The problem is greatly simplified using effects coding, which is available in some procedures via the PARAM=EFFECT option in the CLASS statement. 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. 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. scatter x = age y=dfage / markerchar=id; During the interval [382,385) 1 out of 355 subjects at-risk died, yielding a conditional probability of survival (the probability of survival in the given interval, given that the subject has survived up to the begininng of the interval) in this interval of \(\frac{355-1}{355}=0.9972\). where \(d_i\) is the number who failed out of \(n_i\) at risk in interval \(t_i\). From the plot we can see that the hazard function indeed appears higher at the beginning of follow-up time and then decreases until it levels off at around 500 days and stays low and mostly constant. Chapter 19, The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. 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. You can use the EFFECTPLOT statement to visualize the model. The rows of are specified in order and are separated by commas. In PROC GENMOD or PROC GLIMMIX, use the EXP option in the ESTIMATE statement. Survival analysis models factors that influence the time to an event. It is expected that the model with Bilirubin in the log scale would have a better discriminating power than the model with Bilirubin in the original scale. For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. All produce equivalent results. With effects coding, each row of L can be written to select just one interaction parameter when multiplied by . Graphs are particularly useful for interpreting interactions. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. rights reserved. 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. It is available only for the Bayesian analysis. model lenfol*fstat(0) = gender age;; As we see above, one of the great advantages of the Cox model is that estimating predictor effects does not depend on making assumptions about the form of the baseline hazard function, \(h_0(t)\), which can be left unspecified. hrtime = hr*lenfol; For example, suppose that the model contains effects A and B and their interaction A*B. Several covariates can be evaluated simultaneously. Disease: 1=Disease, 0=No disease Drug: 1=Drug, 0=No drug This make the interaction a "2x2 table" (as below). 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. 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. In PROC LOGISTIC, use the PARAM=GLM option in the CLASS statement to request dummy coding of CLASS variables. All ; While examples in this class provide good examples of the above process for determining coefficients for CONTRAST and ESTIMATE statements, there are other statements available that perform means comparisons more easily. Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject. Particular emphasis is given to proc lifetest for nonparametric estimation, and proc phreg for Cox regression and model evaluation. PROC GENMOD produces the Wald statistic when the WALD option is used in the CONTRAST statement. 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. 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. Institute for Digital Research and Education. histogram lenfol / kernel; 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. Suppose the model contains two interactions: an interaction A*B of CLASS variables A and B, and another interaction A*X of A with a continuous variable X. displays the vector of linear coefficients such that is the log-hazard ratio, with being the vector of regression coefficients. In this interval, we can see that we had 500 people at risk and that no one died, as Observed Events equals 0 and the estimate of the Survival function is 1.0000. 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. The PLOTS= option is not available for the maximum likelihood anaysis. Both proc lifetest and proc phreg will accept data structured this way. (2000). and then i would like to see the trends on age group. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). run; lenfol: length of followup, terminated either by death or censoring. The value number must be between 0 and 1; the default value is 0.05, which results in 95% intervals. 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: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. 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. Other CONTRAST statements involving classification variables with PARAM=EFFECT are constructed similarly. 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. 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. Notice the survival probability does not change when we encounter a censored observation. EXAMPLE 2: A Three-Factor Model with Interactions 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. Below we demonstrate a simple model in proc phreg, where we determine the effects of a categorical predictor, gender, and a continuous predictor, age on the hazard rate: The above output is only a portion of what SAS produces each time you run proc phreg. rights reserved. The exponential function is also equal to 1 when its argument is equal to 0. In some cases, the Laplace or quadrature estimation methods (METHOD=LAPLACE or METHOD=QUAD, first available in SAS 9.2) can be used which compute and report an approximate log likelihood making construction of a LR test possible. The Nelson-Aalen estimator is a non-parametric estimator of the cumulative hazard function and is given by: \[\hat H(t) = \sum_{t_i leq t}\frac{d_i}{n_i},\]. If this option is not specified, PROC PHREG finds all the variables that interact with the variable of interest. Finally, the CONTRAST and ESTIMATE statements use the contrast determined above to compute the AB11 - AB12 difference. 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. SAS provides easy ways to examine the \(df\beta\) values for all observations across all coefficients in the model. Stratification allows each stratum to have its own baseline hazard, which solves the problem of nonproportionality. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. Any serious endeavor into data analysis should begin with data exploration, in which the researcher becomes familiar with the distributions and typical values of each variable individually, as well as relationships between pairs or sets of variables. 