Current Directions in SAS/STAT Software Development

SAS Global Forum 2013
Statistics and Data Analysis
Paper 432-2013
Current Directions in SAS/STAT® Software Development
Maura Stokes
SAS Institute Inc.
Abstract
Recent years brought you SAS/STAT® releases in rapid succession, and another release is targeted for
2013. Which new software features will make a difference in your work? What new statistical trends should
you know about? This paper describes recent areas of development focus, such as Bayesian analysis,
missing data analysis, postfitting inference, quantile modeling, finite mixture models, specialized survival
analysis, and structural equation modeling. This paper introduces you to the concepts and illustrates them
with practical examples.
More Frequent Releases of SAS/STAT Software
In previous years, SAS/STAT software was updated only when Base SAS® software was released, but
SAS/STAT is now released independently of Base SAS along with other SAS analytical products. This means
these products can be released when enhancements are ready, and the goal is to update SAS/STAT every
12 to 18 months. To mark this newfound independence, the release numbering scheme for SAS analytical
products has also changed; the current production release is SAS/STAT 12.1.
With the new release paradigm, SAS provides you with new statistical techniques faster. However, it may be
difficult to keep track of all of the new development. This paper provides a round-up of the current directions
in SAS/STAT development, with emphasis on the areas that have received the most attention.
Developing SAS/STAT Software
SAS/STAT software has a long and rich legacy. Originally, statistical functionality was included in Base
SAS for mainframes, but SAS/STAT became its own product in 1976. Originally written in the PL/I and
Fortran programming languages, it was rewritten in the C language and made portable so it could run on
a variety of platforms (multivendor architecture) including PCs. The Output Delivery System (ODS) was
then incorporated, giving you complete control over the format of the tabular results and making reporting
quite flexible. More recently, the development of ODS Statistical Graphics has resulted in the integration of
graphics into the statistical analyses.
What hasn’t changed over the years has been the goal of providing rich, up-to-date statistical techniques
to SAS customers. SAS/STAT 12.1 includes over 75 procedures, soon to expand again with the 12.3
and 13.1 releases. New development directions are determined in a number of ways: customer input,
company directives, the appearance of important new methodologies, and the drive to constantly refine and
update existing software. Technical support is one channel from customers to development which fosters
greater understanding of customer use and often leads to software enhancements. Other channels include
professional contacts and customer meetings. Often, statistical development directions are influenced by
overall SAS company directives, such as the current initiative to excel in high-performance computing.
Of course, statistical developers, specialized PhD statisticians with computing backgrounds, keep up with
current methodology in the usual ways: journals, professional conferences, expert contacts, and workshops.
Developers attend statistical conferences and SAS user groups, both sources of feedback and suggestions.
When R&D engages in a new area, it often begins with an invited seminar by a renowned expert and then
sustains the contact for feedback as development proceeds.
SAS development focuses on statistical methods that work in practice. SAS customers deal with real-life
data, with its size and pitfalls, and the promising methodology that looked great for a few well-designed test
cases often needs more work before it’s ready for the real world. An important objective is consistency in
syntax so that customers can build on what they already know. The goal is always clear and consistent
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output, including graphics, and a major milestone for each new project is clearing a peer review of tabular
and graphical content and layout. The documentation is written by the developers and undergoes substantial
technical review and editing before it is finalized.
Numerical accuracy and computational performance are essential. Numerical analysts develop the mathematical routines that are incorporated into the statistical procedures. Existing procedures are continously
evaluated to see if they can benefit from cutting-edge algorithms. SAS avoids changing default methods
because the software is used so often in production jobs, but new methods are added regularly and the
documentation states when a newer method has become the predominant practice.
The following sections outline some recent development directions, including model building, Bayesian
analysis, linear model enrichment, analyzing time-to-event data, missing data methods, complex data, and
high-performance computing.
Model Building
Model building has only grown more important in recent times as statisticians and data analysts face the
world of Big Data and its inherent increases in data dimensionality. The value of predictive analytics depends
on finding good models, so model selection is paramount. SAS/STAT model building took a step forward
some years ago with the introduction of the GLMSELECT procedure for linear models selection. PROC
GLMSELECT provides modern methods such as LARS and LASSO, and it also includes a rich array of
selection criteria and numerous graphical displays. More recently, model building in SAS/STAT has expanded
to include model selection for quantile regression and generalized linear models.
New QUANTSELECT Procedure
Quantile regression is a distribution-free method that determines a linear predictor for the conditional
quantile. It is especially useful with data that are heterogeneous such that the tails and central location of the
conditional distributions vary with the covariates. The QUANTREG procedure provides quantile regression
in SAS/STAT software. Beginning with SAS/STAT 12.1, you can also perform model selection for quantile
regression with the new QUANTSELECT procedure. This procedure provides capabilities similar to those
offered by the GLMSELECT procedure, including:
• forward, backward, stepwise, and LASSO selection methods
• variable selection criteria: AIC, SBC, AICC, and so on
• variable selection for both quantiles and the quantile process
• the EFFECT statement for constructed model effects (splines)
PROC QUANTSELECT is multithreaded so that it can take advantage of multiple processors. It is very
efficient and can handle hundreds of variables and thousands of observations. After you have selected a
model with the QUANTSELECT procedure, you can proceed to use the QUANTREG procedure for final
model analysis.
The following example illustrates the use of the QUANTSELECT procedure with baseball data from the 1986
season. The goal is to predict player salary. You can request model selection for any number of quantiles,
and if you do so, you will find that different models are selected. If you are interested only in the model for
those players making the most money, you can base the model on the 90th quantile, which is the analysis
performed here.
