STA429/1007 F 2004 Handout 8
Mixed and Random Effects, nesting (Bread and Health data)
/************* bread.sas ************************
Subsampling Example from NKNW Ch. 28 (Table 28.9)
**************************************************/
title 'Nested mixed model: Bread data';
options linesize=79 pagesize=35 noovp formdlim=' ';
proc format; /* value labels used in data step below */
value tempfmt 1 = 'Low' 2 = 'Medium' 3 = 'High';
data school;
infile 'bread.dat';
input temp batch crusty;
label crusty = 'Crustiness of bread';
format temp tempfmt.;
proc glm;
title2 'Classical F-tests with proc glm';
class temp batch;
model crusty = temp batch(temp); /* batch(temp) = batch within temp */
random batch(temp) / test; /* batch(temp) is a random effect.
Test it. */
means temp batch(temp);
proc varcomp method=type1; /* Just Expected Mean Squares */
title2 'Estimate variance components: Expected Mean Squares ';
class temp batch;
model crusty = temp batch(temp) / fixed=1; /* One fixed effect (Any
fixed effects are first) */
proc varcomp method=ML; /* Maximum Likelihood */
title2 'Maximum Likelihood Estimates of variance components';
class temp batch;
model crusty = temp batch(temp) / fixed=1;
Here is bread.lst
Nested mixed model: Bread data 1
Classical F-tests with proc glm
14:29 Sunday, October 24, 2004
The GLM Procedure
Class Level Information
Class Levels Values
temp 3 High Low Medium
batch 2 1 2
Number of observations 18
Nested mixed model: Bread data 2
Classical F-tests with proc glm
14:29 Sunday, October 24, 2004
The GLM Procedure
Dependent Variable: crusty Crustiness of bread
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 5 284.4444444 56.8888889 21.79 <.0001
Error 12 31.3333333 2.6111111
Corrected Total 17 315.7777778
R-Square Coeff Var Root MSE crusty Mean
0.900774 13.59163 1.615893 11.88889
Source DF Type I SS Mean Square F Value Pr > F
temp 2 235.4444444 117.7222222 45.09 <.0001
batch(temp) 3 49.0000000 16.3333333 6.26 0.0084
Source DF Type III SS Mean Square F Value Pr > F
temp 2 235.4444444 117.7222222 45.09 <.0001
batch(temp) 3 49.0000000 16.3333333 6.26 0.0084
Nested mixed model: Bread data 3
Classical F-tests with proc glm
14:29 Sunday, October 24, 2004
The GLM Procedure
Source Type III Expected Mean Square
temp Var(Error) + 3 Var(batch(temp)) + Q(temp)
batch(temp) Var(Error) + 3 Var(batch(temp))
Nested mixed model: Bread data 4
Classical F-tests with proc glm
14:29 Sunday, October 24, 2004
The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: crusty Crustiness of bread
Source DF Type III SS Mean Square F Value Pr > F
temp 2 235.444444 117.722222 7.21 0.0715
Error 3 49.000000 16.333333
Error: MS(batch(temp))
Source DF Type III SS Mean Square F Value Pr > F
batch(temp) 3 49.000000 16.333333 6.26 0.0084
Error: MS(Error) 12 31.333333 2.611111
Nested mixed model: Bread data 5
Classical F-tests with proc glm
14:29 Sunday, October 24, 2004
Level of ------------crusty-----------
temp N Mean Std Dev
High 6 16.5000000 1.87082869
Low 6 7.6666667 3.01109061
Medium 6 11.5000000 1.87082869
Level of Level of ------------crusty-----------
batch temp N Mean Std Dev
1 High 3 15.3333333 1.52752523
2 High 3 17.6666667 1.52752523
1 Low 3 5.3333333 1.52752523
2 Low 3 10.0000000 2.00000000
1 Medium 3 12.6666667 1.52752523
2 Medium 3 10.3333333 1.52752523
Nested mixed model: Bread data 6
Estimate variance components: Expected Mean Squares
14:29 Sunday, October 24, 2004
Variance Components Estimation Procedure
Class Level Information
Class Levels Values
temp 3 High Low Medium
batch 2 1 2
Number of observations 18
Dependent Variable: crusty
Type 1 Analysis of Variance
Sum of
Source DF Squares Mean Square
temp 2 235.444444 117.722222
Type 1 Analysis of Variance
Source Expected Mean Square
temp Var(Error) + 3 Var(batch(temp)) + Q(temp)
Nested mixed model: Bread data 7
Estimate variance components: Expected Mean Squares
14:29 Sunday, October 24, 2004
Variance Components Estimation Procedure
Type 1 Analysis of Variance
Sum of
Source DF Squares Mean Square
batch(temp) 3 49.000000 16.333333
Error 12 31.333333 2.611111
Corrected Total 17 315.777778 .
