proc phreg estimate statement exampleproc phreg estimate statement example

On the right panel, Residuals at Specified Smooths for martingale, are the smoothed residual plots, all of which appear to have no structure. The following statements show all five ways of computing and testing this contrast. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. 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. 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. After exponentiating, the denominator is not just a simple odds, but rather a geometric mean of the treatment odds.

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. The statements below generate observations from such a model: The following statements fit the main effects and interaction model. If the BAYES statement is specified, the ADJUST=, STEPDOWN, TESTVALUE, LOWER, UPPER, and JOINT options are ignored. 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. run; proc phreg data = whas500; In other words, we would expect to find a lot of failure times in a given time interval if 1) the hazard rate is high and 2) there are still a lot of subjects at-risk. assess var=(age bmi bmi*bmi hr) / resample; However, coefficients for the B effect remain in addition to coefficients for the A*B interaction effect. 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. However, the CONTRAST statement can be used in PROC GENMOD as shown above to produce a score test of the hypothesis. C?1D!^$w"I&#I" NF[cPdn .c@hHa"3IX"P+ !Hp? Institute for Digital Research and Education. It is available only for the Bayesian analysis. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. model lenfol*fstat(0) = gender age;; run; proc phreg data = whas500; The log odds for treatment A in the complicated diagnosis are: The log odds for treatment C in the complicated diagnosis are: Subtracting these gives the difference in log odds, or equivalently, the log odds ratio: The following statements use PROC LOGISTIC to fit model 3c and estimate the contrast. The result is Row1 in the table of LS-means coefficients. Comparing Nonnested Models Here we demonstrate how to assess the proportional hazards assumption for all of our covariates (graph for gender not shown): As we did with functional form checking, we inspect each graph for observed score processes, the solid blue lines, that appear quite different from the 20 simulated score processes, the dotted lines. (Js")*sv1t1} #Hqk*"lf,Rv$"TAlM@e (braP)NP r*$O2H3;0dFik-T'G2\QSDRT2H)!I+M) Run Cox models on intervals of follow up time rather than on its entirety. The hazard function is also generally higher for the two lowest BMI categories. Therneau, TM, Grambsch, PM. 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. This is an extension of the nested effects that you can specify in other procedures such as GLM and LOGISTIC. This subject could be represented by 2 rows like so: This structuring allows the modeling of time-varying covariates, or explanatory variables whose values change across follow-up time. Webproc phreg estimate statement example proc phreg estimate statement example. proc sgplot data = dfbeta; In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. run; proc lifetest data=whas500 atrisk outs=outwhas500; Here we use proc lifetest to graph \(S(t)\). Estimates are formed as linear estimable functions of the form . class gender; The WHAS500 data are stuctured this way.

It is quite powerful, as it allows for truncation, WebIn SAS, we can graph an estimate of the cdf using proc univariate. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. 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. The following statements do the model comparison using PROC LOGISTIC and the Wald test produces a very similar result. 77(1). Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. For such studies, a semi-parametric model, in which we estimate regression parameters as covariate effects but ignore (leave unspecified) the dependence on time, is appropriate. The PHREG Procedure Example 91.12 demonstrated that the log transform is a much improved functional form for Bilirubin in a Cox regression model. The last 10 elements are the parameter estimates for the 10 levels of the A*B interaction, 11 through 52. Estimating and Testing Odds Ratios with Dummy Coding Copyright SAS Institute Inc. All rights reserved. A main effect parameter is interpreted as the difference in the level's effect compared to the reference level. 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). Release is the software release in which the problem is planned to be However, we can still get an idea of the hazard rate using a graph of the kernel-smoothed estimate. 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. During the interval [382,385) 1 out of 355 subjects at-risk died, yielding a conditional probability of survival (the probability of survival in the given interval, given that the subject has survived up to the begininng of the interval) in this interval of \(\frac{355-1}{355}=0.9972\). A Nested Model 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. proc phreg estimate statement example. Models with smaller values of these criteria are considered better models. WebPROC PHREG Statement. 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. 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). We also calculate the hazard ratio between females and males, or \(\frac{HR(gender=1)}{HR(gender=0)}\) at ages 0, 20, 40, 60, and 80.