The following statements input the baseball data:
data baseball;
length name $ 18;
length team $ 12;
input name $ 1-18 nAtBat nHits nHome nRuns nRBI nBB
yrMajor crAtBat crHits crHome crRuns crRbi crBB
league $ division $ team $ position $ nOuts nAssts
nError salary;
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datalines;
Allanson, Andy
293
66
1
30
1
293
66
1
30
29
14
American East Cleveland C 446 33 20 .
Ashby, Alan
315
81
7
24
14 3449
835
69
321
414
375
National West Houston C 632 43 10 475
.....
.....
29
14
38
39
The following statements invoke the QUANTSELECT procedure. The variable SALARY is the response
variable, and a number of explanatory variables are available for selection. The adaptive LASSO method is
used for model selection, with AIC as the stopping criterion. The plot requested is the coefficient panel.
proc quantselect data=baseball plots=(coef);
class league division;
model Salary = nAtBat nHits nHome nRuns nRBI nBB
yrMajor crAtBat crHits crHome crRuns crRbi
crBB league division nOuts nAssts nError /
selection=lasso (adaptive stop=aic)
quantile=.9;
run;
Figure 1 displays the selection summary information. You can see the values of AIC and AICC change as
variables are added to the model. The optimal value of AIC is 2099.23 at the fifth step, which corresponds
to a model with four variables: number of hits, career hits, career home runs, and career RBIs. These
explanatory variables are the main factors in determining salary for the 90th percentile.
Figure 1 Selection Summary
Quantile Level = 0.9
Selection Summary
Step
Effect
Entered
Number
Effects
In
AIC
AICC
SBC
Adjusted
R1
0
Intercept
1
2436.7289
2436.7442
2440.3011
0.0000
1
crHits
2
2197.4349
2197.4811
2204.5792
0.3655
2
crRbi
3
2183.6148
2183.7075
2194.3313
0.3819
3
nHits
4
2113.2757
2113.4308
2127.5643
0.4593
4
crHome
5
2099.2203*
2099.4538*
2117.0811*
0.4735*
* Optimal Value Of Criterion
Figure 2 displays the coefficient panel, which shows the progression of the standardized coefficients and the
SBC throughout the selection process.
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Figure 2 Coefficient Panel
Figure 3 contains the parameter estimates and their standardized versions.
Figure 3 Parameter Estimates
Parameter Estimates
DF
Estimate
Standardized
Estimate
Intercept
1
-100.381451
0
nHits
1
3.895310
0.379872
crHits
1
0.660121
0.949536
crHome
1
3.340492
0.637123
crRbi
1
-0.453248
-0.329973
Parameter
Additional inference can be performed by using this model in the QUANTREG procedure.
New HPGENSELECT Procedure
The GENMOD procedure fits generalized linear models in SAS; these models include normal regression,
logistic regression, Poisson regression, and other analyses where you specify a link function and a distribution
that belongs to the exponential family. The new HPGENSELECT procedure, available with SAS/STAT 12.3
(which runs on Base 9.4), performs model selection for generalized linear models (GLMs). This means that
you can now perform model selection for analyses such as Poisson regression, negative binomial regression,
and any other GLM. Designed for the distributed computing of SAS® High-Performance Statistics, PROC
HPGENSELECT also works in single-machine mode. It provides forward, backward, and stepwise selection
(LASSO-type methods are still a research topic), and includes the AIC, SBC, and AICC selection criteria. It
does not produce graphs with this version, but they will surface in a future release.
Recall the baseball data. You can treat the number of home runs hit during the year as counts that follow
the Poisson distribution, and thus you can employ Poisson regression to model these counts. The following
statements illustrate how you would request model selection for Poisson regression with the HPGENSELECT
procedure.
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proc hpgenselect data=baseball;
class league division;
model nHome = nAtBat nHits nRuns nRBI nBB
yrMajor crAtBat crHits crHome crRuns crRbi
crBB league division nOuts nAssts nError
/ distribution=poisson link=log;
selection method=forward details=all;
run;
You specify exactly the same MODEL statement as you would specify with the GENMOD procedure. You
specify the selection method with the SELECTION statement, a new statement used by the high-performance
procedures. Forward stepwise selection is specified with the METHOD=FORWARD option.
Figure 4 displays information about the execution mode. Two threads were employed on a single machine.
Figure 4 Performance Information
Performance Information
Execution Mode
Single-Machine
Number of Threads
2
Effects are added to the model if they produce improvement as judged by comparing the p-value of a score
test to the entry significance level (SLE), which is 0.05 by default. The forward selection ends when no
additional effect meets this criterion.
Figure 5 provides the final effects that entered the model and the details of effect selection.
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Figure 5 Selection Details
Selection Details
Step
Description
Effects
In Model
0
Initial Model
1
1
nRBI entered
2
1595.3565
2
nAssts entered
3
3
nHits entered
4
Chi-Square
Pr > ChiSq
-2 LogL
AIC
AICC
3419.513
3421.513
3421.525
<.0001
1994.778
1998.778
1998.815
48.6818
<.0001
1943.370
1949.370
1949.445
4
20.2656
<.0001
1922.611
1930.611
1930.737
nRuns entered
5
43.3903
<.0001
1880.196
1890.196
1890.386
5
crHome entered
6
6.3802
0.0115
1873.922
1885.922
1886.189
6
crRbi entered
7
27.3765
<.0001
1845.695
1859.695
1860.052
7
crAtBat entered
8
6.9953
0.0082
1838.769
1854.769
1855.229
8
crRuns entered
9
13.0622
0.0003
1825.626
1843.626
1844.203
9
crHits entered
10
15.2906
<.0001
1810.307
1830.307
1831.014
10
crBB entered
11
4.7299
0.0296
1805.576
1827.576
1828.428
Selection
Details
Step
BIC
0
3425.287
1
2006.327
2
1960.693
3
1945.709
4
1909.069
5
1908.569
6
1886.117
7
1884.965
8
1877.597
9
1868.052
10
1869.096
Figure 6 contains fit statistics for the selected model. Note that the values of the 2 log likelihood and the
various information criteria AIC, BIC, and AICC are smaller than the corresponding values for the full model
(Initial Model) in the Selection Details table, which indicates that the more parsimonious model provides a
better fit. However, note that the value of Pearson chi-square divided by its degrees of freedom is 1.60. This
is an indication of overdispersion, which suggests that Poission regression is not the best technique to apply
to these data.
