Chapter 22
Bivariate Statistical Analysis: Differences Between
Two Variables
AT-A-GLANCE
I. What Is the Appropriate Test of Difference?
II. Cross-Tabulation Tables: The 2 Test for Goodness of Fit
III. The t-Test for Comparing Two Means
A. Independent samples t-Test
Independent samples t-test calculation
Practically speaking
B. Paired samples t-test
IV. The Z-Test for Comparing Two Proportions
V. Analysis of Variance (ANOVA)
A. What is ANOVA?
B. Simple illustration of ANOVA
C. Partitioning variance in ANOVA
Total variability
Between-groups variance
Within-group error
D. The F-Test
Using variance components to compute F-ratios
A different but equivalent representation
E. Practically speaking
VI. Appendix 22A: Manual Calculation of an F-Statistic
VII. Appendix 22B: ANOVA for Complex Experimental Designs
A. Factorial designs
ANOVA for a factorial experiment
Partitioning the sum of squares for a two-way ANOVA
LEARNING OUTCOMES
1. Recognize when a particular bivariate statistical test is appropriate
2. Calculate and interpret a 2 test for a contingency table
3. Calculate and interpret an independent samples t-test comparing two means
4. Understand the concept of analysis of variance (ANOVA)
5. Interpret an ANOVA table
CHAPTER VIGNETTE: Gender Differences and Double Standards in
Ethical Perceptions
Ethical conduct, both of businesses and consumers, is an important issue in the business world. A
research studied examined if there is a difference between women and men in their ethical perceptions
and if there is a double standard—that consumers view an action performed by a customer as more ethical
than the same action performed by a business. Two unethical behavior scenarios were created—one
having a consumer engaging in the unethical act while the other had a business performing the act. While
men rated the activity as more ethical than women, statistical tests show that there is not a significant
difference. However, people significantly perceive the same act as less ethical when performed by a
business than by a consumer.
SURVEY THIS!
The survey data can be analyzed with the techniques discussed in this chapter. Students are asked to
address the following questions:
1. Is there a relationship between student gender and their major? Use cross-tabulations and the
χ2 test.
2. Is there a difference between those respondents that are currently employed and those that are
not regarding their goal achievement and life satisfaction? Use a t-test.
3. Is there a difference among the various student classifications and their attitude regarding
their goal achievement and life satisfaction? Use ANOVA.
RESEARCH SNAPSHOTS
Accurate Information? How About a Chi-Square Test?
When is a cross tabulation with a 2 appropriate? When the answer is “yes” to the following
questions:
Are multiple variables expected to be related to one another?
Is the independent variable nominal or ordinal?
Is the dependent variable nominal or ordinal?
An example of whether or not the adoption of a new information system produced accurate or
inaccurate information is provided, and the 2 X 2 contingency table with the 2 is given and
indicates that the new technology is associated with more incidences of accurate rather than
inaccurate information.
Expert “T-eeze”
When is an independent samples t-test appropriate? When the answer is “yes” to the following
questions:
Is the dependent variable interval or ratio?
Can the dependent variable scores be grouped based upon some categorical variable?
Does the grouping result in scores drawn from independent samples?
Are two groups involved in the research question?
An example looking at the difference in speed for expert and novice salespeople faced with the
same situation is given. Decision speed is a ratio dependent variable and the scores are grouped
based on whether or not the salesperson is an expert or a novice, which produces two groups.
The conclusion is that experts do take less time to make a decision than do novices.
More Than One-Way
An independent samples t-test is a special case of one-way ANOVA. When the independent
variable in ANOVA has only two groups, the results for an independent samples t-test and
ANOVA will be the same, and an example is given to show this. The F-ratio shown in the
ANOVA table is associated with the same p-value as is the t-value, which is no accident since the
F and t are mathematical functions of one another.
OUTLINE
I. WHAT IS THE APPROPRIATE TEST OF DIFFERENCE?
Researchers commonly test hypotheses stating that two groups differ.
Such tests are bivariate tests of differences when they involve only two variables (i.e., a
variable acts like a dependent variable and a variable acts as a classification variable).
Exhibit 22.1 illustrates that both the type of measurement and the number of groups to be
compared influence the type of bivariate statistical analysis.
II. CROSS-TABULATION TABLES: THE 2 TEST FOR GOODNESS-OF-FIT
One of the most widely used and simplest techniques for describing sets of relationships is
the cross-tabulation.
A cross-tabulation, or contingency table, is a joint frequency distribution of observations on
two or more variables.
The 2 distribution provides a means for testing the statistical significance of contingency
tables.
The test involves comparing the observed frequencies (Oi) with the expected frequencies (Ei)
in each cell of the table.
The goodness- (or closeness-) of-fit of the observed distribution with the expected
distribution is captured by this statistic.
The test allows us to conduct tests for significance in the analysis of the R x C contingency
table (where R = row and C = column).
The formula for the 2 statistic is the same as that for one-way frequency tables (see Chapter
21).
