SAS Seminar
Introduction to Survival Analysis in SAS

1. Introduction

Survival analysis models factors that influence the time to an event. Ordinary least squares regression methods fall short because the time to event is typically not normally distributed, and the model cannot handle censoring, very common in survival data, without modification. Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. This seminar introduces procedures and outlines the coding needed in SAS to model survival data through both of these methods, as well as many techniques to evaluate and possibly improve the model. Particular emphasis is given to proc lifetest for nonparametric estimation, and proc phreg for Cox regression and model evaluation.

Note: A slide presentation associated with the live presentation of this seminar can be downloaded here. The code used in this seminar can be found here (right-click to download).

Note: A number of sub-sections are titled Background. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). Provided the reader has some background in surival analysis, these sections are not necessary to understand how to run survival analysis in SAS. These may be either removed or expanded in the future.

Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time.

1.1. Sample dataset

Click here to download the dataset used in this 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)). This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. Follow up time for all participants begins at the time of hospital admission after heart attack and ends with death or loss to follow up (censoring). The variables used in the present seminar are:

The data in the WHAS500 are subject to right-censoring only. That is, for some subjects we do not know when they died after heart attack, but we do know at least how many days they survived.

1.2. Background: Important distributions in survival analysis

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.

1.2.1. Background: The probability density function, $f(t)$

Let T represent survival time, or the time to an event. The function that describes the probability of observing a particular survival time is known as the probability density function (pdf), or $f(t)$. For example, if the survival times were known to be exponentially distributed, then the probability of observing survival time, T, is $Pr(T=t)=f(t)=\lambda e^{-\lambda t}$, where $\lambda$ is the rate parameter of the exponential distribution and is equal to the reciprocal of the mean survival time. 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. 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.


proc univariate data = whas500(where=(fstat=1));
var lenfol;
histogram lenfol / kernel;
run;

In the graph above we see the correspondence between pdfs and histograms. Density functions are esentially histograms comprised of bins of vanishingly small widths. 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. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0)

.

1.2.2. Background: The cumulative distribution function, $F(T)$

Whereas the probability function describes the probability, $Pr(T=t)$, of observing a single surival time $T$, the cumulative distribution function (cdf), $F(t)$, describes the probability of observing a survival time $T$ less than or equal to some time $t$, or $Pr(T<=t)$. In other words we use the pdf to determine the probability of observing a survival time of 100 days and the cdf to determine the probability of observing a survival time of up to 100 days. The pdf and cdf have a simple relationship:

$$f(t) = \frac{dF(t)}{dt}$$

We thus calculate the cdf $F(t)$ by integrating the pdf $f(t)$ over the range of survival times. In SAS, we can graph an estimate of the cdf using proc univariate.

 
proc univariate data = whas500(where=(fstat=1));
var lenfol;
cdfplot lenfol;
run;

In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. Thus, by 200 days, a patient has accumulated quite a bit of risk, which begins to accumulate more slowly after this point. In intervals where event times are more probable (here the beginning intervals), the cdf will increase faster.

1.2.3. Background: The Survival function, S(t)

A simple tranformation of the cumulative distribution function produces the survival function, S(t):

$$S(t) = 1 - F(T)$$

The survivor function, S(t), describes the probability of surviving past time t, or Pr(T > t). If we were to plot the estimate of S(t), we would see that it is a reflection of F(t) (about y=0 and shifted up by 1). Here we use proc lifetest to graph S(t).


proc lifetest data=whas500(where=(fstat=1)) plots=survival(atrisk);
time lenfol*fstat(0);
run; 

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.

1.2.4. Background: The hazard function, $h(t)$

The primary focus of survival anlaysis is typically to model the hazard rate, which has the following relationship with the pdf and S(t):

$$h(t)=\frac{f(t)}{S(t)}$$

The hazard function, then, describes the probability of the event occurring at time $t$ ($f(t)$), conditional on the subject's survival up to that time $t$ ($S(t)$). 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 can estimate the hazard function is SAS as well using proc lifetest:


proc lifetest data=whas500(where=(fstat=1)) plots=hazard(bw=200);
time lenfol*fstat(0);
run;

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. Indeed the hazard rate rigth at the beginning is more than 4 times larger than the hazard 200 days later. Thus, at the begninning 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.

1.2.5. Background: The cumulative hazard function

Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. It is calculated by integrating the hazard function over an interval of time:

$$H(t) = \int_0^th(u)du$$

The cumulative hazard function, then, gives the number of expected number of failures over time interval $t$. Put another way, the hazard function describes the rate at which hazards are accumlated over time. 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:

$$S(t) = exp(-H(t))$$ $$F(t) = 1 - exp(-H(t))$$ $$f(t) = h(t)exp(-H(t))$$

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. As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. 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). 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.

We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. We see a sharper rise in the cumulative hazard at the beginning of analysis time, reflecting the larger hazard rate during this period.


ods output ProductLimitEstimates = ple;
proc lifetest data=whas500(where=(fstat=1))  nelson outs=outwhas500;
time lenfol*fstat(0);
run;
proc sgplot data = ple; series x = lenfol y = CumHaz; run;

2. Data preparation and exploration

2.1. Structure of the data

This seminar covers both proc lifetest and proc phreg, and data can be structured in one of 2 ways for 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. Both proc lifetest and proc phreg will accept data structured this way. The WHAS500 data are stuctured this way. Notice there is one row per subject, with one variable coding the time to event, lenfol:

Obs ID LENFOL FSTAT AGE BMI HR GENDER
1 1 2178 0 83 25.5405 89 Male
2 2 2172 0 49 24.0240 84 Male
3 3 2190 0 70 22.1429 83 Female
4 4 297 1 70 26.6319 65 Male
5 5 2131 0 70 24.4125 63 Male

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. For each subject, the entirety of follow up time is partitioned into intervals, each defined by a "start" and "stop" time. Covariates are permitted to change value between intervals. Additionally, another variable counts the number of events occurring in each interval (either 0 or 1 in Cox regression, same as the censoring variable). 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. This subject could be represented by 2 rows like so:

Obs id start stop status treatment
1 1 0 100 0 1
2 1 100 2178 1 0

This structuring allows the modeling of time-varying covariates, or explanatory variables whose values change across follow-up time. Data that are structured in the first, single-row way can be modified to be structured like the second, multi-row way, but the reverse is typically not true. We will model a time-varying covariate later in the seminar.

