SURVIVAL ANALYSIS USING SAS A PRACTICAL GUIDE PDF
survival analysis using sas pdf. Probability density functions, cumulative distribution functions and the hazard function are central to the analytic techniques. survival analysis using sas a practical guide second edition by survival analysis using sas pdf. Proc LifetestProc Lifetest Estimation of Survival. Survival Analysis Using SAS ®A Practical Guide Second EditionPaul D. Allison with the normal distribution is given by its p.d.f., not its c.d.f. Hazard Function.
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Free Survival Analysis Using Sas: A Practical Guide, Second Edition. Download Users' Guide (pdf | html) . The unconnected in Quantum Mechanics '(PDF). Free Survival Analysis Using Sas A Practical Guide Second Edition. Download · Modules · Projects · Resources Users' Guide (pdf | html). Request PDF on ResearchGate | On Aug 1, , N. E. Rosenberg and others published Survival Analysis Using SAS: A Practical Guide. Second Edition By.
On the other hand, the hazard for death due to ovarian cancer is likely to increase markedly from time since diagnosis. Hence, it is more important to control for time since diagnosis by choosing it as the time origin.
Again, it may be possible to control for other time origins by including them as covariates. Indeed, this structure is fairly standard across many different computer packages for survival analysis. This is straightforward if the covariates are constant over time. The more complex data structure needed for timedependent covariates is discussed in Chapter 5. The basic data structure is illustrated by Output 2.
The DUR variable gives the time in days from the point of randomization to either death or censoring which could occur either by loss to follow up or termination of the observation. An additional covariate, RENAL, is an indicator variable for normal 1 versus impaired 0 renal functioning at the time of randomization. This data set is one of several that are analyzed in the remaining chapters. Output 2.
With its extremely flexible and powerful DATA step, SAS is well-suited to perform the kinds of programming necessary to process such complex data sets. Of particular utility is the rich set of date and time functions available in the DATA step. For example, suppose the origin time for some event is contained in three numeric variables: Once that conversion takes place, simple subtraction suffices to get the duration in days.
Many other functions are also available to convert time data in various formats into SAS date values. For very simple experimental designs, standard tests for comparing survivor functions across treatment groups may suffice for analyzing the data. And in demography, the lifetable method for estimating survivor functions still holds a preeminent place as a means of describing human mortality.
The Kaplan-Meier method is more suitable when event times are measured with precision, especially if the number of observations is small. The life-table or actuarial method may be better for large data sets or when the measurement of event times is crude. Also known as the product-limit estimator, this method was known for many years prior to when Kaplan and Meier showed that it was, in fact, the nonparametric maximum likelihood estimator.
This gave the method a solid theoretical justification. When there are no censored data, the KM estimator is simple and intuitive. The situation is also quite simple in the case of single right censoring that is, when all the censored cases are censored at the same time c and all the observed event times are less than c. Things get more complicated when some censoring times are smaller than some event times.
The solution is as follows. In words, this formula says that for a given time t, take all the event times that are less than or equal to t. Then multiply all of these conditional probabilities together.
For an explanation of the rationale for equation 3. The first variable is the time of the event or censoring, the second variable contains information on whether or not the observation was censored, and the numbers in parentheses there can be more than one are values of the second variable that correspond to censored observations.
These statements produce the results shown in Output 3. The marked survival times are censored observations. The mean survival time and its standard error were underestimated because the largest observation was censored and the estimation was restricted to the largest event time. Censored observations are starred. The crucial column is the second—labeled Survival—which gives the KM estimates. At days, for example, the KM estimate is.
We say, then, that the estimated probability that a patient will survive for days or more is. When there are tied values two or more cases that die at the same reported time , as we have at 8 days and 63 days, the KM estimate is reported only for the last of the tied cases. No KM estimates are reported for the censored times. In fact, however, the KM estimator is defined for any time between 0 and the largest event or censoring time.
Thus, the estimated survival probability for any time from 70 days up to but not including 76 days is. The 1-year days survival probability is. After 2, days the largest censoring time , the KM estimate is undefined.
The third column, labeled Failure, is just 1 minus the KM estimate, which is the estimated probability of a death prior to the specified time. Thus, it is an estimate of the cumulative distribution function. The column labeled Number Left is the number of cases that have neither experienced events nor been censored prior to each point in time. This is the size of the risk set for each time point.
Below the main table, you find the estimated 75th, 50th, and 25th percentiles labeled Quartile Estimates. If these were not given, you could easily determine them from the Failure column. Thus, the 25th percentile 63 in this case is the smallest event time such that the probability of dying earlier is greater than. No value is reported for the 75th percentile because the KM estimator for these data never reaches a failure probability greater than. Of greatest interest is the 50th percentile, which is, of course, the median death time.
