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. Such linear combinations can be estimated and tested using the CONTRAST and/or ESTIMATE statements available in many modeling procedures. Paul Allison’s well-known Survival Analysis Using the SAS System, for instance, gives examples of the use of such programming statements (pp. Be careful to order the coefficients to match the order of the model parameters in the procedure. One of the main purposes of PROC PLM Is to perform postfit estimates and hypothesis tests. One variable is created for each level of the original variable. This is the second reason; it is relatively easy to incorporate time-dependent covariates. EXAMPLE 4: Comparing Models ASSESS statement in SAS includes Plot of randomly generated residual processes to allow for graphic assessment of the observed residuals in terms of what is “too large” Formal hypothesis test based on simulation Checking the functional form proc phreg data=in.short_course ; model intxsurv*dead(0)=yeartx/rl; While examples in this class provide good examples of the above process for determining coefficients for CONTRAST and ESTIMATE statements, there are other statements available that perform means comparisons more easily. For simple pairwise contrasts like this involving a single effect, there are several other ways to obtain the test. The EXP option exponentiates each difference providing odds ratio estimates for each pair. Effects Coding See the example titled "Comparing nested models with a likelihood ratio test" which illustrates using the %VUONG macro to produce the same test as obtained above from the CONTRAST statement in PROC GENMOD. However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. The contrast estimate is exponentiated to yield the odds ratio estimate. in the PROC PHREG model statement numeric. Some procedures allow multiple types of coding. The regression equation is the The CONTRAST statement can also be used to compare competing nested models. 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. 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. Proportional hazards regression with PHREG The SAS procedure PROC PHREG allows us to fit a proportional hazard model to a dataset. 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. Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. The partial results shown below suggest that interactions are not needed in the model: The simpler main-effects-only model can be fit by restricting the parameters for the interactions in the above model to zero. The CONTRAST, ESTIMATE, LSMEANS, RANDOM 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. Appendix 3 contains the output from the procedure. A More Complex Contrast with Effects Coding 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 “GLM” stands for General Linear Model. It is not necessary that the larger model be saturated. There are two PROC PHREG sections to the program. By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. The next section illustrates using the CONTRAST statement to compare nested models. Estimating and Testing a Difference of Means An example of using the LSMEANS and LSMESTIMATE statements to estimate odds ratios in a repeated measures (GEE) model in PROC GENMOD is available. You use model 3e to expand the average treatment effect: So the hypothesis, written in terms of the model parameters, is simply: The following CONTRAST statement used in PROC LOGISTIC estimates and tests this hypothesis, and produces the following output tables: In PROC GENMOD, use this equivalent ESTIMATE statement: The exponentiated contrast estimate, 0.83, is not really an odds ratio. The statements below generate observations from such a model: The following statements fit the main effects and interaction model. The E option shows how each cell mean is formed by displaying the coefficient vectors that are used in calculating the LS-means. The DIFF option estimates and tests each pairwise difference of log odds. PHREG can also make it. We estimate two sets of hazard ratios for age, one for the interval up to 2 years following diagnosis and one set for the interval 2 years or more subsequent to diagnosis. 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. Comparing Nonnested Models As expected, the results show that there is no significant interaction (p=0.3129) or that the reduced model fits as well as the saturated model. Models fit with the GENMOD or GEE procedure using the REPEATED statement are estimated using the generalized estimating equations (GEE) method and not by maximum likelihood so a LR test cannot be constructed. CLR estimates for 1:1 matched studies may be obtained using the PROC LOGISTIC procedure. However, no statistical tests comparing criterion values is possible. Harrell’s Concordance Statistic. Although the coding scheme is different, you still follow the same steps to determine the contrast coefficients. Using effects coding, the model still looks like model 3b, but the design variables for diagnosis and treatment are defined differently as you can see in the following table. You can also duplicate the results of the CONTRAST statement with an ESTIMATE statement. The DIVISOR= option is used to ensure precision and avoid nonestimability. However, to obtain CLR estimates for 1:m and n:m matched studies using SAS, the PROC PHREG procedure must be used. 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. The LSMESTIMATE statement allows you to request specific comparisons. These are the equivalent PROC GENMOD statements: A More Complex Contrast with Effects Coding. Tom In an example from Ries and Smith (1963), the choice of detergent brand (Brand= M or X) is related to three other categorical variables: the softness of the laundry water (Softness= soft, medium, or hard); the temperature of the water (Temperature= high or low); and whether the subject was a previous user of Brand M (Previous= yes or no). 