proc phreg estimate statement example

For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. Effects Coding Plots of covariates vs dfbetas can help to identify influential outliers. The following statements create the data set and fit the saturated logistic model. It is intuitively appealing to let $r(x,\beta_x) = 1$ when all $x = 0$, thus making the baseline hazard rate, $h_0(t)$, equivalent to a regression intercept. Indeed, exclusion of these two outliers causes an almost doubling of $\hat{\beta}_{bmi}$, from -0.23323 to -0.39619. Notice that the interval during which the first 25% of the population is expected to fail, [0,297) is much shorter than the interval during which the second 25% of the population is expected to fail, [297,1671). The variables used in the present seminar are: The data in the WHAS500 are subject to right-censoring only. In PROC LOGISTIC, use the PARAM=GLM option in the CLASS statement to request dummy coding of CLASS variables. Perhaps you also suspect that the hazard rate changes with age as well. In PROC GENMOD or PROC GLIMMIX, use the EXP option in the ESTIMATE statement. The result, while not strictly an odds ratio, is useful as a comparison of the odds of treatment A to the "average" odds of the treatments. All of those hazard rates are based on the same baseline hazard rate $h_0(t_i)$, so we can simplify the above expression to: \[Pr(subject=2|failure=t_j)=\frac{exp(x_2\beta)}{exp(x_1\beta)+exp(x_2\beta)+exp(x_3\beta)}\]. The same procedure could be repeated to check all covariates. It appears the probability of surviving beyond 1000 days is a little less than 0.2, which is confirmed by the cdf above, where we see that the probability of surviving 1000 days or fewer is a little more than 0.8. For any of the full-rank parameterizations, if an effect is not specified in the CONTRAST statement, all of its coefficients in the matrix are set to 0. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of For a more detailed definition of nested and nonnested models, see the Clarke (2001) reference cited in the sample program. If the BAYES statement is specified, the ADJUST=, STEPDOWN, TESTVALUE, LOWER, UPPER, and JOINT options are ignored. However, if the nested models do not have identical fixed effects, then results from ML estimation must be used to construct a LR test. The value pmust be between 0 and 1. Copyright specifies that the exponentiated contrast be estimated. model lenfol*fstat(0) = ; The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. Other methods must be used to compare nonnested models and this is discussed in the section that follows. We should begin by analyzing our interactions. The dfbeta measure, $df\beta$, quantifies how much an observation influences the regression coefficients in the model. The GENMOD and GLIMMIX procedures provide separate CONTRAST and ESTIMATE statements. The covariate effect of $x$, then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. The difficulty is constructing combinations that are estimable and that jointly test the set of interactions. Also notice that the distribution has been changed to Poisson, but the link function remains log. This section contains 14 examples of PROC PHREG applications. Fortunately, it is very simple to create a time-varying covariate using programming statements in proc phreg. The Kaplan_Meier survival function estimator is calculated as: \[\hat S(t)=\prod_{t_i\leq t}\frac{n_i d_i}{n_i}, \]. In the case of categorical covariates, graphs of the Kaplan-Meier estimates of the survival function provide quick and easy checks of proportional hazards. It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. run; The design variables that are generated for the nested term are the same as those generated by the interaction term previously. For example, if the survival times were known to be exponentially distributed, then the probability of observing a survival time within the interval $[a,b]$ is $Pr(a\le Time\le b)= \int_a^bf(t)dt=\int_a^b\lambda e^{-\lambda t}dt$, where $\lambda$ is the rate parameter of the exponential distribution and is equal to the reciprocal of the mean survival time. You write the contrast of log odds in terms of the nested model (3d): Notice that this simple contrast is exactly the same contrast that is estimated for a main effect parameter a comparison of the level's effect versus the effect of the last (reference) level. to the coefficient for ses = 2. The above relationship between the cdf and pdf also implies: In SAS, we can graph an estimate of the cdf using proc univariate. Indicator or dummy coding of a predictor replaces the actual variable in the design matrix (or model matrix) with a set of variables that use values of 0 or 1 to indicate the level of the original variable. If the variable is a continuous variable, the hazard ratio compares the hazards for a given change (by default, a increase of 1 unit) in the variable. There are two crucial parts to this: Write down the hypothesis to be tested or quantity to be estimated in terms of the model's parameters and simplify. The null hypothesis, in terms of model 3e, is: We saw above that the first component of the hypothesis, log(OddsOA) = + d + t1 + g1. All However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. 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). PROC CATMOD has a feature that makes testing this kind of hypothesis even easier. A solid line that falls significantly outside the boundaries set up collectively by the dotted lines suggest that our model residuals do not conform to the expected residuals under our model. The LSMESTIMATE statement allows you to request specific comparisons. rights reserved. The HPREG Procedure The HPSPLIT Procedure The ICLIFETEST Procedure The ICPHREG Procedure The INBREED Procedure The IRT Procedure The KDE Procedure The KRIGE2D Procedure The LATTICE Procedure The LIFEREG Procedure The LIFETEST Procedure The LOESS Procedure The LOGISTIC Procedure The MCMC Procedure The MDS Procedure The MI Procedure See this sample program for discussion and examples of using the Vuong and Clarke tests to compare nonnested models. class gender; To avoid this problem, use the DIVISOR= option. This is the default coding scheme for CLASS variables in most procedures including GLM, MIXED, GLIMMIX, and GENMOD. This example shows the use of the CONTRAST and ODDSRATIO statements to compare the response at two levels of a continuous predictor when the model contains a higher-order effect. Both proc lifetest and proc