2. TRUE or FALSE: Converting the units of one measured variable alters the correlation of between it and a second variable.

Transcription

1 1. The diagnostic plots shown below are from a linear regression that models a patient s score from the SUG-HIGH diabetes risk model as function of their normalized LDL level. a. Based on these plots, can you obtain an estimate for the number of observations considered in the analysis? If so, provide an estimate and explain how you obtained it. If not, explain why not. b. List the underlying assumptions of fitting a linear regression model to this data. c. Can the diagnostic plots be used to assess all of the assumptions listed in part (b)? For the assumptions that can be assessed, determine whether or not the assumption is violated for this data and state which plot(s) you used to make this determination. If there are assumptions that cannot be assessed using these plots, explain why not and what additional information you would need to be to assess the assumption. 2. TRUE or FALSE: Converting the units of one measured variable alters the correlation of between it and a second variable. Page 1 of 9

2 3. TRUE or FALSE: Regardless of sample size, if the residuals from a linear regression model truly come from a normal distribution then a QQ-plot would confirm this with a linear trend along the 45 line between theoretical and sample quantiles. 4. Consider a simple linear regression model that predicts a response variable Y with units AA as a function of the covariate X with units BB. The SAS output from fitting the model is shown below. a. What are the units of the correlation coefficient, the intercept, and the slope? b. Do you have enough information to estimate the correlation coefficient between X and Y? The sign of the correlation coefficient? If so, state it. If not, explain what information is missing from the SAS output. c. Based on the SAS output, would it be appropriate to conclude that the data suggest that X and Y are associated? Explain your response. d. Based on the SAS output, would it be appropriate to conclude that increasing X by 1 unit will cause to Y to increase by about 2.4 units? Explain your response. Consider the residual plots shown below. e. Assuming plot (a) was created to assess the constant error variance assumption, what variables are plotted on the x- and y-axis of the plot? f. Based on plot (a), is the constant error variance assumption violated? Explain your choice. If it is violated, explain how you could update the model fit to resolve this violation. g. Assuming plot (b) was created to assess the linearity assumption, what variables are plotted on the x- and y-axis of the plot? h. Based on plot (b), is the linearity assumption violated? Explain your choice. If it is violated, explain how you could update the model fit to resolve this violation. Page 2 of 9

3 5. Samples of air pollutants were taken at a number of locations in Chicago. Measurements of total suspended particulates (Y; parts per million) were to be related to time of day (T; minutes since midnight), weather condition (R; rain, no rain), and sampling location (L; expressway, lakefront, residential area). a. Using dummy variables, specify the appropriate multivariable regression model relating total suspended particulates to time of day, weather, and location that includes all possible second-order (i.e. two-variable) interactions. b. In terms of regression parameters define in part (a), state the null hypothesis that could be tested to address the following claims: i. Both weather equations are parallel, controlling for sampling location. ii. All sampling location equations have the same intercept, controlling for weather iii. All six regression equations are parallel iv. All six regression equations have the same intercept c. For each of the hypotheses given in part (b), specify the test statistic and the degrees of freedom of its null distribution. 6. Three measurements from Machine A for an item produce y 1 = (6.1,6.2,6.0) while two measurements from Machine B on the same item produce y 2 = (5.8,6.0). a. Suppose the two sets of measurements arise from two normal populations with equal variance. Perform an appropriate non-regression-based statistical test for whether the average measurement from the two machines differ. Be sure to state the null hypothesis and your conclusion. b. Using a dummy variable z to represent the different machine-ids, provide a simple linear regression model representation of the data from this study. That is, identify the vector Y the design matrix Z for linear regression model with the form of y = β 0 + β 1 z + ϵ, where y is the measurement, z is machine-id, and ϵ N(0, σ 2 ). c. Using the data, provide the least-square estimates β. d. How does the estimate of β 1 compare to the point estimated used in the test in part (a)? e. What is the trace of the Hat matrix H? f. Find an estimator for σ. g. Perform a test for H 0 : β 1 = 0. What did you conclude? h. Is the test in part (g) equivalent to the test in part (a)? Explain why or why not. Page 3 of 9

