Question

Show that for the multiple linear regression model, the $F$-value from the analysis of variable table, and $R^2$ are related by the simple formula (9.88), $$ F=\frac{R^2}{1-R^2}\left(\frac{n-k-1}{k}\right) . $$ 

   Show that for the multiple linear regression model, the $F$-value from the analysis of variable table, and $R^2$ are related by the simple formula (9.88),
$$
F=\frac{R^2}{1-R^2}\left(\frac{n-k-1}{k}\right) .
$$
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Arnold O. Allen 2nd Edition
Chapter 9, Problem 16 ↓

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In multiple linear regression, we model the relationship between a dependent variable \( Y \) and \( k \) independent variables \( X_1, X_2, \ldots, X_k \). The model can be expressed as: \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \ldots + \beta_k X_k + \epsilon  Show more…

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Show that for the multiple linear regression model, the $F$-value from the analysis of variable table, and $R^2$ are related by the simple formula (9.88), $$ F=\frac{R^2}{1-R^2}\left(\frac{n-k-1}{k}\right) . $$
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Key Concepts

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Multiple Linear Regression
Multiple linear regression is a statistical method used to model the relationship between a single dependent variable and multiple independent variables. It estimates the effects of each predictor on the outcome by fitting a linear equation, and its analysis is central to understanding how various factors jointly influence the dependent variable.
Coefficient of Determination (R-squared)
The coefficient of determination, R-squared, is a measure that quantifies the proportion of the variance in the dependent variable that is explained by the independent variables in a regression model. It provides insight into the model’s explanatory power and the goodness-of-fit.
F-Statistic in Regression
The F-statistic in regression analysis is used to test the overall significance of the model. It compares the model that includes all the independent variables against a model with no predictors, effectively checking if any of the variables contribute meaningfully to the prediction of the outcome.
Analysis of Variance (ANOVA) in Regression
ANOVA decomposes the total variability of the dependent variable into components attributed to the regression model (explained variation) and the residual or error component (unexplained variation). This breakdown is crucial for determining the statistical significance of the overall regression model through the F-test.
Degrees of Freedom
Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of regression, they are used to adjust the sum of squares estimates for both the regression (model) and residuals, and they play a key role in calculating statistics like the F-value.
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Show that an equivalent way to perform the test for significance of regression in multiple linear regression is to base the test on $R^{2}$ as follows: To test $H_{0}: \beta_{1}=\beta_{2}=\cdots=\beta_{k}$ versus $H_{1}:$ at least one $\beta_{j} \neq 0,$ calculate \[ F_{0}=\frac{R^{2}(n-p)}{k\left(1-R^{2}\right)} \] and to reject $H_{0}$ if the computed value of $F_{0}$ exceeds $F_{\alpha, k, n-p},$ where $p=k+1$
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Show that, in a multiple linear regression model, to test the overall significance of the model, alternative formula of the test statistic is: F = (R^2 / k) / ((1 - R^2) / (n - k - 1))
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Let Y=Bo+ B1X1 +B2X2...+BnXn be a linear multiple regression with Y as dependent variable and X's as independent variable. If B's are the parameters. Show that Bcap=Y'Y (X'X)inverse.
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