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 numerator is the hazard of death for the subject who died The statements below fit the model, estimate each part of the hypothesis, and estimate and test the hypothesis. Copyright SAS Institute, Inc. All Rights Reserved. This is an extension of the nested effects that you can specify in other procedures such as GLM and LOGISTIC. The effect of bmi is significantly lower than 1 at low bmi scores, indicating that higher bmi patients survive better when patients are very underweight, but that this advantage disappears and almost seems to reverse at higher bmi levels. Below, we show how to use the hazardratio statement to request that SAS estimate 3 hazard ratios at specific levels of our covariates. time lenfol*fstat(0); By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. We see a sharper rise in the cumulative hazard right at the beginning of analysis time, reflecting the larger hazard rate during this period. However, despite our knowledge that bmi is correlated with age, this method provides good insight into bmis functional form. 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. where \(n_i\) is the number of subjects at risk and \(d_i\) is the number of subjects who fail, both at time \(t_i\). 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. 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. With any procedure, models that are not nested cannot be compared using the LR test. In the table above, we see that the probability surviving beyond 363 days = 0.7240, the same probability as what we calculated for surviving up to 382 days, which implies that the censored observations do not change the survival estimates when they leave the study, only the number at risk. 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. In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. Biometrika. 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. When the procedure reports a log pseudo-likelihood you cannot construct a LR test to compare models. Examples of Writing CONTRAST and ESTIMATE Statements Introduction EXAMPLE 1: A Two-Factor Model with Interaction Computing the Cell Means Using the ESTIMATE Statement Estimating and Testing a Difference of Means A More Complex Contrast Comparing One Interaction Mean to the Average of All Interaction Means specifies the units of change in the continuous explanatory variable for which the customized hazard ratio is estimated. The Analysis of Maximum Likelihood Estimates table confirms the ordering of design variables in model 3d. %PDF-1.2 % model lenfol*fstat(0) = gender|age bmi|bmi hr; 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. Effects Coding (Js")*sv1t1} #Hqk*"lf,Rv$"TAlM@e (braP)NP r*$O2H3;0dFik-T'G2\QSDRT2H)!I+M) The contrast of the ten LS-means specified in the LSMESTIMATE statement estimates and tests the difference between the AB11 and AB12 LS-means. Models are nested if one model results from restrictions on the parameters of the other model. where a row-description is: effect values <,effect values>. Consider the following medical example in which patients with one of two diagnoses (complicated or uncomplicated) are treated with one of three treatments (A, B, or C) and the result (cured or not cured) is observed. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. This is required so that the probability of being a case is modeled. The calculation of the statistic for the nonparametric Log-Rank and Wilcoxon tests is given by : \[Q = \frac{\bigg[\sum\limits_{i=1}^m w_j(d_{ij}-\hat e_{ij})\bigg]^2}{\sum\limits_{i=1}^m w_j^2\hat v_{ij}},\]. This can be particularly difficult with dummy (PARAM=GLM) coding. "exposure.". We would like to allow parameters, the \(\beta\)s, to take on any value, while still preserving the non-negative nature of the hazard rate. since it is the comparison group. EXAMPLE 2: A Three-Factor Model with Interactions Consider the following data from Kalbeisch and Prentice (1980). model martingale = bmi / smooth=0.2 0.4 0.6 0.8; In each of the graphs above, a covariate is plotted against cumulative martingale residuals. In the simpler case of a main-effects-only model, writing CONTRAST and ESTIMATE statements to make simple pairwise comparisons is more intuitive. class gender; 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. Looking at the table of Product-Limit Survival Estimates below, for the first interval, from 1 day to just before 2 days, \(n_i\) = 500, \(d_i\) = 8, so \(\hat S(1) = \frac{500 8}{500} = 0.984\). 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. Use the Class Level Information table which shows the design variable settings. class gender; Thus, we again feel justified in our choice of modeling a quadratic effect of bmi. For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. The CONTRAST statement can also be used to compare competing nested models. 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. 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 \(S(t)\)). 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. The LSMESTIMATE statement again makes this easier. 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. Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. It is not necessary that the larger model be saturated. 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. In such cases, the correct form may be inferred from the plot of the observed pattern. Here is the syntax for CONTRAST statement. 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. Introduction This example is to illustrate the algorithm used to compute the parameter estimate. The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. The following statements fit the model and compute the AB11 and AB12 cell means by using the LSMEANS statement and equivalent ESTIMATE statements: Suppose you want to test that the AB11 and AB12 cell means are equal. tunes the estimability check. Provided the reader has some background in survival analysis, these sections are not necessary to understand how to run survival analysis in SAS. 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 CONTRAST statement below defines seven rows in L for the seven interaction parameters resulting in a 7 DF test that all interaction parameters are zero. PROC GENMOD can also be used to estimate this odds ratio. 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. In the CONTRAST statement, the rows of L are separated by commas. The statements below generate observations from such a model: The following statements fit the main effects and interaction model. Proc PHREG - Random Statement. 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. Censored observations are represented by vertical ticks on the graph. run; proc phreg data = whas500; Deploy software automatically at the click of a button on the Microsoft Azure Marketplace. 2. It is not at all necessary that the hazard function stay constant for the above interpretation of the cumulative hazard function to hold, but for illustrative purposes it is easier to calculate the expected number of failures since integration is not needed. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. 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. 