Exponentiating this value (exp[.63363] = 1.8845) yields the exponentiated contrast value (the odds ratio estimate) from the CONTRAST statement. /*class exposure*/model period*outcome(0)=exposure / rl;run; Hello@MTeckand welcome to the SAS Support Communities! In the code below, we model the effects of hospitalization on the hazard rate. tunes the estimability check. Biometrics. SAS provides built-in methods for evaluating the functional form of covariates through its assess statement. While only certain procedures are illustrated below, this discussion applies to any modeling procedure that allows these statements. Confidence intervals that do not include the value 1 imply that hazard ratio is significantly different from 1 (and that the log hazard rate change is significanlty different from 0). But the nested term makes it more obvious that you are contrasting levels of treatment within each level of diagnosis. If convergence is not attained in n iterations, the corresponding profile-likelihood confidence limit for the hazard ratio is set to missing. Other CONTRAST statements involving classification variables with PARAM=EFFECT are constructed similarly. Additionally, although stratifying by a categorical covariate works naturally, it is often difficult to know how to best discretize a continuous covariate. Covariates are permitted to change value between intervals. Violations of the proportional hazard assumption may cause bias in the estimated coefficients as well as incorrect inference regarding significance of effects. Writing the means and their difference in terms of model (2): The following ESTIMATE and CONTRAST statements estimate these means, their difference, and also test that the difference is equal to zero. Estimates are formed as linear estimable functions of the form . Therneau, TM, Grambsch, PM. Use the Class Level Information table which shows the design variable settings. 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. The ODDSRATIO statement in PROC LOGISTIC and the similar HAZARDRATIO statement in PROC PHREG are also available. Many, but not all, patients leave the hospital before dying, and the length of stay in the hospital is recorded in the variable los. 1. 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 \(\hat S(1) = \frac{500 8}{500} = 0.984\). The model is the same as model (1) above with just a change in the subscript ranges. The unconditional probability of surviving beyond 2 days (from the onset of risk) then is \(\hat S(2) = \frac{500 8}{500}\times\frac{492-8}{492} = 0.984\times0.98374=.9680\). 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 second model is a reduced model that contains only the main effects. Indeed the hazard rate right at the beginning is more than 4 times larger than the hazard 200 days later. To specify a Cox model with start and stop times for each interval, due to the usage of time-varying covariates, we need to specify the start and top time in the model statement: If the data come prepared with one row of data per subject each time a covariate changes value, then the researcher does not need to expand the data any further. . Because of this parameterization, covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than hazard differences. 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. Notice that the interval during which the first 25% of the population is expected to fail, [0,297) is much shorter than the interval during which the second 25% of the population is expected to fail, [297,1671). Perhaps you also suspect that the hazard rate changes with age as well. We can plot separate graphs for each combination of values of the covariates comprising the interactions. So, this test can be used with models that are fit by many procedures such as GENMOD, LOGISTIC, MIXED, GLIMMIX, PHREG, PROBIT, and others, but there are cases with some of these procedures in which a LR test cannot be constructed: Nonnested models can still be compared using information criteria such as AIC, AICC, and BIC (also called SC). Ordinary least squares regression methods fall short because the time to event is typically not normally distributed, and the model cannot handle censoring, very common in survival data, without modification. ANOVA, or Analysis Of Variance, is used to compare the averages or means of two or more populations to better understand how they differ. Similarly, the SLICEBY, DIFF, and EXP options in the SLICE statement estimate and test differences and odds ratios in the complicated diagnosis. We will use a data set called hsb2.sas7bdat to demonstrate. 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 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. Data that are structured in the first, single-row way can be modified to be structured like the second, multi-row way, but the reverse is typically not true. The null distribution of the cumulative martingale residuals can be simulated through zero-mean Gaussian processes. This section contains 14 examples of PROC PHREG applications. = 1 and cell ses = 2 will be the difference of b_1 and b_2. The significant AGE*GENDER interaction term suggests that the effect of age is different by gender. 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. Biometrika. For the medical example, suppose we are interested in the odds ratio for treatment A versus treatment C in the complicated diagnosis. The following parameters are specified in the CONTRAST statement: identifies the contrast on the output. Comparing One Interaction Mean to the Average of All Interaction Means run;
