July, 2000Guang Jin Statistics in Applied Science and Technology Chapter 12. The Chi-Square Test

July, 2000Guang Jin Key Concepts in this Chapter Assumption involved in  2 test Observed frequency Expected frequency  2 statistics and its distribution Critical  2 value.

July, 2000Guang Jin Rational for the Chi-Square Test The chi-square test is probably the most frequently used test of significance in the health science. This is partly due to the fact most of health science data are not interval-ratio, rather nominal or ordinal. Chi-square test deals with nominal or ordinal variables. Instead of deal with individual score/category, it deals with frequencies in each category.

July, 2000Guang Jin The Basics of a Chi-Square Test The chi-square test compares the observed frequency with the expected frequencies. The expected frequency is calculated from some hypothesis. The chi-square test, is another kind of test of significance. As indicated by it name, the test statistics is chi-square and is approximated well by the Chi-square distribution if the sample sizes and expected numbers are not too small.

July, 2000Guang Jin The Basics of a Chi-Square Test There is, in fact, a family of chi-square distributions. The one to use depends, as in the t or F distribution, on degrees of freedom (df).

July, 2000Guang Jin Application of chi-square test Three types of problems can be addressed using Chi-square test. –Independence (if any) between the two variables (focus of our course) –Whether various subgroups are homogeneous –Whether there is a significant difference in the proportions in the subclasses among the subgroups

July, 2000Guang Jin Test of Independence between two variables (I) Review of Assumptions: –Independent random samples –Level of measurement is nominal –Expected frequencies must not be too small no expected frequency should be less than 1 and no more than 20% of the cells should have an expected frequency of less than 5 merge (“collapse”) some rows or columns if needed.

July, 2000Guang Jin Test of Independence between two variables (II) Stating the hypothesis/research question: –H 0 : The two variables are independent –H 1 : The two variables are dependent (are associated) (Did this remind you anything when we discuss about measure of association between two nominal level variables?)

July, 2000Guang Jin Test of Independence between two variables (III) Test statistics: chi-square (  2 ) with a distribution called  2 distribution. Decision rule: Reject H 0 if  2 fall into the rejection/critical region.

July, 2000Guang Jin Computation of test statistics To calculate  2, you should use the following equation: To find out critical  2, use Table 12.2 (textbook, pg179) and df can be found by: Where: O - observed frequency E - expected frequency df = (r-1)(c-1) Where: r - number of rows c - number of columns

July, 2000Guang Jin Test of Homogeneity and Test of Significance of the Difference between two proportions For the above two tests, the same steps of hypothesis testing should be followed. The only difference here is the calculation of expected frequency is different due to the fact that we now have a different hypothesis.

July, 2000Guang Jin Two-by-Two Contingency Tables (I) The most common chi-square test used in health research involves data presented in a 2 x 2 table in which there are two groups and two possible responses.  2 can be directly calculated from this table.

July, 2000Guang Jin Two-by-Two Contingency Tables (II)

July, 2000Guang Jin Chi-square calculation Based on the notation from previous two-by two contingency table,  2 can be directly calculated according to Disregard the Yate’s correction on text book Pg. 186 &187.

July, 2000Guang Jin Limitation of Chi-Square Test The test statistics approximate the continuous chi- square distribution when the expected frequencies must not be too small. What is “small”? - a general, well-accepted rule is that no expected frequency should be less than 1 and not more than 20% of the cells should have an expected frequency of less than 5.