Question

Data from a sample of \(n=150\) quail eggs were used to fit a multiple regression model relating $$ y=\text { eggshell surface area }\left(\mathrm{mm}^{2}\right) $$ \(x_{1}=\) egg weight \((\mathrm{g})\) \(x_{2}=e g g\) width \((\mathrm{mm})\) $$ x_{3}=\text { egg length }(\mathrm{mm}) $$ (“Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors," International Journal of Poultry Science [2008]: 85-88). The resulting estimated regression function was $$ \begin{array}{l} 10.561+1.535 x_{1}-0.178 x_{2}-0.045 x_{3} \\ \text { and } R^{2}=.996 \end{array} $$ a. Carry out a model utility test to determine if this multiple regression model is useful. b. A simple linear regression model was also used to describe the relationship between \(y\) and \(x_{1}\), resulting in the estimated regression function \(6.254+1.387 x_{1}\). The \(P\) -value for the associated model utility test was reported to be less than .01 , and \(r^{2}=.994 .\) Is the linear model useful? Explain. c. Based on your answers to Parts (a) and (b), which of the two models would you recommend for predicting eggshell surface area? Explain the rationale for your choice.

Short answer

Both the multiple regression model and the simple linear regression model are useful as their R^2 values are significantly high. However, for predicting the eggshell surface area of quail eggs, the multiple regression model would be more recommended due to its slightly higher R^2 value and its account of more related factors.

Step-by-step solution

Carry out a model utility test for the multiple regression model

We start by interpreting the R squared (R^2) value of .996 for the multiple regression model. This high R^2 value shows that 99.6% of the variance in eggshell surface area is explained by the model. Generally, in a model utility test, if the R^2 value is significantly high (like the one we have), we can presume that the model is useful.

Evaluate the usefulness of the simple linear regression model

In this step, we look at the simple linear regression model that describes the relationship between eggshell surface area and egg weight. Given the reported P-value of less than 0.01, we can conclude that this model is statistically significant, hence it is considered useful. Also, we have \(r^2=0.994\), meaning that the model explains 99.4% of the variability in the eggshell surface area, which is reasonably high.

Compare the two models and make a recommendation

Both models have a high R^2 value and are considered useful. However, the multiple regression model with an R^2 of 99.6% marginally outperforms the simple linear regression model with an R^2 of 99.4%. Besides, the multiple regression model takes into account more factors (egg width and egg length) that might affect the eggshell surface area, suggesting a more comprehensive understanding of the factors determining eggshell surface area. Hence, the multiple regression model would usually be recommended.

Key concepts

Model Utility Test

A model utility test is crucial for determining if a regression model is useful in predicting the dependent variable. In this context, the test helps us understand how well the models predict eggshell surface area based on factors like egg weight and dimensions.

Here's how it works:

• We assess the adequacy of a model by looking at its ability to explain the variability in the dependent variable. This is typically measured through the R-squared value.
• If the R-squared value is significantly high, it suggests that the model has strong predictive power, meaning it explains a large portion of the variance in the data.
• An additional aspect is the P-value from a statistical test (like the F-test), which indicates whether the observed model performance could occur by chance.

For our exercise, both the multiple regression and simple linear regression models undergo such a test. The multiple regression model shows an R-squared of 0.996, while the simple linear regression has an R-squared of 0.994. Both values signify high utility in modeling eggshell surface area.

R-squared Value

The R-squared value is a statistical measure that explains the proportion of variance in the dependent variable that's predictable from the independent variables. Simply put, it tells us how well our model represents the data.

In terms of multiple regression:

• R-squared values range from 0 to 1.
• A value closer to 1 indicates that a large proportion of the variance in the outcome variable is explained by the model.
• In this exercise, an R-squared value of 0.996 for the multiple regression suggests that 99.6% of the variability in eggshell surface area is accounted for by the factors included in the model.

For simple regression, though it explains slightly less variability (99.4%), it too has a high R-squared value, indicating excellent model fit. Such high values in both cases affirm the models' capabilities in effectively predicting the outcome.

Simple Linear Regression

Simple linear regression is a statistical method to model the relationship between two variables by fitting a linear equation to observed data. Here, it's used to understand the relationship between eggshell surface area and egg weight.

This technique involves:

• Identifying a straight line that best fits the data points.
• The equation form is usually: \( y = a + bx \), where \( y \) is the dependent variable (eggshell surface area), \( x \) is the independent variable (egg weight), \( a \) is the intercept, and \( b \) is the slope.

In our exercise, the model \(y = 6.254 + 1.387x_1\) signifies how eggshell surface area changes with egg weight. Given the high R-squared (0.994), it shows a strong positive relationship. Furthermore, the P-value below 0.01 confirms the model's significance suggesting it effectively captures the connection between these two variables.

Statistical Significance

Statistical significance helps us determine whether the results observed in the study are likely due to something other than random chance. When talking about regression models, it often involves analyzing P-values.

Key points include:

• A P-value quantifies the probability of observing results as extreme as the actual results, assuming the null hypothesis is true.
• A P-value less than a certain threshold (commonly 0.05 or 0.01) suggests the results are statistically significant, meaning the model is a good fit for the data.
• In our context, the simple linear regression model gives a P-value less than 0.01, indicating it is statistically significant and effectively predicts eggshell surface area using egg weight alone.

This concept is crucial as it supports the validity of the regression models used in our exercise, ensuring confidence in the findings and recommendations based on these models.