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When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods have been designed to screen out the fish. The article “Multiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes" (Journal of Applied Ecology [1983]: 33-42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake: \(\begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed }(\mathrm{knots}) \end{aligned}\) Part of the data given in the article were used to obtain the estimated regression equation $$ \hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4} $$ (based on \(n=26\) ). SSRegr \(=1486.9\) and SSResid = 2230.2 were also calculated. a. Interpret the values of \(b_{1}\) and \(b_{4}\) b. What proportion of observed variation in fish intake can be explained by the model relationship? c. Estimate the value of \(\sigma\). d. Calculate adjusted \(R^{2} .\) How does it compare to \(R^{2}\) itself?

Short Answer

Expert verified
a. For each increase of 1 °C in the water temperature, the fish intake decreases by 2.18 units and for each increase of 1 knot in speed, fish intake increases by 2.32 units. b. The coefficient of determination (\(R^{2}\)) is 0.33 which means that 33% of the observed variation in fish intake can be explained by the model. c. The estimated value of \(\sigma\) is 10.51. d. The adjusted \(R^{2}\) equals to 0.24 which is smaller than \(R^{2}=0.33\), as it takes into account the number of predictors in the model.

Step by step solution

01

Interpretation of \(b_{1}\) and \(b_{4}\)

For \(b_{1}\) and \(b_{4}\), which are the coefficients of the water temperature and speed variables in the regression equation respectively, the interpretation is similar. For each increase by one unit of the variable (one degree Celsius for \(x_{1}\) and one knot for \(x_{4}\)), the amount of fish intake changes by the value of that coefficient, all other things being equal. Thus, for each increase of 1 °C in the temperature, fish intake decreases by 2.18 units. For each increase of 1 knot in speed, fish intake increases by 2.32 units.
02

Calculation of Coefficient of Determination \(R^{2}\)

The coefficient of determination (\(R^{2}\)) is given by 1 - (SSResid/SSRegr). From the problem, we have SSRegr = 1486.9 and SSResid = 2230.2. Thus, \(R^{2}\) = 1 - (2230.2/1486.9) = 0.33.
03

Estimation of \(\sigma\)

The value of \(\sigma\) (sample standard deviation) can be estimated using the formula \(\sigma^2 = SSResid / (n - p - 1)\), where n is the sample size and p is the number of predictors. We have n = 26 and p = 4, so \(\sigma^2 = 2230.2 / (26 - 4 - 1) = 110.45\). Therefore, \(\sigma = \sqrt{110.45} = 10.51\).
04

Calculation of Adjusted \(R^{2}\)

The formula for adjusted \(R^{2}\) is given by \[R_{adj}^{2}= 1-(1-R^{2})\times \frac{n-1}{n-p}\] For this dataset we have \(n=26\), \(p=4\), and \(R^{2}=0.33\). Therefore, the adjusted \(R^{2}\) is 1 - (1 - 0.33) * ((26 - 1) / (26 - 4)) = 0.24. This value is lower than \(R^{2}\) itself, as it penalizes extra predictors added to the model that do not make a significant contribution.

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Key Concepts

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

Coefficient of Determination
The coefficient of determination, often referred to as \(R^2\), is a key metric in multiple regression analysis. It represents the proportion of the variance in the dependent variable that is predictable from the independent variables. In simpler terms, it quantifies how much of the observed data can be explained by the model.

The value of \(R^2\) ranges from 0 to 1. An \(R^2\) of 0 means that the model does not explain any of the variability in the response data around its mean, while an \(R^2\) of 1 means that the model explains all the variability.

In the context of the exercise, the \(R^2\) was calculated as 0.33. This means that 33% of the variation in fish intake can be explained by the predictors (water temperature, number of pumps running, sea state, and speed) included in the model. This leaves 67% of the variation unexplained by the model, indicating that there might be other factors influencing fish intake not captured by the current model.
Adjusted R-Squared
Adjusted \(R^2\) is a modified version of \(R^2\) that penalizes the addition of non-significant predictors to a regression model. While \(R^2\) might increase with the inclusion of more predictors, adjusted \(R^2\) takes into account the number of predictors relative to the number of data points and adjusts the \(R^2\) accordingly.

The formula for adjusted \(R^2\) is given by \[R_{adj}^{2}= 1-(1-R^{2})\times \frac{n-1}{n-p}\]where \(n\) is the sample size and \(p\) is the number of predictors.

For this exercise, the adjusted \(R^2\) was calculated to be 0.24. This is lower than the \(R^2\) value of 0.33, reflecting that the model may still have included predictors that are not improving the accuracy significantly. The adjusted \(R^2\) serves as a more accurate representation of a model's explanatory power when compared to \(R^2\) alone, especially as it discourages overfitting by accounting for the number of predictors used.
Standard Deviation Estimation
In regression analysis, the standard deviation \(\sigma\) is an estimation of the amount by which the observed values deviate from the predicted values in the model. It provides an understanding of the model's accuracy and precision. A smaller standard deviation indicates a better fit, as it means that the predicted values are closer to the actual data.