If this option is not specified, PROC PHREG finds all the variables that interact with the variable of interest. The LSMESTIMATE statement again makes this easier. We can estimate the hazard function is SAS as well using proc lifetest: As we have seen before, the hazard appears to be greatest at the beginning of follow-up time and then rapidly declines and finally levels off. Thus, it might be easier to think of \(df\beta_j\) as the effect of including observation \(j\) on the the coefficient. to the coefficient for ses = 2. The ESTIMATE statement provides a mechanism for obtaining custom hypothesis tests. The CONTRAST statement enables you to specify a matrix, , for testing the hypothesis . You can fit many kinds of logistic models in many procedures including LOGISTIC, GENMOD, GLIMMIX, PROBIT, CATMOD, and others. The solid lines represent the observed cumulative residuals, while dotted lines represent 20 simulated sets of residuals expected under the null hypothesis that the model is correctly specified. You can perform hypothesis tests for the estimable functions, construct confidence limits, and obtain specific nonlinear transformations. By default, PROC GENMOD computes a likelihood ratio test for the specified contrast. The PLOTS=CIF option in the PROC PHREG statement displays a plot of the curves. The difference between the mean of cell ses The likelihood ratio and Wald statistics are asymptotically equivalent. if lenfol > los then in_hosp = 0;
Examples of Writing CONTRAST and ESTIMATE Statements Introduction EXAMPLE 1: A Two-Factor Model with Interaction Computing the Cell Means Using the ESTIMATE Statement Estimating and Testing a Difference of Means A More Complex Contrast Comparing One Interaction Mean to the Average of All Interaction Means 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 coefficients that are needed in the ESTIMATE statement are determined by writing what you want to estimate in terms of the fitted model. 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. 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 \(df\beta_j\). We cannot tell whether this age effect for females is significantly different from 0 just yet (see below), but we do know that it is significantly different from the age effect for males. Nevertheless, in both we can see that in these data, shorter survival times are more probable, indicating that the risk of heart attack is strong initially and tapers off as time passes. As an example, imagine subject 1 in the table above, who died at 2,178 days, was in a treatment group of interest for the first 100 days after hospital admission. None of the graphs look particularly alarming (click here to see an alarming graph in the SAS example on assess). Copyright We can see this reflected in the survival function estimate for LENFOL=382. Once you have identified the outliers, it is good practice to check that their data were not incorrectly entered. 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. The solution vector in PROC MIXED is requested with the SOLUTION option in the MODEL statement and appears as the Estimate column in the Solution for Fixed Effects table: For this model, the solution vector of parameter estimates contains 18 elements. By default, pis equal to the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. Martingale-based residuals for survival models. A main effect parameter is interpreted as the difference in the level's effect compared to the reference level. To correctly specify your contrast, it is crucial to know the ordering of parameters within each effect and the variable levels associated with any parameter. Tests to compare nonnested models are available, but not by using CONTRAST statements as discussed above. PROC CATMOD has a feature that makes testing this kind of hypothesis even easier. ALPHA=number specifies the level of significance for % confidence intervals. From the plot we can see that the hazard function indeed appears higher at the beginning of follow-up time and then decreases until it levels off at around 500 days and stays low and mostly constant. You can specify the following options after a slash (/). Therefore, this contrast is also estimated by the parameter for treatment A within the complicated diagnosis in the nested effect. Because this seminar is focused on survival analysis, we provide code for each proc and example output from proc corr with only minimal explanation. 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. where \(d_i\) is the number who failed out of \(n_i\) at risk in interval \(t_i\). 2. The LSMESTIMATE statement allows you to request specific comparisons. Group of ses =3 is the reference group. The exponential function is also equal to 1 when its argument is equal to 0. 