Additionally, a few heavily influential points may be causing nonproportional hazards to be detected, so it is important to use graphical methods to ensure this is not the case. CONTRAST statement and ESTIMATE statement CONTRAST statement enables you to perform custom hypothesis tests by specifying an L vector or matrix for testing the univariate hypothesis L = 0 or the multivariate hypothesis LBM = 0. Any estimable linear combination of model parameters can be tested using the procedure's CONTRAST statement. run;
The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. This is exactly the contrast that was constructed earlier. We write the null hypothesis this way: The following table summarizes the data within the complicated diagnosis: The odds ratio can be computed from the data as: This means that, when the diagnosis is complicated, the odds of being cured by treatment A are 1.8845 times the odds of being cured by treatment C. The following statements display the table above and compute the odds ratio: To estimate and test this same contrast of log odds using model 3c, follow the same process as in Example 1 to obtain the contrast coefficients that are needed in the CONTRAST or ESTIMATE statement. We can examine residual plots for each smooth (with loess smooth themselves) by specifying the, List all covariates whose functional forms are to be checked within parentheses after, Scaled Schoenfeld residuals are obtained in the output dataset, so we will need to supply the name of an output dataset using the, SAS provides Schoenfeld residuals for each covariate, and they are output in the same order as the coefficients are listed in the Analysis of Maximum Likelihood Estimates table. The default is UNITS=1. | SAS FAQ We will use a data set called hsb2.sas7bdat to demonstrate. 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). Consider the following medical example in which patients with one of two diagnoses (complicated or uncomplicated) are treated with one of three treatments (A, B, or C) and the result (cured or not cured) is observed. The Nelson-Aalen estimator is a non-parametric estimator of the cumulative hazard function and is given by: \[\hat H(t) = \sum_{t_i leq t}\frac{d_i}{n_i},\]. We can remove the dependence of the hazard rate on time by expressing the hazard rate as a product of \(h_0(t)\), a baseline hazard rate which describes the hazard rates dependence on time alone, and \(r(x,\beta_x)\), which describes the hazard rates dependence on the other \(x\) covariates: In this parameterization, \(h(t)\) will equal \(h_0(t)\) when \(r(x,\beta_x) = 1\). Using effects coding, the model still looks like model 3b, but the design variables for diagnosis and treatment are defined differently as you can see in the following table. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. Copyright model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
Censored observations are represented by vertical ticks on the graph. Parameters corresponding to missing level combinations are not included in the model. \[F(t) = 1 exp(-H(t))\] We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. Fortunately, it is very simple to create a time-varying covariate using programming statements in proc phreg. Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. 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. Proc PHREG - Random Statement. The difficulty is constructing combinations that are estimable and that jointly test the set of interactions. 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. requests that each individual contrast (that is, each row, , of ) or exponentiated contrast () be estimated and tested. Thus, both genders accumulate the risk for death with age, but females accumulate risk more slowly. Note: This was the primary reference used for this seminar. There are \(df\beta_j\) values associated with each coefficient in the model, and they are output to the output dataset in the order that they appear in the parameter table Analysis of Maximum Likelihood Estimates (see above). Example Suppose we wish to fit a PH model to the data from . We can similarly calculate the joint probability of observing each of the \(n\) subjects failure times, or the likelihood of the failure times, as a function of the regression parameters, \(\beta\), given the subjects covariates values \(x_j\): \[L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{iin R_j}exp(x_i\beta)}\Bigg\rbrace\]. fstat: the censoring variable, loss to followup=0, death=1, Without further specification, SAS will assume all times reported are uncensored, true failures. Some procedures, like PROC LOGISTIC, produce a Wald chi-square statistic instead of a likelihood ratio statistic. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. model lenfol*fstat(0) = gender|age bmi|bmi hr ;
C?1D!^$w"I&#I" NF[cPdn .c@hHa"3IX"P+ !Hp? This subject could be represented by 2 rows like so: This structuring allows the modeling of time-varying covariates, or explanatory variables whose values change across follow-up time. 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. 80(30). In very large samples the Kaplan-Meier estimator and the transformed Nelson-Aalen (Breslow) estimator will converge. 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 coefficient. Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. Be careful to order the coefficients to match the order of the model parameters in the procedure. The probability of surviving the next interval, from 2 days to just before 3 days during which another 8 people died, given that the subject has survived 2 days (the conditional probability) is \(\frac{492-8}{492} = 0.98374\). Perhaps you also suspect that the hazard rate changes with age as well. 