2.2. Data exploration with proc univariate and proc corr

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 wekk as relationships between pairs or sets of variables. Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. Because this seminar is focused on survival analysis, we provide code for each proc and example output from proc corr with only minimal explanation.


proc univariate data=whas500;
var lenfol age bmi hr;
run;

proc corr data = whas500 plots(maxpoints=none)=matrix(histogram);
var lenfol gender age bmi hr;
run;
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
LENFOL 500 882.43600 705.66513 441218 1.00000 2358
GENDER 500 0.40000 0.49039 200.00000 0 1.00000
AGE 500 69.84600 14.49146 34923 30.00000 104.00000
BMI 500 26.61378 5.40566 13307 13.04546 44.83886
HR 500 87.01800 23.58623 43509 35.00000 186.00000

Pearson Correlation Coefficients, N = 500
  LENFOL GENDER AGE BMI HR
LENFOL 1.00000 -0.06367 -0.31221 0.19263 -0.17974
GENDER -0.06367 1.00000 0.27489 -0.14858 0.11598
AGE -0.31221 0.27489 1.00000 -0.40248 0.14914
BMI 0.19263 -0.14858 -0.40248 1.00000 -0.05579
HR -0.17974 0.11598 0.14914 -0.05579 1.00000


We see in the table above, that the typical subject in our dataset is more likely male, 70 years of age, with a bmi of 26.6 and heart rate of 87. The mean time to event (or loss to followup) is 882.4 days, not a particularly useful quantity. All of these variables vary quite a bit in these data. Most of the variables are at least slightly correlated with the other variables.

3. Nonparametric (Descriptive) Survival Analysis using proc lifetest

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. Non-parametric methods are appealing because no assumption of the shape of the survivor function nor of the hazard function need be made. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. Still, they are useful as exploratory and descriptive tools.

3.1. The Kaplan-Meier estimator of the survival function

3.1.1 Background: The Kaplan Meier Estimator:

We can estimate the probability of survival up to each to each event time in the data (each time an event occurred) using the Kaplan_Meier survival function estimator, which is calculated as:

$$\hat S(t)=\prod_{t_i\leq t}\frac{n_i - d_i}{n_i}, $$

where $n_i$ is the number of subjects at risk and $d_i$ is the number of subjects who fail, both at time $t_i$. Thus, each term in the product is the conditional probability of survival beyond time $t_i$, meaning the probability of surviving beyond time $t_i$, given the subject has survived up to time $t_i$. The survial function estimate of the the unconditional probability of survival beyond time $t$ (the probability of survival beyond time $t$ from the onset of risk) is then obtained by multiplying together these conditional probabilities up to time $t$ together.

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 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 $\frac{492-8}{492} = 0.97374$. The unconditional probability of surviving beyond 2 days (from the onset of risk) then is $\hat S(2) = \frac{500 - 8}{500}\times\frac{492-8}{492} = 0.984\times0.97374=0.9680$

3.1.2. Obtaining and interpreting tables of Kaplan-Meier Estimates from proc lifetest

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:


proc lifetest data=whas500 atrisk outs=outwhas500;
time lenfol*fstat(0);
run;
Product-Limit Survival Estimates
LENFOL   Number
at Risk
Observed
Events
Survival Failure Survival Standard
Error
Number
Failed
Number
Left
0.00   500 0 1.0000 0 0 0 500
1.00   . . . . . 1 499
1.00   . . . . . 2 498
1.00   . . . . . 3 497
1.00   . . . . . 4 496
1.00   . . . . . 5 495
1.00   . . . . . 6 494
1.00   . . . . . 7 493
1.00   500 8 0.9840 0.0160 0.00561 8 492
2.00   . . . . . 9 491
2.00   . . . . . 10 490
2.00   . . . . . 11 489
2.00   . . . . . 12 488
2.00   . . . . . 13 487
2.00   . . . . . 14 486
2.00   . . . . . 15 485
2.00   492 8 0.9680 0.0320 0.00787 16 484
3.00   . . . . . 17 483
3.00   . . . . . 18 482
3.00   484 3 0.9620 0.0380 0.00855 19 481

Above we see the table of Kaplan-Meier estimates of the survival function produced by proc lifetest. 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. For example, the time interval represented by the first row is from 0 days to just before 1 day. 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. 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. It is important to note that the survival probabilities listed in the Survival column are unconditional, and are to be interpreted as the probability of surviving from the beginning of follow up time up to the number days in the LENFOL column.

