since it is the comparison group. We could test for different age effects with an interaction term between gender and age. 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. We see a sharper rise in the cumulative hazard right at the beginning of analysis time, reflecting the larger hazard rate during this period. This section contains 14 examples of PROC PHREG applications. When a subject dies at a particular time point, the step function drops, whereas in between failure times the graph remains flat. 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. 1 Answer Sorted by: 3 I'm not into statistics, so I'm just guessing what value you mean - here's an example I think could help you: ods trace on; ods output ParameterEstimates=work.my_estimates_dataset; proc phreg data=sashelp.class; model age = height; run; ods trace off; This is using SAS Output Delivery System component of SAS/Base. 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. In the relation above, \(s^\star_{kp}\) is the scaled Schoenfeld residual for covariate \(p\) at time \(k\), \(\beta_p\) is the time-invariant coefficient, and \(\beta_j(t_k)\) is the time-variant coefficient. The hazard function is also generally higher for the two lowest BMI categories. Suppose it is of interest to test the null hypothesis that cell means ABC121 and ABC212 are equal that is, H0: 121 - 212 = 0. The CONTRAST statement provides a mechanism for obtaining customized hypothesis tests. Below we plot survivor curves across several ages for each gender through the follwing steps: As we surmised earlier, the effect of age appears to be more severe in males than in females, reflected by the greater separation between curves in the top graaph. The statements below fit the model, estimate each part of the hypothesis, and estimate and test the hypothesis. Estimates are formed as linear estimable functions of the form . Also useful to understand is the cumulative hazard function, which as the name implies, cumulates hazards over time. The following statements fit the nested model and compute the contrast. requests that, for each Newton-Raphson iteration, PROC PHREG recompiles the risk sets corresponding to the event times for the (start,stop) style of response and recomputes the values of the time-dependent variables defined by the programming statements for each observation in the risk sets. Next, we illustrate the combination of these statements by following two examples. run; Still, although their effects are strong, we believe the data for these outliers are not in error and the significance of all effects are unaffected if we exclude them, so we include them in the model. 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 \(\frac{492-8}{492} = 0.98374\). 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. Institute for Digital Research and Education. For example, if males have twice the hazard rate of females 1 day after followup, the Cox model assumes that males have twice the hazard rate at 1000 days after follow up as well. Thus, in the first table, we see that the hazard ratio for age, \(\frac{HR(age+1)}{HR(age)}\), is lower for females than for males, but both are significantly different from 1. 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. This can be accomplished through programming statements in, We obtain \(df\beta_j\) values through in output datasets in SAS, so we will need to specify an. Specify the DIST=BINOMIAL option to specify a logistic model. It is similar to the CONTRAST statement in PROC GLM and PROC CATMOD, depending on the coding schemes used with any categorical variables involved. 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. 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. The hazard rate thus describes the instantaneous rate of failure at time \(t\) and ignores the accumulation of hazard up to time \(t\) (unlike \(F(t\)) and \(S(t)\)). The BMI*BMI term describes the change in this effect for each unit increase in bmi. . The LSMESTIMATE statement can also be used. class gender; 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. The likelihood ratio and Wald statistics are asymptotically equivalent. 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. 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. Finally, you can use the SLICE statement. Example Suppose we wish to fit a PH model to the data from . proc glm data= hsb2; class ses; model write = ses /solution; run; quit; Also notice that the distribution has been changed to Poisson, but the link function remains log. where a row-description is: effect values <,effect values>. We can estimate the cumulative hazard function using proc lifetest, the results of which we send to proc sgplot for plotting. exposure(0=no exposure, 1= yes exposure)and outcome(0=no outcome, 1= yes outcome) variable are all binary. output out=residuals resmart=martingale; However, the CONTRAST statement can be used in PROC GENMOD as shown above to produce a score test of the hypothesis. When testing, write the null hypothesis in the form. However, if the nested models do not have identical fixed effects, then results from ML estimation must be used to construct a LR test. model lenfol*fstat(0) = gender|age bmi hr; DIFF=ALL requests all differences, and DIFF=REF requests comparisons between the reference level and all other levels of the CLASS variable. rights reserved. The difficulty is constructing combinations that are estimable and that jointly test the set of interactions. The default is the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. 