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Introduction to Multilevel Modeling Using SAS

This seminar is based on the paper Using SAS Proc Mixed to Fit Multilevel
Models, Hierarchical Models, and Individual Growth Models

by Judith Singer and can be downloaded from Professor Singer's web site at
http://gseweb.harvard.edu/~faculty/singer/Papers/sasprocmixed.pdf .

SAS data files, hsb12.sas7bdat and willett.sas7bdat and the SAS program code is here.

"The purpose of this paper is to show educational and behavioral
statisticians and researchers how they can use **proc mixed** to fit many common
types of multilevel models."

There are two types of models that this paper has focused on: (a) *school
effects* models and (b) *individual growth* models.

- A school effect model using data file hsb12.sas7bdat
- modeling organizational research;
- students nested within classes, children nested within families, patients nested within hospitals;

- Model 1: Unconditional Means Model
- Model 2: Including Effects of School Level (level 2) Predictors
- Model 3
:Including Effects of Student-Level Predictors- Model 4:
Including Both Level-1 and Level-2 Predictors

- Growth model using data file
willett.sas7bdat
- modeling individual change
- multiple observations on each individual as nested within the person;

- Model 1 :Unconditional Linear Growth Model
- Model 2: A Linear Growth Model with a Person-Level Covariance
- Model 3: Exploring the Structure of Variance Covariance Matrix Within Persons

A segment of the data file:

SCHOOL MATHACH SES MEANSES SECTOR 1296 6.588 -0.178 -0.420 0 1296 11.026 0.392 -0.420 0 1296 7.095 -0.358 -0.420 0 1296 12.721 -0.628 -0.420 0 1296 5.520 -0.038 -0.420 0 1296 7.353 0.972 -0.420 0 1296 7.095 0.252 -0.420 0 1296 9.999 0.332 -0.420 0 1296 10.715 -0.308 -0.420 0 1308 13.233 0.422 0.534 1 1308 13.952 0.562 0.534 1 1308 13.757 -0.058 0.534 1 1308 13.970 0.952 0.534 1 1308 23.434 0.622 0.534 1 1308 9.162 0.832 0.534 1 1308 23.818 1.512 0.534 1 1308 15.998 0.622 0.534 1 1308 16.039 0.332 0.534 1 1308 24.993 0.442 0.534 1 1308 15.657 0.582 0.534 1 1308 16.258 1.102 0.534 1

The data file is a subsample from the 1982 High School and Beyond Survey and
is used extensively in *Hierarchical Linear Models* by Raudenbush and Bryk.
The data file consists of 7185 students nested in 160 schools. The outcome
variable of interest is student-level math achievement score (**MATHACH**).
Variable **SES** is social-economic-status of a student and therefore is a
student-level variable. Variable **MEANSES** is the group mean of **SES**
and therefore is a school-level variable. Both **SES** and **MEANSES** are
centered at the grand mean (they both have means of 0). Variable **SECTOR**
is an indicator variable indicating if a school is public or catholic and is
therefore a school-level variable. There are 90 public schools (SECTOR=0) and 70
catholic schools (SECTOR=1) in the sample.

**Model 1: **Unconditional Means Model

This model is referred as a one-way ANOVA with random effects and is the
simplest possible random effect linear model and is discussed in detail by
Raudenbush and Bryk. The motivation for this model is the question on how
much schools vary in their mean mathematics achievement. In terms of regression
equations, we have the following, where r_{ij}
~ N(0, σ^{2}) and u_{0j }~ N(0, τ^{2}),

MATHACH_{ij }= β_{0j }+ r_{ij}

β_{0j }= γ_{00 }+ u_{0j}

MATHACH_{ij }= γ_{00 }+ u_{0j }+ r_{ij}

proc mixed data = in.hsb12 covtest noclprint; class school; model mathach = / solution; random intercept / subject = school; run;

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept SCHOOL 8.6097 1.0778 7.99 <.0001 Residual 39.1487 0.6607 59.26 <.0001

Fit Statistics

-2 Res Log Likelihood 47116.8 AIC (smaller is better) 47120.8 AICC (smaller is better) 47120.8 BIC (smaller is better) 47126.9

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.6370 0.2443 159 51.72 <.0001

