proc phreg estimate statement example
Notice that the difference in log odds for these two cells (1.02450 0.39087 = 0.63363) is the same as the log odds ratio estimate that is provided by the CONTRAST statement. In this model, this reference curve is for males at age 69.845947 Usually, we are interested in comparing survival functions between groups, so we will need to provide SAS with some additional instructions to get these graphs. A More Complex Contrast with Effects Coding The order of \(df\beta_j\) in the current model are: gender, age, gender*age, bmi, bmi*bmi, hr. class gender; We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. Note that within a set of coefficients for an effect you can leave off any trailing zeros. In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. 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. Parameters corresponding to missing level combinations are not included in the model. EXAMPLE 2: A Three-Factor Model with Interactions \[f(t) = h(t)exp(-H(t))\]. Suppose the model contains two interactions: an interaction A*B of CLASS variables A and B, and another interaction A*X of A with a continuous variable X. For a more detailed definition of nested and nonnested models, see the Clarke (2001) reference cited in the sample program. Comparing Nonnested Models For example, the time interval represented by the first row is from 0 days to just before 1 day. Estimates are formed as linear estimable functions of the form . specifies which differences to consider for the level comparisons of a CLASS variable. 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 likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. Therneau, TM, Grambsch PM, Fleming TR (1990). Any estimable linear combination of model parameters can be tested using the procedure's CONTRAST statement. For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. The degrees of freedom are the number of linearly independent constraints implied by the CONTRAST statementthat is, the rank of . 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. Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. 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. The LSMEANS statement computes the cell means for the 10 A*B cells in this example. 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. have three parameters, the intercept and two parameters for ses =1 and ses 2. Applied Survival Analysis, Second Edition provides a comprehensive and up-to-date introduction to regression modeling for time-to-event Any serious endeavor into data analysis should begin with data exploration, in which the researcher becomes familiar with the distributions and typical values of each variable individually, as well as relationships between pairs or sets of variables. 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 primary focus of survival analysis is typically to model the hazard rate, which has the following relationship with the \(f(t)\) and \(S(t)\): The hazard function, then, describes the relative likelihood of the event occurring at time \(t\) (\(f(t)\)), conditional on the subjects survival up to that time \(t\) (\(S(t)\)). We can similarly calculate the joint probability of observing each of the \(n\) subjects failure times, or the likelihood of the failure times, as a function of the regression parameters, \(\beta\), given the subjects covariates values \(x_j\): \[L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{iin R_j}exp(x_i\beta)}\Bigg\rbrace\]. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. First, write the model, being sure to verify its parameters and their order from the procedure's displayed results: Now write each part of the contrast in terms of the effects-coded model (3e). 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. If ABS is greater than , then is declared nonestimable. Finally, you can use the SLICE statement. and then i would like to see the trends on age group. Other CONTRAST statements involving classification variables with PARAM=EFFECT are constructed similarly. Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. 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 run; In SAS, we can graph an estimate of the cdf using proc univariate. Computing the Cell Means Using the ESTIMATE Statement, Estimating and Testing a Difference of Means, Comparing One Interaction Mean to the Average of All Interaction Means, Example 1: A Two-Factor Model with Interaction, coefficient vectors that are used in calculating the LS-means, Example 2: A Three-Factor Model with Interactions, Example 3: A Two-Factor Logistic Model with Interaction Using Dummy and Effects Coding, Some procedures allow multiple types of coding. These are indeed censored observations, further indicated by the * appearing in the unlabeled second column. 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 DIFF and SLICEBY(A='1') options in the SLICE statement estimate the differences in LS-means at A=1. displays the vector of linear coefficients such that is the log-hazard ratio, with being the vector of regression coefficients. Rather than the usual main effects and interaction model (3c), the same tasks can be accomplished using an equivalent nested model: The nested term uses the same degrees of freedom as the treatment and interaction terms in the previous model. model (start, stop)*status(0) = in_hosp ; 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). 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. Suppose you want to test whether the effect of treatment A in the complicated diagnosis is different from the average effect of the treatments in the complicated diagnosis. We 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. Positive values of \(df\beta_j\) indicate that the exclusion of the observation causes the coefficient to decrease, which implies that inclusion of the observation causes the coefficient to increase. (1994). Printing this document: Because some of the tables in this document are wide, 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. By default, PLMAXITER=25. This test can be done using a CONTRAST statement to jointly test the interaction parameters. 