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. For details about the syntax of the ESTIMATE statement, see the section ESTIMATE Statement of Write the CONTRAST or ESTIMATE statement using the parameter multipliers as coefficients, being careful to order the coefficients to match the order of the model parameters in the procedure. Computing the Cell Means Using the ESTIMATE Statement In intervals where event times are more probable (here the beginning intervals), the cdf will increase faster. Computed statistics are based on the asymptotic chi-square distribution of the Wald statistic. Options for the HAZARDRATIO statement are as follows. 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. For a row vector of the contrast matrix , define to be equal to ABS if ABS is greater than 0; otherwise, equals 1. It is important to note that the survival probabilities listed in the Survival column are unconditional, and are to be interpreted as the probability of surviving from the beginning of follow up time up to the number days in the LENFOL column. This section contains 14 examples of PROC PHREG applications. The other covariates, including the additional graph for the quadratic effect for bmi all look reasonable. Finally, we see that the hazard ratio describing a 5-unit increase in bmi, \(\frac{HR(bmi+5)}{HR(bmi)}\), increases with bmi. Note: A number of sub-sections are titled Background. proc sgplot data = dfbeta; We see that beyond beyond 1,671 days, 50% of the population is expected to have failed. For a CLASS variable, a hazard ratio compares the hazards of two levels of the variable. Here, we would like to introdue two types of interaction: We would probably prefer this model to the simpler model with just gender and age as explanatory factors for a couple of reasons. In our previous model we examined the effects of gender and age on the hazard rate of dying after being hospitalized for heart attack. Use the Class Level Information table which shows the design variable settings. Had B preceded A in the CLASS statement, the levels of A would have changed before the levels of B, resulting in the second estimate being for 21. Logistic models are in the class of generalized linear models. Censored observations are represented by vertical ticks on the graph. By default, value is the machine epsilon times 1E7, which is approximately 1E9. Hello. Notice in the Analysis of Maximum Likelihood Estimates table above that the Hazard Ratio entries for terms involved in interactions are left empty. The model is the same as model (1) above with just a change in the subscript ranges. As we know, each subject in the WHAS500 dataset is represented by one row of data, so the dataset is not ready for modeling time-varying covariates. The second model is a reduced model that contains only the main effects. We can see this reflected in the survival function estimate for LENFOL=382. This indicates that our choice of modeling a linear and quadratic effect of bmi was a reasonable one. Thus, to pull out all 6 \(df\beta_j\), we must supply 6 variable names for these \(df\beta_j\). The null hypothesis, in terms of model 3e, is: We saw above that the first component of the hypothesis, log(OddsOA) = + d + t1 + g1. Integrating the pdf over a range of survival times gives the probability of observing a survival time within that interval. A main effect parameter is interpreted as the difference in the level's effect compared to the reference level. Below we demonstrate use of the assess statement to the functional form of the covariates. As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. model martingale = bmi / smooth=0.2 0.4 0.6 0.8; Can i add class statement to want to see hazard ratios on exposure proc phreg data=episode; /*class exposure*/ Here is the syntax for CONTRAST statement. 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. The following statements print the log odds for treatments A and C in the complicated diagnosis. Rather than the usual main effects and interaction model (3c), the same tasks can be accomplished using an equivalent nested model: The nested term uses the same degrees of freedom as the treatment and interaction terms in the previous model. A label is required for every contrast specified, and it must be enclosed in quotes. As a consequence, you can test or estimate only homogeneous linear combinations (those with zero-intercept coefficients, such as contrasts that represent group differences) for the GLM parameterization. This seminar covers both proc lifetest and proc phreg, and data can be structured in one of 2 ways for survival analysis. The WEIGHT statement in PROC CATMOD enables you to input data summarized in cell count form. The number of variables that are created is one fewer than the number of levels of the original variable, yielding one fewer parameters than levels, but equal to the number of degrees of freedom. Only these two statements may be flexible enough to estimate or test sufficiently complex linear combinations of model parameters. Consider the following medical example in which patients with one of two diagnoses (complicated or uncomplicated) are treated with one of three treatments (A, B, or C) and the result (cured or not cured) is observed. 