**Comments**:

- In proc mixed, the statement
**MODEL**includes intercept as default. Therefore, we can further request that intercept be random in the**random**statement. - There are different estimation methods that proc mixed can use. The default is residual (restricted) maximum likelihood and is the method that we use here. This is also the default for HLM program.
- The option
**solution**in the model statement gives the parameter estimates for the fixed effect. - The option
**covtest**requests for the standard error for the covariance-variance parameter estimates and the corresponding z-test. - The option
**noclprint**requests that SAS not print the class information. - The estimated between variance, τ
^{2}corresponds to the term INTERCEPT in the output of Covariance Parameter Estimates and the estimated within variance, σ^{2}, corresponds to the term RESIDUAL in the same output section. - Based on the covariance estimates, we can compute the intraclass correlation: 8.6097/(8.6097+39.1487) = .18027614. This tells us the portion of the total variance that occurs between schools.
- To measure the magnitude of the variation among schools in their mean
achievement levels, we can calculate the
*plausible values range*for these means, based on the between variance we obtained from the model: 12.637 ± 1.96*(8.61)^{1/2}= (6.89, 18.39).

**Model 2**: Including Effects of School Level (level 2) Predictors --**
**predicting** mathach **from** meanses**

This model is referred as regression with Means-as-Outcomes by Raudenbush and
Bryk. The motivation of this model is the question on if the schools with high
**MEANSES** also have high math achievement. In other words, we want to
understand why there is a school difference on mathematics achievement. In terms
of regression equations, we have the following.

MATHACH_{ij }= β_{0j }+ r_{ij}

β_{0j }= γ_{00 }+ γ_{01}(MEANSES)
+ u_{0j}

Combining the two equations into one by substituting the level-2 equation to level-1 equation, we have

MATHACH_{ij }=
γ_{00 }+ γ_{01}(MEANSES)
+ u_{0j }+ r_{ij}

proc mixed data = in.hsb12 covtest noclprint; class school; model mathach = meanses / solution ddfm = bw; random intercept / subject = school; run;

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept SCHOOL 2.6357 0.4036 6.53 <.0001 Residual 39.1578 0.6608 59.26 <.0001

Fit Statistics

-2 Res Log Likelihood 46961.3 AIC (smaller is better) 46965.3 AICC (smaller is better) 46965.3 BIC (smaller is better) 46971.4

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.6495 0.1492 158 84.77 <.0001 MEANSES 5.8635 0.3613 158 16.23 <.0001

Type 3 Tests of Fixed Effects

Num Den Effect DF DF F Value Pr > F MEANSES 1 158 263.37 <.0001

**Comments**:

- The coefficient for the constant is the predicted math achievement when all predictors are 0, so when the average school SES is 0, the students math achievement is predicted to be 12.65.
- The variance component representing variation between schools decreases
greatly (from 8.6097 to 2.6357). This means that the level-2 variable
**meanses**explains a large portion of the school-to-school variation in mean math achievement. More precisely, the proportion of variance explained by**meanses**is (8.6097 - 2.6357)/8.6097 = .694, that is about 69% of the explainable variation in school mean math achievement scores is explained by**meanses**. - A range of plausible values for school means, given that all schools have
MEANSES of zero, is 12.65 ± 1.96 *(2.64)
^{1/2}= (9.47, 15.83). - We can also calculate the conditional intraclass correlation conditional on the values of MEANSES. 2.64/(2.64 + 39.16) = .06 measures the degree of dependence among observations within schools that are of the same MEANSES.
- Do school achievement means still vary significantly once MEANSES is controlled? From the output of Covariance Parameter Estimates, we see that the test that between variance is zero is highly significant. Therefore, we conclude that after controlling for MEANSES, significant variation among school mean math achievement still remains to be explained.
- Notice though, the standard error used to perform the above hypothesis test is based on large-sample theory of the maximum likelihood estimates and in many cases the normality approximation will be extremely poor. We will only use these results as guidance for further analysis, rather than definitive results. In SAS version 8 and later, SAS uses one-tailed z-test on variance and two-tailed z-test on covariance, trying to avoid misleading results by previously used two-tailed test for both.
- The option
**ddfm = bw**(between and within method) used in the model statement is to request SAS to use between and within method for computing the denominator degrees of freedom for the tests of fixed effects, instead of the default, containment method. This option is especially useful when there are large number of random effects in the model and the design is severely unbalanced. The default, on the other hand, matches the tests performed for balanced split-plot designs and should be adequate for moderately unbalanced designs.

