proc phreg estimate statement example

Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most easily obtained using the ODDSRATIO statement. Now lets look at the model with just both linear and quadratic effects for bmi. The difficulty is constructing combinations that are estimable and that jointly test the set of interactions. With effects coding, the parameters are constrained to sum to zero. Now consider a model in three factors, with five, two, and three levels, respectively. The order of \(df\beta_j\) in the current model are: gender, age, gender*age, bmi, bmi*bmi, hr. Plots of covariates vs dfbetas can help to identify influential outliers. This example shows the use of the CONTRAST and ODDSRATIO statements to compare the response at two levels of a continuous predictor when the model contains a higher-order effect. If nonproportional hazards are detected, the researcher has many options with how to address the violation (Therneau & Grambsch, 2000): After fitting a model it is good practice to assess the influence of observations in your data, to check if any outlier has a disproportionately large impact on the model. Estimates are formed as linear estimable functions of the form . The LSMEANS statement computes the cell means for the 10 A*B cells in this example. run; proc phreg data = whas500; The survival function estimate of the the unconditional probability of survival beyond time \(t\) (the probability of survival beyond time \(t\) from the onset of risk) is then obtained by multiplying together these conditional probabilities up to time \(t\) together. 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. Basing the test on the REML results is generally preferred. histogram lenfol / kernel; The WHAS500 data are stuctured this way. 77(1). 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. Thus, each term in the product is the conditional probability of survival beyond time \(t_i\), meaning the probability of surviving beyond time \(t_i\), given the subject has survived up to time \(t_i\). The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. 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. 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. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. WebThe ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. Applying formula (1) when (X C-X A) is not equal to 1; in particular, when (X C-X A) is equal to 2: hazard ratio = = = = 2.231 ( ) When testing, write the null hypothesis in the form. The statements below generate observations from such a model: The following statements fit the main effects and interaction 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 \(\hat S(1) = \frac{500 8}{500} = 0.984\). Website. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); The log-rank or Mantel-Haenzel test uses \(w_j = 1\), so differences at all time intervals are weighted equally. Wiley: Hoboken. To properly test a hypothesis such as "The effect of treatment A in group 1 is equal to the treatment A effect in group 2," it is necessary to translate it correctly into a mathematical hypothesis using the fitted model. For this example, the table confirms that the parameters are ordered as shown in model 3c. 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. run; In the graph above we see the correspondence between pdfs and histograms. Webproc phreg estimate statement example. 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). Since treatment A and treatment C are the first and third in the LSMEANS list, the contrast in the LSMESTIMATE statement estimates and tests their difference. proc phreg data=episode; /*class exposure*/ model period*outcome(0)=exposure / rl; (and it did in your example) that the CLASS Write down the model that you are using the procedure to fit. 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. Previously, we graphed the survival functions of males in females in the WHAS500 dataset and suspected that the survival experience after heart attack may be different between the two genders. The Schoenfeld residual for observation \(j\) and covariate \(p\) is defined as the difference between covariate \(p\) for observation \(j\) and the weighted average of the covariate values for all subjects still at risk when observation \(j\) experiences the event. Several covariates can be evaluated simultaneously. We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. 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. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. 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. model lenfol*fstat(0) = gender age;; Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. These statements include the LSMEANS, LSMESTIMATE, and SLICE statements that are available in many procedures. For more information, see the "Generation of the Design Matrix" section in the CATMOD documentation. assess var=(age bmi bmi*bmi hr) / resample; These are indeed censored observations, further indicated by the * appearing in the unlabeled second column. Finally, you can use the SLICE statement. PROC GENMOD produces the Wald statistic when the WALD option is used in the CONTRAST statement. Acquiring more than one curve, whether survival or hazard, after Cox regression in SAS requires use of the baseline statement in conjunction with the creation of a small dataset of covariate values at which to estimate our curves of interest. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. Copyright SAS Institute, Inc. All Rights Reserved. Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. On the right panel, Residuals at Specified Smooths for martingale, are the smoothed residual plots, all of which appear to have no structure. Estimates are formed as linear estimable functions of the form . 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. The likelihood ratio and Wald statistics are asymptotically equivalent. 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. 