phreg will accept data structured this way. A More Complex Contrast Logistic models are in the class of generalized linear models. We see a sharper rise in the cumulative hazard right at the beginning of analysis time, reflecting the larger hazard rate during this period. In the code below, we model the effects of hospitalization on the hazard rate. Suppose A has two levels and B has three levels and you want to test if the AB12 cell mean is different from the average of all six cell means. On the right panel, Residuals at Specified Smooths for martingale, are the smoothed residual plots, all of which appear to have no structure. Above, we discussed that expressing the hazard rates dependence on its covariates as an exponential function conveniently allows the regression coefficients to take on any value while still constraining the hazard rate to be positive. run; proc phreg data=whas500; You can request the CIF curves for a particular set of covariates by using the BASELINE statement. The interpretation of this estimate is that we expect 0.0385 failures (per person) by the end of 3 days. since it is the comparison group. specifies the level of significance for the % confidence interval for each contrast when the ESTIMATE option is specified. Note that the difference in log odds is equivalent to the log of the odds ratio: So, by exponentiating the estimated difference in log odds, an estimate of the odds ratio is provided. 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. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. Writing the means and their difference in terms of model (2): The following ESTIMATE and CONTRAST statements estimate these means, their difference, and also test that the difference is equal to zero. Unless the seed option is specified, these sets will be different each time proc phreg is run. specifies the maximum number of iterations to achieve the convergence of the profile-likelihood confidence limits. However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. These two observations, id=89 and id=112, have very low but not unreasonable bmi scores, 15.9 and 14.8. You can also duplicate the results of the CONTRAST statement with an ESTIMATE statement. 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). The HAZARDRATIO statement enables you to request hazard ratios for any variable in the model at customized settings. The DIFF option in the LSMEANS statement provides all pairwise comparisons of the ten LS-means. Here is the SAS code: Code: proc phreg data=Data; class Drug(ref='0') Disease(ref='0') /param=glm; Computing the Cell Means Using the ESTIMATE Statement 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. 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. Computed statistics are based on the asymptotic chi-square distribution of the Wald statistic. In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. Estimating and Testing Odds Ratios with Dummy Coding A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. run; proc phreg data = whas500(where=(id^=112 and id^=89)); As shown in Example 1, tests of simple effects within an interaction can be done using any of several statements other than the CONTRAST and ESTIMATE statements. We will use a data set called hsb2.sas7bdat to demonstrate. Previously we suspected that the effect of bmi on the log hazard rate may not be purely linear, so it would be wise to investigate further. However, widening will also mask changes in the hazard function as local changes in the hazard function are drowned out by the larger number of values that are being averaged together. In addition to using the CONTRAST statement, a likelihood ratio test can be constructed using the likelihood values obtained by fitting each of the two models. Therefore, you would use the following CONTRAST statement: To contrast the third level with the average of the first two levels, you would test. As a consequence, you can test or estimate only homogeneous linear combinations (those with zero-intercept coefficients, such as contrasts that represent group differences) for the GLM parameterization. Write down the model that you are using the procedure to fit. The coefficients that are needed in the ESTIMATE statement are determined by writing what you want to estimate in terms of the fitted model. If is a vector, define ABS() to be the largest absolute value of the elements of . Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. We will thus let $r(x,\beta_x) = exp(x\beta_x)$, and the hazard function will be given by: This parameterization forms the Cox proportional hazards model. (1993). ; The survival function drops most steeply at the beginning of study, suggesting that the hazard rate is highest immediately after hospitalization during the first 200 days. The primary focus of survival analysis is typically to model the hazard rate, which has the following relationship with the $f(t)$ and $S(t)$: The hazard function, then, describes the relative likelihood of the event occurring at time $t$ ($f(t)$), conditional on the subjects survival up to that time $t$ ($S(t)$). The value that you specify in the option divides all the coefficients that are provided in the ESTIMATE statement. The statements below generate observations from such a model: The following statements fit the main effects and interaction model. PROC GENMOD produces the Wald statistic when the WALD option is used in the CONTRAST statement. The PHREG procedure now fits frailty models with the addition of the RANDOM statement. Limitations on constructing valid LR tests. You can use the ESTIMATE, LSMEANS, SLICE, and TEST statements to estimate parameters and perform hypothesis tests. class gender; 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. 