As noted in the table, the confidence intervals are calculated using a log-log transform that preserves the upper bound of 1 and the lower bound of 0 on the survival probabilities. Although other transforms are optionally available, there is rarely any need to use them.
An estimated mean time of death is also reported. This value is calculated directly from the estimated survival function. It is not simply the mean time of death for those who died. As noted in the output, the mean is biased downward when there are censoring times greater than the largest event time.
Even when this is not the case, the upper tail of the distribution will be poorly estimated when a substantial number of the cases are censored, and this can greatly affect estimates of the mean.
Consequently, the median is typically a better measure of central tendency for censored survival data. Usually you will want to see a plot of the estimated survival function. Here I use ODS because it has a much richer array of features.
These marks can be very distracting when the data set is large with lots of censored observations. Another useful option is ATRISK, which adds the number of individuals still at risk not yet dead or censored to the graph. Note that the confidence limits only extend to the largest event time. I prefer the EP method because it tends to produce confidence bands that are more stable in the tails.
As is evident in that graph, the confidence bands are always wider than the pointwise confidence limits. Note that the data set contains one record for each unique event or censoring time, plus a record for time 0.
By default, these limits are calculated by adding and subtracting 1. This method ensures that the confidence limits will not be greater than 1 or less than 0. Other transformations are available, the most attractive being the logit: For the myelomatosis data, it is clearly desirable to test whether the treatment variable TREAT has any effect on the survival experience of the two groups. First, instead of a single table with KM estimates, separate tables not shown here are produced for each of the two treatment groups.
Second, corresponding to the two tables are two graphs of the survivor function, superimposed on the same axes for easy comparison. Look at the graphs in Output 3. Before days, the two survival curves are virtually indistinguishable, with little visual evidence of a treatment effect. The gap that develops after days reflects the fact that no additional deaths occur in treatment group 1 after that time. These are used to compute the chi-square statistics shown near the bottom of Output 3.
In most cases, you can ignore the preliminaries and look only at the p-values. Here they are far from significant, which is hardly surprising given the graphical results and the small sample size. Thus, there is no evidence that would support the rejection of the null hypothesis that the two groups have exactly the same survivor function that is, exactly the same probability distribution for the event times. The p-value for a likelihood-ratio test -2log LR is also reported; this test is usually inferior to the other two because it requires the unnecessary assumption that the hazard function is constant in each group, implying an exponential distribution for event times.
Are there any reasons for choosing one over the other? Each statistic can be written as a function of deviations of observed numbers of events from expected numbers. As shown in Output 3. The chi-square statistic is calculated by squaring this number and dividing by the estimated variance, which is 4. Thus, it is a weighted sum of the deviations of observed numbers of events from expected numbers of events.
As with the log-rank statistic, the chi-square test is calculated by squaring the Wilcoxon statistic for either group —18 for group 1 in this example and dividing by the estimated variance Because the Wilcoxon test gives more weight to early times than to late times nj never increases with time , it is less sensitive than the logrank test to differences between groups that occur at later points in time. To put it another way, although both statistics test the same null hypothesis, they differ in their sensitivity to various kinds of departures from that hypothesis.
This equation defines a proportional hazards model, which is discussed in detail in Chapter 5. In contrast, the Wilcoxon test is more powerful than the log-rank test in situations where event times have log-normal distributions discussed in the next chapter with a common variance but with different means in the two groups. Neither test is particularly good at detecting differences when survival curves cross.
Using the same data set, we can stratify on RENAL, the variable that has a value of 1 if renal functioning was impaired at the time of randomization; otherwise, the variable has a value of 0. Output 3. The three hypothesis tests are unanimous in rejecting the null hypothesis of no difference between the two groups. Like the Wilcoxon and log-rank tests, all of these can be represented as a weighted sum of observed and expected numbers of events: Fleming-Harrington is actually a family of tests in which the weights depend on two parameters, p and q, which can be chosen by the user: When both p and q are 0, you get the log-rank test.
When p is 1 and q is 0, you get something very close to the Peto-Peto test. When q is 1 and p is 0, wi increases with time, unlike any of the other tests. When both q and p are 1, you get a weight function that is maximized at the median and gets small for large or small times.
The tests readily generalize to three or more groups, with the null hypothesis that all groups have the same survivor function. That variable may be either numeric or character. To illustrate this and other options, we need a different data set.
The sample consisted of male inmates who were released from Maryland state prisons in the early s Rossi et al. These men were followed for 1 year after their release, and the dates of any arrests were recorded. The variable ARREST has a value of 1 for those who were arrested during the 1-year follow-up, and it has a value of 0 for those who were not. Only 26 percent of the men were arrested. The data are singly right censored so that all the censored cases have a value of 52 for WEEK.
