## Nested Regression Models: Understanding Likelihood Ratio Test

Regression analysis is a powerful tool that is widely used in various fields of research and applications. It involves a set of statistical techniques that aim to establish the relationship between a response variable and one or more predictor variables.

One important concept in regression models is the notion of nested models. In this article, we will explore nested regression models and the likelihood ratio test, a statistical tool used to compare different models and select the best one for a given problem.

### Nested Models: Definition and Concept

Nested models are a type of regression model where one model is a subset of the other. In other words, the smaller model is nested within the larger model.

The smaller model can be obtained from the larger model by removing one or more predictor variables. For instance, suppose we have a model that includes the predictor variables X1, X2, X3 and a response variable Y.

A nested model of this could be obtained by removing X3 from the model, resulting in a model with X1, X2 and Y only.

The concept of nested models is essential in regression analysis because it allows us to compare models with different numbers of predictor variables and different levels of complexity.

In practice, we can use nested models to investigate which predictor variables are most relevant to the response variable, determine how much predictive power we gain by adding more variables, and choose the best model for our problem. Example: Full Model

Consider the following example in Python that uses the ‘mtcars’ dataset.

Suppose we are interested in understanding the relationship between the fuel consumption (mpg) of cars and other variables such as weight (wt), horsepower (hp), and transmission type (am). Our full model includes all three predictor variables and the response variable mpg.

#Importing libraries

import pandas as pd

import statsmodels.api as sm

#Loading the mtcars dataset

mtcars = sm.datasets.get_rdataset("mtcars").data

#Defining the predictor variables

X = mtcars[['wt', 'hp', 'am']]

#Defining the response variable

y = mtcars['mpg']

#Running the OLS regression

model_full = sm.OLS(y, sm.add_constant(X)).fit()

The resulting model provides us with the coefficients and statistical measures that characterize the relationship between the predictor variables and the response variable. We can also use the log-likelihood value to evaluate the goodness-of-fit of our model.

### Example: Reduced Model

Now, suppose we want to investigate whether the predictor variable ‘am’ is relevant to our model. To test this hypothesis, we can create a reduced model by removing the ‘am’ variable from the full model.

#Defining the predictor variables

X_reduced = mtcars[['wt', 'hp']]

#Running the OLS regression

model_reduced = sm.OLS(y, sm.add_constant(X_reduced)).fit()

The reduced model provides us with a different set of coefficients and statistical measures than the full model. However, we need to determine which model is a better fit for our problem.

### Likelihood Ratio Test: Definition and Concept

The likelihood ratio test is a statistical tool that compares the fit of two nested models and evaluates whether the additional predictor variables in the larger model improve the model fit enough to justify their inclusion. The test compares the log-likelihood values of the full and reduced models and uses a Chi-Squared test statistic to determine if there is a significant difference in the quality of fit between the two models.

## Hypotheses and Significance Level

To conduct the likelihood ratio test, we need to define two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis states that the full model is not significantly better than the reduced model, i.e., the additional predictor variable(s) do not contribute to the model fit.

In contrast, the alternative hypothesis states that the full model is significantly better than the reduced model, i.e., the additional predictor variable(s) improve the model fit. The significance level is the probability of committing a Type I error, i.e., rejecting the null hypothesis when it is true.

In practice, we usually set the significance level at 0.05, which means that we will reject the null hypothesis if the p-value is below 0.05.

### Example: Likelihood Ratio Test in Python

To conduct the likelihood ratio test in Python, we can use the log-likelihood values of the full and reduced models and the Chi-Squared test statistic provided by the statsmodels library.

#Calculating the log-likelihoods of the full and reduced models

loglike_full = model_full.llf

loglike_reduced = model_reduced.llf

#Calculating the Chi-Squared test statistic

LR_statistic = 2*(loglike_full - loglike_reduced)

df = X.shape[1] - X_reduced.shape[1]

pvalue = 1 - stats.chi2.cdf(LR_statistic, df)

The likelihood ratio test in this example yields a p-value less than 0.05, which implies that we can reject the null hypothesis and conclude that the additional predictor variable ‘am’ significantly improves the model fit. Therefore, we can select the full model as our final model.

## Final Model Selection

Finally, we can use the information obtained from nested models and the likelihood ratio test to select the best model for our problem. Typically, we want to select a model that achieves a balance between simplicity and adequacy, i.e., a model that has enough predictive power without overfitting the data.

In practice, we can use a combination of statistical measures such as the coefficient of determination (R-squared), adjusted R-squared, and the Akaike Information Criterion (AIC) to evaluate the goodness-of-fit of the models and choose the one that provides sufficient explanatory power while avoiding overfitting.

## Conclusion

Nested regression models and the likelihood ratio test are essential tools in regression analysis that allow us to compare different models and select the best one for a given problem. The concept of nested models enables us to investigate which predictor variables are most relevant to the response variable, determine how much predictive power we gain by adding more variables, and decide on the final model to use.

The likelihood ratio test is a statistical tool that compares the fit of two nested models and evaluates whether the additional predictor variables significantly improve the model fit. In practice, we can use a combination of statistical measures to evaluate the goodness-of-fit of the models and select the best one based on our problem’s criteria.

In conclusion, the likelihood ratio test is an essential statistical tool in regression analysis that allows us to compare nested models and select the best one for a given problem. This test provides a way to determine whether additional predictor variables improve the model fit and achieve a balance between simplicity and predictive power.

By understanding the concept of nested models and using the likelihood ratio test, we can investigate predictor variables’ relevance and choose the best model based on our problem’s criteria. Overall, the likelihood ratio test is a powerful tool that can help researchers and data analysts make informed decisions and improve their regression analysis results.