Crosstabulation and Measures of Association

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Crosstabulation and Measures of Association
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• Investigating the relationship between two
  variables
• Generally a statistical relationship exists if
  the values of the observations for one variable
  are associated with the values of the
  observations for another variable
• Knowing that two variables are related allows us
  to make predictions.
• If we know the value of one, we can predict the
  value of the other.

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• Determining how the values of one variable are
  related to the values of another is one of the
  foundations of empirical science.
• In making such determinations we must consider
  the following features of the relationship.

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• 1.) The level of measurement of the variables.
  Difference varibles necessitate different
  procedures.
• 2.) The form of the relationship. We can ask if
  changes in X move in lockstep with changes in Y
  or if a more sophisticated relationship exists.
• 3.)The strength of the relationship. Is it
  possible that some levels of X will always be
  associated with certain levels of Y?

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• 4.) Numerical Summaries of the relationship.
  Social scientists strive to boil down the
  different aspects of a relationship to a single
  number that reveals the type and strength of the
  association.
• 5.) Conditional relationships. The variables X
  and Y may seem to be related in some fashion but
  appearances can be deceiving. Spuriousness for
  example. So we need to know if the introduction
  of any other variables into the analysis changes
  the relationship.

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Types of Association

• 1.) General Association simply associated in
  some way.
• 2.) Positive Monotonic Correlation when the
  variables have order (ordinal or continuous) high
  values of one var are associated with high values
  of the other. Converse is also true.
• 3.) Negative Monotonic Correlation Low values
  are associated with high values.

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Types of Association Cont.

• 4.) Positive Linear Association A particular
  type of positive monotonic relationship where the
  plotted values of X-Y fall on a straight line
  that slopes upward.
• 5.) Negative Linear Relaionship Straight line
  that slopes downward.

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Strength of Relationships

• Virtually no relationships between variables in
  Social Science (and largely in natural science as
  well) have a perfect form.
• As a result it makes sense to talk about the
  strength of relationships.

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Strength Cont.

• The strength of a relationship between variables
  can be found by simply looking at a graph of the
  data.
• If the values of X and Y are tied together
  tightly then the relationship is strong.
• If the X-Y points are spread out then the
  relationship is weak.

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Direction of Relationship

• We can also infer direction from a graph by
  simply observing how the values for our variables
  move across the graph.
• This is only true, however, when our variables
  are ordinal or continuous.

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Types of Bivariate Relationships and Associated
Statistics

• 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 Ratio
• 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.
• Crosstabs show values for variables categorized
  by another variable.
• They display the joint distribution of values of
  the variables by listing the categories for one
  along the x-axis and the other along the y-axis

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• Each case is then placed in a cell of the table
  that represents the combination of values that
  corresponds to its scores on the variables.

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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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Measures of Association for Crosstabulations

• These measures of association provide us with
  correlation coefficients that summarize data from
  a table into one number .
• This is extremely useful when dealing with
  several tables or very complex tables.
• These coefficients measure both the strength and
  direction of an association.

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Coefficients for Nominal Data

• When one or both of the variables are nominal,
  ordinal coefficients cannot be used because there
  is no underlying ordering.
• Instead we use PRE tests

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Lambda (PRE coefficient)

• PRE Proportional Reduction in Error
• Two Rules
• 1.) Make a prediction on the value of an
  observation in the absence of no prior
  information
• 2.) Given information on a second variable and
  take it into account in making the prediction.

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Lambda PRE

• If the two variables are associated then the use
  of rule two should lead to fewer errors in your
  predictions than rule one.
• How many fewer errors depends upon how closely
  the variables are associated.
• PRE (E1 E2) / E1
• Scale goes from 0 -1

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Lambda

• Lambda is a PRE coefficient and it relies on
  rules 1 2 above.
• When applying rule one all we have to go on is
  what proportion of the population fit into one
  category as opposed to another.
• So, without any other information, guessing that
  every observation is in the modal category would
  give you the best chance of getting the most
  correct.

