the SAS system. The function that describes likelihood of observing \(Time\) at time \(t\) relative to all other survival times is known as the probability density function (pdf), or \(f(t)\). The graph for bmi at top right looks better behaved now with smaller residuals at the lower end of bmi. assess var=(age bmi bmi*bmi hr) / resample;
model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
The resultant output from the SAS analysis is described in Statistical software output 4. (1995). In a nutshell, these statistics sum the weighted differences between the observed number of failures and the expected number of failures for each stratum at each timepoint, assuming the same survival function of each stratum. The PROC SURVEYPHREG and MODEL statements require. Covariates are permitted to change value between intervals. In the case of categorical covariates, graphs of the Kaplan-Meier estimates of the survival function provide quick and easy checks of proportional hazards. else in_hosp = 1;
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. If, say, a regression coefficient changes only by 1% over time, it is unlikely that any overarching conclusions of the study would be affected. Let us explore it. Survival Analysis: Models and Applications: Presents basic techniques before leading onto some of the most advanced topics in survival analysis. We have already discussed this procedure in SAS/STAT Bayesian Analysis Tutorial. Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. The surface where the smoothing parameter=0.2 appears to be overfit and jagged, and such a shape would be difficult to model. SAS provides built-in methods for evaluating the functional form of covariates through its assess statement. Follow the link to know about SAS/STAT Descriptive Statistics. 77(1). 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). 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 \(\frac{355-1}{355}=0.9972\). 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. Constant multiplicative changes in the hazard rate may instead be associated with constant multiplicative, rather than additive, changes in the covariate, and might follow this relationship: \[HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))\]. 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). For example, if an individual is twice as likely to respond in week 2 as they are in week 4, this information needs to be preserved in the case-control set. Numerous examples of SAS code and output make this an eminently practical resource, ensuring that even the uninitiated becomes a sophisticated user of survival analysis. First, there may be one row of data per subject, with one outcome variable representing the time to event, one variable that codes for whether the event occurred or not (censored), and explanatory variables of interest, each with fixed values across follow up time. Biomedical and social science researchers who want to analyze survival data with SAS will find just what they need with Paul Allison's easy-to-read and comprehensive guide. So, let’s start with SAS Survival Analysis Procedures. 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. The effect of bmi is significantly lower than 1 at low bmi scores, indicating that higher bmi patients survive better when patients are very underweight, but that this advantage disappears and almost seems to reverse at higher bmi levels. For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. Here we demonstrate how to assess the proportional hazards assumption for all of our covariates (graph for gender not shown): As we did with functional form checking, we inspect each graph for observed score processes, the solid blue lines, that appear quite different from the 20 simulated score processes, the dotted lines. 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 \([0,t]\). proc sgplot data = dfbeta;
Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. Survival Analysis in SAS/STAT – PROC LIFETEST, Let’s revise SAS Nonlinear Regression Procedures. Just like LIFETEST procedure, this procedure also tests a linear hypothesis about regression parameters. A common way to address both issues is to parameterize the hazard function as: In this parameterization, \(h(t|x)\) is constrained to be strictly positive, as the exponential function always evaluates to positive, while \(\beta_0\) and \(\beta_1\) are allowed to take on any value. 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, \(H(t)\). The BMI*BMI term describes the change in this effect for each unit increase in bmi. The log-rank or Mantel-Haenzel test uses \(w_j = 1\), so differences at all time intervals are weighted equally. Whereas with non-parametric methods we are typically studying the survival function, with regression methods we examine the hazard function, \(h(t)\). 515-526. run;
Follow DataFlair on Google News & Stay ahead of the game. Wiley: Hoboken. by DarthPathos on 03-11-2016 12:39 PM (1,377 Views) Labels: Analytics U, SAS Studio, Tips and Tricks; A couple of weeks ago, I posted an article on Surviving Survival Analysis that I was really excited about. It is possible that the relationship with time is not linear, so we should check other functional forms of time, such as log(time) and rank(time). The hazard rate thus describes the instantaneous rate of failure at time \(t\) and ignores the accumulation of hazard up to time \(t\) (unlike \(F(t\)) and \(S(t)\)). 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). For example, the hazard rate when time \(t\) when \(x = x_1\) would then be \(h(t|x_1) = h_0(t)exp(x_1\beta_x)\), and at time \(t\) when \(x = x_2\) would be \(h(t|x_2) = h_0(t)exp(x_2\beta_x)\). Still, although their effects are strong, we believe the data for these outliers are not in error and the significance of all effects are unaffected if we exclude them, so we include them in the model. In large datasets, very small departures from proportional hazards can be detected. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. 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. 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. We will also learn important procedures used in Spatial Analysis in SAS/STAT: PROC KRIGE2D, PROC SIM2D, PROC SPP, … 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;
Note: This was the primary reference used for this seminar. 1 Paper SAS4286-2020 Recent Developments in Survival Analysis with SAS® Software G. Gordon Brown, SAS Institute Inc. ABSTRACT Are you interested in analyzing lifetime and survival data in SAS® software?SAS/STAT® and SAS® Visual Statistics offer a suite of procedures and survival analysis methods that enable you to overcome a variety of challenges that are frequently encountered in time … Check that their data were not incorrectly entered for “ LENFOL ” =382 specify the left and right of. Become more complicated when dealing with survival analysis for the two lowest bmi categories name,. Bmi all look reasonable as we did to check functional forms before hypothesis! Grambsch PM, Fleming TR Therneau, TM, Grambsch PM, Therneau, TM, TR... How to run survival analysis Procedures in detail `` event '' change during the beginning intervals ), Department Statistics! 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