Chapter 11, 12, 13, 14 and 16

Association at Nominal and Ordinal Level

The Procedure in Steps

The Procedure in Steps for Nominal Variables

Step 1: Make Tables

Tables must have a title. Cells are intersections of columns and

rows. Subtotals are called marginals. N is reported at the intersection of row

and column marginals.

Step 1: Make Tables

Columns are scores of the independent variable. There will be as many columns as there

are scores on the independent variable.

Rows are scores of the dependent variable. There will be as many rows as there are

scores on the dependent variable.

Step 1: Make TablesTitle

Rows Columns

Row 1 cell a cell b Row Marginal 1

Row 2 cell c cell d Row Marginal 2

Column Marginal 1

Column Marginal 2

N

Example of Table The bivariate table showing the relationship

between gender (columns) and party preference (rows).

Female Male

Labour 8 5 13

Conservatives 4 8 12

12 13 25

Example of Table The bivariate table showing the relationship

between gender (columns) and party preference (rows).

Female Male

Labour 66.7% 38.5%

Conservatives 33.3% 61.5%

100% (N=12)

100% (N=13)

Step 2: What is the pattern/direction of the association?

See percentages in Table. Female voters tend to have party

preference for Labour and male voters have party preference for Conservatives

This relationship does have a clear pattern

But is it also significant?

Step 3: Is the Association between the Variables Significant?

Chi Square is a test of significance based on bivariate tables.

We are looking for significant differences between the actual cell

frequencies in a table (fo) and those

that would be expected by random

chance (fe).

Example of Computation

Use Formula 11.2 to find fe.

Multiply column and row marginals for each cell and divide by N. For Problem above

(13*12)/25 = 156/25 = 6.24 (13*13)/25 = 169/25 = 6.76 (12*12)/25 = 144/25 = 5.76 (12*13)/25 = 156/25 = 6.24

Example of Computation Expected frequencies:

Female Male

Labour 6.24 6.76 13

Conservatives 5.76 6.24 12

12 13 25

Example of Computation Divide each of the squared values by the fe for that

cell. The sum of this column is chi square

fo fe fo - fe (fo - fe)2 (fo - fe)2 /fe

8 6.24 1.76 3.10 .50

5 6.76 -1.76 3.10 .46

4 5.76 -1.76 3.10 .54

8 6.24 1.76 3.10 .50

25 25 0 χ2 = 2.00

Example of Computation See Chapter 11 of Healey (pp. 286-289) for

five-step model for Chi Square Test to find out whether variables are independent/ whether the association between the variables is significant or not

χ2 (critical) = 3.841 χ2 (obtained) = 2.00 The test statistic is not in the Critical

Region. Fail to reject the H0. There is no significant relationship between

gender and party preference

Interpreting Chi Square The chi square test tells us only if the

variables are independent or not. Like all tests of hypothesis, chi square is

sensitive to sample size. As N increases, obtained chi square increases. With large samples, trivial relationships may be

significant.

Remember: significance is not the same thing as importance

Step 4: If an Association does Exist, how Strong is it?

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.

For nominal level variables, there are the following commonly used measures of association: Phi Cramer’s V Lambda

Nominal Measures: Phi See Healey, formula 13.1, p. 342 Phi is used for 2x2 tables. The formula for Phi:

Nominal Measures: Cramer’s V See Healey, formula 13.2, p. 343 Cramer’s V is used for tables larger than 2x2. Formula for Cramer’s V:

Nominal Measures: Lambda See Healey, formula 13.3, p. 348 When dependent and independent variables are clear Formula for Lambda:

Step 5: Is there still an Association, if Control Variables are Added?

See Chapter 16 in Healey See week 10 of this course

The Procedure in Steps for Ordinal Variables

Steps 1 ,2, 3, 5 are similar to those for nominal variables

Only step 4 is different, because you need other measures of association

Step 4: If an Association does Exist, how Strong is it?

For ordinal level variables, there are the following commonly used measures of association: Spearman’s Rho (if there is ranking of

scores, see Healey pp. 376-382) Gamma (Formula 14.1, see Healey, p.

366). Is the strength of the relationship significant? Test whether gamma is significant. See Healey, pp. 380-381

An Ordinal Measure: Gamma

interpret the strength of gamma. e.g. gamma is 0.61. This is a strong association. In

addition to strength, gamma also identifies the direction of the relationship.

This is a positive relationship: e.g. as education increases, income increases.

In a negative relationship, the variables would change in the different direction.

Test whether the strength of Gamma is significant

Value Strength

Between 0.0 and

0.30Weak

Between 0.30 and

0.60Moderate

Greater than 0.60

Strong