This variable was randomly assigned with equal numbers in each category. Next we get overall chi-square tests of the null hypothesis that the survivor functions are identical across the four strata.
All three tests are significant at the. Last, we see the log-rank tests comparing each possible pair of strata. Not shown is a similar table for the Wilcoxon test. Note that except for the two extreme strata, each stratum is identified by the midpoint of the interval. Clearly one must reject the null hypothesis that all age strata have the same survivor function. Three of the six pairwise comparisons are also significant, either with or without the adjustment. The KM method then produces long tables that may be unwieldy for presentation and interpretation.
An alternative solution is to switch to the life-table method, in which event times are grouped into intervals that can be as long or short as you please. In addition, the life-table method also known as the actuarial method can produce estimates and plots of the hazard function. The downside to the life-table method is that the choice of intervals is usually somewhat arbitrary, leading to arbitrariness in the results and possible uncertainty about how to choose the intervals.
There is inevitably some loss of information as well. Note, however, that PROC LIFETEST computes the log-rank and Wilcoxon statistics as well as other optional test statistics from the ungrouped data if available so they are unaffected by the choice of intervals for the life-table method.
We now request a life table for the recidivism data, using the default specification for interval lengths: The intervals will then begin with [0, w and will increment by w. Note that intervals do not have to be the same length. The four statistics displayed in the first panel, while not of major interest in themselves, are necessary for calculating the later statistics.
Number Failed and Number Censored should be self-explanatory. Effective Sample Size is straightforward for the first five intervals because they contain no censored cases. The effective sample size for these intervals is just the number of persons who had not yet been arrested at the start of the interval. For the last interval, however, the effective sample size is only , even though persons made it to the 50th week without an arrest.
The answer is a fundamental property of the life-table method. The method treats any cases censored within an interval as if they were censored at the midpoint of the interval. This treatment is equivalent to assuming that the distribution of censoring times is uniform within the interval.
Because censored cases are only at risk for half of the interval, they only count for half in figuring the effective sample size. The 7 corresponds to the seven men who were arrested in the interval; they are treated as though they were at risk for the whole interval. The Conditional Probability of Failure is an estimate of the probability that a person will be arrested in the interval, given that he made it to the start of the interval.
An estimate of its standard error is given in the next panel. The Survival column is the life-table estimate of the survivor function that is, the probability that the event occurs at a time greater than or equal to the start time of each interval.
For example, the estimated probability that an inmate will not be arrested until week 30 or later is. The survival estimate is calculated from the conditional probabilities of failure in the following way. For interval i, let ti be the start time and qi be the estimated conditional probability of failure. The rationale for equation 3.
Suppose we want an expression for the probability of surviving to t4 or beyond. Extending this argument to other intervals yields the formula in equation 3. Note the similarity between this formula and equation 3.
In equation 3. The major differences between the two formulas are as follows: Thus, each interval for KM estimation extends from one unique event time up to, but not including, the next unique event time.
Continuing with the second panel of Output 3. We are also given the standard errors of the Survival probabilities. The Median Residual Lifetime column is, in principle, an estimate of the remaining time until an event for an individual who survived to the start of the interval.
For this example, however, the estimates are all missing. To calculate this statistic for a given interval, there must be a later interval whose survival probability is less than half the survival probability associated with the interval of interest. It is apparent from Output 3. For many data sets, there will be at least some later intervals for which this statistic cannot be calculated. The PDF Standard Error column gives the estimated value of the probability density function at the midpoint of the interval.
Of greater interest is the Hazard column, which gives estimates of the hazard function evaluated at the midpoint of each interval. For an individual who is known to have survived the whole interval, exposure time is just the interval width bi. For individuals who had events or who withdrew in the interval, exposure time is the time from the beginning of the interval until the event or withdrawal.
Total exposure time is the sum of all the individual exposure times. The denominator in equation 3. Why use an inferior estimator? Well, exact exposure times are not always available see the next section , so the estimator in equation 3. Apparently, the hazard of arrest increases steadily until the week interval, when it drops precipitously from.
This drop is an artifact of the way that the last interval is constructed, however. Although the interval runs from 50 to 60, in fact, no one was at risk of an arrest after week 52 when the study was terminated. As a result, the denominator in equation 3. If we set it at 52, the interval will not include arrests that occurred in week 52 because the interval is open on the right.
At the same time, it is better to recode the censored times from 52 to 53 because they are not censored within the interval but rather at the end. Recoding effectively credits the full interval rather than only half as exposure time for the censored cases. PDF 0. Although this sample is rather small for constructing a life table, it will do fine for illustration. The trick is to create a separate observation for each of the frequency counts in the table. The hazard plot is in Output 3.
