equality. Chi square will determine whether the observed pattern differs significantly from the daily expected 40.
8 CHI SQUARE FORMULA The formula for chi square is the summation for each cell : (O - E) E 2 Where: O = observed frequency - the data observed in our research/survey E = expected frequency, and = the summation over all the cells in the table = Chi 2
9 FORMAT OF CELL • Each cell follows the pattern: Observed Expected O – E (O - E) 2
10 EXAMPLE OF GOODNESS OF FIT observed 200 sick leave absences - the expected frequency in each cell must be 200/5 = 40 Monday Tuesday Wednesday Thursday Friday 64 40 29 40 15 40 20 40 72 40 24 576 11 121 25 625 20 400 32 1024 Chi square = (O - E) 2 = 576 + 121 + 625 + 400 + 1024 E 40 40 40 40 40 Chi square = 68.65 p < .01 i.e. a significant association between absence and particular days of week
11 INTERPRETATION OF GOODNESS OF FIT EXAMPLE • We can reject the null hypothesis with confidence, and accept the alternate hypothesis that sick leave is not randomly distributed through the week. • To specify how it is distributed, you must return to inspect the original data where you can readily appreciate that absences are much higher on Mondays and Fridays and much lower on other days of the week. I leave the interpretation and speculation of why to you!
12 INTERPRETATION OF GOODNESS OF FIT EXAMPLE • A chi square of zero indicates that the observed and expected frequencies match exactly. • Chi square can never be negative since differences between the observed and expected are always squared.
13 SPSS EXAMPLE OF GOODNESS OF FIT Is there any specific preference for one of three drinks? Null hypothesis claims any variation is simply random 1. Click on Analyze and select Nonparametric Tests from the drop-down menu. 2. Choose Chi-square ... which opens the Chi- Square Test dialogue box. 3. Select the variable (in this example ‘drink’) then click on the arrow button which transfers this variable to the Test Variable List : box. 4. Select OK . The results of the analysis are displayed in next slides.
14 SPSS Example
15 SPSS Output Equality of choice Actual choices Residuals are difference between observed and expected
16 SPSS Output Significant as p<.05
17 How to Interpret Output • The observed choice frequencies are presented in the second column. • The expected frequencies of cases are displayed in the third column. The expected frequency for each of the four drinks with 40 personal choices is 40/4, i.e. 10. • The residual column displays the differences between the observed and expected frequencies. • The second box presents the value of chi square, its degrees of freedom and its significance. Chi square is 8.4, its degrees of freedom are 3 (i.e. 4 choices - 1) and its significance level is 0.038. This indicates that there is a statistically significant deviation from the expected distribution of equality beyond p<.05. Coke is most popular while Solo and Sprite are significantly less preferred.
- Summer '14
- Chi-Square Test, Statistical hypothesis testing, Chi-square distribution, Pearson's chi-square test, Fisher's exact test, CHI SQUARE