Also notice that the distribution has been changed to Poisson, but the link function remains log. 2009 by SAS Institute Inc., Cary, NC, USA. Hosmer, DW, Lemeshow, S, May S. (2008). If the BAYES statement is specified, the ADJUST=, STEPDOWN, TESTVALUE, LOWER, UPPER, and JOINT options are ignored. SAS provides built-in methods for evaluating the functional form of covariates through its assess statement. following, where ses1 is the dummy variable for ses =1 and ses2 is the dummy Here are the steps we use to assess the influence of each observation on our regression coefficients: The dfbetas for age and hr look small compared to regression coefficients themselves (\(\hat{\beta}_{age}=0.07086\) and \(\hat{\beta}_{hr}=0.01277\)) for the most part, but id=89 has a rather large, negative dfbeta for hr. One can request that SAS estimate the survival function by exponentiating the negative of the Nelson-Aalen estimator, also known as the Breslow estimator, rather than by the Kaplan-Meier estimator through the method=breslow option on the proc lifetest statement. 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. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. You can use the DIFF option in the LSMEANS statement. model lenfol*fstat(0) = gender age;; This option is not applicable to a Bayesian analysis. SAS provides easy ways to examine the \(df\beta\) values for all observations across all coefficients in the model. The PLMAXITER= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The EXP option provides the odds ratio estimate by exponentiating the difference. The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. So what is the probability of observing subject \(i\) fail at time \(t_j\)? Zeros in this table are shown as blanks for clarity. 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. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. controls the convergence criterion for the profile-likelihood confidence limits. Parameters corresponding to missing level combinations are not included in the model. Thus, if the average is 0 across time, then that suggests the coefficient \(p\) does not vary over time and that the proportional hazards assumption holds for covariate \(p\). 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 \(f(t)\). output out=residuals resmart=martingale; 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. Finally, we calculate the hazard ratio describing a 5-unit increase in bmi, or \(\frac{HR(bmi+5)}{HR(bmi)}\), at clinically revelant BMI scores. Additionally, none of the supremum tests are significant, suggesting that our residuals are not larger than expected. The CONTRAST statement can also be used to compare competing nested models. proc univariate data = whas500(where=(fstat=1)); Earlier in the seminar we graphed the Kaplan-Meier survivor function estimates for males and females, and gender appears to adhere to the proportional hazards assumption. The CONTRAST statement tests the hypothesis L=0, where L is the hypothesis matrix and is the vector of model parameters. Copyright statement to get the L matrix. Computing the Cell Means Using the ESTIMATE Statement In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. EXAMPLE 4: Comparing Models When testing, write the null hypothesis in the form. Here we use proc lifetest to graph \(S(t)\). We then plot each\(df\beta_j\) against the associated coviarate using, Output the likelihood displacement scores to an output dataset, which we name on the, Name the variable to store the likelihood displacement score on the, Graph the likelihood displacement scores vs follow up time using. 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 default is the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. 80(30). We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. 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 \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). This is the default coding scheme for CLASS variables in most procedures including GLM, MIXED, GLIMMIX, and GENMOD. hazardratio 'Effect of 1-unit change in age by gender' age / at(gender=ALL); Can i add class statement to want to see hazard ratios on exposure proc phreg data=episode; /*class exposure*/ This example is to illustrate the algorithm used to compute the parameter estimate. Lin, DY, Wei, LJ, Ying, Z. Still, although their effects are strong, we believe the data for these outliers are not in error and the significance of all effects are unaffected if we exclude them, so we include them in the model. Therneau, TM, Grambsch, PM. hazardratio 'Effect of gender across ages' gender / at(age=(0 20 40 60 80)); Widening the bandwidth smooths the function by averaging more differences together. We will thus let \(r(x,\beta_x) = exp(x\beta_x)\), and the hazard function will be given by: This parameterization forms the Cox proportional hazards model. scatter x = bmi y=dfbmi / markerchar=id; Reference parameterization (using the PARAM=REF option) is also a full-rank parameterization. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. The hazard rate can also be interpreted as the rate at which failures occur at that point in time, or the rate at which risk is accumulated, an interpretation that coincides with the fact that the hazard rate is the derivative of the cumulative hazard function, \(H(t)\). Thus, because many observations in WHAS500 are right-censored, we also need to specify a censoring variable and the numeric code that identifies a censored observation, which is accomplished below with, However, we would like to add confidence bands and the number at risk to the graph, so we add, The Nelson-Aalen estimator is requested in SAS through the, When provided with a grouping variable in a, We request plots of the hazard function with a bandwidth of 200 days with, SAS conveniently allows the creation of strata from a continuous variable, such as bmi, on the fly with the, We also would like survival curves based on our model, so we add, First, a dataset of covariate values is created in a, This dataset name is then specified on the, This expanded dataset can be named and then viewed with the, Both survival and cumulative hazard curves are available using the, We specify the name of the output dataset, base, that contains our covariate values at each event time on the, We request survival plots that are overlaid with the, The interaction of 2 different variables, such as gender and age, is specified through the syntax, The interaction of a continuous variable, such as bmi, with itself is specified by, We calculate the hazard ratio describing a one-unit increase in age, or \(\frac{HR(age+1)}{HR(age)}\), for both genders. We can plot separate graphs for each combination of values of the covariates comprising the interactions. The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. The necessary contrast coefficients are stated in the null hypothesis above: (0 1 0 0 0 0) - (1/6 1/6 1/6 1/6 1/6 1/6) , which simplifies to the contrast shown in the LSMESTIMATE statement below. 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. We compare 2 models, one with just a linear effect of bmi and one with both a linear and quadratic effect of bmi (in addition to our other covariates). You must be familiar with the details of the model parameterization that PROC PHREG uses (for more information, see the PARAM= option in the section CLASS Statement). The PLSINGULAR= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. 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. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. For simple uses, only the PROC PHREG and MODEL statements are required. Instead, you model a function of the response distribution's mean. This note focuses on assessing the effects of categorical (CLASS) variables in models containing interactions. The second three parameters are the effects of the treatments within the uncomplicated diagnosis. However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. The exponential function is also equal to 1 when its argument is equal to 0. 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 \(r(x,\beta_x) = 1\). my dataset includes age, period, outcome, drug age : 1 2 3 (categorical variable) period : 1~365 days ( continuos variable) outcome( :0 1 ( 0 : without outcome, 1: with outcome) drug : 0 . Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. proc sgplot data = dfbeta; 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). The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. Introduction 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. scatter x = bmi y=dfbmibmi / markerchar=id; The PLCONV= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. The HAZARDRATIO statement enables you to request hazard ratios for any variable in the model at customized settings. run; proc phreg data = whas500; The regression equation is the At the beginning of a given time interval \(t_j\), say there are \(R_j\) subjects still at-risk, each with their own hazard rates: The probability of observing subject \(j\) fail out of all \(R_j\) remaing at-risk subjects, then, is the proportion of the sum total of hazard rates of all \(R_j\) subjects that is made up by subject \(j\)s hazard rate. The following statements create the data set and fit the saturated logistic model. 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 \(HR = exp(0.5(1)) = 1.6487\). For example, B*A becomes A*B if A precedes B in the CLASS statement. The LSMEANS statement computes the cell means for the 10 A*B cells in this example. With any procedure, models that are not nested cannot be compared using the LR test. In the simpler case of a main-effects-only model, writing CONTRAST and ESTIMATE statements to make simple pairwise comparisons is more intuitive. ALPHA=number specifies the level of significance for % confidence intervals. Click here to download the dataset used in this seminar. The DIFF option estimates and tests each pairwise difference of log odds. 