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Why?

• Think of it like this. If you knew that I tended
  to make exams where the most often used answer
  was B, then, without any other information, you
  would be best served to pick B every time.

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• But, if you know information about each cases
  value on another variable, rule two directs you
  to only look at the members of that new category
  (variable) and find the modal category (only on
  that var).

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Example

• Suppose a sample of 100 voters and you need to
  predict how they will vote in the general
  election.
• Assume we know that overall 30 voted democrat
  and 30 voted republican and 40 were
  independent.
• Now suppose we take one person out of the group
  (John Smith), our best guess would be that he
  would vote independent.

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• Now suppose we take another person (Larry Mendez)
  and again we would assume he voted independent.
• As a result our best guess is to predict that all
  of the voters (all 100) were independent.
• We are sure to get some wrong but its the best
  we can do over the long run.

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• How many do we get wrong? 60.
• Suppose now that we know something about the
  voters regions (where they are from) and we know
  what proportions various regions voted in the
  election.
• NE-30 , MW 20, SO 30 , WE - 20

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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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Background for c2

• Statistical Independence A property of two
  variables in which the probability that an
  observation is in a particular category of on
  variable and also in a particular category of the
  other variable equals the simple or marginal
  probability of being in those categories.
• Plays a large role in data analysis
• Is another way to view the strength of a
  relaitionship

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Example

• Suppose we have two nominal or categorical
  variables, X and Y. We label the categories for
  the first category (a,b,c) and those of the
  second (r,s,t).
• Let P(X a) stand for the probability that a
  randomly selected case has property a on variable
  X and P(Y r) stand for the probability that a
  randomly selected case has property r on variable
  Y.

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• These two probabilities are called marginal
  distributions and simply refers to the chance
  that an observation has a particular value on a
  particular variable irrespective of its value on
  another variable.

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• Finally, let us assume that P(X a, Y r)
  stands for the joint probability that a randomly
  selected observation has both property a and
  property r simultaneously.
• Statistical Independence The two variables are
  therefore statisitically independent only if the
  chances of observing a combination of categories
  is equal to the marginal probability of choosing
  one category times the marginal probability of
  the other.

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Background for c2

• P(X a, Y r) P(X a) P(Y r)
• For example, if men are as likely to vote as
  women, then the two variables (gender and voter
  turnout) are statistically independent because
  the probability of observing a male nonvoter in
  the sample is equal to the probability of
  observing a male times the probability of
  obseving a nonvoter.

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Example

• If 100/300 are men 210/300 voted then
• The marginal probabilities are
• P(Xm)100/300 .33 and P(Yv) 210/300 .7
• .33 x .7 .23 and is our marginal probability

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• If we know that 70 of the voters are male and
  take that proportion and divide by the total
  number of voters (70/300) we also get .23.
• We can therefore say that the two variables are
  independent.

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• The chi-squared statistic essentially compares an
  observed result (the table produced by the
  sample) with a hypothetical table that would
  occur if (in the population) the variables were
  statistically independent.
• A value of 0 implies statistical independence
  which means no association.

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• Chi-squared increases as the departures of
  observed and expected values grows. There is no
  upper limit to how big the difference can become
  but if it is past a critical value then there is
  reason to reject the null hypothesis that the two
  variables are independent.

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How do we Calc. Chi2

• The observed frequencies are already in the
  crosstab.
• The expected frequencies in each table cell are
  found by multiplying the row and the column
  marginal totals and dividing by the sample size.

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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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• The value 9, is the expected frequency in the
  first cell of the table and is what we would
  expect in a sample of 100 (with 30 Republicans
  and 30 north easterners) if there is statistical
  independence in the population.
• This is more than we have in our sample so there
  is a difference.

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Just Like the Hyp. Test

• Null Statistical Independence between x and Y
• Alt X and Y are not independent.

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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