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Figure 6 Fit Statistics
Fit Statistics
-2 Log Likelihood
1805.57637
AIC (smaller is better)
1827.57637
AICC (smaller is better)
1828.42799
BIC (smaller is better)
1869.09644
Pearson Chi-Square
496.91515
Pearson Chi-Square/DF
1.59780
Negative binomial regression is often an alternative in this situation; it allows the variance to be larger than
the mean, unlike the assumption of equivalence in Poisson regression. The HPGENSELECT procedure also
provides negative binomial regression, and that is requested with the DIST=NB option:
proc hpgenselect data=baseball;
class league division;
model nHome = nAtBat nHits nRuns nRBI nBB
yrMajor crAtBat crHits crHome crRuns crRbi
crBB league division nOuts nAssts nError
/ dist=nb link=log;
selection method=forward details=all;
run;
Figure 7 lists the seven effects that entered the model for negative binomial regression.
Figure 7 Selection Summary
Selection Summary
Step
Effect
Entered
Number
Effects In
p
Value
0
Intercept
1
.
1
nRBI
2
<.0001
2
nAssts
3
<.0001
3
nHits
4
0.0019
4
nRuns
5
<.0001
5
crHome
6
0.0203
6
crRbi
7
<.0001
7
yrMajor
8
0.0464
Figure 8 displays the corresponding fit statistics for the final model; these values are somewhat smaller
than the fit statistics reported for the final Poisson regression model. Note, however, that this is a different
analysis and these measures cannot be directly compared. The Pearson chi-square/degrees of freedom
ratio takes the value 1.01 for this analysis, which indicates no evidence of overdispersion. Thus, this model
is deemed to be satisfactory.
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Figure 8 Fit Statistics
Fit Statistics
-2 Log Likelihood
1785.67309
AIC (smaller is better)
1803.67309
AICC (smaller is better)
1804.25001
BIC (smaller is better)
1837.64405
Pearson Chi-Square
318.66241
Pearson Chi-Square/DF
1.01485
Figure 9 contains the parameter estimates for this model.
Figure 9 Parameter Estimates
Parameter Estimates
DF
Estimate
Standard
Error
Chi-Square
Pr > ChiSq
Intercept
1
1.110918
0.093513
141.1309
<.0001
nHits
1
-0.004750
0.001497
10.0700
0.0015
nRuns
1
0.007781
0.002272
11.7265
0.0006
nRBI
1
0.024132
0.001756
188.8612
<.0001
yrMajor
1
0.024209
0.012201
3.9369
0.0472
crHome
1
0.004504
0.000875
26.4625
<.0001
crRbi
1
-0.001383
0.000326
17.9730
<.0001
nAssts
1
-0.000636
0.000191
11.1047
0.0009
Dispersion
1
0.066758
0.014310
.
.
Parameter
Bayesian Analysis
While the founding principle of Bayesian analysis goes back to the theorem developed by Sir Thomas
Bayes in 1763, modern Bayesian analysis didn’t catch fire until the computing advances of the 20th century.
Bayesian analysis is based on the idea that inferences are based on measures of uncertainty through
probability distributions, and prior knowledge can impact the inferences. In Bayesian analysis, parameters
are random and you must estimate their posterior distributions. Summary statistics produced are posterior
modes and credible intervals as opposed to the point estimates and confidence intervals in frequentist
statistical analysis.
The Bayesian framework provides a straightforward framework for addressing scientific questions. For
example, you can estimate the probability of an interval containing a value, versus the in-the-long run
definition of a confidence interval which confuses so many clients. Thus, the framework is attractive for
all types of situations, not just those in which you have prior information. While every Bayesian analysis
incorporates a prior distribution, you can have noninformative prior distributions which do not influence the
likelihood but do allow you to perform the Bayesian analysis. While there are analytic solutions for a few
analyses, usually a closed form does not exist and simulation methods are required, such as Markov chain
Monte Carlo methods.
SAS provides two avenues to Bayesian analysis: built-in Bayesian analysis in certain modeling procedures
and the MCMC procedure for general-purpose modeling. Adding the BAYES statement generates Bayesian
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analyses without the need to program priors and likelihoods for the GENMOD, PHREG, LIFEREG, and FMM
procedures. Thus, you can obtain Bayesian results for:
• standard regression
• Poisson regression
• logistic regression
• loglinear models
• accelerated failure time models
• Cox proportional models
• piecewise exponential models
• frailty models
• finite mixture models
These procedures are ideal for users beginning to use Bayesian methods and will suffice for many analysis
objectives.
The MCMC procedure is a general-purpose procedure for fitting Bayesian models. It uses a variety of
sampling algorithms to draw from the posterior distribution. It produces the same convergence diagnostics
and posterior summaries that you would find by using the BAYES statement in the modeling procedures.
However, the MCMC procedure allows any likelihood, prior, or hyperprior that can be programmed with the
SAS language. It supports multivariate distributions as well. If you are familiar with the NLMIXED procedure,
you are familiar with the type of programming statements that the MCMC procedure requires.
The RANDOM statement in the MCMC procedure facilitates the specification of random effects in linear
or nonlinear models. You can build nested or nonnested hierarchical models to artbitrary depth. Using
the RANDOM statement can result in reduced simulation time and improved convergence for models that
have a large number of subjects. The MCMC procedure also handles missing data for both responses and
covariates. See papers by Fang Chen in the online conference proceedings for additional information.