Specifically, you need to construct the linear combination of model parameters that corresponds to the hypothesis. Examples of Writing CONTRAST and ESTIMATE Statements Introduction EXAMPLE 1: A Two-Factor Model with Interaction Computing the Cell Means Using the Again, trailing zero coefficients can be omitted. Standard nonparametric techniques do not typically estimate the hazard function directly. However, each of the other 3 at the higher smoothing parameter values have very similar shapes, which appears to be a linear effect of bmi that flattens as bmi increases. The Analysis of Maximum Likelihood Estimates table confirms the ordering of design variables in model 3d. For example, if \(\beta_x\) is 0.5, each unit increase in \(x\) will cause a ~65% increase in the hazard rate, whether X is increasing from 0 to 1 or from 99 to 100, as \(HR = exp(0.5(1)) = 1.6487\). Chapter 19, The PLOTS= option is not available for the maximum likelihood anaysis. 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, 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. EXAMPLE 3: A Two-Factor Logistic Model with Interaction Using Dummy and Effects Coding But the nested term makes it more obvious that you are contrasting levels of treatment within each level of diagnosis. 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.

Finally, we see that the hazard ratio describing a 5-unit increase in bmi, \(\frac{HR(bmi+5)}{HR(bmi)}\), increases with bmi. This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. However, no statistical tests comparing criterion values is possible. 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. proc univariate data = whas500 (where= (fstat=1)); var lenfol; cdfplot lenfol; run; In the graph above we can see WebPROC PHREG syntax is similar to that of the other regression procedures in the SAS System. This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. Although the coding scheme is different, you still follow the same steps to determine the contrast coefficients. Notice that Row2 is the coefficient vector for computing the mean of the AB12 cell. Below we plot survivor curves across several ages for each gender through the follwing steps: As we surmised earlier, the effect of age appears to be more severe in males than in females, reflected by the greater separation between curves in the top graaph. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). model lenfol*fstat(0) = gender|age bmi|bmi hr hrtime; 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). Hosmer, DW, Lemeshow, S, May S. (2008). 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. 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 \(t\). The ESTIMATE statement provides a mechanism for obtaining custom Technical Support can assist you with syntax and other questions that relate to CONTRAST and ESTIMATE statements. These statement essentially look like data step statements, and function in the same way. For simple uses, only the PROC PHREG and MODEL statements are required. 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. One interpretation of the cumulative hazard function is thus the expected number of failures over time interval \([0,t]\). This matches closely with the Kaplan Meier product-limit estimate of survival beyond 3 days of 0.9620. This suggests that perhaps the functional form of bmi should be modified. The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. Some procedures allow multiple types of coding. Proportional hazards tests and diagnostics based on weighted residuals. 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. 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 test can be done using a CONTRAST statement to jointly test the interaction parameters. Unless the seed option is specified, these sets will be different each time proc phreg is run. It is important to note that the survival probabilities listed in the Survival column are unconditional, and are to be interpreted as the probability of surviving from the beginning of follow up time up to the number days in the LENFOL column. The ILINK option in the LSMEANS statement provides estimates of the probabilities of cure for each combination of treatment and diagnosis. Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. For a more detailed definition of nested and nonnested models, see the Clarke (2001) reference cited in the sample program. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. However, if the nested models do not have identical fixed effects, then results from ML estimation must be used to construct a LR test. This can be particularly difficult with dummy (PARAM=GLM) coding. A More Complex Contrast with Effects Coding Such linear combinations can be estimated and tested using the CONTRAST and/or ESTIMATE statements available in many modeling procedures. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. The change in coding scheme does not affect how you specify the ODDSRATIO statement. The difficulty is constructing combinations that are estimable and that jointly test the set of interactions. With any procedure, models that are not nested cannot be compared using the LR test. To accomplish this smoothing, the hazard function estimate at any time interval is a weighted average of differences within a window of time that includes many differences, known as the bandwidth. 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. 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. 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. In the simpler case of a main-effects-only model, writing CONTRAST and ESTIMATE statements to make simple pairwise comparisons is more intuitive. proc phreg estimate statement example 07 Apr. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. Below we demonstrate use of the assess statement to the functional form of the covariates. Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. 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. 