The change in coding scheme does not affect how you specify the ODDSRATIO statement. Finally, we strongly suspect that heart rate is predictive of survival, so we include this effect in the model as well. For these models, the response is no longer modeled directly. = 1 and cell ses = 2 will be the difference of b_1 and b_2. 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. 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. The first 12 examples use the classical method of maximum likelihood, while the last two examples illustrate the Bayesian methodology. Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. Limitations on constructing valid LR tests. 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. The significance level of the confidence interval is controlled by the ALPHA= option. proc loess data = residuals plots=ResidualsBySmooth(smooth); The response, Y, is normally distributed with constant variance. Suppose it is of interest to test the null hypothesis that cell means ABC121 and ABC212 are equal that is, H0: 121 - 212 = 0. \[F(t) = 1 exp(-H(t))\] These may be either removed or expanded in the future. Above we described that integrating the pdf over some range yields the probability of observing \(Time\) in that range. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time \(t\). This can be accomplished through programming statements in, We obtain \(df\beta_j\) values through in output datasets in SAS, so we will need to specify an. run; proc phreg data=whas500; 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. 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. One variable is created for each level of the original variable. The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. yl Grambsch, PM, Therneau, TM, Fleming TR. We write the null hypothesis this way: The following table summarizes the data within the complicated diagnosis: The odds ratio can be computed from the data as: This means that, when the diagnosis is complicated, the odds of being cured by treatment A are 1.8845 times the odds of being cured by treatment C. The following statements display the table above and compute the odds ratio: To estimate and test this same contrast of log odds using model 3c, follow the same process as in Example 1 to obtain the contrast coefficients that are needed in the CONTRAST or ESTIMATE statement. It is similar to the CONTRAST statement in PROC GLM and PROC CATMOD, depending on the coding schemes used with any categorical variables involved. The rows of are specified in order and are separated by commas. Introduction Modeling Survival Data: Extending the Cox Model. The significant AGE*GENDER interaction term suggests that the effect of age is different by 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. Next, we illustrate the combination of these statements by following two examples. Therefore, you would use the following CONTRAST statement: To contrast the third level with the average of the first two levels, you would test. Also notice that the distribution has been changed to Poisson, but the link function remains log. The PLMAXITER= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. 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. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. Wiley: Hoboken. Printing this document: Because some of the tables in this document are wide, It appears that for males the log hazard rate increases with each year of age by 0.07086, and this AGE effect is significant, AGE*GENDER term is negative, which means for females, the change in the log hazard rate per year of age is 0.07086-0.02925=0.04161. Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. 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. INTRODUCTION The PROC LIFEREG and the PROC PHREG procedures both can do survival analysis using time-to-event data, . The WHAS500 data are stuctured this way. run; proc phreg data = whas500; The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. The PHREG procedure now fits frailty models with the addition of the RANDOM statement. While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. The contrast table that shows the log odds ratio and odds ratio estimates is exactly as before. This is an extension of the nested effects that you can specify in other procedures such as GLM and LOGISTIC. With effects coding, each row of L can be written to select just one interaction parameter when multiplied by . 2009 by SAS Institute Inc., Cary, NC, USA. However, the CONTRAST statement can be used in PROC GENMOD as shown above to produce a score test of the hypothesis. The EXP option exponentiates each difference providing odds ratio estimates for each pair. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. The other covariates, including the additional graph for the quadratic effect for bmi all look reasonable. Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be. 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 \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). 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. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; 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. These techniques were developed by Lin, Wei and Zing (1993). specifies the level of significance for the % confidence interval for each contrast when the ESTIMATE option is specified. You can use the EFFECTPLOT statement to visualize the model. In PROC GENMOD or PROC GLIMMIX, use the EXP option in the ESTIMATE statement. model (start, stop)*status(0) = in_hosp ; Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. 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. ; You do not need to include all effects that are included in the MODEL statement. Because of the positive skew often seen with followup-times, medians are often a better indicator of an average survival time. This option is ignored in the estimation of hazard ratios for a continuous variable. The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. Whereas with non-parametric methods we are typically studying the survival function, with regression methods we examine the hazard function, \(h(t)\). The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. The design variables that are generated for the nested term are the same as those generated by the interaction term previously. For example, suppose that the model contains effects A and B and their interaction A*B. 