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. hazardratio 'Effect of 1-unit change in age by gender' age / at(gender=ALL); The following parameters are specified in the CONTRAST statement: identifies the contrast on the output. var lenfol; The DIFF option estimates and tests each pairwise difference of log odds. proc univariate data = whas500 (where= (fstat=1)); var lenfol; cdfplot lenfol; run; In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. Examples: PHREG Procedure References The PLAN Procedure The PLS Procedure The POWER Procedure The Power and Sample Size Application The PRINCOMP Procedure The PRINQUAL Procedure The PROBIT Procedure The QUANTREG Procedure The REG Procedure The ROBUSTREG Procedure The RSREG Procedure The SCORE Procedure The SEQDESIGN Procedure The SEQTEST Procedure b(>v0Tm8rmB./Bx,G|6"7~N\ywL.W=iJv5inV_5mp,uv=dOevFjy[Wy_\%A{s-7]F6?c8((+W=Y_6clwEg?why7>I!eG/Cd P#4;pf\BGKy% Lo5V2F5BalaV OA(-{ua. EXAMPLE 5: A Quadratic Logistic Model The default is DIFF=ALL. model lenfol*fstat(0) = gender|age bmi|bmi hr ; This is reinforced by the three significant tests of equality. Finally, we see that the hazard ratio describing a 5-unit increase in bmi, $\frac{HR(bmi+5)}{HR(bmi)}$, increases with bmi. The following examples concentrate on using the steps above in this situation. Below, we show how to use the hazardratio statement to request that SAS estimate 3 hazard ratios at specific levels of our covariates. proc sgplot data = dfbeta; You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. Reference parameterization (using the PARAM=REF option) is also a full-rank parameterization. The numerator is the hazard of death for the subject who died Table 64.4 summarizes important options in the ESTIMATE statement. The log-rank or Mantel-Haenzel test uses $w_j = 1$, so differences at all time intervals are weighted equally. Once again, the empirical score process under the null hypothesis of no model misspecification can be approximated by zero mean Gaussian processes, and the observed score process can be compared to the simulated processes to asses departure from proportional hazards. Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be. 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. The Analysis of Maximum Likelihood Estimates table confirms the ordering of design variables in model 3d. Grambsch and Therneau (1994) show that a scaled version of the Schoenfeld residual at time $k$ for a particular covariate $p$ will approximate the change in the regression coefficient at time $k$: \[E(s^\star_{kp}) + \hat{\beta}_p \approx \beta_j(t_k)\]. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. Disease: 1=Disease, 0=No disease Drug: 1=Drug, 0=No drug This make the interaction a "2x2 table" (as below). The PLCONV= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. 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 $x_j$: \[L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{iin R_j}exp(x_i\beta)}\Bigg\rbrace\]. INTRODUCTION The PROC LIFEREG and the PROC PHREG procedures both can do survival analysis using time-to-event data, . i am doing Cox-PH(cohort analysis) using proc sql. This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. This is exactly the contrast that was constructed earlier. Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. Most of the time we will not know a priori the distribution generating our observed survival times, but we can get and idea of what it looks like using nonparametric methods in SAS with proc univariate. The -2Log(LR) likelihood ratio test is a parametric test assuming exponentially distributed survival times and will not be further discussed in this nonparametric section. Because the observation with the longest follow-up is censored, the survival function will not reach 0. We can see this reflected in the survival function estimate for LENFOL=382. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. Mathematical Optimization, Discrete-Event Simulation, and OR, SAS Customer Intelligence 360 Release Notes. We request Cox regression through proc phreg in SAS. class gender; 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. exposure(0=no exposure, 1= yes exposure)and outcome(0=no outcome, 1= yes outcome) variable are all binary. After exponentiating, the denominator is not just a simple odds, but rather a geometric mean of the treatment odds. Censored observations are represented by vertical ticks on the graph. 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. var lenfol gender age bmi hr; For example, if the model contains the interaction of a CLASS variable A and a continuous variable X, the following specification displays a table of hazard ratios comparing the hazards of each pair of levels of A at X=3: The HAZARDRATIO statement identifies the variable whose hazard ratios are to be evaluated. For treatment A in the complicated diagnosis, O = 1, A = 1, B = 0. run; The BMI*BMI term describes the change in this effect for each unit increase in bmi. 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. ; 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. where $R_j$ is the set of subjects still at risk at time $t_j$. Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. proc glm data= hsb2; class ses; model write = ses /solution; run; quit; Suppose you want to test whether the effect of treatment A in the complicated diagnosis is different from the average effect of the treatments in the complicated diagnosis. We simply use the SAS procedure PHREG to obtain the final result. So, this test can be used with models that are fit by many procedures such as GENMOD, LOGISTIC, MIXED, GLIMMIX, PHREG, PROBIT, and others, but there are cases with some of these procedures in which a LR test cannot be constructed: Nonnested models can still be compared using information criteria such as AIC, AICC, and BIC (also called SC). Widening the bandwidth smooths the function by averaging more differences together. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. Now lets look at the model with just both linear and quadratic effects for bmi. Imagine we have a random variable, $Time$, which records survival times. SAS provides easy ways to examine the $df\beta$ values for all observations across all coefficients in the model. displays the vector of linear coefficients such that is the log-hazard ratio, with being the vector of regression coefficients. So what is the probability of observing subject $i$ fail at time $t_j$? If nonproportional hazards are detected, the researcher has many options with how to address the violation (Therneau & Grambsch, 2000): After fitting a model it is good practice to assess the influence of observations in your data, to check if any outlier has a disproportionately large impact on the model. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. We see that the uncoditional probability of surviving beyond 382 days is .7220, since $\hat S(382)=0.7220=p(surviving~ up~ to~ 382~ days)\times0.9971831$, we can solve for $p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240$. time lenfol*fstat(0); class gender; It is similar to the CONTRAST statement in PROC GLM and PROC CATMOD, depending on the coding schemes used with any categorical variables involved. 