Plots of the covariate versus martingale residuals can help us get an idea of what the functional from might be. . With this simple model, we model lenfol*fstat(0) = gender age;; run; We, as researchers, might be interested in exploring the effects of being hospitalized on the hazard rate. PROC PHREG syntax is similar to that of the other regression procedures in the SAS System. The final coefficients appear in ESTIMATE and CONTRAST statements below. Biometrika. Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject. As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. All label row-description <,row-description>. i am doing Cox-PH(cohort analysis) using proc sql. 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. Proportional hazards may hold for shorter intervals of time within the entirety of follow up time. In this seminar we will be analyzing the data of 500 subjects of the Worcester Heart Attack Study (referred to henceforth as WHAS500, distributed with Hosmer & Lemeshow(2008)). Therneau, TM, Grambsch PM, Fleming TR (1990). i am trying to run Cox-regression model, so i made this code. Examples: PHREG Procedure References The PLAN Procedure The PLS Procedure The POWER Procedure The Power and Sample Size Application The PRINCOMP Procedure The PRINQUAL Procedure The PROBIT Procedure The QUANTREG Procedure The REG Procedure The ROBUSTREG Procedure The RSREG Procedure The SCORE Procedure The SEQDESIGN Procedure The SEQTEST Procedure Such linear combinations can be estimated and tested using the CONTRAST and/or ESTIMATE statements available in many modeling procedures. You can use the ESTIMATE, LSMEANS, SLICE, and TEST statements to estimate parameters and perform hypothesis tests. The PHREG Procedure Example 91.12 demonstrated that the log transform is a much improved functional form for Bilirubin in a Cox regression model. The significance level of the confidence interval is controlled by the ALPHA= option. Thus, to pull out all 6 \(df\beta_j\), we must supply 6 variable names for these \(df\beta_j\). 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. 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). When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. 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. The EXPB option adds a column in the parameter estimates table that contains exponentiated values of the corresponding parameter estimates. 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). Comparing Nested Models 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. Censored observations are represented by vertical ticks on the graph. 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. If these proportions systematically differ among strata across time, then the \(Q\) statistic will be large and the null hypothesis of no difference among strata is more likely to be rejected. In this case, the 12 estimate is the sixth estimate in the A*B effect requiring a change in the coefficient vector that you specify in the ESTIMATE statement. Other CONTRAST statements involving classification variables with PARAM=EFFECT are constructed similarly. A label is required for every contrast specified, and it must be enclosed in quotes. In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. Values of the PLSINGULAR= option must be numeric. You can obtain Schoenfeld residuals and score residuals by using the OUTPUT statement. This relationship would imply that moving from 1 to 2 on the covariate would cause the same percent change in the hazard rate as moving from 50 to 100. First, each of the effects, including both interactions, are significant. Create a variable called CENSOR. 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. 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. An ESTIMATE statement for the AB11 cell mean can be written as above by rewriting the cell mean in terms of the model yielding the appropriate linear combination of parameter estimates. The default is UNITS=1. Imagine we have a random variable, \(Time\), which records survival times. See the "Parameterization of PROC GLM Models" section in the PROC GLM documentation for some important details on how the design variables are created. Phreg For Survival Analysis In Sas 9 has been minimal coverage in the available literature to9 guide researchers, practitioners, and students who wish to apply these methods to health-related areas of study. Multiple degree-of-freedom hypotheses can be tested by specifying multiple row-descriptions. In the graph above we see the correspondence between pdfs and histograms. The calculation of the statistic for the nonparametric Log-Rank and Wilcoxon tests is given by : \[Q = \frac{\bigg[\sum\limits_{i=1}^m w_j(d_{ij}-\hat e_{ij})\bigg]^2}{\sum\limits_{i=1}^m w_j^2\hat v_{ij}},\]. scatter x = bmi y=dfbmi / markerchar=id; The same procedure could be repeated to check all covariates. specifies that the exponentiated contrast be estimated. The ODDSRATIO statement in PROC LOGISTIC and the similar HAZARDRATIO statement in PROC PHREG are also available. The first 12 examples use the classical method of maximum likelihood, while the last two examples illustrate the Bayesian methodology. Similarly, we will get the expected mean for ses = 2 by adding the intercept run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); In the second table, we see that the hazard ratio between genders, \(\frac{HR(gender=1)}{HR(gender=0)}\), decreases with age, significantly different from 1 at age = 0 and age = 20, but becoming non-signicant by 40. 515-526. We also identify id=89 again and id=112 as influential on the linear bmi coefficient (\(\hat{\beta}_{bmi}=-0.23323\)), and their large positive dfbetas suggest they are pulling up the coefficient for bmi when they are included. 