Transcript Chapter 12 Bivariate Association with Bivariate Tables and

Chapter 13 (1e), (Ch. 11 2/3e)
Association Between Variables
Measured at the Nominal Level:
Phi, Cramer’s V, and Lambda
Nominal Level Measures of Association

It is always useful to compute column
percentages for bivariate tables.

But, it is also useful to have a summary
measure – a single number – to indicate the
strength of the relationship.
Nominal Level Measures of Association

For nominal level variables, there are two
commonly used types of measures of
association:
1. Chi-square based statistics: Phi or Cramer’s V
2. A PRE (Proportional Reduction in Error)
measure known as Lambda
Chi-square Based Measure: Phi (  )


Phi is used for 2x2 tables.
The formula for Phi:
Example of Nominal Measure: Calculating
Phi for #12.1 (1e) or #11.1 (2/3e)
Step 1. Using the 5 step method explained
2

in Chapter 11, calculate
for the table.




2
= 4.696 (critical value = 3.841)
There is a significant relationship between
Authoritarianism and Worker Efficiency
(
)
Example: Calculating Phi for 12.1(11.1)
Step 2: Calculate Phi…
What does this mean?*
Phi = 0.33
There is a strong
association between
authoritarianism
and efficiency.
Value
Strength
0.0 > 0.10
Weak
0.10 > 0.30
Moderate
> 0.30
Strong
*Keep this table handy!
Nominal Measures: Cramer’s V (V)



Cramer’s V is used for tables larger than 2x2.
Formula for Cramer’s V:
Where (min r-1, c-1) is the minimum value of either
the # rows-1 or the # columns-1.
Limitations of Phi and V

Phi is used for 2x2 tables only. For larger
tables, use Cramer’s V.

Phi (or V) is an index of the strength of the
relationship only. It does not identify the
pattern. Phi or V cannot be used to compare
the strength of one relationship to another.

To analyze the pattern of the relationship, see
the column % in the bivariate table.
Nominal Measures: Lambda (λ)




Like Phi and V, Lambda is used to measure the
strength of the relationship between nominal
variables in bivariate tables. It can also be used to
compare one relationship to another.
Unlike Phi and V, Lambda is a PRE measure and
its value has a more direct interpretation.
PRE = Proportional Reduction in Error = how
much less error do you make in your prediction of
the dependent variable (y) when you take into
account the values of the independent variable (x)
Lambda tells us the improvement in predicting Y
while taking X into account.
Lambda (λ) cont.

Formula for Lambda:
E1  E 2

E1


Where E1 = (N – the largest row total)
and E2 = (For each column, subtract the largest
cell frequency from its column total and then add
the differences together)
Calculating Lambda (λ)- 12.1 (1e), 11.1 (2/3e):

To compute λ, we must first find E1 and E2:


E1 = N – largest row total = 44 – 22 = 22
E2 = For each column, subtract the largest cell
frequency from the col. total and add together
= (27 – 17) + (17 – 12) = 10 + 5 = 15
Authoritarianism (x)
Efficiency (y)
Low
High
Total
Low
10
12
22
High
17
5
22
Total
27
17
44
Calculating Lambda (cont.)



Lambda is a PRE measure.
A Lambda of .32 means that knowing something about
authoritarianism (X) reduces our error in prediction of
efficiency by 32%.
We can also say that knowing about X increases our
ability to predict efficiency (Y) by 32%.
More about Lambda (λ)

The values of Lambda vary from 0.0 – +1.00,
where 0.0 is very weak and 1.00 is very strong.

Lambda can be used to compare the strength of
two or more bivariate relationships. Suppose that
you calculate:
1. Prestige X Happiness, λ = .55 and
2. Income X Happiness, λ = .20,
…then you could say that Relationship 1 is
stronger, and that Prestige has a greater effect on
Happiness than Income.



More about Lambda (λ)



Lambda, like Phi and V, tells you about the
strength of the relationship only. It does
not give information about pattern.
To analyze the pattern of the relationship,
use the column % in the bivariate table.
Note that when row totals are very
unequal, lambda can be zero even when
there is an association between the
variables.
Answering the “Three Questions” when
working with Nominal Level Variables.



1. “Is there an association?” can be
answered by looking at the % and by doing
a test of significance like Х2.
2. “How strong is the association?” can be
answered by using a measure of strength
like Phi, Cramer’s V, or Lambda.
3. “What is the pattern of the association?”
can be assessed by looking at the
percentages.
Practice Question: #13.1 ( 1e),11.2 (2/3e)
Work with a partner on parts (a,b,c)
and find an answer to all three of the
above questions
 Compare the three relationships, and
use Lambda to evaluate which
relationship is the strongest.
 Write a brief summary of your findings.
