Measures of Association

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Measures of Association

• Categorical Variables

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Today we will discuss

• Purpose of Measures of Association with
  Categorical Variables
• Different Measures of Association
• When to use
• How to calculate
• How to interpret
• The Different Measures of Association
• Lambda
• Yules Q
• Goodman and Kruskals Gamma
• Chi Square

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Purpose of Measures

• To determine the strength and sometimes the
  direction of relationship between variables.
• Choice of Measures May Depend on
• Level of measurement
• Number of categories in variables
• What you want to know about the relationship
  between the variables

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Lambda

• Used with nominal level variables
• Based on Proportionate Reduction in Error (PRE)
• How much error would be reduced in predicting the
  distribution of the DV, if knew the distribution
  of IV compared to error if have no knowledge
  about IV.
• Interpreting Lambda
• Ranges from 0-1
• 0 means no reduction in error
• 1 means totally reduce PRE if knew IV
  distribution
• Most scores are in between range of 0-1, a .30 or
  better is a good association.

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Calculating Lambda

• Information Needed
• Number of errors knowing distribution of DV only
• Number of fewer errors knowing the distribution
  of DV within categories of IV
• To calculate
• Lambdaitem (2) above/ item (1) above

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Yules Q

• Use in a 2x2 Table
• Indicates strength and direction of association
  between variables
• Interpreting Yules Q
• Range for positive association (0.01) to (1.00)
• Range for negative association (-0.01) to (-1.00)
• A zero indicates no association
• SEE Box on page 401 of Baker to Interpret in words

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Calculating Yules Q

• Arranging Categories of Tables
• For Ordinal Variables arrange as follows

Yules Q ad-bc/adbc
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Goodman and Kruskals Gamma

• Extension of Yules Q
• Used when Table is larger than 2X2
• Interpreted in the same manner
• Especially useful with Ordinal Variables
• Table set-up for ordinal variables
• Extend the set up for Yules Q
• Columns for IV should go from Hi to Lowest value
• Rows with DV should go from Hi to lowest value

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Chi Square

• Use when variables are nominal or ordinal
• Most commonly used test of significance
• Tests of Independence
• Chi square (?2 ) measure if the relationship
  between the variables differs significantly from
  the model of independence or chance.
• Interpretation of (?2 )
• The chances that the observed relationship
  between the variables would occur by chance.
• Examine in combination with other measures of
  association to determine if relationship is
  statistically significant
• Does not speak to direction of association or
  that one causes the other merely that the chances
  of observing such a value of (?2 ) .

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Calculating Chi Square

• Using the following table

a expected frequencies P1n1 b expected
frequencies P2n1 c expected frequencies
P1n2 d expected frequencies P2n2
?2 (observed frequencies expected
frequencies)2 expected frequencies
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Significance of Chi Square

• To determine probability that the value of ?2 by
  chance must look it up in a chi square table (see
  p472 of Baker text)
• Steps
• Calculate ?2
• Calculate Degrees of Freedom for Table
• DF (r-1)(c-1)
• Look up in Chi Square table to see the minimum
  value of ?2 must achieve to assure less .05
  probability that a chi square of the value you
  have occurred by chance.

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Using SPSS

• All of the measures of association we have
  discussed are calculated by SPSS
• Using Frequencies Procedure
• Go to Crosstab command
• Check the box with statistics and click on all of
  the items you wish SPP to calculate