**Model 3: **Including Effects of Student-Level Predictors--predicting**
mathach **from** **centered student-level ses, **cses**

This model is referred as a random-coefficient model by Raudenbush and Bryk.
Pretend that we run regression of **mathach** on centered **ses** on each
school, that is we are going to run 160 regressions.

- What would be the average of the 160 regression equations (both intercept and slope)?
- How much do the regression equations vary from school to school?
- What is the correlation between the intercepts and slopes?

These are some of the questions that motivates the following model.

MATHACH_{ij }= β_{0j }+ β_{1j}
(SES - MEANSES) + r_{ij}

β_{0j }= γ_{00 } + u_{0j
}β_{1j }= γ_{10 } + u_{1j}

Combining the two equations into one by substituting the level-2 equation to level-1 equation, we have

MATHACH_{ij }=
γ_{00 }+ γ_{10}(SES - MEANSES)
+ u_{0j
}+ u_{1j}(SES - MEANSES) + r_{ij}

data hsbc; set in.hsb12; cses = ses - meanses; run; proc mixed data = hsbc noclprint covtest noitprint; class school; model mathach = cses / solution ddfm = bw notest; random intercept cses / subject = school type = un gcorr; run;

Estimated G Correlation Matrix

Row Effect SCHOOL Col1 Col2

1 Intercept 1224 1.0000 0.02068 2 cses 1224 0.02068 1.0000

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) SCHOOL 8.6769 1.0786 8.04 <.0001 UN(2,1) SCHOOL 0.05075 0.4062 0.12 0.9006 UN(2,2) SCHOOL 0.6940 0.2808 2.47 0.0067 Residual 36.7006 0.6258 58.65 <.0001 Fit Statistics

-2 Res Log Likelihood 46714.2 AIC (smaller is better) 46722.2 AICC (smaller is better) 46722.2 BIC (smaller is better) 46734.5

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq

3 1065.70 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.6493 0.2445 159 51.75 <.0001 cses 2.1932 0.1283 7024 17.10 <.0001

**Comments**:

- Specifying level-1 predictor
**cses**as random effect, we formulate that effect of**cses**can vary across schools. - The option
**type = un**in the random statement allows us to estimate the three parameters (the variance of**intercept**and the variance of slopes for**cses**and the covariance between them) from the data. - Option
**gcorr**displays the correlation matrix corresponding to the estimated variance-covariance matrix, called G matrix. - The covariance estimate is 0.05075 with standard error 0.4062. That
yields a p-vlaue of 0.9006. This is saying that there is no evidence that the
effect of
**cses**depending upon the average math achievement in the school. - In the output of Covariance Parameter Estimates, the parameter
corresponding to UN(2,2) is the variability in slopes of
**cses**. The estimate is 0.6940 with standard error 0.2808. That yields a p-value of 0.0067 for 1-tailed test. The test being significant tells us that we can not accept the hypothesis that there is no difference in slopes among schools. - The 95% plausible value range for the school means is 12.65
± 1.96 *(8.68)
^{1/2}= (6.87, 18.41). - The 95% plausible value range for the
SES-achievement slope is 2.19 ± 1.96
*(.69)
^{1/2}= (.56, 3.82). - Notice that the residual variance is now 36.70, comparing with the residual variance of 39.15 in the one-way ANOVA with random effects model. We can compute the proportion variance explained at level 1 by (39.15 - 36.70) / 39.15 = .063. This means using student-level SES as a predictor of math achievement reduced the within-school variance by 6.3%.

**Model 4: **Including Both Level-1 and Level-2 Predictors --predicting**
mathach **from** meanses, sector, cses** and the cross level interaction of
** meanses **and **sector** with **cses**

This model is referred as an intercepts and slopes-as-outcomes model by Raudenbush and Bryk. We have examined the variability of the regression equations across schools. Now we will build an explanatory model to account for the variability. That is we want to model the following:

MATHACH_{ij }= β_{0j }+ β_{1j}
(SES - MEANSES) + r_{ij}

β_{0j }= γ_{00 } + γ_{01}(MEANSES)
+ γ_{02}(SECTOR) + u_{0j
}β_{1j }= γ_{10 } + γ_{11}(MEANSES) + γ_{12}(SECTOR)
+ u_{1j}

Combining the two equations into one by substituting the level-2 equation to level-1 equation, we have

