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Table 3.2, page 46. OLS regression lines over 10 schools.
use http://www.ats.ucla.edu/stat/stata/examples/mlm_imm/imm10, clear
statsby intercept=_b[_cons] se_int=_se[_cons] homework=_b[homework] ///
se_homework =_se[homework] n =e(N) r2=e(r2) , ///
by(schnum) total saving(statby, replace): regress math homework
use statby, clear
gen r = (-1)^((homework>0)-1)*sqrt(r2)
list
+----------------------------------------------------------------------------------+
| schnum interc~t se_int homework se_hom~k n r2 r |
|----------------------------------------------------------------------------------|
1. | 1 50.68354 2.211345 -3.553797 1.249839 23 .2779769 -.5272352 |
2. | 2 49.01229 3.55737 -2.920123 1.330381 20 .2111427 -.4595027 |
3. | 3 38.75 2.943105 7.909091 1.374722 24 .6007232 .7750633 |
4. | 4 34.39382 1.796048 5.592664 .8182317 22 .7002311 .8367981 |
5. | 5 53.93863 2.523248 -4.718411 2.197719 22 .1873034 -.4327856 |
|----------------------------------------------------------------------------------|
6. | 6 49.25896 1.546975 -2.486056 1.107812 20 .2186159 -.4675638 |
7. | 7 59.21022 1.431387 1.09464 .385233 67 .1104923 .3324038 |
8. | 8 36.05535 3.464429 6.49631 1.461505 21 .5097728 .7139838 |
9. | 9 38.52 3.188946 5.86 1.988657 21 .3136609 .5600544 |
10. | 10 37.71392 2.366805 6.335052 1.115723 20 .6417159 .8010717 |
|----------------------------------------------------------------------------------|
11. | . 44.07386 .988641 3.571856 .3882366 260 .2470313 .4970225 |
+----------------------------------------------------------------------------------+
Two equations at the top of page 47.
regress bcons public
Source | SS df MS Number of obs = 10
-------------+------------------------------ F( 1, 8) = 4.20
Model | 232.21294 1 232.21294 Prob > F = 0.0745
Residual | 442.187199 8 55.2733999 R-squared = 0.3443
-------------+------------------------------ Adj R-squared = 0.2624
Total | 674.400139 9 74.9333487 Root MSE = 7.4346
------------------------------------------------------------------------------
bcons | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
public | -16.06283 7.836765 -2.05 0.075 -34.13444 2.008787
_cons | 59.21022 7.434608 7.96 0.000 42.06598 76.35445
------------------------------------------------------------------------------
regress bhmwk public
Source | SS df MS Number of obs = 10
-------------+------------------------------ F( 1, 8) = 0.03
Model | .833856092 1 .833856092 Prob > F = 0.8667
Residual | 222.018254 8 27.7522817 R-squared = 0.0037
-------------+------------------------------ Adj R-squared = -0.1208
Total | 222.85211 9 24.7613455 Root MSE = 5.268
------------------------------------------------------------------------------
bhmwk | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
public | .9625522 5.553005 0.17 0.867 -11.8427 13.76781
_cons | 1.09464 5.268043 0.21 0.841 -11.05349 13.24277
------------------------------------------------------------------------------
Equation near bottom of page 47 and Table 3.3. We will use the xtmixed command (new to Stata 9).
use imm10, clear
xtmixed math homework || schnum: homework, variance covar(un) mle
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -884.69291
Iteration 1: log likelihood = -884.69291
Computing standard errors:
Mixed-effects ML regression Number of obs = 260
Group variable: schnum Number of groups = 10
Obs per group: min = 20
avg = 26.0
max = 67
Wald chi2(1) = 1.94
Log likelihood = -884.69291 Prob > chi2 = 0.1641
------------------------------------------------------------------------------
math | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
homework | 2.048702 1.472231 1.39 0.164 -.8368176 4.934222
_cons | 44.77263 2.603198 17.20 0.000 39.67045 49.8748
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
schnum: Unstructured |
var(homework) | 19.97867 9.808877 7.632283 52.29725
var(_cons) | 61.80612 29.87304 23.96714 159.3847
cov(homework,_cons) | -28.26033 15.48584 -58.61201 2.091356
-----------------------------+------------------------------------------------
var(Residual) | 43.06696 3.92917 36.01519 51.49947
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 146.97 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
estat ic
------------------------------------------------------------------------------
Model | Obs ll(null) ll(model) df AIC BIC
-------------+----------------------------------------------------------------
| 260 . -884.6929 6 1781.386 1802.75
------------------------------------------------------------------------------
display "deviance = " -2*e(ll)
deviance = 1769.3858
Equation near bottom of page 49 and Table 3.4.