Let's take a look at later survival times in the table:

Product-Limit Survival Estimates
LENFOL   Number
at Risk
Observed
Events
Survival Failure Survival Standard
Error
Number
Failed
Number
Left
359.00   . . . . . 136 364
359.00   365 2 0.7260 0.2740 0.0199 137 363
363.00   363 1 0.7240 0.2760 0.0200 138 362
368.00 * 362 0 . . . 138 361
371.00 * . 0 . . . 138 360
371.00 * . 0 . . . 138 359
371.00 * 361 0 . . . 138 358
373.00 * 358 0 . . . 138 357
376.00 * . 0 . . . 138 356
376.00 * 357 0 . . . 138 355
382.00   355 1 0.7220 0.2780 0.0200 139 354
385.00   354 1 0.7199 0.2801 0.0201 140 353

From "LENFOL"=368 to 376, we see that there are several records where it appears no events occurred. These are indeed censored observations, further indicated by the "*" appearing in the unlabeled second column. 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. We can see this reflected in the survival function estimate for "LENFOL"=382. During the interval [382,385) 1 out of 355 subjects at-risk died, yielding a conditional probability of failure (the probability of failure in the given interval, given that the subject has survived up to the begininng of the interval) in this interval of $\frac{355-1}{355}=0.9972$. 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 $p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=0.7240$. In the table above, we see that the probability surviving beyond 363 days = 0.7240, the same probability as what we calculated for surviving up to 382 days, which implies that the censored observations do not change the survival estimates when they leave the study, only the number at risk.

3.1.3. Graphing the Kaplan-Meier estimate

Graphs of the Kaplan-Meier estimate of the survival function allow us to see how the survival function changes over time and are fortunately very easy to generate in SAS:


proc lifetest data=whas500 atrisk plots=survival(cb) outs=outwhas500;
time lenfol*fstat(0);
run;

The step function form of the survival function is apparent in the graph of the Kaplan-Meier estimate. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. 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. Censored observations are represented by vertical ticks on the graph. Notice the survival probability does not change when we encounter a censored observation. Because the observation with the longest follow-up is censored, the survival function will not reach 0. Instead, the survival function will remain at the survival probability estimated at the previous interval. The survival function is undefined past this final interval at 2358 days. The blue-shaded area around the survival curve represents the 95% confidence band, here Hall-Wellner confidence bands. 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. Many transformations of the survivor function are available for alternate ways of calculating confidence intervals through the conftype option, though most transformations should yield very similar confidence intervals.

3.2. Nelson-Aalen estimator of the cumulative hazard function

Because of its simple relationship with the survival function, $S(t)=e^{-H(t)}$, the cumulative hazard function can be used to estimate the survival function. The Nelson-Aalen estimator is a non-parametric estimator of the cumulative hazard function and is given by:

$$\hat H(t) = \sum_{t_i \leq t}\frac{d_i}{n_i},$$

where $d_i$ is the number who failed out of $n_i$ at risk in interval $t_i$. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time $t$.


proc lifetest data=whas500 atrisk nelson;
time lenfol*fstat(0);
run;
Survival Function and Cumulative Hazard Rate
LENFOL   Number
at Risk
Observed
Events
Product-Limit Nelson-Aalen Number
Failed
Number
Left
Survival Failure Survival Standard
Error
Cumulative
Hazard
Cum Haz
Standard
Error
0.00   500 0 1.0000 0 0 0 . 0 500
1.00   . . . . . . . 1 499
1.00   . . . . . . . 2 498
1.00   . . . . . . . 3 497
1.00   . . . . . . . 4 496
1.00   . . . . . . . 5 495
1.00   . . . . . . . 6 494
1.00   . . . . . . . 7 493
1.00   500 8 0.9840 0.0160 0.00561 0.0160 0.00566 8 492
2.00   . . . . . . . 9 491
2.00   . . . . . . . 10 490
2.00   . . . . . . . 11 489
2.00   . . . . . . . 12 488
2.00   . . . . . . . 13 487
2.00   . . . . . . . 14 486
2.00   . . . . . . . 15 485
2.00   492 8 0.9680 0.0320 0.00787 0.0323 0.00807 16 484
3.00   . . . . . . . 17 483
3.00   . . . . . . . 18 482
3.00   484 3 0.9620 0.0380 0.00855 0.0385 0.00882 19 481

Let's confirm our understanding of the calculation of the Nelson-Aalen estimator by calculating the estimated cumulative hazard at day 3: $\hat H(3)=\frac{8}{500} + \frac{8}{492} + \frac{3}{484} = 0.0385$, which matches the value in the table. The interpretation of this estimate is that we expect 0.0385 failures (per person) by the end of 3 days. The estimate of survival beyond 3 days based off this Nelson-Aalen estimate of the cumulative hazard would then be $\hat S(3) = exp(-0.0385) = 0.9623$. This matches closely with the Kaplan Meier product-limit estimate of survival beyond 3 days of 0.9620. 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. In very large samples the Kaplan-Meier estimator and the transformed Nelson-Aalen (Breslow) estimator will converge.

3.3. Calculating median, mean, and other survival times of interest in proc lifetest

Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. Because of the positive skew often seen with followup-times, medians are often a better indicator of an "average" survival time. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. We see that beyond beyond 1,671 days, 50% of the population is expected to have failed. 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). This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time.


proc lifetest data=whas500 atrisk nelson;
time lenfol*fstat(0);
run;
Quartile Estimates
Percent Point
Estimate
95% Confidence Interval
Transform [Lower Upper)
75 2353.00 LOGLOG 2350.00 2358.00
50 1627.00 LOGLOG 1506.00 2353.00
25 296.00 LOGLOG 146.00 406.00

Mean Standard Error
1417.21 48.14

3.4. Comparing survival functions using nonparametric tests

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). One can also use non-parametric methods to test for equality of the survival function among groups in the following manner:


proc lifetest data=whas500 atrisk plots=survival(atrisk cb) outs=outwhas500;
strata gender;
time lenfol*fstat(0);
run;
Test of Equality over Strata
Test Chi-Square DF Pr >
Chi-Square
Log-Rank 7.7911 1 0.0053
Wilcoxon 5.5370 1 0.0186
-2Log(LR) 10.5120 1 0.0012

In the graph of the Kaplan-Meier estimator stratified by gender below, it appears that females generally have a worse survival experience. This is reinforced by the three significant tests of equality.