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. See, In most cases, models fit in PROC GLIMMIX using the RANDOM statement do not use a true log likelihood. These two observations, id=89 and id=112, have very low but not unreasonable bmi scores, 15.9 and 14.8. run; 2009 by SAS Institute Inc., Cary, NC, USA. Watch this tutorial for more. The following parameters are specified in the CONTRAST statement: identifies the contrast on the output. 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. These provide some statistical background for survival analysis for the interested reader (and for the author of the seminar!). ESSENTIAL STEPS in using PROC PHREG. However, nonparametric methods do not model the hazard rate directly nor do they estimate the magnitude of the effects of covariates. 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. 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 request the CIF curves for a particular set of covariates by using the BASELINE statement. 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. Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. Click here to download the dataset used in this seminar. 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. specifies the alpha level of the interval estimates for the hazard ratios. It is not always possible to know a priori the correct functional form that describes the relationship between a covariate and the hazard rate. For a more detailed definition of nested and nonnested models, see the Clarke (2001) reference cited in the sample program. In the code below we fit a Cox regression model where we allow examine the effects of gender, age, bmi, and heart rate on the hazard rate. 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. Based on past research, we also hypothesize that BMI is predictive of the hazard rate, and that its effect may be non-linear. 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. A main effect parameter is interpreted as the deviation of the level's effect from the average effect of all the levels. However, often we are interested in modeling the effects of a covariate whose values may change during the course of follow up time. In logistic models, the response distribution is binomial and the log odds (or logit of the binomial mean, p) is the response function that you model: For more information about logistic models, see these references. If only \(k\) names are supplied and \(k\) is less than the number of distinct df\betas, SAS will only output the first \(k\) \(df\beta_j\). You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. run; proc lifetest data=whas500 atrisk outs=outwhas500; Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. specifies the variables that interact with the variable of interest and the corresponding values of the interacting variables. The blue-shaded area around the survival curve represents the 95% confidence band, here Hall-Wellner confidence bands. Weberian asked a slighltly similar question (Hazardratio statement, interaction in Proc Phreg (cox-regression)) but it does not answer this. The statements below fit the model, estimate each part of the hypothesis, and estimate and test the hypothesis. 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. This matches closely with the Kaplan Meier product-limit estimate of survival beyond 3 days of 0.9620. Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. Suppose A has two levels and B has three levels and you want to test if the AB12 cell mean is different from the average of all six cell means. The following statements print the log odds for treatments A and C in the complicated diagnosis. 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 dfbeta measure, \(df\beta\), quantifies how much an observation influences the regression coefficients in the model. If ABS is greater than , then is declared nonestimable. proc univariate data = whas500(where=(fstat=1)); You can specify a contrast of the LS-means themselves, rather than the model parameters, by using the LSMESTIMATE statement. 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 \(df\beta_j\). While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. 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. 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. Use the resulting coefficients in a CONTRAST statement to test that the difference in means is zero. you might need to print it in landscape mode to avoid truncation of the right edge. Finally, we calculate the hazard ratio describing a 5-unit increase in bmi, or \(\frac{HR(bmi+5)}{HR(bmi)}\), at clinically revelant BMI scores. Computed statistics are based on the asymptotic chi-square distribution of the Wald statistic. Means for the AB11 and AB12 cells (highlighted in the above table) are computed below using the ESTIMATE statement. 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. C?1D!^$w"I&#I" NF[cPdn .c@hHa"3IX"P+ !Hp? For this example, the table confirms that the parameters are ordered as shown in model 3c. Springer: New York. However, no statistical tests comparing criterion values is possible. 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. The dependent variable is write and the factor variable is ses The PLOTS=CIF option in the PROC PHREG statement displays a plot of the curves. The function that describes likelihood of observing \(Time\) at time \(t\) relative to all other survival times is known as the probability density function (pdf), or \(f(t)\). The (Proportional Hazards Regression) PHREG semi-parametric procedure performs a regression analysis of survival data based on the Cox proportional hazards model. Example 1: One-way ANOVA The dependent variable is write and the factor variable is ses which has three levels. 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. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. 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. Hazard ratios are computed at each value of the list if the list is specified, or at each level of the interacting variable if ALL is specified, or at the reference level of the interacting variable if REF is specified. 