91影视

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

The coefficients in a multiple regression analysis represent the change in the dependent variable (in this case, 'fish intake') resulting from a one-unit change in the respective independent variable, assuming all other variables are held constant. Therefore, \(b_{1}\) is -2.18 which means for every one degree increase in water temperature, we would expect the average number of fish intake to decrease by 2.18, assuming all other variables are held constant. Similarly, \(b_{4}\) is 2.32. So, for every one knot increase in speed, we would expect the average number of fish intake to increase by 2.32, assuming all other variables are held constant

02

Calculation of R虏

The coefficient of determination, R虏, which represents the proportion of the variance in the dependent variable that is predictable from the independent variables, is given by 1 - (SSResid / SSTotal). Using the provided sums of squares for the regression (SSRegr = 1486.9) and residual (SSResid = 2230.2) gives R虏 = 1 - SSResid / (SSResid+SSRegr) = 1 - 2230.2 / (2230.2 + 1486.9) = 0.40 (rounded to 2 decimal places). So, the observed model explains about 40% of the variation in fish intake.

03

Estimation of \(\sigma\)

The standard deviation of the residuals, \(\sigma\), can be estimated as \(\sqrt{SSResid/(n-k-1)}\), where \(n\) is the number of observations and \(k\) is the number of predictors. Substituting the provided values gives: \(\sigma = \sqrt{2230.2/(26-4-1)} = \sqrt{111.51} = 10.55\) (rounded to 2 decimal places).

04

Calculation of Adjusted R虏

The adjusted R虏 is calculated as 1 - [(1 - R虏)*(n - 1)/(n - k - 1)]. This is a modified version of R虏 that adjusts for the number of predictors in the model. Substituting the given values gives: Adjusted R虏 = 1 - [(1 - 0.40)*((26-1)/(26 - 4 - 1))] = 0.37 (rounded to 2 decimal places). The adjusted R虏 is slightly lower than the R虏 itself, which is often the case since R虏 tends to slightly overestimate the model's performance.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coefficient of Determination (R虏)

The coefficient of determination, commonly denoted as R虏, is a crucial statistic in multiple regression analysis that measures the proportion of variance in the dependent variable which can be explained by the independent variables in the model. It takes on values between 0 and 1, where a higher R虏 indicates a better fit of the model to the data, meaning that the model explains a larger portion of the variability in the response variable.

In the context of our example with the power station and fish intake, the calculated R虏 value is 0.40. This means that 40% of the variation in the fish intake can be accounted for by the predictors in our model: water temperature, number of pumps running, sea state, and speed. While this value offers some insight, it's worth mentioning that it does not necessarily imply causation and does not indicate which predictors are significant.

Standard Deviation of Residuals (蟽)

The standard deviation of the residuals, or sigma (蟽), represents the average distance the observed values fall from the regression line; in other words, it measures the typical error in predicting the dependent variable from the independent variables. To calculate 蟽, we take the square root of the sum of squared residuals divided by the degrees of freedom, which is the number of observations minus the number of predictors minus one.

In our example, with 蟽 calculated to be approximately 10.55, it tells us that, on average, the actual fish intake numbers deviate from the predicted values obtained through the regression model by about 10.55 fish. Understanding 蟽 helps in assessing the accuracy of the predictions 鈥� the smaller the 蟽, the closer the observed data points fall to the regression line, indicating more reliable predictions.

Adjusted R虏

While R虏 is insightful, it has a limitation: it can increase just by adding more predictors to the model, regardless of whether they are truly significant. This is where adjusted R虏 comes into play as it incorporates the number of predictors and the sample size to 'penalize' for the addition of irrelevant predictors.

The formula for adjusted R虏 takes into account the number of predictors (k) and the total sample size (n). In this fish intake study, the adjusted R虏 of 0.37 is a modified and more balanced measure compared to the R虏 of 0.40. The slight decrease from R虏 to adjusted R虏 indicates the inclusion of all four predictors might not be as beneficial as the R虏 suggested. Students should always consider reporting adjusted R虏, as it gives a more accurate picture of the model's explanatory power, especially in multiple regression analyses with many predictors.