3.4.1. Background: Tests of equality of the survival function

In the output we find three Chi-square based tests of the equality of the survival function over strata, which support our suspicion that survival differs between genders. The calculation of the statistic for the nonparametric "Log-Rank" and "Wilcoxon" tests is given by :

$$Q = \frac{\bigg[\sum\limits_{i=1}^m w_j(d_{ij}-\hat e_{ij})\bigg]^2}{\sum\limits_{i=1}^m w_j^2\hat v_{ij}},$$

where $d_{ij}$ is the observed number of failures in stratum $i$ at time $t_j$, $\hat e_{ij}$ is the expected number of failures in stratum $i$ at time $t_j$, $\hat v_{ij}$ is the estimator of the variance of $d_{ij}$, and $w_i$ is the weight of the difference at time $t_j$ (see Hosmer and Lemeshow(2008) for formulas for $\hat e_{ij}$ and $\hat v_{ij}$). 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. In other words, if all strata have the same survival function, then we expect the same proportion to die in each interval. If these proportions systematically differ among strata across time, then the $Q$ statistic will be large and the null hypothesis of no difference among strata is more likely to be rejected.

The log-rank and Wilcoxon tests in the output table differ in the weights $w_j$ used. The log-rank or Mantel-Haenzel test uses $w_j = 1$, so differences at all time intervals are weighted equally. 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. Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement. The "-2Log(LR)" likelihood ratio test is a parametric test assuming exponentially distributed survival times and will not be further discussed in this nonparametric section.

3.5. Nonparametric estimation of the hazard function

Standard nonparametric techniques do not typically estimate the hazard function directly. However, we can still get an idea of the hazard rate using a graph of the kernel-smoothed estimate. As the hazard function $h(t)$ is the derivative of the cumulative hazard function $H(t)$, we can roughly estimate the rate of change in $H(t)$ by taking successive differences in $\hat H(t)$ between adjacent time points, $\Delta \hat H(t) = \hat H(t_j) - \hat H(t_{j-1})$. SAS computes differences in the Nelson-Aalen estimate of $H(t)$. We generally expect the hazard rate to change smoothly (if it changes) over time, rather than jump around haphazardly. 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. Widening the bandwidth smooths the function by averaging more differences together. 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. Below is an example of obtaining a kernel-smoothed estimate of the hazard function across BMI strata with a bandwidth of 200 days:


proc lifetest data=whas500 atrisk plots=hazard(bw=200) outs=outwhas500;
strata bmi(15,18.5,25,30,40);
time lenfol*fstat(0);
run;

The lines in the graph are labeled by the midpoint bmi in each group. From the plot we can see that the hazard function indeed appears higher at the beginning of follow-up time and then decreases until it levels off at around 500 days and stays low and mostly constant. The hazard function is also generally higher for the two lowest BMI categories. 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. The red curve representing the lowest BMI category is truncated on the right because the last person in that group died long before the end of followup time.

4. Background: The Cox proportional hazards regression model

4.1. Background: Estimating the hazard function, $h(t)$

Whereas with non-parametric methods we are typically studying the survival function, with regression methods we examine the hazard function, $h(t)$. 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. 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, $H(t)$.

In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. We would like to allow parameters, the $\beta$s, to take on any value, while still preserving the non-negative nature of the hazard rate. A common way to address both issues is to parameterize the hazard function as:

$$h(t|x)=exp(\beta_0 + \beta_1x)$$.

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. Notice, however, that $t$ does not appear in the formula for the hazard function, thus implying that in this parameterization, we do not model the hazard rate's dependence on time. 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). However, in many settings, we are much less interested in modeling the hazard rate's relationship with time and are more interested in its dependence on other variables, such as experimental treatment or age. For such studies, a semi-parametric model, in which we estimate regression parameters as covariate effects but ignore (leave unspecified) the dependence on time, is appropriate.

4.2. Background: The Cox proportional hazards model

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:

$$h(t)=h_0(t)r(x,\beta_x)$$.

In this parameterization, $h(t)$ will equal $h_0(t)$ when $r(x,\beta_x) = 1$. It is intuitively appealing to let $r(x,\beta_x) = 1$ when all $x = 0$, thus making the baseline hazard rate, $h_0(t)$, equivalent to a regression intercept. 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. The exponential function is also equal to 1 when its argument is equal to 0. We will thus let $r(x,\beta_x) = exp(x\beta_x)$, and the hazard function will be given by:

$$h(t)=h_0(t)exp(x\beta_x)$$.

This parameterization forms the Cox proportional hazards model. It is called the proportional hazards model because the ratio of hazard rates between two groups with fixed covariates will stay constant over time in this model. 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 $h(t|x_2) = h_0(t)exp(x_2\beta_x)$. The covariate effect of $x$, then is the ratio between these two hazard rates, or a hazard ratio(HR):

$$HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}$$

Notice that the baseline hazard rate, $h_0(t)$ is cancelled out, and that the hazard rate does not depend on time $t$:

$$HR = exp(\beta_x(x_2-x_1))$$

The hazard rate $HR$ will thus stay constant over time with fixed covariates. Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. 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. 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.

