Regression Case

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Chapter 4 Regression Analysis

Case Problem: Alumni Giving

1. Descriptive statistics for graduation rate, % of classes under 20, student-faculty ratio, and alumni giving are shown below.

	Graduation Rate	% of Classes Under 20	Student-Faculty Ratio	Alumni Giving Rate
mean	83.042	55.729	11.542	29.271
median	83.5	59.5	10.5	29.0
standard deviation	8.607	13.194	4.851	13.441
minimum	66	29	3	7
maximum	97	77	23	67
range	31	48	20	60

The correlations for each pair of variables are shown in the table below.

	Graduation Rate	% of Classes Under 20	Student-Faculty Ratio	Alumni Giving Rate
Graduation Rate	1.0000	0.5828	-0.6049	0.7559
% of Classes Under 20	0.5828	1.0000	-0.7856	0.6457
Student-Faculty Ratio	-0.6049	-0.7856	1.0000	-0.7424
Alumni Giving Rate	0.7559	0.6457	-0.7424	1.0000

As would be expected, % of classes under 20 and student-faculty ratio have a relatively strong negative relationship (the correlation between these two variables is -0.7856). Using both of these variables as independent variables in a reg4ession model would introduce multicollinearity into the model.

The relationships the graduation rate has with % of classes under 20 and student-faculty ratio are both weaker.

2. The following Excel output provides the estimated simple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x).

The estimated simple linear regression equation is ., and the coefficient of determination r2 is 0.5715, so this simple linear regression model explains approximately 57% of the variation in the sample values of alumni giving rate.

Before using these results to test the hypothesis of no relationship between the alumni giving rate (y) and the graduation rate (x), we first check the conditions necessary for valid inference in regression. The Excel plot of the residuals and graduation rate follows.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed at any value of displacement. Because there are no apparent severe violations of the conditions necessary for valid inference in regression, we will proceed with our inference. Since the level of significance for use in hypothesis testing has not been given, we will use the standard 0.05 level throughout this problem.

The p-value associated with the estimated regression parameter b1 is 5.23818E-10. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0. We conclude that there is a relationship between the alumni giving rate and the graduation rate, and our best estimate is that a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 1.1805%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -68.7612%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate near 0).

3. The following Excel output provides the estimated multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1), % of Classes Under 20 (x2), and Student-Faculty Ratio (x3).

The multiple simple linear regression equation is ., and the coefficient of determination r2 is 0.6999, so this multiple linear regression model explains approximately 70% of the variation in the sample values of alumni giving rate.

Before using these results to test any hypotheses, we again check the conditions necessary for valid inference in regression. The Excel plots of the residuals and graduate rate, % of classes under 20, and student-faculty ratio follow.

The residuals appear to have a relatively constant variance and do not appear to be badly skewed at any value of displacement. However, the mean residual apears to deviate from zero at several values of graduation rate, suggesting a possible nonlinear relationship between the alumni giving rate and graduation rate. This does not appear to be severe, but we will keep this in mind as we continue.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed at any value of displacement.

Because there are no apparent severe violations of the conditions necessary for valid inference in regression, we will proceed with our inference. Since the level of significance for use in hypothesis testing has not been given, we will again use the standard 0.05 level throughout this problem.

The p-value associated with the F test for an overall regression relationship is 1.43233E-11. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 3 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 4.799E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0 and conclude that there is a relationship between the alumni giving rate and the graduation rate. We estimate that holding % of classes under 20 and student-faculty ratio constant, a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 0.7482%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

The p-value associated with the estimated regression parameter b2 is 0.8358. Because this p-value is greater than the 0.05 level of significance, we do not reject the hypothesis that 2 = 0 and conclude that there is not a relationship between the alumni giving rate and the % of classes under 20 when controlling for the graduation rate and student-faculty ratio.

The p-value associated with the estimated regression parameter b3 is 4.799E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 3 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate and % of classes under 20 constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.1920%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -20.7201%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate, a % of classes under 20, and a student-faculty ratio near 0).

