## SAS PROC GLIMMIX example

The following is a proc glimmix example syntax.  I ran it using a fake dataset, so the results are also fake.  The outcome is an interval variable and the model is a linear model (not a non-linear model like the logistic regression model).  The random statement makes this model "multilevel."  Level-1 units are students and level-2 units are schools (IDs are nces_school_name).  The intercept is being estimated as the grand average of school-specific intercepts.

This model is a multilevel model and different from the OLS model because it estimates intercept as random effects (school-specific intercepts are derived as random effects and the grand average of those are reported as the intercept).

proc glimmix data=temp1 ;
class nces_school_name ;
model y=x1 x2 x3/
random intercept /subject=nces_school_name;
run;

The main result table you will need to look at is the following.  You can interpret this just like you would interpret the OLS regression result.

 Solutions for Fixed Effects Effect Estimate Standard Error DF t Value Pr > |t| Intercept 0.5062 0.01600 1040 31.64 <.0001 x1 0.005776 0.01717 3483 0.34 0.7365 x2 0.000465 0.01684 3486 0.03 0.9780 x3 -0.01117 0.01708 3487 -0.65 0.5133

Another table that you want to look at is this.  Residual (0.54) is Level-1 variance (student-level variance).  Intercept/nces_school_name (0.65) is Level-2 variance (school-level variance).  I made up these numbers.

 Covariance Parameter Estimates Cov Parm Subject Estimate Standard Error Intercept nces_school_name 0.65 0.0181 Residual 0.54 0.2002

I will focus on the first coefficient table when discussing results; however, I would also report information related to the second table.  I would report the following information:

ICC (intraclass correlation):

level2 variance / (level1+level2 variance)

That is:

0.65 / (0.65 + 0.54) =0.546218

This shows the degree to which outcome variance is located between groups (schools, clusters) as opposed to within individuals.

I would also report "variance explained."

I need to run the additional model, which is the analysis-of-variance model where I have no covariate in the model:

proc glimmix data=temp1 ;
class nces_school_name ;
model y=/
random intercept /subject=nces_school_name;
run;

Let's take a look at the covariance table (I made up these values).

 Covariance Parameter Estimates Cov Parm Subject Estimate Standard Error Intercept nces_school_name 0.92 0.181 Residual 0.83 0.200

Let's combine this table with the other one in this way.

 Analysis of variance model Final model Variance explained Level 1 variance 0.83 0.54 0.34939759 Level 2 variance 0.92 0.65 0.293478261

Variance explained was calculated as:

(0.83-0.54)/0.83

and

(0.92-0.65)/0.92

I would make the final table look look like this.   I didn't round numbers, but you should.

 Estimates Standard error p-value statistical test Intercept 0.5062 0.016 <.0001 *** x1 0.00578 0.01717 0.7365 x2 0.00047 0.01684 0.978 x3 -0.0112 0.01708 0.5133 level-1 variance 0.83 level-2 variance 0.92 ICC 0.55 Level-1 variance explained 0.35 Level-2 variance explained 0.3 Notes: *** if p < 0.001, ** if p < 0.01, * if p < 0.05.

How to interpret coefficients in the table

In the table above, x1's coefficient is 0.00578.  This means that one unit increase in X1 will lead to an increase of 0.00578 in Y.   The p-value associated with this is 0.74.  So the coefficient here is not statistically significant at alpha=0.05.

One unit increase in X1 means ... if X is about height in meters (e.g., 1.5 meter, 1.7 meter), then 1 meter is 1 unit increase.  If X is a binary variable (0 or 1), then one unit increase means "0 to 1 increase".

For my work, I almost always have a variable called TREATMENT which is 1 if subjects received treatment/intervention and 0 if the subjects did not.  The coefficient for this is called "program impact effect."  If, for example, the program impact effect is 0.25, I just say that and I also mention that other covariates are in the model and the program impact effect is adjusted for these factors.  If the estimated program effect is 0.25, it means that the difference between the two groups (treatment vs. control) is 0.25 in outcome.

