Exam 13: Nonlinear and Multiple Regression

A multiple regression model with four independent variables to study accuracy in reading liquid crystal displays was used. The variables were y = error percentage for subjects reading a four-digit liquid crystal display $x _ { 1 }$ = level of backlight (ranging from 0 to 122 $\mathrm { cd } / \mathrm { m } ^ { 2 }$ ) $x _ { 2 }$ = character subtense (ranging from $.025 ^ { \circ } \text { to } 1.34 ^ { \circ }$ ) $x _ { 3 }$ = viewing angle (ranging from $0 ^ { \circ } \text { to } 60 ^ { \circ }$ ) $x _ { 4 }$ =level of ambient light (ranging from 20 to 1500 lux) The model fit to data was $Y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \beta _ { 4 } x _ { 4 } + \varepsilon$ The resulting estimated coefficient were $\hat { \beta } _ { 0 } = 1.52 , \hat { \beta } _ { 1 } = .02 , \hat { \beta } _ { 2 } = - 1.40 , \hat { \beta } _ { 3 } = .02 \text {, and } \hat { \beta } _ { 4 } = .0006 \text {. }$ a. Calculate an estimate of expected error percentage when $x _ { 1 } = 10 , x _ { 2 } = .5 , x _ { 3 } = 50 \text {, and } x _ { 4 } = 100$ b. Estimate the mean error percentage associated with a backlight level of 20, character subtense of .5, viewing angle of 10, and ambient light level of 30. c. What is the estimated expected change in error percentage when the level of ambient light is increased by 1 unit while all other variables are fixed at the values given in part (a)? Answer for a 100-unit increase in ambient light level. d. Explain why the answers in part ( c ) do not depend on the fixed values of $x _ { 1 } , x _ { 2 } \text { and } x _ { 3 }$ Under what conditions would there be such a dependence? e. The estimated model was based on n=30 observations, with SST=39.2 and SSE=20.0. Calculate and interpret the coefficient of multiple determination, and then carry out the model utility test using $\alpha = .05$

The transformation __________ of the dependent variable y and the transformation __________ of the independent variable x are used to linearize the power function $y = \alpha x ^ { β }$

With $\hat { y } _ { i } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { i } + \hat { \beta } _ { 2 } x _ { i } ^ { 2 } + \cdots \cdots + \hat { \beta } _ { k } x _ { i } ^ { k }$ , the sum of squared residuals (error sum of squares) is $\operatorname { SSE } = \sum \left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ . Hence the mean square error is MSE =__________/___________.

Consider the following data on mass rate of burning x and flame length y: x 1.7 2.2 2.3 2.6 2.7 3.0 3.2 x 1.3 1.8 1.6 2.0 2.1 2.2 3.0 x 3.3 4.1 4.3 4.6 5.7 6.1 x 2.6 4.1 3.7 5.0 5.8 5.3 a. Estimate the parameters of a power function model. b. Assume that the power function is an appropriate model, test $H _ { 0 } : \beta = 4 / 3 \text { vs. } H _ { \pm } : \beta < 4 / 3 ,$ using a level .05 test. c. Test the null hypothesis that states that the median flame length when burning rate is 5.0 is twice the median flame length when burning rate is 2.5 against the alternative that this is not the case.

If we let $\operatorname { SST } = \sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , and $\operatorname { SSE } = \sum \left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ , then SSE/SST is the proportion of the total variation in the observed $y _ { i}$ 's that is ___________by the polynomial model.

A multiple regression model has

A dichotomous variable, one with just two possible categories, can be incorporated into a regression model via a ___________ or __________ variable x whose possible values 0 and 1 indicate which category is relevant for any particular observations.

Many statisticians recommend __________ for an assessment of model validity and usefulness. These include plotting the residuals $e _ {i }$ or standardized residuals $e _ { i } ^ { * }$ on the vertical axis versus the independent variable $x _ { i }$ or fitted values $\hat { y } _ {i }$ on the horizontal axis.

Which of the following statements are not true?

Which of the following statements are not true?

Which of the following statements are not true?

Which of the following statements are true?

If a data set on at least five predictors is available, regressions involving all possible subsets of the predictors involve at least __________different models

Which of the following statements are not true?

If we let $\operatorname { SST } = \sum \left( y _ {i } - \bar { y } \right) ^ { 2 }$ , and $\operatorname { SSE } = \sum \left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ , then 1-SSE/SST is the proportion of the total variation in the observed $y _ { i}$ 's that is __________ by the polynomial model. It is called the ____________ ,and is denoted by R .

In logistic regression it can be shown that $\frac { p ( x ) } { 1 - p ( x ) } = e ^ { \beta _ { 0 } + \beta _ { 1 } x }$ . The expression on the left-hand side of this equality is well known as the ___________.

Which of the following statements are not true?

A first-order no-interaction model has the form $\hat { Y } = 5 + 3 x _ { 1 } + 2 x _ { 2 }$ . As $x _ { 1 }$ increases by 1-unit, while holding $x _ { 2 }$ fixed, then y will be expected to

Which of the following statements are true?

Which of the following statements are not true?