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. Deploy software automatically at the click of a button on the Microsoft Azure Marketplace. Table 64.4 summarizes important options in the ESTIMATE statement. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. run; proc phreg data = whas500; 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. In this case, the 12 estimate is the sixth estimate in the A*B effect requiring a change in the coefficient vector that you specify in the ESTIMATE statement. 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. Shared Concepts and Topics. 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. There are two crucial parts to this: Write down the hypothesis to be tested or quantity to be estimated in terms of the model's parameters and simplify. The next five elements are the parameter estimates for the levels of A, 1 through 5. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. 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. specifies the level of significance for the % confidence interval for each contrast when the ESTIMATE option is specified. First, each of the effects, including both interactions, are significant. Specify the DIST=BINOMIAL option to specify a logistic model. In this seminar we will be analyzing the data of 500 subjects of the Worcester Heart Attack Study (referred to henceforth as WHAS500, distributed with Hosmer & Lemeshow(2008)). Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. This section contains 14 examples of PROC PHREG applications. Subjects that are censored after a given time point contribute to the survival function until they drop out of the study, but are not counted as a failure. For software releases that are not yet generally available, the Fixed EXAMPLE 5: A Quadratic Logistic Model If this option is not specified, PROC PHREG finds all the variables that interact with the variable of interest. The coefficients for the mean estimates of AB11 and AB12 are again determined by writing them in terms of the model. If we were to plot the estimate of \(S(t)\), we would see that it is a reflection of F(t) (about y=0 and shifted up by 1). 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. Proportional hazards tests and diagnostics based on weighted residuals. We can estimate the hazard function is SAS as well using proc lifetest: As we have seen before, the hazard appears to be greatest at the beginning of follow-up time and then rapidly declines and finally levels off. You can use the ESTIMATE, LSMEANS, SLICE, and TEST statements to estimate parameters and perform hypothesis tests. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; You can specify a contrast of the LS-means themselves, rather than the model parameters, by using the LSMESTIMATE statement. Biometrika. . The XBETA= option in the OUTPUT statement requests the linear predictor, x, for each observation. 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. The contrast table that shows the log odds ratio and odds ratio estimates is exactly as before. You can use the DIFF option in the LSMEANS statement. The default is the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. Notice that id, the individual subject identifier, has been added to the class statement and is also on the repeated statement (with an unstructured correlation matrix), telling proc genmod to calculate the robust errors. The BMI*BMI term describes the change in this effect for each unit increase in bmi. proc glm data= hsb2; class ses; model write = ses /solution; run; quit; Here are the steps we will take to evaluate the proportional hazards assumption for age through scaled Schoenfeld residuals: Although possibly slightly positively trending, the smooths appear mostly flat at 0, suggesting that the coefficient for age does not change over time and that proportional hazards holds for this covariate. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. So the log odds is: The following PROC LOGISTIC statements fit the effects-coded model and estimate the contrast: The same log odds ratio and odds ratio estimates are obtained as from the dummy-coded model. The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time \(t\). This option is ignored when the full-rank parameterization is used. Notice the additional option, We then specify the name of this dataset in the, We request separate lines for each age using, We request that SAS create separate survival curves by the, We also add the newly created time-varying covariate to the, Run a null Cox regression model by leaving the right side of equation empty on the, Save the martingale residuals to an output dataset using the, The fraction of the data contained in each neighborhood is determined by the, A desirable feature of loess smooth is that the residuals from the regression do not have any structure. 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. The second model is a reduced model that contains only the main effects. The Schoenfeld residual for observation \(j\) and covariate \(p\) is defined as the difference between covariate \(p\) for observation \(j\) and the weighted average of the covariate values for all subjects still at risk when observation \(j\) experiences the event. 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. Thus, both genders accumulate the risk for death with age, but females accumulate risk more slowly. SAS computes differences in the Nelson-Aalen estimate of \(H(t)\). However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. 