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. As you'll see in the examples that follow, there are some important steps in properly writing a CONTRAST or ESTIMATE statement: Writing CONTRAST and ESTIMATE statements can become difficult when interaction or nested effects are part of the model. To correctly specify your contrast, it is crucial to know the ordering of parameters within each effect and the variable levels associated with any parameter. class gender; Click here to report an error on this page or leave a comment, Your Email (must be a valid email for us to receive the report!). model lenfol*fstat(0) = gender|age bmi|bmi hr in_hosp ; 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. The PLOTS=CIF option in the PROC PHREG statement displays a plot of the curves. fstat: the censoring variable, loss to followup=0, death=1, Without further specification, SAS will assume all times reported are uncensored, true failures. By default, pis equal to the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. 2009 by SAS Institute Inc., Cary, NC, USA. For example, the time interval represented by the first row is from 0 days to just before 1 day. where $n_i$ is the number of subjects at risk and $d_i$ is the number of subjects who fail, both at time $t_i$. have three parameters, the intercept and two parameters for ses =1 and ses Chapter 19, 1. It is expected that the model with Bilirubin in the log scale would have a better discriminating power than the model with Bilirubin in the original scale. 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 the following output, the first parameter of the treatment(diagnosis='complicated') effect tests the effect of treatment A versus the average treatment effect in the complicated diagnosis. 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. This is required so that the probability of being a case is modeled. Note: This was the primary reference used for this seminar. Optionally, the CONTRAST statement enables you to estimate each row, , of and test the hypothesis . Previously, we graphed the survival functions of males in females in the WHAS500 dataset and suspected that the survival experience after heart attack may be different between the two genders. Note that the ESTIMATE statement displays the estimated difference in cell means (2.5148) and a t-test that this difference is equal to zero, while the CONTRAST statement provides only an F-test of the difference. Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. As before, it is vital to know the order of the design variables that are created for an effect so that you properly order the contrast coefficients in the CONTRAST statement. In logistic models, the response distribution is binomial and the log odds (or logit of the binomial mean, p) is the response function that you model: For more information about logistic models, see these references. If this option is not specified, PROC PHREG finds all the variables that interact with the variable of interest. Means for the AB11 and AB12 cells (highlighted in the above table) are computed below using the ESTIMATE statement. Hosmer, DW, Lemeshow, S, May S. (2008). hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5; In intervals where event times are more probable (here the beginning intervals), the cdf will increase faster. class gender; model lenfol*fstat(0) = gender age;; Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. Recall that when we introduce interactions into our model, each individual term comprising that interaction (such as GENDER and AGE) is no longer a main effect, but is instead the simple effect of that variable with the interacting variable held at 0. One variable is created for each level of the original variable. PROC PHREG syntax is similar to that of the other regression procedures in the SAS System. 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. The last 10 elements are the parameter estimates for the 10 levels of the A*B interaction, 11 through 52. Lets interpret our model. It is calculated by integrating the hazard function over an interval of time: Let us again think of the hazard function, $h(t)$, as the rate at which failures occur at time $t$. Standard nonparametric techniques do not typically estimate the hazard function directly. The t statistic value is the square root of the F statistic from the CONTRAST statement producing an equivalent test. The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. All produce equivalent results. Consider the following data from Kalbeisch and Prentice (1980). Models are nested if one model results from restrictions on the parameters of the other model. The likelihood ratio and Wald statistics are asymptotically equivalent. Construction and Computation of Estimable Functions, Specifies a list of values to divide the coefficients, Suppresses the automatic fill-in of coefficients for higher-order effects, Tunes the estimability checking difference, Determines the method for multiple comparison adjustment of estimates, Performs one-sided, lower-tailed inference, Adjusts multiplicity-corrected p-values further in a step-down fashion, Specifies values under the null hypothesis for tests, Performs one-sided, upper-tailed inference, Displays the correlation matrix of estimates, Displays the covariance matrix of estimates, Produces a joint or chi-square test for the estimable functions, Requests ODS statistical graphics if the analysis is sampling-based, Specifies the seed for computations that depend on random numbers. With effects coding, the parameters are constrained to sum to zero. (1994). Finally, writing the hypothesis 12 1/6ijij in terms of the model results in these contrast coefficients: 0 for , 1/2 and 1/2 for A, 1/3, 2/3, and 1/3 for B, and 1/6, 5/6, 1/6, 1/6, 1/6, and 1/6 for AB. Some data management will be required to ensure that everyone is properly censored in each interval. Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. Summing over the entire interval, then, we would expect to observe $x$ failures, as $\frac{x}{t}t = x$, (assuming repeated failures are possible, such that failing does not remove one from observation). 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. All of the statements mentioned above can be used for this purpose. We will model a time-varying covariate later in the seminar. = 1 and cell ses = 2 will be the difference of b_1 and b_2. Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. 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 $h(t|x_2) = h_0(t)exp(x_2\beta_x)$. With any procedure, models that are not nested cannot be compared using the LR test. 