Above we described that integrating the pdf over some range yields the probability of observing \(Time\) in that range. The same procedure could be repeated to check all covariates. Biometrika. Although the coding scheme is different, you still follow the same steps to determine the contrast coefficients. 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 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. The coefficients for the mean estimates of AB11 and AB12 are again determined by writing them in terms of the model. This confidence band is calculated for the entire survival function, and at any given interval must be wider than the pointwise confidence interval (the confidence interval around a single interval) to ensure that 95% of all pointwise confidence intervals are contained within this band. This seminar covers both proc lifetest and proc phreg, and data can be structured in one of 2 ways for survival analysis. This suggests that perhaps the functional form of bmi should be modified. The Kaplan_Meier survival function estimator is calculated as: \[\hat S(t)=\prod_{t_i\leq t}\frac{n_i d_i}{n_i}, \]. All of those hazard rates are based on the same baseline hazard rate \(h_0(t_i)\), so we can simplify the above expression to: \[Pr(subject=2|failure=t_j)=\frac{exp(x_2\beta)}{exp(x_1\beta)+exp(x_2\beta)+exp(x_3\beta)}\]. Release is the software release in which the problem is planned to be The cumulative distribution function (cdf), \(F(t)\), describes the probability of observing \(Time\) less than or equal to some time \(t\), or \(Pr(Time t)\). 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. A main effect parameter is interpreted as the deviation of the level's effect from the average effect of all the levels. 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. 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). The hazard function is also generally higher for the two lowest BMI categories. If you specify a CONTRAST statement involving A alone, the matrix contains nonzero terms for both A and A*B, since A*B contains A. The 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). For example, suppose an effect coded CLASS variable A has four levels. Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement. model lenfol*fstat(0) = gender age;; The PHREG procedure will produce inverse hazard ratio measuring instead the effect of Standard of Care versus the effect of study Drug Dose Regimen 2. The LSMESTIMATE statement again makes this easier. Suppose it is of interest to test the null hypothesis that cell means ABC121 and ABC212 are equal that is, H0: 121 - 212 = 0. The null hypothesis, in terms of model 3e, is: We saw above that the first component of the hypothesis, log(OddsOA) = + d + t1 + g1. Similarly, because we included a BMI*BMI interaction term in our model, the BMI term is interpreted as the effect of bmi when bmi is 0. 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. class gender; Let us further suppose, for illustrative purposes, that the hazard rate stays constant at \(\frac{x}{t}\) (\(x\) number of failures per unit time \(t\)) over the interval \([0,t]\). Only these two statements may be flexible enough to estimate or test sufficiently complex linear combinations of model parameters. Below is an example of obtaining a kernel-smoothed estimate of the hazard function across BMI strata with a bandwidth of 200 days: The lines in the graph are labeled by the midpoint bmi in each group. Biometrika. class gender; A More Complex Contrast with Effects Coding This example is to illustrate the algorithm used to compute the parameter estimate. The most commonly used test for comparing nested models is the likelihood ratio test, but other tests (such as Wald and score tests) can also be used. Watch this tutorial for more. Notice that the baseline hazard rate, \(h_0(t)\) is cancelled out, and that the hazard rate does not depend on time \(t\): The hazard rate \(HR\) will thus stay constant over time with fixed covariates. For example: When you use the less-than-full-rank parameterization (by specifying PARAM=GLM in the CLASS statement), each row is checked for estimability. The contrast of the ten LS-means specified in the LSMESTIMATE statement estimates and tests the difference between the AB11 and AB12 LS-means. The second model is a reduced model that contains only the main effects. PROC PLM was released with SAS 9.22 in 2010. I am about to use cox-regression to estimate the interaction between two binary variables: Disease (1,0) and Drug (1,0). The contrast table that shows the log odds ratio and odds ratio estimates is exactly as before. class gender; If only \(k\) names are supplied and \(k\) is less than the number of distinct df\betas, SAS will only output the first \(k\) \(df\beta_j\). To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see. 1469-82. 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 \(S(t)\)). If variable exposure is not formatted: If variable exposure is formatted and the formatted value of exposure=0 is 'no': Or, to avoid hardcoding of formatted values: (Among the internal values of exposure, 0 and 1, 0 is the first, regardless of formats. By default, Wald confidence limits are produced. `Pn.bR#l8(QBQ p9@E,IF0QlPC4NC)R- R]*C!B)Uj.$qpa *O'CAI ")7 This section contains 14 examples of PROC PHREG applications. displays the vector of linear coefficients such that is the log-hazard ratio, with being the vector of regression coefficients. specifies the tolerance for testing the singularity of the Hessian matrix in the computation of the profile-likelihood confidence limits. 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: \[HR = exp(\beta_x(log(x_2)-log(x_1)) = exp(\beta_x(log\frac{x_2}{x_1}))\]. proc phreg data=event; While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. Grambsch, PM, Therneau, TM, Fleming TR. Again, trailing zero coefficients can be omitted. 