The value pmust be between 0 and 1. If 3.5 is the average of the sampled values of X, the following two HAZARDRATIO statements are equivalent: specifies whether to create the Wald or profile-likelihood confidence limits, or both for the classical analyis. Here is the code: proc phreg data=Mortality_M3_72 covs (aggregate); class X (ref=first) Y (ref=first); 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. Note that there are 5 2 3 = 30 cell means. It is available only for the Bayesian analysis. This is required so that the probability of being a case is modeled. 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). 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. Finally, we calculate the hazard ratio describing a 5-unit increase in bmi, or \(\frac{HR(bmi+5)}{HR(bmi)}\), at clinically revelant BMI scores. Zeros in this table are shown as blanks for clarity. 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 should begin by analyzing our interactions. Here are the steps we use to assess the influence of each observation on our regression coefficients: The dfbetas for age and hr look small compared to regression coefficients themselves (\(\hat{\beta}_{age}=0.07086\) and \(\hat{\beta}_{hr}=0.01277\)) for the most part, but id=89 has a rather large, negative dfbeta for hr. hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5; However, if the nested models do not have identical fixed effects, then results from ML estimation must be used to construct a LR test. But an equivalent representation of the model is: where Ai and Bj are sets of design variables that are defined as follows using dummy coding: For the medical example above, model 3b for the odds of being cured are: Estimating and Testing Odds Ratios with Dummy Coding. Notice that if you add up the rows for diagnosis (or treatments), the sum is zero. These results are from the SLICE statement: The LSMESTIMATE statement produces these results: Following are the relevant sections of the CONTRAST, ESTIMATE, and LSMEANS statement results: Suppose you want to test the average of AB11 and AB12 versus the average of AB21 and AB22. ALPHA=number specifies the level of significance for % confidence intervals. The dfbeta measure, \(df\beta\), quantifies how much an observation influences the regression coefficients in the model. 81. Because log odds are being modeled instead of means, we talk about estimating or testing contrasts of log odds rather than means as in PROC MIXED or PROC GLM. Lets take a look at later survival times in the table: From LENFOL=368 to 376, we see that there are several records where it appears no events occurred. model lenfol*fstat(0) = gender|age bmi|bmi hr; Basing the test on the REML results is generally preferred. 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. The following statements fit the model and compute the AB11 and AB12 cell means by using the LSMEANS statement and equivalent ESTIMATE statements: Suppose you want to test that the AB11 and AB12 cell means are equal. The tests are equivalent. 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. and then i would like to see the trends on age group. More than one HAZARDRATIO statement can be specified, and an optional label (specified as a quoted string) helps identify the output. 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. The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. We could test for different age effects with an interaction term between gender and age. Table 86.1: PROC PHREG Statement Options You can specify the following options in the PROC PHREG statement. Notice that id, the individual subject identifier, has been added to the class statement and is also on the repeated statement (with an unstructured correlation matrix), telling proc genmod to calculate the robust errors. identifies an effect that appears in the MODEL statement. In the graph above we see the correspondence between pdfs and histograms. A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. The last 10 elements are the parameter estimates for the 10 levels of the A*B interaction, 11 through 52. You can request the CIF curves for a particular set of covariates by using the BASELINE statement. On the right panel, Residuals at Specified Smooths for martingale, are the smoothed residual plots, all of which appear to have no structure. The EXP option provides the odds ratio estimate by exponentiating the difference. The same procedure could be repeated to check all covariates. The numerator is the hazard of death for the subject who died 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). Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. Partial Likelihood The partial likelihood function for one covariate is: where t i is the ith death time, x i is the associated covariate, and R i is the risk set at time t i, i.e., the set of subjects is still alive and uncensored just prior to time t i. = 1 and cell ses = 2 will be the difference of b_1 and b_2. Here are the typical set of steps to obtain survival plots by group: Lets 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. A common way to address both issues is to parameterize the hazard function as: 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. This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. The next five elements are the parameter estimates for the levels of A, 1 through 5. 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. Fortunately, it is very simple to create a time-varying covariate using programming statements in proc phreg. 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. Example 1: One-way ANOVA The dependent variable is write and the factor variable is ses which has three levels. If the interacting variable is a CLASS variable, you can specify, after the equal sign, a list of quoted strings corresponding to various levels of the CLASS variable, or you can specify the keyword ALL or REF. Group of ses =3 is the reference group. The significance level of the confidence interval is controlled by the ALPHA= option. 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. This can be easily accomplished in. 1> Computing from the regression coefficient estimates of PROC PHREG output, 2> Recoding the values of the explanatory variable such that the increase is equal to one unit, 3> Using the CLASS statement to specify the explanatory variable in PROC TPHREG (experimental) procedure. Write down the model that you are using the procedure to fit. The DIVISOR= option is used to ensure precision and avoid nonestimability. run; proc phreg data = whas500; run; proc phreg data = whas500; Positive values of \(df\beta_j\) indicate that the exclusion of the observation causes the coefficient to decrease, which implies that inclusion of the observation causes the coefficient to increase. run; lenfol: length of followup, terminated either by death or censoring. you might need to print it in landscape mode to avoid truncation of the right edge. These statements generate data from the above model: The following statements fit model (2) and display the solution vector and cell means. Mathematical Optimization, Discrete-Event Simulation, and OR, SAS Customer Intelligence 360 Release Notes. The survival function is undefined past this final interval at 2358 days. 