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Types of Bivariate Relationshipsand Associated Statistics
Nominal/Ordinal and Nominal/Ordinal (includingdichotomous) 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 themeans of two populations.
For example, What is the difference in the mean income for blacks
and whites?
What is the difference in the average defenseexpenditure level for Republican and Democraticpresidents?
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Difference of Means
Note that both of these questions areessentially asking if two variables (one ofwhich is interval and the other
dichotomous) are related to one another.
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Difference of Means
The null hypothesis for a difference ofmeans test is:
There is no difference in the mean of Y acrossgroups (m1-m2=0)
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Difference of Means
The alternative hypothesis for a differenceof means test is:
There is a difference in the mean of Y acrossgroups (m1-m20)
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Sampling Distribution for aDifference of Means
The sampling distribution for the difference oftwo means:
1. Is distributed normally (for large N)
2. Has mean m1-m2
3. We can determine the variance of the sampling
distribution of the difference of means (andthus the SE) from information about thepopulation variances (but we use samplevariances to estimate).
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Test Statistic for a Difference ofMeans
The test statistic (used to test the nullhypothesis) for the difference of twomeans (for independent samples) iscalculated as:
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Test Statistic for a Difference ofMeans
After calculating this test statistic, we candetermine the probability of observing a t-value at least this large, assuming thenull hypothesis is true (P-value/sig. level)
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Example: NES and 2000 Election
1. Null hypothesis: there was no differencein age between those who voted for Bushand 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 differenceof means = t statistic (t-test)
3. What would the sampling distributionlook like if the null hypothesis were true?(normal, mean of 0, and SE calculated byresearcher)
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Example: NES and 2000 Election
4. Alpha level (.05) = we will reject the nullhypothesis if the P-value (sig. level) is lessthan .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 sampledifference of at least .033 if in fact there is no differencein 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 blackleaders 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 fromMidwestern city; university students
R (black) MBLACK
R (A-A) MAFRICANAMERICAN
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Zilber and Niven (SSQ)
*p
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Example
NES 2004
Republican Party Feeling Thermometer(537)
Religious importance (51)
Talk Radio (78)
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Analysis of Variance
Purpose ANOVA is used to compare the means of >2groups
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 incomeamong racial/ethnic groups (black, white,
Hispanic, Asian)
Comparing the differences in feeling
thermometer scores for Bush amongRepublicans, Democrats, and Independents
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Analysis of Variance
Essentially, ANOVA partitions the totalvariance in Y (TSS) into two components.
TSS = Total sum of squares totalvariation in Y
_S (Yi Y)2
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Analysis of Variance
BSS = Between Sum of Squares =variation in Y due to differencesbetween groups
_ _
S (Yg Y)2
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Analysis of Variance
WSS = Within Sum of Squares =variation in Y due to differences withingroups
_
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 areequal), 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 than1) we reject the null hypothesis
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Analysis of Variance
Interpreting an ANOVA
We can also calculate a measure of thestrength of the relationship
Eta-squared = the proportion of variation in thedependent variable explained by the independentvariable
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ANOVA Examples
NES 2004
Republican Party Feeling Thermometer(537)
Educ Categ (4)
Religion (8)
Ideology (25)
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Bivariate Statistics
IndependentVariable
Dependent Variable
Nominal or Interval or
Dichotmous Ordinal Ratio
Nominal or
Dichotomous
Lamda,Chi-square,
Cramers V
Lamda,Chi-square,
Cramers V
Difference ofMeans Test (t-test)if IV dichotomous,
ANOVA if IV nominal
Ordinal
Lamda,
Chi-square,
Cramers V
Gamma,
Tau-b, Tau-c,
Somers D
Analysis of
Variance
(ANOVA)
Interval or
Ratio
Difference ofMeans Test (t-test)if DV dichotomous,
ANOVA if DV nominal
Analysis of
Variance
(ANOVA)
Correlation,
Regression
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Bivariate Statistics
IndependentVariable
Dependent Variable
Nominal or Interval or
Dichotmous Ordinal Ratio
Nominal or
Dichotomous
Lamda,Chi-square,
Cramers V
Lamda,Chi-square,
Cramers V
Difference ofMeans Test (t-test)if IV dichotomous,
ANOVA if IV nominal
Ordinal
Lamda,
Chi-square,
Cramers V
Gamma,
Tau-b, Tau-c,
Somers D
Analysis of
Variance
(ANOVA)
Interval or
Ratio
Difference ofMeans Test (t-test)if DV dichotomous,
ANOVA if DV nominal
Analysis of
Variance
(ANOVA)
Correlation,
Regression