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. Several covariates can be evaluated simultaneously. The correct coefficients are determined for the CONTRAST statement to estimate two odds ratios: one for an increase of one unit in X, and the second for a two unit increase. model lenfol*fstat(0) = gender|age bmi|bmi hr; As you'll see in the examples that follow, there are some important steps in properly writing a CONTRAST or ESTIMATE statement: Writing CONTRAST and ESTIMATE statements can become difficult when interaction or nested effects are part of the model. class gender; You write the contrast of log odds in terms of the nested model (3d): Notice that this simple contrast is exactly the same contrast that is estimated for a main effect parameter a comparison of the level's effect versus the effect of the last (reference) level. (1994). This seminar introduces procedures and outlines the coding needed in SAS to model survival data through both of these methods, as well as many techniques to evaluate and possibly improve the model. The degrees of freedom are the number of linearly independent constraints implied by the CONTRAST statementthat is, the rank of . run; lenfol: length of followup, terminated either by death or censoring. Censored observations are represented by vertical ticks on the graph. 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. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. 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. 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. You can specify a contrast of the LS-means themselves, rather than the model parameters, by using the LSMESTIMATE statement. Exponentiating this value (exp[.63363] = 1.8845) yields the exponentiated contrast value (the odds ratio estimate) from the CONTRAST statement. However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. In the second table, we see that the hazard ratio between genders, \(\frac{HR(gender=1)}{HR(gender=0)}\), decreases with age, significantly different from 1 at age = 0 and age = 20, but becoming non-signicant by 40. The contrast estimate is exponentiated to yield the odds ratio estimate. Firths Correction for Monotone Likelihood, Conditional Logistic Regression for m:n Matching, Model Using Time-Dependent Explanatory Variables, Time-Dependent Repeated Measurements of a Covariate, Survivor Function Estimates for Specific Covariate Values, Model Assessment Using Cumulative Sums of Martingale Residuals, Bayesian Analysis of Piecewise Exponential Model. We could thus evaluate model specification by comparing the observed distribution of cumulative sums of martingale residuals to the expected distribution of the residuals under the null hypothesis that the model is correctly specified. From these equations we can see that the cumulative hazard function \(H(t)\) and the survival function \(S(t)\) have a simple monotonic relationship, such that when the Survival function is at its maximum at the beginning of analysis time, the cumulative hazard function is at its minimum. yl Thus, it appears, that when bmi=0, as bmi increases, the hazard rate decreases, but that this negative slope flattens and becomes more positive as bmi increases. The response, Y, is normally distributed with constant variance. In PROC LOGISTIC, the ESTIMATE=BOTH option in the CONTRAST statement requests estimates of both the contrast (difference in log odds or log odds ratio) and the exponentiated contrast (odds ratio). 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. We will use a data set called hsb2.sas7bdat to demonstrate. run; proc phreg data = whas500; The interpretation of this estimate is that we expect 0.0385 failures (per person) by the end of 3 days. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). We will model a time-varying covariate later in the seminar. Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. The survival function is undefined past this final interval at 2358 days. The number of variables that are created is one fewer than the number of levels of the original variable, yielding one fewer parameters than levels, but equal to the number of degrees of freedom. Limitations on constructing valid LR tests. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. Lets interpret our model. Survival analysis often begins with examination of the overall survival experience through non-parametric methods, such as Kaplan-Meier (product-limit) and life-table estimators of the survival function. Not only are we interested in how influential observations affect coefficients, we are interested in how they affect the model as a whole. By default, value is the machine epsilon times 1E7, which is approximately 1E9. 557-72. 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. Here is the SAS code: Code: proc phreg data=Data; class Drug(ref='0') Disease(ref='0') /param=glm; model lenfol*fstat(0) = gender|age bmi hr; To get the expected mean The background necessary to explain the mathematical definition of a martingale residual is beyond the scope of this seminar, but interested readers may consult (Therneau, 1990). The difference between the mean of cell ses format gender gender. Proportional hazards may hold for shorter intervals of time within the entirety of follow up time. Thus, we define the cumulative distribution function as: As an example, we can use the cdf to determine the probability of observing a survival time of up to 100 days. requests that, for each Newton-Raphson iteration, PROC PHREG recompiles the risk sets corresponding to the event times for the (start,stop) style of response and recomputes the values of the time-dependent variables defined by the programming statements for each observation in the risk sets. 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