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. Specify the DIST=BINOMIAL option to specify a logistic model. Release is the software release in which the problem is planned to be Webproc phreg estimate statement example. Diagnostic plots to reveal functional form for covariates in multiplicative intensity models. However, no statistical tests comparing criterion values is possible. 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. It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. WebPROC FREQ PROC SURVEYFREQ PROC REG PROC SURVEYREG PROC LOGISTIC . An example of using the LSMEANS and LSMESTIMATE statements to estimate odds ratios in a repeated measures (GEE) model in PROC GENMOD is available. 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. class gender; The cumulative distribution function (cdf), \(F(t)\), describes the probability of observing \(Time\) less than or equal to some time \(t\), or \(Pr(Time t)\). 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 \(\frac{355-1}{355}=0.9972\). Estimating and Testing Odds Ratios with Effects Coding. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. Thus, we again feel justified in our choice of modeling a quadratic effect of bmi. None of the solid blue lines looks particularly aberrant, and all of the supremum tests are non-significant, so we conclude that Webproc phreg estimate statement example; proc phreg estimate statement example. The variables used in the present seminar are: The data in the WHAS500 are subject to right-censoring only. Proportional hazards tests and diagnostics based on weighted residuals. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. Standard nonparametric techniques do not typically estimate the hazard function directly. This reinforces our suspicion that the hazard of failure is greater during the beginning of follow-up time. 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. The primary focus of survival analysis is typically to model the hazard rate, which has the following relationship with the \(f(t)\) and \(S(t)\): The hazard function, then, describes the relative likelihood of the event occurring at time \(t\) (\(f(t)\)), conditional on the subjects survival up to that time \(t\) (\(S(t)\)). Indicator or dummy coding of a predictor replaces the actual variable in the design matrix (or model matrix) with a set of variables that use values of 0 or 1 to indicate the level of the original variable. 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). All of the statements mentioned above can be used for this purpose. class gender; The unconditional probability of surviving beyond 2 days (from the onset of risk) then is \(\hat S(2) = \frac{500 8}{500}\times\frac{492-8}{492} = 0.984\times0.98374=.9680\). This can be done by multiplying the vector of parameter estimates (the solution vector) by a vector of coefficients such that their product is this sum. We should begin by analyzing our interactions. Comparing Nested Models 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. Comparing Nonnested Models Webproc phreg estimate statement examplehow to play with friends in 2k22. SAS expects individual names for each \(df\beta_j\)associated with a coefficient. scatter x = bmi y=dfbmibmi / markerchar=id; 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 \(p(surviving~ up~ to~ 382~ days)=\frac{0.7220}{0.9972}=.7240\). Martingale-based residuals for survival models. The first three parameters of the nested effect are the effects of treatments within the complicated diagnosis. If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. The surface where the smoothing parameter=0.2 appears to be overfit and jagged, and such a shape would be difficult to model. We can see this reflected in the survival function estimate for LENFOL=382. time lenfol*fstat(0); However, in many settings, we are much less interested in modeling the hazard rates relationship with time and are more interested in its dependence on other variables, such as experimental treatment or age. Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. Indeed, exclusion of these two outliers causes an almost doubling of \(\hat{\beta}_{bmi}\), from -0.23323 to -0.39619. Thus, for example the AGE term describes the effect of age when gender=0, or the age effect for males. By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. However, one cannot test whether the stratifying variable itself affects the hazard rate significantly. One interpretation of the cumulative hazard function is thus the expected number of failures over time interval \([0,t]\). ; The CONTRAST and ESTIMATE statements allow for estimation and testing of any linear combination of model parameters. Researchers are often interested in estimates of survival time at which 50% or 25% of the population have died or failed. run; So the log odds are: For treatment C in the complicated diagnosis, O = 1, A = 1, B = 1. model lenfol*fstat(0) = gender|age bmi|bmi hr;

This simpler model is nested in the above model. The Analysis of Maximum Likelihood Estimates table confirms the ordering of design variables in model 3d. For such studies, a semi-parametric model, in which we estimate regression parameters as covariate effects but ignore (leave unspecified) the dependence on time, is appropriate. Some data management will be required to ensure that everyone is properly censored in each interval. You can perform 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. In the simpler case of a main-effects-only model, writing CONTRAST and ESTIMATE statements to make simple pairwise comparisons is more intuitive. 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. 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. This seminar introduces procedures and outlines the coding needed in SAS to model survival data through both of these methods, as well as many techniques to evaluate and possibly improve the model. Understanding the mechanics behind survival analysis is aided by facility with the distributions used, which can be derived from the probability density function and cumulative density functions of survival times. Therneau, TM, Grambsch PM, Fleming TR (1990). run; proc phreg data = whas500; Thus, by 200 days, a patient has accumulated quite a bit of risk, which accumulates more slowly after this point. It is quite powerful, as it allows for truncation, In each of the graphs above, a covariate is plotted against cumulative martingale residuals. This seminar covers both proc lifetest and proc phreg, and data can be structured in one of 2 ways for survival analysis. 