Outline of Today’s Discussion 1.The Chi-Square Test of Independence 2.The Chi-Square Test of Goodness of Fit

Part 1 Chi-Square Test of Independence

Chi-Square: Independence 1.The chi-square is a non-parametric test - It’s NOT based on a mean, and it does not require that the data are bell-shaped (i.e., Gaussian distributed). 2.We can use the Chi-square test for analyzing data from certain between-subjects designs. Will someone remind us about between-subject versus within-subject designs? 3.Chi-square tests are appropriate for the analysis of categorical data (i.e., on a nominal scale).

Chi-Square: Independence 1.Sometimes a behavior can be described only in an all-or-none manner. 2.Example: Maybe a particular behavior was either observed or not observed. 3.Example: Maybe a participant either completed an assigned task, or did not. 4.Example: Maybe a participant either solved a designated problem, or did not.

Chi-Square: Independence 1.We can use the Chi-square to test data sets that simply reflect how frequently a particular category of behavior is observed. 2.The Chi-square test of independence is also called the two-way chi-square. 3.The Chi-square test of independence requires that two variables are assessed for each participant.

Chi-Square: Independence 1.The Chi-square test is based on a comparison between values that are observed (O), and values that would be expected (E) if the null hypothesis were true. 2.The null hypothesis would state that there is no relationship between the two variables, i.e., that the two variables are independent of each other. 3.The chi-square test allows us to determine if we should reject or retain the null hypothesis…

Chi-Square: Independence 1.To calculate the chi-square statistic, we need to develop a so-called “contingency table”. 2.In the contingency table, the levels of one variable are displayed across rows, and the levels of the other variable are displayed across columns. 3.Let’s see a simple 2 x 2 design…

Chi-Square: Independence Contingency Table: 2 rows by 2 columns The “marginal frequencies” are the row totals and column totals for each level of a particular variable.

Chi-Square: Independence 1.A “cell” in the table is defined as a unique combination of variables (e.g., dark hair, non-professional). 2.For each cell in the contingency table, we need to calculate the expected frequency. 3.To get the expected frequency for a cell, we use the following formula…

Chi-Square: Independence The expected (E) frequency of a cell. Example

Chi-Square: Independence Does everyone now understand where this 28 came from?

Chi-Square: Independence Here’s the Chi-square statistic. Let’s define the components…

Chi-Square: Independence Components of the Chi-Square Statistic

Chi-Square: Independence We’ll need one of these for each cell in our contingency table. Then, we’ll sum those up!

Chi-Square: Independence Check: Be sure to have one of these for each cell in your contingency table. We’ll reduce them, then sum them…

Chi-Square: Independence Finally, for each cell, reduce the parenthetical expression to a single number, and sum those up.

Chi-Square: Independence 1.After calculating the Chi-square statistic, we need to compare it to a “critical value” to determine whether to reject or accept the null hypothesis. 2.The critical value depends on the alpha level. What does the alpha level indicate, again? 3.The critical value also depends on the “degrees of freedom”, which is directly related to the number of levels in being tested…

Chi-Square: Independence Formula for the “degrees of freedom” In our example, we have 2 rows and 2 columns, so df = (2-1) (2-1) df = 1

Chi-Square: Independence 1.We will soon attempt to develop some intuitions about the “degrees of freedom” (df), and why they are important. 2.For now, we will simply compute the df so that we can determine the critical value. 3.For df = 1, and an alpha level of 0.05, what is the critical value? (see the hand-out showing the critical values table). 4.How does the critical value compare to the value of chi-square that we obtained (i.e., 6.43)? 5.So, what do we decide about the null hypothesis?

Chi-Square: Independence 1.In a way, the computations are somewhat similar to the various “r” statistics you’ve previously calculated. 2.However, we had not previously compared our “r” statistics to a critical value. So, we had not previously drawn any conclusions about statistical significance.

Chi-Square: Independence Under what circumstances would the expression that’s circled produce a zero?

Chi-Square: Independence In general, when the observed and expected values are very similar to each other, the chi-square statistic will be small (and we’ll likely retain the null hypothesis).

Chi-Square: Independence By contrast, when the observed and expected values are very different from each other, the chi-square statistic will be large (and we’ll likely reject the null hypothesis).

Chi-Square: Independence 1.The decision to reject or retain the null hypothesis depends, of course, not only on the chi-square value that we obtain, but also on the critical value. 2.Look at the critical values on the Chi-square table: What patterns do you see, and why do those patterns occur?

Part 2 Chi-Square Test: Goodness-of-Fit

Chi-Square: Goodness-of-Fit 1.Good news! The test for the goodness-of-fit is much simpler than that for independence! :-) 2.In the test for goodness-of-fit, each participant is categorized on ONLY ONE VARIABLE. 3.In the test for independence, participants were categorized on two different variables (i.e, affirmative action views and political party: OR say, hair color and career level).

Chi-Square: Goodness-of-Fit 1.The null hypothesis states that the expected frequencies will provide a “good fit” to the observed frequencies. 2.The expected frequencies depend on what the null hypothesis specifies about the population… 3.For example, the null hypothesis might state that all levels of the variable under investigation are equally likely in the population. 4.Example: Let’s consider the factors that go in to choosing a course for next semester…

Chi-Square: Goodness-of-Fit 1.Perhaps we’re identified 4 factors affecting course selection: Time, Instructor, Interest, Ease. 2.The null hypothesis might indicate that, in the population of Denison students, these four factors are equally likely to affect course selection. 3.If we sample 80 Denison students, the expected value for each category would be (80 / 4 categories = 20).

Chi-Square: Goodness-of-Fit 1.Let’s further assume that, after asking students to decide which of the 4 factors most affects their course selection, we obtain the following observed frequencies. 2.Time = 30; Instructor=10; Interest=22; Ease=18. 3.We now have the observed and expected values for all levels being examined…

Chi-Square: Goodness-of-Fit The chi-square computation is simpler than before, since we only have one variable (i.e., only one row).

Chi-Square: Goodness-of-Fit Calculating the degrees of freedom is also simpler than before. (C = # of columns = one for each level of the variable) df = C - 1

Chi-Square: Goodness-of-Fit 1.Let’s now evaluate the null hypothesis using the chi-square test for goodness-of-fit. Again the observed frequencies are… 2.Time = 30; Instructor=10; Interest=22; Ease=18. 3.The expected frequencies are 20 for each category (because the null hypothesis specifies that the four factors are equally likely to effect course selection in the population).

Chi-Square: Goodness-of-Fit 1.Note: There are two assumptions that underlie both chi-square tests. 2.First, each participant can contribute ONLY ONE response to the observed frequencies. 3.Second, each expected frequency must be at least 10 in the 2x2 case, or in the single variable case; each expected frequency must be at least 5 for designs that are 3x2 or higher.

Chi-Square: Goodness-of-Fit 1.Lastly, there are standards by which the chi-square statistics are to be reported, formally. 2.Statistics like this are to be reported in the Method section of an APA style report. 3.APA = American Psychological Association