Diff. Means and ANOVA

8/7/2019 Diff. Means and ANOVA

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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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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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4. Alpha level (.05) = we will reject the nullhypothesis if the P-value (sig. level) is lessthan .05


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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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Hypothesis

Whites will react less favorably to blackleaders who use the label African-American

instead of black.


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Simple 2-group posttest-only

Sample convenience sample fromMidwestern city; university students

R (black) MBLACK

R (A-A) MAFRICANAMERICAN


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*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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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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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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BSS = Between Sum of Squares =variation in Y due to differencesbetween groups

_ _

S (Yg Y)2


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WSS = Within Sum of Squares =variation in Y due to differences withingroups

_

S (Yig Yg)2


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Test statistic:

Fg-1, N-g = [BSS/(g-1)] / [WSS/(N-g)]

[Where g=# groups]


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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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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

Diff. Means and ANOVA

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