Table 5.1 on page 141 using reading_pp data
use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/reading_pp, clear list if inlist(id, 4, 27, 31, 33, 41, 49, 69, 77, 87)+--------------------------------------+ | id wave agegrp age piat | |--------------------------------------| 10. | 4 1 6.5 6 18 | 11. | 4 2 8.5 8.5 31 | 12. | 4 3 10.5 10.66667 50 | 79. | 27 1 6.5 6.25 19 | 80. | 27 2 8.5 9.166667 36 | |--------------------------------------| 81. | 27 3 10.5 10.91667 57 | 91. | 31 1 6.5 6.333333 18 | 92. | 31 2 8.5 8.833333 31 | 93. | 31 3 10.5 10.91667 51 | 97. | 33 1 6.5 6.333333 18 | |--------------------------------------| 98. | 33 2 8.5 8.916667 34 | 99. | 33 3 10.5 10.75 29 | 121. | 41 1 6.5 6.333333 18 | 122. | 41 2 8.5 8.75 28 | 123. | 41 3 10.5 10.83333 36 | |--------------------------------------| 145. | 49 1 6.5 6.5 19 | 146. | 49 2 8.5 8.75 32 | 147. | 49 3 10.5 10.66667 48 | 205. | 69 1 6.5 6.666667 26 | 206. | 69 2 8.5 9.166667 47 | |--------------------------------------| 207. | 69 3 10.5 11.33333 45 | 229. | 77 1 6.5 6.833333 17 | 230. | 77 2 8.5 8.083333 19 | 231. | 77 3 10.5 10 28 | 259. | 87 1 6.5 6.916667 22 | |--------------------------------------| 260. | 87 2 8.5 9.416667 49 | 261. | 87 3 10.5 11.5 64 | +--------------------------------------+
Figure 5.1 on page 143
preserve keep if inlist(id, 4, 27, 31, 33, 41, 49, 69, 77, 87) graph twoway (scatter piat age) (scatter piat agegrp) (lfit piat age, legend(off)) /// (lfit piat agegrp, legend(off)) , by(id) ylabel(0(20)80) xlabel(6(1) 12)
Table 5.2 on page 145.
Part 1: Unconditional growth model on the centered agegrp variable
gen cagegrp = agegrp - 6.5
mixed piat cagegrp || id: cagegrp, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -910.13718
Iteration 1: log likelihood = -909.97622
Iteration 2: log likelihood = -909.97465
Iteration 3: log likelihood = -909.97465
Computing standard errors:
Mixed-effects ML regression Number of obs = 267
Group variable: id Number of groups = 89
Obs per group: min = 3
avg = 3.0
max = 3
Wald chi2(1) = 289.62
Log likelihood = -909.97465 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
piat | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
cagegrp | 5.030899 .2956204 17.02 0.000 4.451494 5.610304
_cons | 21.16292 .6142662 34.45 0.000 19.95898 22.36686
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(cagegrp) | 4.39745 1.271306 2.495273 7.749681
var(_cons) | 11.04583 6.06258 3.767131 32.38815
cov(cagegrp,_cons) | 1.646659 2.062946 -2.396641 5.689959
-----------------------------+------------------------------------------------
var(Residual) | 27.04309 4.053927 20.15839 36.27911
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 86.92 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Part 2: Unconditional growth model on centered age variable
gen cage = age - 6.5
mixed piat cage || id: cage, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -902.4384
Iteration 1: log likelihood = -901.98942
Iteration 2: log likelihood = -901.95011
Iteration 3: log likelihood = -901.9478
Iteration 4: log likelihood = -901.94777
Computing standard errors:
Mixed-effects ML regression Number of obs = 267
Group variable: id Number of groups = 89
Obs per group: min = 3
avg = 3.0
max = 3
Wald chi2(1) = 303.42
Log likelihood = -901.94777 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
piat | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
cage | 4.540021 .2606349 17.42 0.000 4.029186 5.050856
_cons | 21.06082 .5593131 37.65 0.000 19.96458 22.15705
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(cage) | 3.301472 1.014522 1.807751 6.029435
var(_cons) | 5.107369 6.028915 .5051372 51.63986
cov(cage,_cons) | 2.366531 1.802022 -1.165368 5.89843
-----------------------------+------------------------------------------------
var(Residual) | 27.44634 4.371408 20.08689 37.50216
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 88.91 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Table 5.3 on page 147 using wages_pp data
clear
use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/wages_pp
list exper lnw black hgc uerate if id == 206 | id == 332 | id ==1028
