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

Without exact F-distribution critical values, we can't specify for which values of \(k\) the model would be judged useful at 0.05 significance level. A high \(R^{2}\) does not automatically mean a model is useful, as it could also be an indication of overfitting, particularly if the model has many predictors in comparison to the number of observations.

Step-by-step solution

Understand the F-distribution and F-test

The F-distribution is used to test hypotheses about the variance or standard deviation of a population, commonly used in ANOVA and regression analysis. The F-statistic is the test statistic for F-tests. In regression analysis, it tests whether at least one predictor variable's coefficient differs from zero.

Calculate the F-statistic threshold

The degree of freedom for numerator, df1, is \(k\), the number of predictors, and the degree of freedom for denominator, df2, is \(n-k-1\), the number of observations minus the number of predictors minus 1. Since the model will be judged useful at significance level .05, the critical value of F could be looked up in the F-distribution table with df1 = \(k\) and df2 = \(n - k - 1\).

Determine for what values of \(k\) would the model be judged useful

The model's F-statistic should be higher than the calculated F-statistic threshold to be considered useful, given the degree of freedom and \(R^{2} = .90\). To find for which values of \(k\) the model would be judged useful, one would typically need to solve the inequality equation for \(k\). However, without more specific information about the critical F-values, this step cannot be executed exactly.

Discuss whether a high \(R^{2}\) guarantees a useful model

A high \(R^{2}\) value does not necessarily imply a useful model. While a high \(R^{2}\) generally suggests that the model explains a large portion of the variance in the response variable, it could also be a sign of overfitting, especially if the number of predictors is high relative to the number of observations.

Key concepts

F-distribution

The F-distribution is a continuous probability distribution that arises frequently when dealing with ratios of variances. In the context of multiple regression analysis, the variances we compare are typically those of models with and without certain predictors. Imagine you're trying on different pairs of glasses to see which one gives you the clearest vision, the F-distribution would help you to statistically determine which glasses (or model) fit you the best by comparing their effectiveness.

The shape of the F-distribution is impacted by two different types of degrees of freedom: one related to the model's number of predictors and the other associated with the number of data points. It is skewed right, meaning it is not symmetrical and tails off to the right, this is particularly pronounced when the sample size or the number of predictors is small.

F-test

The F-test is like the referee in a game between two competing statistical models. It uses the F-statistic to determine whether the difference in performance between the models is statistically significant. In multiple regression analysis, the F-test checks if at least one of the predictors is useful for explaining variability in the response variable, akin to verifying if any player in a team contributes to scoring goals.

Determining the F-statistic involves calculating the ratio of the variances explained by the models, which follows an F-distribution under the null hypothesis that no predictors are significant. Think of it as comparing a model with your selected predictors to a model without them - if the F-test gives a green light (a statistically significant result), your predictors are likely valuable.

R-squared

R-squared, also known as the coefficient of determination, is a number between 0 and 1 that measures how well the model fits the data. It's like a score for how much of the variability in the response variable can be explained by the model's predictors. A high R-squared value close to 1 suggests a good fit, meaning the model's predictors explain a large portion of the variance.

However, a high R-squared does not always mean the model is useful. It does not account for the number of predictors relative to the number of observations, which could lead to overfitting - this is like memorizing the answers to a test rather than understanding the material.

Model Predictors

Model predictors are the variables in a regression model that 'predict' or explain the variation in the dependent variable. Imagine them as the ingredients in a recipe that contribute to the final taste of the dish. Too few and the dish is bland; too many and the flavors conflict.

Each predictor's coefficient offers insight into the relationship between that predictor and the response variable. The significance of these predictors is tested using statistical tests such as the F-test to determine if they truly contribute to explaining the response variable or if their effects are due to random chance.

Significance Level

The significance level is a critical concept in hypothesis testing used to determine the threshold for rejecting the null hypothesis. It's akin to setting the rules for how strong the evidence must be before you declare a finding. A common significance level used is 0.05, meaning there is a 5% risk of concluding that there is an effect when there is none, which statisticians are willing to accept.

If the calculated p-value in a test is less than the significance level, the results are deemed statistically significant. To put it simply, the significance level helps us avoid jumping to conclusions based on random fluctuations in the data.

Degree of Freedom

Degrees of freedom are often likened to the number of 'choices' available when calculating a statistical estimate. In the context of regression, the degrees of freedom can be divided into two parts: one for the number of predictors (how many variables you're working with), and one for the residuals (the number of observations minus the number of parameters being estimated).

In simplest terms, degrees of freedom help us characterize the shape of the F-distribution and determine the critical values of the F-test. They allow us to attribute the variability in the data to either the model or to randomness, ensuring the validity of our inferences about the model's predictive power.