We

1

• Well now consider contingency tables, a table
  which cross-tablulates two categorical variables.
• See Table 5.4.1 on page 163 for the notation used
  in contingency table analysis
• There are two cases in which this type of data
  arises
• a sample of size n is selected from a population
  and it is cross-tabulated with respect to both of
  the categorical variables
• a fixed number, ni, is selected from the
  population with respect to the ith row
  charactistic, i1,2,,r (r of rows) and then
  classified with respect to the column variable
• both of these cases can be handled with the same
  statistic
• The null hypothesis is that there is no
  association between the row and column variables,
  and even though the two cases above have null
  hypotheses that can be written in different ways,
  the same chi-square statistic is used in the same
  way to test it in both cases
• Lets consider these cases

2

• The expected cell proportion is
• and the row column proportions are
• For case 2, we define the conditional probability
  of column j given row i as
• Case 1 null hypothesis is that the two variables
  are independent of each other.
• Case 2 null hypothesis is that of row
  distribution homogeneity (i.e., for any column,
  the conditional probabilities from row to row are
  all the same)
• We call both cases a test of association between
  the row and column variables and we use the
  so-called chi-square statistic to test the
  hypothesis.

3

• The chi-square statistic is given as
• where is the expected
• frequency in cell (i,j). If the null
  hypothesis of no association between the row and
  column variable is true then this statistic has
  approximately a chi-square distribution with
• (r-1)(c-1) degrees of freedom. When large
  discrepancies exist between what we observe in
  the cells (nij) and what we would expect to see
  in the cells if the null hypothesis is true (eij)
  the chi-square statistic is large and leads to
  small p-values.
• This approximation is best applied when the
  expected cell frequencies are 5 or more. In case
  this is not true, we may use a permutation test
  based on the chi-square statistic.
• The chi-square distribution is tabulated in Table
  A7 on page 344 for various d.f. and various
  upper-tail probabilities...

4

• Lets look at how SAS handles contingency
  tables...
• first consider the organization of the data is
  it in raw form or already cross-tabulated as a
  contingency table?
• if its raw, then PROC FREQ will put the data
  into a table and count the number of observations
  in each cell
• if its already in tabular form, then you must
  use the WEIGHT statement and include a variable
  whose values are the counts in each cell. An
  example follows...
• Raw data
• data table5_4_2_raw
• input trtment satis _at__at_ datalines
• pp not pp not pp somewhat pp somewhat
• sa somewhat sa very sa very
• proc freq datatable5_4_2_raw
• tables trtmentsatis/chisq run quit
• Tabular data already cross-classified
• data table5_4_2
• input trtment satis count _at__at_
• datalines
• pp not 2 pp somewhat 2 pp very 0 sa not 0
• sa somewhat 1 sa very 2
• proc freq datatable 5_4_2
• tables trtmentsatis/chisq

5

• The above code gives the asymptotic p-value for
  the chi-square to get the permutation
  chi-square value, use the exact chisq statement.
  Or for large tables, when the computation time
  might be extensive, you may do random sampling of
  permutations as
• proc freq tables trtmentsatis exact chisq
• or
• proc freq tables trtmentsatis exact
  chisq/n5000
• For Tuesday Read section 5.4.1, especially
  considering the chi-square test in the two
  different formulations. Omit sections 2-4,
  except to do the SAS implementation of the
  permutation test... Do problems 8 and 9a on p.
  190-191. The next set of slides finishes Chapter
  5 with Fishers Exact Test, the Mantel-Haenszel
  Test, and McNemars Test (sections 5.5, 5.7, and
  5.8) so read ahead in these sections if you
  have some time.
• Try these for extra practice
• Injury Level None minimal minor major
• Seat Belt Yes 12,813 647 359 42
• No 65,963 4,000 2,642 303
• 1 2 3 4 5 6
• fair rolls 38 26 26 34 31 45
• biased rolls 12 4 17 17 18 32