Bivariate Data Analysis I: Crosstabulation and Measures of Association

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Bivariate Data Analysis I Crosstabulation and
Measures of Association

• Chapter 12 (JRM)
• Chapters 9 (LeRoy)
• Exercises due Thursday, 4/9

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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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Assessing Relationships between Variables

• 1. Calculate appropriate statistic to measure the
  magnitude of the relationship in the sample
• 2. Calculate additional statistics to determine
  if the relationship holds for the population of
  interest (statistical significance)
• Substantive significance vs. Statistical
  significance

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What is a Crosstabulation?

• Crosstabulations are appropriate for examining
  relationships between variables that are nominal,
  ordinal, or dichotomous.

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What is a Crosstabulation?

• Example We would like to know if presidential
  vote choice in 2000 was related to race.
• Vote choice Gore or Bush
• Race White, Hispanic, Black

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Are Race and Vote Choice Related? Why?
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Are Race and Vote Choice Related? Why?
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Measures of Association for Crosstabulations

• Purpose to determine if nominal/ordinal
  variables are related in a crosstabulation
• At least one nominal variable
• Lamda
• Chi-Square
• Cramers V
• Two ordinal variables
• Tau
• Gamma

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Lamda
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Lamda Rule 1 (prediction based solely on
knowledge of marginal distribution of dependent
variable partisanship)
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Lamda Rule 2(prediction based on knowledge
provided by independent variable )
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Lamda Calculation of Errors

• Errors w/Rule 1 18 12 14 16 60
• Errors w/Rule 2 16 10 14 10 50
• Lamda (Errors R1 Errors R2)/Errors R1
• Lamda (60-50)/6010/60.17

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Lamda

• PRE measure
• Ranges from 0-1
• Potential problems with Lamda
• Underestimates relationship when variables (one
  or both) are highly skewed
• Always 0 when modal category of Y is the same
  across all categories of X

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Chi Square (c2)

• Also appropriate for any crosstabulation with at
  least one nominal variable (and another
  nominal/ordinal variable)
• Based on the difference between the empirically
  observed crosstab and what we would expect to
  observe if the two variables are statistically
  independent

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Chi Square (c2)
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Calculating Expected Frequencies

• To calculate the expected cell frequency for NE
  Republicans
• E/30 30/100, therefore E(3030)/100 9

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Calculating the Chi-Square Statistic

• The chi-square statistic is calculated as
• ? (Obs. Frequencyik - Exp. Frequencyik)2 / Exp.
  Frequencyik
• (25/9)(16/6)(9/9)(16/6)(0)(0)(16/12)(16/8)
  (25/9)16/6)(1/9)(0) 18

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Interpreting the Chi-Square Statistic

• The Chi-Square statistic ranges from 0 to
  infinity
• 0 perfect statistical independence
• Even though two variables may be statistically
  independent in the population, in a sample the
  Chi-Square statistic may be gt 0
• Therefore it is necessary to determine
  statistical significance for a Chi-Square
  statistic (given a certain level of confidence)

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Cramers V

• Problem with Chi-Square not comparable across
  different sample sizes (and their associated
  crosstab)
• Cramers V is a standardization of the Chi-Square
  statistic

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Calculating Cramers V

• V
• Where R rows and C columns
• V ranges from 0-1
• Example (region and partisanship)
• v.09 .30

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Relationships between Ordinal Variables

• There are several measures of association
  appropriate for relationships between ordinal
  variables
• Gamma, Tau-b, Tau-c, Somers d
• All are based on identifying concordant,
  discordant, and tied pairs of observations

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Concordant PairsIdeology and Voting

• Ideology - conserv (1), moderate (2), liberal (3)
• Voting - never (1), sometimes (2), often (3)
• Consider two hypothetical individuals in the
  sample with scores
• Individual A Ideology1, Voting1
• Individual B Ideology2, Voting2
• Pair AB are considered a concordant pair because
  Bs ideology score is greater than As score, and
  Bs voting score is greater than As score

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Concordant Pairs (contd)

• All of the following are concordant pairs
• A(1,1) B(2,2)
• A(1,1) B(2,3)
• A(1,1) B(3,2)
• A(1,2) B(2,3)
• A(2,2) B(3,3)
• Concordant pairs are consistent with a positive
  relationship between the IV and the DV (ideology
  and voting)

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Discordant Pairs

• All of the following are discordant pairs
• A(1,2) B(2,1)
• A(1,3) B(2,2)
• A(2,2) B(3,1)
• A(1,2) B(3,1)
• A(3,1) B(1,2)
• Discordant pairs are consistent with a negative
  relationship between the IV and the DV (ideology
  and voting)

