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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 \(\mathrm{Re}-\) gression 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: $$ \begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed }(\text { knots }) \end{aligned} $$ 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?

Step by step solution

01

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

For every unit increase in water temperature (x鈧�), fish intake (y) decreases by 2.18, assuming all other variables are constant. Similarly, for every unit increase in speed (x鈧�), fish intake (y) increases by 2.32, assuming all other variables are constant. This tells us about the effects of water temperature and speed on fish intake at the power plant.

02

Calculation of \(R^{2}\) and Interpretation

\(R^{2}\), the coefficient of determination, is calculated from the formula \[R^{2} = \frac{SSRegr}{SSTotal}= \frac{SSRegr}{SSRegr + SSResid} \] So, \(R^{2}\) = 1486.9 / (1486.9 + 2230.2) = 0.3998, or approximately 0.4. This means 40% of the variation in fish intake can be explained by this model.

03

Estimation of \(\sigma\)

The value of \(\sigma\) (standard deviation of the residuals) can be estimated by the formula \[ \sigma = \sqrt{\frac{SSResid}{n-p-1}}\] where, n is the sample size and p is the number of predictors. Thus, \(\sigma\) = \(\sqrt{\frac{2230.2}{26 - 4 - 1}}\). Place these values into a calculator to get an estimation of \(\sigma\).

04

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

The adjustment in the \(R^{2}\) value takes the number of predictors into account, which is better for comparison purposes if models have different numbers of predictors. It is calculated as follows: \[Adjusted \ R^{2} = 1 - (1 - R^{2})\frac{n-1}{n-p-1}\] where n is the number of observations and p is the number of predictors. Inserting the known values, the calculation of adjusted \(R^{2}\) can be made, and then it can be compared with \(R^{2}\).

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Key Concepts

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

Regression Coefficients

In a multiple regression analysis, regression coefficients are pivotal in understanding how each predictor variable impacts the dependent variable. These coefficients indicate the expected change in the dependent variable for a one-unit change in the predictor, holding all other predictors constant.

For the given regression equation \[ \hat{y} = 92 - 2.18 x_{1} - 19.20 x_{2} - 9.38 x_{3} + 2.32 x_{4} \]let's interpret the coefficients:

• **Water Temperature (\(x_{1}\)**): The coefficient -2.18 suggests that as the water temperature increases by 1 degree Celsius, the fish intake decreases by 2.18 fish, assuming all other factors remain unchanged.
• **Speed (\(x_{4}\)): The coefficient 2.32 shows that an increase in speed by 1 knot results in an increase of 2.32 fish in the intake, with other conditions held fixed.

These coefficients help predict how each factor influences the fish intake, aiding in strategic planning and operational decisions at the power station.

Coefficient of Determination

The Coefficient of Determination, denoted as \(R^2\), is a measure that provides insights into how well the regression model explains the variability of the dependent variable. In simpler terms, it tells us the proportion of variation in the outcome that is predictable from the predictors.

The formula used for calculating \(R^2\) is:\[R^{2} = \frac{SSRegr}{SSTotal} = \frac{SSRegr}{SSRegr + SSResid} \]For the given dataset, \(R^{2}\) = 0.4, indicating that 40% of the variance in fish intake can be attributed to the model.

This value suggests a moderate fit, meaning that while some variation is captured, there is still a significant portion (60%) that is unexplained by the model. This insight can guide further refinement of the model by potentially introducing new variables or insights.

Standard Deviation of Residuals

The Standard Deviation of Residuals, often represented as \(\sigma\), is a measure of how much the observed values deviate from the values predicted by the regression model. In essence, it informs us about the typical size of the prediction errors.

Calculating \(\sigma\) involves the formula:\[ \sigma = \sqrt{\frac{SSResid}{n-p-1}} \]
where \(n\) equals the sample size, and \(p\) is the number of predictors in the model. In this example, substituting the provided values gives:\[ \sigma = \sqrt{\frac{2230.2}{26 - 4 - 1}} \]

Calculating this yields an estimate for \(\sigma\), which provides a useful metric of the regression model's precision. A smaller \(\sigma\) suggests that the model predictions are close to the actual data, while a larger \(\sigma\) would mean more significant prediction errors.

Adjusted R-squared

The Adjusted \(R^2\) is an improved version of the standard \(R^2\), which takes into account the number of predictors in the model. Adjusted \(R^2\) is particularly useful when comparing models with a different number of predictors, as it provides a more accurate assessment by penalizing the inclusion of unnecessary predictors.

The formula for Adjusted \(R^2\) is: \[Adjusted \ R^{2} = 1 - (1 - R^{2})\frac{n-1}{n-p-1}\]Using this formula and the given values, let's substitute and solve:\[Adjusted \ R^{2} = 1 - (1 - 0.4)\frac{26-1}{26-4-1}\]

The Adjusted \(R^2\) will always be less than or equal to \(R^2\). This example allows us to see how much the predictive power of the model decreases when accounting for the number of predictors used. It becomes a valuable tool in minimizing overfitting and ensuring model simplicity.