### Bivariate Analysis - California State University, Fresno

```Bivariate Analysis
Cross-tabulation and chi-square
So far the statistical methods we
have used only permit us to:
• Look at the frequency in which certain
numbers or categories occur.
• Look at measures of central tendency such
as means, modes, and medians for one
variable.
• Look at measures of dispersion such as
standard deviation and z scores for one
interval or ratio level variable.
Bivariate analysis allows us to:
• Look at associations/relationships among
two variables.
• Look at measures of the strength of the
relationship between two variables.
• Test hypotheses about relationships between
two nominal or ordinal level variables.
For example, what does this table tell us about
opinions on welfare by gender?
Support cutting
welfare benefits
for immigrants
Yes
Male
Female
15
5
No
10
20
Total
25
25
Are frequencies sufficient to
allow us to make comparisons
What other information do we
need?
Benefits for
Immigrants
Males
Yes
15 (60%)
5 (20%)
No
10 (40%)
20 (80%)
25 (100%)
25 (100%)
Total
Female
How would you write a sentence
or two to describe what is in this
table?
Rules for cross-tabulation
• Calculate either column or row percents.
• Calculations are the number of frequencies
in a cell of a table divided by the total
number of frequencies in that column or
row, for example 20/25 = 80.0%
• All percentages in a column or row should
total 100%.
Let’s look at another example –
social work degrees by gender
Social Work
Degree
BA
Male
Female
20 (33.3%)
20 (
%)
MSW
30 (
)
70 (70.0%)
Ph.D.
10 (16.7%)
10 (10.0%)
60 (100.0%)
100 (100.0%
Questions:
What group had the largest percentage of
Ph.Ds?
What are the ways in which you could
find the missing numbers?
Is it obvious why you would use
percentages to make comparisons among
two or more groups?
In the following table, were people with drug,
alcohol, or a combination of both most likely
to be referred for individual treatment?
Services
Individual
Treatment
Group
Treatment
AA
Total
Alcohol
Drugs
Both
10 (25%)
30 (60%)
5 (50%)
10 (25%)
10 (20%)
2 (20%)
20 (50%)
10 (20%)
3 (30%)
40 (100%)
50 (100%)
10 (100%)
Use the same table to answer the
following question:
How much more likely are
people with alcohol problems
alone to be referred to AA than
people with drug problems or a
combination of drug and alcohol
problems?
We use cross-tabulation when:
• We want to look at relationships among two
or three variables.
• We want a descriptive statistical measure to
tell us whether differences among groups
are large enough to indicate some sort of
relationship among variables.
Cross-tabs are not sufficient to:
• Tell us the strength or actually size of the relationships
among two or three variables.
• Test a hypothesis about the relationship between two or
three variables.
• Tell us the direction of the relationship among two or more
variables.
• Look at relationships between one nominal or ordinal
variable and one ratio or interval variable unless the range
of possible values for the ratio or interval variable is small.
What do you think a table with a large number of ratio
values would look like?
We can use cross-tabs to visually
assess whether independent and
dependent variables might be
related. In addition, we also use
cross-tabs to find out if
demographic variables such as
gender and ethnicity are related
to the second variable.
For example, gender may
Democratic or Republican or if
income is high, medium, or low.
Ethnicity might be related to
where someone lives or attitudes
Because we use tables in these ways, we can
set up some decision rules about how to use
tables.
• Independent variables should be column variables.
• If you are not looking at independent and
dependent variable relationships, use the variable
that can logically be said to influence the other as
• Using this rule, always calculate column
percentages rather than row percentages.
• Use the column percentages to interpret your
results.
For example,
• If we were looking at the relationship between
gender and income, gender would be the column
variable and income would be the row variable.
Logically gender can determine income. Income
• If we were looking at the relationship between
ethnicity and location of a person’s home,
ethnicity would be the column variable.
• However, if we were looking at the relationship
between gender and ethnicity, one does not
influence the other. Either variable could be the
column variable.
SPSS will allow you to choose a
column variable and row variable
and whether or not your table
will include column or row
percents.
You must use an additional statistic, chisquare, if you want to:
• Test a hypothesis about two variables.
• Look at the strength of the relationship between an
independent and dependent variable.
• Determine whether the relationship between the
two variables is large enough to rule out random
chance or sampling error as reasons that there
appears to be a relationship between the two
variables.
Chi-square is simply an extension of a
cross-tabulation that gives you more
However, it provides no information
about the direction of the relationship
(positive or negative) between the two
variables.
Let’s use the following table to
test a hypothesis:
Education
Income
High (Above
\$40,000)
High
Low
Total
40
50
Low (\$39,999
or less)
Total
50
50
50
100
I have not filled in all of the information
because we need to talk about two concepts
before we start calculations:
• Degrees of Freedom: In any table, there are
a limited number of choices for the values
in each cell.
• Marginals: Total frequencies in columns and
rows.