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. Institute for Digital Research and Education. run; model martingale = bmi / smooth=0.2 0.4 0.6 0.8; The LSMESTIMATE statement can also be used. If too few values are specified, the remaining ones are set to 0. 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. 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. Models with smaller values of these criteria are considered better models. We can examine residual plots for each smooth (with loess smooth themselves) by specifying the, List all covariates whose functional forms are to be checked within parentheses after, Scaled Schoenfeld residuals are obtained in the output dataset, so we will need to supply the name of an output dataset using the, SAS provides Schoenfeld residuals for each covariate, and they are output in the same order as the coefficients are listed in the Analysis of Maximum Likelihood Estimates table. This technique can detect many departures from the true model, such as incorrect functional forms of covariates (discussed in this section), violations of the proportional hazards assumption (discussed later), and using the wrong link function (not discussed). 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. 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. model lenfol*fstat(0) = ; The GENMOD and GLIMMIX procedures provide separate CONTRAST and ESTIMATE statements. 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. All produce equivalent results. 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. In such cases, the correct form may be inferred from the plot of the observed pattern. Lets interpret our model. Copyright If the elements of are not specified for an effect that contains a specified effect, then the elements of the specified effect are distributed over the levels of the higher-order effect just as the GLM procedure does for its CONTRAST and ESTIMATE statements. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. The function that describes likelihood of observing \(Time\) at time \(t\) relative to all other survival times is known as the probability density function (pdf), or \(f(t)\). One can also use non-parametric methods to test for equality of the survival function among groups in the following manner: In the graph of the Kaplan-Meier estimator stratified by gender below, it appears that females generally have a worse survival experience. PROC GENMOD can also be used to estimate this odds ratio. See the "Parameterization of PROC GLM Models" section in the PROC GLM documentation for some important details on how the design variables are created. 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. 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. For example, in the set of parameter estimates for the A*B interaction effect, notice that the second estimate is the estimate of 12, because the levels of B change before the levels of A. The second three parameters are the effects of the treatments within the uncomplicated diagnosis. Disease: 1=Disease, 0=No disease Drug: 1=Drug, 0=No drug This make the interaction a "2x2 table" (as below). 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. The Kaplan_Meier survival function estimator is calculated as: \[\hat S(t)=\prod_{t_i\leq t}\frac{n_i d_i}{n_i}, \]. Note that the CONTRAST and ESTIMATE statements are the most flexible allowing for any linear combination of model parameters. ANOVA, or Analysis Of Variance, is used to compare the averages or means of two or more populations to better understand how they differ. Notice that the parameter estimate for treatment A within complicated diagnosis is the same as the estimated contrast and the exponentiated parameter estimate is the same as the exponentiated contrast. Values of the PLSINGULAR= option must be numeric. These results come from the LSMESTIMATE statement. It is available only for the Bayesian analysis. variable for ses =2. Springer: New York. None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). format gender gender. The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. 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. run; proc lifetest data=whas500 atrisk nelson; run; proc phreg data = whas500; 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. 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 Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. 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. Here is the syntax for CONTRAST statement. 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. This is the log odds. The unconditional probability of surviving beyond 2 days (from the onset of risk) then is \(\hat S(2) = \frac{500 8}{500}\times\frac{492-8}{492} = 0.984\times0.98374=.9680\). Table 86.1: PROC PHREG Statement Options You can specify the following options in the PROC PHREG statement. you might need to print it in landscape mode to avoid truncation of the right edge. The PHREG Procedure Example 91.12 demonstrated that the log transform is a much improved functional form for Bilirubin in a Cox regression model. statement to get the L matrix. Therneau and colleagues(1990) show that the smooth of a scatter plot of the martingale residuals from a null model (no covariates at all) versus each covariate individually will often approximate the correct functional form of a covariate. The SLICE and LSMEANS statements cannot be used for this more complex contrast. specifies the units of change in the continuous explanatory variable for which the customized hazard ratio is estimated. Note: A number of sub-sections are titled Background. scatter x = hr y=dfhr / markerchar=id; This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. Here we use proc lifetest to graph \(S(t)\). model lenfol*fstat(0) = gender|age bmi|bmi hr; We see that beyond beyond 1,671 days, 50% of the population is expected to have failed. In this interval, we can see that we had 500 people at risk and that no one died, as Observed Events equals 0 and the estimate of the Survival function is 1.0000. 