Transcript Chapter 12 Bivariate Association with Bivariate Tables and

Chapter 14 in 1e
Ch. 12 in 2/3 Can. Ed.
Association Between Variables
Measured at the Ordinal Level
Using the Statistic Gamma and
Conducting a Z-test for Significance
Introduction to Gamma
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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:
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1. Is there an association?
2. How strong is the association?
3. What direction (because level is ordinal) is it?
Introduction to Gamma (cont.)
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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.
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In this case, you would use the 5 step method, as
for χ2 and conduct a hypothesis test.
Introduction to Gamma (cont.)
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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?
Introduction to Gamma (cont.)
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To compute Gamma, two quantities must be
found:
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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).
Computing Gamma
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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.
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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.
Formula for Gamma
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Formula for Gamma:
n s  nd
G
n s  nd
A Simple Example for Gamma using
Healey #12.1 in 1e or #11.1 in 2/3 e
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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
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.
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Ns= 10(5) = 50
Authoritarianism (x)
Efficiency (y)
Low
High
Low
10
12
22
High
17
5
22
Total
27
17
44
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.
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Nd= 12(17) = 204
Authoritarianism (x)
Efficiency (y)
Low
High
Low
10
12
22
High
17
5
22
Total
27
17
44
Simple Example (cont.)
ns  nd 50  204  154
G


 0.61
ns  nd 50  204 254
Using the table, we can
see that G =-0.61 is
a strong negative
association.
Value
Strength
Between 0.0
and 0.30
Weak
Between 0.30
and 0.60
Moderate
Greater than
0.60
Strong
Simple Example (cont.)
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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.
Example: Healey #14.7 (1e), #12.7 in 2/3e)
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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.
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Part a.
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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.
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Is the association significant?
Part A: Calculating Gamma
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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(10+9+9+9) + 6(9+9) + 9(9+9) + 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
Part A: Calculating Gamma (cont.)
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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 (10+9+9+5) + 6 (9+5) + 9(9+5) + 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
Part A: Calculating Gamma (cont.)
ns  nd 767  491 267
G

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 0.22
ns  nd 767  491 1258
Using the table, we can
see that G =+0.22 is
a weak positive
association.
Value
Strength
Between 0.0
and 0.30
Weak
Between 0.30
and 0.60
Moderate
Greater than
0.60
Strong
Part A: Calculating Gamma (cont.)
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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 = + 0.22).
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.
Part B: Testing Gamma for Significance
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The test for significance of Gamma is a
hypothesis test, and the 5 step model should be
used.
Step 1: Assumptions
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Step 2: Null and Alternate hypotheses
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Random sample, ordinal, Sampling Dist. is normal
Ho: γ=0, H1: γ≠0 (Note: γ is the population value of G)
Step 3: Sampling Distribution and Critical Region
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Z-distribution, α = .05, z = +/-1.96
Part B: Testing Gamma for Significance (cont.)
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Part 4: Calculating Test Statistic:
Formula :
Calculate:
ns  nd
zG
2
N 1 G

z  .22
767 491

75 1  .22
2
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 .22(4.19)  0.92
Part B: Testing Gamma for Significance (cont.)
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Step 5: Make Decision and Interpret
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Zobs=.92 < Zcrit= +/-1.96
Fail to reject Ho
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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 significant
association.
Practice Question: Healey #14.8 (1e) or
#12.8 (2/3e)
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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 Spearman’s rho (also
shown in Chapter 14 (12 in 2nd). This
statistic will not be included on the final exam.
Kendall’s Tau b* (not in 1st Can. Ed.)
*do not need to calculate – for SPSS only
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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 Spearman’s rho
Using SPSS to Calculate Gamma
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Go to Analyze>Descriptives>Crosstabs (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 Ztest to test Gamma for significance. Compare
the significance of Gamma (this is the pvalue) to your alpha value. If your p-value is
less than your alpha, then the association is
significant.