Richer Linear Models
Hidden in the SAS/STAT 9.22 release (which had stealth qualities in its own right) is the new PLM procedure.
This new procedure was the end result of a re-architecturing effort which put all of the post-fitting analysis
statements–CONTRAST, LSMEANS, LSMESTIMATES,ESTIMATE–into a common framework. This meant
that:
• 30 postfitting statements added to existing procedures
• faster new procedure development
• more efficient maintenance
• new features get to all relevant procedures faster
In addition, the PLM procedure performs postfitting inference with model fit information saved from a number
of SAS/STAT modeling procedures. These procedures are equipped with the new STORE statement, which
saves model information as a SAS item store. An item store is a special SAS binary file that is used to store
and re-store information that has a hierarchical structure. Ten SAS/STAT procedures now provide the STORE
statement: GENMOD, GLIMMIX, GLM, LOGISTIC, MIXED, ORTHOREG, PHREG, SURVEYLOGISTIC,
SURVEYPHREG, and SURVEYREG.
The PLM procedure takes these item stores as input and performs tasks such as testing hypotheses,
producing effect plots, and scoring a new data set. These tasks are specified through the usual complement
of postfitting statements such as the TEST, LSMEANS, and new EFFECTPLOT and SCORE statements.
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Any procedure that offers the STORE statement can produce the item stores that are necessary for postfitting
processing with the PLM procedure. This allows you to perform additional postfitting inference at a later time
without having to refit your model, which is especially convenient for those models that are computationally
expensive. In addition, with growing concerns for data confidentiality, storing and using intermediate results
for remaining analyses might become a requirement in some organizations.
The EFFECT statement provides a convenient way to add additional modeling terms to your linear model.
These effects can include classification beyond simple grouping (multimember and lag effects), continuous
modeling beyond simple polynomials (polynomial and spline effects), and general terms that you define
yourself (collection effects). These constructed effects are viewed as any other model effect, which means
you can use them in any postfitting analysis that is based on your model. The EFFECT statement also works
well together with the new EFFECTPLOT statement, which provides a way to visualize the impact of the
effects on the response variable. The EFFECT statement is available with a number of SAS/STAT linear
modeling procedures.
Analyzing Time-To-Event Data
Data that measure time until an event, also known as lifetime, failure time, or survival data, occur frequently.
They are often seen in clinical trials for medical treatments, such as survival time for heart transplant patients,
but they also occur in many other settings; consider the lifetime of pedometers or the time until a new real
estate agent makes his first sale. Time-to-event data require special attention. Not only are you measuring
the failure time as a dependent variable, and possibly some covariates so that you can form a statistical
model, but you often have to deal with censoring as well.
Censoring occurs when the event of interest hasn’t occurred by the time data collection ends. It may happen
because patients in a study withdraw, or drop out, before the study concludes. Or the experiment may simply
be terminated before the event has occurred for some experimental units. In any event, you only know
the lower bound of the failure time; and the observations are right-censored. You can have data that are
left-censored, or only known to be smaller than a given value, and you can also have interval-censored data,
or failure times that are only known to fall within a certain interval. You have to take censoring into account
because, for example, in the clinical trial, the longer-lived individuals are more likely to be right-censored.
SAS/STAT has provided tools for the analysis of time-to-event data for years. They include the LIFETEST
procedure for estimating the survivor function and comparing the survival curves of various groups. In
addition, the LIFEREG procedure provides parametric regression methods for modeling the distribution of
survival times with a set of covariates, and the PHREG procedure provides proportional hazards regression
(Cox regression).
With recent releases, the survival analysis tools have been extended in a number of ways, including
• SURVEYPHREG procedure provides Cox regression for data collected from a complex survey
• Bayesian methods are available with the BAYES statement in the PHREG and LIFEREG procedures
• QUANTLIFE procedure provides quantile regression for right-censored data
• macro for interval censoring
• piecewise exponential regression available via PROC PHREG
• frailty analysis available via PROC PHREG
The SURVEYPHREG procedure is useful for data collected in large government surveys; for example, you
might use Cox regression to model the time until the onset of depression in a national mental health care
survey with a number of socioeconomic covariates. Bayesian methods are widely used in survival data
analysis, and the BAYES statement had been added to the PHREG and LIFEREG procedures so that
you can now perform Bayesian analysis for Cox regression and accelerated lifetime models. Note that the
PIECEWISE= option has been added to the BAYES statement in the PHREG procedure so that you can
now perform piecewise exponential modeling easily with SAS (both the Bayesian analysis and the maximum
likelihood analysis are produced).
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When experimental units are clustered, the failure times of those units within a cluster tend to be correlated.
You need to account for the within-cluster correlation, and one way of doing that is the shared frailty model,
in which the cluster effects are incorporated in the model as normally distributed random variables. Stokes,
Chen, and So (2011) describe the new PHREG functionality to fit shared frailty models via the specification
of a RANDOM statement in the SAS/STAT 9.3 release. The penalized partial likelihood approach is used,
and that first implementation assumed that the frailties were distributed as lognormal. With SAS/STAT 12.1,
the frailties can also be assumed to be distributed as gamma. PROC PHREG also provides Bayesian frailty
analysis.
Competing risks develop when subjects are exposed to more than one cause of failure: for example, the
cause of death in a bone marrow transplant could be relapse, death during remission, or death due to another
cause. In that case, the cumulative incidence function is more appropriate than the standard Kaplan-Meier
method of survival analysis. The SAS macro %CIF implements nonparametric methods for implementing
this method and also provides Gray’s method for testing differences between these functions in multiple
groups. See Lin et al. (2012) for more information about this method and this macro.