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. use eventcode option in proc phreg, model statement. Examples Stepwise Regression ... Table 66.4 summarizes important options in the ESTIMATE statement. USING THE NATIVE PHREG PROCEDURE . Y is vector of dependent variable values while X is the matrix of independent coeffcients, I is the identity matrix and σ… The PROC PHREG statement is simply a call and specifies the data set. The DIFF and SLICEBY(A='1') options in the SLICE statement estimate the differences in LS-means at A=1. 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 αβ21. Institute for Digital Research and Education. 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. The null hypothesis, in terms of model 3e, is: We saw above that the first component of the hypothesis, log(OddsOA) = μ + d + t1 + g1. 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. Consider the following medical example in which patients with one of two diagnoses (complicated or uncomplicated) are treated with one of three treatments (A, B, or C) and the result (cured or not cured) is observed. The next two elements are the parameter estimates for the levels of B, β1 and β2. variable for ses =2. The likelihood ratio and Wald statistics are asymptotically equivalent. The dependent variable is write and the factor variable is ses 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. The change in coding scheme does not affect how you specify the ODDSRATIO statement. All of the statements mentioned above can be used for this purpose. You can specify a contrast of the LS-means themselves, rather than the model parameters, by using the LSMESTIMATE statement. diagnosis. The following statements show all five ways of computing and testing this contrast. Left panel: Survival estimates from PROC PHREG, using a BY statement to get curves for different levels of a strata variable; right panel: survival estimates from PROC PHREG using the covariates = option in the BASELINE statement. For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. The CONTRAST statement tests the hypothesis Lβ=0, where L is the hypothesis matrix and β is the vector of model parameters. The LSMESTIMATE statement again makes this easier. proc phreg data=Rats; model Days*Status(0)=Group; run; The statements below fit the model, estimate each part of the hypothesis, and estimate and test the hypothesis. Group of ses =3 is the reference group. Cite. INTRODUCTION We begin by defining a time-dependent variable and use Stanford heart transplant study as example. for ses = 1, we will add the coefficient for ses1 to the intercept. Therefore, the estimate of the last level of an effect, A, is αa= â(α1 + α2 + ... + αaâ1). Note that the CONTRAST statement in PROC LOGISTIC provides an estimate of the contrast as well as a test that it equals zero, so an ESTIMATE statement is not provided. 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. At last, we also learn SAS mixe… 138-154) but does not discuss counting process format at all. In our following figure, y is dependent variable while x1, x2, x3 … are independent variables. 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. The first observation has survival time 0 and survivor function estimate 1.0. The design variables that are generated for the nested term are the same as those generated by the interaction term previously. The final coefficients appear in ESTIMATE and CONTRAST statements below. Based on the theory behind Cox proportional hazard model, I need the 95% CI. Models with smaller values of these criteria are considered better models. To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see. The following statements do the model comparison using PROC LOGISTIC and the Wald test produces a very similar result. Variables in this statement that are not specified in a CLASS statement are assumed to be continuous. Note that the CONTRAST and ESTIMATE statements are the most flexible allowing for any linear combination of model parameters. These statistics are provided in most procedures using maximum likelihood estimation. In our previous article we have seen Longitudinal Data Analysis Procedures, today we will discuss what is SAS mixed model. Estimating and Testing Odds Ratios with Dummy Coding In this case, the αβ12 estimate is the sixth estimate in the A*B effect requiring a change in the coefficient vector that you specify in the ESTIMATE statement. Again, trailing zero coefficients can be omitted. Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. Means for the AB11 and AB12 cells (highlighted in the above table) are computed below using the ESTIMATE statement. = 1 and cell ses = 2 will be the difference of b_1 and b_2. You can use the DIFF option in the LSMEANS statement. The CONTRAST and ESTIMATE statements allow for estimation and testing of any linear combination of model parameters. Note that there are 5 à 2 à 3 = 30 cell means. It is important to know how variable levels change within the set of parameter estimates for an effect. An estimate statement corresponds to an L-matrix, which corresponds to a The log odds for treatment A in the complicated diagnosis are: The log odds for treatment C in the complicated diagnosis are: Subtracting these gives the difference in log odds, or equivalently, the log odds ratio: The following statements use PROC LOGISTIC to fit model 3c and estimate the contrast. Suppose it is of interest to test the null hypothesis that cell means ABC121 and ABC212 are equal â that is, H0: μ121 - μ212 = 0. Note that these are the fourth and eighth cell means in the Least Squares Means table. So