Types of Bivariate Relationships and Associated Statistics

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Types of Bivariate Relationships and Associated
Statistics

• Nominal/Ordinal and Nominal/Ordinal (including
  dichotomous)
• Crosstabulation (Lamda, Chi-Square Gamma, etc.)
• Interval and Dichotomous
• Difference of means test
• Interval and Nominal/Ordinal
• Analysis of Variance
• Interval and Interval
• Regression and correlation

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Difference of Means

• Often, we are interested in the difference in the
  means of two populations.
• For example,
• What is the difference in the mean income for
  blacks and whites?
• What is the difference in the average defense
  expenditure level for Republican and Democratic
  presidents?

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Difference of Means

• Note that both of these questions are essentially
  asking if two variables (one of which is interval
  and the other dichotomous) are related to one
  another.

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Difference of Means

• The null hypothesis for a difference of means
  test is
• There is no difference in the mean of Y across
  groups.

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Sampling Distribution for a Difference of Means

• The sampling distribution for the difference of
  two means
• Is distributed normally
• Has mean m1-m2
• 3. We can determine the variance of the sampling
  distribution of the difference of means (and thus
  the SE) from information about the population
  variances.

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Test Statistic for a Difference of Means

• The test statistic (used to test the null
  hypothesis) for the difference of two means (for
  independent samples) is calculated as
• 
  

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Test Statistic for a Difference of Means

• After calculating this test statistic, we can
  determine the probability of observing a t-value
  at least this large, assuming the null hypothesis
  is true (P-value/sig. level)

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Example NES and 2000 Election

• 1. Null hypothesis there was no difference in
  age between those who voted for Bush and those
  who voted for Gore (alternative hypothesis there
  WAS a difference)

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Example NES and 2000 Election

• 2. Appropriate test statistic for difference of
  means t statistic (t-test)
• 3. What would the sampling distribution look
  like if the null hypothesis were true? (normal,
  mean of 0, and SE calculated by researcher)

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Example NES and 2000 Election

• 4. Alpha level (.05) we will reject the null
  hypothesis if the P-value (sig. level) is less
  than .05

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Example NES and 2000 Election

• 5. Calculate test statistic
• Mean for Gore voters 49.63
• Mean for Bush voters 49.60
• Difference .033
• SE .98
• T-statistic 0.0337
• P-value 0.9732 (the probability of obtaining a
  sample difference of at least .033 if in fact
  there is no difference in the population)
• Conclusion ???

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Zilber and Niven (SSQ)
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Zilber and Niven (SSQ)

• Hypothesis
• Whites will react less favorably to black leaders
  who use the label African-American instead of
  black.

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Zilber and Niven (SSQ)

• Simple 2-group posttest-only
• Sample convenience sample from Midwestern city
  university students
• R (black) MBLACK
• R (A-A) MAFRICANAMERICAN

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Zilber and Niven (SSQ)
plt.05 plt.10
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Examples

• NES 2000
• GWB Feeling Thermometer (147)
• Death Penalty (92)
• Gay/Lesbian Adoption (95)
• English Only (90)
• School Vouchers (89)
• Late-Term Abortion (43)

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Analysis of Variance

• Purpose ANOVA is used to compare the means of
  gt2 groups
• More specifically, ANOVA is used to test
• Null Hypothesis m1 m2 m3 ... mg
• against
• Alternative Hypothesis At least one mean is
  different

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Analysis of Variance

• Examples
• Comparing the differences in mean income among
  racial/ethnic groups (black, white, Hispanic,
  Asian)
• Comparing the differences in feeling thermometer
  scores for Bush among Republicans, Democrats, and
  Independents

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Analysis of Variance

• Essentially, ANOVA partitions the total variance
  in Y (TSS) into two components.
• TSS Total sum of squares total variation in Y
• _
• S (Yi Y)2

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Analysis of Variance

• BSS Between Sum of Squares variation in Y due
  to differences between groups
• _ _
• S (Yg Y)2

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Analysis of Variance

• WSS Within Sum of Squares variation in Y due
  to differences within groups
• _
• S (Yig Yg)2

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Analysis of Variance

• Test statistic
• Fg-1, N-g BSS/(g-1) / WSS/(N-g)
• Where g groups

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Analysis of Variance

• Interpreting an ANOVA
• If the null hypothesis is true (i.e. all means
  are equal), the F-statistic will be equal to 1
  (in the population)
• If the F-statistic is judged to be statistically
  significant (and thus sufficiently greater than
  1) we reject the null hypothesis

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Analysis of Variance

• Interpreting an ANOVA
• We can also calculate a measure of the strength
  of the relationship
• Eta-squared the proportion of variation in the
  dependent variable explained by the independent
  variable

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ANOVA Examples

• NES 2000
• GWB Feeling Thermometer (147)
• Educ Categ (4)
• Religion (17)