The most striking fact about this example is the rapid decline in the hazard of death from the origin to about days after surgery. After that, the hazard is fairly stable. This decline is reflected in the Median Residual Lifetime column.
At time 0, the median residual lifetime of However, of those patients still alive at 50 days, the median residual lifetime rises to The median remaining life continues to rise until it reaches a peak of Conditional Probability Standard Error 0.
Evaluated at the Midpoint of the Interval Interval [Lower, Upper 0 50 50 Median Standard Error It also treats them as a set, testing the null hypothesis that they are jointly unrelated to survival time and also testing for certain incremental effects of adding variables to the set.
The statistics are generalizations of the log-rank and Wilcoxon tests discussed earlier in this chapter. However, in most cases, you are better off switching to the regression procedures, for two reasons. Second, the incremental tests do not really test the effect of each variable controlling for all the others. Instead, you get a test of the effect of each variable controlling for those variables that have already been included. Because you have no control over the order of inclusion, these tests can be misleading.
Because the log-rank and Wilcoxon tests do not require iterative calculations, they use relatively little computer time. The covariate tests are invoked by listing the variable names in a TEST statement: I have omitted the Wilcoxon statistics because they are nearly identical to the log-rank statistics for this example.
I also omitted the variance-covariance matrix for the statistics because it is primarily useful as input to other analyses.
The signs of the log-rank test statistics tell you the direction of the relationship. The negative sign for PRIO indicates that inmates with more prior convictions tend to have shorter times to arrest.
On the other hand, the positive coefficient for AGE indicates that older inmates have longer times to arrest. As already noted, none of these tests controls or adjusts for any of the other covariates. The lower panel displays results from a forward inclusion procedure.
The joint chi-square of The chi-square increment of This process is repeated until all the variables are added. For each variable, we get a test of the hypothesis that the variable is unrelated to survival time controlling for all the variables above it but none of the variables below it. For variables near the beginning of the sequence, however, the results can be quite different. For this example, the forward inclusion procedure leads to some substantially different conclusions from the univariate procedure.
While WEXP has a highly significant effect on survival time when considered by itself, there is no evidence of such an effect when other variables are controlled.
The reason is that work experience is moderately correlated with age and the number of prior convictions, both of which have substantial effects on survival time. Marital status also loses its statistical significance in the forward inclusion test. Both produce a test of the null hypothesis that the survivor functions are the same for the two categories of FIN. In fact, if there are no ties in the data no cases with exactly the same event time , the two statements will produce identical chi-square statistics and p-values.
In the recidivism data, for example, the arrests occurred at only 49 unique arrest times, so the number of ties was substantial. The TEST statement produces a log-rank chi-square of 3. Obviously the differences are minuscule in this case.
While the STRATA statement produces separate tables and graphs of the survivor function for the two groups, the TEST statement produces only the single table and graph for the entire sample. In other words, they are stratified statistics that control for whatever variable or variables are listed in the STRATA statement. Suppose, for example, that for the myelomatosis data, we want to test the effect of the treatment while controlling for renal functioning.
We can submit these statements: This result is in sharp contrast with the unstratified chi-square of only 1.
Free Survival Analysis Using Sas: A Practical Guide, Second Edition
The S gives us the now-familiar survival curve. To explain how this plot is useful, we need a little background. From equation 2. Moreover, an examination of the logsurvival plot can tell us whether the hazard is constant, increasing, or decreasing with time. Instead of a straight line, the graph appears to increase at a decreasing rate. This fact suggests that the hazard is not constant but rather declines with time. If the plot had curved upward rather than downward, it would suggest that the hazard was increasing with time.
Of course, because the sample size is quite small, caution is advisable in drawing any conclusions. A formal test, such as the one described in the next chapter, might not show a significant decrease in the hazard. Examining Output 3. Again, however, the data are so sparse that this is probably not sufficient evidence for rejecting the Weibull distribution.
What we really want to see is a graph of the hazard function. We got that from the life-table method, but at the cost of grouping the event times into arbitrarily chosen intervals. There are several ways to smooth these estimates by calculating some sort of moving average. One method, known as kernel smoothing, has been shown to have good properties for hazard functions RamlauHansen, The graph bears some resemblance to the grouped hazard plot in Output 3.
This means that, when calculating the hazard for any specified point in time, the smoothing function only uses data within The bandwidth has a big impact on the appearance of the graph: But sometimes you may want to experiment with different bandwidths. But the procedure is not adequate for two-factor designs because there is no way to test for interactions.
