1. Chi-Square Analyses

2. Outline of Today’s Discussion
Please refrain from typing, surfing or printing during our conversation! 
Outline of Today’s Discussion
The Chi-Square Test of Independence – Introduction
The Chi-Square Test of Independence – Excel
The Chi-Square Test of Independence – SPSS
The Chi-Square Test for Goodness of Fit - Introduction
The Chi-Square Test for Goodness of Fit – Excel

3. Chi-Square Test of Independence
Part 1
Chi-Square Test of Independence
(Introduction)

4. Chi-Square: Independence
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).
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?
Chi-square tests are appropriate for the analysis of categorical data (i.e., on a nominal scale).

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

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

7. Chi-Square: Independence
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.
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.
The chi-square test allows us to determine if we should reject or retain the null hypothesis…

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

9. Chi-Square: Independence
Contingency Table: 2 rows by 2 columns
Political Party
City
Democrat Republican
Minneapolis
Atlanta
The “marginal frequencies” are the
row totals and column totals for each
level of a particular variable.

10. Chi-Square: Independence
A “cell” in the table is defined as a unique combination of variables (e.g., city, political party).
For each cell in the contingency table, we need to calculate the expected frequency.
To get the expected frequency for a cell, we use the following formula…

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

12. Chi-Square: Independence
Political Party
City
Democrat Republican
Minneapolis
Atlanta
Does everyone now
understand where this 28
came from?

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

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

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

16. 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…

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

18. Chi-Square: Independence
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.
The critical value depends on the alpha level. What does the alpha level indicate, again?
The critical value also depends on the “degrees of freedom”, which is directly related to the number of levels in being tested…

19. 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

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

21. Chi-Square: Independence
Congratulations! You’ve completed your first try at hypothesis testing!
In a way, the computations are somewhat similar to the various “r” statistics you’ve previously calculated.
However, we had not previously compared our “r” statistics to a critical value. So, we had not previously drawn any conclusions about statistical significance.
Questions so far?

22. Chi-Square: Independence
Before we move on, I’d like you to develop some intuitions about the computations…
Let’s look at a portion of the computation that you just completed, and really understand it…

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

24. 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).

25. 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).

26. Chi-Square: Independence
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.
Look at the critical values on the Chi-square table that was handed out. What patterns do you see, and why do those patterns occur?
Questions or comments?

27. Part 2
Chi-Square
Test of Independence
In Excel

28. Chi-Square Test of Independence
Part 3
Chi-Square Test of Independence
In SPSS

29. Chi-Square in SPSS
Here’s the sequence of steps for Chi-Square in SPSS.
Analyze --> Descriptive Statistics --> CrossTabs (yeah, it’s weird).
Select the two variables of interest by moving one into the ROWS box, and the other into COLUMNS box.
Statistics --> check off the chi-square
Cell display --> check off observed, expected, row & column
In the output, look for a large value of Pearson Chi square, we need “asymp sig (2 sided)” to be < 0.05, our alpha level.

30. Chi-Square in SPSS When the “asymp sig (2 sided)” value is < 0.05,
reject the null hypothesis.
In practice, there are 2 alpha levels:
There’s the criterion alpha level (usually 0.05),
and the observed alpha level (shown in SPSS output)

31. Chi-Square in SPSS
For a given degree-of-freedom level, there is an inverse relationship between the observed chi-square statistic and observed alpha level.
The higher the observed chi-square value, the smaller the observed alpha level, i.e., “sig” value.
Probability by Chance

32. Chi-Square in SPSS There is a low probability of large c2 values.
Probability by Chance
Large chi-square values are unlikely to occur just by chance,
So….large chi-square values correspond to low alpha levels.
(Note: Alpha levels are called “sig” values in SPSS)

33. Part 4
Chi-Square Test:
Goodness-of-Fit

34. Chi-Square: Goodness-of-Fit
Good news! The test for the goodness-of-fit is much simpler than that for independence! :-)
In the test for goodness-of-fit, each participant is categorized on ONLY ONE VARIABLE.
In the test for independence, participants were categorized on two different variables (i.e, city and political party)).

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

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

37. Chi-Square: Goodness-of-Fit
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.
Time = 30; Instructor=10; Interest=22; Ease=18.
We now have the observed and expected values for all levels being examined…

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

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

40. Chi-Square: Goodness-of-Fit
Let’s now evaluate the null hypothesis using the chi-square test for goodness-of-fit. Again the observed frequencies are…
Time = 30; Instructor=10; Interest=22; Ease=18.
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).

41. Chi-Square: Goodness-of-Fit
Note: There are two assumptions that underlie both chi-square tests.
First, each participant can contribute ONLY ONE response to the observed frequencies.
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.

42. Chi-Square: Goodness-of-Fit
Lastly, there are standards by which the chi-square statistics are to be reported, formally.
Statistics like this are to be reported in the Method section of an APA style report.
APA = American Psychological Association
p = 0.033

43. Chi-Square: Goodness-of-Fit
Memorize This!
All APA-style manuscripts
consist of the following sections,
in this order:
Abstract
Introduction
Method (singular, not not Methods)
Results
Discussion
References

44. Part 5
Chi-Square Test for
Goodness-of-Fit
In Excel

45. Goodness of Fit in Excel
We’ve already seen how we can use a Chi-square table to find the critical value (“the number to beat”).
We can also use the following Excel command:
=chiinv( probability, degrees of freedom)
where probability = criterion alpha level
i.e., 0.05 in most cases.
The output of “=Chiinv()” is the critical value
“the number to beat”

46. Goodness of Fit in Excel
We can also use Excel to find the observed alpha level, given an observed c2 value.
Here’ the Excel command: =chidist(c2 , degrees of freedom)
The output of “=Chidist()” is the observed alpha level (“sig value in SPSS”).
The observed alpha level must be less than 0.05 (or the criterion alpha level) to reject the null hypothesis.

47. Goodness of Fit in Excel
We can also use Excel to find the observed alpha level, given an observed c2 value.
Here’ the Excel command: =chidist(c2 , degrees of freedom)
The output of “=Chidist()” is the observed alpha level (“sig value in SPSS”).
The observed alpha level must be less than 0.05 (or the criterion alpha level) to reject the null hypothesis.