Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence by Judith D. Singer and John B. Willett Chapter 5: Treating TIME More Flexibly

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)

Image fig5_1-2

Table 5.2 on page 145.

Part 1: Unconditional growth model on centered agegrp variable.

gen cagegrp = agegrp - 6.5
xtmixed piat cagegrp || id: cagegrp, mle cov(un) var

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
xtmixed piat cage || id: cage, mle cov(un) var

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

xtmixed lnw exper || id: exper, mle cov(un) var

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
xi: xtmixed lnw exper black hgc_9 hgc_exper bxexp || id: exper, mle cov(un) var

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

xi: xtmixed lnw exper hgc_9 bxexp || id: exper, mle cov(un) var

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 have to create a data set first 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 xi: xtmixed lnw exper hgc_9 bxexp || id: exper, mle cov(un) var

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

xtmixed lnw hgc_9 exper bexper || id: exper, var mle 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:

xi: xtmixed lnw exper hgc_9 bxexp || id: , mle var

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:

xi: xtmixed cesd months || id: months, mle cov(un) var

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0: log likelihood = -2567.3171 Iteration 1: log likelihood = -2566.5828 Iteration 2: log likelihood = -2566.5688 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 | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- months | -.4219943 .0829791 -5.09 0.000 -.5846302 -.2593583 _cons | 17.66936 .7755634 22.78 0.000 16.14929 19.18944 ------------------------------------------------------------------------------

------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ id: Unstructured | var(months) | .3550396 .1844914 .1282217 .9830869 var(_cons) | 86.8483 14.96308 61.95951 121.7348 cov(months,_cons) | -3.057268 1.384603 -5.771041 -.3434951 -----------------------------+------------------------------------------------ var(Residual) | 68.85002 6.60262 57.05254 83.087 ------------------------------------------------------------------------------ LR test vs. linear regression: chi2(3) = 106.52 Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference

Model B: Main effect of unemployment

xi: xtmixed cesd months unemp || id: months, mle cov(un) var

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0: log likelihood = -2554.2517 Iteration 1: log likelihood = -2553.8054 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 | 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

Model C: Effect of unemployment on initial status and growth rate

gen unempxmth = unemp*months

xi: xtmixed cesd months unemp unempxmth|| id: months, mle cov(un) var

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.

xi: xtmixed cesd unemp unempxmth || id: unemp unempxmth , mle cov(un) var

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0: log likelihood = -2552.1214 Iteration 1: log likelihood = -2548.5862 Iteration 2: log likelihood = -2548.1007 Iteration 3: log likelihood = -2547.7787 Iteration 4: log likelihood = -2547.6959 Iteration 5: log likelihood = -2547.6587 Iteration 6: log likelihood = -2547.6504 Iteration 7: log likelihood = -2547.6481 (not concave) Iteration 8: log likelihood = -2547.648 Iteration 9: log likelihood = -2547.6475 Iteration 10: log likelihood = -2547.6474 Iteration 11: log likelihood = -2547.6474 Iteration 12: log likelihood = -2547.6474 (not concave) Iteration 13: log likelihood = -2547.6474 Iteration 14: log likelihood = -2547.6474 (not concave) Iteration 15: log likelihood = -2547.6474

Computing standard errors: standard error calculation failed

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) = 55.93 Log likelihood = -2547.6474 Prob > chi2 = 0.0000

------------------------------------------------------------------------------ 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

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

xtmixed cesd months unemp || id: months, cov(un) var mle 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:

xi: xtmixed cesd months unemp || id: months, mle cov(un) var

(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:

xi: xtmixed cesd months unemp unempxmth || id: months, mle cov(un) var

(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:

xi: xtmixed cesd unemp unempxmth || id: unemp unempxmth , mle cov(un) var

(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

xtmixed lnw hgc_9 ue_7 exper bexper || id: exper, mle cov(un) var

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

xtmixed lnw hgc_9 ue_mean ue_person_centered exper bexper || id: exper, mle cov(un) var

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

xtmixed lnw hgc_9 ue1 ue_centert1 exper bexper || id: exper, mle cov(un) var

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

xtmixed pos treat time ttime || id: time, cov(un) mle var

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

xtmixed pos treat time333 ttime333 || id: time333, cov(un) mle var

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

xtmixed pos treat time667 ttime667 || id: time667, cov(un) mle var

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

xtmixed pos treat time ttime || id: time, cov(un) mle var

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.

xtmixed pos initial tinit final tfinal, nocons || id: initial final, cov(un) mle var 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