Exam 13: Nonlinear and Multiple Regression

A study reported data on y-tensile strength (MPa), $x _ { 1 }$ = slab thickness (cm), $x _ { 2 }$ = load (kg), $x _ { 3 }$ = age at loading (days), and $x _ { 4 }$ = time under test (days) resulting from stress tests of n=9 reinforced concrete slabs. The results of applying the BE elimination method of variable selection are summarized in the accompanying tabular format. Explain what occurred at each step of the procedure.

The principle__________selects $\hat { \beta } _ { 0 }$ and $\hat { \beta } _ { 1 }$ to minimize $\sum \left| y _ { i } - \left( b _ { 0 } + b _ { 1 } x _ {i } \right) \right|$ .

The additive exponential and power models, $Y = α e ^ { \beta _ { x } } + \varepsilon$ and $Y = \alpha x ^ { \beta } + \varepsilon$ are ___________ linear.

An investigation of the influence of sodium benzoate concentration on the critical minimum pH necessary for the inhibition of Fe yielded the accompanying data, which suggests that expected critical minimum pH is linearly related to the natural logarithm of concentrate: Concentration .01 .025 .1 .95 5.1 5.5 6.1 7.3 a. What is the implied probabilistic model, and what are the estimates of the model parameters? b. What critical minimum pH would you predict for a concentration of 1.0? Obtain a 95% PI for critical minimum pH when concentration is 1.0.

Quite frequently, residual plots as well as other plots of the data will suggest some difficulties or abnormality in the data. Which of the following statements are not considered difficulties?

The function $p ( x ) = e ^ { β _ { 0 } + \beta _ { 1 } x } / \left[ 1 + e ^ {β _ { 0 } + \beta _ { 1 } x } \right]$ has been found quite useful in many applications. This function is well known as the ___________function.

The transformation __________ is used to linearize the reciprocal function $y = \alpha + \beta \cdot \frac { 1 } { x }$

Multiple regression analysis involves building models for relating dependent variable y to __________or more independent variables.

For the case of two independent variables $x _ { 1 }$ and $x _ { 2 }$ , which of the following statements are not true?

A multiple regression model with k predictors will include __________ regression parameters, because $\beta _ { 0 }$ will always be included.

The transformation __________ is used to linearize the function $y = \alpha + \beta \cdot \log ( x )$

A study reports the accompanying data on discharge amount ( $q , \text { in } m ^ { 3 } / \mathrm { sec }$ ), flow area ( $a , \text { in } m ^ { 2 }$ ), and slope of the water surface (b, in m/m) obtained at a number of floodplain stations. The study proposed a multiplicative power model $Q = \alpha a ^ { \beta } b ^ { y} \in$ . q 17.6 23.8 5.7 3.0 7.5 a 8.4 31.6 5.7 1.0 3.3 b .0048 .0073 .0037 .0412 .0413 q 89.2 60.9 27.5 13.2 12.2 a 41.1 26.2 16.4 6.7 9.7 b .0063 .0061 .0036 .0039 .0025 a. Use an appropriate transformation to make the model linear and then estimate the regression parameters for the transformed model. Finally, estimate $\alpha , \beta \text {, and } \gamma$ (the parameters of the original model). What would be your prediction of discharge amount when flow area is 10 and slope is .01? b. Without actually doing any analysis, how would you fit a multiplicative exponential model ? c. After the transformation to linearity in part (a), a 95% CI for the value of the transformed regression function when a = 3.3 and b = .0046 was obtained from computer output as (.217, 1.755). Obtain a 95% CI for $\alpha a ^ { \mathcal { \beta } } b ^ { y }$ when a = 3.3 and b = .0046.

Which of the following statements are true?

The coefficient of multiple determination R is

A trucking company considered a multiple regression model for relating the dependent variable y=total daily travel time for one of its drivers (hours) to the predictors $x _ { 1 }$ =distance traveled (miles) and $x _ { 2 }$ the number of deliveries made. Suppose that the model equation is $Y = - .800 + .062 x _ { 1 } + .900 x _ { 2 } + \varepsilon$ a. What is the mean value of travel time when distance traveled is 50 miles and three deliveries are made? b. How would interpret $\beta _ { 1 } = .060 \text {, }$ the coefficient of the predictor $x _ { 1 }$ ? What is the interpretation of $\beta _ { 2 } = .900 ?$ c. If $\sigma = .5$ hour, what is the probability that travel time will be at most 6 hours when three deliveries are made and the distance traveled is 50 miles?

If the value of the coefficient of multiple determination $R ^ { 2 }$ is .80 for a quadratic regression model, and that n = 11, then the adjusted $R ^ { 2 }$ value is

In the accompanying table, we give the smallest SSE for each number of predictors k (k = 1,2,3,4) for a regression problem in which y=cumulative heat of hardening in cement, $x _ { 1 }$ =% tricalcium aluminate, $x _ { 2 }$ = % tricalcium silicate, $x _ { 3 }$ = % aluminum ferrate, and $x _ { 4 }$ = % dicalcium silicate. In addition, n=13, and SST=2715.16. a. Use the criteria discussed in the text to recommend the use of a particular regression model. b. Would forward selection result in the best two-predictor model? Explain.

If $S S E _ {k }$ is the error sum of squares computed from a model with k predictors and n observations, then the mean squared error for the model is $M S E_{k}$ = __________/__________.

The kth -degree polynomial regression model equation is $Y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } + \cdots \cdots + \beta _ {k} x ^ { k } + \varepsilon$ , where $\varepsilon$ is a normally distributed random variable with $\mu _ { z }$ = ___________ and $\sigma _ { z } ^ { 2 }$ = ___________

Answer the following questions. a. Could a linear regression result in residuals 25, -25, 7, 19, -6, 11, and 17? Why or why not? b. Could a linear regression result in residuals 25, -25, 7, 19, -6, -10, and 4 corresponding to x values 4, -3, 9, 13, -13, -19, and 26? Why or why not?