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Chapter 6: Problem 9

Exercise 6.3 provides regression output for the full model (including all explanatory variables available in the data set) for predicting birth weight of babies. In this exercise we consider a forward-selection algorithm and add variables to the model one-at-a-time. The table below shows the p-value and adjusted \(R^{2}\) of each model where we include only the corresponding predictor. Based on this table, which variable should be added to the model first? $$ \begin{array}{lcccccc} \hline \text { variable } & \text { gestation } & \text { parity } & \text { age } & {\text { height }} & \text { weight } & \text { smoke } \\ \hline \text { p-value } & 2.2 \times 10^{-16} & 0.1052 & 0.2375 & 2.97 \times 10^{-12} & 8.2 \times 10^{-8} & 2.2 \times 10^{-16} \\ R_{a d j}^{2} & 0.1657 & 0.0013 & 0.0003 & 0.0386 & 0.0229 & 0.0569 \\ \hline \end{array} $$

Short Answer

Expert verified
Add 'gestation' to the model first based on its low p-value and highest adjusted \( R^2 \).

Step by step solution

01

Understand the Selection Criteria

In forward selection, we add the predictor variable to the model that has the most significant impact on improving the model's fit. We consider both p-value significance and adjusted \( R^2 \) for this purpose. Lower p-values indicate a stronger relationship with the response variable, and a higher adjusted \( R^2 \) suggests better model fit.
02

Analyze P-values for Significance

List the p-values for each variable: gestation (\(2.2 \times 10^{-16}\)), parity (0.1052), age (0.2375), height (\(2.97 \times 10^{-12}\)), weight (\(8.2 \times 10^{-8}\)), smoke (\(2.2 \times 10^{-16}\)). Note that lower p-values (<0.05) are considered statistically significant. Therefore, gestation, smoke, height, and weight are candidates based on p-value.
03

Compare Adjusted \(R^2\) Values

Evaluate the adjusted \( R^2 \) values for the significant variables identified in Step 2: gestation (0.1657), smoke (0.0569), height (0.0386), weight (0.0229). Higher values of adjusted \( R^2 \) indicate a better fit, so we use this to decide which of these variables contributes most to the model.
04

Select the Best Variable Based on Both Criteria

Gestation has both the lowest p-value and the highest adjusted \( R^2 \) among the significant predictors (0.1657). Hence, it should be added to the model first as it shows the greatest potential improvement in model fit.

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Key Concepts

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

Forward Selection
Forward selection is a method used in regression analysis to simplify and improve a model by adding predictor variables gradually. This approach starts with an empty model and includes the predictor that best enhances the model's performance. The selection process is typically guided by statistical measures like p-value and adjusted \( R^2 \).

Forward selection aims to identify which variables provide the most value in predicting the response variable. By focusing on one variable at a time, we can periodically assess the impact each predictor has on the model's accuracy.

The main benefit of forward selection is that it helps avoid overfitting. By gradually adding variables, only those that truly enhance the model's fit are included. This results in a simpler, more generalizable model that can perform better on new data.
P-Value
In statistics, the p-value is a measure that helps determine the significance of a predictor in a regression model. It represents the probability of observing the data given that the null hypothesis is true. The null hypothesis typically states that there is no relationship between the predictor and the response variable.

A lower p-value indicates a stronger evidence against the null hypothesis, implying that there is a meaningful relationship between the predictor and the response variable.

  • P-values less than 0.05 are generally considered statistically significant, suggesting a reliable association with the dependent variable.
  • In the context of forward selection, predictors with lower p-values are preferred as they suggest a more significant impact on improving the model's fit.
In the given exercise, predictors like 'gestation' and 'smoke' have very low p-values, indicating their strong potential to model the birth weight effectively.
Adjusted R-Squared
Adjusted \( R^2 \) is a modified version of the \( R^2 \) statistic that accounts for the number of predictors in the model relative to the number of data points. Unlike \( R^2 \), which can only increase as more variables are added, adjusted \( R^2 \) can decrease if the additional predictors do not improve the model enough to offset the complexity of having more predictors.

