Question

This exercise requires the use of a computer package. The accompanying data resulted from a study of the relationship between \(y=\) brightness of finished paper and the independent variables \(x_{1}=\) hydrogen peroxide \((\%\) by weight), \(x_{2}=\) sodium hydroxide (\% by weight), \(x_{3}=\) silicate (\% by weight), and \(x_{4}=\) process temperature (“Advantages of CE-HDP Bleaching for High Brightness Kraft Pulp Production," TAPPI [1964]: \(107 \mathrm{~A}-173 \mathrm{~A})\). a. Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. b. Using a .05 significance level, perform the model utility test. c. Interpret the values of the following quantities: SSResid, \(R^{2},\) and \(s_{e}\)

Short answer

The estimated regression equation will depend on the data obtained from software and will have all variables included with quadratic terms and interaction. For model utility test, based on the F statistics and p-value, the null hypothesis will be rejected or failed to reject. SSResid is the variation of y not explained by predictors and small value indicates a better fit to the data. \(R^{2}\) explains the variability of the model, a value close to 1 indicates better explanatory power. \(s_{e}\) is the standard error of the estimation, a small value indicates better predictive accuracy.

Step-by-step solution

Model Formulation

Start by setting up the general form of a multiple regression, which includes all linear, quadratic, and interaction terms. The form should look like this: \(y = \beta_0 + \beta_1x_{1} + \beta_2x_{2} + \beta_3x_{3} + \beta_4x_{4} + \beta_5x_{1}^{2} + \beta_6x_{2}^{2} + \beta_7x_{3}^{2} + \beta_8x_{4}^{2} + \beta_9x_{1}x_{2} + \beta_{10}x_{1}x_{3} + \beta_{11}x_{1}x_{4} + \beta_{12}x_{2}x_{3} + \beta_{13}x_{2}x_{4} + \beta_{14}x_{3}x_{4} + \epsilon\)

Fit the Regression Model

Next, use your computer software package to perform the multiple regression analysis, following the the form in Step 1. This software will give you the estimated coefficients for the variables.

Model Utility Test

Perform a model utility test using a .05 level of significance to determine if the model has significant predictive utility. This will involve calculating an F statistic for the regression and comparing it to a critical value or p-value, which your software can also provide. If the F statistic is greater than the critical value or the p-value is less than the .05 significance level, reject the null hypothesis, otherwise fail to reject it.

Interpret the Values

Interpret the following quantities:SSResid: This is the sum of the squares of the residuals, which represents the total variation in y that is not explained by the predictors. A smaller residual sum of squares indicates a better fit to the data.\(R^{2}\): This is the coefficient of determination, which provides a measure of how well observed outcomes are replicated by the model. An \(R^{2}\) close to 1 indicates a model that explains a large amount of the variability.\(s_{e}\): This is the standard error of the estimate, the standard deviation of the error term \(\epsilon\), and it is a measure of the accuracy of predictions.

Key concepts

Model Formulation

Model formulation in multiple regression analysis is akin to laying the foundation for a building — it sets the stage for the entire analytical process. In essence, it involves specifying the model that best represents the relationship between the dependent variable (in this case, the brightness of finished paper) and a set of independent variables (variables like hydrogen peroxide, sodium hydroxide, etc.).

To establish a comprehensive model, one must consider not only the primary effects (linear terms) but also the quadratic (squared) terms and the interaction terms between these independent variables. This robust form encapsulates complex relationships that might exist, where the effect of one variable could depend on the level of another. For instance, a term like \(\beta_9x_{1}x_{2}\) explores how the combined influence of hydrogen peroxide and sodium hydroxide affects the paper's brightness.

Model Utility Test

Think of the model utility test as a reality check for the created regression model. It essentially answers the question: 'Does this model provide useful insights?' To conduct this test, one computes an F-statistic and compares it against a critical value, or checks the p-value against the chosen significance level, usually 0.05.

These statistical procedures help determine whether the data provides sufficient evidence to suggest that at least one of the predictor variables is significantly related to the dependent variable. If the p-value is less than 0.05, we reject the null hypothesis — typically that the model has no utility — and conclude that our model has predictive capabilities.

SSResid

SSResid or the sum of squares due to error represents the difference between the observed and predicted values — something akin to the amount of 'unexplained' variation in the data. Imagine shooting arrows at a target; SSResid measures how far the arrows (predicted values) are from the bullseye (actual values).

In the context of our regression model, a smaller SSResid signifies that the model's predictions are close to the actual data points, implying a better fit. It's important for students and researchers alike to aim for a more precise model which is indicated by a lower value of SSResid.

Coefficient of Determination

The coefficient of determination, denoted as \(R^2\), is a statistic that gauges the proportion of variance in the dependent variable that's predictable from the independent variables. It's a powerful number that ranges from 0 to 1, where a value closer to 1 indicates that a large fraction of the variance in the dependent variable is captured by the model.

It's the statistical equivalent of an accuracy rating for the model's predictions. When explaining to students, it's like assessing the effectiveness of a study guide in helping them ace a test — a higher \(R^2\) means the guide (model) is highly effective.

Standard Error of the Estimate

The standard error of the estimate, denoted as \(s_{e}\), measures the typical distance between the observed data points and the model's predicted values — it's like gauging the precision of a measuring tool. A smaller \(s_{e}\) indicates that the model's predictions are, on average, closer to the actual observed values, signifying higher prediction accuracy.

I often describe \(s_{e}\) to my students as the ruler with which we measure how 'off' our predictions might be on average. A lower standard error is akin to using a fine-point pen instead of a chunky marker to draw the line of best fit — it represents higher precision in the model.