Type 1 Analysis of Variance
Source Expected Mean Square
batch(temp) Var(Error) + 3 Var(batch(temp))
Error Var(Error)
Corrected Total .
Type 1 Estimates
Variance Component Estimate
Var(batch(temp)) 4.57407
Var(Error) 2.61111
Nested mixed model: Bread data 8
Maximum Likelihood Estimates of variance components
14:29 Sunday, October 24, 2004
Variance Components Estimation Procedure
Class Level Information
Class Levels Values
temp 3 High Low Medium
batch 2 1 2
Number of observations 18
Dependent Variable: crusty
Maximum Likelihood Iterations
Var(batch
Iteration Objective (temp)) Var(Error)
0 24.9947701601 3.8117283951 2.1759259259
1 24.1609121270 2.1819629992 2.4911480395
2 24.1177557504 1.8652543177 2.6055996112
3 24.1176751083 1.8517086457 2.6111703288
4 24.1176750991 1.8518518356 2.6111111178
Nested mixed model: Bread data 9
Maximum Likelihood Estimates of variance components
14:29 Sunday, October 24, 2004
Variance Components Estimation Procedure
Convergence criteria met.
Maximum Likelihood Estimates
Variance
Component Estimate
Var(batch(temp)) 1.85185
Var(Error) 2.61111
Asymptotic Covariance Matrix of Estimates
Var(batch(temp)) Var(Error)
Var(batch(temp)) 2.59642 -0.37877
Var(Error) -0.37877 1.13632
/************* health.sas ************************
Subsampling Example from NWK prob 26.11
**************************************************/
title 'Pure random effects with subsampling: Health Awareness data';
options linesize=79 pagesize=35 noovp formdlim=' ';
data final;
infile 'health.dat';
input state city househld aware;
proc glm;
title2 'Classical F-test with proc glm';
class state city;
model aware = state city(state);
random state city(state) / test;
proc varcomp method=type1; /* Expected Mean Squares */
title2 'Estimate variance components: Expected Mean Squares ';
class state city;
model aware = state city(state);
proc varcomp method=ML; /* Maximum Likelihood */
title2 'Maximum Likelihood Estimates of variance components';
class state city;
model aware = state city(state);
proc sort; /* May be needed by proc nested */
by state city;
proc nested;
title2 'Estimate % of variance with proc nested';
class state city; /* Assume all IVs random and nested in order given */
var aware; /* Just name the DV(s) */
Here is health.lst.
Pure random effects with subsampling: Health Awareness data 1
Classical F-test with proc glm
14:36 Sunday, October 24, 2004
The GLM Procedure
Class Level Information
Class Levels Values
state 3 1 2 3
city 3 1 2 3
Number of observations 45
Pure random effects with subsampling: Health Awareness data 2
Classical F-test with proc glm
14:36 Sunday, October 24, 2004
The GLM Procedure
Dependent Variable: aware
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 8 7144.44444 893.05556 8.26 <.0001
Error 36 3893.20000 108.14444
Corrected Total 44 11037.64444
R-Square Coeff Var Root MSE aware Mean
0.647280 24.94490 10.39925 41.68889
Source DF Type I SS Mean Square F Value Pr > F
state 2 6976.844444 3488.422222 32.26 <.0001
city(state) 6 167.600000 27.933333 0.26 0.9526
Source DF Type III SS Mean Square F Value Pr > F
state 2 6976.844444 3488.422222 32.26 <.0001
city(state) 6 167.600000 27.933333 0.26 0.9526
Pure random effects with subsampling: Health Awareness data 3
Classical F-test with proc glm
14:36 Sunday, October 24, 2004
The GLM Procedure
Source Type III Expected Mean Square
state Var(Error) + 5 Var(city(state)) + 15 Var(state)
city(state) Var(Error) + 5 Var(city(state))
Pure random effects with subsampling: Health Awareness data 4
Classical F-test with proc glm
14:36 Sunday, October 24, 2004
The GLM Procedure
Tests of Hypotheses for Random Model Analysis of Variance
Dependent Variable: aware
