Chapter 9
Comparing Two Groups
 Learn
….
How to Compare Two Groups On
a Categorical or Quantitative
Outcome Using Confidence
Intervals and Significance Tests
Agresti/Franklin Statistics, 1 of 111
Bivariate Analyses

The outcome variable is the response
variable

The binary variable that specifies the
groups is the explanatory variable
Agresti/Franklin Statistics, 2 of 111
Bivariate Analyses

Statistical methods analyze how the
outcome on the response variable
depends on or is explained by the
value of the explanatory variable
Agresti/Franklin Statistics, 3 of 111
Independent Samples

The observations in one sample are
independent of those in the other
sample
• Example:
•
Randomized experiments that
randomly allocate subjects to two
treatments
Example: An observational study that
separates subjects into groups according
to their value for an explanatory variable
Agresti/Franklin Statistics, 4 of 111
Dependent Samples

Data are matched pairs – each subject
in one sample is matched with a
subject in the other sample
• Example:
•
set of married couples, the men
being in one sample and the women in the
other.
Example: Each subject is observed at two
times, so the two samples have the same
people
Agresti/Franklin Statistics, 5 of 111
 Section 9.1
Categorical Response: How Can We
Compare Two Proportions?
Agresti/Franklin Statistics, 6 of 111
Categorical Response
Variable


Inferences compare groups in terms
of their population proportions in a
particular category
We can compare the groups by the
difference in their population
proportions:
(p1 – p2)
Agresti/Franklin Statistics, 7 of 111
Example: Aspirin, the Wonder
Drug

Recent Titles of Newspaper Articles:
• “Aspirin cuts deaths after heart attack”
• “Aspirin could lower risk of ovarian
•
•
cancer”
“New study finds a daily aspirin lowers the
risk of colon cancer”
“Aspirin may lower the risk of Hodgkin’s”
Agresti/Franklin Statistics, 8 of 111
Example: Aspirin, the Wonder
Drug

The Physicians Health Study
Research Group at Harvard Medical
School
• Five year randomized study
• Does regular aspirin intake reduce
deaths from heart disease?
Agresti/Franklin Statistics, 9 of 111
Example: Aspirin, the Wonder
Drug

Experiment:
•
•
•
•
Subjects were 22,071 male physicians
Every other day, study participants took either an
aspirin or a placebo
The physicians were randomly assigned to the
aspirin or to the placebo group
The study was double-blind: the physicians did
not know which pill they were taking, nor did
those who evaluated the results
Agresti/Franklin Statistics, 10 of 111
Example: Aspirin, the Wonder
Drug
Results displayed in a contingency table:
Agresti/Franklin Statistics, 11 of 111
Example: Aspirin, the Wonder
Drug

What is the response variable?

What are the groups to compare?
Agresti/Franklin Statistics, 12 of 111
Example: Aspirin, the Wonder
Drug

The response variable is whether the
subject had a heart attack, with
categories ‘yes’ or ‘no’

The groups to compare are:
• Group 1:
• Group 2:
Physicians who took a placebo
Physicians who took aspirin
Agresti/Franklin Statistics, 13 of 111
Example: Aspirin, the Wonder
Drug

Estimate the difference between the
two population parameters of interest
Agresti/Franklin Statistics, 14 of 111
Example: Aspirin, the Wonder
Drug


p1: the proportion of the population
who would have a heart attack if they
participated in this experiment and
took the placebo
p2: the proportion of the population
who would have a heart attack if they
participated in this experiment and
took the aspirin
Agresti/Franklin Statistics, 15 of 111
Example: Aspirin, the Wonder
Drug
Sample Statistics:
pˆ  189 / 11034  0.017
1
pˆ  104 / 11037  0.009
2
( pˆ  pˆ )  0.017  0.009  0.008
1
2
Agresti/Franklin Statistics, 16 of 111
Example: Aspirin, the Wonder
Drug

To make an inference about the
difference of population proportions,
(p1 – p2), we need to learn about the
variability of the sampling distribution of:
( pˆ  pˆ )
1
2
Agresti/Franklin Statistics, 17 of 111
Standard Error for Comparing
Two Proportions



