Chapter 14 (Ch. 12 in 2nd Can. Ed.)

1
Chapter 14 (Ch. 12 in 2nd Can. Ed.)

• Association Between Variables Measured at the
  Ordinal Level
• Using the Statistic Gamma and Conducting a Z-test
  for Significance

2
Introduction to Gamma

• Gamma is the preferred measure to test strength
  and direction of two ordinal-level variables that
  have been arrayed in a bivariate table.
• Before computing and interpreting Gamma, it is
  always useful to find and interpret the column
  percentages.
• Gamma can answer the questions
• 1. Is there an association?
• 2. How strong is the association?
• 3. What direction (because level is ordinal) is
  it?

3
Introduction to Gamma (cont.)

• Gamma can also be tested for significance using a
  Z or t-test to see if the association
  (relationship) between two ordinal level
  variables is significant.
• In this case, you would use the 5 step method, as
  for ?2 and conduct a hypothesis test.

4
Introduction to Gamma (cont.)

• Like Lambda, Gamma is a PRE (Proportional
  Reduction in Error) measure it tells us how much
  our error in predicting y is reduced when we take
  x into account.
• With Gamma, we try to predict the order of pairs
  of cases (predict whether one case will have a
  higher or lower score than another)
• For example, if case A scores High on Variable1
  and High on Variable 2, will case B also score
  High-High on both variables?

5
Introduction to Gamma (cont.)

• To compute Gamma, two quantities must be found
• Ns is the number of pairs of cases ranked in the
  same order on both variables.
• Nd is the number of pairs of cases ranked
  differently on the variables.
• Gamma is calculated by finding the ratio of cases
  that are ranked the same on both variables minus
  the cases that are not ranked the same (Ns Nd)
  to the total number of cases (Ns Nd).

6
Computing Gamma

• This ratio can vary from 1.00 for a perfect
  positive relationship to -1.00 for a perfect
  negative relationship. Gamma 0.00 means no
  association or no relationship between two
  variables.
• Note that when Ns is greater than Nd, the ratio
  with be positive, and when Ns is less than Nd the
  ratio will be negative.

7
Formula for Gamma

• Formula for Gamma

8
A Simple Example for Gamma using Healey 12.1
(11.1 in 2nd Can.)

• As previously seen, this table shows the
  relationship between authoritarianism of bosses
  (X) and the efficiency of workers (Y) for 44
  workplaces. Since the variables are at the
  ordinal level, we can measure the association
  using the statistic Gamma.

Authoritarianism (x)
Efficiency (y) Low High
Low 10 12 22
High 17 5 22
Total 27 17 44
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Simple Example (cont.)For Ns, start with the
Low-Low cell (upper left) and multiply the cell
frequency by the cell frequency below and to the
right.

• Ns 10(5) 50

Authoritarianism (x)
Efficiency (y) Low High
Low 10 12 22
High 17 5 22
Total 27 17 44
10
Simple Example (cont.)For Nd, start with the
High-Low cell (upper right) and multiply each
cell frequency by the cell frequency below and to
the left.

• Nd 12(17) 204

Authoritarianism (x)
Efficiency (y) Low High
Low 10 12 22
High 17 5 22
Total 27 17 44
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Simple Example (cont.)Using the table, we
cansee that G -0.61 isa strong
association.Also, Gamma tells us that we will
make 61 fewer errors predicting efficiency
when we take authoritarianism into account.
Value Strength
Between 0.0 and 0.30 Weak
Between 0.30 and 0.60 Moderate
Greater than 0.60 Strong
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Simple Example (cont.)

• In addition to strength, gamma also identifies
  the direction of the relationship. We can look at
  the sign of Gamma ( or -). In this case, the
  sign is negative (G - 0.61).
• This is a negative relationship as
  Authoritarianism increases, Efficiency decreases.
• In a negative relationship, the variables change
  in opposite directions.