+--------------------------------------+
| exper lnw black hgc uerate |
|--------------------------------------|
75. | 1.874 2.028 0 10 9.2 |
76. | 2.814 2.297 0 10 11 |
77. | 4.314 2.482 0 10 6.295 |
197. | .125 1.63 0 8 7.1 |
198. | 1.625 1.476 0 8 9.6 |
|--------------------------------------|
199. | 2.413 1.804 0 8 7.2 |
200. | 3.393 1.439 0 8 6.195 |
201. | 4.47 1.748 0 8 5.595 |
202. | 5.178 1.526 0 8 4.595 |
203. | 6.082 2.044 0 8 4.295 |
|--------------------------------------|
204. | 7.043 2.179 0 8 3.395 |
205. | 8.197 2.186 0 8 4.395 |
206. | 9.092 4.035 0 8 6.695 |
466. | .004 .872 1 8 9.3 |
467. | .035 .903 1 8 7.4 |
|--------------------------------------|
468. | .515 1.389 1 8 7.3 |
469. | 1.483 2.324 1 8 7.4 |
470. | 2.141 1.484 1 8 6.295 |
471. | 3.161 1.705 1 8 5.895 |
472. | 4.103 2.343 1 8 6.9 |
+--------------------------------------+
Table 5.4 on page 149
Model A: Unconditional growth model
mixed lnw exper || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2460.7432
Iteration 1: log likelihood = -2460.6972
Iteration 2: log likelihood = -2460.6971
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(1) = 380.59
Log likelihood = -2460.6971 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
exper | .0456807 .0023416 19.51 0.000 .0410913 .0502701
_cons | 1.715604 .0107965 158.90 0.000 1.694443 1.736765
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .001726 .00022 .0013444 .0022158
var(_cons) | .0542681 .0050012 .0453001 .0650114
cov(exper,_cons) | -.0029149 .000869 -.004618 -.0012118
-----------------------------+------------------------------------------------
var(Residual) | .0951047 .0019442 .0913694 .0989927
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1580.05 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model B: Conditional growth model
gen hgc_exper = hgc_9*exper
gen bxexp = black*exper
mixed lnw exper black hgc_9 hgc_exper bxexp || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2436.9369
Iteration 1: log likelihood = -2436.8758
Iteration 2: log likelihood = -2436.8757
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(5) = 441.56
Log likelihood = -2436.8757 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
exper | .0493428 .0026317 18.75 0.000 .0441848 .0545008
black | .0153954 .0239265 0.64 0.520 -.0314996 .0622905
hgc_9 | .0349201 .0078815 4.43 0.000 .0194727 .0503675
hgc_exper | .0012794 .0017232 0.74 0.458 -.0020979 .0046568
bxexp | -.0182129 .0054991 -3.31 0.001 -.0289911 -.0074348
_cons | 1.717139 .0125424 136.91 0.000 1.692556 1.741721
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .0016358 .0002139 .001266 .0021137
var(_cons) | .0517481 .0048685 .0430341 .0622265
cov(exper,_cons) | -.0028508 .0008441 -.0045052 -.0011965
-----------------------------+------------------------------------------------
var(Residual) | .0951938 .0019462 .0914547 .0990857
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1465.80 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model C: conditional growth model
mixed lnw exper hgc_9 bxexp || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2437.4097
Iteration 1: log likelihood = -2437.3518
Iteration 2: log likelihood = -2437.3518
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(3) = 439.40
Log likelihood = -2437.3518 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
exper | .048847 .0025134 19.43 0.000 .0439208 .0537732
hgc_9 | .0383608 .0064334 5.96 0.000 .0257516 .05097
bxexp | -.016115 .0045114 -3.57 0.000 -.0249571 -.0072729
_cons | 1.721475 .0106971 160.93 0.000 1.700509 1.742441
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .0016467 .0002141 .0012762 .0021247
var(_cons) | .0518309 .0048732 .0431081 .0623188
cov(exper,_cons) | -.0028798 .0008455 -.0045369 -.0012227
-----------------------------+------------------------------------------------
var(Residual) | .0951735 .0019453 .0914361 .0990637
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1474.50 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Figure 5.2 on page 150
We will first have to create a data set for generating predicted values for different groups.
