Statistical Analysis for Education and Psychology Researchers

The null hypothesis tested with this sampling design is that the proportions of subjects
in each category of the row variable, examination achievement, is the same in each of the
three independent samples (column variable).
We can conclude that there is no significant difference in examination achievements

between the three populations, of maternal employment, (with 4df). This is
not significant at the 5 per cent level.

Worked Example

Computation of an r×k χ^2 statistic follows the same procedure as that presented earlier in
this chapter when discussing the two-sample χ^2 test (section 6.1). The procedure is
outlined only briefly here. χ^2 is calculated as the difference between each observed and
corresponding expected cell count squared and then divided by the expected count
(O−E)^2 /E). The total χ^2 value for the contingency table is the sum of the χ^2 values
calculated for each cell in the contingency table. The degrees of freedom are calculated as
(rows−1×columns−1).

Interpretation

A non-significant χ^2 value would indicate either no significant difference between the k
populations with respect to the categorical response variable (homogeneity test) or no
significant association between the two categorical variables (test of independence).

Computer Analysis

Data from the study on maternal employment is used to illustrate an r×k Chi-square
analysis. The following lines of SAS code show how the data is entered and how the
procedure PROC FREQ is used:

data chi;

input row col celln @@;

cards; /* Data entered on

next line*/

1 1 19 1 2 32 1 3 10 2 19 2 2 10 2 3 5 3 1 13 3 2 23 3

3 15

proc freq data=chi;

weight celln;

table row* col/chisq expected;

title 'Chi square test';

run;

The option expected is used to obtain the expected frequencies for each cell in the
contingency table. Output from this SAS programme is shown in Figure 6.6:
Table of row by column
Row column^
Frequency^

Statistical analysis for education and psychology researchers 194