Association

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Association

• Relationships More Than You Ever Wanted To Know
  About Them!

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Three Types of Variable

• Dependent that which we are trying to explain
  (voting decisions, fiscal problems, political
  instability)
• Independent variables which might explain the
  dependent variables behavior (media,
  overspending, ethnic clashes)
• Control variables which might influence the
  relationship between the dependent and the
  independent variable)

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Analyzing Tables

• A useful convention place independent variable
  on columns
• Convert cell frequencies (raw numbers) into
  percentages
• Identify independent dependent variables
• Calculate s in direction of the independent
  variable

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Analyzing Tables, cont.

• Compare percentages in direction of the dependent
  variable
• Look for 5 or greater difference(s) across cells

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Controlling Crosstabulation Tables

• By holding constant
• Create a sub-table for each category of the
  control variable
• Compare percentages within sub-tables
• Compare percentages across sub-tables

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What We Are Measuring

• When two variables change (vary) with regards to
  each other in a predictable manner, they are said
  to co-vary or associate
• Association (co-variation) is also commonly
  referred to as correlation and relationship

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What We Are Measuring, cont.

• Association may be identified and measured by
• Constructing tables
• Cross tabulations
• Contingency tables
• Elaboration tables
• Calculating summary statistics, such as
• Cramers V
• Gamma
• Tau

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What We Are Measuring, cont.

• Association has two important properties
• Direction
• Strength

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Direction

• Positive or direct
• As one grows larger, the other grows larger
• Negative or Inverse
• As one grows larger, the other grows smaller
• No identifiable direction indicates that no
  association exists

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Strength

• Some associations are quite pronounced
• Others are rather weak
• Again, we use tables and/or summary statistics to
  measure it

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What We Are NOT Measuring Cause

• To INFER cause we must establish
• Association (covariation)
• Time sequence
• Plausible explanation
• Consistency with evidence
• No plausible competing explanations

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Crosstabulation

• Association and Controls

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Terms, etc.

• Table number
• Title
• Categories category labels
• Cell frequencies cell percentages
• Marginals (marginal totals)

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Analyzing Tables

• Is relatively easy with small tables (2 x 2)
• Becomes more difficult
• As tables grow larger (more rows and/or columns)
• As controls become more complex
• More categories
• More controls
• So, we turn to summary association measures for
  assistance

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Summary Statistics

• a.k.a. Summary Measures of Association
• Single numbers
• Summarize strength of association

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There Are Many Summary Association Measures

• Which to use depends upon
• Level of measure
• Technical considerations (number of rows
  columns, etc)

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Summary Association Desirable Characteristics

• Direction by sign
• Range from -1.0 to 1.0 (no negatives for
  nominal data)
• Ease of computation
• Readily interpreted
• Common usage
• Sensitivity to data (e.g. monotonicity)

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Summary Association Nominal Measures

• Chi-squared based measures
• Contingency Coefficient (square tables)
• Cramers V (larger tables)
• Phi (2 x 2) tables
• Proportional reduction in error (PRE) measures
• lambda

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Summary Association Ordinal Measures

• Concordance, discordance, ties
• PRE
• Gamma
• Somers D
• Kendalls Tau
• All others
• Spearmans Rho

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Positive
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Negative
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None
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Table 1
Family Income and Education
4
LT
12
1-3
12
Years
Coll
Yrs Col
LT 17.5
51.6
25.2
19.1
9.2
17.5 35.0
27.9
33.3
29.1
21.5
35.1 60.0
26.3
15.3
25.7
29.3
60.1
5.2
15.8
22.4
43.0
Total
405
739
700
651
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Nominal Statistics For Table 1
Chi-Square 421.851 DF 9
(Prob. 0.000)
V 0.237 C 0.380
Lambda (DV 7, income) 0.132 Lambda (DV 3,
education) 0.114
Lambda (symmetric) 0.123
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Ordinal Statistics For Table 1
Gamma 0.424
Dyx 0.322 (row variable is dependent)
Dxy 0.318 (column variable is dependent)
Tau-b 0.320 Tau-c 0.318
Prob. 0.000
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