Association between Variables Measured at the Nominal Level

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Association between
Variables Measured at the
Nominal Level
Introduction
► The
measures of association are more
efficient methods of expressing an
association than calculating percentages for
bivariate tables—they express the
relationship in a single number
► However, you always need to look at the
bivariate tables (crosstabs), since a single
number loses some information
Many Different Measures of
Association
► Different
ones are used for different levels of
measurement (nominal, ordinal, or interval/ratio)
► When selecting measures of association for
assessing the relationship between variables
measured at different levels, social scientists
generally choose the measure that is appropriate
for the lower of the levels
 So if one variable is nominal, and the other interval, you
would use a level of association appropriate for the
nominal variable
Chi-Square-Based Measures of
Association
► These
have been commonly used, since you
already have calculated chi square for
inferential statistics; it is simple to transform
it into a measure of association
► We can see from the percentages in a
bivariate table that two variables are
associated, and know from chi square that
the differences are statistically significant
Phi
► To
find the strength of the association, will
compute a phi
► This statistic is used as a measure of
association appropriate for tables with only
two rows and two columns
► Formula 14.1 for phi:

2

N
Phi, cont.
► Phi
is the square root of the value of the
obtained chi square divided by the sample
size
► For a 2 x 2 table, phi ranges in value from 0
(no association) to 1.00 (perfect
association)
► A phi of .33 indicates a weak to moderate
relationship between the two variables
► This measure does not reveal the pattern of
the association, so need to look at the table
Cramer’s V
► For
tables with three or more columns or
three or more rows, phi has an upper limit
that can exceed 1.00
 Cramer’s V is used for tables that are larger
than 2 x 2, is based on chi square, and is also
easy to calculate
Formula for Cramer’s V

V
 N min . of r  1, c  1
2
Interpretation of Cramer’s V
► It
has an upper limit of 1.00 for any size
table
► Like phi, it can be interpreted as an index
that measures the strength of the
association between two variables
► A major problem with phi and Cramer’s V is
the absence of a direct or meaningful
interpretation for values between the
extremes of 0.00 and 1.00
 Both indicate the strength of the association
 But it is only an index of relative strength
Proportional Reduction in Error (PRE)
► For
nominal-level variables, the logic of PRE
involves first attempting to guess or predict
the category into which each case will fall
on the dependent variable (Y) while
ignoring the independent variable (X)
 Will be predicting blindly in this case, and will
make many errors
► The
second step would be to predict again
the category of each case on the dependent
variable, but take the independent variable
into account
PRE, cont.
► If
the two variables are associated, the additional
information from the independent variable should
enable us to reduce our errors of prediction
► The stronger the association between the
variables, the more we will reduce our errors
 In the case of a perfect association, we would make no
errors at all when predicting scores on Y from scores on
X
 When there is no association between the variables,
knowledge of the independent will not improve the
accuracy of our predictions—we would make just as
many errors of prediction
Lambda
► Lambda
is a PRE measure for nominal-level
variables
► We know that gender and height are associated
by looking at the percentages
► To measure the strength of this association, a PRE
measure called lambda will be calculated
 First need to find the number of prediction errors made
while ignoring the independent variable (gender)
 Then will find the number of prediction errors made
while taking gender into account
 These two sums will be compared in order to derive the
statistic
Example of Height by Gender (Table
12.15)
► Can
ignore information given by the independent
variable (gender) by working only with the row
marginals
 Two different predictions can be made using these
marginals
► We
can predict either that all subjects are tall or that all
subjects are short (these are the only two permitted by lambda)
 For the first prediction (all subjects are tall), 48 errors
will be made
► For
this prediction, all 100 cases would be placed in the first
row
► Since only 52 cases belong in this row, this prediction would
result in (100 – 52) or 48 errors
Example, cont.
► If
we had predicted that all subjects were
short, we would have made 52 errors (100 48 = 52)
► We will use the lesser of these two numbers
and refer to this quantity as E sub 1 for the
number of errors made while ignoring the
independent variable
 So, E sub 1 = 48 [N – (largest row total)]
Second Step
► The
second step in computing lambda is to again
predict scores on Y (height), this time taking X
(gender) into account
 Follow the same procedure as in the first step, but this
time move from column to column
► Since
each column is a category of X, we take X into account in
making our predictions
 For the left-hand column (males), we predict that all 50
cases will be tall and make six errors
 For the second column (females), our prediction is that
all females are short, and eight errors will be made
 We have made a total of 14 errors of prediction, a
quantity we will label E sub 2
Logic of Lambda
► The
logic of lambda is that, if the variables
are associated, fewer errors will be made
under the second procedure than under the
first (want E sub 2 to be less than E sub 1)
 Clearly, gender and height are associated, since
we made fewer errors of prediction while
considering gender (E sub 2 = 14) than while
ignoring gender (E sub 1 = 48)
Computing Lambda
► To
find the proportional reduction in error,
use Formula 12.3:
E1  E 2

E1
Interpretation of Lambda
► For
the above example, lambda equals .71
► Lambda has a possible range of 0 to 1
 A lambda of 0 would indicate that the
information given by the independent variable
does not improve our ability to predict the
dependent and therefore, that there is no
association between the variables
 A lambda of 1.00 would mean that it was
possible to predict Y without error from X
PRE Interpretation
► Additionally,
lambda allows a direct and
meaningful interpretation of the numbers in
between
 When multiplied by 100, the value of lambda indicates
directly the proportional reduction in error—the strength
of the association
 So, a lambda of .71 tells us that knowledge of gender
improves our ability to predict height by a factor of 71%
► Of,
we are 71% better off knowing gender when attempting to
predict height than we are not knowing gender
Other Examples
► If
lambda = .20, this indicates that we are 20%
better off knowing the independent variable when
attempting to predict a person’s score or value on
the dependent variable
► If we make 75 errors when predicting Y without
knowledge of X, and 60 errors when predicting Y
with knowledge of X, then X and Y are associated
► If the value of lambda is relatively low, we may
conclude that other variables are importantly
associated with the dependent variable
Problems with Lambda
► It
changes if you reverse the independent
and dependent variables
 Need to follow the convention of putting the
independent variable in the columns and
compute lambda as done above
 You also need to be confident which variable is
the independent one and which is the
dependent one
Second Problem
► If
one of the row totals is much larger than the
others, lambda can take on a value of 0 even
when other measures of association would not be
0, and calculating percentages for the table
indicates some association between the variables
 Suggests that you use great caution in interpretation of
lambda when the row marginals are very unequal
 If the row totals are unequal, you should use a chisquare-based measure of association (phi or Cramer’s
V)
 For the same bivariate table, Cramer’s V is .27 and
lambda is zero, we can conclude that the variables may
be associated even if lambda is zero—need to disregard
lambda if the row marginals are very unequal

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