Quantile regression provides an alternative and flexible technique for the analysis of survival data. You
can apply this technique to right-censored responses, which allows you to explore the covariate effects on
the quantiles of interest. Two approaches are implemented: one is based on the idea of the Kaplan-Meier
estimator, and the other is based on the Nelson-Aalen estimator of the cumulative hazard function. The
new QUANTLIFE procedure provides interior point algorithms for estimation, Wald tests for the parameter
estimates, survival plots, conditional quantile plots, and quantile process plots. It also supports the EFFECT
statement so that it can fit regression quantile spline curves, and it is multithreaded to take advantage of
multiple processors when they are available.
Consider a study of primary biliary cirrhosis disease discussed in Lin, Wei, and Ying (1993). Prognostic
factors studied included age, edema, bilirubin, albumin, and prothrombin. Researchers followed 418 patients
between 1974 and 1984. The patients had a median follow-up time of 4.74 years and a censoring rate of
61.5%. The following SAS statements create the SAS data set PBC:
data pbc;
input Time Status Age Albumin Bilirubin Edema Protime @@;
label Time="Follow-up Time in Days";
logAlbumin
=log(Albumin);
logBilirubin =log(Bilirubin);
logProtime
=log(Protime);
datalines;
400 1 58.7652 2.60 14.5 1.0 12.2 4500 0 56.4463 4.14 1.1 0.0
1012 1 70.0726 3.48 1.4 0.5 12.0 1925 1 54.7406 2.54 1.8 0.5
1504 0 38.1054 3.53 3.4 0.0 10.9 2503 1 66.2587 3.98 0.8 0.0
1832 0 55.5346 4.09 1.0 0.0 9.7 2466 1 53.0568 4.00 0.3 0.0
2400 1 42.5079 3.08 3.2 0.0 11.0
51 1 70.5599 2.74 12.6 1.0
3762 1 53.7139 4.16 1.4 0.0 12.0 304 1 59.1376 3.52 3.6 0.0
...
...
10.6
10.3
11.0
11.0
11.5
13.6
The syntax for the MODEL statement for the QUANTLIFE procedure is similar to that used in other SAS
survival procedures. The LOG option requests that the log response values be analyzed, the METHOD=NA
option specifies the Nelson-Aalen method, and the PLOT=(QUANTPLOT SURVIVAL QUANTILE) option
requests the estimated parameter by quantiles plot, the survival plot, and the predicted quantiles plot. The
QUANTILE=(.1 .4 .5 .85) option requests that those quantiles be modeled.
ods graphics on;
proc quantlife data=pbc LOG method=na plot=(quantplot survival quantile) seed=1268;
model Time*Status(0)=logBilirubin logProtime logAlbumin Age Edema
/quantile=(.1 .4 .5 .85);
run;
ods graphics off;
Figure 10 contains the parameter estimates. Each of the requested quantiles has its own set of parameter
estimates. The confidence limits are computed by resampling methods.
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Figure 10 Parameter Estimates
Parameter Estimates
Quantile
0.1000
0.4000
0.5000
0.8500
DF
Estimate
Standard
Error
Intercept
1
14.8030
4.0967
6.7736
logBilirubin
1
-0.4488
0.1485
logProtime
1
-3.6378
logAlbumin
1
Age
Parameter
95%
Confidence Limits
t Value
Pr > |t|
22.8325
3.61
0.0003
-0.7398
-0.1578
-3.02
0.0027
1.4560
-6.4915
-0.7841
-2.50
0.0129
1.9286
0.9756
0.0165
3.8408
1.98
0.0487
1
-0.0244
0.0107
-0.0455
-0.00334
-2.27
0.0237
Edema
1
-1.0712
0.6688
-2.3820
0.2396
-1.60
0.1100
Intercept
1
13.4716
3.0874
7.4204
19.5228
4.36
<.0001
logBilirubin
1
-0.6047
0.0846
-0.7705
-0.4389
-7.15
<.0001
logProtime
1
-2.1632
1.1726
-4.4615
0.1351
-1.84
0.0658
logAlbumin
1
0.9819
0.7191
-0.4274
2.3912
1.37
0.1728
Age
1
-0.0255
0.00681
-0.0389
-0.0122
-3.74
0.0002
Edema
1
-1.0589
0.3104
-1.6672
-0.4506
-3.41
0.0007
Intercept
1
10.9205
2.8047
5.4235
16.4175
3.89
0.0001
logBilirubin
1
-0.5315
0.0904
-0.7087
-0.3543
-5.88
<.0001
logProtime
1
-1.2222
1.2142
-3.6020
1.1577
-1.01
0.3148
logAlbumin
1
1.5700
0.6284
0.3383
2.8016
2.50
0.0129
Age
1
-0.0318
0.00883
-0.0491
-0.0145
-3.60
0.0004
Edema
1
-0.7316
0.3743
-1.4653
0.00202
-1.95
0.0513
Intercept
1
11.0778
.
.
.
.
.
logBilirubin
1
-0.5615
.
.
.
.
.
logProtime
1
-1.2711
.
.
.
.
.
logAlbumin
1
1.4471
.
.
.
.
.
Age
1
-0.0144
.
.
.
.
.
Edema
1
-0.4339
.
.
.
.
.
This behavior of the covariate coefficients is illustrated in the plot in Figure 11. This is a scatter plot of the
estimated regression parameter against the quantiles. In the plot for logPROTIME, the parameter estimate
grows smaller from its value of –3.6456 for the 0.1 quantile and levels off around –1.0 for the 0.5 and higher
quantiles.
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Figure 11 Estimated Parameter by Quantiles Plot
See Lin and Rodriguez (2013) for more information about the QUANTLIFE procedure.
Missing Data Methods
While most of the data sets analyzed in classroom settings (or displayed in SAS documentation) are chock
full of data, practicing statisticians rarely find this situation in the wild. Standard practice in software packages
is to analyze complete case data by default, also known as casewise deletion, and SAS follows this practice.