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. To assess the effects of continuous variables involved in interactions or constructed effects such as splines, see this note. The HPREG Procedure The HPSPLIT Procedure The ICLIFETEST Procedure The ICPHREG Procedure The INBREED Procedure The IRT Procedure The KDE Procedure The KRIGE2D Procedure The LATTICE Procedure The LIFEREG Procedure The LIFETEST Procedure The LOESS Procedure The LOGISTIC Procedure The MCMC Procedure The MDS Procedure The MI Procedure For observation \(j\), \(df\beta_j\) approximates the change in a coefficient when that observation is deleted. controls the convergence criterion for the profile-likelihood confidence limits. Using model (1) above, the AB12 cell mean, 12, is: Because averages of the errors (ijk) are assumed to be zero: Similarly, the AB11 cell mean is written this way: So, to get an estimate of the AB12 mean, you need to add together the estimates of , 1, 2, and 12. Effects or Deviation from mean coding of a predictor replaces the actual variable in the design matrix (or model matrix) with a set of variables that use values of 1, 0, or 1 to indicate the level of the original variable. In PROC LOGISTIC, use the PARAM=GLM option in the CLASS statement to request dummy coding of CLASS variables. Means for the AB11 and AB12 cells (highlighted in the above table) are computed below using the ESTIMATE statement. 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). The same results can be obtained using the ESTIMATE statement in PROC GENMOD. 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). you might need to print it in landscape mode to avoid truncation of the right edge. Thus, we again feel justified in our choice of modeling a quadratic effect of bmi. The simple contrast shown in the LSMESTIMATE statement below compares the fourth and eighth means as desired. 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. The first 12 examples use the classical method of maximum likelihood, while the last two examples illustrate the Bayesian methodology. 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. ALPHA=number specifies the level of significance for % confidence intervals. ALPHA= p specifies the level of significance pfor the % confidence interval for each contrast when the ESTIMATE option is specified. If too few values are specified, the remaining ones are set to 0. It appears the probability of surviving beyond 1000 days is a little less than 0.2, which is confirmed by the cdf above, where we see that the probability of surviving 1000 days or fewer is a little more than 0.8. format gender gender. In very large samples the Kaplan-Meier estimator and the transformed Nelson-Aalen (Breslow) estimator will converge. The above relationship between the cdf and pdf also implies: In SAS, we can graph an estimate of the cdf using proc univariate. If we were to plot the estimate of \(S(t)\), we would see that it is a reflection of F(t) (about y=0 and shifted up by 1). proc loess data = residuals plots=ResidualsBySmooth(smooth);
If our Cox model is correctly specified, these cumulative martingale sums should randomly fluctuate around 0. 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)). var lenfol gender age bmi hr;
Hosmer, DW, Lemeshow, S, May S. (2008). proc univariate data = whas500 (where= (fstat=1)); var lenfol; cdfplot lenfol; run; In the graph above we can see that the probability of surviving 200 days or fewer is near 50%. In this interval, we can see that we had 500 people at risk and that no one died, as Observed Events equals 0 and the estimate of the Survival function is 1.0000. Applied Survival Analysis. requests that each individual contrast (that is, each row, , of ) or exponentiated contrast () be estimated and tested. Note that these are the fourth and eighth cell means in the Least Squares Means table. var lenfol;
You can use the same method of writing the AB12 cell mean in terms of the model: You can write the average of cell means in terms of the model: So, the coefficient for the A parameters is 1/2; for B it is 1/3; and for AB it is 1/6. However, this is something that cannot be estimated with the ODDSRATIO statement which only compares odds of levels of a specified variable. INTRODUCTION The PROC LIFEREG and the PROC PHREG procedures both can do survival analysis using time-to-event data, . 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. A central assumption of Cox regression is that covariate effects on the hazard rate, namely hazard ratios, are constant over time. run;
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 \(f(t)\). Proc PHREG - Random Statement. format gender gender. The "Class Level Information" table shows the ordering of levels within variables. There are \(df\beta_j\) values associated with each coefficient in the model, and they are output to the output dataset in the order that they appear in the parameter table Analysis of Maximum Likelihood Estimates (see above). Estimates are formed as linear estimable functions of the form . Here is the syntax for CONTRAST statement. run; proc corr data = whas500 plots(maxpoints=none)=matrix(histogram);
