Survey of Statistical Methods We will be meeting in the Computer Lab today

1. Survey of Statistical MethodsWe will be meeting in the Computer Lab today April 27, 2005

2. Chi Square Test of Independence • Purpose • To determine if two variables of interest independent (not related) or are related (dependent)? • When the variables are independent, we are saying that knowledge of one gives us no information about the other variable. When they are dependent, we are saying that knowledge of one variable is helpful in predicting the value of the other variable. • The chi-square test of independence is a test of the influence or impact that a subject’s value on one variable has on the same subject’s value for a second variable. • Some examples where one might use the chi-squared test of independence are: • Is level of education related to level of income? • Is the level of price related to the level of quality in production? • Is one party affiliation related to the person's preferred television network? • Hypotheses • The null hypothesis is that the two variables are independent. This will be true if the observed counts in the sample are similar to the expected counts. • H0: X and Y are independent • H1: X and Y are dependent

3. Chi Square Test of Independence • Wording of Research questions • Are X and Y independent? • Are X and Y related? • The research hypothesis states that the two variables are dependent or related. This will be true if the observed counts for the categories of the variables in the sample are different from the expected counts. • Level of Measurement • Both X and Y are categorical

4. AssumptionsChi Square Test of Independence • Each subject contributes data to only one cell • Finite values • Observations must be grouped in categories. No assumption is made about level of data. Nominal, ordinal, or interval data may be used with chi-square tests. • A sufficiently large sample size • In general N > 20. • No one accepted cutoff – the general rules are • No cells with observed frequency = 0 • No cells with the expected frequency < 5 • Applying chi-square to small samples exposes the researcher to an unacceptable rate of Type II errors. Note: chi-square must be calculated on actual count data, not substituting percentages, which would have the effect of pretending the sample size is 100.

5. Post Hoc Strategyfor the Chi Square Test of Independence • If there is at least one cell with a significant standardized residual • Formulate your conclusion based on a comparison of all of the cells containing significant standardized residuals. • If none of the cells have a significant standardized residual • Interpret the findings based on a comparison of the ‘sign (+ or -)’ of the largest values for the standardized residuals. • Apply caution when this is the case!

6. Chi Square Test of Goodness of Fit • Purpose • To determine whether an observed frequency distribution departs significantly from a hypothesized frequency distribution. • This test is sometimes called a One-sample Chi Square Test • Hypotheses • The null hypothesis is that the two variables are independent. This will be true if the observed counts in the sample are similar to the expected counts. • H0: X follows the hypothesized distribution • H1: X deviates from the hypothesized distribution

7. Chi Square Test of Goodness of Fit • Sample Research Questions • Do students buy more Coke, Gatoraide or Coffee at the CHS coffee cart? • Does my sample contain a disproportionate amount of Hispanics as compared to the population of the county from which they were sampled? • Has the ethnic composition of the city of Ithaca changed since 1990? • Level of Measurement • X is categorical

8. AssumptionsChi Square Test of Goodness of Fit • The research question involves the comparison of the observed frequency of one categorical variable within a sample to the expected frequency of that variable. • The observed and theoretical distributions must contain the same divisions (i.e. ‘levels’ or ‘classes’) • The expected frequency in each division must be >5 • There must be a sufficient sample (in general N>20)

9. SPSS AnalysisChi Square Test of Goodness of Fit X variable goes here • Analyze – Nonparametric Tests – Chi Square If all expected frequencies are the same, click this box If all expectedfrequenciesare not the same, enter the expected value for each division here

10. Examples(using the Montana.sav data) All expected frequencies are the same All expectedfrequenciesare not the same

11. Sample Problem • A researcher in Montana wanted to determine if people in Montana perceived themselves to be financially better, worse or the same as they were last year. A random sample of 400 residents were sent a questionnaire at the end of April in 1999. A total of 209 complete surveys were returned to the researcher.

12. Research Questions • Did an equal number of men and women reply to the questionnaire? • Were the respondents equally distributed across the state? • Do Montana residents perceive themselves to be financially better, worse, or the same as last year? • Are the perceptions the same for men and women? • Are the perceptions different depending on where you live? • Are the perceptions the same across income levels? • Are the perceptions the same for individuals who make under $20K vs. those that make over $35K?

13. Task for Class Today(due at the end of class today) • You can work by yourself or with one other person (if you work with someone, you must sit next to each other) • I will answer any questions you might have, however, if you ask a question that I have already answered, I will refer you to the person who asked it first to answer • Write your answers on the ‘assignment sheet’ provided in class • Print out and attach the graph you have created • You need to hand in the assignment to me before you leave

15. Are there more Chi Square Tests? • X2 Test for Independence • X2 Test for Goodness of Fit • YES! Here are a few… • X2 Test for a Population Variance • X2 Test for an Assumed Population Variance • X2 Test for Compatibility of K Counts • X2 Test for Consistency in a 2 X 2 Table • X2 Test for Consistency in a K X 2 Table • X2 Test for Consistency in a 2 X K Table • X2 Test for a Suitable Probabilistic Model