Question

Suppose that a multiple regression data set consists of \(n=15\) observations. For what values of \(k\), the number of model predictors, would the corresponding model with \(R^{2}=.90\) be judged useful at significance level \(.05 ?\) Does such a large \(R^{2}\) value necessarily imply a useful model? Explain.

Short answer

To determine the number of predictors a model can handle given a set of parameters, use an F-distribution given the \(R^{2}\) and significance level. While a high \(R^{2}\) value does imply a potentially useful model as it captures a high percentage of the variance in the dependent variable, it does not automatically guarantee it since we also need to validate the assumptions of the model and check against overfitting.

Step-by-step solution

Understanding Coefficient of Determination \(R^{2}\)

The coefficient of determination, represented as \(R^{2}\), is a key measure used to assess the quality of a regression model. It provides the proportion of response variation that is captured by the regression model. In other words, an \(R^{2}\) value of .90 means that 90% of the variation in the dependent variable can be explained by the independent variables present in the model.

Determine Values of \(k\) Using F-Distribution and Significance Level

Since we want to judge if the model is statistically useful at a significance level of .05, we have to involve the use of F-distribution, specifically the upper quartile of the F-distribution. Given that we have the values for \(R^{2}\), \(n\), and significance level, we can obtain the threshold F-value. From there, we can isolate \(k\) by using the formula for F-value in multiple regression which is: \( F = \frac{R^{2}/k}{(1-R^{2})/(n-k-1)} \)

Implication of High \(R^{2}\)

A high \(R^{2}\) value does imply a potentially useful model, as it suggests that a high percentage of the variance in the dependent variable can be explained by the independent variables in the model. However, the deemed usefulness of the model that yields a high \(R^{2}\) value also depends on the validity of any assumptions made in the creation of the model and if the model is not overfitting the sample data (i.e., the model also performs well on unseen data).