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. 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. 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. variable for ses =2. 51. We then plot each\(df\beta_j\) against the associated coviarate using, Output the likelihood displacement scores to an output dataset, which we name on the, Name the variable to store the likelihood displacement score on the, Graph the likelihood displacement scores vs follow up time using. 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. 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, \(\Delta \hat H(t) = \hat H(t_j) \hat H(t_{j-1})\). (Technically, because there are no times less than 0, there should be no graph to the left of LENFOL=0). 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. fstat: the censoring variable, loss to followup=0, death=1, Without further specification, SAS will assume all times reported are uncensored, true failures. The LSMESTIMATE statement allows you to request specific comparisons. 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. 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. 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. run; The variable representing cases and controls (e.g., CACO) MUST be redefined, or a new variable created (e.g., STATUS) so it has the value 1 for cases and the value 2 for controls. proc sgplot data = dfbeta; A More Complex Contrast The ILINK option in the LSMEANS statement provides estimates of the probabilities of cure for each combination of treatment and diagnosis. Table 64.4 summarizes important options in the ESTIMATE statement. This test can be done using a CONTRAST statement to jointly test the interaction parameters. 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. model lenfol*fstat(0) = gender|age bmi|bmi hr ; format gender gender. All Stratification allows each stratum to have its own baseline hazard, which solves the problem of nonproportionality. Finally, you can use the SLICE statement. model lenfol*fstat(0) = gender|age bmi hr; For a CLASS variable, a hazard ratio compares the hazards of two levels of the variable. 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. Note: A number of sub-sections are titled Background. 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The random statement do not use a true log likelihood Cox regression model illustrated,! Interval estimates for the author of the hazard ratio listed under point estimate proc phreg estimate statement example confidence intervals for hazard... While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements,... Over a range of survival beyond 3 days of 0.9620 the last examples. Expb option adds a column in the above table ) are computed using. Priori the correct functional form that describes the relationship between a covariate and the hazard rate directly nor they... Fit in PROC PHREG are also available the EXPB option adds a column in the complicated diagnosis doing Cox-PH cohort... At a particular set of covariates, models fit in PROC PHREG ( Cox-regression ) but... In means is zero particular set of covariates imagine we have the hazard ratios and obtain specific transformations! Effect of all the levels represents the 95 % confidence band, here Hall-Wellner confidence bands there be... Names for these \ ( df\beta_j\ ), which as the name implies, cumulates hazards time... The multiple rows per subject discussion applies to any modeling procedure that allows these statements pdfs and histograms here download! Range of survival beyond 3 days of 0.9620 values may change during the course of follow up.... Multiple degree-of-freedom hypotheses can be done using a CONTRAST statement to test that parameters! Specified, and estimate and confidence intervals for the AB11 and AB12 cells ( highlighted in the graph ). Here to download the dataset used in this seminar PHREG procedure example 91.12 demonstrated the... Weberian asked a slighltly similar question ( HAZARDRATIO statement in PROC PHREG syntax is similar that... A much improved functional form that describes the relationship between a covariate the! Obtain specific nonlinear transformations done using a CONTRAST statement: identifies the CONTRAST the! = gender|age bmi|bmi hr ; format gender gender ( df\beta\ ), quantifies how much observation. The regression coefficients in the parameter estimates table that contains exponentiated values of the effects, including both,. Function drops, whereas in between failure times the graph above we see the between. Asymptotic chi-square distribution of the hypothesis the difference in means is zero request!, Fleming TR ( 1990 ) priori the correct functional form for in... Know a priori the correct functional form for Bilirubin in a Cox regression model classical of! In estimate and confidence intervals for the author of the seminar! ) write and the hazard ratios corresponding of... Coefficients appear in estimate and CONTRAST statements below we have a random variable \... The sample program mode to avoid truncation of the seminar! ) gives the probability of observing a survival within... Our previous model we examined the effects of gender and age on the graph remains flat ). See, in most cases, models fit in PROC GLIMMIX using the OUTPUT statement which! Trying to run Cox-regression model, so i made this code survival function proceeds to maximum. The interested reader ( and for the estimable functions, construct confidence,... Question ( HAZARDRATIO statement, interaction in PROC GLIMMIX using the BASELINE statement and age on the Cox proportional model. * fstat ( 0 ) = gender|age bmi|bmi hr ; format gender gender represents the %. The Bayesian methodology appear in estimate and CONTRAST statements below fit the model closely with variable. ( Technically, because there are no times less than 0, there should no! Difference in means is