input exper hgc_9 black group exper hgc_9 black group 0 0 0 1 10 0 0 1 0 0 1 2 10 0 1 2 0 3 0 3 10 3 0 3 0 3 1 4 10 3 1 4 end gen bxexp = black*exper save fig5_2 use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/wages_pp, clear gen bxexp = black*exper mixed lnw exper hgc_9 bxexp || id: exper, cov(un) Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = -2437.4097 Iteration 1: log likelihood = -2437.3518 Iteration 2: log likelihood = -2437.3518 Computing standard errors: Mixed-effects ML regression Number of obs = 6402 Group variable: id Number of groups = 888 Obs per group: min = 1 avg = 7.2 max = 13 Wald chi2(3) = 439.40 Log likelihood = -2437.3518 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- exper | .048847 .0025134 19.43 0.000 .0439208 .0537732 hgc_9 | .0383608 .0064334 5.96 0.000 .0257516 .05097 bxexp | -.016115 .0045114 -3.57 0.000 -.0249571 -.0072729 _cons | 1.721475 .0106971 160.93 0.000 1.700509 1.742441 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(exper) | .0016467 .0002141 .0012762 .0021247 var(_cons) | .0518309 .0048732 .0431081 .0623188 cov(exper,_cons) | -.0028798 .0008455 -.0045369 -.0012227 -----------------------------+------------------------------------------------ var(Residual) | .0951735 .0019453 .0914361 .0990637 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 1474.50 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference use fig5_2, clear predict p (option xb assumed) twoway scatter p exper, c(L) ytitle("predicted lnw")
Table 5.5 on page 154 using wages_small_pp data. Stata does not have problem estimating this model.
use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/wages_small_pp
gen bexper = black*exper
mixed lnw hgc_9 exper bexper || id: exper, cov(un)
Performing EM optimization:<
Performing gradient-based optimization:
Iteration 0: log likelihood = -144.13188
Iteration 1: log likelihood = -142.21488
Iteration 2: log likelihood = -142.01812
Iteration 3: log likelihood = -141.9574
Iteration 4: log likelihood = -141.9398
Iteration 5: log likelihood = -141.93609
Iteration 6: log likelihood = -141.93515
Iteration 7: log likelihood = -141.93492
Iteration 8: log likelihood = -141.93487
Iteration 9: log likelihood = -141.93486
Computing standard errors:
Mixed-effects ML regression Number of obs = 257
Group variable: id Number of groups = 124
Obs per group: min = 1
avg = 2.1
max = 3
Wald chi2(3) = 11.96
Log likelihood = -141.93486 Prob > chi2 = 0.0075
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hgc_9 | .0461057 .024471 1.88 0.060 -.0018565 .0940679
exper | .0516136 .0210767 2.45 0.014 .010304 .0929232
bexper | -.0596725 .0347673 -1.72 0.086 -.1278152 .0084701
_cons | 1.737337 .0475981 36.50 0.000 1.644046 1.830628
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | 3.57e-06 .0000829 5.94e-26 2.14e+14
var(_cons) | .0821216 .0320226 .0382418 .1763503
cov(exper,_cons) | .0005401 .0061749 -.0115624 .0126427
-----------------------------+------------------------------------------------
var(Residual) | .1149779 .0147141 .0894713 .1477561
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 23.71 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Fixed rates of change approach
mixed lnw exper hgc_9 bxexp || id:
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -141.94137
Iteration 1: log likelihood = -141.93861
Iteration 2: log likelihood = -141.93861
Computing standard errors:
Mixed-effects ML regression Number of obs = 257
Group variable: id Number of groups = 124
Obs per group: min = 1
avg = 2.1
max = 3
Wald chi2(3) = 12.09
Log likelihood = -141.93861 Prob > chi2 = 0.0071
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
exper | .0517798 .0209332 2.47 0.013 .0107516 .0928081
hgc_9 | .0457585 .0244959 1.87 0.062 -.0022525 .0937696
bxexp | -.0600723 .0345782 -1.74 0.082 -.1278443 .0076996
_cons | 1.737344 .0477509 36.38 0.000 1.643754 1.830934
