### Generalized Latent Variable Modeling by Skrondal and Rabe-Hesketh Section 14.4: Job training and depressionA complier average causal effect model

The data set used in this section is wjobs.dat. We would like to thank Amiram Vinokur and Bengt Muthén for making the data available.

The do-file is cace.do.

Variables (names as in book):
 depress depression (response variable) risk baseline risk score Tx randomized to receive job training (1:yes, 0:no). This is r_j basedep baseline depression score age age in years motivate motivation to attend training educ school grade completed assert assertiveness single dummy for being single econ economic hardship nonwhite dummy for not being white x10 not used c1 complier in treatment group (1: complied, 0: did not comply). This is c_j c2 not used

```
infile depress risk r depbase age motivate educ /*
*/ assert single econ nonwhite x10 c c0 using wjobs.dat, clear

gen y1 = c if r==1  /* compliance missing in control group */
gen y2 = depress

gen id=_n
reshape long y, i(id) j(var)
drop if y==.
tab var, gen(d)     /* create dummies d1 and d2 */

```

List some data
```
list id var d1 d2 y r c if id==1 | id==2 | id==175 | id==176, clean

id   var   d1   d2       y   r   c
1.     1     2    0    1     .45   0   1
2.     2     2    0    1    -.72   0   1
182.   175     1    1    0       0   1   0
183.   175     2    0    1   -1.37   1   0
184.   176     1    1    0       1   1   1
185.   176     2    0    1     .54   1   1
```

Path diagram of CACE model

Interactions and equations (for model without covariates)
```
gen nr_d2 = (1-r)*d2
gen c_r_d2 = c*r*d2

eq load: nr_d2         /* for beta_1(1-r_j)d_2i */

```

Constraints
```
cons def 1 [p2_1]_cons = [y]d1    /* constraint for varrho */
cons def 2 [z2_1_1]nr_d2 = 1      /* e_1 = 1 */
cons def 3 [z2_1_2]nr_d2 = 0      /* e_2 = 0 */

```

gllamm command for model without covariates in Table 14.6 (iteration log not shown)
```
gllamm y d1 d2 c_r_d2, i(id) eqs(load) l(logit ident) /*
*/ f(binom gauss) lv(var) fv(var) ip(fn) nip(2)     /*

number of level 1 units = 837
number of level 2 units = 502

Condition Number = 5.8885922

gllamm model with constraints:
( 1) - [y]d1 + [p2_1]_cons = 0
( 2)  [z2_1_1]nr_d2 = 1
( 3)  [z2_1_2]nr_d2 = 0

log likelihood = -815.1493933028294

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
d1 |   .1855983   .1097431     1.69   0.091    -.0294942    .4006907
d2 |  -.3909498   .0651724    -6.00   0.000    -.5186854   -.2632143
c_r_d2 |  -.1224928   .0867746    -1.41   0.158    -.2925679    .0475822
------------------------------------------------------------------------------

Variance at level 1
------------------------------------------------------------------------------

.60067675 (.03791846)

Probabilities and locations of random effects
------------------------------------------------------------------------------

***level 2 (id)

loc1: 1, 0
var(1): .24785938

nr_d2: .01526969 (.17004298)

prob: 0.5463, 0.4537
------------------------------------------------------------------------------
```

Estimate of Complier average causal effect:
```
lincom [y]c_r_d2 - [id1_1l]nr_d2

( 1)  [y]c_r_d2 - [id1_1l]nr_d2 = 0

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) |  -.1377625    .141096    -0.98   0.329    -.4143056    .1387806
------------------------------------------------------------------------------
```

Define more interactions to add covariates to the model
Make interactions with d1 (dummy for compliance response), for compliance model
```
gen age_d1 = age*d1
gen motivate_d1 = motivate*d1
gen educ_d1 = educ*d1
gen assert_d1 = assert*d1
gen single_d1 = single*d1
gen econ_d1 = econ*d1
gen nonwhite_d1 = nonwhite*d1

```

Add predictors of depression: make interactions with d2 (dummy for depression response)
```
gen depbase_d2 = depbase*d2
gen risk_d2 = risk*d2

```

New constraints: effects of covariates on compliance same in treatment and control groups
```
cons def 4 [p2_1]age = [y]age_d1
cons def 5 [p2_1]motivate = [y]motivate_d1
cons def 6 [p2_1]educ = [y]educ_d1
cons def 7 [p2_1]assert = [y]assert_d1
cons def 8 [p2_1]single = [y]single_d1
cons def 9 [p2_1]econ = [y]econ_d1
cons def 10 [p2_1]nonwhite = [y]nonwhite_d1