4 7. A political scientist developed a questionnaire to determine political tolerance scores for a random sample of faculty members at her university. (The higher the political tolerance score, the more tolerant the individual.) For each faculty member, she also recorded age (in years) and faculty rank. The variable RANK equals 1 for full professors, 2 for associate professors, and 3 for assistant professors. The next two pages give SAS code, a printout of the data, and output for an analysis of these data. a. It is your job to describe the fitted model to a group of faculty members who know very little statistics. In a few sentences, what would you tell them? (Talk about fitting lines to the data.) b. Use the fitted model to give a point estimate for the mean tolerance score of 40-year-old associate professors. Show your work c. The ANOVA table includes the F-value 5.02 (P-value= ). What null and alternative hypotheses are tested by this F-statistic? State the hypotheses in words, not symbols. d. Explain to the faculty members from part (a) the meaning of a P-value. What does the P- value of in part (c) tell us about tolerance scores, age, and rank? e. Does the relationship between age and tolerance score depend on rank? Explain. What test statistic and P-value are you using to make this decision? f. Suppose we ignore rank and fit a simple linear regression of tolerance score on age. What proportion of the variability in tolerance score would be explained by that simple linear regression? g. Does the analysis on page 4 explain significantly more variability than the simple linear regression of tolerance score on age? Compute a test statistic for comparing these two models and carry out the test using α = Page 4 of 9

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7 8. A multiple regression (with intercept) of the usual form Y = Xβ + ε is run where E[Y i ] = β 0 + β 1 X 1i + β 2 X 2i + ε i and β = (β 0, β 1, β 2 ). The errors are independent and identically distributed normal random variables with mean zero. The elements of X are fixed constants where X X = ( ) and (X X) 1 = ( ) The resulting estimated linear equation using the usual notation is and the error mean square is 4. Y i = X 1i + 2X 2i a. Fill in the missing entries in the inverse matrix. b. How many rows and columns does the matrix X have? What is the rank of the matrix X? c. Find, if possible, the response vector Y and the vector X Y. If you think either is impossible, so state and move to the other. d. What is the correlation between the estimate of the intercept and the estimate of the partial slope for X 1? How much would that correlation change if 20 were added to each element of Y? 9. These data are from a study of sleep in mammals ranging from African elephant to yellow-bellied marmot. Some summary statistics on Y = ln(brain weight) and X = ln(body weight) are given in the table below. Variable N Mean Standard Deviation X (ln(kg)) Y (ln(g)) The sample correlation coefficient is r = a. Compute the slope in the simple linear regression of Y on X. b. Gray seals have average ln(body weight) = 4.44 and average ln(brain weight) = Does the regression model described in part (a) over or under predict for this X-value? c. Compute the MSE from the regression of Y on X. Page 7 of 9

8 10. Consider the two plots shown below. Use these plots answer the following questions. Plot A Plot B a. TRUE or FALSE: If y was regressed on x for the data in both plot, both estimated slope coefficients would be positive. b. TRUE or FALSE: The red dot in Plot A is an outlier in Y. c. TRUE or FALSE: The red dot in Plot A is an influential point. d. TRUE or FALSE: The red dot in Plot B is an outlier in Y. e. TRUE or FALSE: The red dot in Plot B is an influential point. f. TRUE or FALSE: If y was regressed on x for the data in Plot B, the estimated slope coefficient would be larger if the regression line was fit without including the red dot in the analysis cohort than if it was fit including the red dot. 11. A researcher was studying the relationship between hours of sleep per day and brain weight, body weight, and gestation time. In the regression of Y on X1, X2, and X3 where Y = ln(hours of sleep per day) = LNSLEEP, X1 = ln(gestation time) = LNGEST, X2 = ln(brain weight) = LNBRAIN. And X3 = ln(bodyweight) = LNBODY. The next page gives SAS output from a PROC GLM analysis of these data. a. Fill in the three blanks in the analysis of variance table. b. Test the hypothesis that there is no additional relationship between ln(body weight) and ln(hours of sleep) given that ln(gestation time) and ln(brain weight} are in the model, against the one-sided alternative that there is a negative relationship (H 0 : β 3 < 0). Use α = State the rejection criterion and your conclusion. c. Test the hypothesis that ln(brain weight) and ln(body weight) together have no effect on ln(hours of sleep) given that ln(gestation time} is in the model (H 0 : β 2 = β 3 = 0). Use α = State the rejection criterion and your conclusion. Page 8 of 9

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