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Chapter 12: Problem 22

A sample of 25 observations is used to fit a regression model in seven variables. The estimate of \(\sigma^{2}\) for this full model is \(M S_{E}=10\). (a) A forward selection algorithm has put three of the original seven regressors in the model. The error sum of squares for the three-variable model is \(S S_{E}=300 .\) Based on \(C_{p}\) would you conclude that the three-variable model has any remaining bias? (b) After looking at the forward selection model in part (a), suppose you could add one more regressor to the model. This regressor will reduce the error sum of squares to 275 Will the addition of this variable improve the model? Why?

Short Answer

Expert verified
(a) Yes, the three-variable model shows bias since \( C_p = 13 > p = 4 \). (b) Yes, adding the new regressor improves the model as \( C_p \) moves closer to \( p \).

Step by step solution

01

Understanding Criteria for C_p

The criterion \( C_p \) is calculated as: \( C_p = \frac{SSE}{MSE} + 2p - n \) where \( SSE \) is the sum of squared errors for the model, \( MSE \) is the mean square error for the full model, \( p \) is the number of model parameters (including the intercept), and \( n \) is the sample size.
02

Calculate C_p for the Three-Variable Model

For the three-variable model, the number of parameters \( p = 4 \) (3 regressors + 1 intercept). Given \( SSE = 300 \), \( MSE = 10 \) from the full model, and \( n = 25 \), we substitute into the formula: \[ C_p = \frac{300}{10} + 2(4) - 25 \] which simplifies to \[ C_p = 30 + 8 - 25 = 13. \] Since \( C_p > p \), it suggests potential bias in the model.
03

New Model Calculation with Four Variables

Adding another regressor changes \( SSE \) to 275 and \( p \) to 5. Now, calculate \( C_p \) for the new model: \[ C_p = \frac{275}{10} + 2(5) - 25 = 27.5 + 10 - 25 = 12.5. \] Since the new \( C_p \) (12.5) is closer to \( p = 5 \), adding the variable potentially reduces bias.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Forward Selection Algorithm
The Forward Selection Algorithm is a step-by-step method used in regression analysis to find a subset of predictor variables that best predict the outcome variable. It begins without any predictors in the model and adds them one by one. The choice of which predictor to add is based on a certain criterion, typically related to minimizing error.
  • Start with an empty model with no predictors.
  • Add predictors one at a time, selecting the one that results in the most significant improvement in model performance at each step.
  • Stop adding predictors when no significant improvement is observed or a predefined criterion is met.
By using this method, we efficiently explore various combinations of variables to identify the most useful predictors. This approach helps prevent overfitting and develops a simpler, more interpretable model.
Cp Criterion
The Cp Criterion is a statistical measure used to assess the fit of a regression model. It helps to determine whether a model has potential bias and how well it predicts new data.
The formula for the Cp Criterion is given by: \[ C_p = \frac{SSE}{MSE} + 2p - n \]where:
  • \(SSE\) is the error sum of squares for the model.
  • \(MSE\) is the mean square error from the full model.
  • \(p\) is the number of parameters, including the intercept.
  • \(n\) is the sample size.
The goal is to achieve a \(C_p\) value close to the number of parameters \(p\). If \(C_p > p\), it suggests that the model might contain some bias. Conversely, if \(C_p\approx p\), the model likely has minimal bias and is a good predictor.
Error Sum of Squares
Error Sum of Squares (SSE) measures the total deviation of observed values from the estimated values predicted by a model. It is a crucial concept in understanding how well a model fits the data.
SSE is calculated by:\[ SSE = \sum(y_i - \hat{y}_i)^2 \]where:
  • \(y_i\) is the observed value.
  • \(\hat{y}_i\) is the predicted value from the regression model.
SSE represents the unexplained variance by the model. A smaller SSE implies a better fit, as the predicted values are closer to the observed values. It is used in calculating important model metrics, such as the Cp Criterion, to determine the quality of the fit and possible improvements.
Model Bias
Model Bias refers to the error introduced by approximating a real-world problem, which may be complex, by a simpler model. It is one of the key factors in assessing model performance alongside variance.
Model bias can occur when the chosen model is too simple and cannot capture the underlying patterns in the data. This leads to systematic error.
Factors leading to model bias include:
  • Omitting important predictors, which leads to underfitting.
  • Choosing inappropriate models that do not match the data structure.
  • Incorrect assumptions about the relationship between predictors and response.
Minimizing bias is essential for creating a model that not only fits the given data well but also generalizes effectively to unseen data. Techniques such as adding more predictors, or using more complex models help in reducing bias and improving prediction accuracy.