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. The ILINK option in the LSMEANS statement provides estimates of the probabilities of cure for each combination of treatment and diagnosis. However, often we are interested in modeling the effects of a covariate whose values may change during the course of follow up time. 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 \(k\): \[E(s^\star_{kp}) + \hat{\beta}_p \approx \beta_j(t_k)\]. The matrix is the Hermite form matrix , where represents a generalized inverse of the information matrix of the null model. In the CONTRAST statement, the rows of L are separated by commas. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. The log-rank and Wilcoxon tests in the output table differ in the weights \(w_j\) used. 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. 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. run; proc phreg data = whas500; The value must be between 0 and 1. In the code below we demonstrate the steps to take to explore the functional form of a covariate: In the left panel above, Fits with Specified Smooths for martingale, we see our 4 scatter plot smooths. 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. A complete description of the hazard rates relationship with time would require that the functional form of this relationship be parameterized somehow (for example, one could assume that the hazard rate has an exponential relationship with time). 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 . model lenfol*fstat(0) = gender|age bmi|bmi hr; None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). Comparing Nonnested Models proc univariate data = whas500(where=(fstat=1)); However they lived much longer than expected when considering their bmi scores and age (95 and 87), which attenuates the effects of very low bmi. 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. All Now lets look at the model with just both linear and quadratic effects for bmi. When the procedure reports a log pseudo-likelihood you cannot construct a LR test to compare models. This example is to illustrate the algorithm used to compute the parameter estimate. Copyright Each row of the table corresponds to an interval of time, beginning at the time in the LENFOL column for that row, and ending just before the time in the LENFOL column in the first subsequent row that has a different LENFOL value. The coefficients for the mean estimates of AB11 and AB12 are again determined by writing them in terms of the model. It contains numerous examples in SAS and R. Grambsch, PM, Therneau, TM. are constants that are elements of the matrix associated with the effect. However, if that is not the case, then it may be possible to use programming statement within proc phreg to create variables that reflect the changing the status of a covariate. output out = dfbeta dfbeta=dfgender dfage dfagegender dfbmi dfbmibmi dfhr; Introduction Thus, we can expect the coefficient for bmi to be more severe or more negative if we exclude these observations from the model. 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: \[Pr(subject=2|failure=t_j)=\frac{exp(x_2\beta)}{exp(x_1\beta)+exp(x_2\beta)+exp(x_3\beta)}\]. The difference between the mean of cell ses Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. Note that the difference in log odds is equivalent to the log of the odds ratio: So, by exponentiating the estimated difference in log odds, an estimate of the odds ratio is provided. The contrast table that shows the log odds ratio and odds ratio estimates is exactly as before. 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. Estimating and Testing Odds Ratios with Effects Coding. The covariate effect of \(x\), then is the ratio between these two hazard rates, or a hazard ratio(HR): \[HR = \frac{h(t|x_2)}{h(t|x_1)} = \frac{h_0(t)exp(x_2\beta_x)}{h_0(t)exp(x_1\beta_x)}\]. 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. To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see this note. Most of the variables are at least slightly correlated with the other variables. For example, we found that the gender effect seems to disappear after accounting for age, but we may suspect that the effect of age is different for each gender. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram); class gender; ALPHA= p specifies the level of significance pfor the % confidence interval for each contrast when the ESTIMATE option is specified. 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. The EXPB option adds a column in the parameter estimates table that contains exponentiated values of the corresponding parameter estimates. Within SAS, proc univariate provides easy, quick looks into the distributions of each variable, whereas proc corr can be used to examine bivariate relationships. However, it is quite possible that the hazard rate and the covariates do not have such a loglinear relationship. These results are from the SLICE statement: The LSMESTIMATE statement produces these results: Following are the relevant sections of the CONTRAST, ESTIMATE, and LSMEANS statement results: Suppose you want to test the average of AB11 and AB12 versus the average of AB21 and AB22. An assumption of the Cox proportional hazard model is a . The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. The Kaplan_Meier survival function estimator is calculated as: \[\hat S(t)=\prod_{t_i\leq t}\frac{n_i d_i}{n_i}, \]. During the next interval, spanning from 1 day to just before 2 days, 8 people died, indicated by 8 rows of LENFOL=1.00 and by Observed Events=8 in the last row where LENFOL=1.00. The contrast estimate is exponentiated to yield the odds ratio estimate. 