To compute an expected number for each cell use the formula
E
ij
= R
i
C
j
n
where Ri= total observed frequency in the ith row
Cj= total observed frequency in the jth column
n = sample size
To compute a chi-square statistic the same formula as before is used, except that we calculate
degrees of freedom as the number of rows minus one (R – 1) times the number of columns
minus one (C – 1).
Testing the hypothesis involves two key steps:
1. Examine the statistical significance of the observed contingency table.
2. Examine to see if the differences between the observed and expected values are
consistent with the hypothesized prediction.
Proper use of the chi-square test requires that each expected cell frequency (Eij) have a value
of at least 5.
If this sample size requirement is not met, the researcher should take a larger sample or
combine (collapse) response categories.
III. THE t-TEST FOR COMPARING TWO MEANS
Independent Samples t-test
A t-test is appropriate for when a researcher needs to compare means for a variable
grouped into two categories based on some less than interval variable.
One way to think about this is as testing the way a dichotomous (two-level) independent
variable is associated with changes in a continuous dependent variable.
Most typically, the researcher will apply the independent samples t-test which tests the
differences between means taken from two independent samples or groups.
This test assumes the two samples are drawn from normal distributions and that the
variances of the two populations are approximately equal (homoscedasticity).
Independent Samples t-test Calculation
The t-test actually tests whether or not the differences between two means is zero.
The null hypothesis is normally stated as:
m
1
= m
2
or m
1
– m
2
= 0
However, since this is inferential statistics, we test the idea by comparing two sample
means (
21
XX
).
Thus, the t-value is a ratio with information about the differences between means
(provided by the sample) in the numerator and the standard error in the denominator.
The question is whether the observed differences have occurred by chance alone.
A pooled estimate of the standard error is a better estimate of the standard error
than one based on the variance from either sample.
A higher t-value is associated with a lower p-value, and as the t gets higher and the
p-value gets lower, the researcher has more confidence that the means are truly
different.
In a test of two means, the degrees of freedom are calculated as follows:
d.f. = n – k (where n = n1 + n2 and k = number of groups)
Table A.3 in the appendix yields the critical t-values.
Practically Speaking
In practice, computer software is used to compute the t-test results.
Exhibit 22-2 displays a typical t-test printout.
These particular results examine the following research question:
RQ: Does religion relate to price sensitivity?
Because no direction of the relationship is stated (no hypothesis is offered), a
two-tailed test is appropriate.
The interpretation of the t-test is made simply by focusing on either the p-value or the
confidence interval and the group means.
Basic steps:
1. Examine the difference in means to find the “direction” of any difference.
2. Compute or locate the computed t-test value.
3. Find the p-value associated with this t and the corresponding degrees of freedom.
4. The difference can also be examined using the 95 percent confidence interval,
and if the interval does not include 0, we lack sufficient confidence that the true
difference between the population mean is 0.
Note:
1. Strictly speaking, the t-test assumes that the two population variances are equal –
a slightly more complicated formula exists which will compute the t-statistic
assuming they are not equal, and SPSS provides both results when an
independent samples t-test is performed.
The rule of thumb in business research is to use the equal variance
assumption, and in the vast majority of cases, the same conclusion will be
drawn using either assumption.
2. Even though the means appear to be not so close to each other, the statistical
conclusion could be that they are the same due to the variance because the
t-statistic is a function of the standard error, which is a function of the standard
deviation.
Check for outliers.
Consider increasing the sample size and test again.
3. As samples get larger, the t-test and Z-test will tend to yield the same result.
A t-test can be used with large samples.
A Z-test should not be used with small samples.
Also, a Z-test can be used in instances where the population variance is
known ahead of time.
Paired Samples t-Test
A paired samples t-test is appropriate when means that need to be compared are not
from independent samples (i.e., the same respondent is measured twice).
When a paired samples t-test is appropriate, the two numbers being compared are usually
scored as separate variables.
IV. THE Z-TEST FOR COMPARING TWO PROPORTIONS
Testing whether the population proportion for one group equals the population proportion for
another group is conceptually the same as the t-test of two means, and the Z-test for
differences of proportions is used to test the hypothesis that the two proportions will be
significantly different for two independent samples or groups.
This test requires a sample size greater than 30.
The test is appropriate for a hypothesis of this form:
H0: 1 2
or
H0: 1 2 = 0
The comparison of the observed sample proportions p1 and p2 allows the researcher to ask
whether the differences from two large random samples occurred due to chance alone.
The Z-test statistic can be computed using the following formula:
Z =
21
2121
pp
S
pp
where
p1 = sample proportion of successes in group 1
p2 = sample proportion of successes in group 2
π 1 – π 2 = hypothesized population proportion 1 minus hypothesized population
proportion 2
Sp1–p2 = pooled estimate of the standard error of differences in proportions
The statistic normally works on the assumption that the value of π 1 – π 2 is zero, so this formula
is actually much simpler than it looks at first inspection.