The difference, ( pˆ  pˆ ) , is obtained from
sample data
It will vary from sample to sample
1
2
This variation is the standard error of the
sampling distribution of ( pˆ  pˆ ) :
1
se 
2
pˆ (1  pˆ ) pˆ (1  pˆ )

n
n
1
1
1
2
2
2
Agresti/Franklin Statistics, 18 of 111
Confidence Interval for the
Difference between Two Population
Proportions
pˆ (1  pˆ ) pˆ (1  pˆ )
( pˆ  pˆ )  z

n
n
1
1
1
2
2
2
1
2

The z-score depends on the confidence level

This method requires:
• Independent random samples for the two groups
•
Large enough sample sizes so that there are at
least 10 “successes” and at least 10 “failures” in
each group
Agresti/Franklin Statistics, 19 of 111
Confidence Interval Comparing
Heart Attack Rates for Aspirin and
Placebo

95% CI:
.017(1  .017) .009(1  .009)
(.017  .009)  1.96


11034
11037
0.008  0.003, or (0.005, 0.011)
Agresti/Franklin Statistics, 20 of 111
Confidence Interval Comparing
Heart Attack Rates for Aspirin and
Placebo

Since both endpoints of the confidence
interval (0.005, 0.011) for (p1- p2) are
positive, we infer that (p1- p2) is positive

Conclusion: The population proportion of
heart attacks is larger when subjects take
the placebo than when they take aspirin
Agresti/Franklin Statistics, 21 of 111
Confidence Interval Comparing
Heart Attack Rates for Aspirin and
Placebo



The population difference (0.005, 0.011) is
small
Even though it is a small difference, it may
be important in public health terms
For example, a decrease of 0.01 over a 5
year period in the proportion of people
suffering heart attacks would mean 2
million fewer people having heart attacks
Agresti/Franklin Statistics, 22 of 111
Confidence Interval Comparing
Heart Attack Rates for Aspirin and
Placebo

The study used male doctors in the
U.S
• The inference applies to the U.S.
population of male doctors

Before concluding that aspirin
benefits a larger population, we’d
want to see results of studies with
more diverse groups
Agresti/Franklin Statistics, 23 of 111
Interpreting a Confidence Interval
for a Difference of Proportions





Check whether 0 falls in the CI
If so, it is plausible that the population
proportions are equal
If all values in the CI for (p1- p2) are
positive, you can infer that (p1- p2) >0
If all values in the CI for (p1- p2) are
negative, you can infer that (p1- p2) <0
Which group is labeled ‘1’ and which is
labeled ‘2’ is arbitrary
Agresti/Franklin Statistics, 24 of 111
Interpreting a Confidence Interval
for a Difference of Proportions


The magnitude of values in the
confidence interval tells you how
large any true difference is
If all values in the confidence interval
are near 0, the true difference may be
relatively small in practical terms
Agresti/Franklin Statistics, 25 of 111
Significance Tests Comparing
Population Proportions
1. Assumptions:


Categorical response variable for two
groups
Independent random samples
Agresti/Franklin Statistics, 26 of 111
Significance Tests Comparing
Population Proportions
Assumptions (continued):

Significance tests comparing proportions use
the sample size guideline from confidence
intervals: Each sample should have at least
about 10 “successes” and 10 “failures”

Two–sided tests are robust against violations of
this condition
•
At least 5 “successes” and 5 “failures” is adequate
Agresti/Franklin Statistics, 27 of 111
Significance Tests Comparing
Population Proportions
2. Hypotheses:
 The null hypothesis is the hypothesis of
no difference or no effect:
H0: (p1- p2) =0
• Under the presumption that p1= p2, we
create a pooled estimate of the common
value of p1and p2
• This pooled estimate is p̂
Agresti/Franklin Statistics, 28 of 111
Significance Tests Comparing
Population Proportions
2. Hypotheses (continued):
Ha: (p1- p2) ≠ 0 (two-sided test)
Ha: (p1- p2) < 0 (one-sided test)
Ha: (p1- p2) > 0 (one-sided test)
Agresti/Franklin Statistics, 29 of 111
Significance Tests Comparing
Population Proportions
3. The test statistic is:
z
( pˆ  pˆ )  0
pˆ (1  pˆ ) pˆ (1  pˆ )

n
n
1
2
1
Agresti/Franklin Statistics, 30 of 111
2
Significance Tests Comparing
Population Proportions
4. P-value: Probability obtained from
the standard normal table
5. Conclusion: Smaller P-values give
stronger evidence against H0 and
supporting Ha
Agresti/Franklin Statistics, 31 of 111
Example: Is TV Watching
Associated with Aggressive
Behavior?

