Association of Attributes

Introduction:

In social sciences, we come across certain phenomena which are incapable of quantitative
measurement. Blindness, deafness, religion; juvenile delinquency, marital status etc are some
phenomena which are not measurable. Such characteristics are called attributes. In these
cases, one can make only counting of individuals who possess or do not possess these
attributes. In other words what can do is to state so many individuals are blind or so many non-
blind. While dealing with one attribute the classification of data is done on the basis of
presence or absence of the attribute. It is also absolutely essential that a clear-cut definition of
the attribute under study is made because only such a definition paves way for the counting of
the individuals possessing or not possessing the attribute. Two attributes are said to be
associated only if they appear together in a great number of cades than is to be expected if
they are independent. On the other hand, if the number of observed cases is less than the
expected, under assumption of independence, attributes are associated. In order to ascertain
whether the attributes are associated or not the following methods can be used.

1. Comparison of observed and expected frequencies.

2. Proportion method
3. Yule's coefficient of Association
4. Coefficient of colligation
5. Coefficient of contingency

Example of Attributes and Contingency Table

Suppose that we have two variables, sex (male or female) and handedness (right- or left-
handed). Further suppose that 100 individuals are randomly sampled from a very large
population as part of a study of sex differences in handedness. A contingency table can be
created to display the numbers of individuals who are male and right-handed, male and left-
handed, female and right-handed, and female and left-handed. Such a contingency table is
shown below.

Right-handed Left-handed TOTALS

Males 43 9 52

Females 44 4 48
TOTALS 87 13 100

40
The numbers of the males, females, and right- and left-handed individuals are called marginal
totals. The grand total, i.e., the total number of individuals represented in the contingency
table, is the number in the bottom right corner.

The table allows us to see at a glance that the proportion of men who are right-handed is about
the same as the proportion of women who are right-handed although the proportions are not
identical. The significance of the difference between the two proportions can be assessed with
a variety of statistical tests including Pearson's chi-square test, the G-test, Fisher's exact test,
and Barnard's test, provided the entries in the table represent individuals randomly sampled
from the population about which we want to draw a conclusion. If the proportions of individuals
in the different columns vary significantly between rows (or vice versa), we say that there is a
contingency between the two variables. In other words, the two variables are not independent.
If there is no contingency, we say that the two variables are independent.

The example above is the simplest kind of contingency table, a table in which each variable
has only two levels; this is called a 2 x 2 contingency table. In principle, any number of rows
and columns may be used. There may also be more than two variables, but higher order
contingency tables are difficult to represent on paper. The relation between ordinal variables,or
between ordinal and categorical variables, may also be represented in contingency tables,
although such a practice is rare.

Measurement of Association

Measures of association

The degree of association between the two variables can be assessed by a number of
coefficients: the simplest is the phi coefficient defined by

where χ2is derived from Pearson's chi-square test, and N is the grand total of observations. φ
varies from 0 (corresponding to no association between the variables) to 1 or -1 (complete
association or complete inverse association). This coefficient can only be calculated for
frequency data represented in 2 x 2 tables. φ can reach a minimum value -1.00 and a
maximum value of 1.00 only when every marginal proportion is equal to .50 (and two diagonal
cells are empty). Otherwise, the phi coefficient cannot reach those minimal and maximal
values.[1]

Alternatives include the tetrachoric correlation coefficient (also only applicable to 2 ×2 tables),
the contingency coefficient C, and Cramér'sV.

C suffers from the disadvantage that it does not reach a maximum of 1 or the minimum of -1;
the highest it can reach in a 2 x 2 table is .707; the maximum it can reach in a 4 x 4 table is
0.870. It can reach values closer to 1 in contingency tables with more categories. It should,
therefore, not be used to compare associations among tables with different numbers of
categories.[2]Moreover, it does not apply to asymmetrical tables (those where the numbers of
row and columns are not equal).

The formulae for the Cand V coefficients are:

and

K being the number of rows or the number of columns, which ever is less.
C can be adjusted so it reaches a maximum of 1 when there is complete association in a table

Of any number of rows and columns by dividing C by (recall that C only applies to
tables in which the number of rows is equal to the number of columns and therefore equal to
k).

The tetrachoric correlation coefficient assumes that the variable underlying each dichotomous
measure is normally distributed. The tetrachoric correlation coefficient provides "a convenient
measure of [the Pearson product-moment] correlation when graduated measurements have
been reduced to two categories." The tetrachoric correlation should not be confused with the
Pearson product-moment correlation coefficient computed by assigning, say, values0 and 1 to
represent the two levels of each variable (which is mathematically equivalent to the phi
coefficient). An extension of the tetrachoric correlation to tables involving variables with more
than two levels is the polychoric correlation coefficient.

The Lambda coefficient is a measure the strength of association of the cross tabulations
when the variables are measured at the nominal level. Values range from 0 (no association) to
1 (the theoretical maximum possible association). Asymmetric lambda measures the
percentage improvement in predicting the dependent variable. Symmetric lambda measures
the percentage improvement when prediction is done in both directions.

The uncertainty coefficient is an other measure for variables at the nominal level.

All of the following measures are used for variables at the ordinal level. The values range from
-1 (100% negative association, or perfect inversion) to +1 (100% positive association, or
perfect agreement). A value of zero indicates the absence of association.

 Gammatest: No adjustment for either table size or ties.

 Kendalltau: Adjustment forties.
o Taub: For square tables.
o Tauc: For rectangular tables.
Calculating the test-statistic

The value of the test-statisticis

where

Χ2=Pearson's cumulative test statistic, which asymptotically approaches a χ2distribution.

Oi=an observed frequency;

Ei=an expected(theoretical) frequency, asserted by the null hypothesis;

n =the number of cells in the table.

The chi-square statistic can then be used to calculate a p-value by comparing the value of the
statistic to a chi-squared distribution. The number of degrees of freedom is equal to the
number of cells n, minus the reduction in degrees of freedom, p.

The result about the number of degrees of freedom is valid when the original data was
multinomial and hence the estimated parameters are efficient for minimizing the chi-square
statistic. More generally however, when maximum likelihood estimation does not coincide with
minimum chi-square estimation, the distribution will lie somewhere between a chi-square
distribution with n − 1 − p and n − 1 degrees of freedom.