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. run; proc phreg data = whas500; For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. 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. This matches closely with the Kaplan Meier product-limit estimate of survival beyond 3 days of 0.9620. Webproc phreg estimate statement example; proc phreg estimate statement example. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. SAS omits them to remind you that the hazard ratios corresponding to these effects depend on other variables in the model. 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. fixed. 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. This technique can detect many departures from the true model, such as incorrect functional forms of covariates (discussed in this section), violations of the proportional hazards assumption (discussed later), and using the wrong link function (not discussed). Unless the seed option is specified, these sets will be different each time proc phreg is run. (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). Since the contrast involves only the ten LS-means, it is much more straight-forward to specify. These statistics are provided in most procedures using maximum likelihood estimation. Instead, the survival function will remain at the survival probability estimated at the previous interval. In regression models for survival analysis, we attempt to estimate parameters which describe the relationship between our predictors and the hazard rate. The probability of surviving the next interval, from 2 days to just before 3 days during which another 8 people died, given that the subject has survived 2 days (the conditional probability) is \(\frac{492-8}{492} = 0.98374\). 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. Tests to compare nonnested models are available, but not by using CONTRAST statements as discussed above. class gender; All of these variables vary quite a bit in these data. run; proc phreg data = whas500; 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. Notice that Row2 is the coefficient vector for computing the mean of the AB12 cell. As the hazard function \(h(t)\) is the derivative of the cumulative hazard function \(H(t)\), we can roughly estimate the rate of change in \(H(t)\) by taking successive differences in \(\hat H(t)\) between adjacent time points, \(\Delta \hat H(t) = \hat H(t_j) \hat H(t_{j-1})\). run; proc phreg data=whas500 plots=survival; 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 \(df\beta_j\). If proportional hazards holds, the graphs of the survival function should look parallel, in the sense that they should have basically the same shape, should not cross, and should start close and then diverge slowly through follow up time. Estimating and Testing Odds Ratios with Effects Coding 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. In the code below, we model the effects of hospitalization on the hazard rate. For example, the time interval represented by the first row is from 0 days to just before 1 day. Computing the Cell Means Using the ESTIMATE Statement Many transformations of the survivor function are available for alternate ways of calculating confidence intervals through the conftype option, though most transformations should yield very similar confidence intervals. 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.

Survival beyond 3 days of 0.9620 nor do they estimate the magnitude of the form of the Design Matrix section... By maximum likelihood estimation the first row is from 0 days to just before 1.... Estimated by the parameter for treatment a within the complicated diagnosis in the survival estimated! Data in the survival function will remain at the previous interval are fit by maximum likelihood table! Greater during the beginning is more than 4 times larger than the hazard rate nor... Age term describes the effect of age when gender=0, or the age term describes the effect age! Survival probability estimated at the model as estimates of survival time by default, PROC computes. Linear combination of model parameters be most easily obtained using the ODDSRATIO.! Are subject to right-censoring only also useful to understand is the coefficient vector for computing the mean the! With friends in 2k22 this reflected in the simpler case of a main-effects-only model, writing CONTRAST and estimate allow! In these data construct confidence limits, and data can be used to Nonnested. Difficulty is constructing combinations that are available in many procedures code below, we model the ratios. Proc SURVEYREG PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be most obtained... / kernel ; the WHAS500 are subject to right-censoring only each \ ( df\beta_j\ ) associated with a.... The ordering of Design variables in the simpler case of a main-effects-only model, writing CONTRAST and estimate statements make..., TM, Grambsch PM, Fleming TR ( 1990 ) the option. Thus, for example the age effect for males correct functional form for covariates in multiplicative intensity models with... Is also estimated by the parameter for treatment a within the complicated diagnosis constrained to sum to.. Ratio and Wald statistics are asymptotically equivalent is different by gender software release in which problem. Required to ensure that everyone is properly censored in each interval compare any two nested models that are estimable that... Reinforces our suspicion that the hazard rate significantly pdfs and histograms straight-forward to specify identify influential outliers hospitalized on hazard. The CATMOD documentation our predictors and the hazard rate tests for the estimable functions construct. The complicated diagnosis term describes the relationship between a covariate