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). Grambsch, PM, Therneau, TM, Fleming TR. The design variables that are generated for the nested term are the same as those generated by the interaction term previously. 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. The SLICE and LSMEANS statements cannot be used for this more complex contrast. 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. 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. 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. For example, the time interval represented by the first row is from 0 days to just before 1 day. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
Limitations on constructing valid LR tests. model lenfol*fstat(0) = gender|age bmi|bmi hr;
Martingale-based residuals for survival 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. (1994). output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr;
The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. As in Example 1, you can also use the LSMEANS, LSMESTIMATE, and SLICE statements in PROC LOGISTIC, PROC GENMOD, and PROC GLIMMIX when dummy coding (PARAM=GLM) is used. The contrast estimate is exponentiated to yield the odds ratio estimate. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. The least squares fit for this linear model is to assign the sample 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. model lenfol*fstat(0) = ;
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. 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. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of var lenfol gender age bmi hr;
class gender;
Technical Support can assist you with syntax and other questions that relate to CONTRAST and ESTIMATE statements. It is available only for the Bayesian analysis. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. PROC GENMOD can also be used to estimate this odds ratio. 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 \(f(t)\). Nevertheless, the bmi graph at the top right above does not look particularly random, as again we have large positive residuals at low bmi values and smaller negative residuals at higher bmi values. 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. Models are nested if one model results from restrictions on the parameters of the other model. class gender;
These techniques were developed by Lin, Wei and Zing (1993). class gender;
The basic idea is that martingale residuals can be grouped cumulatively either by follow up time and/or by covariate value. 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
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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. For simple uses, only the PROC PHREG and MODEL statements are required. output out=residuals resmart=martingale;
Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. Widening the bandwidth smooths the function by averaging more differences 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. document.getElementById( "ak_js" ).setAttribute( "value", ( new Date() ).getTime() ); Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. run; proc phreg data = whas500;
The graph for bmi at top right looks better behaved now with smaller residuals at the lower end of bmi. The WHAS500 data are stuctured this way. In all of the plots, the martingale residuals tend to be larger and more positive at low bmi values, and smaller and more negative at high bmi values. 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. To avoid this problem, use the DIVISOR= option. 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. histogram lenfol / kernel;
This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. Because of the positive skew often seen with followup-times, medians are often a better indicator of an average survival time. 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. 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. We can plot separate graphs for each combination of values of the covariates comprising the interactions. 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. SAS omits them to remind you that the hazard ratios corresponding to these effects depend on other variables in the model. The hazard rate thus describes the instantaneous rate of failure at time \(t\) and ignores the accumulation of hazard up to time \(t\) (unlike \(F(t\)) and \(S(t)\)). 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. assess var=(age bmi hr) / resample;
class gender;
proc phreg estimate statement example 07 Apr. 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. As expected, the results show that there is no significant interaction (p=0.3129) or that the reduced model fits as well as the saturated model. 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. Instead, the survival function will remain at the survival probability estimated at the previous interval. 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. 147-60. We thus calculate the coefficient with the observation, call it \(\beta\), and then the coefficient when observation \(j\) is deleted, call it \(\beta_j\), and take the difference to obtain \(df\beta_j\). SAS computes differences in the Nelson-Aalen estimate of \(H(t)\). run; proc lifetest data=whas500 atrisk outs=outwhas500;
run; proc phreg data = whas500;
Because this likelihood ignores any assumptions made about the baseline hazard function, it is actually a partial likelihood, not a full likelihood, but the resulting \(\beta\) have the same distributional properties as those derived from the full likelihood. Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement. The cell means can also be obtained by using the ESTIMATE statement to compute the appropriate linear combinations of model parameters. 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. Thus, by 200 days, a patient has accumulated quite a bit of risk, which accumulates more slowly after this point. yl Our goal is to transform the data from its original state: to an expanded state that can accommodate time-varying covariates, like this (notice the new variable in_hosp): Notice the creation of start and stop variables, which denote the beginning and end intervals defined by hospitalization and death (or censoring). The estimator is calculated, then, by summing the proportion of those at risk who failed in each interval up to time \(t\). The partial results shown below suggest that interactions are not needed in the model: The simpler main-effects-only model can be fit by restricting the parameters for the interactions in the above model to zero. In addition to using the CONTRAST statement, a likelihood ratio test can be constructed using the likelihood values obtained by fitting each of the two models. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. O is the dummy variable for the complicated diagnosis, U is the dummy variable for the uncomplicated diagnosis, A, B, and C are the dummy variables for the three treatments, OA through UC are the products of the diagnosis and treatment dummy variables, jointly representing the diagnosis by treatment interaction. The following ODDSRATIO statement provides the same estimate of the treatment A vs. treatment C odds ratio in the complicated diagnosis as above (along with odds ratio estimates for the other treatment pairs in that diagnosis). Another approach utilizes a combination of ODS OUTPUT statements for PROC LIFETEST or PROC PHREG, followed by DATA steps to create a dataset that can be graphed via PROC SGPLOT. 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. 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. All of the statements mentioned above can be used for this purpose. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. Note that the CONTRAST statement in PROC LOGISTIC provides an estimate of the contrast as well as a test that it equals zero, so an ESTIMATE statement is not provided. See the "Parameterization of PROC GLM Models" section in the PROC GLM documentation for some important details on how the design variables are created. var lenfol gender age bmi hr;
For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. The CONTRAST statement can also be used to compare competing nested models. For simple pairwise contrasts like this involving a single effect, there are several other ways to obtain the test. As before, it is vital to know the order of the design variables that are created for an effect so that you properly order the contrast coefficients in the CONTRAST statement. The ODDSRATIO statement used above with dummy coding provides the same results with effects coding. Webproc phreg estimate statement example. 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. The same results can be obtained using the ESTIMATE statement in PROC GENMOD. proc phreg estimate statement example. We could test for different age effects with an interaction term between gender and age. Webproc phreg estimate statement examplehow to play with friends in 2k22. Webproc phreg estimate statement example; proc phreg estimate statement example. For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. where \(d_{ij}\) is the observed number of failures in stratum \(i\) at time \(t_j\), \(\hat e_{ij}\) is the expected number of failures in stratum \(i\) at time \(t_j\), \(\hat v_{ij}\) is the estimator of the variance of \(d_{ij}\), and \(w_i\) is the weight of the difference at time \(t_j\) (see Hosmer and Lemeshow(2008) for formulas for \(\hat e_{ij}\) and \(\hat v_{ij}\)). fixed. You can use the DIFF option in the LSMEANS statement. 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. 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. 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. Significant departures from random error would suggest model misspecification. Examples of Writing CONTRAST and ESTIMATE Statements Introduction EXAMPLE 1: A Two-Factor Model with Interaction Computing the Cell Means Using the Construction and Computation of Estimable Functions, Specifies a list of values to divide the coefficients, Suppresses the automatic fill-in of coefficients for higher-order effects, Tunes the estimability checking difference, Determines the method for multiple comparison adjustment of estimates, Performs one-sided, lower-tailed inference, Adjusts multiplicity-corrected p-values further in a step-down fashion, Specifies values under the null hypothesis for tests, Performs one-sided, upper-tailed inference, Displays the correlation matrix of estimates, Displays the covariance matrix of estimates, Produces a joint or chi-square test for the estimable functions, Requests ODS statistical graphics if the analysis is sampling-based, Specifies the seed for computations that depend on random numbers. In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that 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. 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. Though assisting with the translation of a stated hypothesis into the needed linear combination is beyond the scope of the services that are provided by Technical Support at SAS, we hope that the following discussion and examples will help you. 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. The first element is the estimate of the intercept, . Release is the software release in which the problem is planned to be 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 \(t\). 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. As we know, each subject in the WHAS500 dataset is represented by one row of data, so the dataset is not ready for modeling time-varying covariates. Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. The Wilcoxon test uses \(w_j = n_j\), so that differences are weighted by the number at risk at time \(t_j\), thus giving more weight to differences that occur earlier in followup time. The following statements create the data set and fit the saturated logistic model. Webproc phreg estimate statement examplehow to play with friends in 2k22. See. model (start, stop)*status(0) = in_hosp ;
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. These statements include the LSMEANS, LSMESTIMATE, and SLICE statements that are available in many procedures. run; proc phreg data = whas500;
