Comparing Contingency Tables

 See section 19.5 of text.

 Although the following data is hypothetical, it based a real data set
discussed by Bickel and O'Connell, "Is there a sex bias in graduate
admissions?", Science 187 (1975), p. 398-404.

 Upper Wabash Tech has 2 professional schools: business and law. To
investigate sex bias at this school, a 2-way table was constructed for
admission decision and gender:

                Admission Decision

 Gender            Admit   Deny
           -------------------------
  Male              490     210
 Female             280     220

 Find the odds ratio to compare the chance of a male being admitted
to the chance of a female being admitted. What does it suggest?

 Now we will construct a 3-way table, categorized by gender, school and
admission decision.

Business:           
                Admission Decision 

 Gender		   Admit   Deny
           ------------------------- 
  Male              480     120
 Female             180      20

Law:   
                Admission Decision 

 Gender            Admit   Deny
           -------------------------
  Male               10      90
 Female             100     200

 For each school, find the odds ratio to compare the chance of a male
being admitted to the chance of a female being admitted. What do we see?

 SIMPSON'S PARADOX: The reversal of the direction of a comparison or an
association when data from several groups are combined to form a single
group.

 One way to analyze 3-way tables is the MANTEL-HAENSZEL TEST. In this
example, it would test if the odds of a male getting admission are
greater than a female getting admission, GIVEN that the type of school
is accounted for.

 Assumptions:
	    1. The odds ratio (odds male admitted/odds female admitted)
is the same for all tables. If this doesn't hold, the test doesn't make
sense.
            2. Test assumes that there is no 3-way dependence of
admission, gender and school.















 One way to save this data in R is to use a 3-dimensional array,
which attaches several matrices together. 

 # Enter the data COLUMNWISE in the "array" command, beginning with
 # your first matrix.
 # When the data is entered, we must tell R the size of our
 # array. Here it is 2x2x2, where the first number is the number of rows
 # in each matrix, followed by the number of columns, followed by 
 # the number of matrices we have.

> admit.status <- array(c(480,180,120,20,10,100,90,200),c(2,2,2))
> admit.status

, , 1
     [,1] [,2] 
[1,]  480  120
[2,]  180   20

, , 2
     [,1] [,2] 
[1,]   10   90
[2,]  100  200

 # In creating names for the rows, etc., we start with names for the
 # rows, then columns, then each matrix

> dimnames(admit.status) <- list(c("Male", "Female"),c("Admit", "Deny"),
                                c("Business", "Law"))
> admit.status

, , Business
       Admit Deny 
  Male   480  120
Female   180   20

, , Law
       Admit Deny 
  Male    10   90
Female   100  200
  
# Assuming that odds ratio of (male admitted)/(female admitted) is the
# same for both schools. Not sure how reasonable this is.

> mantelhaen.test(admit.status)

        Mantel-Haenszel chi-squared test with continuity correction

data:  admit.status 
Mantel-Haenszel X-squared = 27.9217, df = 1, p-value = 1.263e-07
alternative hypothesis: true common odds ratio is not equal to 1 
95 percent confidence interval:
 0.2288500 0.5153906 
sample estimates:
common odds ratio 
        0.3434343