In PROC LOGISTIC, use the PARAM=GLM option in the CLASS statement to request dummy coding of CLASS variables. run; Then, as before, subtracting the two coefficient vectors yields the coefficient vector for testing the difference of these two averages. Though assisting with the translation of a stated hypothesis into the needed linear combination is beyond the scope of the services that are provided by Technical Support at SAS, we hope that the following discussion and examples will help you. class gender; 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. First, there may be one row of data per subject, with one outcome variable representing the time to event, one variable that codes for whether the event occurred or not (censored), and explanatory variables of interest, each with fixed values across follow up time. format gender gender. The ODDSRATIO statement in PROC LOGISTIC and the similar HAZARDRATIO statement in PROC PHREG are also available. Note that some functions, like ratios, are nonlinear combinations and cannot generally be obtained with these statements. In some cases, the Laplace or quadrature estimation methods (METHOD=LAPLACE or METHOD=QUAD, first available in SAS 9.2) can be used which compute and report an approximate log likelihood making construction of a LR test possible. The t statistic value is the square root of the F statistic from the CONTRAST statement producing an equivalent test. This coding scheme is used by default by PROC CATMOD and PROC LOGISTIC and can be specified in these and some other procedures such as PROC GENMOD with the PARAM=EFFECT option in the CLASS statement. PROC PHREG displays the point estimate, its standard error, a Wald confidence interval, and a Wald chi-square test for each contrast. Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. The PHREG procedure now fits frailty models with the addition of the RANDOM statement. Notice the survival probability does not change when we encounter a censored observation. Diagnostic plots to reveal functional form for covariates in multiplicative intensity models. 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. Note that the CONTRAST and ESTIMATE statements are the most flexible allowing for any linear combination of model parameters. Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. If ABS is greater than , then is declared nonestimable. 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. For example, if there were three subjects still at risk at time \(t_j\), the probability of observing subject 2 fail at time \(t_j\) would be: \[Pr(subject=2|failure=t_j)=\frac{h(t_j|x_2)}{h(t_j|x_1)+h(t_j|x_2)+h(t_j|x_3)}\]. class gender; /*class exposure*/model period*outcome(0)=exposure / rl;run; Hello@MTeckand welcome to the SAS Support Communities! Ordinary least squares regression methods fall short because the time to event is typically not normally distributed, and the model cannot handle censoring, very common in survival data, without modification. See the "Parameterization of PROC GLM Models" section in the PROC GLM documentation for some important details on how the design variables are created. 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. 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. exposure(0=no exposure, 1= yes exposure)and outcome(0=no outcome, 1= yes outcome) variable are all binary. The CONTRAST statement enables you to specify a matrix, , for testing the hypothesis . To properly test a hypothesis such as "The effect of treatment A in group 1 is equal to the treatment A effect in group 2," it is necessary to translate it correctly into a mathematical hypothesis using the fitted model. Some procedures, like PROC LOGISTIC, produce a Wald chi-square statistic instead of a likelihood ratio statistic. Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. 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. However, coefficients for the B effect remain in addition to coefficients for the A*B interaction effect. I would use the CLASS statement (because exposure is a classification variable) and explicitly specify the reference level so that the intended results are clear. The EXP option provides the odds ratio estimate by exponentiating the difference. An estimate statement corresponds to an L-matrix, which corresponds to a ALPHA= p specifies the level of significance pfor the % confidence interval for each contrast when the ESTIMATE option is specified. Models with smaller values of these criteria are considered better models. Other methods must be used to compare nonnested models and this is discussed in the section that follows. Example 3: using the CONTRAST statement to do comparison: When we set the reference levels to be REF='NEV' for TOBHX and REF='GP' for RND, we need to manually set the contrast parameters for each comparison in the CONTRAST statement. We generally expect the hazard rate to change smoothly (if it changes) over time, rather than jump around haphazardly. However, the CONTRAST statement can be used in PROC GENMOD as shown above to produce a score test of the hypothesis. Thus, each term in the product is the conditional probability of survival beyond time \(t_i\), meaning the probability of surviving beyond time \(t_i\), given the subject has survived up to time \(t_i\). 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. class gender; This is an extension of the nested effects that you can specify in other procedures such as GLM and LOGISTIC. 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). If the MULTIPASS option is not specified, PROC PHREG . 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. At the beginning of a given time interval \(t_j\), say there are \(R_j\) subjects still at-risk, each with their own hazard rates: The probability of observing subject \(j\) fail out of all \(R_j\) remaing at-risk subjects, then, is the proportion of the sum total of hazard rates of all \(R_j\) subjects that is made up by subject \(j\)s hazard rate. The null hypothesis, in terms of model 3e, is: We saw above that the first component of the hypothesis, log(OddsOA) = + d + t1 + g1. 