The likelihood ratio (lr) test and Wald test test are commonly used to evaluate the difference between nested models.
One model is considered nested in another if the first model can be
generated by imposing restrictions on the parameters of the second. Most often,
the restriction is that the parameter is equal to zero. In a regression model
restricting a parameters to zero is accomplished
by removing the predictor variables from the model. For example, in the models below, the
model with the predictor variables **female**, and **read**, is nested within
the model with the predictor variables **female**, **read**, **math**, and
**science**. The lr and Wald ask the same basic question, which is, does constraining these
parameters to zero (i.e., leaving out these predictor variables) significantly reduce the fit
of the model? To perform a likelihood ratio test, one must estimate both of the
models one wishes to compare. The
advantage of the Wald test is that it approximates the lr test but
require that only one model be estimated. When computing power was much more limited, and many models took
a long time to run, this was a fairly major advantage. Today, for most of the models
researchers are likely to want to compare, this is not an issue, and we generally
recommend
running the likelihood ratio test in most situations. This is not to say that
one should never use the Wald test. For example, the Wald test is
commonly used to perform multiple degree of freedom tests on sets of dummy
variables used to model categorical variables in regression (for more
information see our webbook on Regression with Stata, specifically
Chapter 3 – Regression with Categorical Predictors).

As we mentioned above, the lr test requires that two models be run, one of which has a set of parameters (variables), and a second model with all of the parameters from the first, plus one or more other variables. The Wald test examines a model with more parameters and assess whether restricting those parameters (generally to zero, by removing the associated variables from the model) seriously harms the fit of the model. In general, both tests should come to the same conclusion (because the Wald test, at least in theory, approximate the lr test). As an example, we will test for a statistically significant difference between two models, using both tests.

The dataset for this example includes demographic data, as well as standardized
test scores for 200 high school students. We will compare two models. The dependent
variable for both models is **hiwrite** (to be nested two models must share the
same dependent variable), which is a dichotomous variable indicating that the
student had a writing score that was above the mean. There are four possible
predictor variables, **female**, a dummy variable which indicates that the student
is female, and the continuous variables **read**, **math**, and **science**,
which are the student’s standardized test scores in reading, math, and science, respectively.
We will test a model containing just the predictor variables **female** and **read**,
against a model that contains the predictor variables **female** and **read**,
as well as, the additional predictor variables, **math** and **science**.

## Example of a likelihood ratio test.

As discussed above, the lr test involves estimating two models and comparing them. Fixing one or more parameters to zero, by removing the variables associated with that parameter from the model, will almost always make the model fit less well, so a change in the log likelihood does not necessarily mean the model with more variables fits significantly better. The lr test compares the log likelihoods of the two models and tests whether this difference is statistically significant. If the difference is statistically significant, then the less restrictive model (the one with more variables) is said to fit the data significantly better than the more restrictive model. The lr test statistic is calculated in the following way:

LR = -2 ln(L(m1)/L(m2)) = 2(ll(m2)-ll(m1))

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, predicting **hiwrite** based
on students’ gender (**female**), and reading scores (**read**). The
third line of code stores the value of the log likelihood for the model, which
is temporarily stored as the returned estimate **e(ll)** (for more
information type **help return** in the Stata command window), in the scalar named **
m1**.

use https://stats.idre.ucla.edu/stat/stata/faq/nested_tests, clear logit hiwrite female read scalar m1 = e(ll)

Below 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.

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

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 (
**e(ll)** ), in the scalar named **m2**. Again, we won’t say much about the output except to note that the coefficients for both **math** and
**science** are both statistically significant. So we know that, individually, they are statistically significant
predictors of **hiwrite**.

logit 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 ------------------------------------------------------------------------------

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 adding **math** and **science** as predictor variables together (not just
individually) results in a statistically significant improvement in model fit. Note that if we performed a likelihood ratio test for adding a single variable to the model, the results would be the same as the significance test for the coefficient for that variable presented in the
above table.

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

## Using Stata’s postestimation commands to calculate a likelihood ratio test

As you have seen, it is easy enough to calculate a likelihood ratio
test “by hand.” However, you can also use Stata to store the estimates and run
the test for you. This method is easier still, and probably less error prone. The first line of syntax runs a logistic regression model, predicting **hiwrite** based on students’ gender
(**female**), and reading scores (**read**). The second line of syntax asks Stata to store
the estimates from the model we just ran, and instructs Stata that we want to call the estimates
**m1**.
It is necessary to give the estimates a name, since Stata allows users to store
the estimates from more than one analysis, and we will be storing more than one
set of estimates.

use https://stats.idre.ucla.edu/stat/stata/faq/nested_tests, clear logit hiwrite female read estimates store m1

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.

Iteration 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 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 them **m2**. Below the syntax is the output generated. Again, we won’t say much about
the output except to note that the coefficients for both **math** and **science** are both
statistically significant. So we know that, individually, they are statistically significant predictors
of **hiwrite**. The tests below will allow us to test whether adding both of these variables to the model
significantly improves the fit of the model, compared to a model that contains just **female** and **read**.

logit 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 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 as **m1** to those we have saved as **m2**. The output reminds us that this test assumes that
A is nested in B, which it is. It also gives us the chi-squared value for the test (36.05) as well as the
p-value for a chi-squared of 36.05 with two degrees of freedom. Note that the degrees of freedom for
the lr test, along with the other two tests, is equal to the number of parameters that are constrained (i.e., removed from the model), in our
case, 2. Note that the results are the same as when we calculated the lr test by
hand above. Adding **math** and **science** as predictor variables together (not just
individually) results in a statistically significant improvement in model fit.
As noted when we calculated the likelihood ratio test by hand, if we performed a likelihood ratio test for adding a single variable to the model, the results would be the same as the significance test for the coefficient for that variable presented in the table above.

lrtest m1 m2Likelihood-ratio test LR chi2(2) = 36.05 (Assumption: A nested in B) Prob > chi2 = 0.0000

The entire syntax for a likelihood ratio test, all in one block, looks like this:

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

## Example of a Wald test

As was mentioned above, the Wald test approximates the lr test, but with the advantage that it only requires estimating one model. The Wald test works by testing that the parameters of interest are simultaneously equal to zero. If they are, this strongly suggests that removing them from the model will not substantially reduce the fit of that model, since a predictor whose coefficient is very small relative to its standard error is generally not doing much to help predict the dependent variable.

The first step in performing a Wald test is to run the full model (i.e., the model containing all four predictor variables). The first line of syntax below does this (but uses the quietly
prefix so that the output from the regression is not shown). The second line of syntax below instructs Stata to run a Wald test
in order to test whether the coefficients for the variables **math** and **science**
are simultaneously equal to zero. The output first gives the null hypothesis.
Below that we see the chi-squared value generated by the Wald test, as well as
the p-value associated with a chi-squared of 27.53 with two degrees of freedom.
Based on the p-value, we are able to reject the null hypothesis, again
indicating that the coefficients for **math** and **science** are not simultaneously equal to zero,
meaning that including
these variables create a statistically significant improvement in the fit of the model.

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