4. Given our recognition in (1) of possible multicollinearity between the graduation rate and % of classes under 20, and the results from (2) and (3), it is reasonable to estimate a multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1) and Student-Faculty Ratio (x2). The following Excel output provides this estimated multiple linear regression model.

The estimated multiple linear regression equation is ., and the coefficient of determination r2 is 0.6996, so removing the % of classes under 20 from the multiple linear regression estimated in (3) resulted in practically no loss in the ability of the regression model to explain variation in the sample values of alumni giving rate.

Before using these results to test any hypotheses, we again check the conditions necessary for valid inference in regression. The Excel plots of the residuals and the independent variables (graduate rate and student-faculty ratio) follow.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed at any value of displacement.

The p-value associated with the F test for an overall regression relationship is 1.76525E-12. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 2.34782E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0 and conclude that there is a relationship between the alumni giving rate and the graduation rate. We estimate that holding student-faculty ratio constant, a 1% increase in the graduation rate corresponds to an increase in the alumni giving rate of 0.7557%. The alumni giving rate is expected to increase as the graduation rate increases, so this result is consistent with what is expected.

The p-value associated with the estimated regression parameter b2 is 6.95424E-05. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 2 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.2460%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate is zero, the alumni giving rate is -19.1063%. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate and a student-faculty ratio near 0).

Our results have suggested that the relationship between the alumni giving rate and the graduation rate may be nonlinear, so we will also estimate a model with a second order quadratic relationship between these two variables, i.e.,

where x1 is the graduate rate and x2 is the Student-Faculty Ratio. The following Excel output provides this estimated regression model.

The estimated multiple linear regression equation is ., and the coefficient of determination r2 is 0.7513, so adding the squared graduation rate as an independent variable resulted in an increase of 5% in the ability of the regression model to explain variation in the sample values of alumni giving rate.

The residuals appear to have a relatively constant variance and do not appear to be badly skewed at any value of displacement. Also note that the deviation of the mean residuals from zero at several values of graduation rate has been reduced.

The residuals appear to have a relatively constant variance with a mean of zero, and do not appear to be badly skewed at any value of displacement.

The p-value associated with the F test for an overall regression relationship is 2.38729E-13. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 2 = 3 = 0 and conclude that there is an overall regression relationship at the 0.05 level of significance.

The p-value associated with the estimated regression parameter b1 is 0.0093. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 1 = 0. Similarly, the p-value associated with the estimated regression parameter b2 is 0.0042. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 2 = 0. On the basis of the results of these two hypothesis tests, we conclude that there is a nonlinear relationship between the alumni giving rate and the graduation rate. We estimate that holding student-faculty ratio constant, a 1% increase in the graduation rate from a value of x1 to a value of x1 + 1corresponds to an increase in the alumni giving rate of

-6.9200 [(x1 + 1) – x1] + 0.0467 [(x1 + 1)2 –]

= -6.9200 (x1 – x1 +1) + 0.0467 (+ 2x1 + 1 –)

= -6.9200 + 0.0467 (2x1 + 1)

= -6.8733 + 0.0933x1

That is, estimated alumni giving rate initially decreases as graduation rate increases, and then eventually increases as graduation increases. Solving this result for x

-6.8733 + 0.0933x1= 0

-6.8733 = 0.0933x1

x1 = -6.8733 / - 0.0933 = 73.6532.

or 73.7%. The alumni giving rate decreases as the graduation rate increases until the graduation rate reaches 73.7%, at which point the alumni giving rate increases as the graduation rate increases. Perhaps alumni give more at low graduation rates alumni because they feel the university is in greater need of support, and they give more at high graduation rates because they are recognizing the university’s exceptional performance. If this explanation is reasonable, it also suggests that when the graduation rate is at an intermediate level, alumni are less motivated to give.

The p-value associated with the estimated regression parameter b3 is 6.68201E-06. Because this p-value is less than the 0.05 level of significance, we reject the hypothesis that 3 = 0 and conclude that there is a relationship between the alumni giving rate and the student-faculty ratio. We estimate that holding the graduation rate constant, a 1 unit increase in the student-faculty ratio corresponds to decrease in the alumni giving rate of 1.3484%. The alumni giving rate is expected to decrease as the student-faculty ratio increases, so this result is consistent with what is expected.