However, unless the missing observations magically reflect a random sample from the complete observations,
the resulting inference is almost sure to be biased. Thus, managing missing data is an important aspect of
statistical analysis.
One strategy for handling missing data is to impute the missing value, or substitute a value, and then analyze
the data as if they were complete. Single imputation substitutes a single value, but the resulting results are
biased toward zero since they don’t reflect the uncertainty about the predictions of the unknown missing
values (Rubin 1987). Multiple imputation does incorporate that uncertainty by replacing each missing value
with a set of plausible values. You generate m complete data sets with m replacement values, and then you
combine the results. This produces unbiased results.
The MI procedure in SAS/STAT produces multiply imputed data sets for incomplete multivariate data sets.
The imputation method depends on the pattern of missingness and the type of the imputed variable. You then
use standard SAS procedures to analyze the m imputed data sets, and you use the MIANALYZE procedure
to combine the results and generate valid statistical inference.
The pattern of missing data determines the imputation method. There are many choices when you have
monotone missing data: for missing continuous variables, you can use a regression method, predictive
mean matching, or a propensity scoring method. For categorical missing variables, you can apply a logistic
regression method or a discriminant function method. When you have arbitrary missing data patterns, you
can use an MCMC method that assumes multivariate normality or a fully functional specification method
(FCS) that assumes the existence of a joint distribution for all variables. The FCS method is a recent addition
to the MI procedure, and it offers additional flexibility.
SAS/STAT offers other techniques for managing missing data. The CALIS procedure for fitting structural
equations model now provides full information maximum likelihood (FIML), which is an estimation method
that uses information from both the incomplete and the complete observations. Besides FIML estimation,
PROC CALIS also provides features for analyzing data coverage and missing patterns. Since structural
equations modeling includes measurement error models, regression models, and factor analyses as subset
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analyses, the FIML method it includes has a wide range of applications. See Yung and Zhang (2011) for
more information.
One of the benefits of the Bayesian approach is that it can handle missing data in a straightforward manner.
It treats the missing values as unknown parameters and estimates their posterior distributions. It is a
model-based solution, and the additional parameters don’t add that much additional complexity to the
analysis; they are simply sampled sequentially in the MCMC simulation. This approach does take into
account the uncertainty about the missing values so you can estimate the posterior marginal distributions of
the parameters of interest conditional on the observed and partially observed data. The MCMC procedure
automatically samples all missing values and incorporates them in the Markov chain for the parameters.
You can use PROC MCMC to handle various types of missing data, including data that are missing at
random (MAR) and missing not at random (MNAR). PROC MCMC can also perform joint modeling of missing
responses and covariates. See Chen (2013) for more information.
Managing missing data continues to be an important area of research and application, and providing
additional techniques is a major focus of current SAS/STAT research and development.
High-Performance Computing
SAS/STAT software has taken advantage of multithreading algorithms to improve performance in several
procedures, including the REG, GLM, LOESS, and GLMSELECT procedures, for years. See Cohen (2002)
for information on how this works and how the data configuration affects the resulting computing performance.
Recent new procedures in SAS/STAT come equipped with multithreading if it would benefit their performance.
These procedures include the FMM, QUANTSELECT, QUANTLIFE, and ADAPTIVEREG procedures.
Most recently, SAS has focused on meeting the challenges of Big Data with more attention to highperformance computing. New software products designed specifically for distributed computing include
the high-performance analytical products, which operate on data stored in databases such as Teradata,
Greenplum, and Hadoop and uses multiple parallel processing techniques across a grid of servers. New
statistical procedures were designed for this software, including those that perform logistic regression, linear
regression, mixed models, and model selection for both linear models and generalized linear models.
Beginning with Base SAS 9.4/SAS/STAT 12.3, released in the summer of 2013, the SAS/STAT product
includes the high-performance procedures for use in single-machine mode. These procedures were designed
to provide specific functionality required for Big Data analysis in a distributed environment, such as predictive
modeling. Not all features of the traditional procedure are included or are relevant. The high-performance
procedures are evolving and additional functionality such as graphics and BY-group processing will surface
in later releases. However, these procedures may provide benefit for the typical SAS/STAT user:
• New functionality is included in some of these procedures. For example, model selection for generalized
linear models is available with the HPGENSELECT procedure.
• Depending on the characteristics of the data and the complexity of the analysis, users may find
performance gains in single-machine mode with these procedures. However, note that if you compare
a high-performance procedure with a traditional procedure that is multithreaded (for example, PROC
HPREG compared to PROC REG), you are unlikely to see performance differences in the singlemachine mode of execution.
• Users with Big Data who would benefit from using the high-performance analytics products in a
distributed environment can exercise the procedures in single-machine mode and assess their functionality. When the customer needs to process Big Data (more observations and larger numbers of
variables), she can license the high-performance analytics products and execute the same procedures
in a distributed, in-memory environment.
The high-performance procedures are in the first stages of development. While production software, they
will be enhanced as time goes on and will include some additional features, such as ODS graphics and
BY-group processing. These procedures are documented in SAS/STAT 12.3 User’s Guide: High-Performance
Procedures. See Cohen (2013) for more detail.
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Complex Data
Data today are increasingly complex, or perhaps that complexity is being acknowledged because there are
more sophisticated methods available to analyze them. A major growth area for SAS/STAT has always been
statistical techniques that address data complexity. Recently, additional methods for handling complex data
include quantile regression, finite mixture models, and adaptive regression splines.
As previously discussed, quantile regression extends the basic regression model to the relationship between
the conditional quantiles of a response variable with one or more covariates. It makes no distributional
assumptions about the error term, and so it offers model robustness. It is a semiparametric method that can
provide a more complete picture of your data based on these conditional distributions. Linear programming
algorithms are used to produce the quantile regression estimates. See Koenker (2005) for further detail.