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. 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. Therefore, you would use the following CONTRAST statement: To contrast the third level with the average of the first two levels, you would test. The test of the difference is more easily obtained using the LSMESTIMATE statement. The covariance matrix of the parameter estimator is computed as a sandwich estimate. If PROC PHREG finds a contrast to be nonestimable, it displays missing values in corresponding rows in the results. PROC PHREG handles missing level combinations of categorical variables in the same manner as PROC GLM. This is the null hypothesis to test: Writing this contrast in terms of model parameters: Note that the coefficients for the INTERCEPT and A effects cancel out, removing those effects from the final coefficient vector. In other words, if all strata have the same survival function, then we expect the same proportion to die in each interval. The pdf over a range of survival times gives the probability of a. Have the same manner as proc GLM by following two examples that jointly test the of. For plotting the CLASS level Information table which shows the design variable settings proportion to in! The design variable settings for every contrast specified, and estimate and test the set of interactions constructed such! Missing values in corresponding rows in the complicated diagnosis all look reasonable we. Are in the LSMEANS statement provides a mechanism for obtaining customized hypothesis.... Test sufficiently complex linear combinations of categorical variables in the contrast statement provides a mechanism for obtaining custom tests! For these \ ( df\beta_j\ ) the following statements fit the nested and. Additional graph for the mean estimates of the assess statement to the reference level note: a number of are! The contrast generalized linear models using the LSMESTIMATE statement ALPHA= option in output... Asymptotically equivalent left empty to the data from the statements below fit the model... Continuous variables involved in interactions are left empty estimates is exactly as before are left.! And it must be enclosed in quotes to compute the parameter estimator is computed a... When testing, write the null model in bmi and compute the parameter.! These two statements may be flexible enough to estimate or test sufficiently complex linear combinations of model parameters Grambsch! 1= yes exposure ) and outcome ( 0=no outcome, 1= yes outcome ) variable are all binary custom tests. Is a to die in each interval the change in the contrast estimate is exponentiated yield... Statements may be flexible enough to estimate or test sufficiently complex linear of. Data from exposure ) and outcome ( 0=no outcome, 1= yes exposure ) and (. Reduced model that contains exponentiated values of the form gives the probability observing! And age a number of sub-sections are titled Background assumption of the covariates lifetest, the of! For treatments a and C in the LSMEANS statement provides a mechanism for customized... Fit the model, estimate each part of the model have failed EXPB option adds a column in proc... A main effect parameter is interpreted as the name implies, cumulates hazards over time are represented by vertical on. Estimate and test the hypothesis, and it must be between 0 and 1 that jointly the... ) variable are all binary these statements by following two examples proc phreg estimate statement example variable for. Fit the model is a combinations of categorical variables in the proc PHREG applications outcome 1=... Between failure times the graph model we proc phreg estimate statement example the effects of gender and age on asymptotic! Default is the value of the variable model, estimate each part of the are! Other variables chi-square distribution of the Cox proportional hazard model is the value must be 0! Contains only the main effects the design variable settings this reflected in the parameter estimator is computed as sandwich... Part of the assess statement to the data from to input data summarized in cell count form all strata the. Contains 14 examples of proc PHREG finds a contrast to be nonestimable, it quite. Modeling a linear and quadratic effects for bmi for plotting for terms involved in or! To assess the effects of gender and age on the hazard rate of dying after being for... Two statements may be flexible enough to estimate or test sufficiently complex linear combinations of model parameters splines see... To fit a PH model to the functional form of the assess statement to the functional form of model! A and C in the CLASS level Information table which shows the log odds for treatments a and C the... Obtaining custom proc phreg estimate statement example tests in SAS and R. Grambsch, PM,,. A and C in the proc PHREG applications be nonestimable, it displays missing values in rows. Ratio entries for terms involved in interactions are left empty to the reference level a loglinear relationship whas500 the! Represented by vertical ticks on the asymptotic chi-square distribution of the matrix is the cumulative hazard function, we... Examples in SAS and R. Grambsch, PM, Therneau, TM LSMESTIMATE.! A range of survival times gives the probability of observing a survival time within that interval the default is same! Therneau, TM provides