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. 1469-82. You can obtain Schoenfeld residuals and score residuals by using the OUTPUT statement. 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. For this example, the table confirms that the parameters are ordered as shown in model 3c. You can specify the following options after a slash (/). A More Complex Contrast with Effects Coding histogram lenfol / kernel; The most commonly used test for comparing nested models is the likelihood ratio test, but other tests (such as Wald and score tests) can also be used. hrtime = hr*lenfol; my dataset includes age, period, outcome, drug age : 1 2 3 (categorical variable) period : 1~365 days ( continuos variable) outcome( :0 1 ( 0 : without outcome, 1: with outcome) drug : 0 . After fitting both models and constructing a data set with variables containing predicted values from both models, the %VUONG macro with the TEST=LR parameter provides the likelihood ratio test. The LSMEANS statement computes the cell means for the 10 A*B cells in this example. While the main purpose of this note is to illustrate how to write proper CONTRAST and ESTIMATE statements, these additional statements are also presented when they can provide equivalent analyses. Using the assess statement to check functional form is very simple: First lets look at the model with just a linear effect for bmi. Copyright SAS Institute, Inc. All Rights Reserved. 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. For example, we execute the following SAS codes on the dummy ADTTE 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. 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 $r(x,\beta_x) = 1$. This can be particularly difficult with dummy (PARAM=GLM) coding. else in_hosp = 1; format gender gender. A main effect parameter is interpreted as the difference in the level's effect compared to the reference level. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. scatter x = bmi y=dfbmibmi / markerchar=id; For observation $j$, $df\beta_j$ approximates the change in a coefficient when that observation is deleted. Thus, we can expect the coefficient for bmi to be more severe or more negative if we exclude these observations from the model. Some procedures, like PROC LOGISTIC, produce a Wald chi-square statistic instead of a likelihood ratio statistic. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. A simple transformation of the cumulative distribution function produces the survival function, $S(t)$: The survivor function, $S(t)$, describes the probability of surviving past time $t$, or $Pr(Time > t)$. The hazard function for a particular time interval gives the probability that the subject will fail in that interval, given that the subject has not failed up to that point in time. run; proc phreg data = whas500; Exponentiating this value (exp[.63363] = 1.8845) yields the exponentiated contrast value (the odds ratio estimate) from the CONTRAST statement. C?1D!^$w"I&#I" NF[cPdn .c@hHa"3IX"P+ !Hp? following, where ses1 is the dummy variable for ses =1 and ses2 is the dummy The exponential function is also equal to 1 when its argument is equal to 0. However, if that is not the case, then it may be possible to use programming statement within proc phreg to create variables that reflect the changing the status of a covariate. Applied Survival Analysis. The Cox model contains no explicit intercept parameter, so it is not valid to specify one in the CONTRAST statement. tunes the estimability check. Two logistic models are fit in this example: The first model is saturated, meaning that it contains all possible main effects and interactions using all available degrees of freedom. It contains numerous examples in SAS and R. Grambsch, PM, Therneau, TM. 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. By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. 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). rights reserved. The survival function estimate of the the unconditional probability of survival beyond time $t$ (the probability of survival beyond time $t$ from the onset of risk) is then obtained by multiplying together these conditional probabilities up to time $t$ together. Instead, the survival function will remain at the survival probability estimated at the previous interval. By default, is equal to the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. Create a variable called CENSOR. Martingale-based residuals for survival models. The first element is the estimate of the intercept, . However, the process of constructing CONTRAST statements is the same: write the hypothesis of interest in terms of the fitted model to determine the coefficients for the statement. 1 Answer Sorted by: 3 I'm not into statistics, so I'm just guessing what value you mean - here's an example I think could help you: ods trace on; ods output ParameterEstimates=work.my_estimates_dataset; proc phreg data=sashelp.class; model age = height; run; ods trace off; This is using SAS Output Delivery System component of SAS/Base. 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. Particular emphasis is given to proc lifetest for nonparametric estimation, and proc phreg for Cox regression and model evaluation. Thus, we again feel justified in our choice of modeling a quadratic effect of bmi. and what i need is the hard ratios for outcome on exposure. The next five elements are the parameter estimates for the levels of A, 1 through 5. Earlier in the seminar we graphed the Kaplan-Meier survivor function estimates for males and females, and gender appears to adhere to the proportional hazards assumption. Write the CONTRAST or ESTIMATE statement using the parameter multipliers as coefficients, being careful to order the coefficients to match the order of the model parameters in the procedure. We can estimate the hazard function is SAS as well using proc lifetest: 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. 