This study investigates the factors that influence alumni donations. We use exploratory data analysis and multi factor linear regression techniques to predict alumni giving rate. After analyzing relationships between the variables, we fit linear regression models and perform residual diagnosis to ensure that assumptions of linear regression are not violated. We find that lower student-faculty ratio leads to higher alumni donations and smaller class size is insignificant in predicting alumni donations. We include an additional parameter graduation rate, which is highly correlated with alumni giving rate and interaction between student-faculty ratio and dummy variable private in our final model. The final mode has an adjusted R-squared of 75.76%.

1. Alumni GivingRate LinearRegressionAnalysis
Introduction:
Thisstudyinvestigatesthe factorsthatinfluencealumnidonations. We use exploratorydataanalysisand
multi factor linear regression techniques to predict alumni giving rate. After analyzing relationships
between the variables, we fit linear regression models and perform residual diagnosis to ensure that
assumptionsof linearregressionare notviolated.We findthatlowerstudent-faculty ratioleadstohigher
alumni donations and smaller class size is insignificant in predicting alumni donations. We include an
additional parameter graduation rate, which is highlycorrelated with alumni giving rate and interaction
between student-faculty ratio and dummy variable private in our final model. The final mode has an
adjusted R-squared of 75.76%.
Data Description:
In thisstudywe use data from 48 US universities collected from America’s Best Colleges, Year 2000 Ed.
 School: University Name
 % of Classes Under 20: Percentage of classes offered with fewer than 20 students
 Student/Faculty Ratio: Ratio of the students to the faculty
 Alumni Giving Rate: Percentage of alumni that donated to the university
 Private:A categorical variable forprivate orpublicuniversitieswith1for private and 0 for public
We observed that there are no null values and outlier in the dataset and descriptive statistics of all the
variables is presented in table 1.
TABLE 1: Summary statistics for the response and predictor variables
Variable Minimum 1st
Quantile Median Mean 3rd
Quantile Maximum
Alumni GivingRate 7.00 18.75 29.00 29.27 38.50 67.00
PercentClassSize under20 29.00 44.75 59.50 55.73 66.25 77.00
StudentFacultyRatio 3.00 8.00 10.50 11.54 13.50 23.00
Private 0.00 0.00 1.00 0.688 1.00 1.00
We observe that 𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 has positive correlation with 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20 and
strong negative correlation with 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 with correlation coefficients of 0.646 and

2. Alumni GivingRate LinearRegressionAnalysis
−0.742 respectively. Further, 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20 and 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 have a
correlation of −0.786 which can lead to multicollinearity issues.
Figure 1: Pairwise scatter plots of all the variables
Figure 2: Box plot for Alumni Giving Rate, Percent class under 20 and Student-Faculty Ratio
Methodology
Researchshowsthat studentswhoare more satisfiedwiththeircontactwithteachersare more likelyto
graduate. As a result, one might suspect that smaller class sizes and lower student-faculty ratios might
leadtoa higherpercentageof satisfiedgraduates,whichinturnmightleadtoincreasesinthe percentage
of alumni who donate.
First,we fita linearregressionmodel withclasssizeandstudent-facultyratioaspredictors andour fitted
regression equation is:
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒̂ = 39.66 + 0.17 ∗ 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶𝑙𝑎𝑠𝑠 𝑆𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20̂ − 1.7 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂

3. Alumni GivingRate LinearRegressionAnalysis
From the results of hypothesis tests (𝐻0: 𝛽0 = 0; 𝛽1 = 0 ; 𝛽2 = 0 𝑎𝑛𝑑 𝐻 𝑎:𝛽0 ≠ 0; 𝛽1 ≠ 0 ; 𝛽2 ≠ 0 :
significance of the coefficients) we conclude 𝛽0 and 𝛽2 are significant and 𝛽1is insignificant.
Figure 3: Scatter plots grouped by public and private universities
From figure 3, we observe that alumni donationforprivate andpublicschoolsare significantlydifferent.
Now,we include the dummyvariable 𝑝𝑟𝑖𝑎𝑣𝑡𝑒 inour initial model. The fitted regression equations are:
Private Schools:
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒̂ = 43.07 + 0.08 ∗ 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶𝑙𝑎𝑠𝑠 𝑆𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20̂ − 1.4 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂
Public Schools:
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒̂ = 36.78 + 0.08 ∗ 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶𝑙𝑎𝑠𝑠 𝑆𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20̂ − 1.4 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂
From the results of hypothesis tests (𝐻0: 𝛽0 = 0; 𝛽1 = 0 ; 𝛽2 = 0; 𝛽3 = 0 𝑎𝑛𝑑 𝐻 𝑎: 𝛽0 ≠ 0; 𝛽1 ≠
0 ; 𝛽2 ≠ 0 ; 𝛽3 ≠ 0 ), we conclude that only 𝛽0 and 𝛽2 are significant. The model Adjusted R squared is
.5747, indicating the model is not much better than flipping a coin in terms of predicting power.
Residual analysis:
The QQ plot of model residuals vs fitted values indicate that normality assumption of error term is
violated. When the residuals are plotted against 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20, the residuals show
increasingvariance and against 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 show constant variance. Thisis alignedwiththe
results of NCV Test (𝐻 𝑜: 𝜎𝑒
2 = 𝜎2 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ; 𝐻 𝑎: 𝜎𝑒
2 ≠ 𝜎2), with a large 𝑝-𝑣𝑎𝑙𝑢𝑒 (0.29). We conclude
that there is no issue of heteroscedasticity. The results of Durbin Watson Test indicate that there is no
first order autocorrelation (𝐷 − 𝑊 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ∶ 1.61378 ; 𝑝𝑣𝑎𝑙𝑢𝑒 ∶ 0.172).

