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Questions and Answers
Adjusted R Square (Ra2) is computed by the following ______: Ra2 = 1 - (SSE/SST) * (n-1)/(n-k-1).
Adjusted R Square (Ra2) is computed by the following ______: Ra2 = 1 - (SSE/SST) * (n-1)/(n-k-1).
formula
The coefficient of determination (R2) is the proportion of the total variation of ______ that is explained by the relationship between ______ and independent variables X’s.
The coefficient of determination (R2) is the proportion of the total variation of ______ that is explained by the relationship between ______ and independent variables X’s.
Y
The additional ______ may not contribute significantly to the explanation of the dependent variable y, but they do increase R2.
The additional ______ may not contribute significantly to the explanation of the dependent variable y, but they do increase R2.
independent variables
Indicator variables are assigned the values ______.
Indicator variables are assigned the values ______.
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Multiple regression model is used to estimate the ______ for employees.
Multiple regression model is used to estimate the ______ for employees.
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The general linear regression model in matrix terms is Yn1 = Xn(k+1) (k+1)1 + n1 or simply Y = X + where is a vector of independent ______ variables.
The general linear regression model in matrix terms is Yn1 = Xn(k+1) (k+1)1 + n1 or simply Y = X + where is a vector of independent ______ variables.
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The least squares normal equations X’Xb = X’Y where b = (b0, b1, b2 , … , bk)’. The estimated regression coefficients LSE and MLE: b = (X’X)-1 X’Y are minimum variance unbiased, consistent, and ______.
The least squares normal equations X’Xb = X’Y where b = (b0, b1, b2 , … , bk)’. The estimated regression coefficients LSE and MLE: b = (X’X)-1 X’Y are minimum variance unbiased, consistent, and ______.
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The variance covariance matrix Var(b) = 2 (X’X)-1 The estimated variance-covariance matrix of b is s2(b) = MSE (X’X)-1 = s2(X’X)-1. Inferences bi is normally distributed random variable for the normal model. The (1 - ) 100% Confidence interval for i bi - t/2 s(bi) < i < b0 + t/2 s(bi) where t/2 is a value of the ______ - distribution with df = (n -k-1).
The variance covariance matrix Var(b) = 2 (X’X)-1 The estimated variance-covariance matrix of b is s2(b) = MSE (X’X)-1 = s2(X’X)-1. Inferences bi is normally distributed random variable for the normal model. The (1 - ) 100% Confidence interval for i bi - t/2 s(bi) < i < b0 + t/2 s(bi) where t/2 is a value of the ______ - distribution with df = (n -k-1).
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Example 2 (cont) (a) The model for executive salary is Y = 0 + 1 X + The mean salary is E(Y) = 0+1X (b) The advantage of using a 0-1 coding scheme is that the coefficients are easily interpreted. if X = 1 (male) M =E(Y) = 0+ 1(1) = 0+1 F = E(Y) = 0 + 1(0) = 0 if X = 0 (female), then 1 = M - F. That is, 0 represents the mean salary for ______, and 1 represents the difference between the mean salary for males and the mean salary for ______. Therefore, when a 0-1 coding convention is used, 0 will always represent the mean response associated with the level of the qualitative variable assigned the value 0 (called the base level), and 1 will always represent the difference between the mean response for the level assigned the value 1 and the mean for the base level.
Example 2 (cont) (a) The model for executive salary is Y = 0 + 1 X + The mean salary is E(Y) = 0+1X (b) The advantage of using a 0-1 coding scheme is that the coefficients are easily interpreted. if X = 1 (male) M =E(Y) = 0+ 1(1) = 0+1 F = E(Y) = 0 + 1(0) = 0 if X = 0 (female), then 1 = M - F. That is, 0 represents the mean salary for ______, and 1 represents the difference between the mean salary for males and the mean salary for ______. Therefore, when a 0-1 coding convention is used, 0 will always represent the mean response associated with the level of the qualitative variable assigned the value 0 (called the base level), and 1 will always represent the difference between the mean response for the level assigned the value 1 and the mean for the base level.
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Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the ______ model (or full model) and the simpler of the two is called the ______ model.
Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the ______ model (or full model) and the simpler of the two is called the ______ model.
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Multiple linear regression model is used to estimate the ______ for employees.
Multiple linear regression model is used to estimate the ______ for employees.
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In the general linear regression model, the parameters (partial coefficients) are denoted as ______.
In the general linear regression model, the parameters (partial coefficients) are denoted as ______.
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The first-order model in general linear regression does not have any ______ effects between the predictor variables.
The first-order model in general linear regression does not have any ______ effects between the predictor variables.
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Extra sums of squares measure the marginal reduction in the error sum of squares when one or several predictor variables are added to the regression model, given that other predictor variables are already in the model. Equivalently, one can view an extra sum of squares as measuring the marginal increase in the regression sum of squares when one or several predictor variables are added to the regression model. The reason for the equivalence of the marginal reduction in the error sum of squares and the marginal increase in the regression sum of squares is $SST = ______ + SSE$. That is, $SST$ does not depend on the regression model fitted, any reduction in $SSE$ implies an identical increase in $______$.
Extra sums of squares measure the marginal reduction in the error sum of squares when one or several predictor variables are added to the regression model, given that other predictor variables are already in the model. Equivalently, one can view an extra sum of squares as measuring the marginal increase in the regression sum of squares when one or several predictor variables are added to the regression model. The reason for the equivalence of the marginal reduction in the error sum of squares and the marginal increase in the regression sum of squares is $SST = ______ + SSE$. That is, $SST$ does not depend on the regression model fitted, any reduction in $SSE$ implies an identical increase in $______$.