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Identity |
var(_cons) | .0842504 .0211917 .0514597 .1379356
-----------------------------+------------------------------------------------
var(Residual) | .1147999 .0145473 .0895526 .1471652
------------------------------------------------------------------------------
LR test vs. linear regression: chibar2(01) = 23.71 Prob >= chibar2 = 0.0000
Table 5.6 on page 161 using unemployment_pp data
use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/unemployment_pp, clear
list id months cesd unemp if id == 7589 | id == 55697 | id == 67641 | id ==65441 | id == 53782
+---------------------------------+
| id months cesd unemp |
|---------------------------------|
212. | 7589 1.314168 36 1 |
213. | 7589 5.092402 40 1 |
214. | 7589 11.79466 39 1 |
454. | 53782 .4271047 22 1 |
455. | 53782 4.238193 15 0 |
|---------------------------------|
456. | 53782 11.07187 21 1 |
504. | 55697 1.347023 7 1 |
505. | 55697 5.782341 4 1 |
623. | 65441 1.084189 27 1 |
624. | 65441 4.698152 15 1 |
|---------------------------------|
625. | 65441 11.26899 7 0 |
647. | 67641 .3285421 32 1 |
648. | 67641 4.106776 9 0 |
649. | 67641 10.94045 10 0 |
+---------------------------------+
Table 5.7 on page 163
Model A: Initial growth model
mixed cesd months || id: months,cov(un)
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -2567.3171
Iteration 1: Log likelihood = -2566.571
Iteration 2: Log likelihood = -2566.5687
Iteration 3: Log likelihood = -2566.5687
Computing standard errors ...
Mixed-effects ML regression Number of obs = 674
Group variable: id Number of groups = 254
Obs per group:
min = 1
avg = 2.7
max = 3
Wald chi2(1) = 25.86
Log likelihood = -2566.5687 Prob > chi2 = 0.0000
-----------------------------------------------------------------------------------
cesd | Coefficient Std. err. z P>|z| [95% conf. interval]
------------------+----------------------------------------------------------------
months | -.4219941 .0829789 -5.09 0.000 -.5846297 -.2593585
_cons | 17.66936 .7755637 22.78 0.000 16.14929 19.18944
-----------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(months) | .3550302 .1844942 .128213 .9831023
var(_cons) | 86.84826 14.96314 61.95938 121.7349
cov(months,_cons) | -3.057233 1.384615 -5.771029 -.343438
-----------------------------+------------------------------------------------
var(Residual) | 68.85021 6.6027 57.0526 83.08738
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 106.52 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.
Model B: Main effect of unemployment
mixed cesd months unemp || id: months, cov(un)
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -2554.2517
Iteration 1: Log likelihood = -2553.8032
Iteration 2: Log likelihood = -2553.8016
Iteration 3: Log likelihood = -2553.8016
Computing standard errors ...
Mixed-effects ML regression Number of obs = 674
Group variable: id Number of groups = 254
Obs per group:
min = 1
avg = 2.7
max = 3
Wald chi2(2) = 52.78
Log likelihood = -2553.8016 Prob > chi2 = 0.0000
-----------------------------------------------------------------------------------
cesd | Coefficient Std. err. z P>|z| [95% conf. interval]
------------------+----------------------------------------------------------------
months | -.2019834 .0933163 -2.16 0.030 -.38488 -.0190869
unemp | 5.111307 .9888444 5.17 0.000 3.173208 7.049407
_cons | 12.6656 1.242071 10.20 0.000 10.23118 15.10001
-----------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(months) | .4647051 .1797869 .2177015 .9919586
var(_cons) | 93.51887 14.82016 68.54998 127.5825
cov(months,_cons) | -3.894106 1.370256 -6.579758 -1.208453
-----------------------------+------------------------------------------------
var(Residual) | 62.38752 6.013225 51.64813 75.35999
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 114.84 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.
