Question

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?

Short answer

Part a: \(b_1\) represents a decrease of 2.18 in fish intake for every increase in water temperature while keeping other factors constant, and \(b_4\) represents an increase of 2.32 in fish intake for every increment in speed, keeping other factors constant. Part b: Approximately 40% of the variation in fish intake can be explained by the variables in this regression model. Part c: \(\sigma\) can be estimated using the provided formula and given SSResid value. Part d: Adjusted \(R^{2}\) can be estimated using the given formula and then compared to \(R^{2}\). The detailed values for \(\sigma\) and Adjusted \(R^{2}\) can be calculated using a calculator or software.

Step-by-step solution

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

For every unit increase in water temperature (x₁), fish intake (y) decreases by 2.18, assuming all other variables are constant. Similarly, for every unit increase in speed (x₄), 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.

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.

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 \(\sigma\).

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 \(R^{2}\).