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. The function that describes likelihood of observing \(Time\) at time \(t\) relative to all other survival times is known as the probability density function (pdf), or \(f(t)\). specifies which differences to consider for the level comparisons of a CLASS variable. class gender; The model is the same as model (1) above with just a change in the subscript ranges. The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. run; Thus, if the average is 0 across time, then that suggests the coefficient \(p\) does not vary over time and that the proportional hazards assumption holds for covariate \(p\). See the "Parameterization of PROC GLM Models" section in the PROC GLM documentation for some important details on how the design variables are created. The default is DIFF=ALL. Tests to compare nonnested models are available, but not by using CONTRAST statements as discussed above. Because of the positive skew often seen with followup-times, medians are often a better indicator of an average survival time. It is quite powerful, as it allows for truncation, time-varying covariates and . Proportional hazards may hold for shorter intervals of time within the entirety of follow up time. 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 \(\frac{492-8}{492} = 0.98374\). 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. "exposure.". 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, \(\Delta \hat H(t) = \hat H(t_j) \hat H(t_{j-1})\). The LSMEANS, LSMESTIMATE, and SLICE statements cannot be used with effects coding. The following statements create the data set and fit the saturated logistic model. For a row vector of the contrast matrix , define to be equal to ABS if ABS is greater than 0; otherwise, equals 1. The surface where the smoothing parameter=0.2 appears to be overfit and jagged, and such a shape would be difficult to model. The next section illustrates using the CONTRAST statement to compare nested models. run; proc phreg data=whas500 plots=survival; (Js")*sv1t1} #Hqk*"lf,Rv$"TAlM@e (braP)NP r*$O2H3;0dFik-T'G2\QSDRT2H)!I+M) This paper will discuss this question by using some examples. Particular emphasis is given to proc lifetest for nonparametric estimation, and proc phreg for Cox regression and model evaluation. All produce equivalent results. Then there are three parameters () representing the first three levels, and the fourth parameter is represented by, To test the first versus the fourth level of A, you would test. In PROC GENMOD or PROC GLIMMIX, use the EXP option in the ESTIMATE statement. If proportional hazards holds, the graphs of the survival function should look parallel, in the sense that they should have basically the same shape, should not cross, and should start close and then diverge slowly through follow up time. ; The necessary contrast coefficients are stated in the null hypothesis above: (0 1 0 0 0 0) - (1/6 1/6 1/6 1/6 1/6 1/6) , which simplifies to the contrast shown in the LSMESTIMATE statement below. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. As in Example 1, you can also use the LSMEANS, LSMESTIMATE, and SLICE statements in PROC LOGISTIC, PROC GENMOD, and PROC GLIMMIX when dummy coding (PARAM=GLM) is used. Modeling Survival Data: Extending the Cox Model. (2000). since it is the comparison group. 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. Hosmer, DW, Lemeshow, S, May S. (2008). The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. proc phreg data=event; You can specify nested-by-value effects in the MODEL statement to test the effect of one variable within a particular level of another variable. model lenfol*fstat(0) = gender|age bmi|bmi hr ; 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. =2. 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). It is not necessary that the larger model be saturated. This is critical for properly ordering the coefficients in the CONTRAST or ESTIMATE statement. You can use the EFFECTPLOT statement to visualize the model. of the mean for cell ses =1 and the cell ses =3. With effects coding, the parameters are constrained to sum to zero. Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. See the documentation for more details.). exposure(0=no exposure, 1= yes exposure) and outcome(0=no outcome, 1= yes outcome) variable are all binary. (1993). It is important to know how variable levels change within the set of parameter estimates for an effect. Thus, to pull out all 6 \(df\beta_j\), we must supply 6 variable names for these \(df\beta_j\). PROC GENMOD can also be used to estimate this odds ratio. The t statistic value is the square root of the F statistic from the CONTRAST statement producing an equivalent test. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. 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. Because the observation with the longest follow-up is censored, the survival function will not reach 0. Notice that the difference in log odds for these two cells (1.02450 0.39087 = 0.63363) is the same as the log odds ratio estimate that is provided by the CONTRAST statement. Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be. It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. See the Analysis of Maximum Likelihood Estimates table to verify the order of the design variables. Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. 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 \(h(t|x_2) = h_0(t)exp(x_2\beta_x)\). We see a sharper rise in the cumulative hazard right at the beginning of analysis time, reflecting the larger hazard rate during this period. However, a common subclass of interest involves comparison of means and most of the examples below are from this class. fstat: the censoring variable, loss to followup=0, death=1, Without further specification, SAS will assume all times reported are uncensored, true failures. 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)). Not only are we interested in how influential observations affect coefficients, we are interested in how they affect the model as a whole. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; The cumulative distribution function (cdf), \(F(t)\), describes the probability of observing \(Time\) less than or equal to some time \(t\), or \(Pr(Time t)\). Survivor Function Estimates for Specific Covariate Values; Analysis of Residuals; proc sgplot data = dfbeta; The PHREG Procedure Example 91.12 demonstrated that the log transform is a much improved functional form for Bilirubin in a Cox regression model. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that proportional hazards holds for all of our covariates. Here are the steps we will take to evaluate the proportional hazards assumption for age through scaled Schoenfeld residuals: 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. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); Here is the syntax for CONTRAST statement. The PLSINGULAR= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. 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. Thus, at the beginning of the study, we would expect around 0.008 failures per day, while 200 days later, for those who survived we would expect 0.002 failures per day. model lenfol*fstat(0) = ; It contains numerous examples in SAS and R. 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