One can also use non-parametric methods to test for equality of the survival function among groups in the following manner: In the graph of the Kaplan-Meier estimator stratified by gender below, it appears that females generally have a worse survival experience. 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). One variable is created for each level of the original variable. Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. class gender; 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. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. rights reserved. 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. The SAS procedure PROC PHREG allows us to fit a proportional hazard model to a dataset. It is intuitively appealing to let \(r(x,\beta_x) = 1\) when all \(x = 0\), thus making the baseline hazard rate, \(h_0(t)\), equivalent to a regression intercept. Estimates are formed as linear estimable functions of the form . For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. The PHREG Procedure: Examples: PHREG Procedure. The DIVISOR= option is used to ensure precision and avoid nonestimability. On the right panel, Residuals at Specified Smooths for martingale, are the smoothed residual plots, all of which appear to have no structure. 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 simple contrast shown in the LSMESTIMATE statement below compares the fourth and eighth means as desired. For simple pairwise contrasts like this involving a single effect, there are several other ways to obtain the test. 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. We can similarly calculate the joint probability of observing each of the \(n\) subjects failure times, or the likelihood of the failure times, as a function of the regression parameters, \(\beta\), given the subjects covariates values \(x_j\): \[L(\beta) = \prod_{j=1}^{n} \Bigg\lbrace\frac{exp(x_j\beta)}{\sum_{iin R_j}exp(x_i\beta)}\Bigg\rbrace\]. The contrast estimate is exponentiated to yield the odds ratio estimate. proc sgplot data = dfbeta; 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 \(df\beta_j\). Printing this document: Because some of the tables in this document are wide, While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. 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. have three parameters, the intercept and two parameters for ses =1 and ses EXAMPLE 4: Comparing Models format gender gender. For a more detailed definition of nested and nonnested models, see the Clarke (2001) reference cited in the sample program. class gender; The ODDSRATIO statement used above with dummy coding provides the same results with effects coding. A More Complex Contrast 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. We should begin by analyzing our interactions. 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 survival curves for females is slightly higher than the curve for males, suggesting that the survival experience is possibly slightly better (if significant) for females, after controlling for age. Particular emphasis is given to proc lifetest for nonparametric estimation, and proc phreg for Cox regression and model evaluation. The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. model lenfol*fstat(0) = gender|age bmi|bmi hr; 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. We will model a time-varying covariate later in the seminar. Estimating and Testing Odds Ratios with Effects Coding Here we see the estimated pdf of survival times in the whas500 set, from which all censored observations were removed to aid presentation and explanation. The DIFF and SLICEBY(A='1') options in the SLICE statement estimate the differences in LS-means at A=1. Both proc lifetest and proc phreg will accept data structured this way. Several covariates can be evaluated simultaneously. Copyright The hazard rate can also be interpreted as the rate at which failures occur at that point in time, or the rate at which risk is accumulated, an interpretation that coincides with the fact that the hazard rate is the derivative of the cumulative hazard function, \(H(t)\). To accomplish this smoothing, the hazard function estimate at any time interval is a weighted average of differences within a window of time that includes many differences, known as the bandwidth. 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: \[Pr(subject=2|failure=t_j)=\frac{exp(x_2\beta)}{exp(x_1\beta)+exp(x_2\beta)+exp(x_3\beta)}\]. Note: The terms event and failure are used interchangeably in this seminar, as are time to event and failure time. model lenfol*fstat(0) = gender age;; 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 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. If convergence is not attained in n iterations, the corresponding profile-likelihood confidence limit for the hazard ratio is set to missing. All produce equivalent results. 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. This is exactly the contrast that was constructed earlier. The graph for bmi at top right looks better behaved now with smaller residuals at the lower end of bmi. It is called the proportional hazards model because the ratio of hazard rates between two groups with fixed covariates will stay constant over time in this model. SAS provides built-in methods for evaluating the functional form of covariates through its assess statement. run; lenfol: length of followup, terminated either by death or censoring. For example, the hazard rate when time \(t\) when \(x = x_1\) would then be \(h(t|x_1) = h_0(t)exp(x_1\beta_x)\), and at time \(t\) when \(x = x_2\) would be \(h(t|x_2) = h_0(t)exp(x_2\beta_x)\). The PLCONV= option has no effect if profile-likelihood confidence intervals (CL=PL) are not requested. Here is the syntax for CONTRAST statement. 