MATHACH_{ij }=
γ_{00 } + γ_{01}(MEANSES)
+ γ_{02}(SECTOR) + γ_{10}
(SES - MEANSES) +

γ_{11}(MEANSES)*
(SES - MEANSES) + γ_{12}(SECTOR)*
(SES - MEANSES) +

u_{0j} +u_{1j}(SES-MEANSES)
+ r_{ij}

The questions that we are interested in are:

- Do MEANSES and SECTOR significantly predict the intercept?
- Do MEANSES and SECTOR significantly predict the within-school slopes?
- How much variation in the intercepts and the slopes is explained by MEANSES and SECTOR?

proc mixed data = hsbc noclprint covtest noitprint; class school; model mathach = meanses sector cses meanses*cses sector*cses / solution ddfm = bw notest; random intercept cses / subject = school type = un; run;

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) SCHOOL 2.3817 0.3717 6.41 <.0001 UN(2,1) SCHOOL 0.1926 0.2045 0.94 0.3464 UN(2,2) SCHOOL 0.1014 0.2138 0.47 0.3177 Residual 36.7212 0.6261 58.65 <.0001 Fit Statistics

-2 Res Log Likelihood 46503.7 AIC (smaller is better) 46511.7 AICC (smaller is better) 46511.7 BIC (smaller is better) 46524.0

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq 3 220.57 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.1136 0.1988 157 60.93 <.0001 MEANSES 5.3391 0.3693 157 14.46 <.0001 SECTOR 1.2167 0.3064 157 3.97 0.0001 cses 2.9388 0.1551 7022 18.95 <.0001 MEANSES*cses 1.0389 0.2989 7022 3.48 0.0005 SECTOR*cses -1.6426 0.2398 7022 -6.85 <.0001

**Comments**:

- First take a look at the output of Solutions for Fixed Effects. The first three parameters are about the intercept, or more precisely about the mean math achievement across schools. We see that MEANSES is positively related to math achievement and catholic schools have significantly higher mean math achievement than public schools, controlling for other effects.
- The last three parameters in the output are about the slopes. Schools of high MEANSES tend to have larger slopes and catholic schools have significantly weaker slopes, on the average, than public schools.
- Variable
**sector**and its interaction with**cses**are significant in the model, indicating that the intercepts and the slopes for**cses**are different for Catholic and public schools. This can also be shown by plotting the predicted math achievement scores constraining the meanses to low, medium and high. We use 25th/50th/75th percentiles to define the strata of low, medium and high.**proc univariate data = hsbc; var meanses; run;**/* 90% 0.523 75% Q3 0.333 50% Median 0.038 25% Q1 -0.317 10% -0.579 5% -0.696 1% -1.043 0% Min -1.188 */**data toplot; set hsbc; if meanses <= -0.317 then do; ms = -0.317; strata = "Low"; end; else if meanses >= 0.333 then do; ms = 0.333; strata = "Hig"; end; else do; ms = 0.038; strata = "Med" ; end; predicted = 12.1136 + 5.3391*ms + 1.2167*sector + 2.9388*cses + 1.0389*ms*cses - 1.6426*sector*cses; run; proc sort data = toplot; by strata; run; goptions reset = all; symbol1 v = none i = join c = red ; symbol2 v = none i = join c = blue ; axis1 order = (-4 to 3 by 1) minor = none label=("Group Centered SES"); axis2 order = (0 to 22 by 2) minor = none label=(a = 90 "Math Achievement Score"); proc gplot data = toplot; by strata; plot predicted*cses = sector / vaxis = axis2 haxis = axis1; run; quit;** - Possibly there would be two-way interaction between
**meanses**and**sector**and a three way interaction between**meanses**,**cses**and**sector**. We can test it by adding the interaction into the model. For example,**proc mixed data = hsbc noclprint covtest noitprint; class school; model mathach = meanses sector cses meanses*sector meanses*cses sector*cses meanses*sector*cses / solution ddfm = bw notest; random intercept cses / subject = school type = un; run;**Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.1842 0.2030 156 60.01 <.0001 MEANSES 5.8732 0.5065 156 11.60 <.0001 SECTOR 1.2430 0.3052 156 4.07 <.0001 cses 2.9513 0.1616 7021 18.26 <.0001

