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How can I perform the likelihood ratio, Wald, and Lagrange multiplier (score) test in Stata?

Where L(m*) denotes the likelihood of the respective model, and ll(m*) the natural log of the models' likelihood.

This statistic is distributed chi-squared with degrees of freedom equal to the difference in the number of degrees of freedom between the two models (i.e., the number of variables added to the model). In order to perform the likelihood ratio test we will need to run both models and make note of their final log likelihoods. We will run the models using Stata and use commands to store the log likelihoods. We could also just copy the likelihoods down (i.e., by writing them down, or cutting and pasting), but using commands is a little easier and is less likely to result in errors. The first line of syntax below reads in the dataset from our website. The second line of syntax runs a logistic regression model, predictingBelow is the output. In order to perform the likelihood ratio test we will need to keep track of the log likelihood (-102.44), the syntax for this example (above) does this by storing the value in a scalar. Since it is not our primary concern here, we will skip the interpretation of the rest logistic regression model. Note that storing the returned estimate does not produce any output.use http://www.ats.ucla.edu/stat/stata/faq/nested_tests, clear logit hiwrite female read scalar m1 = e(ll)

The first line of syntax below runs the second model, that is, the model with all four predictor variables. The second line of code stores the value of the log likelihood for the model (-84.4), which is temporarily stored as the returned estimate (Iteration 0: log likelihood = -137.41698 Iteration 1: log likelihood = -104.79885 Iteration 2: log likelihood = -102.52269 Iteration 3: log likelihood = -102.44531 Iteration 4: log likelihood = -102.44518 Logistic regression Number of obs = 200 LR chi2(2) = 69.94 Prob > chi2 = 0.0000 Log likelihood = -102.44518 Pseudo R2 = 0.2545 ------------------------------------------------------------------------------ hiwrite | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.403022 .3671964 3.82 0.000 .6833301 2.122713 read | .1411402 .0224042 6.30 0.000 .0972287 .1850517 _cons | -7.798179 1.235685 -6.31 0.000 -10.22008 -5.376281 ------------------------------------------------------------------------------

Now that we have the log likelihoods from both models, we can perform a likelihood ratio test. The first line of syntax below calculates the likelihood ratio test statistic. The second line of syntax below finds the p-value associated with our test statistic with two degrees of freedom. Looking below we see that the test statistic is 36.05, and that the associated p-value is very low (less than 0.0001). The results show that addinglogit hiwrite female read math science scalar m2 = e(ll)Iteration 0: log likelihood = -137.41698 Iteration 1: log likelihood = -90.166892 Iteration 2: log likelihood = -84.909776 Iteration 3: log likelihood = -84.42653 Iteration 4: log likelihood = -84.419844 Iteration 5: log likelihood = -84.419842 Logistic regression Number of obs = 200 LR chi2(4) = 105.99 Prob > chi2 = 0.0000 Log likelihood = -84.419842 Pseudo R2 = 0.3857 ------------------------------------------------------------------------------ hiwrite | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.805528 .4358101 4.14 0.000 .9513555 2.6597 read | .0529536 .0275925 1.92 0.055 -.0011268 .107034 math | .1319787 .0318836 4.14 0.000 .069488 .1944694 science | .0577623 .027586 2.09 0.036 .0036947 .1118299 _cons | -13.26097 1.893801 -7.00 0.000 -16.97275 -9.549188 ------------------------------------------------------------------------------

di "chi2(2) = " 2*(m2-m1) di "Prob > chi2 = "chi2tail(2, 2*(m2-m1))chi2(2) = 36.050677 Prob > chi2 = 1.485e-08

Below is the output. Since it is not our primary concern here, we will skip the interpretation of the logistic regression model. Note that storing the estimates does not produce any output.use http://www.ats.ucla.edu/stat/stata/faq/nested_tests, clear logit hiwrite female read estimates store m1

