Slides to accompany Weathington, Cunningham & Pittenger (2010), Chapter 16: Research with Categorical Data

Objectives Goodness-of-Fit test χ2 test of Independence χ2 test of Homogeneity Reporting χ2 Assumptions of χ2 Follow-up tests for χ2 McNemar Test

Background Sometimes we want to know how people fit into categories Typically involves nominal and ordinal scales Person only fits one classification The DV in this type of research is a frequency or count e.g., sex, age, year in college

Goodness-of-Fit Test Do frequencies of different categories match (fit) what would be hypothesized in a broader population? χ2 will be large if nonrandom difference between Oi and Ei If χ2 < critical value, distributions match Sum of differences of Observed and Expected frequencies, divided by Expected frequencies Nonsignificant chi square = Oi distribution matches Ei distribution fairly well If Ei are not known there are other techniques for comparing the Oi – one of the most common is to hypothesize that the frequencies will be equal across all conditions

Figure 16.1 Random sample of 600 marriage licenses from county courthouse In this case, expected proportions come from existing government records

Table 16.2

Calculation Example From Table 16.3 The chi-square value is < critical value, thus the distribution of observed marriage licenses appears to match/fit the national trends  there is no reason to assume differences between these trends are due to anything other than minor chance fluctuations

Another Example – Table 16.4 In this example, the E come from dividing the total number of observations by 4, assuming equal proportions across the 4 seasons (otherwise calculations are the same as the first example

Goodness-of-Fit Test χ2 is nondirectional (like F) Assumptions: Categories are mutually exclusive Conditions are exhaustive Observations are independent N is large enough Each participant fits only one category There is a category available for each person in the sample (ensures each participant is categorized) Independence = classification of participants into one category has no effect on other categories Try to keep Ei > 5 These assumptions also generally apply to the other types of χ2 tests discussed in this chapter

χ2 Test of Independence Are two categorical variables independent of each other? If so, Oij for one variable should have nothing to do with Eij for other variable and the difference between them will be 0. Ri = row total Cj = column totals T = total number of observations Eij = expected frequency for each cell if there is no relationship between the two variables If no relation then overall difference between O and E should be minimal Χ2 test requires calculating a ratio for each cell = number of rows (r) and number of columns (c) determine df = (r-1)(c-1) If observed Χ2 greater than critical value, you can reject Ho and infer that the two variables do correspond, and are not fully independent (related)

Table 16.5 Test of Independence example

Table 16.6

Computing χ2 Test Statistic This statistic exceeds the critical value for this test and df so we reject the null hypothesis and infer that the risk of suicide attempts corresponds with the experience of sexual abuse as a child  Recommend having the class generate this interpretation before you share this with them fully.

Interpreting χ2 Test of Independence Primary purpose is to identify independence If Ho retained, then we cannot assume the two variables are related (independence) If Ho rejected, the two variables are somehow related, but not necessarily cause-and-effect Perhaps the two variables are randomly distributed C-e depends on the variables and data collection method used

χ2 Test of Homogeneity Can be used to test cause-effect relationships Categories indicate level of change and χ2 statistic tests whether pattern of Oi deviates from chance levels If significant χ2, can assume c-e relation Type of therapy is a manipulated IV and the outcome is the condition at end of study  in this type of Χ2 design there is the basic structure of an experiment and not simply two variables that you are considering side by side. This type of Χ2 test requires equal n within each level of one variable, assessing the DV using some sort of outcome measure (here, effectiveness of treatment).

χ2 Test of Homogeneity Example Table 16.7 Calculation procedures are identical to those used for the test of independence  Recommend walking through them with the students in-class or asking them to do it as homework and hand it in for evaluation

Reporting χ2 Results Typical standard is to include the statistic, df, sample size, and significance levels at a minimum: χ2 (df, N = n) = #, p < α χ2(6, N = 240) = 23.46, p < .05

Follow-up Tests to χ2 Cramér’s coefficient phi (Φ) Indicates degree of association between two variables analyzed with χ2 Values between 0 and 1 Does not assume linear relationship between the variables Good estimate of magnitude of relationship, helpful for discussing practical significance.

Post-Hoc Tests to χ2 Standardized residual, e Converts differences between Oi and Ei to a statistic Shows relative difference between frequencies Highlights which cells represent statistically significant differences and which show chance findings Details on this more advanced post-hoc method are found at the end of Chapter 16 in the text – may be overly advanced for presentation in class, but is included in the text for reference if necessary or desired.

Follow-up Tests to χ2 McNemar Test For comparing correlated samples in a 2 x 2 table Table 16.9 illustrates  special form of χ2 test Ho: differences between groups are due to chance Example presented in text and Table 16.10 provides an application Details on this more advanced post-hoc method are found at the end of Chapter 16 in the text – may be overly advanced for presentation in class, but is included in the text for reference if necessary or desired.

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