I also want to provide a standardized version of the program effect.  I would run the same statistical model with the z-score version of the outcome variable.  To do this, I usually use proc standard:

data abc2; set abc1;

Z_Y=Y;

run;

proc standard data=abc2 out=abc3 mean=0 std=1;

var Z_Y;

run;

Another approach would be to code this by hand in a datastep.  if the mean of Y is -0.42 and SD is 0.5:

data abc2; set abc1;

Z_Y= (Y - 0.42)/0.5 ;

run;

## Anova test using PROC GLIMMIX

%macro clem(var1=,var2=);
proc glimmix data=main.Final_year4;
class status;
model &var1=status
ods output
/*ModelInfo=x1var1 ParameterEstimates=x2var1 CovParms=x3var1*/
Diffs=DIF_RESULT1 LSMeans=LS1;
run;

data &var1.;
set LS1 DIF_RESULT1;
outcome_name="&var1";
v="vs.";
if Mu ne . then analysis_type="Descriptive Stats";
if Mu = . then do;
analysis_type="Anova";
if Probt < 0.05 then sig="* ";
if Probt < 0.01 then sig="** ";
if Probt < 0.001 then sig="***";
if Probt =. then sig="";
end;
drop effect;
run;

data &var2;
retain outcome_name analysis_type status v _status;
set &var1.;
run;

%mend clem;
%clem(var1=year4_outcome1,var2=a);
%clem(var1=year4_outcome2,var2=b);
%clem(var1=year4_outcome3,var2=c);
%clem(var1=year4_outcome4,var2=d);

data em;
run;

data allresults;
set a em b em c em d ;
run;

## Statistical joint test of categorical variables when expressed as a series of dummy variables

When I have multiple subgroup represented in a series of dummy variables (e.g.,  race groups, grade levels, etc.), I want to know if dummy variables as a system  contribute to the model with statistical significance.   This may be called a joint test because I want to know if, for example, race groups together (not separately)  make a differences to the model.

The easiest way to do this is to treat those variables as classification variables.  You will get a joint statistical test in one of the result tables.

proc glimmix ..;

....

run;

In my applications I almost always use numeric version of variables, i.e., dummy variables (coded as 0 or 1).  I like this approach because I can use PROC MEANS on them to create a descriptive statistics table.

The question is how I get joint statistical tests when  all of my predictors are numerically coded and thus I can't rely on the class statement (shown above in the syntax example).

The GLIMMIX syntax below treats race groups and grade levels as numerically coded dummy variables (if YES 1, else 0).

The parameter estimate tables will show coefficients derived for each of the numeric variables; however, I wouldn't know if race groups as a group matters to the model or grade levels as a system matters to the model.   For example, even when  the coefficient derived for subjects being black is statistically significant, that is only about how black students are different from white students (reference group in this example).  We don't know if race as a group matters and race groups jointly make a statistically significant contribution to the model.

<Again this can be done easily by using class variables instead (as shown earlier); however, I like using numeric variables in my models.>

Contrast statements will do the trick.

proc glimmix data=usethis namelen=32;
class groupunit;
output out=&outcome.gmxout residual=resid;
random intercept /subject=groupunit;
CONTRAST 'Joint F-Test Race groups ' Black 1, Hispanic 1, other 1;

ods output
ParameterEstimates=_3_&outcome.result covparms=_3_&outcome.cov
Contrasts=cont&outcome;
run;

## PROC GLIMMIX's option: lsmeans group / ilink diff;

proc glimmix data=asdf METHOD=RSPL;

class CAMPUS_14 subgroup;

model y=x1 x2 x3 subgroup

ods output  ModelInfo=x1var1 ParameterEstimates=x2var1 CovParms=x3var1

Diffs=DIF_RESULT1 LSMeans=LS1;

run;

## CHISQ test using SAS PROC FREQ

proc freq data=kaz.institution ;
weight weights;
tables PERSISTANY*COHORT /CHISQ ;run;

## HLM: What happens if I enter level2 variables at level 1?

The goal of this exercise is to find out what happens if I intentionally use a level2 variable at level 1 in HLM.   I found that the coefficients and standard errors remain about the same.  The parameter that differed was just the degree of freedom, which was consistent with my expectation.

Using my old NELS dataset, I ran two different HLM models using Bryk and Raudenbush’s software (See model 1 and model 2 equations in the table below).

• The URBAN as level 1 covariate model (I entered the level2 variable URBAN wrongly at level 1)
• The URBAN as level 2 covariate model (I entered the level2 variable URBAN correctly at level 2)

The outcome variable is the achievement composite (POSTTEST), students are level 1 and schools are level 2.  When expressed as mixed models, the two models are identical, which is why I expected most parameters to come out the same.

POSTTESTij = γ00
γ10*URBANij  + u0jrij

The first model (MODEL 1; see below) included URBAN (students are in urban school) as a level 1 predictor.  Of course this is a wrong specification because urban is a school characteristic.  In the second model (MODEL 2), I used it at the expected level, which is at level 2 (school level).

These models look different, but AGAIN when expressed as mixed models, they are identical.  As the third model (MODEL 3), I replicated the same HLM model using SAS PROC GLIMMIX.  SAS requires that the equation be expressed as a mixed model.