As explained in Chapter 8, uncontrolled heterogeneity tends to make hazard functions look as though they are declining, even when there is no real decline for any individual in the sample. As we will see in Chapter 5, one way to reduce the effect of heterogeneity is to estimate and plot baseline survivor functions after fitting Cox regression models.
In its most general form, the AFT model describes a relationship between the survivor functions of any two individuals. This model says, in effect, that what makes one individual different from another is the rate at which they age. A good example is the conventional wisdom that a year for a dog is equivalent to 7 years for a human.
This relationship can be represented by equation 4. Let Ti be a random variable denoting the event time for the ith individual in the sample, and let xi1, …, xik be the values of k covariates for that same individual. Exponentiating both sides of equation 4. This notational strategy could also be used for linear models.
One member of the AFT class, the log-normal model, has exactly these assumptions. It is called the log-normal model because if log T has a normal distribution, then T has a log-normal distribution. If there are no censored data, we can readily estimate this model by ordinary least squares OLS. Simply create a new variable in a DATA step that is equal to the log of the event time and use the REG procedure with the transformed variable as the dependent variable.
But survival data typically have at least some censored observations, and these are difficult to handle with OLS. Instead, we can use maximum likelihood estimation.
Later, this chapter examines the mathematics of maximum likelihood ML for censored regression models in some detail. In this recidivism example, the variable WEEK contains the week of the first arrest or censoring. There are seven covariates.
However, it is now followed by an equal sign and a list of covariates. Here, we have merely indicated our choice of the lognormal distribution. Output 4. The output first provides some preliminary information: Then the output shows that the log-likelihood for the model is — This is an important statistic that we will use later to test various hypotheses.
The first is simply the log-likelihood multiplied by —2. The models being compared do not have to be nested, in the sense of one model being a special case of another.
However, these statistics cannot be used to construct a formal hypothesis test, so the comparison is only informal. But keep in mind that the overall magnitude of these statistics depends heavily on sample size. Finally, we get a table of estimated coefficients, their standard errors, chi-square statistics for the null hypothesis that each coefficient is 0, and p-values associated with those statistics.
The chi-squares are calculated by dividing each coefficient by its estimated standard error and squaring the result. The signs of the coefficients tell us the direction of the relationship. The positive coefficient for FIN indicates that those who received financial aid had longer times to arrest than those who did not.
The negative coefficient for PRIO indicates that additional convictions were associated with shorter times to arrest. As in any regression procedure, these coefficients adjust or control for the other covariates in the model. The numerical magnitudes of the coefficients are not very informative in the reported metrics, but a simple transformation leads to a more intuitive interpretation.
Thus, e. This statement also applies to the median time to arrest, or any other percentile for that matter. Thus, exp —. According to the model, then, each additional prior conviction is associated with a 6. We can also interpret the coefficients for any of the other AFT models discussed in this chapter in this way. For some distributions, changes in the value of this parameter can produce qualitative differences in the shape of the hazard function. For each of these distributions, there is a corresponding distribution for T: Weibull extreme value 1 par.
Because the logistic and normal lead to the log-logistic and log-normal, you might expect that the gamma will lead to the log-gamma. But it is just the reverse. This is one of those unfortunate inconsistencies in terminology that we just have to live with. The main reason for allowing other distributions is that they have different implications for hazard functions that may, in turn, lead to different substantive interpretations. The remainder of this section explores each of these alternatives in some detail.
This implies that T itself has an exponential distribution, which is why we call it the exponential model. The standard extreme value distribution is also known as a Gumbel distribution or a double exponential distribution. It has a p.
Like the normal distribution, this is a unimodal distribution defined on the entire real line. Unlike the normal, however, it is not symmetrical, being slightly skewed to the left.
However, equation 2. Although the dependent variable in equation 4. It turns out that the two models are completely equivalent. Furthermore, there is a simple relationship between the coefficients in equation 4. The change in signs makes intuitive sense. If the hazard is high, then events occur quickly and survival times are short. On the other hand, when the hazard is low, events are unlikely to occur and survival times are long. It is important to be able to shift back and forth between these two ways of expressing the model so that you can compare results across different computer programs.
You may wonder why there is no disturbance term in equation 4. Even if two individuals have exactly the same covariate values and, therefore, the same hazard , they will not have the same event time. Comparing this with the log-normal results in Output 4. The coefficient for AGE is about twice as large in the exponential model, and its p-value declines from. Similarly, the coefficient for PRIO increases somewhat in magnitude, and its p-value also goes down substantially.
On the other hand, the p-value for FIN increases to slightly above the. Later, this chapter considers some criteria for choosing among these and other models.
