Introduction to Categorical Data Analysis

5.2 MODEL CHECKING 149
5.2.5 Example: Graduate Admissions at University of Florida
Table 5.5 refers to graduate school applications to the 23 departments in the College of
Liberal Arts and Sciences at the University of Florida, during the 1997–98 academic
year. It cross-classifies whether the applicant was admitted (Y ), the applicant’s gender
(G), and the applicant’s department (D). For the nik applications by gender i in
department k, let yik denote the number admitted and let πik denote the probability
of admission. We treat {Yik} as independent binomial variates for {nik} trials with
success probabilities {πik}.
Other things being equal, one would hope the admissions decision is independent
of gender. The model with no gender effect, given department, is
logit(πik) = α + β D k
However, the model may be inadequate, perhaps because a gender effect exists in
some departments or because the binomial assumption of an identical probability
of admission for all applicants of a given gender to a department is unrealistic. Its
goodness-of-fit statistics are G 2 = 44.7 and X 2 = 40.9, both with df = 23. This
model fits rather poorly (P -values = 0.004 and 0.012).
Table 5.5 also reports standardized residuals for the number of females who
were admitted, for this model. For instance, the Astronomy department admitted
six females, which was 2.87 standard deviations higher than predicted by the model.
Each department has df = 1 (the df for independence in a 2 × 2 table) and only a
single nonredundant standardized residual. The standardized residuals are identical
Table 5.5. Table Relating Whether Admitted to Graduate School at Florida to
Gender and Department, Showing Standardized Residuals for Model with no
Gender Effect
Females Males Std. Res Females Males Std. Res
Dept Yes No Yes No (Fem,Yes) Dept Yes No Yes No (Fem,Yes)
anth 32 81 21 41 −0.76 ling 21 10 7 8 1.37
astr 6 0 3 8 2.87 math 25 18 31 37 1.29
chem 12 43 34 110 −0.27 phil 3 0 9 6 1.34
clas 3 1 4 0 −1.07 phys 10 11 25 53 1.32
comm 52 149 5 10 −0.63 poli 25 34 39 49 −0.23
comp 8 7 6 12 1.16 psyc 2 123 4 41 −2.27
engl 35 100 30 112 0.94 reli 3 3 0 2 1.26
geog 9 1 11 11 2.17 roma 29 13 6 3 0.14
geol 6 3 15 6 −0.26 soci 16 33 7 17 0.30
germ 17 0 4 1 1.89 stat 23 9 36 14 −0.01
hist 9 9 21 19 −0.18 zool 4 62 10 54 −1.76
lati 26 7 25 16 1.65
Note: Thanks to Dr. James Booth for showing me these data.