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13.12.2014 admin
Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. 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)). 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. As an example, we can use the cdf to determine the probability of observing a survival time of up to 100 days. In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. The survivor function, $S(t)$, describes the probability of surviving past time $t$, or $Pr(Time > t)$. The hazard function, then, describes the relative likelihood of the event occurring at time $t$ ($f(t)$), conditional on the subject's survival up to that time $t$ ($S(t)$). 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. Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time.
Let us again think of the hazard function, $h(t)$, as the rate at which failures occur at time $t$.
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. We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. This seminar covers both proc lifetest and proc phreg, and data can be structured in one of 2 ways for survival analysis. 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.
This structuring allows the modeling of time-varying covariates, or explanatory variables whose values change across follow-up time. 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.
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.
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 $\hat S(1) = \frac{500 - 8}{500} = 0.984$.
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. At a minimum proc lifetest requires specification of a failure time variable, here lenfol, on the time statement.
Without further specification, SAS will assume all times reported are uncensored, true failures. We also specify the option atrisk on the proc lifetest statement to display the number at risk in our sample at various time points. Above we see the table of Kaplan-Meier estimates of the survival function produced by proc lifetest. From "LENFOL"=368 to 376, we see that there are several records where it appears no events occurred.
By default, proc lifetest graphs the Kaplan Meier estimate, even without the plot= option on the proc lifetest statement, so we could have used the same code from above that produced the table of Kaplan-Meier estimates to generate the graph.


However, we would like to add confidence bands and the number at risk to the graph, so we add plots=survival(atrisk cb). The step function form of the survival function is apparent in the graph of the Kaplan-Meier estimate. 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 requested in SAS through the nelson option on the proc lifetest statement.
Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. 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).
When provided with a grouping variable in a strata statement in proc lifetest, SAS will produce graphs of the survival function (unless other graphs are requested) stratified by the grouping variable as well as tests of equality of the survival function across strata. In the graph of the Kaplan-Meier estimator stratified by gender below, it appears that females generally have a worse survival experience. 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. Whereas with non-parametric methods we are typically studying the survival function, with regression methods we examine the 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.
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.
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. We also would like survival curves based on our model, so we add plots=survival to the proc phreg statement, although as we shall see this specification is probably insufficient for what we want.
On the model statement, on the left side of the equation, we provide the follow up time variable, lenfol, and the censoring variable, fstat, with all censoring values listed in parentheses. Model Fit Statistics: Displays fit statistics which are typically used for model comparison and selection. Analysis of Maximum Likelihood Estimates: Displays model coefficients, tests of significance, and exponentiated coefficient as hazard ratio.
When only plots=survival is specified on the proc phreg statement, SAS will produce one graph, a "reference curve" of the survival function at the reference level of all categorical predictors and at the mean of all continuous predictors. 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. 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. This expanded dataset can be named and then viewed with the out= option, but obtaining the out= dataset is not at all necessary to generate the survival plots.
Both survival and cumulative hazard curves are available using the plots= option on the proc phreg statement, with the keywords survival and cumhaz, respectively. 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.
We request survival plots that are overlaid with the plot(overlay)=(survival) specification on the proc phreg statement.
We also add the rowid=option on the baseline statement, which tells SAS to label the curves on our graph using the variable gender.


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. In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. 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.
Proportional hazard model including the design variables for age using deviation from mean coding. Feel free to take a look around, meet the Waverunners, and see how the foundation is being set in place, by building a softball powerhouse in Indiana! This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. 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. 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 $t_i$. 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. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. SAS will output both Kaplan Meier estimates of the survival function and Nelson-Aalen estimates of the cumulative hazard function in one table. 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. 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. We pride ourselves on hard work, dedication, and improvement; while enjoying the game of fastpitch softball.
Our goal as a team is to develop as softball players and build character within ourselves as well as represent our communities as responsible and classy individuals. Additionally, another variable counts the number of events occurring in each interval (either 0 or 1 in Cox regression, same as the censoring variable). Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement.
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. 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. 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. 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. In very large samples the Kaplan-Meier estimator and the transformed Nelson-Aalen (Breslow) estimator will converge. We have been established in the area for several years and have helped to teach young ladies the skills needed to become champions on and off the field.



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