Also notice the similarity between this and the paired-samples t-test.
To calculate the standard error of the differences in proportions, use the formula:
Sp1–p2 =
21
11
nn
qp
where
p
= pooled estimate of proportion of successes in a sample
q
= 1 –
p
, or pooled estimate of proportion of failures in a sample
n1= sample size for group 1
n2= sample size for group 2
To calculate the pooled estimator,
p
, use the formula:
21
2211
nn
pnpn
p
V. ANALYSIS OF VARIANCE (ANOVA)
What is ANOVA?
When the means of more than two groups or populations are to be compared, one-way
analysis of variance (ANOVA) is the appropriate statistical tool.
ANOVA involving only one grouping variable is often referred to as one-way ANOVA
because only one independent variable is involved.
Another way to define ANOVA is as the appropriate statistical technique to examine the
effect of a less than interval independent variable on an at least interval dependent
variable.
An independent samples t-test can be thought of as a special case of ANOVA in which the
independent variable has only two levels.
When more levels exist, the t-test alone cannot handle the problem.
The null hypothesis in such a test is that all the means are equal—that is,
m
1 = m
2 = m
3
… up to K where K is the number of groups or categories for an independent variable.
The substantive hypothesis tests in ANOVA is: “At least one group mean is not equal to
another group mean.”
As the term analysis of variance suggests, the problem requires comparing variances to
make inferences about the means.
Simple Illustration of ANOVA
Data are given describing how much coffee respondents report drinking each day based
on which shift they work (i.e., day shift, second shift, or nights).
A table displaying the means for each group and the overall mean is given, and Exhibit
22.5 plots each observation with a bar and lines corresponding to the variances.
Partitioning Variance in ANOVA
Total Variability
An implicit question with the use of ANOVA is “How can the dependent variable best
be predicted?”
Absent any additional information, the error in predicting an observation is
minimized by choosing the central tendency, or mean, for an interval variable.
The total error (or variability) that would result from using the grand mean, meaning
the mean over all observations, can be thought of as:
SST = Total of (observed value – grand mean)2
Although the term error is used, this really represents how much total variation exists
among the measures.
Between-Groups Variance
ANOVA tests whether “grouping” observations explains variance in the dependent
variable.
The between groups variance can be found by taking the total sum of the weighted
difference between group means and the overall mean:
SSB = Total of ngroup(Group Mean – Grand Mean)2
The weighting factor (ngroup) is the specific group sample size.
The total SSB represents the variation explained by the experimental or independent
variable.
Within-Group Error
Finally, error within each group would remain.
While the group means explain the variation between the total mean and the group
mean, the distance from the group mean and each individual observation remains
unexplained, and this distance is called within-group error or variance.
The values for each observation can be found by:
SSE = Total of (Observed Mean – Group Mean)2
The term total error variance is sometimes used to refer to SSE since it is variability
not accounted for by the group means.
The F-Test
The F-Test is the key statistical test for an ANOVA model.
Determines whether there is more variability in the scores of one sample than in the
scores of another sample.
The key question is whether the two sample variances are different from each other or
whether they are from the same population.
Thus, the test breaks down the variance in a total sample and illustrates why ANOVA is
analysis of variance.
The F-statistic (of F-ratio) can be obtained by taking the larger sample variance and
dividing by the smaller sample variance.
Table A.5 or A.6 (used much like using the tables of the Z– and t-distributions) indicates
that the distribution of F is actually a family of distributions that change quite drastically
with changes in sample sizes.
Degrees of freedom must be specified.
Using Variance Components to Compute F-Ratios
In ANOVA, the basic consideration for the F-test is identifying the relative size of
variance components.
The three forms of variation described briefly are:
1. SSE – variation of scores due to random error or within-group variation due to
individual differences from the group mean. This is the error of prediction.
2. SSB – systematic variation of scores between groups due to manipulation of an
experimental variable or group classifications of a measured independent
variable or between-groups variance.
3. SST – the total observed variation across all groups and individual observations.
Thus, total variability can be partitioned into within-group variance and
between-group variance.
The F-distribution is a function of the ratio of these two sources of variance:
SSE
SSB
fF
A larger ratio of variance between groups to variance within groups implies a greater
value of F.
If the F-value is large, the results are likely to be statistically significant.
A Different but Equivalent Representation
F also can be thoughts of as a function of the between group variance and total
variance.
SSBSST
SSB
fF
Appendix 22A explains the calculations in more detail with an illustration.
Practically Speaking
The first thing to check is whether or not the overall model F is significant.
Second, the researcher must remember to examine the actual means from each group to
properly interpret the result.
A later chapter describes ways of examining specifically which group means are different
form one another.
in gaining a thorough understanding of ANOVA. The process to calculate an F-ratio is
explained through an example.
VII. APPENDIX 22B: ANOVA FOR COMPLEX EXPERIMENTAL DESIGNS
Appendix 22B explains the use of ANOVA for more complex experimental designs such as
the randomized block design and the factorial design.