Example L3 Continuous response growth model

Snijders & Bosker, (1999) analysed the development over time of teacher evaluations.  Starting from the first year of their career, teachers were evaluated on their interpersonal behaviour in the classroom.  This happened repeatedly, at intervals of about one year.  In this example, results are presented about the 'proximity' dimension, representing the degree of cooperation or closeness between a teacher and his or her students.  The higher the proximity score of a teacher, the more cooperation is perceived by his or her students.

There are four measurement occasions: after 0, 1, 2, and 3 years of experience.  A total of 51 teachers were studied. The number of observations for the 4 moments decreased from 46 at time i=0 to 32 at time i=3. The non-response at various is treated as ignorable.

Reference

Snijders, T. A. B. and Bosker, R. J. (1999). Multilevel Analysis. London: Sage.

Data description

Number of observations (rows): 153

Number of variables (columns): 8

The subset of variables we use are:

teacher =  teacher identifier

time = 0,1,2,3 the year at which the teacher evaluation was made

proximity = degree of cooperation or closeness between a teacher and his or her students

gender = 1 if teacher is female, 0 otherwise

d1 =  1 if time =0, 0 otherwise

d2 =  1 if time =1, 0 otherwise

d3 =  1 if time =2, 0 otherwise

d4 =  1 if time =3, 0 otherwise

The first few lines of growth.dat look like

Sabre commands

out growth.log

trace growth.trace

data teacher time proximity gender d1 d2 d3 d4

case teacher

yvar proximity

family g

constant cons

lfit cons

dis m

dis e

mass 64

scale 0.5

fit cons

dis m

dis e

lfit d1 d2 d3 d4

dis m

dis e

sigma 0.25

scale 0.5

fit d1 d2 d3 d4

dis m

dis e

stop

Sabre log file

<S> trace growth.trace

<S> data teacher time proximity gender d1 d2 d3 d4

153 observations in dataset

<S> case teacher

<S> yvar proximity

<S> family g

<S> constant cons

<S> lfit cons

Iteration       Log. lik.       Difference

__________________________________________

1          -88.715008

<S> dis m

X-vars            Y-var

______________________________

cons              proximity

Univariate model

Standard linear

Number of observations             =     153

X-var df           =     1

Log likelihood =     -88.715008     on     152 residual degrees of freedom

<S> dis e

Parameter              Estimate         Std. Err.

___________________________________________________

cons                   0.63856          0.35048E-01

sigma                  0.43352

<S> mass 64

<S> scale 0.5

<S> fit cons

Initial Homogeneous Fit:

Iteration       Log. lik.       Difference

__________________________________________

1          -88.715008

Iteration       Log. lik.         Step      End-points     Orthogonality

length    0          1      criterion

________________________________________________________________________

1          -84.581007        1.0000    fixed  fixed       136.94

2          -75.772623        0.5000    fixed  fixed       1298.9

3          -64.493890        1.0000    fixed  fixed       741.33

4          -61.916672        1.0000    fixed  fixed       606.92

5          -61.737341        1.0000    fixed  fixed       1198.4

6          -61.702731        1.0000    fixed  fixed       661.65

7          -61.688023        1.0000    fixed  fixed       1557.1

8          -61.688018        1.0000    fixed  fixed       1843.3

9          -61.688018        1.0000    fixed  fixed

<S> dis m

X-vars            Y-var             Case-var

________________________________________________

cons              proximity         teacher

Univariate model

Standard linear

Gaussian random effects

Number of observations             =     153

Number of cases                    =      51

X-var df           =     1

Sigma df           =     1

Scale df           =     1

Log likelihood =     -61.688018     on     150 residual degrees of freedom

<S> dis e

Parameter              Estimate         Std. Err.

___________________________________________________

cons                   0.64795          0.53346E-01

sigma                  0.27155          0.19025E-01

scale                  0.34388          0.42213E-01

<S> lfit d1 d2 d3 d4

Iteration       Log. lik.       Difference

__________________________________________

1          -87.536537

<S> dis m

X-vars            Y-var

______________________________

d1                proximity

d2

d3

d4

Univariate model

Standard linear

Number of observations             =     153

X-var df           =     4

Log likelihood =     -87.536537     on     149 residual degrees of freedom

<S> dis e

Parameter              Estimate         Std. Err.

___________________________________________________

d1                     0.58652          0.64064E-01

d2                     0.72395          0.70485E-01

d3                     0.64378          0.71431E-01

d4                     0.60594          0.76810E-01

sigma                  0.43450

<S> sigma 0.25

<S> scale 0.5

<S> fit d1 d2 d3 d4

Initial Homogeneous Fit:

Iteration       Log. lik.       Difference

__________________________________________

1          -87.536537

Iteration       Log. lik.         Step      End-points     Orthogonality

length    0          1      criterion

________________________________________________________________________

1          -64.095200        1.0000    fixed  fixed       94.548

2          -63.262786        0.5000    fixed  fixed       773.48

3          -60.084462        1.0000    fixed  fixed       507.61

4          -59.421825        1.0000    fixed  fixed       182.88

5          -59.263588        1.0000    fixed  fixed       310.69

6          -59.204119        1.0000    fixed  fixed       204.81

7          -59.146517        1.0000    fixed  fixed       335.13

8          -59.146452        1.0000    fixed  fixed       348.43

9          -59.146452        1.0000    fixed  fixed

<S> dis m

X-vars            Y-var             Case-var

________________________________________________

d1                proximity         teacher

d2

d3

d4

Univariate model

Standard linear

Gaussian random effects

Number of observations             =     153

Number of cases                    =      51

X-var df           =     4

Sigma df           =     1

Scale df           =     1

Log likelihood =     -59.146452     on     147 residual degrees of freedom

<S> dis e

Parameter              Estimate         Std. Err.

___________________________________________________

d1                     0.58508          0.62624E-01

d2                     0.71760          0.66132E-01

d3                     0.67158          0.66631E-01

d4                     0.63893          0.69505E-01

sigma                  0.26500          0.18573E-01

scale                  0.34532          0.41967E-01

<S> stop