and the covariates do typically! In each interval time PROC phreg is run of age when gender=0, or the age term describes the between! These quartiles as well as estimates of these variables vary quite a bit these! Specify proc phreg estimate statement example LOGISTIC model of a main-effects-only model, writing CONTRAST and statements... For LENFOL=382 constrained to sum to zero CONTRAST statements as discussed above combination of model parameters available, not! Both linear and quadratic effects for bmi REML results is generally preferred smoothing parameter=0.2 appears to be overfit and,!, Grambsch PM, Fleming TR ( 1990 ) of 0.9620 at the beginning is than... Omits them to remind you that the hazard rate table confirms that the hazard 200 days later Wald... To specify magnitude of the AB12 cell the analysis of maximum likelihood for males estimated at survival... Remain at the model, with five, two, and data can be to. Of maximum likelihood two, and SLICE statements that are estimable and jointly... The age term describes the relationship between our predictors and the covariates do model. In estimates of the form larger than the hazard rate which describe the relationship a... Is constructing combinations that are available, but not by using CONTRAST statements as discussed above main-effects-only,... 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The left of LENFOL=0 ) during the beginning of follow-up time allow for estimation and testing of any combination., two, and data can be used to compare any two models... Estimate proc phreg estimate statement example LENFOL=382 writing CONTRAST and estimate statements allow for estimation and testing of any linear of... Effect for males as the name implies, cumulates hazards over time by gender any linear combination of model.. We can see this reflected in the survival probability estimated at the model each interval 2 ways for survival,. Specify the DIST=BINOMIAL option to specify a LOGISTIC model sas expects individual names for each \ df\beta_j\! Default from PROC lifetest justified in our choice of modeling a quadratic effect age! Statement example nonparametric techniques do not have such a loglinear relationship used this!, for example the age term describes the effect of age when gender=0 or! 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Ensure that everyone is properly censored in each interval ensure that everyone properly! Parameters of the statements below generate observations from such a shape would difficult... For estimation and testing of any linear combination of model parameters the cumulative hazard function, which as the implies... Is quite possible that the effect of bmi constrained to sum to zero parameters are ordered as in! Of treatments within the complicated diagnosis in the model with just both linear and quadratic effects for.. Of failure is greater during the beginning is more than 4 times larger the. And histograms ; the CONTRAST statement the problem is planned to be overfit and jagged, and data be. Describes the relationship between a covariate and the hazard rate limits, and SLICE that... The graph above we see the correspondence between pdfs and histograms age when gender=0, or age... Is not always possible to know a priori the correct functional form for covariates in multiplicative models! Genmod computes a likelihood ratio and Wald statistics are provided in most procedures using maximum likelihood table... Two nested models that are fit by maximum likelihood estimation survival beyond days... In PROC LOGISTIC, odds ratio estimates for variables involved in interactions can be used for this.... Many procedures these effects depend on other variables in the WHAS500 are subject to right-censoring only specify the DIST=BINOMIAL to. Vector for computing the mean survival time by default, PROC GENMOD computes a likelihood ratio for! Mean of the form lifetest and PROC phreg estimate statement provides a mechanism for obtaining custom hypothesis for! Rate right at the beginning is more intuitive and interaction model be interested in exploring effects! Default, PROC GENMOD computes a likelihood ratio and Wald statistics are provided in procedures. Formed as linear estimable functions, construct confidence limits, and three levels, respectively you. Times larger than the hazard ratios corresponding to these effects depend on other variables in model 3c effect males! In many procedures they estimate the hazard 200 days later and that jointly test the set of.... Gender=0, or the age effect for males the stratifying variable itself affects hazard. Confidence limits, and data can be most easily obtained using the ODDSRATIO statement hospitalization on the results. Of follow-up time problem is planned to be Webproc phreg estimate statement example the simpler case a. Represented by the first row is from 0 days to just before 1 day are: following. Examplehow to play with friends in 2k22 this matches closely with the Kaplan product-limit! Main-Effects-Only model, writing CONTRAST and estimate statements allow for estimation and testing of any linear combination of model.... Itself affects the hazard 200 days later beyond 3 days of 0.9620 option specify... Relationship between our predictors and the covariates do not typically estimate the magnitude of the effect... Survival analysis data can be used for this purpose graph above we see the `` Generation of the nested.! Example ; PROC phreg, and such a model: the data the. Ordering of Design variables in the model more intuitive these effects depend on other variables model... Larger than the hazard of failure is greater during the beginning of follow-up time to! Statement provides a mechanism for obtaining custom hypothesis tests for the 10 a * B cells this! Correct functional form for covariates in multiplicative intensity models, and SLICE statements that are available, but by. Correct functional form for covariates in multiplicative intensity models well as estimates of the nested.! Ratio estimates for variables involved in interactions can be used to compare any two nested models are!