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Most popular questions from this chapter

An article in Optical Engineering ["Operating Curve Extraction of a Corrclator's Filter" (2004, Vol. \(43,\) pp. \(2775-2779)]\) reported on use of an optical correlator to perform an experiment by varying brightness and contrast. The resulting modulation is characterized by the useful range of gray levels. The data are shown below: \(\begin{array}{llllrrrrr}\text { Brightncss (\%): } & 54 & 61 & 65 & 100 & 100 & 100 & 50 & 57 & 54 \\ \text { Contrast (\%): } & 56 & 80 & 70 & 50 & 65 & 80 & 25 & 35 & 26 \\ \text { Useful range (ng): } & 96 & 50 & 50 & 112 & 96 & 80 & 155 & 144 & 255\end{array}\) (a) lit a multiple linear regression model to these data. (b) Iistimate \(\sigma^{2}\) (c) Compute the standard errors of the regression coefficicnts. (d) Prediet the uscful range when brightness \(=80\) and contrast \(=75\)

A study was performed to investigate the shear strength of soil \((y)\) as it related to depth in feet \(\left(x_{1}\right)\) and \(\%\) moisture content \(\left(x_{2}\right)\). Ten observations were collected, and the following summary quantities obtained: \(n=10, \sum x_{i 1}=223,\) \(\sum x_{2}=553, \Sigma y_{i}=1,916, \sum x_{i 1}^{2}=5,200.9, \sum x_{R}^{2}=31,729\) \(\sum x_{i 1} x_{i 2}=12,352, \sum x_{i 1} y_{i}=43,550.8, \sum x_{i 2} y_{i}=104,736.8\) and \(\sum y_{i}^{2}=371,595.6\) (a) Set up the least squares normal equations for the model \(Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\epsilon\) (b) Estimate the parameters in the model in part (a). (c) What is the predicted strength when \(x_{1}=18\) feet and \(x_{2}=43 \% ?\)

An article entitled "A Method for Improving the Accuracy of Polynomial Regression Analysis" in the Journal of Quality Technology \((1971,\) pp. \(149-155)\) reported the fol lowing data on \(y=\) ultimate shear strength of a rubber compound (psi) and \(x=\) cure temperature \(\left({ }^{\circ} F\right) .\) \( \begin{array}{c|c|c|c|c|c} y & 770 & 800 & 840 & 810 \\ \hline x & 280 & 284 & 292 & 295 \\ y & 735 & 640 & 590 & 560 \\ \hline x & 298 & 305 & 308 & 315 \end{array} \) (a) Fit a second-order polynomial to these data. (b) Test for significance of regression using \(\alpha=0.05 .\) (c) Test the hypothesis that \(\beta_{11}=0\) using \(\alpha=0.05\). (d) Compute the residuals from part (a) and use them to evaluate model adequacy.

An article in the Journal of the American Ceramics Society (1992, Vol. 75, pp. \(112-116\) ) describes a process for immobilizing chemical or nuclear wastes in soil by dissolving the contaminated soil into a glass block. The authors mix \(\mathrm{CaO}\) and \(\mathrm{Na}_{2} \mathrm{O}\) with soil and model viscosity and electrical conductivity. The electrical conductivity model involves six regressors, and the sample consists of \(n=14\) observations. (a) For the six-regressor model, suppose that \(S S_{r}=0.50\) and \(R^{2}=0.94 .\) Find \(S S_{E}\) and \(S S_{R},\) and use this information to test for significance of regression with \(\alpha=0.05 .\) What are your conclusions? (b) Suppose that one of the original regressors is deleted from the model, resulting in \(R^{2}=0.92 .\) What can you conclude about the contribution of the variable that was removed? Answer this question by calculating an \(F\) -statistic. (c) Does deletion of the regressor variable in part (b) result in a smaller value of \(M S_{E}\) for the five-variable model, in comparison to the original six-variable model? Comment on the significance of your answer.

Consider the linear regression model $$ Y_{j}=\beta_{0}^{\prime}+\beta_{1}\left(x_{i 1}-\bar{x}_{1}\right)+\beta_{2}\left(x_{i 2}-\bar{x}_{2}\right)+\epsilon_{i} $$ where \(\bar{x}_{1}=\sum x_{i 1} / n\) and \(\bar{x}_{2}=\sum x_{i} / n .\) (a) Write out the least squares normal equations for this model. (b) Verify that the least squares estimate of the intercept in this model is \(\hat{\beta}_{0}^{\prime}=\sum y_{d} / n=\bar{y}\) (c) Suppose that we use \(y_{i}-\bar{y}\) as the response variable in the model above What effect will this have on the least squares estimate of the intercept?
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