Association Between Variables

This following material is derived from SPSS eTutor by Dee Britton ^[1] and is used under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Licence.

While analysing data, researchers are interested in finding out the association (or relationship) between different types of variables. Two types of  association are of particular interest:

• Association between categorical/nominal/ordinal variables
• Association between interval/continuous variables

Association between categorical variables

Sometimes a researcher wants to understand the relationship between categorical variables, which are also referred to as nominal or ordinal variables. Examples of such variables include gender, age categories, occupation, and brand preferences. Such associations can are explored via:

a) cross-tabulation and/or,

b) the chi-square test

a) Cross-tabulation

A cross-tabulation table is simply a frequency distribution for two variables together. A crosstab is a matrix that shows the distribution of one variable for each category of a second variable.^[2]

To run a crosstab on SPSS, go to: Analyze, Descriptive Statistics, Crosstabs

Highlight your dependent variable name, click on arrow pointing toward Row box

Highlight your independent variable name, click on arrow pointing toward Column box.

Your screen should look like this:

Click on Cells and then click the Column percentage box:

Click Continue, then OK. Your output should look like this:

What does this crosstab mean? This is relatively easy to interpret. 28.7% of males and 29.4% of females claimed that they are very happy.  But what about a crosstab that has many attributes?  For example, what if you wanted to analyze the relationship between the number of children that you have and your general happiness? This is the output of that analysis:

b) Chi-square test

While cross-tabulation is simply a frequency/percentage table for two variables, the Chi-Square test is undertaken to examine if the results of the cross-tabulation are statistically significant. The Chi-square test is a non-parametric test used to determine whether there is a statistically significant association between two categorical variables. ^[3]

Source: EZPSS ^[4]

Click Analyze, Descriptive Statistics, Crosstabs.  Click on your dependent variable name and place it in the “row” box and then select your independent variable and place it in the “column” box. For this example, our independent variable is SEX and dependent variable is PARTYID.

Your screen should look like this:

Now click the Statistics button and select Chi-Square.

Click OK. Then click Cells. Check to ensure that Observed is in the “count” box and that Row, Column and Total boxes are all checked in the “percentage” box.

Click OK. Your output should look like this:

Interpreting Chi-square test for independence

One of the requirements for Chi-Square is that each and every cell has a frequency of 5 or greater. You first need to check to see if the data in your table meet this requirement. Look for footnote underneath the Chi-square Tests box. Our output includes this information in footnote ‘a’; none of our cells have a frequency less than 5 and therefore we have not violated this chi-square assumption.

Now look at the “Pearson Chi-Square Asymp. Sig (2 sided)”*. Since Chi-Square is testing the null hypothesis, the Sig value must be .05 or less for there to be a significant statistical for the relationship between the variables. In this example, the Sig. is .001, so there is very strong statical significance for the relationship between gender and political party identification.

Association between interval/continuous variables

When the aim is to explore the relationship between two variables which are measured on an interval/ratio scale, then researchers use the correlation test.

Pearson’s correlation

A hypothesis test of Pearson’s correlation coefficient is used to determine whether there is a statistically significant linear correlation between two continuous variables (for example, dollars spent on groceries and a consumer’s age – in years). ^[5]

To calculate Pearson’s r, go to Analyze, Correlate, Bivariate. Enter your two variables. For example, we can examine the correlation between two continuous variables, “Age” and “TVhours” (the number of tv viewing hours per day). Your screen should look like this:

Click OK. Look in your output for the following:

Note that the Pearson’s r value for comparing age to age is 1, suggesting perfect correlation. If you think about this, that makes logical sense. What you are truly interested in examining is the Pearson’s r value of the 2 different variables (in this case, the value is .139). This suggests that someone ages, they watch more television.

Here are guidelines for interpreting the strength of association for Lambda,  Gamma, and Pearson’s r (remember, lambda can only have a positive value):

Source: How2stats^[6]