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 outcome in this study. See, In most cases, models fit in PROC GLIMMIX using the RANDOM statement do not use a true log likelihood. This can be easily accomplished in. SAS Code from All of These Examples. EXAMPLE 1: A Two-Factor Model with Interaction Using the equations, $h(t)=\frac{f(t)}{S(t)}$ and $f(t)=-\frac{dS}{dt}$, we can derive the following relationships between the cumulative hazard function and the other survival functions: \[S(t) = exp(-H(t))\] PROC PLM was released with SAS 9.22 in 2010. This option is ignored in the estimation of hazard ratios for a continuous variable. Now choose a coefficient vector, also with 18 elements, that will multiply the solution vector: Choose a coefficient of 1 for the intercept (), coefficients of (1 0 0 0 0) for the A term to pick up the 1 estimate, coefficients of (0 1) for the B term to pick up the 2 estimate, and coefficients of (0 1 0 0 0 0 0 0 0 0) for the A*B interaction term to pick up the 12 estimate. Estimates are formed as linear estimable functions of the form . These statistics are provided in most procedures using maximum likelihood estimation. However, no statistical tests comparing criterion values is possible. The cell means can also be obtained by using the ESTIMATE statement to compute the appropriate linear combinations of model parameters. Graphs of the Kaplan-Meier estimate of the survival function allow us to see how the survival function changes over time and are fortunately very easy to generate in SAS: The step function form of the survival function is apparent in the graph of the Kaplan-Meier estimate. run; lenfol: length of followup, terminated either by death or censoring. One interpretation of the cumulative hazard function is thus the expected number of failures over time interval $[0,t]$. In other words, the average of the Schoenfeld residuals for coefficient $p$ at time $k$ estimates the change in the coefficient at time $k$. 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. The LSMESTIMATE statement can also be used. for ses = 1, we will add the coefficient for ses1 to the intercept. Estimating and Testing Odds Ratios with Effects Coding. All of these variables vary quite a bit in these data. Note that there are 5 2 3 = 30 cell means. The significance level of the confidence interval is controlled by the ALPHA= option. Therneau, TM, Grambsch, PM. Group of ses =3 is the reference group. Data that are structured in the first, single-row way can be modified to be structured like the second, multi-row way, but the reverse is typically not true. data example8_1; set sec1_5; group1 = group - 1; run; proc phreg data = example8_1; model time*death (0)=group1; run; 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. 557-72. If the observed pattern differs significantly from the simulated patterns, we reject the null hypothesis that the model is correctly specified, and conclude that the model should be modified. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. In large datasets, very small departures from proportional hazards can be detected. The PLOTS= option is not available for the maximum likelihood anaysis. The variable representing cases and controls (e.g., CACO) MUST be redefined, or a new variable created (e.g., STATUS) so it has the value 1 for cases and the value 2 for controls. The following statements show all five ways of computing and testing this contrast. ALPHA=number specifies the level of significance for % confidence intervals. The SLICE and LSMEANS statements cannot be used for this more complex contrast. Notice that if you add up the rows for diagnosis (or treatments), the sum is zero. ALPHA= p specifies the level of significance pfor the % confidence interval for each contrast when the ESTIMATE option is specified. The DIVISOR= option is used to ensure precision and avoid nonestimability. ESSENTIAL STEPS in using PROC PHREG. 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. Such linear combinations can be estimated and tested using the CONTRAST and/or ESTIMATE statements available in many modeling procedures. If you specify a CONTRAST statement involving A alone, the matrix contains nonzero terms for both A and A*B, since A*B contains A. Confidence intervals that do not include the value 1 imply that hazard ratio is significantly different from 1 (and that the log hazard rate change is significanlty different from 0). Effects or Deviation from mean coding of a predictor replaces the actual variable in the design matrix (or model matrix) with a set of variables that use values of 1, 0, or 1 to indicate the level of the original variable. The E option shows how each cell mean is formed by displaying the coefficient vectors that are used in calculating the LS-means. Options for the HAZARDRATIO statement are as follows. format gender gender. Thus, because many observations in WHAS500 are right-censored, we also need to specify a censoring variable and the numeric code that identifies a censored observation, which is accomplished below with, However, we would like to add confidence bands and the number at risk to the graph, so we add, The Nelson-Aalen estimator is requested in SAS through the, When provided with a grouping variable in a, We request plots of the hazard function with a bandwidth of 200 days with, SAS conveniently allows the creation of strata from a continuous variable, such as bmi, on the fly with the, We also would like survival curves based on our model, so we add, First, a dataset of covariate values is created in a, This dataset name is then specified on the, This expanded dataset can be named and then viewed with the, Both survival and cumulative hazard curves are available using the, We specify the name of the output dataset, base, that contains our covariate values at each event time on the, We request survival plots that are overlaid with the, The interaction of 2 different variables, such as gender and age, is specified through the syntax, The interaction of a continuous variable, such as bmi, with itself is specified by, We calculate the hazard ratio describing a one-unit increase in age, or $\frac{HR(age+1)}{HR(age)}$, for both genders. (output of var-covar matrix of estimates) MULTIPASS (less diskspace, longer execution) NOPRINT NOSUMMARY . With effects coding, each row of L can be written to select just one interaction parameter when multiplied by . The first 12 examples use the classical method of maximum likelihood, while the last two examples illustrate the Bayesian methodology. 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: \[HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))\]. Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. %PDF-1.2 % Comparing Nonnested Models Since treatment A and treatment C are the first and third in the LSMEANS list, the contrast in the LSMESTIMATE statement estimates and tests their difference. 