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. 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). This relationship would imply that moving from 1 to 2 on the covariate would cause the same percent change in the hazard rate as moving from 50 to 100. The Wilcoxon test uses \(w_j = n_j\), so that differences are weighted by the number at risk at time \(t_j\), thus giving more weight to differences that occur earlier in followup time. 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. The Nelson-Aalen estimator is a non-parametric estimator of the cumulative hazard function and is given by: \[\hat H(t) = \sum_{t_i leq t}\frac{d_i}{n_i},\]. The default is UNITS=1. Notice that Row2 is the coefficient vector for computing the mean of the AB12 cell. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. 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. Survival analysis models factors that influence the time to an event. Shared Concepts and Topics. Consider the following data from Kalbeisch and Prentice (1980). The hazard rate thus describes the instantaneous rate of failure at time \(t\) and ignores the accumulation of hazard up to time \(t\) (unlike \(F(t\)) and \(S(t)\)). This section contains 14 examples of PROC PHREG applications. rights reserved. Survivor Function Estimates for Specific Covariate Values; Analysis of Residuals; The next section illustrates using the CONTRAST statement to compare nested models. This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. In the Cox proportional hazards model, additive changes in the covariates are assumed to have constant multiplicative effects on the hazard rate (expressed as the hazard ratio (\(HR\))): In other words, each unit change in the covariate, no matter at what level of the covariate, is associated with the same percent change in the hazard rate, or a constant hazard ratio. You can fit many kinds of logistic models in many procedures including LOGISTIC, GENMOD, GLIMMIX, PROBIT, CATMOD, and others. (1994). Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. The EXPB option adds a column in the parameter estimates table that contains exponentiated values of the corresponding parameter estimates. In PROC LOGISTIC, the ESTIMATE=BOTH option in the CONTRAST statement requests estimates of both the contrast (difference in log odds or log odds ratio) and the exponentiated contrast (odds ratio). Consider a model for two factors: A with five levels and B with two levels: where i=1,2,,5, j=1,2, k=1, 2,,nij. Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. 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 E option shows how each cell mean is formed by displaying the coefficient vectors that are used in calculating the LS-means. (1995). However, it can happen (and it did in your example) that the CLASS statement uses level '1' of that explanatory variable as the reference level so that the sign of the corresponding parameter estimate changes and the inverse hazard ratio and confidence limits are computed,here: the hazard ratio of "no exposure" vs. Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. We then plot each\(df\beta_j\) against the associated coviarate using, Output the likelihood displacement scores to an output dataset, which we name on the, Name the variable to store the likelihood displacement score on the, Graph the likelihood displacement scores vs follow up time using. 77(1). 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 PLSINGULAR= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The calculation of the statistic for the nonparametric Log-Rank and Wilcoxon tests is given by : \[Q = \frac{\bigg[\sum\limits_{i=1}^m w_j(d_{ij}-\hat e_{ij})\bigg]^2}{\sum\limits_{i=1}^m w_j^2\hat v_{ij}},\]. We simply use the SAS procedure PHREG to obtain the final result. histogram lenfol / kernel; time lenfol*fstat(0); For example, if males have twice the hazard rate of females 1 day after followup, the Cox model assumes that males have twice the hazard rate at 1000 days after follow up as well. However, a common subclass of interest involves comparison of means and most of the examples below are from this class. run; The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. exposure(0=no exposure, 1= yes exposure) and outcome(0=no outcome, 1= yes outcome) variable are all binary. since it is the comparison group. 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. specifies that both the contrast and the exponentiated contrast be estimated. Run Cox models on intervals of follow up time rather than on its entirety. However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. Introduction The regression equation is the The outcome in this study. In particular we would like to highlight the following tables: Handily, proc phreg has pretty extensive graphing capabilities.< Below is the graph and its accompanying table produced by simply adding plots=survival to the proc phreg statement. specifies the level of significance for the % confidence interval for each contrast when the ESTIMATE option is specified. model lenfol*fstat(0) = gender age;; The simple contrast shown in the LSMESTIMATE statement below compares the fourth and eighth means as desired. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time \(t\). If, say, a regression coefficient changes only by 1% over time, it is unlikely that any overarching conclusions of the study would be affected. With this simple model, we The cell means can also be obtained by using the ESTIMATE statement to compute the appropriate linear combinations of model parameters. As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. Comparing One Interaction Mean to the Average of All Interaction Means This is required so that the probability of being a case is modeled. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. The ESTIMATE statement syntax enables you to specify the coefficient vector in sections as just described, with one section for each model effect: Note that this same coefficient vector is given in the table of LS-means coefficients, which was requested by the E option in the LSMEANS statement. Other CONTRAST statements involving classification variables with PARAM=EFFECT are constructed similarly. Weberian asked a slighltly similar question (Hazardratio statement, interaction in Proc Phreg (cox-regression)) but it does not answer this. An ESTIMATE statement for the AB11 cell mean can be written as above by rewriting the cell mean in terms of the model yielding the appropriate linear combination of parameter estimates. The design variables that are generated for the nested term are the same as those generated by the interaction term previously. else in_hosp = 1; you might need to print it in landscape mode to avoid truncation of the right edge. Thus, in the first table, we see that the hazard ratio for age, \(\frac{HR(age+1)}{HR(age)}\), is lower for females than for males, but both are significantly different from 1. Parameters corresponding to missing level combinations are not included in the model. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. This example is to illustrate the algorithm used to compute the parameter estimate. The following statements create the data set and fit the saturated logistic model. Specify the DIST=BINOMIAL option to specify a logistic model. The following examples concentrate on using the steps above in this situation. It is not at all necessary that the hazard function stay constant for the above interpretation of the cumulative hazard function to hold, but for illustrative purposes it is easier to calculate the expected number of failures since integration is not needed. 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). This seminar introduces procedures and outlines the coding needed in SAS to model survival data through both of these methods, as well as many techniques to evaluate and possibly improve the model. Wiley: Hoboken. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. C?1D!^$w"I&#I" NF[cPdn .c@hHa"3IX"P+ !Hp? If PROC PHREG finds a contrast to be nonestimable, it displays missing values in corresponding rows in the results. The following parameters are specified in the CONTRAST statement: identifies the contrast on the output. Whereas with non-parametric methods we are typically studying the survival function, with regression methods we examine the hazard function, \(h(t)\). One variable is created for each level of the original variable. run; proc phreg data = whas500; We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. Notice, however, that \(t\) does not appear in the formula for the hazard function, thus implying that in this parameterization, we do not model the hazard rates dependence on time. From the plot we can see that the hazard function indeed appears higher at the beginning of follow-up time and then decreases until it levels off at around 500 days and stays low and mostly constant. The variables used in the present seminar are: The data in the WHAS500 are subject to right-censoring only. 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 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. Had B preceded A in the CLASS statement, the levels of A would have changed before the levels of B, resulting in the second estimate being for 21. This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. 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 PLMAXITER= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The parameter for ses1 is the difference Suppose it is of interest to test the null hypothesis that cell means ABC121 and ABC212 are equal that is, H0: 121 - 212 = 0. 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. var lenfol; Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. In large datasets, very small departures from proportional hazards can be detected. PROC PLM was released with SAS 9.22 in 2010. The next five elements are the parameter estimates for the levels of A, 1 through 5. 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. model lenfol*fstat(0) = ; 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. Ignore the nonproportionality if it appears the changes in the coefficient over time are very small or if it appears the outliers are driving the changes in the coefficient. 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. requests that, for each Newton-Raphson iteration, PROC PHREG recompiles the risk sets corresponding to the event times for the (start,stop) style of response and recomputes the values of the time-dependent variables defined by the programming statements for each observation in the risk sets. Beside using the solution option to get the parameter estimates, In the second table, we see that the hazard ratio between genders, \(\frac{HR(gender=1)}{HR(gender=0)}\), decreases with age, significantly different from 1 at age = 0 and age = 20, but becoming non-signicant by 40. hazardratio 'Effect of 5-unit change in bmi across bmi' bmi / at(bmi = (15 18.5 25 30 40)) units=5; Comparing Nonnested Models This article emphasizes four features of PROC PLM: You can use the SCORE statement to score the model on new data. 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. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. 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. 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. 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. The test of the difference is more easily obtained using the LSMESTIMATE statement. Notice that the baseline hazard rate, \(h_0(t)\) is cancelled out, and that the hazard rate does not depend on time \(t\): The hazard rate \(HR\) will thus stay constant over time with fixed covariates. 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. Copyright SAS Institute, Inc. All Rights Reserved. Copyright For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of During the next interval, spanning from 1 day to just before 2 days, 8 people died, indicated by 8 rows of LENFOL=1.00 and by Observed Events=8 in the last row where LENFOL=1.00. run; proc phreg data = whas500; The tests are equivalent.

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