Model C: Effect of unemployment on initial status and growth rate
gen unempxmth = unemp*months
mixed cesd months unemp unempxmth|| id: months, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2551.9898
Iteration 1: log likelihood = -2551.5272
Iteration 2: log likelihood = -2551.5235
Iteration 3: log likelihood = -2551.5235
Computing standard errors:
Mixed-effects ML regression Number of obs = 674
Group variable: id Number of groups = 254
Obs per group: min = 1
avg = 2.7
max = 3
Wald chi2(3) = 57.88
Log likelihood = -2551.5235 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
cesd | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
months | .162036 .1936623 0.84 0.403 -.2175351 .5416072
unemp | 8.529059 1.877875 4.54 0.000 4.848492 12.20963
unempxmth | -.4652222 .2172146 -2.14 0.032 -.890955 -.0394895
_cons | 9.616744 1.889309 5.09 0.000 5.913766 13.31972
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(months) | .4512058 .1773447 .2088397 .9748464
var(_cons) | 93.71305 14.77709 68.79849 127.6501
cov(months,_cons) | -3.873144 1.358796 -6.536335 -1.209953
-----------------------------+------------------------------------------------
var(Residual) | 62.03126 5.965555 51.37484 74.8981
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 116.31 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model D: This model is problematic. The problem with this example is that the variance-covariance matrix of the three random effects is singular. The determinant is negative. Different algorithms and packages react differently to this situation. Many researchers might consider this an unacceptable model. Stata was not able to estimate standard errors for the random effects variance components using the default NR algorithm. Experimentation with other estimators produced no results at all. So we decided to show the solution using the default algorithm even though the standard errors of the random effects are missing.
mixed cesd unemp unempxmth || id: unemp unempxmth, cov(un)
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -2552.1214
Iteration 1: Log likelihood = -2548.306
Iteration 2: Log likelihood = -2547.758
Iteration 3: Log likelihood = -2547.6515
Iteration 4: Log likelihood = -2547.6474
Iteration 5: Log likelihood = -2547.6474 (not concave)
Iteration 6: Log likelihood = -2547.6474 (backed up)
Computing standard errors ...
Mixed-effects ML regression Number of obs = 674
Group variable: id Number of groups = 254
Obs per group:
min = 1
avg = 2.7
max = 3
Wald chi2(2) = 55.92
Log likelihood = -2547.6474 Prob > chi2 = 0.0000
--------------------------------------------------------------------------------------
cesd | Coefficient Std. err. z P>|z| [95% conf. interval]
---------------------+----------------------------------------------------------------
unemp | 6.927268 .9300356 7.45 0.000 5.104432 8.750104
unempxmth | -.3029835 .1120896 -2.70 0.007 -.5226751 -.0832919
_cons | 11.19492 .7898842 14.17 0.000 9.646772 12.74306
--------------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(unemp) | 44.96896 . . .
var(unempxmth) | .7534248 . . .
var(_cons) | 45.2535 . . .
cov(unemp,unempxmth) | -5.629083 . . .
cov(unemp,_cons) | 6.543357 . . .
cov(unempxmth,_cons) | .6511535 . . .
-----------------------------+------------------------------------------------
var(Residual) | 59.01791 . . .
------------------------------------------------------------------------------
LR test vs. linear model: chi2(6) = 124.60 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.
Figure 5.3 on page 165 based on model B from the previous example. We will have to create a small data set for generating predicted values.
input months unemp group 0 1 1 5 1 1 14 1 1 0 1 2 5 1 2 5 0 2 14 0 2 0 1 3 10 1 3 10 0 3 14 0 3 0 1 4 5 1 4 5 0 4 10 0 4 10 1 4 14 1 4 end save fig5_3 file fig5_3.dta saved use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/unemployment_pp, clear mixed cesd months unemp || id: months, cov(un) nolog Mixed-effects ML regression Number of obs = 674 Group variable: id Number of groups = 254 Obs per group: min = 1 avg = 2.7 max = 3 Wald chi2(2) = 52.78 Log likelihood = -2553.8016 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ cesd | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- months | -.2019835 .0933162 -2.16 0.030 -.38488 -.019087 unemp | 5.111305 .9888446 5.17 0.000 3.173205 7.049405 _cons | 12.6656 1.242071 10.20 0.000 10.23118 15.10001 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(months) | .4647019 .1797873 .2176985 .9919587 var(_cons) | 93.51871 14.82016 68.54982 127.5824 cov(months,_cons) | -3.894087 1.370257 -6.579741 -1.208433 -----------------------------+------------------------------------------------ var(Residual) | 62.38761 6.013247 51.64818 75.36013 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 114.84 Prob > chi2 = 0.0000 use fig5_3, clear predict p (option xb assumed) twoway scatter p month, by(group) c(L)
Figure 5.4 on page 167 based on Model B, C and D from previous example.