Here the null hypothesis is soundly rejected, indicating that the hazard function is not constant over time. While this might suggest that the log-normal model is superior, things are not quite that simple. There are other models to consider as well. The Weibull Model The Weibull model is a slight modification of the exponential model, with big consequences.
Second, in addition to being an AFT model, the Weibull model is also a proportional hazards model. This means that its coefficients when suitably transformed can be interpreted as relative hazard ratios.
In fact, the Weibull model and its special case, the exponential model is the only model that is simultaneously a member of both these classes. On the other hand, standard errors and confidence intervals for coefficients in the log-survival time format are not so easily converted to the log-hazard format. Collett gives formulas for accomplishing this. Compared with the exponential model in Output 4. But the standard errors are also smaller, so the chi-square statistics and p-values are hardly affected at all.
These coefficients are much closer to the log-hazard coefficients for the exponential model which differ only in sign from the log-survival time coefficients. It is included in the output because some statisticians prefer this way of parameterizing the model. The Log-Normal Model Although we have already discussed the log-normal model and applied it to the recidivism data, we have not yet considered the shape of its hazard function. Unlike the Weibull model, the log-normal model has a nonmonotonic hazard function.
It rises to a peak and then declines toward 0 as t goes to infinity. The log-normal is not a proportional hazards model, and its hazard function cannot be expressed in closed form it involves the c. It can, however, be expressed as a regression model in which the dependent variable is the logarithm of the hazard. This equation also applies to the log-logistic and gamma models to be discussed shortly, except that h0. Some typical log-normal hazard functions are shown in Figure 4.
All three functions correspond to distributions with a median of 1. The inverted U-shape of the log-normal hazard is often appropriate for repeatable events. Suppose, for example, that the event of interest is a residential move.
Immediately after a move, the hazard of another move is likely to be extremely low. People need to rest and recoup the substantial costs involved in moving. The hazard will certainly rise with time, but much empirical evidence indicates that it eventually begins to decline. One explanation is that, as time goes by, people become increasingly invested in a particular location or community. However, Chapter 8 shows how the declining portion of the hazard function may also be a consequence of unobserved heterogeneity.
It is well known to students of the logistic logit regression model, which can be derived by assuming a linear model with a logistically distributed error term and a dichotomization of the dependent variable.
It follows that T has a log-logistic distribution. This produces the characteristic shapes shown in Figure 4. Despite the complexity of its hazard function, the log-logistic model has a rather simple survivor function: This is just a logistic regression model for the probability that an event occurs prior to t. Thus, for the recidivism data, the log-logistic model can be estimated by fitting a logistic regression model to the dichotomy arrested versus not arrested in the first year after release.
Of course, this estimation method is not fully efficient because we are not using the information on the exact timing of the arrests, and we certainly will not get the same estimates. The point is that the two apparently different methods are actually estimating the same underlying model.
Given what I just said about the similarity of the lognormal and log-logistic hazards, you might expect the other results to be similar to the log-normal output in Output 4. But the coefficients and test statistics actually appear to be closer to those for the Weibull model in Output 4. In particular, the exponential, Weibull, and log-normal models but not the log-logistic are all special cases of the generalized gamma model.
This fact is exploited later in this chapter when we consider likelihood ratio tests for comparing different models. But the generalized gamma model can also take on shapes that are unlike any of these special cases. Most important, it can have hazard functions with U or bathtub shapes in which the hazard declines, reaches a minimum, and then increases. It is well known that the hazard for human mortality, considered over the whole life span, has such a shape.
On the other hand, the generalized gamma model cannot represent hazard functions that have more than one reversal of direction. There are two reasons. First, the formula for the hazard function for the generalized gamma model is rather complicated, involving the gamma function and the incomplete gamma function. Consequently, you may often find it difficult to judge the shape of the hazard function from the estimated parameters.
By contrast, hazard functions for the specific submodels can be rather simply described, as we have already seen. Second, computation for the generalized gamma model is more difficult. For example, it took more than five times as much computer time to estimate the generalized gamma model for the recidivism data as compared with the exponential model. This fact can be an important consideration when you are working with very large data sets.
When it is 1. In Output 4. For categorical covariates with more than two categories, the standard approach is to create a set of indicator variables, one for each category except for one. Another covariate in the recidivism data set is education, which was originally coded like this: This test has two degrees of freedom df , corresponding to the two coefficients that are estimated for EDUC.
What is particularly attractive about this test is that it does not depend on which category of EDUC is the omitted category. In the table of estimates, there are three lines for the EDUC variable.
The first two lines contain coefficients, standard errors, and hypothesis tests for levels 3 and 4 of EDUC, while the last line merely informs us that level 5 is the omitted category.
Hence, each of the estimated coefficients is a contrast with level 5. Methods for constructing such a test by hand calculation are described in the Hypothesis Tests section below.