This makes adjusted \( R^2 \) a more reliable measure when comparing models with a differing number of predictors. A higher adjusted \( R^2 \) suggests a better balance between model fit and complexity. It's particularly useful in techniques like forward selection where choosing the most impactful predictors is crucial.

For example, in the given exercise, the variable 'gestation' not only has a low p-value but also the highest adjusted \( R^2 \), making it the best candidate to enhance the model's accuracy effectively.
Predictor Variables
Predictor variables, also known as independent variables, are the inputs in a regression model that are used to predict the response or dependent variable. In the context of regression analysis, choosing the right set of predictor variables is essential for building a robust and accurate model.

When employing methods like forward selection, analysts start with identifying potential predictor variables that could contribute to explaining the variability in the response variable.

Here are a few considerations regarding predictor variables:
  • Select predictors based on their ability to significantly influence the response variable, typically evaluated through p-values and adjusted \( R^2 \).
  • Avoid multicollinearity, where predictor variables are highly correlated with each other, as this can distort the model and confuse interpretations.
  • Consider the practical significance besides statistical significance to ensure the model remains meaningful in real-world applications.
In the exercise, the goal is to determine which predictor variables best project birth weight, aiding in better understanding and predictions.

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Most popular questions from this chapter

Exercise 6.14 introduced us to O-rings that were identified as a plausible explanation for the breakup of the Challenger space shuttle 73 seconds into takeoff in 1986. The investigation found that the ambient temperature at the time of the shuttle launch was closely related to the damage of O-rings, which are a critical component of the shuttle. See this earlier exercise if you would like to browse the original data. (a) The data provided in the previous exercise are shown in the plot. The logistic model fit to these data may be written as $$ \log \left(\frac{\hat{p}}{1-\hat{p}}\right)=11.6630-0.2162 \times \text { Temperature } $$ where \(\hat{p}\) is the model-estimated probability that an O-ring will become damaged. Use the model to calculate the probability that an O-ring will become damaged at each of the following ambient temperatures: \(51,53,\) and 55 degrees Fahrenheit. The model-estimated probabilities for several additional ambient temperatures are provided below, where subscripts indicate the temperature: $$ \begin{array}{llll} \hat{p}_{57}=0.341 & \hat{p}_{59}=0.251 & \hat{p}_{61}=0.179 & \hat{p}_{63}=0.124 \\ \hat{p}_{65}=0.084 & \hat{p}_{67}=0.056 & \hat{p}_{69}=0.037 & \hat{p}_{71}=0.024 \end{array} $$ (b) Add the model-estimated probabilities from part (a) on the plot, then connect these dots using a smooth curve to represent the model-estimated probabilities. (c) Describe any concerns you may have regarding applying logistic regression in this application, and note any assumptions that are required to accept the model's validity.

The Child Health and Development Studies investigate a range of topics. One study considered all pregnancies between 1960 and 1967 among women in the Kaiser Foundation Health Plan in the San Francisco East Bay area. Here, we study the relationship between smoking and weight of the baby. The variable smoke is coded 1 if the mother is a smoker, and 0 if not. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, based on the smoking status of the mother. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 123.05 & 0.65 & 189.60 & 0.0000 \\ \text { smoke } & -8.94 & 1.03 & -8.65 & 0.0000 \\ \hline \end{array} $$ The variability within the smokers and non-smokers are about equal and the distributions are symmetric. With these conditions satisfied, it is reasonable to apply the model. (Note that we don't need to check linearity since the predictor has only two levels.) (a) Write the equation of the regression line. (b) Interpret the slope in this context, and calculate the predicted birth weight of babies born to smoker and non-smoker mothers. (c) Is there a statistically significant relationship between the average birth weight and smoking?