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. Graphs are particularly useful for interpreting interactions. In other words, if all strata have the same survival function, then we expect the same proportion to die in each interval. 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. 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. (2000). This section contains 14 examples of PROC PHREG applications. 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. 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. 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. 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. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. A Nested Model data example8_1; set sec1_5; group1 = group - 1; run; proc phreg data = example8_1; model time*death (0)=group1; run; So what is the probability of observing subject \(i\) fail at time \(t_j\)? Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. 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. Instead, you model a function of the response distribution's mean. 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. 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. Thus, we can expect the coefficient for bmi to be more severe or more negative if we exclude these observations from the model. For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. 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). Hosmer, DW, Lemeshow, S, May S. (2008). controls the convergence criterion for the profile-likelihood confidence limits. This section contains 14 examples of PROC PHREG applications. This option is not applicable to a Bayesian analysis. However, no statistical tests comparing criterion values is possible. 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. Alternatively, the data can be expanded in a data step, but this can be tedious and prone to errors (although instructive, on the other hand). 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. How do I write an estimate statement in proc glm? /*class exposure*/model period*outcome(0)=exposure / rl;run; Hello@MTeckand welcome to the SAS Support Communities! Estimating and Testing Odds Ratios with Effects Coding In an example from Ries and Smith (1963), the choice of detergent brand (Brand= M or X) is related to three other categorical variables: the softness of the laundry water (Softness= soft, medium, or hard); the temperature of the water (Temperature= high or low); and whether the subject was a previous user of Brand M (Previous= yes or no). 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. Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. See this sample program for discussion and examples of using the Vuong and Clarke tests to compare nonnested models. SAS Code from All of These Examples. 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. The survival function is undefined past this final interval at 2358 days. model lenfol*fstat(0) = ; If is a vector, define ABS() to be the largest absolute value of the elements of . The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. The EXPB option adds a column in the parameter estimates table that contains exponentiated values of the corresponding parameter estimates. run; proc phreg data = whas500; Had B preceded A in the CLASS statement, the levels of A would have changed before the levels of B, resulting in the second estimate being for 21. 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). Note that the difference in log odds is equivalent to the log of the odds ratio: So, by exponentiating the estimated difference in log odds, an estimate of the odds ratio is provided. SAS omits them to remind you that the hazard ratios corresponding to these effects depend on other variables in the model. You can use the EFFECTPLOT statement to visualize the model. All of the statements mentioned above can be used for this purpose. An estimate statement corresponds to an L-matrix, which corresponds to a output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. Significant departures from random error would suggest model misspecification. PROC GENMOD produces the Wald statistic when the WALD option is used in the CONTRAST statement. This is the default coding scheme for CLASS variables in most procedures including GLM, MIXED, GLIMMIX, and GENMOD. model lenfol*fstat(0) = gender|age bmi|bmi hr hrtime; While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. I am about to use cox-regression to estimate the interaction between two binary variables: Disease (1,0) and Drug (1,0). where \(d_i\) is the number who failed out of \(n_i\) at risk in interval \(t_i\). class gender; In a nutshell, these statistics sum the weighted differences between the observed number of failures and the expected number of failures for each stratum at each timepoint, assuming the same survival function of each stratum. However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. The above relationship between the cdf and pdf also implies: In SAS, we can graph an estimate of the cdf using proc univariate. The second three parameters are the effects of the treatments within the uncomplicated diagnosis. Some procedures allow multiple types of coding. In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. First, there may be one row of data per subject, with one outcome variable representing the time to event, one variable that codes for whether the event occurred or not (censored), and explanatory variables of interest, each with fixed values across follow up time. SAS provides easy ways to examine the \(df\beta\) values for all observations across all coefficients in the model. Finally, the CONTRAST and ESTIMATE statements use the contrast determined above to compute the AB11 - AB12 difference. Table 64.4 summarizes important options in the ESTIMATE statement. If the interacting variable is a CLASS variable, you can specify, after the equal sign, a list of quoted strings corresponding to various levels of the CLASS variable, or you can specify the keyword ALL or REF. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. proc univariate data = whas500 (where= (fstat=1)); var lenfol; cdfplot lenfol; run; In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. Estimates are formed as linear estimable functions of the form . Example Suppose we wish to fit a PH model to the data from . In particular we would like to highlight the following tables: Handily, proc phreg has pretty extensive graphing capabilities.< Below is the graph and its accompanying table produced by simply adding plots=survival to the proc phreg statement. Checking the Cox model with cumulative sums of martingale-based residuals. For a CLASS variable, a hazard ratio compares the hazards of two levels of the variable. EXAMPLE 5: A Quadratic Logistic 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. Beside using the solution option to get the parameter estimates, The value must be between 0 and 1. In the case of a dichotomous explanatory variable with values 0 and 1 (like exposure in your data) the results with vs. without a CLASS statement are essentially the same. model lenfol*fstat(0) = gender|age bmi|bmi hr ; 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. 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.
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