SAS 9. This section explores some of the basics of ML estimation, with an emphasis on how it handles censored observations. The discussion is not intended to be rigorous. If you want a more complete and careful treatment of ML, you should consult one of the many texts available on the subject. For example, Kalbfleisch and Prentice give a more detailed introduction in the context of survival analysis.
ML is a quite general approach to estimation that has become popular in many different areas of application. There are two reasons for this popularity. First, ML produces estimators that have good large-sample properties.
Provided that certain regularity conditions are met, ML estimators are consistent, asymptotically efficient, and asymptotically normal. Consistency means that the estimates converge in probability to the true values as the sample gets larger, implying that the estimates will be approximately unbiased in large samples. Asymptotically efficient means that, in large samples, the estimates will have standard errors that are approximately at least as small as those for any other estimation method. And, finally, asymptotically normal means that the sampling distribution of the estimates will be approximately normal in large samples, which implies that you can use the normal and chi-square distributions to compute confidence intervals and p-values.
All these approximations get better as the sample size gets larger. The fact that these desirable properties have been proven only for large samples does not mean that ML has bad properties for small samples.
And in the absence of attractive alternatives, researchers routinely use ML estimation for both large and small samples. Despite the temptation to accept larger p-values as evidence against the null hypothesis in small samples, it is actually more reasonable to demand smaller values to compensate for the fact that the approximation to the normal or chi-square distributions may be poor.
As we will see, one case that ML handles nicely is data with censored observations. Although you can use least squares with certain adjustments for censoring Lawless, , such estimates often have much larger standard errors, and there is little available theory to justify the construction of hypothesis tests or confidence intervals.
The basic principle of ML is to choose as estimates those values that will maximize the probability of observing what we have, in fact, observed. There are two steps to this: The first step is known as constructing the likelihood function. To accomplish this, you must specify a model, which amounts to choosing a probability distribution for the dependent variable and choosing a functional form that relates the parameters of this distribution to the values of the covariates.
We have already considered those two choices. The second step—maximization—typically requires an iterative numerical method that involves successive approximations. In the next section, I work through the basic mathematics of constructing and maximizing the likelihood function. Maximum Likelihood Estimation: For each individual i, the data consist of three parts: For simplicity, we treat xi as fixed rather than random.
But that would just complicate the notation. We also assume that censoring is non-informative. For the moment, suppose that all the observations are uncensored. Because we are assuming independence, it follows that the probability of the entire data is found by taking the product of the probabilities of the data for every individual. Because ti is assumed to be measured on a continuum, the probability that it will take on any specific value is 0. To proceed further, we need to substitute an expression for fi ti that involves the covariates and the unknown parameters.
But the probability of an event time greater than ti is given by the survivor function S t evaluated at time ti. Now suppose that we have r uncensored observations and n — r censored observations. As a result, we do not need to order the observations by censoring status. Once we choose a particular model, we can substitute appropriate expressions for the p. Because the logarithm is an increasing function, whatever maximizes the logarithm also maximizes the original function.
There are many different methods for maximizing functions like this. Consequently, except in special cases like a single dichotomous x variable , there is no explicit solution.
Instead, we have to rely on iterative methods, which amount to successive approximations to the solution until the approximations converge to the correct value. Again, there are many different methods for doing this. All give the same solution, but they differ in such factors as speed of convergence, sensitivity to starting values, and computational difficulty at each iteration.
The method is named after Sir Isaac Newton, who devised it for a single equation and a single unknown. But who was Raphson?
Joseph Raphson was a younger contemporary of Newton who generalized the algorithm to multiple equations with multiple unknowns. The Newton-Raphson algorithm can be described as follows. These starting values are substituted into the right side of equation 4. This process is repeated until the maximum change in the parameter estimates from one step to the next is less than. This is an absolute change if the current parameter value is less than.
This matrix, which can be printed by listing COVB as an option in the MODEL statement, is often useful for constructing hypothesis tests about linear combinations of coefficients.
For the most part, the iterative methods used to accomplish this task work quite well with no attention from the data analyst. When the iterations are complete, the final gradient vector and the negative of the Hessian matrix will also be printed see the preceding section for definitions of these quantities. When the exponential model was fitted to the recidivism data, the ITPRINT output revealed that it took six iterations to reach a solution.
The log-likelihood for the starting values was — Examination of the coefficient estimates showed only slight changes after the fourth iteration. By comparison, the generalized gamma model took 13 iterations to converge.
Occasionally the algorithm fails to converge, although this seems to occur much less frequently than it does with logistic regression. In general, nonconvergence is more likely to occur when samples are small, when censoring is heavy, or when many parameters are being estimated. There is one situation, in particular, that guarantees nonconvergence at least in principle.