Source DF Type III SS Mean Square F Value Pr > F
state 2 6976.844444 3488.422222 124.88 <.0001
Error 6 167.600000 27.933333
Error: MS(city(state))
Source DF Type III SS Mean Square F Value Pr > F
city(state) 6 167.600000 27.933333 0.26 0.9526
Error: MS(Error) 36 3893.200000 108.144444
Pure random effects with subsampling: Health Awareness data 5
Estimate variance components: Expected Mean Squares
14:36 Sunday, October 24, 2004
Variance Components Estimation Procedure
Class Level Information
Class Levels Values
state 3 1 2 3
city 3 1 2 3
Number of observations 45
Dependent Variable: aware
Type 1 Analysis of Variance
Sum of
Source DF Squares Mean Square
state 2 6976.844444 3488.422222
Type 1 Analysis of Variance
Source Expected Mean Square
state Var(Error) + 5 Var(city(state)) + 15 Var(state)
Pure random effects with subsampling: Health Awareness data 6
Estimate variance components: Expected Mean Squares
14:36 Sunday, October 24, 2004
Variance Components Estimation Procedure
Type 1 Analysis of Variance
Sum of
Source DF Squares Mean Square
city(state) 6 167.600000 27.933333
Error 36 3893.200000 108.144444
Corrected Total 44 11038 .
Type 1 Analysis of Variance
Source Expected Mean Square
city(state) Var(Error) + 5 Var(city(state))
Error Var(Error)
Corrected Total .
Type 1 Estimates
Variance Component Estimate
Var(state) 230.69926
Var(city(state)) -16.04222
Var(Error) 108.14444
Pure random effects with subsampling: Health Awareness data 7
Maximum Likelihood Estimates of variance components
14:36 Sunday, October 24, 2004
Variance Components Estimation Procedure
Class Level Information
Class Levels Values
state 3 1 2 3
city 3 1 2 3
Number of observations 45
Dependent Variable: aware
Maximum Likelihood Iterations
Var(city
Iteration Objective Var(state) (state)) Var(Error)
0 215.3846762821 202.5274227843 0 94.9383872844
1 215.2568105655 149.3097905207 0 96.6541361806
2 215.2567768466 148.5955055944 0 96.6857039721
3 215.2567768466 148.5955055944 0 96.6857039721
Convergence criteria met.
Pure random effects with subsampling: Health Awareness data 8
Maximum Likelihood Estimates of variance components
14:36 Sunday, October 24, 2004
Variance Components Estimation Procedure
Maximum Likelihood Estimates
Variance
Component Estimate
Var(state) 148.59551
Var(city(state)) 0
Var(Error) 96.68570
Asymptotic Covariance Matrix of Estimates
Var(state) Var(city(state)) Var(Error)
Var(state) 16027.2 0 -29.67765
Var(city(state)) 0 0 0
Var(Error) -29.67765 0 445.14883
Pure random effects with subsampling: Health Awareness data 9
Estimate % of variance with proc nested
21:10 Sunday, October 24, 2004
The NESTED Procedure
Coefficients of Expected Mean Squares
Source state city Error
state 15 5 1
city 0 5 1
Error 0 0 1
Nested Random Effects Analysis of Variance for Variable aware
Variance Sum of Error
Source DF Squares F Value Pr > F Term
Total 44 11038
state 2 6976.844444 124.88 <.0001 city
city 6 167.600000 0.26 0.9526 Error
Nested Random Effects Analysis
of Variance for Variable aware
Variance Variance Percent
Source Mean Square Component of Total
Total 250.855556 338.843704 100.0000
state 3488.422222 230.699259 68.0843
city 27.933333 -16.042222 0.0000
Pure random effects with subsampling: Health Awareness data 10
Estimate % of variance with proc nested
21:10 Sunday, October 24, 2004
The NESTED Procedure
Nested Random Effects Analysis of Variance for Variable aware
Variance Sum of Error
Source DF Squares F Value Pr > F Term
Error 36 3893.200000
Nested Random Effects Analysis
of Variance for Variable aware
Variance Variance Percent
Source Mean Square Component of Total
Error 108.144444 108.144444 31.9157
aware Mean 41.68888889
Standard Error of aware Mean 8.80457233