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- An experiment to evaluate the effects of certain variables on soil erosion was performed on 10-foot-square plot of sloped farmland subjected to 2 inches of artificial rain applied over a 20-minute period.^1 The data are as follows:
Plot 1 2 3 4 5 6 SL (Y ) 27.1 35.6 31.4 37.8 40.2 39. SG (X 1 ) 0.43 0.47 0.44 0.48 0.48 0. LOBS (X 2 ) 1.95 5.13 3.98 6.25 7.12 6. PGC (X 3 ) 0.34 0.32 0.29 0.30 0.25 0. Plot 7 8 9 10 11 SL (Y ) 55.5 43.6 52.1 43.8 35. SG (X 1 ) 0.53 0.50 0.55 0.51 0. LOBS (X 2 ) 10.67 7.08 9.88 8.72 4. PGC (X 3 ) 0.10 0.16 0.19 0.18 0.
(a) Find the least-square estimate of the regression model when Y regressed on X 1. Determine the R^2 value for this model. Find r Y X^21 and compare it to the value of R^2. Are they equal? Should it be?
(b) Find the least-square estimate of the regression model when Y regressed on X 1 and X 2. Determine the R^2 value for this model. Find r^2 Y X 1 and r^2 Y X 2 and compare them to the value of R^2. Are they equal? Should they be? Does adding X 2 to a model that already contains X 1 appear to be useful in predicting Y?
(c) Find the least-square estimate of the regression model when Y regressed on X 1 , X 2 and X 3. Determine the R^2 value for this model. Based on the values of R^2 obtained in parts (a)-(c), Which model would you conclude is superior?
(d) For the regression model Y regressed on X 1 , X 2 and X 3 , compute and compare observed and predicted values of Y for plots 1, 5, and 7.
(e) Find the ANOVA table for the model with independent variables X 1 , X 2 and X 3. What does the p-value imply?
(^1) Adapted from a study by Packer (1951), “An Approach to Watershed Protection Criteria”, Journal of Forestry, 9 , 638-644.
- An engineer performed an experiment to determine the effect of CO 2 pressure, CO 2 temperature, peanut moisture, CO 2 flow rate, and peanut particle size on the yield of oil per batch of peanuts.^2 The following table summarizes the experimental results.
Pressure Temperature Moisture Flow Rate Particle Size Yield (bar) (oC) (% by weight) (L/min) (mm) 415 25 5 40 1.28 63 550 25 5 40 4.05 21 415 95 5 40 4.05 36 550 95 5 40 1.28 99 415 25 15 40 4.05 24 550 25 15 40 1.28 66 415 95 15 40 1.28 71 550 95 15 40 4.05 54 415 25 5 60 4.05 23 550 25 5 60 1.28 74 415 95 5 60 1.28 80 550 95 5 60 4.05 33 415 25 15 60 1.28 63 550 25 15 60 4.05 21 415 95 15 60 4.05 44 550 95 15 60 1.28 96
(a) Prepare a stem-and-leaf plot for each predictor variable. What information do these plots provide?
(b) Fit a multiple linear regression model relating yield to these regressors.
(c) Construct the analysis of variance table and the test for significance of regression. What conclusion can you draw?
(d) Use t-tests to assess the contribution of each regressor to the model. Discuss your finding.
(e) Calculate R^2 and R^2 Adj for this model. Compare these values to R^2 and R^2 Adj for the multiple linear regression model relating yield to temperature and particle size. Use a F -test to compare these two models. Discuss your results.
(^2) See Kilgo, “An Application of Fractional Experimental Design”, Quality Engineering, 1 , 19-23.
effect associated with the variables X 1 and X 2 , which concerns whether the relationship between Y and one of these two variables depends on the levels of other variable.^3 Find the ANOVA table for this model. Does the addition of X 4 lead to large improvement in fit over the model with X 1 and X 2 as the only independent variables? Explain.
(d) Find the least-square estimate of the following multiple linear regression model:
E(Y ) = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X^21 + β 5 X 22 + β 6 X 1 X 2
Comparing models in part (c) and (d), which one is superior? Use R^2 and F -test.
(^3) A more detailed discussion of the concept of interaction will be given in future.