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 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. It is similar to the CONTRAST statement in PROC GLM and PROC CATMOD, depending on the coding schemes used with any categorical variables involved. Not only are we interested in how influential observations affect coefficients, we are interested in how they affect the model as a whole. The ILINK option in the LSMEANS statement provides estimates of the probabilities of cure for each combination of treatment and diagnosis. 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; Optionally, the CONTRAST statement enables you to estimate each row, , of and test the hypothesis . 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. 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. run; proc phreg data = whas500; Include covariate interactions with time as predictors in the Cox model. In the case of categorical covariates, graphs of the Kaplan-Meier estimates of the survival function provide quick and easy checks of proportional hazards. EXAMPLE 4: Comparing Models 81. The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. Here we see the estimated pdf of survival times in the whas500 set, from which all censored observations were removed to aid presentation and explanation. See, In most cases, models fit in PROC GLIMMIX using the RANDOM statement do not use a true log likelihood. 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. Estimating and Testing Odds Ratios with Effects Coding. Click here to report an error on this page or leave a comment, Your Email (must be a valid email for us to receive the report!). We will use a data set called hsb2.sas7bdat to demonstrate. However, we have decided that there covariate scores are reasonable so we retain them in the model. All The EXPB option adds a column in the parameter estimates table that contains exponentiated values of the corresponding parameter estimates. See the documentation for more details.). Logistic models are in the class of generalized linear models. Institute for Digital Research and Education. In the graph above we see the correspondence between pdfs and histograms. Only as many residuals are output as names are supplied on the, We should check for non-linear relationships with time, so we include a, As before with checking functional forms, we list all the variables for which we would like to assess the proportional hazards assumption after the. Suppose that you suspect that the survival function is not the same among some of the groups in your study (some groups tend to fail more quickly than others). The tests are equivalent. 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. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). Applied Survival Analysis. Because of the positive skew often seen with followup-times, medians are often a better indicator of an average survival time. The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. Expressing the above relationship as \(\frac{d}{dt}H(t) = h(t)\), we see that the hazard function describes the rate at which hazards are accumulated over time. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. This can be easily accomplished in. 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. Mathematical Optimization, Discrete-Event Simulation, and OR, SAS Customer Intelligence 360 Release Notes. run; proc phreg data = whas500; assess var=(age bmi bmi*bmi hr) / resample; The regression equation is the To correctly specify your contrast, it is crucial to know the ordering of parameters within each effect and the variable levels associated with any parameter. Thus, by 200 days, a patient has accumulated quite a bit of risk, which accumulates more slowly after this point. The next section illustrates using the CONTRAST statement to compare nested models. The dfbeta measure, \(df\beta\), quantifies how much an observation influences the regression coefficients in the model. Acquiring more than one curve, whether survival or hazard, after Cox regression in SAS requires use of the baseline statement in conjunction with the creation of a small dataset of covariate values at which to estimate our curves of interest. PROC PHREG provides the possibility to compute the Breslow estimator of the baseline cumulative hazard function based on the estimates from a conventional Cox model. Graphs are particularly useful for interpreting interactions. It appears the probability of surviving beyond 1000 days is a little less than 0.2, which is confirmed by the cdf above, where we see that the probability of surviving 1000 days or fewer is a little more than 0.8. Additionally, another variable counts the number of events occurring in each interval (either 0 or 1 in Cox regression, same as the censoring variable). The survival function is undefined past this final interval at 2358 days. data example8_1; set sec1_5; group1 = group - 1; run; proc phreg data = example8_1; model time*death (0)=group1; run; A Nested Model The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. 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. In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most easily obtained using the ODDSRATIO statement. hazardratio 'Effect of 1-unit change in age by gender' age / at(gender=ALL); Another common mistake that may result in inverse hazard ratios is to omit the CLASS statement in the PHREG procedure altogether. There is no limit to the number of CONTRAST statements that you can specify, but they must appear after the MODEL statement. All of these variables vary quite a bit in these data. 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 ODDSRATIO statement used above with dummy coding provides the same results with effects coding. Here, we would like to introdue two types of interaction: We would probably prefer this model to the simpler model with just gender and age as explanatory factors for a couple of reasons. which has three levels. Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement. With any procedure, models that are not nested cannot be compared using the LR test. 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