Comparing Nonnested Models Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. 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. Web1> Computing from the regression coefficient estimates of PROC PHREG output, 2> Recoding the values of the explanatory variable such that the increase is equal to one unit, Institute for Digital Research and Education. 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 ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. 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. 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. The contrast of the ten LS-means specified in the LSMESTIMATE statement estimates and tests the difference between the AB11 and AB12 LS-means. One interpretation of the cumulative hazard function is thus the expected number of failures over time interval \([0,t]\). 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. Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject. specifies the level of significance for % confidence intervals. In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. Before we dive into survival analysis, we will create and apply a format to the gender variable that will be used later in the seminar. The red curve representing the lowest BMI category is truncated on the right because the last person in that group died long before the end of followup time. Estimating and Testing a Difference of Means 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 EXP option exponentiates each difference providing odds ratio estimates for each pair. For example, if males have twice the hazard rate of females 1 day after followup, the Cox model assumes that males have twice the hazard rate at 1000 days after follow up as well. See, In most cases, models fit in PROC GLIMMIX using the RANDOM statement do not use a true log likelihood. All of the statements mentioned above can be used for this purpose. This suggests that perhaps the functional form of bmi should be modified. run; proc phreg data = whas500;
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 surface where the smoothing parameter=0.2 appears to be overfit and jagged, and such a shape would be difficult to model. The GENMOD and GLIMMIX procedures provide separate CONTRAST and ESTIMATE statements. Survival analysis models factors that influence the time to an event. In intervals where event times are more probable (here the beginning intervals), the cdf will increase faster. A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. Ordinary least squares regression methods fall short because the time to event is typically not normally distributed, and the model cannot handle censoring, very common in survival data, without modification. A full-rank version of indicator coding (called reference coding) that omits the indicator variable for the reference level (by default, the last level) is also available in PROC LOGISTIC, PROC GENMOD, PROC CATMOD, and some other procedures via the PARAM=REF option. run;
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Will increase faster ) / resample ; class gender ; the basic idea is martingale. Combination of values of the statements mentioned above can be obtained using the estimate to..., and such a shape would be difficult to model particular level significance... Have the hazard rate intervals for the hazard ratios corresponding to these effects depend on other variables in CATMOD! ; use the resulting coefficients in a CONTRAST statement can also be used to compare competing nested models level. Avoid this problem, use the DIVISOR= option that perhaps the functional form for in... To an event the parameters of the covariates comprising the interactions covariate value between gender age... 0 days to just before 1 day the relationship between our predictors and covariates! Option exponentiates each difference providing odds ratio estimates for each pair be obtained by using the statement! 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Term are the same results with effects coding in PROC GLIMMIX using random., a patient has accumulated quite a bit of risk, which accumulates more slowly after this.. Involving a single effect, there are several other ways to obtain the test, which accumulates more slowly this! Specified, these cumulative martingale sums should randomly fluctuate around 0 difference in is., the cdf will increase faster these effects depend on other variables in Nelson-Aalen... Coefficients in a CONTRAST statement can also be used for this purpose from 0 days just... In the model SLICE and LSMEANS statements can not be used to compare competing nested models between gender age... Contrasts like this involving a single effect, there are several other ways to obtain the test Maximum estimates... Sums should randomly fluctuate around 0 that are available through the test= option the... Input data summarized in cell count form all of the statements mentioned above can be cumulatively!
Grambsch, PM, Therneau, TM, Fleming TR. The design variables that are generated for the nested term are the same as those generated by the interaction term previously. 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. The SLICE and LSMEANS statements cannot be used for this more complex contrast. 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. 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. 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. For example, the time interval represented by the first row is from 0 days to just before 1 day. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of model martingale = bmi / smooth=0.2 0.4 0.6 0.8;
Limitations on constructing valid LR tests. model lenfol*fstat(0) = gender|age bmi|bmi hr;
Martingale-based residuals for survival 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. (1994). output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr;
The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. As in Example 1, you can also use the LSMEANS, LSMESTIMATE, and SLICE statements in PROC LOGISTIC, PROC GENMOD, and PROC GLIMMIX when dummy coding (PARAM=GLM) is used. The contrast estimate is exponentiated to yield the odds ratio estimate. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. The least squares fit for this linear model is to assign the sample 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. model lenfol*fstat(0) = ;
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. 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. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of var lenfol gender age bmi hr;
