Measures of Association

1
Measures of Association

• Political Science 102
• Introduction to Political Inquiry
• Lecture 20

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Why Use Measures of Association?

• Cross-tabs and scatter plots are flexible tools
  for exploring relationships between variables
• Chi-squared test evaluates statistical
  significance
• Neither method provides a summary measure of the
  relationship
• What is the direction?
• How strong is the relationship?
• Measures of Association seek to provide this
  information

3
Ordinal Linear Measures

• Coefficient compares pairs of cases record them
  as concordant, discordant, or tied
• Concordant case 1 is higher (or lower) than
  case 2 on both X and Y
• Discordant case 1 is lower than case 2 on X,
  but higher than case 2 on Y (or vice versa)
• Tied case 1 and case 2 are equal on either X,
  or Y, or both
• Positive coefficient indicates more concordant
  than discordant pairs negative coefficient
  indicates more discordant pairs than condordant

4
Ordinal Linear Measures

• Coefficients vary in how they weight and account
  for ties
• Gamma ignores ties (may ignore much of the data)
• Tau-b uses a weighted average of ties on X and Y
• All of these coefficients focus on linear
  relationships (or at least monotonic)
• Curvilinear and contingent relationships may be
  masked by these procedures

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Goodman Kruskals Gamma
C Concordant pairs D Discordant pairs
C 23 x 68 1,564 D 5 x 3 15 Tx (23 x 5)
(3 x 68) Ty (23 x 3) (5 x 68)
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Goodman Kruskals Gamma
C 23 x 68 1,564 D 5 x 3 15 Tx (23 x 5)
(3 x 68) 319 Ty (23 x 3) (5 x 68) 409
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Kendalls Tau-B
C 23 x 68 1,564 D 5 x 3 15 Tx (23 x 5)
(3 x 68) 319 Ty (23 x 3) (5 x 68) 409
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Linearity and the Limits of Gama and Tau-b
Level of Interest in Politics/Current
Ideology Events
Very Libe Liberal Moderate Conservat
Very Cons Total ---------------------------
-------------------------------------------------
---------- Not much interested 15
46 100 62 21 244
4.69 7.01
8.25 6.78 4.36 6.81
------------------------------------------------
-------------------------------------- Somewhat
Interested 62 232 461
272 96 1,123
19.38 35.37 38.04 29.73
19.92 31.32 -----------------------------
-------------------------------------------------
-------- Very Much Interested 243
378 651 581 365 2,218
75.94 57.62
53.71 63.50 75.73 61.87
------------------------------------------------
--------------------------------------
Total 320 656 1,212
915 482 3,585
100.00 100.00 100.00 100.00
100.00 100.00 Pearson chi2(8)
106.8563 Pr 0.000 gamma
0.0748 ASE 0.023 Kendall's tau-b
0.0466 ASE 0.014
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Correlation Coefficients

• For ratio data we can construct measures of
  association with more information about distance
  between categories
• Gamma and tau-b make only ordinal comparisons
• Analysts sought to construct summary statistic
  that would allow comparison of the strength of
  the relationship despite different units of
  measure
• The correlation coefficient!

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Origins of the Correlation Coefficient

• Analysts wanted to summarize how much changes in
  X and Y are associated with one another
• But X and Y are on different scales with
  different levels of variation
• Step 1 Measure the association of variation in X
  and Y by subtracting out the mean level of each
  variable

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The Origins of the Correlation Coefficient

• This formula focuses on deviations in X and Y,
  but X and Y are still measured in different units
• Solution Divide deviations in X and Y by their
  respective standard deviations
• Puts deviations in units of standard deviations

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Aspirations of the Correlation Coefficient

• Aims to be a unit-free measure of association
  that allows comparison of degrees of association
  across variables measured in different units
• It FAILS on all counts!
• Cannot compare correlation of X and Y to Z and Y
  because X and Z have different standard
  deviations
• The goal of unit-free comparisons is
  wrong-headed
• Cannot generalize correlations between the same
  variables across different samples because the
  standard deviations of the samples differ
• Instead of being universally comparable,
  correlations are universally incommensurable!

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What is the Solution?

• DONT use correlation coefficients to make
  generalizable claims about the association
  between variables
• Assess the strength of relationships by looking
  at
• Variation across categories in cross-tabs
• Difference of means or proportions tests
• Scatter plots
• Rely on chi-squared and t-tests for statistical
  significance
• Rely on regression analysis to summarize the
  strength of relationships between variables

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The Sample Dependence of Correlation Coefficients

• I created a dataset with these characteristics
• X1 varies from 20 to 20 where sx 10
• Y is defined as Y132X1e
• E is a random error term such that eN(0,20)
• Thus we KNOW the true relationship between X
  and Y
• We can change the sample to see if correlations
  are generalizable

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Correlation Coefficients Depend on the Sample
. corr y1 x1 (obs100) y1
x1 --------------------------- y1
1.0000 x1 0.6401 1.0000
Analysis of Full Sample
. corr y1 x1 if x1gt-10 x1lt10 (obs70)
y1 x1 ---------------------------
y1 1.0000 x1 0.4655 1.0000
Analysis of Restricted Variation in X
Correlation coefficient drops by 1/3 due to
arbitrary changes in the sample
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Regression Coefficients Are Generalizable
. reg y1 x1 Source SS df
MS Number of obs
100 ---------------------------------------
F( 1, 98) 68.03 Model
30717.8966 1 30717.8966 Prob gt
F 0.0000 Residual 44248.6995 98
451.517342 R-squared
0.4098 ---------------------------------------
Adj R-squared 0.4037 Total
74966.5961 99 757.238344 Root
MSE 21.249 ------------------------------
------------------------------------------------
y1 Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- x1 1.92611 .2335192
8.248 0.000 1.462699 2.389522 _cons
4.27945 2.130969 2.008 0.047
.0506116 8.508289 -----------------------------
-------------------------------------------------
Analyzing the Full Sample
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Regression Coefficients Are Generalizable
. reg y1 x1 if x1gt-10 x1lt10 Source
SS df MS Number of
obs 70 -----------------------------------
---- F( 1, 68) 18.81
Model 8038.90856 1 8038.90856
Prob gt F 0.0000 Residual 29063.7331
68 427.407839 R-squared
0.2167 ---------------------------------------
Adj R-squared 0.2051 Total
37102.6416 69 537.719444 Root
MSE 20.674 ------------------------------
------------------------------------------------
y1 Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- x1 1.971043 .4544842
4.337 0.000 1.064134 2.877952 _cons
5.884096 2.517394 2.337 0.022
.8607142 10.90748 -----------------------------
-------------------------------------------------
Analyzing the Restricted Sample