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Identifying Concordant Pairs

• Concordant Pairs for Never - Conserv (1,1)
• Concordant 8070 8010 8020 8080
  
• 14,400

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Identifying Concordant Pairs

• Concordant Pairs for Never - Moderate (1,2)
• Concordant 1010 1080 900

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Identifying Discordant Pairs

• Discordant Pairs for Often - Conserv (1,3)
• Discordant 010 010 070 010 0

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Identifying Discordant Pairs

• Discordant Pairs for Often - Moderate (2,3)
• Discordant 2010 2010

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Gamma

• Gamma is calculated by identifying all possible
  pairs of individuals in the sample and
  determining if they are concordant or discordant
• Gamma (C - D) / (C D)

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Interpreting Gamma

• Gamma 21400/24400 .88
• Gamma ranges from -1 to 1
• Gamma does not account for tied pairs
• Tau (b and c) and Somers d account for tied
  pairs in different ways

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Square tables
Non-Square tables
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Example

• NES 2004 What explains variation in ones
  political Ideology?
• Income?
• Education?
• Religion?
• Race?

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Bivariate Relationships and Hypothesis Testing
(Significance Testing)

• 1. Determine the null and alternative hypotheses
• Null There is no relationship between X and Y (X
  and Y are statistically independent and test
  statistic 0).
• Alternative There IS a relationship between X
  and Y (test statistic does not equal 0).

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Bivariate Relationships and Hypothesis Testing

• 2. Determine Appropriate Test Statistic (based on
  measurement levels of X and Y)
• 3. Identify the type of sampling distribution for
  test statistic, and what it would look like if
  the null hypothesis were true.

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Bivariate Relationships and Hypothesis Testing

• 4. Calculate the test statistic from the sample
  data and determine the probability of observing a
  test statistic this large (in absolute terms) if
  the null hypothesis is true.
• P-value (significance level) probability of
  observing a test statistic at least as large as
  our observed test statistic, if in fact the null
  hypothesis is true

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Bivariate Relationships and Hypothesis Testing

• 5. Choose an alpha level a decision rule to
  guide us in determining which values of the
  p-value lead us to reject/not reject the null
  hypothesis
• When the p-value is extremely small, we reject
  the null hypothesis (why?). The relationship is
  deemed statistically significant,
• When the p-value is not small, we do not reject
  the null hypothesis (why?). The relationship is
  deemed statistically insignificant.
• Most common alpha level .05

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Bottom Line

• Assuming we will always use an alpha level of
  .05
• Reject the null hypothesis if P-valuelt.05
• Do not reject the null hypothesis if P-valuegt.05

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An Example

• Dependent variable Vote Choice in 2000
• (Gore, Bush, Nader)
• Independent variable Ideology
• (liberal, moderate, conservative)

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An Example

• 1. Determine the null and alternative hypotheses.

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An Example

• Null Hypothesis There is no relationship between
  ideology and vote choice in 2000.
• Alternative (Research) Hypothesis There is a
  relationship between ideology and vote choice
  (liberals were more likely to vote for Gore,
  while conservatives were more likely to vote for
  Bush).

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An Example

• 2. Determine Appropriate Test Statistic (based on
  measurement levels of X and Y)
• 3. Identify the type of sampling distribution for
  test statistic, and what it would look like if
  the null hypothesis were true.

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Sampling Distributions for the Chi-Squared
Statistic(under assumption of perfect
independence)df (rows-1)(columns-1)
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Bivariate Relationships and Hypothesis Testing

• 4. Calculate the test statistic from the sample
  data and determine the probability of observing a
  test statistic this large (in absolute terms) if
  the null hypothesis is true.
• P-value (significance level) probability of
  observing a test statistic at least as large as
  our observed test statistic, if in fact the null
  hypothesis is true

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Bivariate Relationships and Hypothesis Testing

• 5. Choose an alpha level a decision rule to
  guide us in determining which values of the
  p-value lead us to reject/not reject the null
  hypothesis
• When the p-value is extremely small, we reject
  the null hypothesis (why?). The relationship is
  deemed statistically significant,
• When the p-value is not small, we do not reject
  the null hypothesis (why?). The relationship is
  deemed statistically insignificant.
• Most common alpha level .05

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In-Class Exercise

• For some years now, political commentators have
  cited the importance of a gender gap in
  explaining election outcomes. What is the source
  of the gender gap?
• Develop a simple theory and corresponding
  hypothesis (where gender is the independent
  variable) which seeks to explain the source of
  the gender gap.
• Specifically, determine
• Theory
• Null and research hypothesis
• Test statistic for a cross-tabulation to test
  your hypothesis