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. So what is the probability of observing subject $i$ fail at time $t_j$? At the beginning of a given time interval $t_j$, say there are $R_j$ subjects still at-risk, each with their own hazard rates:

$$h(t_j|x_i)=h_0(t_j)exp(x_i\beta)$$

The probability of observing subject $j$ fail out of all $R_j$ remaing at-risk subjects, then, is the proportion of the sum total of hazard rates of all $R_j$ subjects that is made up by subject $j$'s hazard rate. 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:

$$Pr(subject=2|failure=t_j)=\frac{h(t_j|x_2)}{h(t_j|x_1)+h(t_j|x_2)+h(t_j|x_3)}$$

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 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 $x_j$:

$$L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{i\in R_j}exp(x_i\beta)}\Bigg\rbrace$$

where $R_j$ is the set of subjects still at risk at time $t_j$. 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. 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.

.

5. Cox proportional hazards regression in SAS using proc phreg

5.1. Fitting a simple Cox regression model

We request Cox regression through proc phreg in SAS. 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. Perhaps you also suspect that the hazard rate changes with age as well. Below we demonstrate a simple model in proc phreg, where we determine the effects of a categorical predictor, gender, and a continuous predictor, age on the hazard rate:


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender age;
run;
Model Fit Statistics
Criterion Without
Covariates
With
Covariates
-2 LOG L 2455.158 2313.140
AIC 2455.158 2317.140
SBC 2455.158 2323.882

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 142.0177 2 <.0001
Score 126.6381 2 <.0001
Wald 119.3806 2 <.0001

Type 3 Tests
Effect DF Wald Chi-Square Pr > ChiSq
GENDER 1 0.2175 0.6410
AGE 1 116.3986 <.0001

Analysis of Maximum Likelihood Estimates
Parameter   DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Label
GENDER Female 1 -0.06556 0.14057 0.2175 0.6410 0.937 GENDER Female
AGE   1 0.06683 0.00619 116.3986 <.0001 1.069  

The above output is only a portion of what SAS produces each time you run proc phreg. In particular we would like to highlight the following tables:

5.2. Producing graphs of the survival function after Cox regression

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./p>


proc phreg data=whas500 plots=survival;
class gender;
model lenfol*fstat(0) = gender age;;
run;

Reference Set of Covariates for
Plotting
AGE GENDER
69.845947 Male

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.

5.2.1. Use the baseline statement to generate survival plots by group

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. 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.


proc format library=mylib;
value gender
	  0 = "Male"
	  1 = "Female";
run;

options fmtsearch = (mylib);

data covs;
format gender gender.;
input gender age;
datalines;
0 69.845947
1 69.845947
;
run;

proc phreg data = whas500 plots(overlay)=(survival);
class gender;
model lenfol*fstat(0) = gender age;
baseline covariates=covs out=base / rowid=gender;
run;

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. The estimated hazard ratio of .937 comparing females to males is not significant.

5.3. Expanding and interpreting the Cox regression model with interaction terms

In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. At this stage we might be interested in expanding the model with more predictor effects. For example, we found that the gender effect seems to disappear after accounting for age, but we may suspect that the effect of age is different for each gender. We could test for different age effects with an interaction term between gender and age. Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well.

In the code below we fit a Cox regression model where we allow examine the effects of gender, age, bmi, and heart rate on the hazard rate. Here, we would like to introdue two types of interaction:


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr ;
run;
Model Fit Statistics
Criterion Without
Covariates
With
Covariates
-2 LOG L 2455.158 2276.150
AIC 2455.158 2288.150
SBC 2455.158 2308.374

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 179.0077 6 <.0001
Score 174.7924 6 <.0001
Wald 154.9497 6 <.0001

Type 3 Tests
Effect DF Wald Chi-Square Pr > ChiSq
GENDER 1 4.5117 0.0337
AGE 1 72.0368 <.0001
AGE*GENDER 1 5.4646 0.0194
BMI 1 7.0382 0.0080
BMI*BMI 1 4.8858 0.0271
HR 1 21.4528 <.0001

Analysis of Maximum Likelihood Estimates
Parameter   DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Label
GENDER Female 1 2.10986 0.99330 4.5117 0.0337 . GENDER Female
AGE   1 0.07086 0.00835 72.0368 <.0001 .  
AGE*GENDER Female 1 -0.02925 0.01251 5.4646 0.0194 . GENDER Female * AGE
BMI   1 -0.23323 0.08791 7.0382 0.0080 .  
BMI*BMI   1 0.00363 0.00164 4.8858 0.0271 . BMI * BMI
HR   1 0.01277 0.00276 21.4528 <.0001 1.013  

We would probably prefer this model to the simpler model with just gender and age as explanatory factors for a couple of reasons. First, each of the effects, including both interactions, are significant. 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.

Let's interpret our model. We should begin by analyzing our interactions. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. Recall that when we introduce interactions into our model, each individual term comprising that interaction (such as GENDER and AGE) is no longer a main effect, but is instead the simple effect of that variable with the interacting variable held at 0. Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. It appears that for males the log hazard rate increases with each year of age by 0.07086, and this AGE effect is significant, p<0.0001. The age effect is less severe for females, as the 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. 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. 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. The BMI*BMI term describes the change in this effect for each unit increase in bmi. 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.

5.4. Using the hazardratio statement and graphs to interpret effects, particularly interactions

Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. SAS omits them to remind you that the hazard ratios corresponding to these effects depend on other variables in the model.

Below, we show how to use the hazardratio statement to request that SAS estimate 3 hazard ratios at specific levels of our covariates.