Considers a model that predicts a newborn's weight using several predictors. Use the regression table below, which summarizes the model, to answer the following questions. If necessary, refer back to Exercise 6.3 for a reminder about the meaning of each variable. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.41 & 14.35 & -5.60 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.26 & 0.0000 \\ \text { parity } & -3.33 & 1.13 & -2.95 & 0.0033 \\ \text { age } & -0.01 & 0.09 & -0.10 & 0.9170 \\ \text { height } & 1.15 & 0.21 & 5.63 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 1.99 & 0.0471 \\ \text { smoke } & -8.40 & 0.95 & -8.81 & 0.0000 \\ \hline \end{array} $$ (a) Determine which variables, if any, do not have a significant linear relationship with the outcome and should be candidates for removal from the model. If there is more than one such variable, indicate which one should be removed first. (b) The summary table below shows the results of the model with the age variable removed. Determine if any other variable(s) should be removed from the model. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.64 & 14.04 & -5.74 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.28 & 0.0000 \\ \text { parity } & -3.29 & 1.06 & -3.10 & 0.0020 \\ \text { height } & 1.15 & 0.20 & 5.64 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 2.00 & 0.0459 \\ \text { smoke } & -8.38 & 0.95 & -8.82 & 0.0000 \\ \hline \end{array} $$

We considered the variables smoke and parity, one at a time, in modeling birth weights of babies in Exercises 6.1 and \(6.2 .\) A more realistic approach to modeling infant weights is to consider all possibly related variables at once. Other variables of interest include length of pregnancy in days (gestation), mother's age in years (age), mother's height in inches (height), and mother's pregnancy weight in pounds (weight). Below are three observations from this data set. $$ \begin{array}{rccccccc} \hline & \text { bwt } & \text { gestation } & \text { parity } & \text { age } & \text { height } & \text { weight } & \text { smoke } \\ \hline 1 & 120 & 284 & 0 & 27 & 62 & 100 & 0 \\ 2 & 113 & 282 & 0 & 33 & 64 & 135 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1236 & 117 & 297 & 0 & 38 & 65 & 129 & 0 \\ \hline \end{array} $$ The summary table below shows the results of a regression model for predicting the average birth weight of babies based on all of the variables included in the data set. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.41 & 14.35 & -5.60 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.26 & 0.0000 \\ \text { parity } & -3.33 & 1.13 & -2.95 & 0.0033 \\ \text { age } & -0.01 & 0.09 & -0.10 & 0.9170 \\ \text { height } & 1.15 & 0.21 & 5.63 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 1.99 & 0.0471 \\ \text { smoke } & -8.40 & 0.95 & -8.81 & 0.0000 \\ \hline \end{array} $$ (a) Write the equation of the regression line that includes all of the variables. (b) Interpret the slopes of gestation and age in this context. (c) The coefficient for parity is different than in the linear model shown in Exercise 6.2 . Why might there be a difference? (d) Calculate the residual for the first observation in the data set. (e) The variance of the residuals is \(249.28,\) and the variance of the birth weights of all babies in the data set is 332.57. Calculate the \(R^{2}\) and the adjusted \(R^{2}\). Note that there are 1,236 observations in the data set.

Considers a model that predicts the number of days absent using three predictors: ethnic background (eth), gender (sex), and learner status (lrn). Use the regression table below to answer the following questions. If necessary, refer back to Exercise 6.4 for additional details about each variable. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \mathrm{t} \text { value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 18.93 & 2.57 & 7.37 & 0.0000 \\ \text { eth } & -9.11 & 2.60 & -3.51 & 0.0000 \\ \text { sex } & 3.10 & 2.64 & 1.18 & 0.2411 \\ \text { lrn } & 2.15 & 2.65 & 0.81 & 0.4177 \\ \hline \end{array} $$ (a) Determine which variables, if any, do not have a significant linear relationship with the outcome and should be candidates for removal from the model. If there is more than one such variable, indicate which one should be removed first. (b) The summary table below shows the results of the regression we refit after removing learner status from the model. Determine if any other variable(s) should be removed from the model. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 19.98 & 2.22 & 9.01 & 0.0000 \\ \text { eth } & -9.06 & 2.60 & -3.49 & 0.0006 \\ \text { sex } & 2.78 & 2.60 & 1.07 & 0.2878 \\ \hline \end{array} $$
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