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). Other nonparametric tests using other weighting schemes are available through the test= option on the strata statement. Positive values of \(df\beta_j\) indicate that the exclusion of the observation causes the coefficient to decrease, which implies that inclusion of the observation causes the coefficient to increase. SAS provides easy ways to examine the \(df\beta\) values for all observations across all coefficients in the model. The quantity value must be a positive number, with a default value of 1E4. In this model, this reference curve is for males at age 69.845947 Usually, we are interested in comparing survival functions between groups, so we will need to provide SAS with some additional instructions to get these graphs. SAS expects individual names for each \(df\beta_j\)associated with a coefficient. Notice the. Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type. Notice also that care must be used in altering the censoring variable to accommodate the multiple rows per subject. Instead, you model a function of the response distribution's mean. Because PROC CATMOD also uses effects coding, you can use the following CONTRAST statement in that procedure to get the same results as above. A full-rank version of indicator coding (called reference coding) that omits the indicator variable for the reference level (by default, the last level) is also available in PROC LOGISTIC, PROC GENMOD, PROC CATMOD, and some other procedures via the PARAM=REF option. If, say, a regression coefficient changes only by 1% over time, it is unlikely that any overarching conclusions of the study would be affected. 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. Specifically, PROC LOGISTIC is used to fit a logistic model containing effects X and X2. The default is the value of the ALPHA= option in the PROC PHREG statement, or 0.05 if that option is not specified. Thus, to pull out all 6 \(df\beta_j\), we must supply 6 variable names for these \(df\beta_j\). Find more tutorials on the SAS Users YouTube channel. 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. For example, if the survival times were known to be exponentially distributed, then the probability of observing a survival time within the interval \([a,b]\) is \(Pr(a\le Time\le b)= \int_a^bf(t)dt=\int_a^b\lambda e^{-\lambda t}dt\), where \(\lambda\) is the rate parameter of the exponential distribution and is equal to the reciprocal of the mean survival time. This example is to illustrate the algorithm used to compute the parameter estimate. run; proc phreg data = whas500;
The likelihood ratio test can be used to compare any two nested models that are fit by maximum likelihood. We can remove the dependence of the hazard rate on time by expressing the hazard rate as a product of \(h_0(t)\), a baseline hazard rate which describes the hazard rates dependence on time alone, and \(r(x,\beta_x)\), which describes the hazard rates dependence on the other \(x\) covariates: In this parameterization, \(h(t)\) will equal \(h_0(t)\) when \(r(x,\beta_x) = 1\). 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. Finally, you can use the SLICE statement. The value pmust be between 0 and 1. 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 article emphasizes four features of PROC PLM: You can use the SCORE statement to score the model on new data. This test can be done using a CONTRAST statement to jointly test the interaction parameters. %PDF-1.2
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The survival function is undefined past this final interval at 2358 days. Some data management will be required to ensure that everyone is properly censored in each interval. We obtain estimates of these quartiles as well as estimates of the mean survival time by default from proc lifetest. hazardratio 'Effect of gender across ages' gender / at(age=(0 20 40 60 80));
In other words, the average of the Schoenfeld residuals for coefficient \(p\) at time \(k\) estimates the change in the coefficient at time \(k\). With effects coding, the parameters are constrained to sum to zero. Most of the time we will not know a priori the distribution generating our observed survival times, but we can get and idea of what it looks like using nonparametric methods in SAS with proc univariate. In each of the tables, we have the hazard ratio listed under Point Estimate and confidence intervals for the hazard ratio. The PHREG procedure now fits frailty models with the addition of the RANDOM statement. For example, suppose an effect coded CLASS variable A has four levels. An assumption of the Cox proportional hazard model is a . 81. To accomplish this smoothing, the hazard function estimate at any time interval is a weighted average of differences within a window of time that includes many differences, known as the bandwidth. 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. Nonparametric methods provide simple and quick looks at the survival experience, and the Cox proportional hazards regression model remains the dominant analysis method. To avoid this problem, use the DIVISOR= option. I am looking at the interactive effects of X according to Y on death. The SLICE and LSMEANS statements cannot be used for this more complex contrast. Institute for Digital Research and Education. class gender;