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr ;
hazardratio 'Effect of 1-unit change in age by gender' age / at(gender=ALL); 
hazardratio 'Effect of gender across ages' gender / at(age=(0 20 40 60 80));
hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5;
run;
Effect of 1-unit change in age by gender: Hazard Ratios for AGE
Description Point Estimate 95% Wald Confidence Limits
AGE Unit=1 At GENDER=Female 1.042 1.022 1.063
AGE Unit=1 At GENDER=Male 1.073 1.056 1.091

Effect of gender across ages: Hazard Ratios for GENDER
Description Point Estimate 95% Wald Confidence Limits
GENDER Female vs Male At AGE=0 8.247 1.177 57.783
GENDER Female vs Male At AGE=20 4.594 1.064 19.841
GENDER Female vs Male At AGE=40 2.559 0.955 6.857
GENDER Female vs Male At AGE=60 1.426 0.837 2.429
GENDER Female vs Male At AGE=80 0.794 0.601 1.049

Effect of 5-unit change in bmi across bmi: Hazard Ratios for BMI
Description Point Estimate 95% Wald Confidence Limits
BMI Unit=5 At BMI=15 0.588 0.428 0.809
BMI Unit=5 At BMI=18.5 0.668 0.535 0.835
BMI Unit=5 At BMI=25 0.846 0.733 0.977
BMI Unit=5 At BMI=30 1.015 0.797 1.291
BMI Unit=5 At BMI=40 1.459 0.853 2.497

In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. 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). Thus, in the first table, we see that the hazard ratio for age, $\frac{HR(age+1)}{HR(age)}$, is lower for females than for males, but both are significantly different from 1. Thus, both genders accumulate the risk for death with age, but females accumulate risk more slowly. 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. 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. Finally, we see that the hazard ratio describing a 5-unit increase in bmi, $\frac{HR(bmi+5)}{HR(bmi)}$, increases with bmi. 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.

Graphs are particularly useful for interpreting interactions. We can plot separate graphs for each combination of values of the covariates comprising the interactions. Below we plot survivor curves across several ages for each gender through the follwing steps:




data covs2;
format gender gender.;
input gender age bmi hr;
datalines;
0 40 26.614 23.586
0 60 26.614 23.586
0 80 26.614 23.586
1 40 26.614 23.586
1 60 26.614 23.586
1 80 26.614 23.586
;
run;

proc phreg data = whas500 plots(overlay=group)=(survival);
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr ;
baseline covariates=covs2  / rowid=age group=gender;
run;

As we surmised earlier, the effect of age appears to be more severe in males than in females, reflected by the greater separation between curves in the top graaph.

5.5. Create time-varying covariates with programming statements

Thus far in this seminar we have only dealt with covariates with values fixed across follow up time. With such data, each subject can be represented by one row of data, as each covariate only requires only value. However, often we are interested in modeling the effects of a covariate whose values may change during the course of follow up time. For example, patients in the WHAS500 dataset are in the hospital at the beginnig of follow-up time, which is defined by hospital admission after heart attack. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. 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. Our goal is to transform the data from its original state:

Obs ID LENFOL FSTAT LOS
1 1 2178 0 5
2 2 2172 0 5
3 3 2190 0 5
4 4 297 1 10
5 5 2131 0 6
6 6 1 1 1
7 7 2122 0 5

to an expanded state that can accommodate time-varying covariates, like this (notice the new variable in_hosp):

Obs ID start stop status in_hosp
1 1 0 5 0 1
2 1 5 2178 0 0
3 2 0 5 0 1
4 2 5 2172 0 0
5 3 0 5 0 1
6 3 5 2190 0 0
7 4 0 10 0 1
8 4 10 297 1 0
9 5 0 6 0 1
10 5 6 2131 0 0
11 6 0 1 1 1
12 7 0 5 0 1
13 7 5 2122 0 0

Notice the creation of start and stop variables, which denote the beginning and end intervals defined by hospitalization and death (or censoring). Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject.

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. 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. 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).

Fortunately, it is very simple to create a time-varying covariate using programming statements in proc phreg. These statement essentially look like data step statements, and function in the same way. In the code below, we model the effects of hospitalization on the hazard rate. To do so:

   
proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr in_hosp ;
if lenfol > los then in_hosp = 0;
else in_hosp = 1; 
run;

Analysis of Maximum Likelihood Estimates
Parameter   DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Label
GENDER Female 1 2.16143 1.00426 4.6322 0.0314 . GENDER Female
AGE   1 0.07301 0.00851 73.6642 <.0001 .  
AGE*GENDER Female 1 -0.03025 0.01266 5.7090 0.0169 . GENDER Female * AGE
BMI   1 -0.22302 0.08847 6.3548 0.0117 .  
BMI*BMI   1 0.00348 0.00166 4.4123 0.0357 . BMI * BMI
HR   1 0.01222 0.00277 19.4528 <.0001 1.012  
in_hosp   1 2.09971 0.39617 28.0906 <.0001 8.164  

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.

6. Exploring functional form of covariates

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 ($HR$)):

$$HR = exp(\beta_x(x_2-x_1))$$

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. 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 $HR = exp(0.5(1)) = 1.6487$. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. 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:

$$HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))$$

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.

It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be.

6.1 Plotting cumulative martingale residuals against covariates to determine the functional form of covariates

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). For this seminar, it is enough to know that the martingale residual can be interpreted as a measure of excess observed events, or the difference between the observed number of events and the expected number of events under the model:

$$martingale~ residual = excess~ observed~ events = observed~ events - (expected~ events|model)$$

Therneau and colleagues(1990) show that the smooth of a scatter plot of the martingale residuals from a null model (no covariates at all) versus each covariate individually will often approximate the correct functional form of a covariate. Previously we suspected that the effect of bmi on the log hazard rate may not be purely linear, so it would be wise to investigate further. In the code below we demonstrate the steps to take to explore the functional form of a covariate:


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = ;
output out=residuals resmart=martingale;
run;

proc loess data = residuals plots=ResidualsBySmooth(smooth);
model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
run;

In the left panel above, "Fits with Specified Smooths for martingale", we see our 4 scatter plot smooths. 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. 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). On the right panel, "Residuals at Specified Smooths for martingale", are the smoothed residual plots, all of which appear to have no structure. The surface where the smoothing parameter=0.2 appears to be overfit and jagged, and such a shape would be difficult to model. However, each of the other 3 at the higher smoothing parameter values have very similar shapes, which appears to be a linear effect of bmi that flattens as bmi increases. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. One caveat is that this method for determining functional form is less reliable when covariates are correlated. Howevever, despite our knowledge that bmi is correlated with age, this method provides good insight into bmi's functional form.