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: \[HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))\]. Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be. This paper will discuss this question by using some examples. For software releases that are not yet generally available, the Fixed Springer: New York. Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. The coefficients that are needed in the ESTIMATE statement are determined by writing what you want to estimate in terms of the fitted model. This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. These may be either removed or expanded in the future. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. This is the null hypothesis to test: Writing this contrast in terms of model parameters: Note that the coefficients for the INTERCEPT and A effects cancel out, removing those effects from the final coefficient vector. As expected, the results show that there is no significant interaction (p=0.3129) or that the reduced model fits as well as the saturated model. Phreg For Survival Analysis In Sas 9 has been minimal coverage in the available literature to9 guide researchers, practitioners, and students who wish to apply these methods to health-related areas of study. In the code below we demonstrate the steps to take to explore the functional form of a covariate: In the left panel above, Fits with Specified Smooths for martingale, we see our 4 scatter plot smooths. 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. proc loess data = residuals plots=ResidualsBySmooth(smooth);
Note that these are the fourth and eighth cell means in the Least Squares Means table. The "Class Level Information" table shows the ordering of levels within variables. 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, \(H(t)\). Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. It contains numerous examples in SAS and R. Grambsch, PM, Therneau, TM. Not only are we interested in how influential observations affect coefficients, we are interested in how they affect the model as a whole. We can plot separate graphs for each combination of values of the covariates comprising the interactions. This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. In the code below, we model the effects of hospitalization on the hazard rate. The statements below generate observations from such a model: The following statements fit the main effects and interaction model. Biometrika. You do not need to include all effects that are included in the MODEL statement. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. Hazard ratios are computed at each value of the list if the list is specified, or at each level of the interacting variable if ALL is specified, or at the reference level of the interacting variable if REF is specified. All Such linear combinations can be estimated and tested using the CONTRAST and/or ESTIMATE statements available in many modeling procedures. Similarly, the SLICEBY, DIFF, and EXP options in the SLICE statement estimate and test differences and odds ratios in the complicated diagnosis. 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. . 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. Using dummy coding, the right-hand side of the logistic model looks like it does when modeling a normally distributed response as in Example 1: where i=1,2,,5, j=1,2, k=1, 2,,Nij. The log odds for treatment A in the complicated diagnosis are: The log odds for treatment C in the complicated diagnosis are: Subtracting these gives the difference in log odds, or equivalently, the log odds ratio: The following statements use PROC LOGISTIC to fit model 3c and estimate the contrast. Unless the seed option is specified, these sets will be different each time proc phreg is run. (1995). Some procedures allow multiple types of coding. The PHREG Procedure Example 91.12 demonstrated that the log transform is a much improved functional form for Bilirubin in a Cox regression model. See the documentation for more details.). specifies that both the contrast and the exponentiated contrast be estimated. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. The DIFF option estimates and tests each pairwise difference of log odds. However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. The likelihood displacement score quantifies how much the likelihood of the model, which is affected by all coefficients, changes when the observation is left out. 2. Disease: 1=Disease, 0=No disease Drug: 1=Drug, 0=No drug This make the interaction a "2x2 table" (as below). So the log odds is: The following PROC LOGISTIC statements fit the effects-coded model and estimate the contrast: The same log odds ratio and odds ratio estimates are obtained as from the dummy-coded model. EXAMPLE 1: A Two-Factor Model with Interaction Lets interpret our model. Estimates are formed as linear estimable functions of the form . These statement essentially look like data step statements, and function in the same way. 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. specifies the maximum number of iterations to achieve the convergence of the profile-likelihood confidence limits. However, if the nested models do not have identical fixed effects, then results from ML estimation must be used to construct a LR test. Only these two statements may be flexible enough to estimate or test sufficiently complex linear combinations of model parameters. The estimate of survival beyond 3 days based off this Nelson-Aalen estimate of the cumulative hazard would then be \(\hat S(3) = exp(-0.0385) = 0.9623\). 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