**MEANSES*SECTOR -1.1276 0.7355 156 -1.53 0.1273 MEANSES*SECTOR*cses -0.1888 0.5997 7021 -0.31 0.7528**MEANSES*cses 1.1289 0.4232 7021 2.67 0.0077 SECTOR*cses -1.6407 0.2406 7021 -6.82 <.0001 - Since the variance component for slopes is very small and its
corresponding p-value is 0.3177. We cannot reject the hypothesis that the
slopes do not differ across schools. Similarly, we can not reject the
hypothesis that the covariance between intercepts and slopes is zero.
Therefore, a simpler model can be used:
**proc mixed data = hsbc noclprint covtest noitprint; class school; model mathach = meanses sector cses meanses*cses sector*cses / solution ddfm = bw notest; random intercept / subject = school; run;**Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept SCHOOL 2.3752 0.3709 6.40 <.0001 Residual 36.7661 0.6207 59.24 <.0001

Fit Statistics

-2 Res Log Likelihood 46504.8 AIC (smaller is better) 46508.8 AICC (smaller is better) 46508.8 BIC (smaller is better) 46514.9

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 12.1138 0.1986 157 60.98 <.0001 MEANSES 5.3429 0.3690 157 14.48 <.0001 SECTOR 1.2146 0.3061 157 3.97 0.0001 cses 2.9358 0.1507 7022 19.48 <.0001 MEANSES*cses 1.0441 0.2910 7022 3.59 0.0003 SECTOR*cses -1.6421 0.2331 7022 -7.04 <.0001

To compare the original model with this simplified one, we can compare their -2LL's, since the fixed portion of these two models are the same.

Model Number of parameters -2 LL restricted 2 46504.8 Unrestricted 4 46503.7 Approximately, the difference in -2LL's is a χ

^{2}distribution with two degrees of freedom (corresponding to the difference in the number of parameters). The p-value is .577. This justifies the use of the simpler model. The SAS program is shown below.**data pvalue; df = 2; chisq = 46504.8 - 46503.7; pvalue = 1 - probchi(chisq, df); run; proc print data = pvalue noobs; run;**df chisq pvalue

2 1.1 0.57695

A segment of the data file:

id time cons covar y 1 0 1 137 205 1 1 1 137 217 1 2 1 137 268 1 3 1 137 302 2 0 1 123 219 2 1 1 123 243 2 2 1 123 279 2 3 1 123 302 3 0 1 129 142 3 1 1 129 212 3 2 1 129 250 3 3 1 129 289 4 0 1 125 206 4 1 1 125 230 4 2 1 125 248 4 3 1 125 273 5 0 1 81 190 5 1 1 81 220 5 2 1 81 229 5 3 1 81 220

**Model 1**: Unconditional Linear Growth Model -- page 340

proc mixed data = willett noclprint covtest; class id; model y = time /solution ddfm = bw notest; random intercept time / subject = id type = un; run;

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 1198.78 318.38 3.77 <.0001 UN(2,1) id -179.26 88.9634 -2.01 0.0439 UN(2,2) id 132.40 40.2107 3.29 0.0005 Residual 159.48 26.9566 5.92 <.0001

Fit Statistics

-2 Res Log Likelihood 1266.8 AIC (smaller is better) 1274.8 AICC (smaller is better) 1275.1 BIC (smaller is better) 1281.0

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq

3 120.90 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.37 6.1188 34 26.86 <.0001 time 26.9600 2.1666 104 12.44 <.0001

**Comments**:

- Notice that variable
**time**is coded 0, 1, 2 and 3. Therefore, the intercept is the estimate of the initial value and the slope is the estimate of the rate of change across occasions. - We may want to visually see the relationship between the dependent
variable and time by subject. This gives us a good sense if the linear
relationship holds across all the subjects and if the slopes vary across all
the subjects.
**proc gplot data = willett; plot y*time = id; where id <=20; run; quit;**

**Model 2**: A Linear Growth Model with a Person-Level Covariance --
predicting y by **time** and centered **covar** -- page 344

data willett; set in.willett; wave = time; ccovar = covar - 113.4571429; run;

proc mixed data = willett noclprint covtest; class id; model y = time ccovar time*ccovar /solution ddfm = bw notest; random intercept time / subject = id type = un gcorr; run;

Estimated G Correlation Matrix

Row Effect id Col1 Col2 1 Intercept 1 1.0000 -0.4895 2 time 1 -0.4895 1.0000

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 1236.41 332.40 3.72 <.0001 UN(2,1) id -178.23 85.4298 -2.09 0.0370 UN(2,2) id 107.25 34.6767 3.09 0.0010 Residual 159.48 26.9566 5.92 <.0001