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. A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. PROC PHREG syntax is similar to that of the other regression procedures in the SAS System. The CONTRAST statement provides a mechanism for obtaining customized hypothesis tests. To correctly specify your contrast, it is crucial to know the ordering of parameters within each effect and the variable levels associated with any parameter. Martingale-based residuals for survival models. specifies the tolerance for testing the singularity of the Hessian matrix in the computation of the profile-likelihood confidence limits. 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. 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. We generally expect the hazard rate to change smoothly (if it changes) over time, rather than jump around haphazardly. Imagine we have a random variable, \(Time\), which records survival times. run; proc phreg data = whas500; Partial Likelihood The partial likelihood function for one covariate is: where t i is the ith death time, x i is the associated covariate, and R i is the risk set at time t i, i.e., the set of subjects is still alive and uncensored just prior to time t i. The t statistic value is the square root of the F statistic from the CONTRAST statement producing an equivalent test. Thus, by 200 days, a patient has accumulated quite a bit of risk, which accumulates more slowly after this point. For a CLASS variable, a hazard ratio compares the hazards of two levels of the variable. We see in the table above, that the typical subject in our dataset is more likely male, 70 years of age, with a bmi of 26.6 and heart rate of 87. (1995). Therefore, you would use the following CONTRAST statement: To contrast the third level with the average of the first two levels, you would test. 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. 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. controls the convergence criterion for the profile-likelihood confidence limits. The estimate of survival beyond 3 days based off this Nelson-Aalen estimate of the cumulative hazard would then be \(\hat S(3) = exp(-0.0385) = 0.9623\). EXAMPLE 1: A Two-Factor Model with Interaction Therefore, the estimate of the last level of an effect, A, is a= (1 + 2 + + a1). 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. For this example, the table confirms that the parameters are ordered as shown in model 3c. The HPREG Procedure The HPSPLIT Procedure The ICLIFETEST Procedure The ICPHREG Procedure The INBREED Procedure The IRT Procedure The KDE Procedure The KRIGE2D Procedure The LATTICE Procedure The LIFEREG Procedure The LIFETEST Procedure The LOESS Procedure The LOGISTIC Procedure The MCMC Procedure The MDS Procedure The MI Procedure exposure(0=no exposure, 1= yes exposure)and outcome(0=no outcome, 1= yes outcome) variable are all binary. You can use the EFFECTPLOT statement to visualize the model. fstat: the censoring variable, loss to followup=0, death=1, Without further specification, SAS will assume all times reported are uncensored, true failures. Stated another way, are any of the interaction parameters not equal to zero as implied by the main-effects model? In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most easily obtained using the ODDSRATIO statement. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. Biometrics. Note that there are 5 2 3 = 30 cell means. 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. This is the log odds. 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 could test for different age effects with an interaction term between gender and age. In other words, we would expect to find a lot of failure times in a given time interval if 1) the hazard rate is high and 2) there are still a lot of subjects at-risk. Subjects that are censored after a given time point contribute to the survival function until they drop out of the study, but are not counted as a failure. You can fit many kinds of logistic models in many procedures including LOGISTIC, GENMOD, GLIMMIX, PROBIT, CATMOD, and others. 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. It appears that for males the log hazard rate increases with each year of age by 0.07086, and this AGE effect is significant, AGE*GENDER term is negative, which means for females, the change in the log hazard rate per year of age is 0.07086-0.02925=0.04161. Since the contrast involves only the ten LS-means, it is much more straight-forward to specify. 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. Finally, we see that the hazard ratio describing a 5-unit increase in bmi, \(\frac{HR(bmi+5)}{HR(bmi)}\), increases with bmi. All Another common mistake that may result in inverse hazard ratios is to omit the CLASS statement in the PHREG procedure altogether. We request Cox regression through proc phreg in SAS. The dependent variable is write and the factor variable is ses The hazard function for a particular time interval gives the probability that the subject will fail in that interval, given that the subject has not failed up to that point in time. SAS Code from All of These Examples. Here is the SAS code: Code: proc phreg data=Data; class Drug(ref='0') Disease(ref='0') /param=glm; The parameter for ses1 is the difference Grambsch and Therneau (1994) show that a scaled version of the Schoenfeld residual at time \(k\) for a particular covariate \(p\) will approximate the change in the regression coefficient at time \(k\): \[E(s^\star_{kp}) + \hat{\beta}_p \approx \beta_j(t_k)\]. 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