the log odds are: For treatment C in the complicated diagnosis, O = 1, A = â1, B = â1. Beside using the solution option to get the parameter estimates, we can also use the option "e" following the estimate statement to get the L matrix. All produce equivalent results. With any procedure, models that are not nested cannot be compared using the LR test. Comparing One Interaction Mean to the Average of All Interaction Means In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most easily obtained using the ODDSRATIO statement. The value must be between 0 and 1. The CONTRAST statement below defines seven rows in L for the seven interaction parameters resulting in a 7 DF test that all interaction parameters are zero. After exponentiating, the denominator is not just a simple odds, but rather a geometric mean of the treatment odds. Words in italic are new statements added to SAS version 9.22. Note that the ESTIMATE statement displays the estimated difference in cell means (â2.5148) and a t-test that this difference is equal to zero, while the CONTRAST statement provides only an F-test of the difference. 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 . we can also use the option "e" following the estimate Finally, writing the hypothesis μ12 â 1/6 Σijμij in terms of the model results in these contrast coefficients: 0 for μ, 1/2 and â1/2 for A, â1/3, 2/3, and â1/3 for B, and â1/6, 5/6, â1/6, â1/6, â1/6, and â1/6 for AB. This test can be done using a CONTRAST statement to jointly test the interaction parameters. The same results can be obtained using the ESTIMATE statement in PROC GENMOD. 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. Technical Support can assist you with syntax and other questions that relate to CONTRAST and ESTIMATE statements. A More Complex Contrast ALPHA= number specifies the alpha level of the interval estimates for the hazard ratios. In the MODEL statement, the response variable, Days, is crossed with the censoring variable, Status, with the value that indicates censoring enclosed in parentheses (0). The parameter for the intercept is the expected cell mean for ses =3 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. You can fit many kinds of logistic models in many procedures including LOGISTIC, GENMOD, GLIMMIX, PROBIT, CATMOD, and others. 1 Recommendation. Using dummy coding, the right-hand side of the logistic model looks like it does when modeling a normally distributed response as in Example 1: where i=1,2,...,5, j=1,2, k=1, 2,...,Nij . These results come from the LSMESTIMATE statement. 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 â a comparison of the level's effect versus the effect of the last (reference) level. This simpler model is nested in the above model. For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. For simple analyses, only the PROC LIFETEST and TIME statements are required. 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. Comparing Nested Models 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. 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. Other methods must be used to compare nonnested models and this is discussed in the section that follows. With mixed models fit in PROC MIXED, if the models are nested in the covariance parameters and have identical fixed effects, then a LR test can be constructed using results from REML estimation (the default) or from ML estimation. The following statements print the log odds for treatments A and C in the complicated diagnosis. 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. 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). Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. to the coefficient for ses = 2. proc phreg data=surv(where=(trt in (0,2)); model survtime*survcen(1)=trt_cd; run; (4) The partial SAS output with the estimates for β and hazard ratio is: Output 4. trt_cd=1 vs. trt_cd=0, partial print out from PROC PHREG Analysis of Maximum Likelihood Estimates Parameter Standard Hazard Use the Class Level Information table which shows the design variable settings. For treatment A in the complicated diagnosis, O = 1, A = 1, B = 0. The EXP option provides the odds ratio estimate by exponentiating the difference. Potential Issues The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. following, where ses1 is the dummy variable for ses =1 and ses2 is the dummy Dummy Coding A main effect parameter is interpreted as the deviation of the level's effect from the average effect of all the levels. Introduction The MODEL statement must appear after the CLASS statement if CLASS statement is used. The following ODDSRATIO statement provides the same estimate of the treatment A vs. treatment C odds ratio in the complicated diagnosis as above (along with odds ratio estimates for the other treatment pairs in that diagnosis). To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see this note. See. A full-rank version of indicator coding (called reference coding) that omits the indicator variable for the reference level (by default, the last level) is also available in PROC LOGISTIC, PROC GENMOD, PROC CATMOD, and some other procedures via the PARAM=REF option. To get the expected mean It is quite powerful, as it allows for truncation, time-varying covariates and provides us with a few model selection algorithms and model diagnostics. 3. Specify the DIST=BINOMIAL option to specify a logistic model. This is the default coding scheme for CLASS variables in most procedures including GLM, MIXED, GLIMMIX, and GENMOD. The numerator is the hazard of death for the subject who died The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. The test of the difference is more easily obtained using the LSMESTIMATE statement. 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