The estimated regression parameter b0 suggests that when the graduation rate and the student-faculty ratio are zero, the alumni giving rate is 294.3311 %. This result is obviously not realistic, but this parameter estimate and the test of the hypothesis that 0 = 0 are meaningless because the y-intercept has been estimated through extrapolation (there is no university in the sample with a graduation rate and a student-faculty ratio near 0).

5. The residuals for the quadratic regression model, which were generated as part of the Excel output, are sorted and given in the table that follows.

University	Predicted Alumni Giving Rate	Residuals
U. of Washington	22.38665484	-10.3867
Stanford University	43.18507469	-9.1851
Columbia University	40.04079983	-9.0408
Johns Hopkins University	35.91193735	-8.9119
Northwestern University	38.69244742	-8.6924
Georgetown University	37.52122011	-8.5212
U. of Michigan–Ann Arbor	21.18859365	-8.1886
New York University	20.44978354	-7.4498
U. of Wisconsin–Madison	20.29550386	-7.2955
U. of Virginia	35.09496024	-7.0950
U. of California–San Diego	13.73838728	-5.7384
Harvard University	51.33050668	-5.3305
U. of California–Davis	12.14442958	-5.1444
College of William and Mary	31.86688013	-4.8669
Tulane University	21.79813595	-4.7981
Tufts University	33.32758156	-4.3276
Brown University	43.54883947	-3.5488
Washington University–St. Louis	34.87208824	-1.8721
Rice University	41.83672228	-1.8367
U. of California–Irvine	10.79607717	-1.7961
U. of Rochester	24.43764111	-1.4376
U. of California–Los Angeles	14.18214108	-1.1821
U. of Chicago	36.89270873	-0.8927
Boston College	25.72309551	-0.7231
Massachusetts Inst. of Technology	44.53342709	-0.5334
U. of California–Berkeley	18.49188883	-0.4919
U. of California–Santa Barbara	11.59983558	0.4002
U. of Texas–Austin	12.54835908	0.4516
U. of Pennsylvania	40.04079983	0.9592
U. of Southern California	21.03830244	0.9617
Cornell University	33.47616289	1.5238
Vanderbilt University	28.49978934	2.5002
Duke University	41.83672228	3.1633
Yale University	46.70262892	3.2974
Carnegie Mellon University	24.31196126	3.6880
Case Western Reserve Univ.	27.19154558	3.8085
Emory University	32.84765151	4.1523
Brandeis University	28.07130454	4.9287
U. of Illinois–Urbana Champaign	17.91487876	5.0851
California Institute of Technology	39.20661958	6.7934
U. of North Carolina–Chapel Hill	19.06132248	6.9387
Pennsylvania State Univ.	13.73838728	7.2616
U. of Florida	9.137455524	9.8625
Dartmouth College	42.65757169	10.3424
U. of Notre Dame	38.61251447	10.3875
Wake Forest University	25.80308452	12.1969
Lehigh University	25.11748561	14.8825
Princeton University	51.29809061	15.7019

These residuals suggest that the alumni giving rate several schools (U. of Florida, Dartmouth College, U. of Notre Dame, Wake Forest University, Lehigh University, and Princeton University) is at least 9% above what the model predicts. The Presidents of these universities should be pleased with the efforts of their Offices of Alumni Affairs.

These residuals also suggest that the alumni giving rate several schools (U. of Washington, Stanford University, Columbia University, Johns Hopkins University, Northwestern University, Georgetown University, U. of Michigan–Ann Arbor) is at least 8% below what the model predicts. Perhaps the Alumni Affairs offices at these universities should review the efforts of U. of Florida, Dartmouth College, U. of Notre Dame, Wake Forest University, Lehigh University, and Princeton University.

Other independent variables that could be included in the model include some measure of the success of the university’s football or basketball team and proportion of students on scholarship or student aid.

