PPA 415

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PPA 415 Research Methods in Public
Administration

• Lecture 8 Chi-square

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Introduction

• The chi-square (?2) has probably been the most
  frequently used test of hypothesis in the social
  sciences, largely because its assumptions are the
  easiest to satisfy.
• It assumes only that the sample is randomly
  selected and the variable is measured at the
  nominal level.
• No assumptions about shape of distribution or the
  sampling distribution.

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Bivariate Tables

• Bivariate tables are used to find if there is a
  significant relationship between the independent
  and dependent variable.
• Rows (dependent variables), columns (independent
  variables), and cells (independent by dependent
  variable).

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Bivariate Tables
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The Logic of Chi-Square

• In two-sample tests, independence means that the
  two samples are randomly selected.
• In chi-square, independence means that the
  classification of a case on one variable has no
  influence on how cases are classified on the
  other variable.

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The Logic of Chi-Square
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The Computation of Chi-Square

• Chi-square is calculated by comparing the
  observed frequency (fo) each cell to the expected
  frequency (fe)if there were no relationship.
• The larger the difference between observed and
  expected frequency, the larger is chi-square.

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The Computation of Chi-Square
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The Computation of Chi-Square

• Example

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Five-Step Hypothesis Test

• Step 1. Making assumptions.
• Independent random samples.
• Nominal level of measurement.
• Step 2. Stating the null hypothesis.
• H0 The two variables are independent.
• H1 The two variables are dependent (related).

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Five-Step Hypothesis Test

• Step 3. Selecting the sampling distribution and
  establishing the critical region.
• Sampling distribution ?2 distribution.
• Alpha0.05.
• Df(r-1)(c-1)(3-1)(2-1)2(1)2.
• ?2(critical)Appendix C, p. 4665.991.

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Five-Step Hypothesis Test

• Step 4.
• ?2(obtained)5.87 (from slide 9).
• Step 5. Making a decision.
• ?2(obtained) lt ?2(critical), therefore do not
  reject the null hypothesis that there is no
  relationship between presidential administration
  and disaster recommendations.

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Example 2
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Example 2

• Step 1. Making assumptions.
• Independent random samples.
• Nominal level of measurement.
• Step 2. Stating the null hypothesis.
• H0 The two variables are independent.
• H1 The two variables are dependent (related).

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Example 2

• Step 3. Selecting the sampling distribution and
  establishing the critical region.
• Sampling distribution ?2 distribution.
• Alpha0.05.
• Df(r-1)(c-1)(2-1)(3-1)1(2)2.
• ?2(critical)Appendix C, p. 4665.991.

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Example 2

• Step 4.

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Example 2

• Step 5. Making a decision.
• ?2(obtained) gt?2(critical), therefore reject the
  null hypothesis that there is no relationship
  between ethnicity and whether or not Birmingham
  residents have experienced discrimination.
  Minorities are significantly more likely to
  report that they have experienced discrimination
  than whites.

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Limitations of Chi-Square

• The chi-square becomes difficult to interpret if
  the table is larger than 4 x 4.
• When sample size is small, expected frequencies
  can be less than five per cell. If the table is
  2 x 2, use the following correction.

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Limitations of Chi-Square

• If table is more than 2 x 2, there is no
  correction for small sample size.
• Chi-square is also sensitive to large sample
  sizes. For example, doubling the sample size,
  doubles chi-square. For large samples, very
  small (and meaningless) differences can be
  significant. You should also calculate a measure
  of association. See the next lecture.