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. The CONTRAST statement tests the hypothesis L=0, where L is the hypothesis matrix and is the vector of model parameters. These statement essentially look like data step statements, and function in the same way. model lenfol*fstat(0) = ; Follow up time for all participants begins at the time of hospital admission after heart attack and ends with death or loss to follow up (censoring). We previously saw that the gender effect was modest, and it appears that for ages 40 and up, which are the ages of patients in our dataset, the hazard rates do not differ by gender. Imagine we have a random variable, \(Time\), which records survival times. The following examples concentrate on using the steps above in this situation. In each of the graphs above, a covariate is plotted against cumulative martingale residuals. The LSMESTIMATE statement again makes this easier. Notice, however, that \(t\) does not appear in the formula for the hazard function, thus implying that in this parameterization, we do not model the hazard rates dependence on time. Note: A number of sub-sections are titled Background. rights reserved. The individual AB11 and AB12 cell means are: The coefficients for the average of the AB21 and AB22 cells are determined in the same fashion. With any procedure, models that are not nested cannot be compared using the LR test. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. This test can be done using a CONTRAST statement to jointly test the interaction parameters. 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 \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). Deploy software automatically at the click of a button on the Microsoft Azure Marketplace. In such cases, the correct form may be inferred from the plot of the observed pattern. Applied Survival Analysis, Second Edition provides a comprehensive and up-to-date introduction to regression modeling for time-to-event By default, Wald confidence limits are produced. Estimating and Testing Odds Ratios with Dummy Coding 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. Note that some functions, like ratios, are nonlinear combinations and cannot generally be obtained with these statements. time lenfol*fstat(0); If the elements of are not specified for an effect that contains a specified effect, then the elements of the specified effect are distributed over the levels of the higher-order effect just as the GLM procedure does for its CONTRAST and ESTIMATE statements. 80(30). This study examined several factors, such as age, gender and BMI, that may influence survival time after heart attack. The numerator is the hazard of death for the subject who died 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 \(df\beta_j\). 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). If too few values are specified, the remaining ones are set to 0. 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; 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). 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. Second, all three fit statistics, -2 LOG L, AIC and SBC, are each 20-30 points lower in the larger model, suggesting the including the extra parameters improve the fit of the model substantially. If the BAYES statement is specified, the ADJUST=, STEPDOWN, TESTVALUE, LOWER, UPPER, and JOINT options are ignored. Again, trailing zero coefficients can be omitted. The ODDSRATIO statement used above with dummy coding provides the same results with effects coding. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. 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. These statements fit the restricted, main effects model: This partial output summarizes the main-effects model: The question is whether there is a significant difference between these two models. Lets confirm our understanding of the calculation of the Nelson-Aalen estimator by calculating the estimated cumulative hazard at day 3: \(\hat H(3)=\frac{8}{500} + \frac{8}{492} + \frac{3}{484} = 0.0385\), which matches the value in the table. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. 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. In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. Copyright SAS Institute, Inc. All Rights Reserved. Specifically, you need to construct the linear combination of model parameters that corresponds to the hypothesis. The SLICE and LSMEANS statements cannot be used for this more complex contrast. model lenfol*fstat(0) = gender|age bmi hr; Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see this note. 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. (1994). 2009 by SAS Institute Inc., Cary, NC, USA. Write down the model that you are using the procedure to fit. for ses = 1, we will add the coefficient for ses1 to the intercept. 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. and what i need is the hard ratios for outcome on exposure. The first three parameters of the nested effect are the effects of treatments within the complicated diagnosis. Widening the bandwidth smooths the function by averaging more differences together. Because PROC CATMOD also uses effects coding, you can use the following CONTRAST statement in that procedure to get the same results as above. The LSMESTIMATE statement can also be used. Plots of covariates vs dfbetas can help to identify influential outliers. Estimating and Testing Odds Ratios with Effects Coding. are constants that are elements of the matrix associated with the effect. 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. Proportional hazards tests and diagnostics based on weighted residuals. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; A main effect parameter is interpreted as the deviation of the level's effect from the average effect of all the levels. This is reinforced by the three significant tests of equality. class gender; Nevertheless, in both we can see that in these data, shorter survival times are more probable, indicating that the risk of heart attack is strong initially and tapers off as time passes. 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 second model is a reduced model that contains only the main effects. The following statements show all five ways of computing and testing this contrast. Group of ses =3 is the reference group. If an interacting variable is a CLASS variable, variable= ALL is the default; if the interacting variable is continuous, variable= is the default, where is the average of all the sampled values of the continuous variable. This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. Notice that if you add up the rows for diagnosis (or treatments), the sum is zero.
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