```

Estimate model with covariates for Table 14.6 using gllamm (iteration log not shown)
```
eq p: age educ motivate econ assert single nonwhite
gllamm y d1 age_d1 educ_d1 motivate_d1 econ_d1 assert_d1 single_d1 nonwhite_d1 /*
*/ d2 c_r_d2 depbase_d2 risk_d2, i(id) eqs(load) /*
*/ peqs(p) l(logit ident) f(binom gauss) lv(var) fv(var) ip(fn) nip(2)  /*

number of level 1 units = 837
number of level 2 units = 502

Condition Number = 508.29708

gllamm model with constraints:
( 1) - [y]d1 + [p2_1]_cons = 0
( 2)  [z2_1_1]nr_d2 = 1
( 3)  [z2_1_2]nr_d2 = 0
( 4) - [y]age_d1 + [p2_1]age = 0
( 5) - [y]motivate_d1 + [p2_1]motivate = 0
( 6) - [y]educ_d1 + [p2_1]educ = 0
( 7) - [y]assert_d1 + [p2_1]assert = 0
( 8) - [y]single_d1 + [p2_1]single = 0
( 9) - [y]econ_d1 + [p2_1]econ = 0
(10) - [y]nonwhite_d1 + [p2_1]nonwhite = 0

log likelihood = -729.4141540404285

------------------------------------------------------------------------------
|      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
d1 |  -8.740012   1.581554    -5.53   0.000     -11.8398   -5.640222
age_d1 |   .0790446   .0140223     5.64   0.000     .0515615    .1065277
educ_d1 |   .2997689   .0675165     4.44   0.000     .1674391    .4320987
motivate_d1 |   .6668722   .1598218     4.17   0.000     .3536272    .9801172
econ_d1 |  -.1586014   .1596605    -0.99   0.321    -.4715302    .1543274
assert_d1 |   -.375871   .1464048    -2.57   0.010    -.6628191   -.0889229
single_d1 |    .540194   .2754362     1.96   0.050     .0003489    1.080039
nonwhite_d1 |  -.4985877   .3123484    -1.60   0.110    -1.110779    .1136038
d2 |   1.632537   .2791255     5.85   0.000     1.085461    2.179613
c_r_d2 |  -.1299147   .0755598    -1.72   0.086    -.2780092    .0181798
risk_d2 |   .9117567   .2624528     3.47   0.001     .3973587    1.426155
depbase_d2 |  -1.463379   .1826867    -8.01   0.000    -1.821438    -1.10532
------------------------------------------------------------------------------

Variance at level 1
------------------------------------------------------------------------------

.50639704 (.0322776)

Probabilities and locations of random effects
------------------------------------------------------------------------------

***level 2 (id)

loc1: 1, 0
var(1): .00016

nr_d2: .1799526 (.1329182)

prob: 01.6e-04, 0.9998

log odds parameters
class 1
age: .0790446 (.01402225)
educ: .2997689 (.06751647)
motivate: .66687221 (.15982181)
econ: -.15860142 (.15966048)
assert: -.37587104 (.14640478)
single: .54019404 (.27543623)
nonwhite: -.49858773 (.31234837)
_cons: -8.7400116 (1.5815542)
------------------------------------------------------------------------------
```

Complier Average Causal Effect
```
lincom [y]c_r_d2 - [id1_1l]nr_d2

( 1)  [y]c_r_d2 - [id1_1l]nr_d2 = 0

------------------------------------------------------------------------------
y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) |  -.3098673   .1173219    -2.64   0.008    -.5398141   -.0799205
------------------------------------------------------------------------------
```

References

Vinokur, A. D., Price, R. H. and Schul, Y. (1995). Impact of JOBS intervention on unemployed workers varying in risk for depression. American Journal of Community Psychology 19, 543-562.

Little, R. J. A. and Yau, L. H. Y. (1998). Statistical techniques for analyzing data from prevention trials. Psychological Methods 3, 147-159.

Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized Latent Variable Modeling: Multilevel, Longitudinal and Structural Equation Models. Boca Raton, FL: Chapman & Hall/ CRC Press.

Outline
Datasets and do-files