Leftmost panel using Model B:
mixed cesd months unemp || id: months, cov(un) (output omitted) ------------------------------------------------------------------------------ cesd | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- months | -.2019835 .0933162 -2.16 0.030 -.38488 -.019087 unemp | 5.111305 .9888446 5.17 0.000 3.173205 7.049405 _cons | 12.6656 1.242071 10.20 0.000 10.23118 15.10001 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(months) | .4647019 .1797873 .2176985 .9919587 var(_cons) | 93.51871 14.82016 68.54982 127.5824 cov(months,_cons) | -3.894087 1.370257 -6.579741 -1.208433 -----------------------------+------------------------------------------------ var(Residual) | 62.38761 6.013247 51.64818 75.36013 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 114.84 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference predict p (option xb assumed) sort unemp months scatter p months, c(L)
Panel in the middle using Model C
mixed cesd months unemp unempxmth || id: months, cov(un) (output omitted) ------------------------------------------------------------------------------ cesd | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- months | .162036 .1936623 0.84 0.403 -.2175351 .5416072 unemp | 8.529059 1.877875 4.54 0.000 4.848492 12.20963 unempxmth | -.4652222 .2172146 -2.14 0.032 -.890955 -.0394895 _cons | 9.616744 1.889309 5.09 0.000 5.913766 13.31972 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(months) | .4512058 .1773447 .2088397 .9748464 var(_cons) | 93.71305 14.77709 68.79849 127.6501 cov(months,_cons) | -3.873144 1.358796 -6.536335 -1.209953 -----------------------------+------------------------------------------------ var(Residual) | 62.03126 5.965555 51.37484 74.8981 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 116.31 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference predict pc (option xb assumed) sort unemp months scatter pc months, c(L)
Rightmost panel using Model D
mixed cesd unemp unempxmth || id: unemp unempxmth, cov(un) (output omitted) ------------------------------------------------------------------------------ cesd | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- unemp | 6.927277 .930033 7.45 0.000 5.104445 8.750108 unempxmth | -.3029831 .1120893 -2.70 0.007 -.5226741 -.0832921 _cons | 11.19491 .7898805 14.17 0.000 9.646771 12.74305 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(unemp) | 44.96733 . . . var(unempx~h) | .753401 . . . var(_cons) | 45.25229 . . . cov(unemp,unempx~h) | -5.628865 . . . cov(unemp,_cons) | 6.544479 . . . cov(unempx~h,_cons) | .6510678 . . . -----------------------------+------------------------------------------------ var(Residual) | 59.01838 . . . ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(6) = 124.60 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference predict pd (option xb assumed) sort unemp months scatter pd months, c(L)
Table 5.8 on page 175 using data set wages_pp
Model A: Unemployment rate centered around 7
mixed lnw hgc_9 ue_7 exper bexper || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2415.3186
Iteration 1: log likelihood = -2415.2596
Iteration 2: log likelihood = -2415.2595
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(4) = 488.69
Log likelihood = -2415.2595 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hgc_9 | .040011 .0063627 6.29 0.000 .0275403 .0524816
ue_7 | -.0119504 .0017916 -6.67 0.000 -.015462 -.0084389
exper | .0440539 .0026034 16.92 0.000 .0389513 .0491564
bexper | -.0181832 .0044837 -4.06 0.000 -.0269711 -.0093953
_cons | 1.748989 .0113993 153.43 0.000 1.726646 1.771331
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .0016317 .0002126 .001264 .0021064
var(_cons) | .0506369 .0048085 .0420374 .0609955
cov(exper,_cons) | -.0029129 .0008386 -.0045565 -.0012693
-----------------------------+------------------------------------------------
var(Residual) | .0947952 .0019382 .0910714 .0986711
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1423.34 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model B Within context centering
mixed lnw hgc_9 ue_mean ue_person_centered exper bexper || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2413.5457
Iteration 1: log likelihood = -2413.4893
Iteration 2: log likelihood = -2413.4892
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(5) = 494.17