Results showed that coefficients and standard errors are more or less the same across three models.  The only one thing that was different was degree of freedom.

Conclusion: As long as variables enter the model as fixed effects as done here, there is nothing magical about the HLM model.  HLM software or SAS PROC GLIMMIX (option ddfm=kr) adjust degree of freedom values, accounting for the fact that URBAN is a school-level variable and thus should not be awarded a value that is too large.  Notice that under the correct specification (MODEL 2 and MODEL 3), the degree of freedom for URBAN is close to the number of schools, not to the number of students.

Thanks for any comments you may have.

MODEL 1 MODEL 2 MODEL 3
URBAN as LEVEL 1 covariate URBAN as level 2 covariate SAS PROC GLIMMIX
Level-1 Model

POSTTESTij = β0j + β1j*(URBANij) + rij

Level-2 Model

β0j = γ00 + u0j
β1j = γ10

Mixed Model

POSTTESTij = γ00
γ10*URBANij  + u0jrij

Level-1 Model

POSTTESTij = β0j + rij

Level-2 Model

β0j = γ00 + γ01*(URBAN_LEj) + u0j

Mixed Model

POSTTESTij = γ00 + γ01*URBAN_LEj  + u0jrij

proc glimmix data=kaz.level1;

class schoolID;

model

posttest =

urban

/solution ddfm=kr dist=normal link=identity s ;

random schoolID;

run;

Final estimation of fixed effects
(with robust standard errors)

 Fixed Effect Coefficient Standard error t-ratio Approx. d.f. p-value For INTRCPT1, β0 INTRCPT2, γ00 52.643432 0.526139 100.056 125 <0.001 For URBAN slope, β1 INTRCPT2, γ10 -0.450022 1.157924 -0.389 692 0.698

Final estimation of variance components

 Random Effect Standard Deviation Variance Component d.f. χ2 p-value INTRCPT1, u0 3.76951 14.20923 125 292.48369 <0.001 level-1, r 8.39004 70.39271

Statistics for current covariance components model

Final estimation of fixed effects
(with robust standard errors)

 Fixed Effect Coefficient Standard error t-ratio Approx. d.f. p-value For INTRCPT1, β0 INTRCPT2, γ00 52.643459 0.526140 100.056 124 <0.001 URBAN_LE, γ01 -0.449983 1.157920 -0.389 124 0.698

Final estimation of variance components

 Random Effect Standard Deviation Variance Component d.f. χ2 p-value INTRCPT1, u0 3.76919 14.20678 124 292.48068 <0.001 level-1, r 8.39007 70.39334

Statistics for current covariance components model

 Solutions for Fixed Effects Effect Estimate Standard Error DF t Value Pr > |t| Intercept 52.6434 0.5460 95.01 96.41 <.0001 urban -0.4501 1.1095 132.4 -0.41 0.6856 Covariance Parameter Estimates Cov Parm Estimate Standard Error schoolID 14.2150 3.4610 Residual 70.3913 3.7455

Datasets:

www.nippondream.com/file/datafiles_HLM.zip

Results from Model 1

Results from Model 2

Results from PROC GLIMMIX

## PROG GLM syntax -- contrasting groups using LSMEANS

proc glimmix data=kaz2.Year2AnalysisSample;
model
&z_post = treat
/solution ddfm=kr dist=normal link=identity s ;
random SCHOOL_NAME;
covtest /wald;
run;

## Cronbach Coefficient Alpha

SAS's proc cor procedure produces two types of cronbach coefficient alpha: raw value and standardized value.

proc corr alpha data=dataname_here;
var  item1 item2 item3 item4 item5 item6 item7;
run;

The result table includes two values:

Cronbach Coefficient Alpha

Variables Alpha
--------------------------------
Raw 0.74
Standardized 0.75

Standardized version is based on standardized values of all variables included in the analysis.  If you standardize the variables yourself by creating z-score version of items and apply the same procedure, you will get the same value for both raw and standardized values.

proc standardize data=dataname_here out=dataname_here_B mean=0 std=1;
var  item1 item2 item3 item4 item5 item6 item7;
run;
proc corr alpha data=dataname_here_B;
var  item1 item2 item3 item4 item5 item6 item7;
run;

Cronbach Coefficient Alpha

Variables Alpha
--------------------------------
Raw 0.75
Standardized 0.75

## How to edit a PROC GLIMMIX syntax into a PROC MIXED syntax

I encountered a situation where somehow PROC GLIMMIX can't produce results while PROC MIXED can.  In this WORD document, I showed you how I edited the PROC GLIMMIX syntax to be a PROC MIXED syntax. Please ignore my macro %w.