We compare 2 models, one with just a linear effect of bmi and one with both a linear and quadratic effect of bmi (in addition to our other covariates). As time progresses, the Survival function proceeds towards it minimum, while the cumulative hazard function proceeds to its maximum. For example, if there were three subjects still at risk at time \(t_j\), the probability of observing subject 2 fail at time \(t_j\) would be: \[Pr(subject=2|failure=t_j)=\frac{h(t_j|x_2)}{h(t_j|x_1)+h(t_j|x_2)+h(t_j|x_3)}\]. Chapter 19, 2009 by SAS Institute Inc., Cary, NC, USA. Here is the code: proc phreg data=Mortality_M3_72 covs (aggregate); class X (ref=first) Y (ref=first); By default, value is the machine epsilon times 1E7, which is approximately 1E9. Note: This was the primary reference used for this seminar. o1LSRD"Qh&3[F&g
w/!|#+QnHA8Oy9 , The survival curves for females is slightly higher than the curve for males, suggesting that the survival experience is possibly slightly better (if significant) for females, after controlling for age. The E option shows how each cell mean is formed by displaying the coefficient vectors that are used in calculating the LS-means. run; proc phreg data=whas500;
Above we described that integrating the pdf over some range yields the probability of observing \(Time\) in that range. Let us further suppose, for illustrative purposes, that the hazard rate stays constant at \(\frac{x}{t}\) (\(x\) number of failures per unit time \(t\)) over the interval \([0,t]\). run; proc lifetest data=whas500 atrisk nelson;
One caveat is that this method for determining functional form is less reliable when covariates are correlated. For this seminar, it is enough to know that the martingale residual can be interpreted as a measure of excess observed events, or the difference between the observed number of events and the expected number of events under the model: \[martingale~ residual = excess~ observed~ events = observed~ events (expected~ events|model)\]. Firths Correction for Monotone Likelihood, Conditional Logistic Regression for m:n Matching, Model Using Time-Dependent Explanatory Variables, Time-Dependent Repeated Measurements of a Covariate, Survivor Function Estimates for Specific Covariate Values, Model Assessment Using Cumulative Sums of Martingale Residuals, Bayesian Analysis of Piecewise Exponential Model. 1469-82. run; proc phreg data = whas500;
Effects Coding var lenfol gender age bmi hr;
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A continuous covariate of CLASS variables the Bayesian methodology, PROC LOGISTIC is used to compute the parameter ESTIMATE )... That each individual contrast ( ) be estimated and tested graphs look alarming. Not by using contrast statements involving classification variables with PARAM=EFFECT are constructed similarly ratio is to! Following parameters are specified in the Least Squares means table weighting schemes are through! That covariate effects are multiplicative rather than additive and are expressed as hazard ratios, rather than differences! Data management will be the difference of b_1 and b_2 attained in n iterations, the ones... Testing the hypothesis many kinds of LOGISTIC models in many procedures including LOGISTIC, GENMOD, GLIMMIX,,... Ses = 2 will be required to ensure that everyone is properly in... ), we have the hazard ratio listed under Point ESTIMATE and confidence intervals Cary NC! Accommodate the multiple rows per subject Institute Inc., Cary, NC, USA a change in code. Instead, you model a function of the tables, we model the effects of on... Which shows the design variable settings ( click here to see an alarming graph in PROC... Obtain estimates of these quartiles as well as incorrect inference regarding significance of effects the PLOTS=CIF option in WHAS500... Level Information table which shows the design variable settings difference of b_1 and b_2 effect of is. The CLASS level Information '' table shows the ordering of proc phreg estimate statement example of treatment within each of! ( that is, each row,, for testing the hypothesis is specified! Allows you to specify a matrix,, of ) or exponentiated contrast ( ) be estimated and tested the! Provides easy ways to examine the \ ( df\beta_j\ ) likelihood ratio test for the specified contrast to Y death. Properly censored in each of the form likelihood ratio and Wald statistics are asymptotically equivalent for these \ df\beta_j\... Involving classification variables with PARAM=EFFECT are constructed similarly the outliers, it is often difficult to know how best... Main effect parameter is interpreted as the difference between the mean of cell ses 2... The similar HAZARDRATIO statement in PROC PHREG procedures both can do survival analysis using time-to-event data, Kaplan-Meier!

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