Log likelihood = -2413.4892 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hgc_9 | .0401695 .0063507 6.33 0.000 .0277224 .0526166
ue_mean | -.0177091 .0035202 -5.03 0.000 -.0246085 -.0108097
ue_person_~d | -.0099015 .0020973 -4.72 0.000 -.0140121 -.0057909
exper | .0450568 .0026498 17.00 0.000 .0398632 .0502503
bexper | -.0188696 .0044769 -4.21 0.000 -.0276441 -.010095
_cons | 1.87426 .0295235 63.48 0.000 1.816395 1.932125
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .0016286 .0002123 .0012615 .0021026
var(_cons) | .0510114 .0048412 .042353 .0614399
cov(exper,_cons) | -.0030248 .0008431 -.0046772 -.0013724
-----------------------------+------------------------------------------------
var(Residual) | .0948018 .0019384 .0910777 .0986782
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1407.18 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model C: Centering on time-1
mixed lnw hgc_9 ue1 ue_centert1 exper bexper || id: exper, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -2412.9798
Iteration 1: log likelihood = -2412.9209
Iteration 2: log likelihood = -2412.9208
Computing standard errors:
Mixed-effects ML regression Number of obs = 6402
Group variable: id Number of groups = 888
Obs per group: min = 1
avg = 7.2
max = 13
Wald chi2(5) = 493.23
Log likelihood = -2412.9208 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnw | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hgc_9 | .0399276 .0063491 6.29 0.000 .0274835 .0523717
ue1 | -.0161774 .0026484 -6.11 0.000 -.0213682 -.0109866
ue_centert1 | -.010309 .0019445 -5.30 0.000 -.0141201 -.006498
exper | .0447649 .0026249 17.05 0.000 .0396202 .0499096
bexper | -.0183238 .0044848 -4.09 0.000 -.0271138 -.0095338
_cons | 1.869345 .0260316 71.81 0.000 1.818324 1.920366
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(exper) | .0016354 .0002129 .0012672 .0021107
var(_cons) | .0502771 .0047894 .0417143 .0605976
cov(exper,_cons) | -.0029024 .0008381 -.004545 -.0012598
-----------------------------+------------------------------------------------
var(Residual) | .0947669 .0019371 .0910452 .0986406
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 1423.82 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Table 5.10 on page 184 using medication_pp data
Model A: TIME
use https://stats.idre.ucla.edu/stat/stata/examples/alda/data/medication_pp, clear
gen ttime = treat*time
mixed pos treat time ttime || id: time, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -6340.2258
Iteration 1: log likelihood = -6340.2258
Computing standard errors:
Mixed-effects ML regression Number of obs = 1242
Group variable: id Number of groups = 64
Obs per group: min = 2
avg = 19.4
max = 21
Wald chi2(3) = 7.05
Log likelihood = -6340.2258 Prob > chi2 = 0.0702
------------------------------------------------------------------------------
pos | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
treat | -3.109294 12.33256 -0.25 0.801 -27.28067 21.06208
time | -2.418128 1.730852 -1.40 0.162 -5.810535 .9742796
ttime | 5.53681 2.277846 2.43 0.015 1.072314 10.00131
_cons | 167.4635 9.32626 17.96 0.000 149.1843 185.7426
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(time) | 63.73588 14.27148 41.09476 98.8511
var(_cons) | 2111.401 420.2165 1429.428 3118.739
cov(time,_cons) | -121.6214 59.03397 -237.3259 -5.916964
-----------------------------+------------------------------------------------
var(Residual) | 1229.93 52.09137 1131.955 1336.384
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 971.31 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model B: TIME-3.33
gen ttime333 = treat*time333
mixed pos treat time333 ttime333 || id: time333, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -6340.2258
Iteration 1: log likelihood = -6340.2258
Computing standard errors:
Mixed-effects ML regression Number of obs = 1242
Group variable: id Number of groups = 64
Obs per group: min = 2
avg = 19.4
max = 21
Wald chi2(3) = 7.05
Log likelihood = -6340.2258 Prob > chi2 = 0.0702
------------------------------------------------------------------------------
pos | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
treat | 15.34674 11.54466 1.33 0.184 -7.280383 37.97386
time333 | -2.418128 1.730852 -1.40 0.162 -5.810535 .9742798
ttime333 | 5.536811 2.277846 2.43 0.015 1.072314 10.00131
_cons | 159.403 8.764519 18.19 0.000 142.2249 176.5812