6.2. Using the assess statement to explore functional forms

SAS provides built-in methods for evaluating the functional form of covariates through its assess statement. These techniques were developed by Lin, Wei and Zing (1993). The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. If our Cox model is correctly specified, these cumulative martingale sums should randomly flucatuate around 0. Significant departures from random error would suggest model misspecification. We could thus evaluate model specification by comparing the observed distribution of cumulative sums of martinglae residuals to the expected distribution of the residuals under the null hypthesis that the model is correctly specified. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. 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 such cases, the correct form may be inferred from the plot of the observed pattern. This technique can detect many departures from the true model, such as incorrect functional forms of covariates (discussed in this section), violations of the proportional hazards assumption (discussed later), and using the wrong link function (not discussed).

Below we demonstrate use of the assess statement to the functional form of the covariates. Several covariates can be evaluated simulatneously. 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). 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.


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi hr;
assess var=(age bmi hr) / resample;
run;

Supremum Test for Functional Form
Variable Maximum Absolute
Value
Replications Seed Pr >
MaxAbsVal
AGE 11.2240 1000 164727001 0.1510
BMI 11.0212 1000 164727001 0.2440
HR 9.3459 1000 164727001 0.3940

In each of the graphs above, a covariate is plotted against cumulative martingale residuals. 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. Unless the seed option is specified, these sets will be different each time proc phreg is run. 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. None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. 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. This suggests that perhaps the functional form of bmi should be modified.

Now let's look at the model with just both linear and quadratic effects for bmi.


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
assess var=(age bmi bmi*bmi hr) / resample;
run;

Supremum Test for Functional Form
Variable Maximum Absolute
Value
Replications Seed Pr >
MaxAbsVal
AGE 9.7412 1000 179001001 0.2820
BMI 7.8329 1000 179001001 0.6370
BMIBMI 7.8329 1000 179001001 0.6370
HR 9.1548 1000 179001001 0.4200

The graph for bmi at top right looks better behaved now with smaller residuals at the lower end of bmi. The other covariates, including the additional graph for the quadratic effect for bmi all look reasonable. Thus, we again feel justified in our choice of modeling a quadratic effect of bmi.

7. Assessing the proportional hazards assumption

A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. 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. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects.

7.1. Graphing Kaplan-Meier survival function estimates to assess proportional hazards for categorical covarites

In the case of categorical covariates, graphs of the Kaplan-Meier estimates of the survival function provide quick and easy checks of proportional hazards. If proportional hazards holds, the graphs of the surival 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. Earlier in the seminar we graphed the Kaplan-Meier survivor function estimates for males and females, and gender appears to adhere to the proportional hazards assumption.


proc lifetest data=whas500 atrisk plots=survival(atrisk cb) outs=outwhas500;
atrata gender;
time lenfol*fstat(0);
run;

7.2. Plotting scaled Schoenfeld residuals vs functions of time to assess proportional hazards of a continuous covariate

A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. 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. Grambsch and Therneau (1994) show that a scaled version of the Schoenfeld residual at time $k$ for a particular covariate $p$ will approximate the change in the regression coefficient at time $k$:

$$E(s^\star_{kp}) + \hat{\beta}_p \approx \beta_j(t_k)$$

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 coefficent. In other words, the mean of the Schoenfeld residuals for coefficient $p$ at time $k$ estimates the change in the coefficient at time $k$. Thus, if the mean is 0 across time, that suggests that coefficient $p$ does not vary over time and that the proportional hazards assumption holds for covariate $p$. 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).

We will use scatterplot smooths to explore the scaled Schoenfeld residuals' relationship with time, as we did to check functional forms before. Here are the steps we will take to evaluate the proportional hazards assumption for age through scaled Schoenfeld residuals:



proc phreg data=whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
output out=schoen ressch=schgender schage schgenderage
   schbmi schbmibmi schhr;
run;

data schoen;
set schoen;
loglenfol = log(lenfol);
run;

proc loess data = schoen;
model schage=lenfol / smooth=(0.2 0.4 0.6 0.8);
run;
proc loess data = schoen;
model schage=loglenfol / smooth=(0.2 0.4 0.6 0.8);
run;



In the graphs above we see smooths of the relationship between the scaled Schoenfeld residuals and time (top panel) and log(time) (bottom panel). Although possibly slightly positively trending, the smooths appear mostly flat at 0, suggesting that the coefficient for age does not change over time and that proportional hazards holds for this covariate. The same procedure could be repeated to check all covariates.

7.3. Using assess with the ph option to check proportional hazards

The procedure Lin, Wei, and Zing(1990) developed that we previously introduced to explore covariate functional forms can also detect violations of proportional hazards by using a transform of the martingale residuals known as the empirical score process. Once again, the empirical score process under the null hypothesis of no model misspecification can be approximated by zero mean Guassian processes, and the observed score process can be compared to the simulated processes to asses departure from proportional hazards.

