cdanote087.txt
-------------
Section 4.2.3
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. infile LI n r using cda088.dat
(14 observations read)
. blogit r n LI
Logit Estimates Number of obs = 27
chi2(1) = 8.30
Prob > chi2 = 0.0040
Log Likelihood = -13.036482 Pseudo R2 = 0.2414
------------------------------------------------------------------------------
_outcome | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
LI | .1448632 .0593412 2.441 0.015 .0285567 .2611697
_cons | -3.77714 1.378628 -2.740 0.006 -6.479202 -1.075078
------------------------------------------------------------------------------
. blogit, or
Logit Estimates Number of obs = 27
chi2(1) = 8.30
Prob > chi2 = 0.0040
Log Likelihood = -13.036482 Pseudo R2 = 0.2414
------------------------------------------------------------------------------
_outcome | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
LI | 1.155881 .0685913 2.441 0.015 1.028968 1.298448
------------------------------------------------------------------------------
Estimating the dose that produces 50% remission
= finding value for dose that makes logit=0 (that is, prob=0.5)
= - alpha/beta
= -3.777/0.145
. display -3.777/0.145
-26.048276
Constructing Table 4.2 requires us to calculate fitted probabilities for
the logit, probit, and linear probability models. We have just
calculated the logistic regression model, so the predict command will do
exactly the calculation we need for the logit model. Following that, we
fit the probit model and have Stata create the fitted probabilities from
that model.
Finally, we calculate the linear probability model and the fitted values
corresponding to that model. To accomplish that, we must first estimate
the probabilities from each row, and then fit these probabilities using
linear regression. Note that the number of cases at each value of the
Labelling Index is not the same; the [fweight=n] portion of the regress
command lets Stata know that the variable n contains the frequency
corresponding to that particular observation.
. predict logithat
. bprobit r n LI
. predict prbithat
. generate phat = r/n
. regress phat LI [fweight=n]
Source | SS df MS Number of obs = 27
---------+------------------------------ F( 1, 25) = 18.33
Model | 1.76246588 1 1.76246588 Prob > F = 0.0002
Residual | 2.40420086 25 .096168034 R-squared = 0.4230
---------+------------------------------ Adj R-squared = 0.3999
Total | 4.16666675 26 .160256413 Root MSE = .31011
------------------------------------------------------------------------------
phat | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
LI | .0278284 .0065004 4.281 0.000 .0144405 .0412163
_cons | -.2252962 .1434906 -1.570 0.129 -.5208206 .0702282
------------------------------------------------------------------------------
. predict wlinhat
. list LI n r logithat prbithat wlinhat
LI n r logithat prbithat wlinhat
1. 8 2 0 .0679741 .0531573 -.0026689
2. 10 2 0 .0887893 .0750353 .0529879
3. 12 3 0 .1151908 .1031899 .1086447
4. 14 3 0 .1481664 .1383233 .1643015
5. 16 3 0 .18857 .1808359 .2199583
6. 18 1 1 .2369268 .2307179 .2756152
7. 20 3 2 .2932034 .287472 .331272
8. 22 2 1 .3566005 .3500869 .3869288
9. 24 1 0 .425454 .4170732 .4425856
10. 26 1 1 .4973257 .4865633 .4982424
11. 28 1 1 .5693082 .5564648 .5538992
12. 32 1 0 .7023434 .6891388 .6652129
13. 34 1 1 .7591835 .7482874 .7208697
14. 38 3 2 .849113 .8462564 .8321833
For Table 4.3, the following illustration shows how to calculate the
empirical logits and the predicted number of remissions.
. input LI
LI
1. 10
2. 16
3. 22
4. 28
5. 36
. regress emplogit LI [fw=cases]
Source | SS df MS Number of obs = 27
---------+------------------------------ F( 1, 25) = 235.22
Model | 43.5440066 1 43.5440066 Prob > F = 0.0000
Residual | 4.62796322 25 .185118529 R-squared = 0.9039
---------+------------------------------ Adj R-squared = 0.9001
Total | 48.1719698 26 1.85276807 Root MSE = .43025
------------------------------------------------------------------------------
emplogit | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
LI | .1456771 .0094984 15.337 0.000 .1261147 .1652395
_cons | -3.824297 .2078753 -18.397 0.000 -4.252424 -3.39617
------------------------------------------------------------------------------
. predict fitlogit
. gen predrem = cases * exp(fitlogit) / (1+ exp(fitlogit))
. format emplogit fitlogit predrem %6.2f
. list LI cases remits emplogit fitlogit predrem
LI cases remits emplogit fitlogit predrem
1. 10 7 0 -2.71 -2.37 0.60
2. 16 7 1 -1.47 -1.49 1.28
3. 22 6 3 0.00 -0.62 2.10
4. 28 3 2 0.51 0.25 1.69
5. 36 4 3 0.85 1.42 3.22
Note that the predicted number of remissions disagrees slightly with those
of Table 4.3. The predicted numbers in the last column correspond to an
estimated beta of 0.1473 instead of the 0.1457 value obtained above. For
practical purposes, the differences are of no importance, but it would be
nice to know just how the last column of the table was calculated.
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Section 4.2.4 The Wald statistic
-------------
Returning to the top of the page, here is an excerpt from the logistic
regression calculation based on the full table
Logit Estimates Number of obs = 27
chi2(1) = 8.30
_outcome | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
LI | .1448632 .0593412 2.441 0.015 .0285567 .2611697
_cons | -3.77714 1.378628 -2.740 0.006 -6.479202 -1.075078
------------------------------------------------------------------------------
The z-statistic is exactly what Agresti refers to in the last paragraph of
the section. Note that the chi-squared statistic at the upper right of
the display is the likelihood-ratio statistic.