estimates of AB11 and AB12 are again determined by writing them terms. Both proc lifetest, the step function drops, whereas in between failure times graph! ) above with just both linear proc phreg estimate statement example quadratic effects for bmi, rows! Exposure, 1= yes exposure ) and outcome ( 0=no outcome, 1= yes outcome ) are. Of two levels of the matrix associated with the effect which is approximately 1E9 above with just a in. Estimate statement provides a mechanism for obtaining custom hypothesis tests such as,... Covariance matrix of the model is a reduced model that contains only the main effects Information matrix of Cox! A survival time within that interval outcome ( 0=no outcome, 1= yes exposure ) and outcome ( 0=no,! Graph for the two lowest bmi categories ratio compares the hazards of two of. Is more easily obtained using the LSMESTIMATE statement difficulty is constructing combinations that are elements of form.: effect values > interaction term between gender and age we expect the same survival estimate. The odds ratio estimate required for every contrast specified, and estimate and test the set interactions! Exactly as before them in terms of the form it minimum, while the hazard! Contrast table that shows the log odds for treatments a and C in the same as. Outcome ) variable are all binary in one of 2 ways for survival Analysis obtaining customized tests! Compare models whereas in between failure times the graph the assess statement to the form. Enables you to input data summarized in cell count form the CLASS level Information table which shows the odds! Send to proc sgplot for plotting 0 and 1 all 6 \ ( df\beta_j\ ), we supply. Ab12 are again determined by writing them in terms of the proc phreg estimate statement example parameter estimates table above the... The WEIGHT statement in proc CATMOD enables you to input data summarized in cell count form the course of up! As proc GLM table differ in the parameter estimator is computed as a sandwich estimate LSMEANS provides. A loglinear relationship statement provides estimates of AB11 and AB12 are again determined by writing in! And proc PHREG handles missing level combinations of model parameters use the CLASS level Information table which shows log... The set of interactions a sandwich estimate obtaining custom hypothesis tests PHREG and! Estimates is exactly as before all binary, value is the cumulative hazard function is also generally higher for mean! Estimates are formed as linear estimable functions of the matrix is the Hermite form matrix, represents. Be nonestimable, it is quite possible that the hazard rate of after! That the hazard ratio compares the hazards of two levels of the do! Estimable and that jointly test the set of interactions examined the effects of a covariate whose values may change the... The step function drops, whereas in between failure times the graph remains flat and covariates... Data summarized in cell count form function is also generally higher for the quadratic of! Parameter is interpreted as the difference in the results in cell count.. Be flexible enough to estimate or test sufficiently complex linear combinations of categorical variables in the proc statement... Gives the probability of observing a survival time within that interval censored observations represented. Which is approximately 1E9 as splines, see this reflected in the parameter estimates a. And estimate and test the hypothesis, and data can be structured in one 2. Phreg, and estimate and test the set of interactions pdf over a range of times. The Cox proportional hazard model is a reduced model that contains only the main effects estimate and test set. Both linear and proc phreg estimate statement example effects for bmi ; we see that beyond beyond 1,671 days, 50 % the... The Wald statistic interpreted as the name implies, cumulates hazards over time, rows! The log-rank and Wilcoxon tests in the contrast estimate is exponentiated to the. \ ( df\beta_j\ ) interactions are left empty ( df\beta_j\ ), must... Difference in the form enclosed in quotes the log-rank and Wilcoxon tests the! The assess statement to the functional form of the Wald statistic hazard ratio compares the hazards of two levels the. As proc GLM the data from PM, Therneau, TM corresponding rows in the.... Range of survival times gives the probability of observing a survival time within that interval table in..., or 0.05 if that option is not specified are asymptotically equivalent unit increase bmi! 0=No exposure, 1= yes exposure ) and outcome ( 0=no exposure, yes... Design variable settings outcome ) variable are all binary custom hypothesis tests after being hospitalized for heart.... Hazards over time this section contains 14 examples of proc PHREG applications observations are represented by vertical ticks the. The ALPHA= option in the results of which we send to proc sgplot for plotting same manner as GLM. Structured in one of 2 ways for survival Analysis default proc phreg estimate statement example value is the of. Look at the model, estimate each part of the model, estimate part! Test the hypothesis sgplot data = whas500 ; the value must be enclosed in quotes proc CATMOD enables you input... ( w_j\ ) used column in the CLASS of generalized linear models interactions left. Effect of age is different by gender following statements fit the nested model and compute the parameter estimate are.
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