Lecture Notes

Chapter 12 The Chi Square Distribution

Unlike previous sampling distributions, the chi square distribution analyzes frequency counts not scores. You should use the t test and the ANOVA when you have scores. But, when you do not, you can make the same kind of logical inferences on frequencies by using the chi square distribution. The chi square can be used to test null hypothesis about "goodness of fit' or about independence.

Introduction
- Another sampling distribution
- Helps you make decisions about probabilities about frequency counts
- Discovered by Karl Pearson to make decisions about goodness of fit, but chi square has other uses too
The Chi Square Distribution
- Another family of curves (see Figure 12.1) whose shape changes with the df
- Unlike the t and F distributions, the critical values of chi square increase as the df increase
- Formula:
  - c² = S[(O - E)² / E]
Goodness of Fit
- Null hypothesis is that the actual data fit the expected data
- Thus, the H₀ is retained when the actual data and the expected data do not differ
  - in the case above, you are hoping NOT TO REJECT
- You may also be interested in demonstrating that data do not fit
- In such a case, you are hoping TO REJECT
- Example (p. 291)
  - Run an experiment to see who is "hired" men or women. You expect a 50:50 ratio (the null hypothesis) but you want to REJECT because you suspect discrimination
  - Data: the 120 Ss "hired" 75 men and 45 women. Is discrimination operating (can you reject null hypothesis?)
  - c² = S[(O - E)² / E]
  - c² = (75 - 60)² / 60 + (45 - 60)² / 60
  - c² = 3.75 + 3.75
  - c² = 7.50
  - df = # of categories (2) - 1 = 1
  - Critical value from Table = 6.64
  - Thus we reject, discrimination is an alternative hypothesis
Test of Independence
- Null hypothesis:
  - The variables in the table are independent
  - Rejecting the null hypothesis means that the variables are NOT INDEPENDENT
  - What you are really looking for is an interaction between the variables
- Degrees of freedom
  - df = (R - 1)(C - 1)
  - Where:
  - R = # of rows
  - C = # of columns
- Calculating expected frequencies
  - (Row Total)(Column Total) / N
- Example (p. 293 ff.)
  - Is there a relationship between attitudes about abortion and gender?
  - Null hypothesis:
    - there is no relationship, the two variables are independent
  - Data:

Notice how the expected values are calculated from the row totals, column totals, and N

c² = S[(O - E)² / E]

c² = (59 - 46.51)² / 46.51 + (15 - 27.49)² / 27.49 + (29 - 41.49)² / 41.49 + (27 - 24.51)² / 24.51

c² = 3.35 + 5.67 + 3.76 + 6.36

c² = 19.14

Shortcut for 2 x 2 Table
- For a 2 x 2 table you can use the following computational formula

Extending the Chi Square
- Chi squares can be used to test several variables at a time
- Because the chi square is additive, you may examine the source of significant effects
- The formula remains the same, simply decide how to find your expected values (i.e., goodness of fit, or independent)
Small Expected Frequencies and Yates's Correction
- In the past, small expected frequencies were subjected to Yates's correction (adding .5 to frequencies less than 5)
- Recent studies have shown such a step is usually unnecessary
- However, use large Ns to reduce the probability of committing Type II errors
Combining Categories
- Another strategy to avoid Type II errors is to combine categories to create larger frequency counts
- Again, because of the additive nature of the chi square, this is permissible
Requirements for Conducting a Chi Square
- Random sampling
- Data are frequency counts
- Each event must be independent
- Use for goodness of fit or test of independence

URLs
- Chi Square-Goodness of Fit--Text, discusses assumptions and use of this type of chi square
  - http://www-prophet.bbn.com/statguide/gf-dist.html
- Chi Square for Analyzing Contingency Tables--Text, example problem to test null hypothesis of indepencence
  - http://www.sct.edu/sct/departments/cs/classes/cs610/cs610chi.html
- Web Chi Square Calculator--Interactive, calculates chi squares for values entered
  - http://www.georgetown.edu/cball/webtools/web_chi.html
- Chi Square Tutorial--Text, contents page for teaching chi square
  - http://www.georgetown.edu/cball/webtools/web_chi_tut.html
- Chi Square Probabilities--Text, table of chi square critical values
  - http://www.richland.cc.il.us/james/lecture/m170/tbl-chi.html
- Using SYSTAT to Calculate Chi Square--Text, photos, explains how to calculate
  - http://research.med.umkc.edu/tlwbiostats/chi_hmwrk.html
- Chi Square: The Basic Statistical Method Used in Inheritance Studies--Text, HS level discussion and example
  - http://www.uswcl.ars.ag.gov/exper/chi_sqr.htm
- Quiz: Chi Square--Text, two problems, no answers given
  - http://neors.cat.cc.md.us/~dlinksz/q10.html
- Analysis of Frequency Data--Text, example of chi square problem
  - http://research.med.umkc.edu/tlwbiostats/chi_square.html

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