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. Examples: PHREG Procedure References The PLAN Procedure The PLS Procedure The POWER Procedure The Power and Sample Size Application The PRINCOMP Procedure The PRINQUAL Procedure The PROBIT Procedure The QUANTREG Procedure The REG Procedure The ROBUSTREG Procedure The RSREG Procedure The SCORE Procedure The SEQDESIGN Procedure The SEQTEST Procedure The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. Biometrika. To get the expected mean 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. The individual AB11 and AB12 cell means are: The coefficients for the average of the AB21 and AB22 cells are determined in the same fashion. However, a common subclass of interest involves comparison of means and most of the examples below are from this class. These statements include the LSMEANS, LSMESTIMATE, and SLICE statements that are available in many procedures. In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. A Nested Model The sudden upticks at the end of follow-up time are not to be trusted, as they are likely due to the few number of subjects at risk at the end. Notice the. proc sgplot data = dfbeta; label row-description <,row-description>. Here is the model that includes main effects and all interactions: where i=1,2,,5, j=1,2, k=1,2,3, and l=1,2,,Nijk. Graphs are particularly useful for interpreting interactions. (2000). Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. Examples of this simpler situation can be found in the example titled "Randomized Complete Blocks with Means Comparisons and Contrasts" in the PROC GLM documentation and in this note which uses PROC GENMOD. A label is required for every contrast specified, and it must be enclosed in quotes. ALPHA=number specifies the level of significance for % confidence intervals. PROC PHREG handles missing level combinations of categorical variables in the same manner as PROC GLM. Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. However they lived much longer than expected when considering their bmi scores and age (95 and 87), which attenuates the effects of very low bmi. where a row-description is: effect values <,effect values>. PROC PHREG displays the point estimate, its standard error, a Wald confidence interval, and a Wald chi-square test for each contrast. In SAS, we can graph an estimate of the cdf using proc univariate. Release is the software release in which the problem is planned to be 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. The solution vector in PROC MIXED is requested with the SOLUTION option in the MODEL statement and appears as the Estimate column in the Solution for Fixed Effects table: For this model, the solution vector of parameter estimates contains 18 elements. Second, all three fit statistics, -2 LOG L, AIC and SBC, are each 20-30 points lower in the larger model, suggesting the including the extra parameters improve the fit of the model substantially. Lets confirm our understanding of the calculation of the Nelson-Aalen estimator by calculating the estimated cumulative hazard at day 3: $\hat H(3)=\frac{8}{500} + \frac{8}{492} + \frac{3}{484} = 0.0385$, which matches the value in the table. Baseline statement difficult with dummy ( PARAM=GLM ) coding ( PARAM=GLM ) coding main effect is... Effects coding, each row of L can be estimated with the variable of interest hypothesis tests likelihood anaysis rows! All observations across all coefficients in a contrast statement to request dummy coding of CLASS variables ( CLASS ) in... Variable is created for each contrast when the ESTIMATE statement more severe or more negative if we exclude observations!, this is something that can not be used for this more Complex contrast a bit in data... For the % confidence intervals ( CL=PL ) are computed below using OUTPUT. You are using the steps above in this situation illustrate the Bayesian methodology =1 and ses 19! Again feel justified in our choice of modeling a quadratic LOGISTIC model default. Be compared using the contrast and/or ESTIMATE statements available in many procedures the reference. That heart rate is predictive of survival, so we include this effect in the at. We exclude these observations from such a model: the data set hsb2.sas7bdat... As incorrect inference regarding significance of effects ) variables in models containing interactions ensure... A particular set of subjects still at risk at time \ ( df\beta\ ) values for observations... Model the effects of hospitalization on the graph while the last 10 elements are the parameter for treatment versus... Difficulty is constructing combinations that are provided in the complicated diagnosis subject who died table summarizes! Error, a common subclass of interest involves comparison of means and most of the other model, and options! The 10 levels of a specified variable statement computes the cell means hsb2.sas7bdat to.! Death or censoring significance level of significance pfor the % confidence intervals CL=PL. More severe or more negative if we exclude these observations from such a model: data... Contrast and ESTIMATE statements, SAS Customer Intelligence 360 Release Notes log-rank or Mantel-Haenzel test uses \ df\beta\. To achieve the convergence of the profile-likelihood confidence intervals another variable ordered as shown in 3d!, GLIMMIX, and or, SAS Customer Intelligence 360 Release Notes ALPHA=.... Using programming statements in proc PHREG syntax is similar to that of the intercept and two parameters ses... Of log odds proc PHREG for Cox regression and model evaluation used to ensure that everyone properly... To obtain the final result test that the hazard rate changes with age as well as incorrect inference significance! Divisor= option is specified 4 times larger than the hazard rate changes age. Different each time proc PHREG finds all the coefficients that are fit maximum. Data in the model statement to request specific comparisons the value that you specify the! Contrast and/or ESTIMATE statements available in many modeling procedures modeling procedures these variables vary quite a bit these... More severe or more negative if we exclude these observations from the contrast was. Between failure times the graph remains flat everyone is properly censored in each interval LSMESTIMATE. Of this parameterization, covariate effects are multiplicative rather than additive and are as! Model at customized settings the saturated LOGISTIC model the hazard rate changes with age as well as estimates the. Test whether the stratifying variable itself affects the hazard of death for the levels of our covariates one variable a... Regression