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. Enter terms to search videos. 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. Survival Analysis (also known as Kaplan-Meier curve or Time-to-event analysis) is one of my favourite forms of analysis; this type of analysis can be used for most data that has a time-based component. Currently loaded videos are 1 through 15 of 15 total videos. 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. 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. The event can be anything like birth, death, an occurrence of a disease, divorce, marriage etc. 515-526. The exponential function is also equal to 1 when its argument is equal to 0. 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. In the code below, we model the effects of hospitalization on the hazard rate. This suggests that perhaps the functional form of bmi should be modified. Graphs are particularly useful for interpreting interactions. Applied Survival Analysis. run; proc phreg data = whas500; In our last tutorial, we studied SAS survival analysis Procedure. In particular we would like to highlight the following tables: Handily, proc phreg has pretty extensive graphing capabilities.< Below is the graph and its accompanying table produced by simply adding plots=survival to the proc phreg statement. ), Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. Below, we show how to use the hazardratio statement to request that SAS estimate 3 hazard ratios at specific levels of our covariates. Today, we will discuss SAS Survival Analysis in this SAS/STAT Tutorial. The event can be anything like birth, death, an occurrence of a disease, divorce, marriage etc. Notice the. Wiley: Hoboken. Hosmer, DW, Lemeshow, S, May S. (2008). 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. 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. In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. These are indeed censored observations, further indicated by the “*” appearing in the unlabeled second column. Censored observations are represented by vertical ticks on the graph. 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. In this interval, we can see that we had 500 people at risk and that no one died, as “Observed Events” equals 0 and the estimate of the “Survival” function is 1.0000. 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. In other words, the average of the Schoenfeld residuals for coefficient \(p\) at time \(k\) estimates the change in the coefficient at time \(k\). Objective. 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. In other words, if all strata have the same survival function, then we expect the same proportion to die in each interval. model lenfol*fstat(0) = gender age;; 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. Therneau, TM, Grambsch, PM. 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. We can plot separate graphs for each combination of values of the covariates comprising the interactions. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. Fortunately, it is very simple to create a time-varying covariate using programming statements in proc phreg. 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. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). run; proc phreg data = whas500; else in_hosp = 1; Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. Survival analysis models factors that influence the time to an event. Because this seminar is focused on survival analysis, we provide code for each proc and example output from proc corr with only minimal explanation. Any serious endeavor into data analysis should begin with data exploration, in which the researcher becomes familiar with the distributions and typical values of each variable individually, as well as relationships between pairs or sets of variables. The BMI*BMI term describes the change in this effect for each unit increase in bmi. If, say, a regression coefficient changes only by 1% over time, it is unlikely that any overarching conclusions of the study would be affected. 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. run; proc phreg data = whas500; Thus, both genders accumulate the risk for death with age, but females accumulate risk more slowly. Lin, DY, Wei, LJ, Ying, Z. proc sgplot data = dfbeta; Additionally, another variable counts the number of events occurring in each interval (either 0 or 1 in Cox regression, same as the censoring variable). 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). The probability of surviving the next interval, from 2 days to just before 3 days during which another 8 people died, given that the subject has survived 2 days (the conditional probability) is \(\frac{492-8}{492} = 0.98374\). time lenfol*fstat(0); We can estimate the hazard function is SAS as well using proc lifetest: As we have seen before, the hazard appears to be greatest at the beginning of follow-up time and then rapidly declines and finally levels off. class gender; class gender; 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. 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. Survival Analysis Using SAS: A Practical Guide, Second Edition (English Edition) eBook: Allison, Paul D.: Amazon.it: Kindle Store The background necessary to explain the mathematical definition of a martingale residual is beyond the scope of this seminar, but interested readers may consult (Therneau, 1990). Here we use proc lifetest to graph \(S(t)\). hazardratio 'Effect of 1-unit change in age by gender' age / at(gender=ALL); 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. Acquiring more than one curve, whether survival or hazard, after Cox regression in SAS requires use of the baseline statement in conjunction with the creation of a small dataset of covariate values at which to estimate our curves of interest. 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. Unless the seed option is specified, these sets will be different each time proc phreg is run. View more in. Each row of the table corresponds to an interval of time, beginning at the time in the “LENFOL” column for that row, and ending just before the time in the “LENFOL” column in the first subsequent row that has a different “LENFOL” value. All of those hazard rates are based on the same baseline hazard rate \(h_0(t_i)\), so we can simplify the above expression to: \[Pr(subject=2|failure=t_j)=\frac{exp(x_2\beta)}{exp(x_1\beta)+exp(x_2\beta)+exp(x_3\beta)}\]. We can see this reflected in the survival function estimate for “LENFOL”=382. Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); time lenfol*fstat(0); These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). It is calculated by integrating the hazard function over an interval of time: Let us again think of the hazard function, \(h(t)\), as the rate at which failures occur at time \(t\). hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5; We can remove the dependence of the hazard rate on time by expressing the hazard rate as a product of \(h_0(t)\), a baseline hazard rate which describes the hazard rates dependence on time alone, and \(r(x,\beta_x)\), which describes the hazard rates dependence on the other \(x\) covariates: In this parameterization, \(h(t)\) will equal \(h_0(t)\) when \(r(x,\beta_x) = 1\). output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; However, we can still get an idea of the hazard rate using a graph of the kernel-smoothed estimate. Martingale-based residuals for survival models. Proportional hazards tests and diagnostics based on weighted residuals. Trending. Therneau, TM, Grambsch PM, Fleming TR (1990). model lenfol*fstat(0) = gender|age bmi|bmi hr; The PROC SURVEYPHREG and MODEL statements require. 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 \(f(t)\). We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. We request Cox regression through proc phreg in SAS. 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). Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. The solid lines represent the observed cumulative residuals, while dotted lines represent 20 simulated sets of residuals expected under the null hypothesis that the model is correctly specified. However, often we are interested in modeling the effects of a covariate whose values may change during the course of follow up time. Institute for Digital Research and Education. We could test for different age effects with an interaction term between gender and age. At this stage we might be interested in expanding the model with more predictor effects. ; It makes use of full likelihood instead of a partial likelihood for estimating regression coefficients. Survival analysis often begins with examination of the overall survival experience through non-parametric methods, such as Kaplan-Meier (product-limit) and life-table estimators of the survival function. Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. 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). 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. Data that measure lifetime or the length of time until the occurrence of an event are called lifetime, failure time, or survival data. We can use the TEST statement to test whether the underlying survival functions are the same between the groups. Biometrika. Survival Analysis: Models and Applications: Presents basic techniques before leading onto some of the most advanced topics in survival analysis. 51. Springer: New York. We can examine residual plots for each smooth (with loess smooth themselves) by specifying the, List all covariates whose functional forms are to be checked within parentheses after, Scaled Schoenfeld residuals are obtained in the output dataset, so we will need to supply the name of an output dataset using the, SAS provides Schoenfeld residuals for each covariate, and they are output in the same order as the coefficients are listed in the “Analysis of Maximum Likelihood Estimates” table. Moreover, we will discuss SAS/STAT survival analysis example for better understanding. 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. Several covariates can be evaluated simultaneously. Follow the link to know about SAS/STAT Descriptive Statistics. The time for the event to occur or survival time can be measured in days, weeks, months, years, etc. 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 \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). Diagnostic plots to reveal functional form for covariates in multiplicative intensity models. Previously, we graphed the survival functions of males in females in the WHAS500 dataset and suspected that the survival experience after heart attack may be different between the two genders. We cannot tell whether this age effect for females is significantly different from 0 just yet (see below), but we do know that it is significantly different from the age effect for males. 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. Provided the reader has some background in survival analysis, these sections are not necessary to understand how to run survival analysis in SAS. To accomplish this smoothing, the hazard function estimate at any time interval is a weighted average of differences within a window of time that includes many differences, known as the bandwidth. 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. 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. The LIFETEST procedure in SAS/STAT is a nonparametric procedure for analyzing survival data. model lenfol*fstat(0) = gender|age bmi|bmi hr ; This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. 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 \(S(t)\)). Indeed, exclusion of these two outliers causes an almost doubling of \(\hat{\beta}_{bmi}\), from -0.23323 to -0.39619. Note: A number of sub-sections are titled Background. These statement essentially look like data step statements, and function in the same way. model lenfol*fstat(0) = gender age;; 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. One can also use non-parametric methods to test for equality of the survival function among groups in the following manner: In the graph of the Kaplan-Meier estimator stratified by gender below, it appears that females generally have a worse survival experience. SAS computes differences in the Nelson-Aalen estimate of \(H(t)\). This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. However they lived much longer than expected when considering their bmi scores and age (95 and 87), which attenuates the effects of very low bmi. Perform search. We have already discussed this procedure in SAS/STAT Bayesian Analysis Tutorial. Understanding the mechanics behind survival analysis is aided by facility with the distributions used, which can be derived from the probability density function and cumulative density functions of survival times. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. This procedure in SAS/STAT is specially designed to perform nonparametric or statistical analysis of interval-censored data. class gender; Notice that the interval during which the first 25% of the population is expected to fail, [0,297) is much shorter than the interval during which the second 25% of the population is expected to fail, [297,1671). A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. Thus, at the beginning of the study, we would expect around 0.008 failures per day, while 200 days later, for those who survived we would expect 0.002 failures per day. On the right panel, “Residuals at Specified Smooths for martingale”, are the smoothed residual plots, all of which appear to have no structure. 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 \(Pr(Time > t)\). For example, if the survival times were known to be exponentially distributed, then the probability of observing a survival time within the interval \([a,b]\) is \(Pr(a\le Time\le b)= \int_a^bf(t)dt=\int_a^b\lambda e^{-\lambda t}dt\), where \(\lambda\) is the rate parameter of the exponential distribution and is equal to the reciprocal of the mean survival time. 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. In other words, we would expect to find a lot of failure times in a given time interval if 1) the hazard rate is high and 2) there are still a lot of subjects at-risk. The same procedure could be repeated to check all covariates. 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. 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. What we most often associate with this approach to survival analysis and what we generally see in practice are the Kaplan-Meier curves — a plot of the Kaplan-Meier estimator over time. format gender gender. 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. 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. In the above example, the time variable is height and the censoring variable is weight with value 4 indicating censored observations. The main topics presented include censoring, survival curves, Kaplan-Meier estimation, accelerated failure time models, Cox regression models, and discrete-time analysis. scatter x = age y=dfage / markerchar=id; It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. 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. Survival analysis is a set of methods for analyzing data in which the outcome variable is the time until an event of interest occurs. Subjects that are censored after a given time point contribute to the survival function until they drop out of the study, but are not counted as a failure. Positive values of \(df\beta_j\) indicate that the exclusion of the observation causes the coefficient to decrease, which implies that inclusion of the observation causes the coefficient to increase. 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 \(HR = exp(0.5(1)) = 1.6487\). 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