SAS/STAT provides the QUANTREG procedures for quantile regression analysis, the QUANTSELECT
procedure for model selection for quantile regression, and the QUANTLIFE procedure for right-censored
data.
New ADAPTIVEREG Procedure
SAS/STAT software provides various tools for nonparametric regression, including the LOESS, TPSPLINE,
and GAM procedures. Typical nonparametric regression methods involve a large number of parameters
to capture nonlinear trends, so the model space is fairly large. The sparsity of data in high dimensions
is another issue, often resulting in slow convergence or even failure for many nonparametric regression
methods.
The LOESS and TPSPLINE procedures are limited to problems in low dimensions. The GAM procedure fits
generalized additive models with the assumption of additivity. It can handle data sets, but the computation
time for its local scoring algorithm (Hastie and Tibshirani 1990) to converge increases quickly with the size of
the data set.
The new ADAPTIVEREG procedure provides a nonparametric modeling approach for high-dimensional data.
PROC ADAPTIVEREG fits multivariate adaptive regression splines as introduced by Friedman (1991b). The
method is a nonparametric regression technique that combines both regression splines and model selection
methods. It does not assume parametric model forms, and it does not require knot values for constructing
regression spline terms. Instead, it constructs spline basis functions in an adaptive way by automatically
selecting appropriate knot values for different variables; it performs model reduction by applying model
selection techniques. Thus, the ADAPTIVEREG procedure is both a nonparametric regression procedure
and a predictive modeling procedure.
The ADAPTIVEREG procedure:
• supports classification variables with different ordering options
• enables you to force effects into the final model or restrict variables in linear forms
• supports options for fast forward selection
• supports partitioning of data into training, validation, and testing roles
• provides leave-one-out and k-fold cross validation
• produces graphical representations of the selection process, model fit, functional components and fit
diagnostics
For more detail, see Kuhfeld and Cai (2013).
The following example illustrates the use of the ADAPTIVEREG procedure. Researchers collected data on
city-cycle fuel efficiency and automobile characteristics for 361 vehicle models manufactured from 1970 to
1982. The data can be downloaded from the UCI Machine Learning Repository (Asuncion and Newman
2007). The following DATA step creates the data set AUTOMPG:
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title 'Automobile MPG study';
data autompg;
input mpg cylinders displacement horsepower weight
acceleration year origin name $35.;
datalines;
18.0
8
307.0
130.0
3504
12.0
70
1
chevrolet chevelle malibu
15.0
8
350.0
165.0
3693
11.5
70
1
buick skylark 320
18.0
8
318.0
150.0
3436
11.0
70
1
plymouth satellite
16.0
8
304.0
150.0
3433
12.0
70
1
amc rebel sst
17.0
8
302.0
140.0
3449
10.5
70
1
ford torino
...
...
;
There are nine variables in the data set. The response variable MPG is city-cycle mileage per gallon (mpg).
Seven predictor variables (number of cylinders, displacement, weight, acceleration, horsepower, year and
origin) are created. The variables for number of cylinders, year, and origin are categorical.
The dependency of vehicle fuel efficiency on these factors might be nonlinear. Dependency structures
within the predictors might also mean that some of the predictors are redundant. For example, a model with
more cylinders is likely to have more horsepower. The object of this analysis is to explore the nonlinear
dependency structure and to find a parsimonious model that does not overfit the data. A more parsimonious
model has better predictive ability.
The following PROC ADAPTIVEREG statements fit an additive model with linear spline terms of continuous
predictors. The variable transformations and the model selection based on the transformed terms are
performed in an adaptive and automatic way. If the ADDITIVE option is not supplied, PROC ADAPTIVEREG
will fit a model with both main effects and two-way interaction terms.
ods graphics on;
proc adaptivereg data=autompg plots=all;
class cylinders year origin;
model mpg = cylinders displacement horsepower
weight acceleration year origin / additive;
run;
ods graphics off;
Figure 12 displays information on how the bases are constructed.
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Figure 12 Bases
Automobile MPG Study
Basis Information
Name
Transformation
Basis0
None
Basis1
Basis0*MAX(Weight -
Basis2
Basis0*MAX(
Basis3
Basis0*NOT(MISSING(HorsePower))
Basis4
Basis0*MISSING(HorsePower)
Basis5
Basis3*MAX(HorsePower -
Basis6
Basis3*MAX(
Basis7
Basis3*(Year = 80 OR Year = 82 OR Year = 81 OR Year = 79 OR Year = 78 OR Year = 77 OR
Year = 73)
Basis8
Basis3*NOT(Year = 80 OR Year = 82 OR Year = 81 OR Year = 79 OR Year = 78 OR Year = 77
OR Year = 73)
Basis9
Basis0*MAX(Acceleration -
Basis10
Basis0*MAX(
Basis11
Basis0*(Cylinders = 3 OR Cylinders = 6)
Basis12
Basis0*NOT(Cylinders = 3 OR Cylinders = 6)
Basis13
Basis4*(Origin = 1)
Basis14
Basis4*NOT(Origin = 1)
Basis15
Basis0*(Origin = 3)
Basis16
Basis0*NOT(Origin = 3)
Basis17
Basis0*(Cylinders = 6)
Basis18
Basis0*NOT(Cylinders = 6)
Basis19
Basis0*(Year = 73 OR Year = 80 OR Year = 82 OR Year = 81 OR Year = 79)
Basis20
Basis0*NOT(Year = 73 OR Year = 80 OR Year = 82 OR Year = 81 OR Year = 79)
3139,0)
3139 - Weight,0)
158,0)
158 - HorsePower,0)
21,0)
21 - Acceleration,0)
The “Parameter Estimates” table in Figure 13 displays parameter estimates for constructed basis functions
in addition to each function’s construction component. For example, BASIS1 has an estimate of –0.003242.
