Statistics for the Behavioral Sciences (5 th ed.) Gravetter & Wallnau Chapter 17 The Chi-Square Statistic: Tests for Goodness of Fit and Independence University of Guelph Psychology 3320 — Dr. K. Hennig Winter 2003 Term

Example: Table 17-1 (p. 578) The number of people littering or not littering depending on the amount of litter already on the ground.

Parametric and nonparametric tests compared Parametric tests Parametric tests test specific population parameters (  1 ≠  2 ) test specific population parameters (  1 ≠  2 ) “a numerical value that describes a population” (p. 5) “a numerical value that describes a population” (p. 5) assumptions about the population: normal, homogeneity of variances assumptions about the population: normal, homogeneity of variances require a numerical score (by definition - see above) require a numerical score (by definition - see above) Nonparametric tests (alternative) Nonparametric tests (alternative) hypotheses not stated in terms of a specific parameter hypotheses not stated in terms of a specific parameter make few assumptions re. populations, i.e., distribution- free tests make few assumptions re. populations, i.e., distribution- free tests use frequencies (most obvious difference) use frequencies (most obvious difference) less sensitive less sensitive

1. Chi-square goodness of fit Uses sample data to test hypotheses about the shape or proportions of a population distribution Uses sample data to test hypotheses about the shape or proportions of a population distribution Examples: Examples: to what extent are different ethnics groups represented in the population of your college or university? to what extent are different ethnics groups represented in the population of your college or university? of the three leading softdrinks, which is preferred by most Americans? of the three leading softdrinks, which is preferred by most Americans?

Figure 17-1 (p. 580) Distribution of eye colors for a sample of n = 40 individuals. The same frequency distribution is shown as a bar graph, as a table, and with the frequencies written in a series of boxes. Note. nominal scale used

The null hypothesis for goodness of fit No preference - No preference - No difference from a comparison population No difference from a comparison population Brand X Brand Y Brand Z Ho: 1/31/31/3 FavorOpposeHo:60%40%

The data Observed frequencies (∑/ 0 = n) Observed frequencies (∑/ 0 = n) A measure of the discrepancy between expected and observed frequencies A measure of the discrepancy between expected and observed frequencies Brand X Brand Y Brand Z n=40

Figure 17-2 (p. 584) Chi-square distributions are positively skewed. The critical region is placed in the extreme tail, which reflects large chi-square values.

The shape of the chi-square distribution for different values of df. As the number of categories increases, the peak (mode) of the distribution has a larger chi-square value. df = no. of free choices, df = C - 1 df = no. of free choices, df = C - 1 Brand X Brand Y Brand Z 1519??? n=40

A portion of the table of critical values for the chi- square distribution.

Abstract art: e.g., Fielding Step 1: state H o H 0 : in general population, no preference for orientation H 0 : in general population, no preference for orientation H 1 : one or more orientations is preferred in the general population H 1 : one or more orientations is preferred in the general population Top up (correct) Bottom up Left side up Right side up 25%25%25%25%

Step 2: Locate χ 2 (crit) = 7.81 (for alpha=.05). df = C - 1 = = 3 df = C - 1 = = 3

Calculate χ 2 (p. 588) f e = pn = ¼ (50) = 12.5 f e = pn = ¼ (50) = 12.5 reporting results χ 2 (3, n = 50) = 8.08, p <.05 reporting results χ 2 (3, n = 50) = 8.08, p <.05

II. Chi-square test for independence Color preferences according to personality types. Null Ho: two variables being measured are independent Null Ho: two variables being measured are independent H 0 (vers. 1): no relationship between color preference and personality (like a correlation) H 0 (vers. 1): no relationship between color preference and personality (like a correlation) H 0 (vers. 2): (treat as two different populations), the distribution of color is same for both catergories H 0 (vers. 2): (treat as two different populations), the distribution of color is same for both catergories Obtained:

Computing expected frequencies

Expected frequencies corresponding to the data in Table (This is the distribution predicted by the null hypothesis.)

Obtained:

Degrees of freedom and expected frequencies. (Once three values have been selected, all the remaining expected frequencies are determined by the row totals and the column totals. This example has only three free choices, so df = 3.) - df = (R - 1)(C - 1) = (2-1)(3-1) = 1*3 = 3

What to do when? Data set A: What is the appropriate analyses? Data set A: What is the appropriate analyses? ParticipantAcademicperformanceSelf-esteem A9431 B7826 C8127 D6523 ………

What to do when? What is the appropriate analysis where the dependent variable is self-esteem. What is the appropriate analysis where the dependent variable is self-esteem. independent-measures t-test independent-measures t-test Academic Performance LowHigh ……

What to do when? n = 150 n = 150 level of self-esteem Academic Perform’ highmedlow high low134334

Example - Table 17-8 (p. 596) level of self-esteem Academic Perform’ highmedlow high?????? low?????? n = 150

The expected frequencies (f e value) that would be predicted if academic performance and self-esteem were completely independent. vers. 1 - there is a relationship between academic performance and self-esteem vers. 2 - there is a significant difference between the two distributions

X 2 (2, n = 150) = 8.22 X 2 (2, n = 150) = 8.22 Computations

Assumptions & restrictions independence of observations (≠ independence of variables; e.g., violence and TV where participants come from same family) independence of observations (≠ independence of variables; e.g., violence and TV where participants come from same family) fe  5 fe  5 cell contribution to X 2 where difference =4 cell contribution to X 2 where difference =4 small cell size inflate small cell size inflate

Hypothetical data showing the first four individuals in a sample where each person is classified on two dichotomous variables. The original data (top of figure) can be reorganized into a form suitable for computing the phi-coefficient (lower left) or into a form suitable for computing a chi-square test for independence (lower right). Φ 2 = reports % of variance accounted for, same as r 2 -if > 2X2 use Cramer’s V

Table (p. 606) Standards for interpreting Cramér’s V as proposed by Cohen (1988).