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. PROC GENMOD produces the Wald statistic when the WALD option is used in the CONTRAST statement. This option is ignored in the estimation of hazard ratios for a continuous variable. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. Note that some functions, like ratios, are nonlinear combinations and cannot generally be obtained with these statements. It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. class gender; (Js")*sv1t1} #Hqk*"lf,Rv$"TAlM@e (braP)NP r*$O2H3;0dFik-T'G2\QSDRT2H)!I+M) The unconditional probability of surviving beyond 2 days (from the onset of risk) then is \(\hat S(2) = \frac{500 8}{500}\times\frac{492-8}{492} = 0.984\times0.98374=.9680\). 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. 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. EXAMPLE 5: A Quadratic Logistic Model The sudden upticks at the end of follow-up time are not to be trusted, as they are likely due to the few number of subjects at risk at the end. Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. Notice the survival probability does not change when we encounter a censored observation. Copyright See the Analysis of Maximum Likelihood Estimates table to verify the order of the design variables. 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. With this simple model, we specifies the level of significance for the % confidence interval for each contrast when the ESTIMATE option is specified. 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 . The order of \(df\beta_j\) in the current model are: gender, age, gender*age, bmi, bmi*bmi, hr. 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). and what i need is the hard ratios for outcome on exposure. 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 degrees of freedom are the number of linearly independent constraints implied by the CONTRAST statementthat is, the rank of . 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. run; proc phreg data = whas500; This option is ignored when the full-rank parameterization is used. The "Class Level Information" table shows the ordering of levels within variables. A popular method for evaluating the proportional hazards assumption is to examine the Schoenfeld residuals. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. If the variable is a continuous variable, the hazard ratio compares the hazards for a given change (by default, a increase of 1 unit) in the variable. Lin, DY, Wei, LJ, Ying, Z. Here we use proc lifetest to graph \(S(t)\). We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. Limitations on constructing valid LR tests. This analysis proceeds in much the same was as dfbeta analysis, in that we will: We see the same 2 outliers we identifed before, id=89 and id=112, as having the largest influence on the model overall, probably primarily through their effects on the bmi coefficient. We will use scatterplot smooths to explore the scaled Schoenfeld residuals relationship with time, as we did to check functional forms before. Be careful to order the coefficients to match the order of the model parameters in the procedure. of the mean for cell ses =1 and the cell ses =3. scatter x = bmi y=dfbmibmi / markerchar=id; linear combination of the parameter estimates. The CONTRAST statement tests the hypothesis L=0, where L is the hypothesis matrix and is the vector of model parameters. Therneau, TM, Grambsch, PM. controls the convergence criterion for the profile-likelihood confidence limits. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. The dfbeta measure, \(df\beta\), quantifies how much an observation influences the regression coefficients in the model. 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. Most of the variables are at least slightly correlated with the other variables. Because this seminar is focused on survival analysis, we provide code for each proc and example output from proc corr with only minimal explanation. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); The DIVISOR= option is used to ensure precision and avoid nonestimability. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most easily obtained using the ODDSRATIO statement. Introduction 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. have three parameters, the intercept and two parameters for ses =1 and ses There is no limit to the number of CONTRAST statements that you can specify, but they must appear after the MODEL statement. If ABS is greater than , then is declared nonestimable. The documentation for the procedure lists all ODS tables that the procedure can create, or you can use the ODS TRACE ON statement to display the table names that are produced by PROC REG. The cell means can also be obtained by using the ESTIMATE statement to compute the appropriate linear combinations of model parameters. ; Comparing One Interaction Mean to the Average of All Interaction Means Because the observation with the longest follow-up is censored, the survival function will not reach 0. 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. 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. However, despite our knowledge that bmi is correlated with age, this method provides good insight into bmis functional form. For simple uses, only the PROC PHREG and MODEL statements are required. In other words, if all strata have the same survival function, then we expect the same proportion to die in each interval. This paper is not limited to any particular operating system. Checking the Cox model with cumulative sums of martingale-based residuals. A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. 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. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). Covariates are permitted to change value between intervals. R$3T\T;3b'P,QM$?LFm;tRmPsTTc+Rk/2ujaAllaD;DpK.@S!r"xJ3dM.BkvP2@doUOsuu8wuYu1^vaAxm Some data management will be required to ensure that everyone is properly censored in each interval. Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. where \(n_i\) is the number of subjects at risk and \(d_i\) is the number of subjects who fail, both at time \(t_i\). These are the equivalent PROC GENMOD statements: A More Complex Contrast with Effects Coding. 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. So the log odds are: For treatment C in the complicated diagnosis, O = 1, A = 1, B = 1. Therneau and colleagues(1990) show that the smooth of a scatter plot of the martingale residuals from a null model (no covariates at all) versus each covariate individually will often approximate the correct functional form of a covariate. The CONTRAST statement enables you to specify a matrix, , for testing the hypothesis . 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. This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. 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 following parameters are specified in the CONTRAST statement: identifies the contrast on the output. For example, if there were three subjects still at risk at time \(t_j\), the probability of observing subject 2 fail at time \(t_j\) would be: \[Pr(subject=2|failure=t_j)=\frac{h(t_j|x_2)}{h(t_j|x_1)+h(t_j|x_2)+h(t_j|x_3)}\]. scatter x = bmi y=dfbmi / markerchar=id; The result, while not strictly an odds ratio, is useful as a comparison of the odds of treatment A to the "average" odds of the treatments. We cannot tell whether this age effect for females is significantly different from 0 just yet (see below), but we do know that it is significantly different from the age effect for males. 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). 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. 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. You can specify a contrast of the LS-means themselves, rather than the model parameters, by using the LSMESTIMATE statement. Since the contrast involves only the ten LS-means, it is much more straight-forward to specify. For a CLASS variable, a hazard ratio compares the hazards of two levels of the variable. However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. (1995). var lenfol gender age bmi hr; The next five elements are the parameter estimates for the levels of A, 1 through 5. The HAZARDRATIO statement enables you to request hazard ratios for any variable in the model at customized settings. Two groups of rats received different pretreatment regimes and then were exposed to a carcinogen. All produce equivalent results. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of However, no statistical tests comparing criterion values is possible. Martingale-based residuals for survival models. Instead, we need only assume that whatever the baseline hazard function is, covariate effects multiplicatively shift the hazard function and these multiplicative shifts are constant over time. See this sample program for discussion and examples of using the Vuong and Clarke tests to compare nonnested models. ALPHA=number specifies the level of significance for % confidence intervals. since it is the comparison group. These two observations, id=89 and id=112, have very low but not unreasonable bmi scores, 15.9 and 14.8. 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. \[F(t) = 1 exp(-H(t))\] 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. Beside using the solution option to get the parameter estimates, run; lenfol: length of followup, terminated either by death or censoring. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. requests that each individual contrast (that is, each row, , of ) or exponentiated contrast () be estimated and tested. 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 \(HR = exp(0.5(1)) = 1.6487\). There are \(df\beta_j\) values associated with each coefficient in the model, and they are output to the output dataset in the order that they appear in the parameter table Analysis of Maximum Likelihood Estimates (see above). 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. However they lived much longer than expected when considering their bmi scores and age (95 and 87), which attenuates the effects of very low bmi. If 3.5 is the average of the sampled values of X, the following two HAZARDRATIO statements are equivalent: specifies whether to create the Wald or profile-likelihood confidence limits, or both for the classical analyis. class gender; Finally, the CONTRAST and ESTIMATE statements use the contrast determined above to compute the AB11 - AB12 difference. run; proc phreg data=whas500 plots=survival; 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. 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. This seminar covers both proc lifetest to graph \ ( Time\ ) in that range matrix! The estimate statement to compute the parameter estimates, a hazard ratio compares hazards. Gender and bmi, that may influence survival time after heart attack, may... Are illustrated below, this method provides good insight into bmis functional form that describes the relationship a! It is not always possible to know a priori the correct functional proc phreg estimate statement example that describes relationship... The deviation of the Hessian matrix in the contrast statement tests the between... Lfm ; tRmPsTTc+Rk/2ujaAllaD ; DpK estimates table to verify the order of the parameter estimates for involved... Shows the ordering of levels within variables equivalent proc GENMOD produces the Wald option is in... 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