If all the cases at one value of a dichotomous covariate are censored, the coefficient for that variable becomes larger in magnitude at each iteration. But if all the cases in a group are censored, the ML estimate for the hazard in that group is 0. If the 0 is in the denominator of the ratio, then the coefficient tends toward plus infinity. By extension, if a covariate has multiple values that are treated as a set of dichotomous variables for example, with a CLASS statement and all cases are censored for one or more of the values, nonconvergence should result.
When this happens, there is no ideal solution. You can remove the offending variable from the model, but that variable may actually be one of the strongest predictors. When the variable has more than two values, you can combine adjacent values or treat the variable as quantitative. If the number of iterations exceeds the maximum allowed the default is 50 , SAS issues the message: Convergence not attained in 50 iterations.
The procedure is continuing but the validity of the model fit is questionable. The negative of the Hessian is not positive definite. The convergence is questionable. The only indication of a problem is a coefficient that is large in magnitude together with a huge standard error. It automatically reports a chi-square test for the hypothesis that each coefficient is 0.
These are Wald tests that are calculated simply by dividing each coefficient by its estimated standard error and squaring the result. For models like the exponential that restrict the scale parameter to 1.
To test other hypotheses, you have to construct the appropriate statistic yourself. For all the regression models considered in this book, there are three general methods for constructing test statistics: Wald statistics, score statistics, and likelihood-ratio statistics.
Allison P.D. Survival Analysis Using SAS: A Practical Guide
Wald statistics are calculated using certain functions quadratic forms of parameter estimates and their estimated variances and covariances. Score statistics are based on similar functions of the first and second derivatives of the log-likelihood function. Finally, likelihood-ratio statistics are calculated by maximizing the likelihood twice: The statistic is then twice the positive difference in the two loglikelihoods. You can use all three methods to test the same hypotheses, and all three produce chi-square statistics with the same number of degrees of freedom.
Furthermore, they are asymptotically equivalent, meaning that their approximate large-sample distributions are identical. Hence, asymptotic theory gives no basis for preferring one method over another.
There is some evidence that likelihood-ratio statistics may more closely approximate a chi-square distribution in small- to moderate-sized samples, however, and some authors for example, Collett, express a strong preference for these statistics.
This is analogous to the usual Ftest that is routinely reported for linear regression models. To calculate this statistic, we need only to fit a null model that includes no covariates. For a Weibull model, we can accomplish that with the following statement: By contrast, the Weibull model with seven covariates displayed in Output 4.
Taking twice the positive difference between these two values yields a chi-square value of With seven degrees of freedom the number of covariates excluded from the null model , the p-value is less than. So we reject the null hypothesis and conclude that at least one of the coefficients is nonzero.
You can also test the same hypothesis with a Wald statistic, but that involves the following steps: These calculations include inverting the appropriate submatrix of the covariance matrix and premultiplying and postmultiplying that matrix by a vector containing appropriate linear combinations of the coefficients. Wald statistics for testing the equality of any two coefficients are simple to calculate.
The method is particularly useful for doing post-hoc comparisons of the coefficients of CLASS variables. As shown in Output 4. But there is no test reported for comparing category 3 with category 4. The covariance is. Combining these numbers with the coefficient estimates in Output 4. With 1 degree of freedom, the chi-square value is far from the.
We conclude that there is no difference in the time to arrest between those with 9th grade or less and those with 10th or 11th grade education. This should not be surprising because the overall chi-square test is not significant, nor is the more extreme comparison of category 3 with category 5. Of course, another way to get this same test statistic is simply to re-run the model with category 4 as the omitted category rather than category 5 which can be accomplished by recoding the variable so that category 4 has the highest value.
The chi-square statistic for category 3 will then be equal to. When performing post-hoc comparisons like this, it is generally advisable to adjust the alpha level for multiple comparisons. The simplest approach is the well-known Bonferroni method: Twice the positive difference is. When we tried out those models on the recidivism data, we found that they produced generally similar coefficient estimates and p-values. A glaring exception is the log-normal model, which yields qualitatively different conclusions for some of the covariates.
Clearly, we need some way of deciding between the log-normal and the other models. Even if all the models agree on the coefficient estimates, they still have markedly different implications for the shape of the hazard function.
Again we may need methods for deciding which of these shapes is the best description of the true hazard function.
Here, we examine a simple and often decisive method based on the likelihood-ratio statistic. In general, likelihood-ratio statistics can be used to compare nested models. A model is said to be nested within another model if the first model is a special case of the second. More precisely, model A is nested within model B if A can be obtained by imposing restrictions on the parameters in B.
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