4. Alumni GivingRate LinearRegressionAnalysis
Figure 4: Residual plots
Outliers and Influential Points:
Table 2:
Measure Condition Outlier
Y Outlier Studentizedresidual | 𝑟𝑗| ≥ 3 Princeton University
X Outlier Leverage ℎ 𝑖 𝑗 ≥ 2𝑝/𝑛 Boston College, U. of Washington, UCB
Influential Point Cook’s D 𝐷𝑖~𝐹𝑝,𝑛−𝑝 NYU, Princeton,U of Florida,U. of Norte
Dame, U. of Washington
Afterremovingthe outliersandinfluencepoints,the adjustedR-squaredimproved from57.47% to 63.67.
However, 𝛽1, 𝛽3 are still insignificant. Since, 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20 and 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜
are highlycorrelatedandthe coefficientof 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20 isinsignificant,we remove this
predictorandfita newregressionmodel,withadjustedR-squaredimprovedto64.25%. Resultsof Partial
F-test (table 3) also confirm that 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒 𝑢𝑛𝑑𝑒𝑟 20 is insignificant as the p-value is large.
Table 3: Partial F-Test results
Model 1: Alumni.Giving.rate ~SFratio + Private
Model 2: Alumni.Giving.rate ~Per.under.20+ SFratio + Private
Res.Df RSS Df Sum of sq F Pr(>F)
38 1991.6
37 1971.2 1 20.477 0.3844 0.5391

5. Alumni GivingRate LinearRegressionAnalysis
The regression equation is:
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒̂ = 43.36 − 1.61 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂ + 5.53 ∗ 𝑃𝑟𝑖𝑣𝑎𝑡𝑒̂
Discussion:
For the final model, we included an additional variable 𝑔𝑟𝑎𝑑_𝑟𝑎𝑡𝑒, a variable part of original America’s
Best Colleges,Year2000 Ed dataset (withcorrelationcoefficientof 0.756 with 𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒) to
the initial data set and fitted a linear regression model with all other variables.
The model adjusted R-squaredimproves to 69.79%.
By adding an interaction parameter
𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 ∶ 𝑝𝑟𝑖𝑣𝑎𝑡𝑒 model adjusted
R-squared improves to 71.74%. All the regression
coefficients are significant except for
𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜.
Fitted Model
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖 𝑛 𝑔 𝑟𝑎𝑡𝑒̂
= −24.39 − 0.04 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂ + 26.27 ∗ 𝑃𝑟𝑖𝑣𝑎𝑡𝑒̂ − 1.52 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂
∗ 𝑃𝑟𝑖𝑣𝑎𝑡𝑒̂ + 0.53 ∗ 𝑔𝑟𝑎𝑑_𝑟𝑎𝑡𝑒
Now,we fitaregressionmodelwithout 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜,withmuchimprovedadjustedR-squared
value of 75.76%. Residual diagnosis show that error terms follow normal distribution with constant
variance (Shapiro-Wilk Test: p-value 0.16).
Our final regression model is:
𝐴𝑙𝑢𝑚𝑛𝑖 𝑔𝑖𝑣𝑖𝑛𝑔 𝑟𝑎𝑡𝑒̂ = −25.32 + 26.92∗ 𝑃𝑟𝑖𝑣𝑎𝑡𝑒̂ − 1.55 ∗ 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑟𝑎𝑡𝑖𝑜̂ ∗ 𝑃𝑟𝑖𝑣𝑎𝑡𝑒̂ + 0.54 ∗ 𝑔𝑟𝑎𝑑_𝑟𝑎𝑡𝑒