It is constructed from a parent basis function BASIS0 (intercept) and a linear spline function of WEIGHT with
a single knot placed at 3139. BASIS3 is constructed from a parent basis function BASIS0 and an indicator
function of YEAR. The indicator is set to 1 when a class level of YEAR falls into the subset of levels listed in
the “Levels” column and set to 0 otherwise.
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Figure 13 Parameter Estimates
Regression Spline Model after Backward Selection
Name
Coefficient
Parent
Variable
Knot
Levels
Basis0
29.4394
Intercept
Basis2
0.004412
Basis0
Weight
Basis3
-21.2899
Basis0
HorsePower
.
Basis6
0.1534
Basis3
HorsePower
158.00
Basis7
2.3920
Basis3
Year
Basis9
1.6658
Basis0
Acceleration
21.0000
Basis10
0.4672
Basis0
Acceleration
21.0000
Basis11
-8.1766
Basis0
Cylinders
03
Basis13
-10.0976
Basis4
Origin
0
Basis15
2.1354
Basis0
Origin
2
Basis17
6.7675
Basis0
Cylinders
3
Basis19
1.4987
Basis0
Year
3 10 12 11 9
3139.00
10 12 11 9 8 7 3
During the model construction and selection process, some basis function terms are removed.
Variable importance is another criterion that focuses on the contribution of each individual. Variable
importance is defined to be the square root of the GCV value of a submodel with all basis functions that
involve a removed variable, minus the square root of the GCV value of the selected model, then scaled to
have the largest importance value of 100. Figure 14 lists importance values for four variables that comprise
the selected model. Similar to the ANOVA decomposition results, WEIGHT and YEAR are two dominant
factors that determine vehicle mpg values, while DISPLACEMENT and ACCELERATION are less important.
Figure 14 Variable Importance
Variable Importance
Number of
Bases
Importance
HorsePower
1
100.00
Year
2
85.46
Weight
1
21.10
Cylinders
2
19.08
Origin
2
18.67
Acceleration
2
16.38
Variable
The component panel in Figure 15 displays the fitted functional components against their forming variables.
When a vehicle model’s displacement is less than 85, its mpg value increases with its displacement. The
displacement does not matter much once it exceeds 85. The shape of the functional component strongly
suggests a logarithm transformation. The component of WEIGHT shows that vehicle weight has negative
impact on its mpg value. The trend suggests a possible reciprocal transformation. When a model’s
acceleration value is larger than 20.7, it affects the mpg value in a positive manner. It does not matter much if
it is less than 20.7. Although YEAR is treated as a classification variable, its values are ordinal. The general
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trend is quite clear: more recent models tend to have higher mpg values. Automobile companies apparently
paid more attention to improving vehicle fuel efficiency after 1976.
Figure 15 Component Panel
Finite Mixture Models
Another type of complexity occurs when data can be viewed as coming from a mixture of different distributions.
Finite mixture models enable you to fit statistical models to data when the distribution of the response is
a finite mixture of univariate distributions. These models are useful for applications such as estimating
multimodal or heavy-tailed densities, fitting zero-inflated or hurdle models to count data with excess zeros,
modeling overdispersed data, and fitting regression models with complex error distributions. Many wellknown statistical models for dealing with overdispersed data are members of the finite mixture model family
(for example, zero-inflated Poisson models and zero-inflated negative binomial models.)
PROC FMM performs maximum likelihood estimation for all models, and it provides Markov chain Monte
Carlo estimation for many models, including zero-inflated Poisson models. The procedure includes many
built-in link and distribution functions, including the beta, shifted, Weibull, beta-binomial, and generalized
Poisson distributions, as well as standard members of the exponential family of distributions. In addition,
several specialized built-in mixture models are provided, such as the binomial cluster model (Morel and
Nagaraj, 1993).
The results of a finite mixture models analysis is displayed in Figure 16. The FMM procedure was used to fit
a three-component model–two normal components and a Weibull component– to log feeding time for cattle.
The plot shows the observed and estimated distributions for the response.
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Figure 16 Density Plot
Summary
Of course, recent SAS/STAT releases include numerous other updates. The STDRATE procedure computes
direct and indirect standardized rates and proportions, measures key in epidemiology. The survey data
analysis procedures continue to be a major development focus, with post-stratification estimation now
available with the SURVEYMEANS procedure and Poisson sampling included in PROC SURVEYSELECT.
Other new features include:
• WEIGHT statement in PROC LIFETEST
• partial R-square for relative importance of parameters in PROC LOGISTIC
• Miettinen-Nurminen confidence limits for the difference of proportions in PROC FREQ
• group sequential design with nonbinding acceptance boundary in the SEQDESIGN and SEQTEST
procedures
• REF= option added to the CLASS statement for GLM, MIXED, GLIMMIX, and ORTHOREG procedures
SAS/STAT software provides its customers with a rich array of current statistical techniques. Released
every 12-18 months, it provides modern statistical methodology via software that is geared towards user
expectations and shaped for today’s data.
For Further Information
A good place to start for further information is the “What’s New in SAS/STAT 12.1” chapter in the online
documentation. In addition, the Statistics and Operations Focus Area includes substantial information
about the statistical products, and you can find it at support.sas.com/statistics/. The quarterly
e-newsletter for that site is available on its home page. And of course, complete information is available in
the online documentation located here: support.sas.com/documentation/onlinedoc/stat/.
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Acknowledgments
The authors are grateful to Fang Chen for his contributions to the manuscript.
Contact Information
Your comments and questions are valued and encouraged. Contact the author:
Maura Stokes
SAS Institute Inc.
SAS Campus Drive
Cary, NC 27513
[email protected]
Version
1.0
SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of
SAS Institute Inc. in the USA and other countries. ® indicates USA registration.
Other brand and product names are trademarks of their respective companies.
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