Fit Statistics

-2 Res Log Likelihood 1260.3 AIC (smaller is better) 1268.3 AICC (smaller is better) 1268.6 BIC (smaller is better) 1274.5

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq

3 120.72 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.37 6.2061 33 26.49 <.0001 time 26.9600 1.9939 103 13.52 <.0001 ccovar -0.1136 0.5040 33 -0.23 0.8231 time*ccovar 0.4329 0.1619 103 2.67 0.0087

**Comments**:

- Variable
**wave**created in the data step will be used in our next model. - Estimated correlation matrix among the random effect is requested by using
the option
**gcorr**. - Comparing with the model of unconditional growth, this model

**Model 3**: Exploring the Structure of Variance Covariance Matrix Within
Persons

**A. Compound Symmetry**

proc mixed data = willett covtest noitprint; class id wave; model y = time / s notest; repeated wave /type = cs subject = id r; run;

Estimated R Matrix for id 1

Row Col1 Col2 Col3 Col4 1 1280.71 904.81 904.81 904.81 2 904.81 1280.71 904.81 904.81 3 904.81 904.81 1280.71 904.81 4 904.81 904.81 904.81 1280.71

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z CS id 904.81 242.59 3.73 0.0002 Residual 375.90 52.1281 7.21 <.0001

Fit Statistics

-2 Res Log Likelihood 1300.3 AIC (smaller is better) 1304.3 AICC (smaller is better) 1304.4 BIC (smaller is better) 1307.5

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq

1 87.39 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.37 5.7766 34 28.45 <.0001 time 26.9600 1.4656 104 18.40 <.0001

**B.Unstructured**

proc mixed data = willett covtest noitprint; class id wave; model y = time / s notest; repeated wave /type = un subject = id r; run;

Estimated R Matrix for id 1

Row Col1 Col2 Col3 Col4 1 1307.96 977.17 921.87 563.54 2 977.17 1120.32 1018.97 855.53 3 921.87 1018.97 1289.47 1081.77 4 563.54 855.53 1081.77 1415.03

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z UN(1,1) id 1307.96 316.95 4.13 <.0001 UN(2,1) id 977.17 266.55 3.67 0.0002 UN(2,2) id 1120.32 270.69 4.14 <.0001 UN(3,1) id 921.87 272.81 3.38 0.0007 UN(3,2) id 1018.97 269.55 3.78 0.0002 UN(3,3) id 1289.47 312.07 4.13 <.0001 UN(4,1) id 563.54 252.45 2.23 0.0256 UN(4,2) id 855.53 260.70 3.28 0.0010 UN(4,3) id 1081.77 296.64 3.65 0.0003 UN(4,4) id 1415.03 343.17 4.12 <.0001

Fit Statistics

-2 Res Log Likelihood 1263.4 AIC (smaller is better) 1283.4 AICC (smaller is better) 1285.2 BIC (smaller is better) 1299.0

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq

9 124.30 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 165.83 5.8668 34 28.27 <.0001 time 26.5846 2.1215 34 12.53 <.0001

**C. AR(1)**

proc mixed data = willett covtest noitprint; class id wave; model y = time / s notest; repeated wave /type = ar(1) subject = id r; run;

Estimated R Matrix for id 1

Row Col1 Col2 Col3 Col4 1 1323.77 1092.07 900.93 743.24 2 1092.07 1323.77 1092.07 900.93 3 900.93 1092.07 1323.77 1092.07 4 743.24 900.93 1092.07 1323.77

Covariance Parameter Estimates

Standard Z Cov Parm Subject Estimate Error Value Pr Z AR(1) id 0.8250 0.03937 20.96 <.0001 Residual 1323.77 258.56 5.12 <.0001

Fit Statistics -2 Res Log Likelihood 1273.5 AIC (smaller is better) 1277.5 AICC (smaller is better) 1277.6 BIC (smaller is better) 1280.6

Null Model Likelihood Ratio Test

DF Chi-Square Pr > ChiSq 1 114.26 <.0001

Solution for Fixed Effects

Standard Effect Estimate Error DF t Value Pr > |t| Intercept 164.34 6.1371 34 26.78 <.0001 time 27.1979 1.9198 104 14.17 <.0001

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