Bivariate AssociationBivariate Association

Introduction Introduction

This chapter is about This chapter is about measures of measures of association association These are designed to quantify the strength These are designed to quantify the strength

(or importance) of a relationship(or importance) of a relationshipThey can increase understanding of the They can increase understanding of the causalcausal relationships among variables relationships among variables

Can improve our ability to predict from one Can improve our ability to predict from one variable to anothervariable to another

Association Between Variables and Association Between Variables and the Bivariate Tablethe Bivariate Table

If the distribution of the scores of one If the distribution of the scores of one variable changes across the categories of variable changes across the categories of another variable, the variables are another variable, the variables are associated to some extentassociated to some extent

The independent variable is in the The independent variable is in the columns (on top of the table)columns (on top of the table) In a bivariate table, the categories of the In a bivariate table, the categories of the

dependent variable are placed in the rowsdependent variable are placed in the rows

Association, cont.Association, cont.

If you read the table from column to If you read the table from column to column, you can observe the effects of the column, you can observe the effects of the independent variable on the dependent independent variable on the dependent variable (provided that the table is variable (provided that the table is constructed with the independent variable constructed with the independent variable in the columns)in the columns)These These ““within-columnwithin-column”” frequency distributions frequency distributions

are called the are called the conditional distribution of Yconditional distribution of YSince they display the distribution of scores on the Since they display the distribution of scores on the

dependent variable for each condition (or score) of dependent variable for each condition (or score) of the independent variablethe independent variable

Chi SquareChi Square

Another way to find the existence of an Another way to find the existence of an association between two variables organized association between two variables organized into table format is the chi square statisticinto table format is the chi square statistic This is a test of significance, but can also be used as This is a test of significance, but can also be used as

an indicator of associationan indicator of association Any nonzero value for the obtained chi square Any nonzero value for the obtained chi square

statistic indicates the existence of an associationstatistic indicates the existence of an association At least in the sampleAt least in the sample But, a large chi square will not indicate a strong relationship, But, a large chi square will not indicate a strong relationship,

since all significance tests are affected by sample sizesince all significance tests are affected by sample size

Three Characteristics of Three Characteristics of Bivariate AssociationsBivariate Associations

Does an association exist?Does an association exist?

Because the column totals are not equal, you will Because the column totals are not equal, you will need to calculate percentages for each cell in the need to calculate percentages for each cell in the tabletable The association between two variables can be seen The association between two variables can be seen

when the variables are arranged in a bivariate table when the variables are arranged in a bivariate table and percentages are computed in the direction of the and percentages are computed in the direction of the independent variableindependent variable

So, you percentage down, and read acrossSo, you percentage down, and read across You divide the number in each cell by the total at the bottom You divide the number in each cell by the total at the bottom

of the column only (not the total of the sample)of the column only (not the total of the sample)

If two variables are not associated, then the If two variables are not associated, then the conditional distributions of Y will not change across conditional distributions of Y will not change across the columnsthe columns

How strong is the association?How strong is the association?

At one extreme is the case of no associationAt one extreme is the case of no association At the other is the case of At the other is the case of ““perfect associationperfect association””

The strongest possible relationship between X and Y The strongest possible relationship between X and Y A perfect association exists between two variables if A perfect association exists between two variables if

each value of Y is associated with one and only one each value of Y is associated with one and only one value of Xvalue of X

In a bivariate table, all cases in each column would be In a bivariate table, all cases in each column would be located in a single celllocated in a single cell

There would be no variation in Y for a given value of X (the There would be no variation in Y for a given value of X (the independent variable)independent variable)

Strength of Association, cont.Strength of Association, cont.

A perfect relationship would be taken as strong A perfect relationship would be taken as strong evidence of a causal relationship between the evidence of a causal relationship between the variables, for the samplevariables, for the sample

In the case of a perfect association, predictions In the case of a perfect association, predictions from one variable to another can be made from one variable to another can be made without error. However, most associations are without error. However, most associations are much less extreme, since humans are not that much less extreme, since humans are not that predictablepredictable

For intermediate relationships that show a For intermediate relationships that show a tendency for two variables to be associated, tendency for two variables to be associated, measures of association quantify the association measures of association quantify the association by showing how close it is to a perfect by showing how close it is to a perfect association or to no associationassociation or to no association

Results Indicating StrengthResults Indicating Strength

Most all of the statistics are designed so that Most all of the statistics are designed so that they have a lower limit of 0 and an upper limit of they have a lower limit of 0 and an upper limit of 1 (plus or minus 1 for ordinal and interval ratio 1 (plus or minus 1 for ordinal and interval ratio measures of association)measures of association)

A measure that equals 0 indicates no A measure that equals 0 indicates no association between the variablesassociation between the variables If the conditional distributions of Y do not change If the conditional distributions of Y do not change

across the categories of X, any measure of across the categories of X, any measure of association would have a value of 0.00association would have a value of 0.00

And the closer the value of a measure of association And the closer the value of a measure of association is to 1, the stronger the relationshipis to 1, the stronger the relationship

For ordinal and interval-ratio measures, would be a plus or For ordinal and interval-ratio measures, would be a plus or minus oneminus one

Strength and DirectionStrength and Direction

-1 Perfect negative association-1 Perfect negative association -.9, -.8, -.7 Strong negative association-.9, -.8, -.7 Strong negative association -.6, -.5, -.4 Moderate negative association-.6, -.5, -.4 Moderate negative association -.3, -.2- -.1 Weak negative association-.3, -.2- -.1 Weak negative association 0 No association between variables0 No association between variables .1, .2, .3 Weak positive association.1, .2, .3 Weak positive association .4, .5, .6 Moderate positive association.4, .5, .6 Moderate positive association .7, .8, .9 Strong positive association.7, .8, .9 Strong positive association + 1 Perfect positive association+ 1 Perfect positive association

What is the pattern or direction of What is the pattern or direction of the association?the association?

When both variables are at least ordinal in level When both variables are at least ordinal in level of measurement, the pattern of association may of measurement, the pattern of association may also have a direction to italso have a direction to it No direction for nominal variables, because No direction for nominal variables, because

categories are all the samecategories are all the same If there is a positive association between two If there is a positive association between two

variables, as one variable increases in value, the variables, as one variable increases in value, the other also increasesother also increases High scores on one variable are associated with high High scores on one variable are associated with high

scores on the other variable, and low scores on one scores on the other variable, and low scores on one variable are associated with low scores on the othervariable are associated with low scores on the other

Interpretation, cont.Interpretation, cont. The measures of association find if the variables The measures of association find if the variables

are related by looking at each individual in the are related by looking at each individual in the samplesample We have to look at one personWe have to look at one person’’s answer to one s answer to one

question, and their answer to the other questionquestion, and their answer to the other question Then, we put it together with all the other people to Then, we put it together with all the other people to

see if there is a patternsee if there is a pattern When variables vary in opposite directions, the When variables vary in opposite directions, the

association between variables is negativeassociation between variables is negative Measures of association are designed so they will be Measures of association are designed so they will be

positive for positive associations and negative values positive for positive associations and negative values for negative associationsfor negative associations

So, we need to look at the existence, the So, we need to look at the existence, the strength, and the direction of the association to strength, and the direction of the association to complete the analysiscomplete the analysis