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Coefficients of partial determination measure the proportionate reduction in the variation of $Y$ achieved by the introduction of the entire set of $X$ variables considered in the model. A coefficient of partial determination, in contrast, measures the marginal contribution of one $X$ variable when all others are already included in the model.
Coefficients of partial determination measure the proportionate reduction in the variation of $Y$ achieved by the introduction of the entire set of $X$ variables considered in the model. A coefficient of partial determination, in contrast, measures the marginal contribution of one $X$ variable when all others are already included in the model.
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SSE($X_1$) measures the variation in $Y$ when $X_1$ is included in the model. SSE($X_1, X_2$) measures the variation in $Y$ when both $X_1$ and $X_2$ are included in the model. Hence, the relative marginal reduction in the variation in $Y$ associated with $X_2$ when $X_1$ is already in the model is $______ = rac{SSR(X_2|X_1)}{SSE(X_1)} = rac{SSE(X_1)-SSE(X_1, X_2)}{SSE(X_1)}$. The above is the coefficient of partial determination between $Y$ and $X_2$, given that $X_1$ is in the model.
SSE($X_1$) measures the variation in $Y$ when $X_1$ is included in the model. SSE($X_1, X_2$) measures the variation in $Y$ when both $X_1$ and $X_2$ are included in the model. Hence, the relative marginal reduction in the variation in $Y$ associated with $X_2$ when $X_1$ is already in the model is $______ = rac{SSR(X_2|X_1)}{SSE(X_1)} = rac{SSE(X_1)-SSE(X_1, X_2)}{SSE(X_1)}$. The above is the coefficient of partial determination between $Y$ and $X_2$, given that $X_1$ is in the model.
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When the independent variables are highly correlated, we say that ________ exists.
When the independent variables are highly correlated, we say that ________ exists.
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Highly correlated independent variables produce greater sampling ______ for the LS coefficients.
Highly correlated independent variables produce greater sampling ______ for the LS coefficients.
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The mean ______ values = (3.75 + 3.75)/2 = 3.75 is considerably larger than 1
The mean ______ values = (3.75 + 3.75)/2 = 3.75 is considerably larger than 1
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One of the commonly used simple methods to solve the ______ is to drop one or more of the highly correlated independent variables from the multiple regression model.
One of the commonly used simple methods to solve the ______ is to drop one or more of the highly correlated independent variables from the multiple regression model.
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Interaction Model with Two Independent Variables ______ where (\beta_1 + \beta_3 X2) represents the change in E(Y) for every 1-unit increase in X1, holding X2 fixed. (\beta_2 + \beta_3 X1) represents the change in E(Y) for every 1-unit increase in X2, holding X1 fixed. \beta_0 is the intercept of the model, the value of E(Y) when X1=X2 =0 The cross-product term, \beta_3 X1X2 , is called an interaction term.
Interaction Model with Two Independent Variables ______ where (\beta_1 + \beta_3 X2) represents the change in E(Y) for every 1-unit increase in X1, holding X2 fixed. (\beta_2 + \beta_3 X1) represents the change in E(Y) for every 1-unit increase in X2, holding X1 fixed. \beta_0 is the intercept of the model, the value of E(Y) when X1=X2 =0 The cross-product term, \beta_3 X1X2 , is called an interaction term.
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Is there evidence that X1 and X2 interact. Test at \alpha = 0.05. Solution H0: \beta_3 = 0 vs Ha: \beta_3 \neq 0 The p-value = 0.169 is greater than \alpha = 0.05, H0 is not rejected. There is ______ evidence to indicate X1 (experience) and X2 (performance) interact at the 5% level.
Is there evidence that X1 and X2 interact. Test at \alpha = 0.05. Solution H0: \beta_3 = 0 vs Ha: \beta_3 \neq 0 The p-value = 0.169 is greater than \alpha = 0.05, H0 is not rejected. There is ______ evidence to indicate X1 (experience) and X2 (performance) interact at the 5% level.
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Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the complete model (or full model) and the simpler of the two is called the reduced model.
Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the complete model (or full model) and the simpler of the two is called the reduced model.
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Partial F-Test for Comparing Nested Models Y= \beta_0 + \beta_1 X1 + \beta_2 X2 +… + \beta_g Xg + \epsilon -- Reduced model Y = \beta_0 + \beta_1 X1 + \beta_2 X2 + … + \beta_g Xg + \beta_g+1 Xg+1 + … + \beta_k Xk + \epsilon H0: \beta_g+1 = \beta_g+2 =. = \beta_k = 0 The test statistic is vs -- Complete model Ha: H0 is not true SSER = Sum of squared errors for the reduced model. SSEC = Sum of squared errors for the complete model. MSEC= Mean square error for the complete model. k - g = Number of \beta parameters tested (in H0). k = Number of independent variables in the complete model. For the partial F-test, df1 = (k-g) and df2 = (n-k-1).
Partial F-Test for Comparing Nested Models Y= \beta_0 + \beta_1 X1 + \beta_2 X2 +… + \beta_g Xg + \epsilon -- Reduced model Y = \beta_0 + \beta_1 X1 + \beta_2 X2 + … + \beta_g Xg + \beta_g+1 Xg+1 + … + \beta_k Xk + \epsilon H0: \beta_g+1 = \beta_g+2 =. = \beta_k = 0 The test statistic is vs -- Complete model Ha: H0 is not true SSER = Sum of squared errors for the reduced model. SSEC = Sum of squared errors for the complete model. MSEC= Mean square error for the complete model. k - g = Number of \beta parameters tested (in H0). k = Number of independent variables in the complete model. For the partial F-test, df1 = (k-g) and df2 = (n-k-1).
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