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -6.5826771653543261 8.5004188306248949 -1.0268051599932875 14.417322834645674 8.2224828279443756 14.764030825934 -9.3047411626738068 -6.485257162003677 -3.6657731613335471 10.792678840676842 5.1537108393365827 6.597838833975544 -9.6657731613335471 0.25113084268721764 -9.3047411626738068 13.139386831965155 4.1537108393365827 -3.2359691740659997 -7.485257162003677 -4.6800971687049753 23 .612162841346972 0.15371083933658269 -5.8462891606634173 -4.9437091640140665 0.76403082593400029 -11.221645166694586 -11.59700117272574 -9.59700117272574 -10.319065170045235 -17.680097168704975 -1.8749371754062594 5.597838833975544 8.6666108225833511 0.86145082928463523 -16.221645166694586 -2.0411291673647156 6.7926788406768424 3.514742837996323 2.0419668286145054 8.1250628245937406 3.8471268219132213 -11.846289160663417 -1.8749371754062594 -4.4164851733958699 2.9588708326352844 9.9588708326352844 0.23680683531580371 7.7926788406768424

Graduation Rate

Residuals

Graduation Rate Residual Plot

85 79 93 85 75 72 89 90 91 94 92 84 91 97 89 81 92 72 90 80 95 92 92 87 72 83 74 74 78 80 70 84 67 77 83 82 94 90 76 70 66 92 70 73 82 82 86 94 -3.5118498755356811 2.1750146512850321 -1.067219353946065 4.8129863288677939 2.5808078853063527 5.8769446183006835 -5.8708356564693958 -9.2760474840442271 1.0407119370650832 13.539578586852699 4.4486382596504015 1.3292119049748621 -8.012588477537733 1.5625210217678216 -9.9986400552124124 11.632209561994998 1.1517388159351469 -6.4824490411811837 -8.9969148451965779 3.5844128239127429 20.628057779017958 -0.37711785102956696 -7.7724130827572395 -6.5893964011594051 -3.471175197828785 -4.7992138162522053 -5.9264903799481985 -3.0248861679606165 -4.372598105541929 -9.8802375476073436 2.8807230848054033 -3.2468201775880559 16.107872640517932 3.148040389891726 -11.979950667087444 3.2796899568767586 13.347935855175663 0.84011510883579632 -3.0512123211701372 4.3043230397318908 8.2397088657391144 -5.8943327121312734 -6.4230372829356597 -6.4755749868175556 -0.87752305259686381 8.7678641697584681 -4.3994788794981972 6.4988961327697012

Graduation Rate

Residuals

% of Classes Under 20 Residual Plot

39 68 60 65 67 52 45 69 72 61 68 65 54 73 64 55 65 63 66 32 68 62 69 67 56 58 32 42 41 48 45 65 31 29 51 40 53 65 63 53 39 44 37 37 68 59 73 77 -3.5118498755356811 2.1750146512850321 -1.067219353946065 4.8129863288677939 2.5808078853063527 5.8769446183006835 -5.8708356564693958 -9.2760474840442271 1.0407119370650832 13.539578586852699 4.4486382596504015 1.3292119049748621 -8.012588477537733 1.5625210217678216 -9.9986400552124124 11.632209561994998 1.1517388159351469 -6.4824490411811837 -8.9969148451965779 3.5844128239127429 20.6280577790 17958 -0.37711785102956696 -7.7724130827572395 -6.5893964011594051 -3.471175197828785 -4.7992138162522053 -5.9264903799481985 -3.0248861679606165 -4.372598105541929 -9.8802375476073436 2.8807230848054033 -3.2468201775880559 16.107872640517932 3.148040389891726 -11.979950667087444 3.2796899568767586 13.347935855175663 0.84011510883579632 -3.0512123211701372 4.3043230397318908 8.2397088657391144 -5.8943327121312734 -6.4230372829356597 -6.4755749868175556 -0.87752305259686381 8.7678641697584681 -4.3994788794981972 6.4988961327697012

% of Classes Under 20