To estimate \(\sigma\), the formula used is \[\sigma = \sqrt{\frac{SS_{Resid}}{n - p - 1}}\]where \(SS_{Resid}\) is the sum of squares of the residuals, \(n\) is the number of observations, and \(p\) is the number of predictors.

In the exercise provided, with \(SS_{Resid} = 2230.2\), \(n = 26\), and \(p = 4\), the standard deviation is computed as \(\sigma = 10.51\). This gives a measure of how spread out the residuals are and helps assess the effectiveness of the regression model in making accurate predictions.
Linear Regression Coefficients
Linear regression coefficients are essential elements in determining the relationship between the independent variables and the dependent variable. Each coefficient represents the change in the dependent variable for one unit change in the independent variable, while all other variables in the model are held constant.

In the given regression equation:
\[\hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4}\]* \(b_1 = -2.18\): For every 1 degree Celsius increase in water temperature, fish intake decreases by 2.18 units, assuming all other factors remain constant.

* \(b_4 = 2.32\): For every 1 knot increase in speed, fish intake increases by 2.32 units, assuming all other factors remain constant.

These coefficients help in understanding the influence of each predictor variable on the response variable, thereby enabling predictions and insights about the relationship dynamics within the data.

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Most popular questions from this chapter

Consider the dependent variable \(y=\) fuel efficiency of a car (mpg). a. Suppose that you want to incorporate size class of car, with four categories (subcompact, compact, midsize, and large), into a regression model that also includes \(x_{1}=\) age of car and \(x_{2}=\) engine size. Define the necessary indicator variables, and write out the complete model equation. b. Suppose that you want to incorporate interaction between age and size class. What additional predictors would be needed to accomplish this?

A number of investigations have focused on the problem of assessing loads that can be manually handled in a safe manner. The article "Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors in the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: \(1033-1044\) ) proposed using a regression model to relate the dependent variable \(y=\) individual's rating of acceptable load \((\mathrm{kg})\) to \(k=3\) independent (predictor) variables: \(x_{1}=\) extent of left lateral bending \((\mathrm{cm})\) \(x_{2}=\) dynamic hand grip endurance (seconds) $$ x_{3}=\text { trunk extension ratio }(\mathrm{N} / \mathrm{kg}) $$ Suppose that the model equation is $$ \begin{array}{l} y=30+.90 x_{1}+.08 x_{2}-4.50 x_{3}+e \\ \text { and that } \sigma=5 \end{array} $$ a. What is the population regression function? b. What are the values of the population regression coefficients? c. Interpret the value of \(\beta_{1}\). d. Interpret the value of \(\beta_{3}\). e. What is the mean rating of acceptable load when extent of left lateral bending is \(25 \mathrm{~cm}\), dynamic hand grip endurance is 200 seconds, and trunk extension ratio is \(10 \mathrm{~N} / \mathrm{kg}\) ? f. If repeated observations on rating are made on different individuals, all of whom have the values of \(x_{1}\), \(x_{2},\) and \(x_{3}\) specified in Part (e), in the long run approximately what percentage of ratings will be between \(13.5 \mathrm{~kg}\) and \(33.5 \mathrm{~kg}\) ?

Suppose that the variables \(y, x_{1},\) and \(x_{2}\) are related by the regression model \(y=1.8+.1 x_{1}+.8 x_{2}+e\) a. Construct a graph (similar to that of Figure 14.5\()\) showing the relationship between mean \(y\) and \(x_{2}\) for fixed values \(10,20,\) and 30 of \(x_{1}\). b. Construct a graph depicting the relationship between mean \(y\) and \(x_{1}\) for fixed values \(50,55,\) and 60 of \(x_{2}\). c. What aspect of the graphs in Parts (a) and (b) can be attributed to the lack of an interaction between \(x_{1}\) and \(x_{2}\) ? d. Suppose the interaction term \(.03 x_{3}\) where \(x_{3}=x_{1} x_{2}\) is added to the regression model equation. Using this new model, construct the graphs described in Parts (a) and (b). How do they differ from those obtained in Parts (a) and (b)?

The article "The Influence of Temperature and Sunshine on the Alpha-Acid Contents of Hops" (Agricultural Meteorology [1974]: 375-382) used a multiple regression model to relate \(y=\) yield of hops to \(x_{1}=\) average temperature \(\left({ }^{\circ} \mathrm{C}\right)\) between date of coming into hop and date of picking and \(x_{2}=\) average percentage of sunshine during the same period. The model equation proposed is $$ y=415.11-6.60 x_{1}-4.50 x_{2}+e $$ a. Suppose that this equation does indeed describe the true relationship. What mean yield corresponds to an average temperature of 20 and an average sunshine percentage of \(40 ?\) b. What is the mean yield when the average temperature and average percentage of sunshine are 18.9 and 43, respectively? c. Interpret the values of the population regression coefficients.

The following statement appeared in the article “Dimensions of Adjustment Among College Women” (Journal of College Student Development [1998]: 364): Regression analyses indicated that academic adjustment and race made independent contributions to academic achievement, as measured by current GPA. Suppose \(\begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race }(\text { with white }=0, \text { other }=1) \end{aligned}\) What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

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