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Example using Healey 14.7 (12.7 in 2nd)

• This question involves a more complicated
  calculation for Gamma. The question asks if
  aptitude test scores are related to job
  performance rating for 75 city employees.
• Part a.
• Are the two variables, Aptitude, (measured as
  Low, Medium and High) and Job Performance (Low,
  Medium, and High) associated?
• How strong is this association?
• What direction is the association?
• Part b.
• Is the association significant?

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Part A Calculating Gamma

• For Ns, start with the Low-Low cell (upper left)
  and multiply each cell frequency by total of all
  cell frequencies below and to the right and add
  together.
• For this table, Ns is 11(10999) 6(99)
  9(99) 10 (9) 767

Test Scores (x)
Efficiency (y) Low Moderate High Total
Low 11 6 7 24
Moderate 9 10 9 28
High 5 9 9 23
Totals 25 25 25 75
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Part A Calculating Gamma (cont.)

• For Nd, start with High-Low cell (upper right)
  and multiply each cell frequency by total of all
  cell frequencies below and to the left and add
  together.
• For this table, Nd 7 (10995) 6 (95)
  9(95) 10 (5) 491

Test Scores (x)
Efficiency (y) Low Moderate High Total
Low 11 6 7 24
Moderate 9 10 9 28
High 5 9 9 23
Totals 25 25 25 75
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Part A Calculating Gamma (cont.)Using the
table, we cansee that G 0.21 isa weak
association.We make only 21less error.
Value Strength
Between 0.0 and 0.30 Weak
Between 0.30 and 0.60 Moderate
Greater than 0.60 Strong
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Part A Calculating Gamma (cont.)

• As noted before, gamma also identifies the
  direction of the relationship. We can look at the
  sign of Gamma ( or -). In this case, the sign is
  positive (G 0.21).
• This is a positive relationship as Aptitude Test
  Scores increase, Job Performance increases.
• Next, we test the association for significance,
  using the 5 step method.

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Part B Testing Gamma for Significance

• The test for significance of Gamma is a
  hypothesis test, and the 5 step model should be
  used.
• Step 1 Assumptions
• Random sample, ordinal, Sampling Dist. is normal
• Step 2 Null and Alternate hypotheses
• Ho ?0, H1 ??0 (Note ? is the population
  value of G)
• Step 3 Sampling Distribution and Critical Region
• Z-distribution, a .05, z /-1.96

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Part B Testing Gamma for Significance (cont.)

• Part 4 Calculating Test Statistic
• Formula
• Calculate

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Part B Testing Gamma for Significance (cont.)

• Step 5 Make Decision and Interpret
• Zobt.88 lt Zcrit /-1.96
• Fail to reject Ho
• The association between aptitude tests and job
  performance is not significant.
• Part C No, the aptitude test should not be
  continued, because there is no association.

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Practice Question Healey 14.8 (12.8 in 2nd
Can. Ed.)

• Try this question as a homework assignment.
• The solution to the question can be found in the
  Final Review powerpoint.
• Note We will not cover Spearmans rho (also
  shown in Chapter 14 (12 in 2nd). This statistic
  will not be included on the final exam.

22
Kendalls Tau b (not in 1st Can. Ed.)do not
need to calculate for SPSS only

• The statistic Tau b is the preferred measure of
  strength to report when a bivariate table has
  many tied pairs (when cases are scored the same
  on both variables in a table)
• In this case, gamma will tend to overestimate the
  strength of the association.
• Rule of thumb when the value of gamma is double
  that of Tau b, report Tau b instead, because it
  will be a better measure of strength.
• omit Tau c and Spearmans rho

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Using SPSS to Calculate Gamma

• Go to AnalyzegtDescriptivesgtCrosstabs (as with
  Chi-square). Click on Cells for column and on
  Statistics, asking for both Gamma and Tau b.
• Note that SPSS uses a t-test rather than a Z-test
  to test Gamma for significance. Compare the
  significance of Gamma (this is the p-value) to
  your alpha value. If your p-value is less than
  your alpha, then the association is significant.