Let’s look at the number of choices
we have in the previous table:
Education
Income
High (Above
\$40,000)
High
Low
Total
40
50
Low (\$39,999
or less)
Total
50
50
50
100
So the table becomes:
Education
Income
High
Low
Total
High (Above
\$40,000)
40
10
50
Low (\$39,999
or less)
10
40
50
Total
50
50
100
The rules for determining degrees of freedom
in cross-tabulations or contingency tables:
• In any two by two tables (two columns, two
rows, excluding marginals) DF = 1.
• For all other tables, calculate DF as:
(c -1 ) * (r-1) where c = columns and r =
rows.
( So for a table with 3 columns and 4 rows,
DF = ____. )
Importance of Degrees of Freedom
• You will see degrees of freedom on your SPSS
print out.
• Most types of inferential statistics use DF in
calculations.
• In chi-square, we need to know DF if we are
calculating chi-square by hand. You must use the
value of the chi-square and DF to determine if the
chi-square value is large enough to be statistically
significant (consult chi-square table in most
statistics books).
Steps in testing a hypothesis:
• State the research hypothesis
• State the null hypothesis
• Choose a level of statistical significance
(alpha level)
• Select and compute the test statistic
• Make a decision regarding whether to
accept or reject the null hypothesis.
Calculating Chi-Square
• Formula is [0 - E]2
E
Where 0 is the observed value in a cell
E is the expected value in the same
cell we would see if there was no
association
First steps
Alternative hypothesis is: There is a relationship
between income level and education for
respondents in a survey of BA students.
Null hypothesis is: There is no relationship between
income level and education for respondents in a
survey of BA students
Confidence level set at .05
Rules for determining whether the chi-square
statistic and probability are large enough to verify a
relationship.
• For hand calculations, use the degree(s) of
freedom and the confidence level you set to check
the Chi-square table found in most statistics
books. For the chi-square to be statistically
significant, it must be the same size or larger than
the number in the table.
• On an SPSS print out, the p. or significance value
must be the same size or smaller than your
significance level.
The formula for expected values are
E = R*C
Education
Income
High
Low
Total
High (Above
\$40,000)
25
25
50
Low (\$39,999
or less)
25
25
50
Total
50
50
100
Go back to our first table
Education
Income
High
Low
Total
High (Above
\$40,000)
40
10
50
Low (\$39,999
or less)
10
40
50
Total
50
50
100
Chi-square calculation is
Cell 1
Cell 2
Expected
Values
50 * 50/100
50*50/100
Chi-square
25 (40-25)2/25
25 (10-25)2/25
9
9
Cell 3
Cell 4
50 * 50/100
50*50/100
25 (10-25)2/25
25 (40-25)2/25
9
9
36
At .05, 1 = df, chi-square must be larger
than 3.84 to be statistically significant
Let’s calculate another chi-square- service
receipt by location of residence
Service
Urban
Rural
Total
Yes
20
40
60
No
30
10
40
Total
50
50
100
For this table,
• DF = 1
• Alternative hypothesis:
Receiving service is associated with
location of residence.
Null hypothesis:
There is no association between receiving
service and location of residence.
Calculations for chi-square are
Cell 1
Cell 2
Expected
Values
50 * 60/100
50*40/100
Chi-square
30 (20-30)2/30
20 (30-20)2/20
3.33
5.00
Cell 3
Cell 4
50*60/100
50*40/100
30 (40-30)2/30
20 (10-20)2/20
3.33
5.00
16.67
At 1 DF at .01 chi-square must be greater than 6.64. Do
we accept or reject the null hypothesis?
Running chi-square in SPSS
•
•
•
•
•
•
•
•
•
•
•
Select descriptive statistics
Select cross-tabulation
Highlight your independent variable and click on the arrow.
Highlight your dependent variable and click on the arrow.
Select Cells
Choose column percents
Click continue
Select statistics
Select chi-square
Click continue
Click ok
SPSS print out
Chi-Square Tests
Pearson Chi-Square
Likel ihood Ratio
Linear-by-Linear
Association
N of Val id Cases
Value
2.569a
2.590
.087
5
5
Asymp. Sig.
(2-sided)
.766
.763
1
.768
df
336
a. 2 cells (16.7%) have expected count less than 5. The
mi nimum expected count is 1.57.
Recode
• To run ratio or interval level variables into SPSS
you need to recode or change the variable into a
categorical or nominal or ordinal variable.
You first need to decide how you will set up
categories and assign a number to them.
For example if your ratio variables for Age are: 25,
37, 42, 50, and 64, you might decide on two
categories: 1 = under 50
2 = 50 and over
Recode Instructions
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Go to Recode
Select different variable
Type in new variable name
Click continue
Enter range of ratio numbers for first category (25 to 49)
Enter number for first category (1) in right hand screen.
Enter range of ratio numbers (50 to 54) for category two
Enter number for second category (2)
Click Continue
Click Change
Click o.k.
```