The first line of syntax below this paragraph runs the second model, that is the model with all four predictor variables. The second line of syntax saves the estimates from this model, and names themIteration 0: log likelihood = -137.41698-137.41698-137.41698 Iteration 1: log likelihood = -104.79885 Iteration 2: log likelihood = -102.52269 Iteration 3: log likelihood = -102.44531 Iteration 4: log likelihood = -102.44518 Logistic regression Number of obs = 200 LR chi2(2) = 69.94 Prob > chi2 = 0.0000 Log likelihood = -102.44518 Pseudo R2 = 0.2545 ------------------------------------------------------------------------------ hiwrite | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.403022 .3671964 3.82 0.000 .6833301 2.122713 read | .1411402 .0224042 6.30 0.000 .0972287 .1850517 _cons | -7.798179 1.235685 -6.31 0.000 -10.22008 -5.376281 ------------------------------------------------------------------------------

The first line of syntax below tells Stata that we want to run an lr test, and that we want to compare the estimates we have saved aslogit hiwrite female read math science estimates store m2Iteration 0: log likelihood = -137.41698 Iteration 1: log likelihood = -90.166892 Iteration 2: log likelihood = -84.909776 Iteration 3: log likelihood = -84.42653 Iteration 4: log likelihood = -84.419844 Iteration 5: log likelihood = -84.419842 Logistic regression Number of obs = 200 LR chi2(4) = 105.99 Prob > chi2 = 0.0000 Log likelihood = -84.419842 Pseudo R2 = 0.3857 ------------------------------------------------------------------------------ hiwrite | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.805528 .4358101 4.14 0.000 .9513555 2.6597 read | .0529536 .0275925 1.92 0.055 -.0011268 .107034 math | .1319787 .0318836 4.14 0.000 .069488 .1944694 science | .0577623 .027586 2.09 0.036 .0036947 .1118299 _cons | -13.26097 1.893801 -7.00 0.000 -16.97275 -9.549188 ------------------------------------------------------------------------------

The entire syntax for a likelihood ratio test, all in one block, looks like this:lrtest m1 m2Likelihood-ratio test LR chi2(2) = 36.05 (Assumption: A nested in B) Prob > chi2 = 0.0000

logit hiwrite female read estimates store m1 logit hiwrite female read math science estimates store m2 lrtest m1 m2

quietly: logit hiwrite female read math science test math science( 1) math = 0 ( 2) science = 0 chi2( 2) = 27.53 Prob > chi2 = 0.0000

*Please note that the user-written testomit is no longer
available in Stata. *

*Please note that the user-written testomit is no longer
available in Stata. *

The first part of the output gives the type of model run, followed by a table of results. The results of the score test are distributed chi-squared with degrees of freedom equal to the number of variables added to the model. The table has three columns, the first giving the value of the test statistic, the second the number of degrees of freedom for the test, and the third giving the p-value associated with a chi-squared of a given value with a given number of degrees of freedom. The variablesquietly: logit hiwrite female read predict test, score testomit math science, score(test)logit: score tests for omitted variables Term | score df p ---------------------+---------------------- math | 28.94 1 0.0000 science | 15.39 1 0.0001 ---------------------+---------------------- simultaneous test | 35.51 2 0.0000 ---------------------+----------------------

The command **testomit** behaves somewhat differently for different estimation
commands. Below are examples of how to use **testomit** with several other regression commands.
Most multiple equation commands will use a syntax similar to the syntax for **mlogit**. Two exceptions
are **ologit** and **oprobit**, and **regress**, which are shown separately.

*Please note that the user-written testomit is no longer
available in Stata. *

For **mlogit** and many other multiple-equation commands:

mlogit perform read write predict mslo mshi, score testomit (low: science math) (high: science math), score(mslo mshi)

For **ologit** and **oprobit**:

ologit perform read write predict coef cut1 cut2, score testomit (LP: math science), score(coef)

For **regress**:

reg read write math predict regs, score testomit (mean: science female), score(regs)

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