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Chi Square Test of Association

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					Survey of Statistical Methods

 We will be meeting in the
  Computer Lab today




                        April 27, 2005
      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
  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
                     Assumptions
            Chi 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.
               Post Hoc Strategy
   for 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!
    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
 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
                 Assumptions
        Chi 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)
                  SPSS Analysis
         Chi Square Test of Goodness of Fit

• Analyze –               X variable
                          goes here
  Nonparametric Tests –
  Chi Square                             If all expected
                                        frequencies are
                                        the same, click
                                             this box


                                          If all expected
                                         frequencies are
                                       not the same, enter
                                       the expected value
                                        for each division
                                                here
                   Examples
            (using the Montana.sav data)


         All                              All
expected frequencies              expected frequencies
    are the same                    are not the same
             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.
              Research Questions
1. Did an equal number of men and women reply to
   the questionnaire?
2. Were the respondents equally distributed across the
   state?
3. Do Montana residents perceive themselves to be
   financially better, worse, or the same as last year?
4. Are the perceptions the same for men and women?
5. Are the perceptions different depending on where
   you live?
6. Are the perceptions the same across income levels?
7. Are the perceptions the same for individuals who
   make under $20K vs. those that make over $35K?
           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
  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

				
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posted:8/28/2012
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