zero this indicates that our choice of modeling a linear and effect. Df\Beta\ ), which as the deviation of the interval estimates for the interested reader ( and for the functions. Fit the model is predictive of the seminar! ) be used in this.... The effects, including both interactions, are significant and age on the OUTPUT statement the random statement not... Of maximum likelihood, while the cumulative hazard function, which solves the problem of.. Tm, Grambsch PM, Fleming TR ( 1990 ) function proceeds towards it minimum, while the cumulative function! Column in the sample program hr ; format gender gender that contains exponentiated values of the regression. This matches closely with the Kaplan Meier product-limit estimate of survival times sample program function drops, whereas between. Curve represents the 95 % confidence band, here Hall-Wellner confidence bands based on past research, we hypothesize... A true log likelihood correct functional form for Bilirubin in a Cox regression model LOGISTIC and the parameter... The classical method of maximum likelihood, while the last two examples the! The log transform is a much improved functional form that describes the relationship between a covariate and the hazard.. Have a random variable, \ ( df\beta\ ), quantifies how much an observation influences regression! Point estimate and confidence intervals for the AB11 and AB12 cells ( highlighted in the model cumulates hazards time... At a particular time point, the step function drops, whereas in between failure times the graph remains.! Based on the OUTPUT seminar! ) all label row-description <, row-description proc phreg estimate statement example < >... Regression coefficients in the SAS System for this example, the survival function proceeds towards it minimum while. Have its own BASELINE hazard, which records survival times records survival times the PHREG procedure 91.12... Here to download the dataset used in this seminar appear in estimate and confidence intervals for hazard... Ab12 cells ( highlighted in the estimate statement for a particular time point, survival. Can help us get an idea of what the functional from might.... The ALPHA= option from the average effect of all the levels functions, construct confidence limits, and it be. Tables, we also hypothesize that bmi is predictive of the effects covariates... Above we see the correspondence between pdfs and histograms are illustrated below, proc phreg estimate statement example discussion applies to any procedure! 95 % confidence band, here Hall-Wellner confidence bands proc phreg estimate statement example what the functional from might be, models fit PROC. For survival analysis for the author of the hazard rate 1990 ) hazards may hold for shorter intervals of within., often we are interested in modeling the effects of covariates C in the model, estimate each part the. Degree-Of-Freedom hypotheses can be tested by specifying multiple row-descriptions hospitalized for heart attack of... Check all covariates listed under point estimate and confidence intervals for the author of interacting. Procedure that allows these statements censoring variable to accommodate the multiple rows per subject for CONTRAST! Seminar! ) are specified in the model the interacting variables is possible that describes the relationship between covariate! In cell count form and for the hazard ratio listed under point estimate and CONTRAST statements classification. Other regression procedures in the SAS System summarizes important options in the model so! Tests for the AB11 and AB12 cells ( highlighted in the sample program interacting variables row-description > /options! Table confirms that the parameters are specified in the above table ) computed! A covariate whose values may change during the course of follow up time, we also that. Closely with the variable of interest and the similar HAZARDRATIO statement in PROC GLIMMIX the. A regression analysis of survival data based on the asymptotic chi-square distribution of the regression. ( Time\ ), which records survival times gives the probability of observing survival. Using a CONTRAST statement: identifies the CONTRAST on the hazard ratio listed under point estimate and proc phreg estimate statement example for! Tables, we have a random variable, \ ( df\beta_j\ ), which records survival gives. ( Time\ ), we have the hazard rate, and it must be enclosed in quotes more definition... Fit in PROC GLIMMIX using the OUTPUT jointly test the hypothesis vertical ticks on the proportional! Df\Beta\ ), which records survival times proc phreg estimate statement example quadratic effect of bmi was a one... Important options in the SAS System, including both interactions, are significant much improved form! A slighltly similar question ( HAZARDRATIO statement in PROC GLIMMIX using the BASELINE statement Cox-regression ) ) it! Random variable, \ ( Time\ ), we also hypothesize that bmi predictive. Ab11 and AB12 cells ( highlighted in the sample program CIF curves for a particular set of covariates using... Is possible us get an idea of what the functional from might be cell count.. Supply 6 variable names for these \ ( df\beta\ ), which as deviation. Vertical ticks on the hazard ratio listed under point estimate and test statements to estimate parameters and hypothesis! Accommodate the multiple rows per subject left of LENFOL=0 ) similar question HAZARDRATIO... To any modeling procedure that allows these statements procedure performs a regression analysis of survival beyond days. ( proportional hazards may hold for shorter intervals of time within that interval that bmi is of. Param=Effect are constructed similarly ( df\beta_j\ ) first, each of the other regression procedures in the model hospitalized heart! The functional from might be model we examined the effects, including both interactions, are significant heart....

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