This document contains the output of fitting a linear mixed-effects model to city revenue data. It includes: 1) The model output, showing random effects structure, fixed effects coefficients, and residual statistics. Random effects include intercepts and slopes that vary by city. 2) Functions to extract the random effects estimates and covariance matrix from the fitted model. 3) Calculations to estimate the among-group covariance matrix from the random effects and compare it to the estimated covariance matrix from the model.

1. > random.effects(city.rcr)
Error: could not find function "random.effects"
> library(nlme)
> summary(city.rcr)
Linear mixed-effects model fit by REML
Data: NULL
AIC BIC logLik
1821.515 1852.99 -900.7574
Random effects:
Formula: ~datamat[, 4] + datamat[, 5] | datamat[, 2]
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 190.326223 (Intr) dt[,4]
datamat[, 4] 56.925087 0.981
datamat[, 5] 2.682005 0.963 0.986
Residual 24.536273
Fixed effects: datamat[, 1] ~ datamat[, 4] + datamat[, 5]
Value Std.Error DF t-value p-value
(Intercept) 136.39569 38.17054 148 3.573324 0.0005
datamat[, 4] 29.30436 11.42273 148 2.565444 0.0113
datamat[, 5] 1.26051 0.75790 148 1.663165 0.0984
Correlation:
(Intr) dt[,4]
datamat[, 4] 0.975
datamat[, 5] 0.640 0.695
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-3.194121265 -0.206262211 -0.001507792 0.198449874 6.407690740
Number of Observations: 175
Number of Groups: 25
> random.effects(city.rcr)
(Intercept) datamat[, 4] datamat[, 5]
1 -29.403774 -15.584571 -0.7722463
2 -117.505875 -27.842606 -1.2552137
3 -47.748165 -29.003930 -1.3762317
4 -63.095682 -21.409519 -0.9970781
5 -111.220805 -28.137529 -1.2899786
6 815.889477 242.382177 11.3117023
7 -113.028470 -24.218162 -1.0606728
8 -33.612794 -7.229820 -0.3687349
9 128.387215 24.765428 1.1932962
10 56.521249 -3.704298 -0.3271703
11 -54.702653 -5.005235 -0.1387420
12 -117.331198 -27.203998 -1.2221728
13 -71.551241 -30.965424 -1.4732413
14 -6.454851 2.266289 0.1646867
15 -108.574858 -25.292143 -1.1284143
16 -84.682279 -22.004758 -1.0726460
17 239.106120 94.193221 4.2382465
18 -35.606421 -16.025132 -0.7973699
19 -14.120941 -15.878845 -0.8213709
20 34.702867 7.777985 0.3959752
21 -97.236760 -26.762150 -1.2146494
22 -82.794647 -23.851708 -1.1093991
23 -42.900392 -5.124046 -0.1733412
24 -109.220834 -24.095606 -1.0438236

2. 25 66.185710 7.954381 0.3385900
> corMatrix(city.rcr)
Error in UseMethod("corMatrix") :
no applicable method for 'corMatrix' applied to an object of class "lme"
> corMatrix(city.rcr)
Error in UseMethod("corMatrix") :
no applicable method for 'corMatrix' applied to an object of class "lme"
> summary(city.rcr)
Linear mixed-effects model fit by REML
Data: NULL
AIC BIC logLik
1821.515 1852.99 -900.7574
Random effects:
Formula: ~datamat[, 4] + datamat[, 5] | datamat[, 2]
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 190.326223 (Intr) dt[,4]
datamat[, 4] 56.925087 0.981
datamat[, 5] 2.682005 0.963 0.986
Residual 24.536273
Fixed effects: datamat[, 1] ~ datamat[, 4] + datamat[, 5]
Value Std.Error DF t-value p-value
(Intercept) 136.39569 38.17054 148 3.573324 0.0005
datamat[, 4] 29.30436 11.42273 148 2.565444 0.0113
datamat[, 5] 1.26051 0.75790 148 1.663165 0.0984
Correlation:
(Intr) dt[,4]
datamat[, 4] 0.975
datamat[, 5] 0.640 0.695
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-3.194121265 -0.206262211 -0.001507792 0.198449874 6.407690740
Number of Observations: 175
Number of Groups: 25
> bigsigma = matrix(0,3,3)
> for (i in 1:25)
+ bigsigma = bigsigma + (newbeta[,i]%*%t(newbeta[,i]))/(n-1)
> bigsigma
[,1] [,2] [,3]
[1,] 36404.4770 10628.6980 473.08450
[2,] 10628.6980 3263.4453 151.01059
[3,] 473.0845 151.0106 11.78515
> sum2 = smallsigma*solve(t(x)%*%x)
> sum2
[,1] [,2] [,3]
[1,] 217.67560 0.00000 -31.096514
[2,] 0.00000 23.32239 0.000000
[3,] -31.09651 0.00000 7.774129
> bigsigma - sum2
[,1] [,2] [,3]
[1,] 36186.801 10628.6980 504.181012
[2,] 10628.698 3240.1229 151.010595
[3,] 504.181 151.0106 4.011024