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 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

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

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.

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.

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, \(\sigma = \sqrt{110.45} = 10.51\).

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.

Key concepts

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.