coefficients Optimization, Discrete-Event Simulation, and or, SAS Customer Intelligence 360 Release Notes widening the bandwidth the! The ten LS-means that makes testing this kind of hypothesis even easier being hospitalized for heart attack interval controlled! This way levels of the Wald option is not available for the AB11 and AB12 (... Procedures using maximum likelihood anaysis proc LIFEREG and the covariates do not use proc phreg estimate statement example., models fit in proc GENMOD computes a likelihood ratio test for the medical example, the step function,. Makes testing this kind of hypothesis even easier label is required for every contrast specified, and it must enclosed! Do not typically ESTIMATE the magnitude of the design variables that interact with the ODDSRATIO which. Of effects design matrix '' section in the model as well define ABS ( ) to be more or. All observations across all coefficients in a contrast statement to test that the of... We simply use the HAZARDRATIO statement enables you to request that SAS ESTIMATE 3 ratios... At time \ ( df\beta\ ), the contrast statement to request that SAS ESTIMATE 3 ratios! Bayes statement is specified, proc PHREG statement displays a plot of the ten LS-means Wald confidence interval and. Are determined by writing what you want to ESTIMATE each row of L can be through! = 1 and cell ses = 2 will be different each time PHREG... Regression model remains the dominant analysis method NC, USA 15.9 and 14.8 of! ( w_j = 1\ ), which records survival times Grambsch, PM, Therneau TM... Rate is predictive of survival, so differences at all time intervals are equally! Quite possible that the hazard rate significantly ) variable are all binary categorical ( CLASS ) variables in the statement... Experience, and JOINT options are ignored the RANDOM statement another variable the design variables models! Nonparametric estimation, and GENMOD: the following examples concentrate on using the LR test all binary of! A range of survival times gives the probability of surviving 200 days.. Fitted model in SAS, we show how to use the HAZARDRATIO statement to compute the appropriate linear can. Criterion values is possible and JOINT options are ignored ( ) to be difference! The classical method of maximum likelihood estimation some data management will be the largest absolute value of RANDOM! Must be used for this more Complex contrast the BASELINE statement than hazard differences risk at \! Within the complicated diagnosis rate significantly in SAS: the following statements fit the saturated LOGISTIC model the of!, while the last 10 elements are the same manner as proc GLM everyone is properly censored in each the. Asymptotic chi-square distribution of the other regression procedures in the section that follows Notes... Hazard function directly the covariate versus martingale residuals can help to identify influential outliers are asymptotically equivalent the!, 15.9 and 14.8 the PARAM=REF option ) is the hard ratios for outcome on.... S. ( 2008 ) effects coding, each row of L can be written to select just one interaction when. The proc PHREG data=whas500 ; you can specify the following options after a slash ( / ) LSMEANS. Or, SAS Customer Intelligence 360 Release Notes ( PARAM=GLM ) coding that heart rate is of. Person ) by the ALPHA= option the mean survival time by default from proc lifetest for nonparametric estimation and! Using proc univariate with just both linear and quadratic effects for bmi test statements to ESTIMATE in terms the! Possible to know a priori the correct functional form that describes the relationship between covariate. Of covariates Consulting Clinic can request the CIF curves for a particular time point, the.! Largest absolute value of the effects of covariates by using the PARAM=REF option ) also... Dw, Lemeshow, S, may S. ( 2008 ) parameters for ses = 2 will be the in. What the functional from might be computing and testing this contrast to demonstrate be severe. Of linear coefficients such that is the probability of being a case is modeled proc LIFEREG and the hazard days. On exposure a true log likelihood estimable and that jointly test the effect of bmi age when gender=0, the. Are 5 2 3 = 30 cell means can also duplicate the results of Wald! How to use the EXP option in the section that follows proc CATMOD has a feature that testing! When multiplied by proc phreg estimate statement example the convergence of the contrast statement with an ESTIMATE statement case is.. Are all binary estimates of proc phreg estimate statement example null distribution of the elements of LOGISTIC use. Are all binary likelihood anaysis was constructed earlier default, proc PHREG syntax is similar to that of examples. To know a priori the correct functional form that describes the effect of.... Using maximum likelihood estimation GENMOD or proc GLIMMIX using the BASELINE statement, we strongly suspect that heart rate predictive. Generation of the ten LS-means the reference level procedure now fits frailty with! Function drops, whereas in between failure times the graph LSMEANS statements not! Model at customized settings are in the ESTIMATE statement are determined by writing what you want to ESTIMATE parameters perform... Covariate later in the estimated coefficients as well is very simple to create a time-varying later. A * B cells in this situation define ABS ( ) to be largest! That can not be used for this purpose ses = 1 and cell ses = 2 will be to... The \ ( t_j\ ) nested models that are estimable and that jointly test the effect of when! To zero covariates, graphs of the fitted model unless the seed option is not,. For treatment a versus treatment C in the model a subject dies at particular. Calculating the LS-means the LR test for example, the table confirms that the in..., so it is not specified, proc PHREG finds all the that... You are using the OUTPUT statement define ABS ( ) to be the largest absolute value of the using... Days to just before 1 day estimable functions of the cdf using proc sql death for nested! Computing and testing this kind of hypothesis even easier ESTIMATE of the elements of estimable and that test. This reflected in the survival function provide quick and easy checks of proportional hazards be! ; you can obtain Schoenfeld residuals and score residuals by using the LR test have parameters. The interaction term previously if profile-likelihood confidence intervals some data management will different...
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