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. Here we demonstrate how to assess the proportional hazards assumption for all of our covariates (graph for gender not shown):


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
assess var=(age bmi bmi*bmi hr) ph / resample;
run;


Supremum Test for Proportionals Hazards Assumption
Variable Maximum Absolute
Value
Replications Seed Pr >
MaxAbsVal
GENDERFemale 4.4713 1000 350413001 0.7860
AGE 0.6418 1000 350413001 0.9190
GENDERFemaleAGE 5.3466 1000 350413001 0.5760
BMI 5.9498 1000 350413001 0.3170
BMIBMI 5.8837 1000 350413001 0.3440
HR 0.8979 1000 350413001 0.2690

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. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are nonsignificant, so we conclude that proportional hazards holds for all of our covariates.

7.4. Dealing with nonproportionality

If nonproportional hazards are detected, the researcher has many options with how to address the violation:


proc phreg data=whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr hrtime;
hrtime = hr*lenfol;
run;

Analysis of Maximum Likelihood Estimates
Parameter   DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Label
GENDER Female 1 2.10602 0.99264 4.5013 0.0339 . GENDER Female
AGE   1 0.07099 0.00835 72.2933 <.0001 .  
AGE*GENDER Female 1 -0.02927 0.01250 5.4805 0.0192 . GENDER Female * AGE
BMI   1 -0.23297 0.08788 7.0276 0.0080 .  
BMI*BMI   1 0.00363 0.00164 4.8856 0.0271 . BMI * BMI
HR   1 0.01174 0.00350 11.2671 0.0008 1.012  
hrtime   1 2.84574E-6 5.88882E-6 0.2335 0.6289 1.000  

8. Influence Diagnostics

8.1. Inspecting dfbetas to assess influence of observations on individual regression coefficients

After fitting a model it is good practice to assess the influence of observations in your data, to check if any outlier has a disproportionately large impact on the 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.

The dfbeta measure quantifies how much an observation influences the regression coefficients in the model. For observation $j$, $dfbeta_j$ approximates the change in a coefficient when that observation is deleted. We thus calculate the coefficient with the observation, call it $\beta$, and then the coefficient when observation $j$ is deleted, call it $\beta_j$, and take the difference to obtain $dfbeta_j$.

$$dfbeta_j \approx \hat{\beta} - \hat{\beta_j}$$

Positive values of $dfbeta_j$ indicate that the exclusion of the observation causes the coefficient to decrease, which implies that inclusion of the observation causes the coefficient to increase. Thus, it might be easier to think of $dfbeta_j$ as the effect of including observation $j$ on the the coefficient.

SAS provides easy ways to examine the dfbeta values for all observations across all coefficients in the model. Plots of covariates vs dfbetas can help to identify influential outliers. Here are the steps we use to assess the influence of each observation on our regression coefficients:


proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr;
run;

proc sgplot data = dfbeta;
scatter x = age y=dfage / markerchar=id;
run;
proc sgplot data = dfbeta;
scatter x = bmi y=dfbmi / markerchar=id;
run;
proc sgplot data = dfbeta;
scatter x = bmi y=dfbmibmi / markerchar=id;
run;
proc sgplot data = dfbeta;
scatter x = hr y=dfhr / markerchar=id;
run; 

The dfbetas for age and hr look small compared to regression coeffients 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 also identify id=89 again and id=112 as influential on the linear bmi coefficient ($\hat{\beta}_{bmi}=-0.23323$), and their large positive dfbetas suggest they are pulling up the coefficient for bmi when they are included.

Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. These two observations, id=89 and id=112, have very low but not unreasonable bmi scores, 15.9 and 14.8. 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. Thus, we can expect the coefficient for bmi to be more severe or more negative if we exclude these observations from the model. Indeed, exclusion of these two outliers causes an almost doubling of $\hat{\beta}_{bmi}$, from -0.23323 to -0.39619. 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.


proc print data = whas500(where=(id=112 or id=89));
var lenfol gender age bmi hr;
run;
Obs LENFOL GENDER AGE BMI HR
89 1553 Male 95 15.9270 62
112 2123 Female 87 14.8428 105

proc phreg data = whas500(where=(id^=112 and id^=89));
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr;
run;
Analysis of Maximum Likelihood Estimates
Parameter   DF Parameter
Estimate
Standard
Error
Chi-Square Pr > ChiSq Hazard
Ratio
Label
GENDER Female 1 2.07605 1.01218 4.2069 0.0403 . GENDER Female
AGE   1 0.07412 0.00855 75.2370 <.0001 .  
AGE*GENDER Female 1 -0.02959 0.01277 5.3732 0.0204 . GENDER Female * AGE
BMI   1 -0.39619 0.09365 17.8985 <.0001 .  
BMI*BMI   1 0.00640 0.00171 14.0282 0.0002 . BMI * BMI
HR   1 0.01244 0.00279 19.9566 <.0001 1.013  

8.2. Plotting likelihood displacement scores to assess influence on the overall model

Not only are we interested in how influential observations affect individual coefficients, we are interested in how they affect the model as a whole. The likelihood displacement score quantifies how much the likelihood of the model, which is dependent on all coefficients, changes when an observation is left out. This analysis proceeds in much the same was as dfbeta analysis, in that we will:



proc phreg data = whas500;
class gender;
model lenfol*fstat(0) = gender|age bmi|bmi hr;
output out=ld ld=ld;
run;

proc sgplot data=ld;
scatter x=lenfol y=ld / markerchar=id;
run;
      

We see the same 2 outliers we identifed before, id=89 and id=112, as having the largest influence on the model overall, probably primarily through their effects on the bmi coefficient. However, we have decided that there covariate scores are reasonable so we retain them in the model.

9. References

Hosmer, DW, Lemeshow, S, May, S. (2008). Applied Survival Analysis. Wiley: Hoboken, NJ.

Therneau, TM, Grambsch, PM. (2000). Modeling Survival Data. Springer: New York, NY.
     The primary reference used for this seminar. Contains many examples worked in SAS.

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