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. Follow up time for all participants begins at the time of hospital admission after heart attack and ends with death or loss to follow up (censoring). This indicates that omitting bmi from the model causes those with low bmi values to modeled with too low a hazard rate (as the number of observed events is in excess of the expected number of events). This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. Many transformations of the survivor function are available for alternate ways of calculating confidence intervals through the conftype option, though most transformations should yield very similar confidence intervals. proc sgplot data = dfbeta; 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. These statistics are provided in most procedures using maximum likelihood estimation. In the case of categorical covariates, graphs of the Kaplan-Meier estimates of the survival function provide quick and easy checks of proportional hazards. Widening the bandwidth smooths the function by averaging more differences together. The cumulative distribution function (cdf), \(F(t)\), describes the probability of observing \(Time\) less than or equal to some time \(t\), or \(Pr(Time t)\). An example of using the LSMEANS and LSMESTIMATE statements to estimate odds ratios in a repeated measures (GEE) model in PROC GENMOD is available. data example8_1; set Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. Estimating and Testing Odds Ratios with Effects Coding 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: \[HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))\]. In the code below, we model the effects of hospitalization on the hazard rate. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. This is reinforced by the three significant tests of equality. Significant departures from random error would suggest model misspecification. For example, we found that the gender effect seems to disappear after accounting for age, but we may suspect that the effect of age is different for each gender. Reference parameterization (using the PARAM=REF option) is also a full-rank parameterization. format gender gender. Webproc phreg estimate statement example. This is exactly the contrast that was constructed earlier. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that proportional hazards holds for all of our covariates. The model is the same as model (1) above with just a change in the subscript ranges. 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. The first element is the estimate of the intercept, . Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. If the BAYES statement is specified, the ADJUST=, STEPDOWN, TESTVALUE, LOWER, UPPER, and JOINT options are ignored. if lenfol > los then in_hosp = 0; hazardratio 'Effect of gender across ages' gender / at(age=(0 20 40 60 80)); Indeed, exclusion of these two outliers causes an almost doubling of \(\hat{\beta}_{bmi}\), from -0.23323 to -0.39619. 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. Biometrika. 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. model lenfol*fstat(0) = gender|age bmi|bmi hr; So the log odds are: For treatment C in the complicated diagnosis, O = 1, A = 1, B = 1. Click here to download the dataset used in this seminar. run; proc phreg data = whas500; This simpler model is nested in the above model. Using model (1) above, the AB12 cell mean, 12, is: Because averages of the errors (ijk) are assumed to be zero: Similarly, the AB11 cell mean is written this way: So, to get an estimate of the AB12 mean, you need to add together the estimates of , 1, 2, and 12. Grambsch, PM, Therneau, TM, Fleming TR. Notice there is one row per subject, with one variable coding the time to event, lenfol: A second way to structure the data that only proc phreg accepts is the counting process style of input that allows multiple rows of data per subject. run; proc phreg data = whas500; It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. The following statements print the log odds for treatments A and C in the complicated diagnosis. 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. The above relationship between the cdf and pdf also implies: In SAS, we can graph an estimate of the cdf using proc univariate. SAS computes differences in the Nelson-Aalen estimate of \(H(t)\). These are indeed censored observations, further indicated by the * appearing in the unlabeled second column. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. class gender; Modeling Survival Data: Extending the Cox Model. 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.

The unconditional probability of surviving beyond 2 days (from the onset of risk) then is \(\hat S(2) = \frac{500 8}{500}\times\frac{492-8}{492} = 0.984\times0.98374=.9680\). Density functions are essentially histograms comprised of bins of vanishingly small widths. EXAMPLE 5: A Quadratic Logistic Model Table 66.4 summarizes important options in the ESTIMATE statement. PROC PHREG handles missing level combinations of As we see above, one of the great advantages of the Cox model is that estimating predictor effects does not depend on making assumptions about the form of the baseline hazard function, \(h_0(t)\), which can be left unspecified. The GENMOD and GLIMMIX procedures provide separate CONTRAST and ESTIMATE statements. Perhaps you also suspect that the hazard rate changes with age as well. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that Shared Concepts and Topics. o1LSRD"Qh&3[F&g w/!|#+QnHA8Oy9 , 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. Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. 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. Webproc phreg estimate statement example; proc phreg estimate statement example. The tests are equivalent. 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. Survival analysis models factors that influence the time to an event. 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. %PDF-1.2 % Thus, we again feel justified in our choice of modeling a quadratic effect of bmi. 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.