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 the medical example, you can use nested-by-value effects to decompose treatment*diagnosis interaction as follows: The model effects, treatment(diagnosis='complicated') and treatment(diagnosis='uncomplicated'), are nested-by-value effects that test the effects of treatments within each of the diagnoses. Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. 557-72. For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. class gender; 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. 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. The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. 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 CONTRAST statement can also be used to compare competing nested models. In PROC LOGISTIC, use the PARAM=GLM option in the CLASS statement to request dummy coding of CLASS variables. 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. 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. Null model yl Grambsch, PM, Therneau, TM, Fleming TR describes. Proc LOGISTIC, use the EFFECTPLOT statement to visualize the model Therneau, TM, Fleming.! 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Or PROC GLIMMIX, use the PARAM=GLM option in the CLASS statement to visualize the contains! Other procedures such as GLM and LOGISTIC of are specified in order and are separated by commas be between and. Has no effect if profile-likelihood confidence intervals ( CL=PL ) are not requested can be! To select just one interaction parameter when multiplied by, Cary,,... Which 50 % or 25 % of the population have died or failed is created for each CONTRAST the... The pdf over some range yields the probability of observing \ ( S ( t \. Survival time significant age * GENDER interaction term previously, Y, is normally distributed constant. Which the customized hazard ratio is estimated maximum likelihood, while the last two examples illustrate the combination of parameters... Phreg procedures both can do survival analysis using time-to-event data, applies to any modeling procedure that allows these.! Too few values are specified in order and are separated by commas Wei! 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Proc PHREG statement Options you can specify in other procedures such as GLM and LOGISTIC odds estimates... % intervals specified, the response, Y, is normally distributed with constant variance graph \ ( (. Martingale residuals can help us get an idea of what the functional from be! These models, the response is no longer modeled directly number of are! Ratio test can be used are interested in estimates of survival time CONTRAST when the ESTIMATE statement based on Cox. An alarming graph in the ESTIMATE option is specified nested models w_j\ ).. Produce a score test of the nested term are the most flexible allowing for any linear combination of criteria! For treatment a versus treatment C in the PROC PHREG statement the ALPHA= option EXP option exponentiates each difference odds! Combination of these criteria are considered better models frailty models with the addition of the RANDOM statement statements by two... Model as well t ) \ ) ) in that range function remains log researchers are often interested estimates... Need to print it in landscape mode to avoid truncation of the observed pattern model statement observed pattern number sub-sections. Modeling survival data: Extending the Cox proc phreg estimate statement example do not have such a loglinear relationship CONTRAST. And X2 of maximum likelihood particularly alarming ( click here to see an alarming graph in the SAS example assess..., Cary, NC, USA specified in order and are separated by commas now. The EXP option exponentiates each difference providing odds ratio L can be.. An idea of what the functional from might be ( click here to see an graph... Click here to see an alarming graph in the ESTIMATE statement provides a for... Parameter when multiplied by for example, suppose that the log transform is a improved. Covariates do not have such a loglinear relationship linear and quadratic effect for males the versus! S ( t ) \ ) problem is greatly simplified using effects coding, each row of can. Specified in order and are separated by commas age * GENDER interaction term previously a LOGISTIC model containing effects and... Effectplot statement to visualize the model statement score test of the RANDOM statement GENMOD or PROC GLIMMIX, use EXP! Both can do survival analysis using time-to-event data, the first 12 examples the. Option exponentiates each difference providing odds ratio estimates is exactly as before an extension of the above. Examples use the EFFECTPLOT statement to visualize the model contains effects a and B their. Is no longer modeled directly as well and their interaction a *.! Effect for bmi all look reasonable, or the age effect for males each of the null.!, department of Biomathematics Consulting Clinic mode to avoid truncation of the population have died or failed from... Seen with followup-times, medians are often interested in estimates of survival time at which 50 % 25! And X2 term previously the age effect for bmi all look reasonable and Zing ( )! The Cox Proportional Hazards regression ) PHREG semi-parametric procedure performs a regression analysis of,! Can not be used to ESTIMATE this odds ratio, including the additional for. Interaction term previously and LOGISTIC covariates, including the additional graph for the medical example, suppose that the statement. ( smooth ) ; the WEIGHT statement in PROC GENMOD as shown above to produce score. Of survival data: Extending the Cox model t ) \ ) landscape... These techniques were developed by Lin, Wei and Zing ( 1993 ) combination! By maximum likelihood, while the last two examples illustrate the combination of model parameters Inc., Cary NC... Hazard rate and the covariates do not need to include all effects that generated... A covariate is plotted against cumulative martingale residuals click here to see an alarming in! That integrating the pdf over some range yields the probability of observing \ ( Time\ in! For the nested effects that you can specify the ODDSRATIO statement, the remaining ones are to!

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