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(time333) | 63.73589 14.27147 41.09478 98.85109
var(_cons) | 2008.768 367.2589 1403.804 2874.44
cov(time333,_cons) | 90.8317 52.45916 -11.98636 193.6498
-----------------------------+------------------------------------------------
var(Residual) | 1229.929 52.09137 1131.955 1336.384
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 971.31 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Model C: TIME-6.67
gen ttime667 = treat*time667
mixed pos treat time667 ttime667 || id: time667, cov(un)
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -6340.2258
Iteration 1: log likelihood = -6340.2258
Computing standard errors:
Mixed-effects ML regression Number of obs = 1242
Group variable: id Number of groups = 64
Obs per group: min = 2
avg = 19.4
max = 21
Wald chi2(3) = 7.05
Log likelihood = -6340.2258 Prob > chi2 = 0.0702
------------------------------------------------------------------------------
pos | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
treat | 33.80278 15.1581 2.23 0.026 4.093444 63.51211
time667 | -2.418128 1.730852 -1.40 0.162 -5.810535 .9742797
ttime667 | 5.536811 2.277846 2.43 0.015 1.072314 10.00131
_cons | 151.3426 11.54247 13.11 0.000 128.7198 173.9654
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(time667) | 63.73588 14.27144 41.09481 98.85099
var(_cons) | 3322.485 632.1168 2288.328 4824.004
cov(time667,_cons) | 303.2845 80.90138 144.7207 461.8483
-----------------------------+------------------------------------------------
var(Residual) | 1229.93 52.09138 1131.955 1336.384
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 971.31 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
Figure 5.5 on page 185
mixed pos treat time ttime || id: time, cov(un) Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = -6340.2258 Iteration 1: log likelihood = -6340.2258< Computing standard errors: Mixed-effects ML regression Number of obs = 1242 Group variable: id Number of groups = 64 Obs per group: min = 2 avg = 19.4 max = 21 Wald chi2(3) = 7.05 Log likelihood = -6340.2258 Prob > chi2 = 0.0702 ------------------------------------------------------------------------------ pos | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- treat | -3.109294 12.33256 -0.25 0.801 -27.28067 21.06208 time | -2.418128 1.730852 -1.40 0.162 -5.810535 .9742796 ttime | 5.53681 2.277846 2.43 0.015 1.072314 10.00131 _cons | 167.4635 9.32626 17.96 0.000 149.1843 185.7426 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(time) | 63.73588 14.27148 41.09476 98.8511 var(_cons) | 2111.401 420.2165 1429.428 3118.739 cov(time,_cons) | -121.6214 59.03397 -237.3259 -5.916964 -----------------------------+------------------------------------------------ var(Residual) | 1229.93 52.09137 1131.955 1336.384 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 971.31 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference predict pf (option xb assumed) scatter pf time, c(L)
Page 188 on modeling initial and final status
mixed pos initial tinit final tfinal, nocons || id: initial final, cov(un) nocons Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = -6340.2258 Iteration 1: log likelihood = -6340.2258 Computing standard errors: Mixed-effects ML regression Number of obs = 1242 Group variable: id Number of groups = 64 Obs per group: min = 2 avg = 19.4 max = 21 Wald chi2(4) = 912.52 Log likelihood = -6340.2258 Prob > chi2 = 0.0000------------------------------------------------------------------------------ pos | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- initial | 167.4635 9.326258 17.96 0.000 149.1843 185.7426 tinit | -3.109296 12.33256 -0.25 0.801 -27.28066 21.06207 final | 151.3426 11.54248 13.11 0.000 128.7198 173.9654 tfinal | 33.80278 15.1581 2.23 0.026 4.093439 63.51211 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(initial) | 2111.399 420.2156 1429.428 3118.735 var(final) | 3322.486 632.1178 2288.328 4824.009 cov(initial,final) | 1300.589 392.7855 530.7431 2070.434 -----------------------------+------------------------------------------------ var(Residual) | 1229.93 52.09137 1131.955 1336.384 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 971.31 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference estat recov, corr Random-effects correlation matrix for level id | initial final -------------+---------------------- initial | 1 final | .4910471 1