xtmixed math homework public || schnum: homework, variance covar(un) mle
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -875.41796
Iteration 1: log likelihood = -875.41775
Iteration 2: log likelihood = -875.41775
Computing standard errors:
Mixed-effects ML regression Number of obs = 260
Group variable: schnum Number of groups = 10
Obs per group: min = 20
avg = 26.0
max = 67
Wald chi2(2) = 65.83
Log likelihood = -875.41775 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
math | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
homework | 1.940879 1.524946 1.27 0.203 -1.04796 4.929718
public | -14.65118 1.831514 -8.00 0.000 -18.24088 -11.06148
_cons | 58.05564 2.694585 21.55 0.000 52.77435 63.33693
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
schnum: Unstructured |
var(homework) | 21.68283 10.53731 8.364752 56.20548
var(_cons) | 40.67683 20.62146 15.05995 109.8679
cov(homework,_cons) | -29.15938 14.48164 -57.54288 -.7758841
-----------------------------+------------------------------------------------
var(Residual) | 42.95451 3.910387 35.93512 51.34502
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 95.17 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
estat ic
------------------------------------------------------------------------------
Model | Obs ll(null) ll(model) df AIC BIC
-------------+----------------------------------------------------------------
| 260 . -875.4177 7 1764.835 1789.76
------------------------------------------------------------------------------
display "deviance = " -2*e(ll)
deviance = 1750.8355
Equation at the bottom of page 50 and Table 3.5. The negative value for the interaction coefficient in the book is probably a typo error, it should be positive.
/* create interaction */
generate homepub = homework*public
xtmixed math homework public homepub || schnum: homework, variance covar(un) mle
Performing EM optimization:
Performing gradient-based optimization:
Iteration 0: log likelihood = -875.39961
Iteration 1: log likelihood = -875.39941
Iteration 2: log likelihood = -875.39941
Computing standard errors:
Mixed-effects ML regression Number of obs = 260
Group variable: schnum Number of groups = 10
Obs per group: min = 20
avg = 26.0
max = 67
Wald chi2(3) = 65.80
Log likelihood = -875.39941 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
math | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
homework | 1.09464 4.66862 0.23 0.815 -8.055687 10.24497
public | -15.9419 6.97755 -2.28 0.022 -29.61765 -2.266157
homepub | .947246 4.938212 0.19 0.848 -8.731471 10.62596
_cons | 59.21022 6.597578 8.97 0.000 46.2792 72.14123
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
schnum: Unstructured |
var(homework) | 21.57689 10.49938 8.313648 55.99976
var(_cons) | 40.50285 20.54612 14.9862 109.4661
cov(homework,_cons) | -29.02209 14.42962 -57.30362 -.7405487
-----------------------------+------------------------------------------------
var(Residual) | 42.95437 3.910361 35.93504 51.34483
------------------------------------------------------------------------------
LR test vs. linear regression: chi2(3) = 92.27 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference
estat ic
------------------------------------------------------------------------------
Model | Obs ll(null) ll(model) df AIC BIC
-------------+----------------------------------------------------------------
| 260 . -875.3994 8 1766.799 1795.284
------------------------------------------------------------------------------
display "deviance = " -2*e(ll)
deviance = 1750.7988
Table 3.6, page 52.
/* quietly rerun model from page 47 */ quietly xtmixed math homework || schnum: homework, variance covar(un) mle /* predict random effects */ predict u1 u0, reffects /* generate posterior intercept and slope */ generate b0 = _b[_cons] + u0 generate b1 = _b[homework] + u1 tablist schnum b0 b1, sort(v) /* findit tablist */ +--------------------------------------+ | schnum b0 b1 Freq | |--------------------------------------| | 1 50.2709 -3.143404 23 | | 2 48.88624 -2.754 20 | | 3 39.19527 7.56635 24 | | 4 35.15662 5.394136 22 | | 5 53.08111 -3.73824 22 | |--------------------------------------| | 6 48.58598 -1.765202 20 | | 7 58.05541 1.335182 67 | | 8 37.15231 6.060019 21 | | 9 39.16966 5.430479 21 | | 10 38.17278 6.101702 20 | +--------------------------------------+
Figure 3.8, page 53.
twoway (function 50.2709-3.143404*x, range(0 7)) /// (function 48.88624-2.754*x, range(0 7)) /// (function 39.19527+7.56635*x, range(0 7)) /// (function 35.15662+5.394136*x, range(0 7)) /// (function 53.08111-3.73824*x, range(0 7)) /// (function 48.58598-1.765202*x, range(0 7)) /// (function 58.05541+1.335182*x, range(0 7)) /// (function 37.15231+6.060019*x, range(0 7)) /// (function 39.16966+5.430479*x, range(0 7)) /// (function 38.17278+6.101702*x, range(0 7)), /// xlabel(0(1)7) yline(0) legend(off) aspectratio(1.15)
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