Various studies have examined a link
between TV violence and aggressive
behavior by those who watch a lot of TV

A study sampled 707 families in two
counties in New York state and made
follow-up observations over 17 years

The data shows levels of TV watching
along with incidents of aggressive acts
Agresti/Franklin Statistics, 32 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?
Agresti/Franklin Statistics, 33 of 111
Example: Is TV Watching
Associated with Aggressive
Behavior?
Test the Hypotheses:
H0: (p1- p2) = 0
Ha: (p1- p2) ≠ 0
• Using a significance level of 0.05
• Group 1:
• Group 2:
less than 1 hr. of TV per day
at least 1 hr. of TV per day
Agresti/Franklin Statistics, 34 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?
Agresti/Franklin Statistics, 35 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?



Conclusion: Since the P-value is less
than 0.05, we reject H0
We conclude that the population
proportions of aggressive acts differ
for the two groups
The sample values suggest that the
population proportion is higher for
the higher level of TV watching
Agresti/Franklin Statistics, 36 of 111
In 2002, the median net worth was estimated as
$89,000 for white households and $6000 for
black households.
What is the response variable?
a.
Net worth
b. Households: white or black
Agresti/Franklin Statistics, 37 of 111
In 2002, the median net worth was estimated as
$89,000 for white households and $6000 for
black households.
What is the explanatory variable?
a.
Net worth
b. Households: white or black
Agresti/Franklin Statistics, 38 of 111
In 2002, the median net worth was estimated as
$89,000 for white households and $6000 for black
households.
Identify the two groups that are the categories of
the explanatory variable.
a. White and Black households
b. Net worth and households
Agresti/Franklin Statistics, 39 of 111
In 2002, the median net worth was estimated as
$89,000 for white households and $6000 for black
households.
The estimated medians were based on a sample
of households. Were the samples of white
households and black households independent
samples or dependent samples?
a. Independent samples
b. Dependent samples
Agresti/Franklin Statistics, 40 of 111
Section 9.2
Quantitative Response: How
Can We Compare Two Means?
Agresti/Franklin Statistics, 41 of 111
Comparing Means

We can compare two groups on a
quantitative response variable
by comparing their means
Agresti/Franklin Statistics, 42 of 111
Example: Teenagers Hooked on
Nicotine

A 30-month study:
• Evaluated the degree of addiction that
teenagers form to nicotine
• 332 students who had used nicotine
were evaluated
• The response variable was constructed
using a questionnaire called the
Hooked on Nicotine Checklist (HONC)
Agresti/Franklin Statistics, 43 of 111
Example: Teenagers Hooked on
Nicotine


The HONC score is the total number
of questions to which a student
answered “yes” during the study
The higher the score, the more
hooked on nicotine a student is
judged to be
Agresti/Franklin Statistics, 44 of 111
Example: Teenagers Hooked on
Nicotine

The study considered explanatory
variables, such as gender, that might be
associated with the HONC score
Agresti/Franklin Statistics, 45 of 111
Example: Teenagers Hooked on
Nicotine

How can we compare the sample
HONC scores for females and males?

We estimate (µ1 - µ2) by (x1 - x2):
2.8 – 1.6 = 1.2

On average, females answered “yes”
to about one more question on the
HONC scale than males did
Agresti/Franklin Statistics, 46 of 111
Example: Teenagers Hooked on
Nicotine

To make an inference about the
difference between population means,
(µ1 – µ2), we need to learn about the
variability of the sampling distribution of:
(x  x )
1
2
Agresti/Franklin Statistics, 47 of 111
Standard Error for Comparing
Two Means

The difference, (x  x ) , is obtained from
sample data. It will vary from sample to
sample.

This variation is the standard error of the
sampling distribution of (x  x ) :
1
2
1
2
se 
s
s

n n
1
1
2
2
2
2
Agresti/Franklin Statistics, 48 of 111
Confidence Interval for the
Difference between Two Population
Means

A 95% CI:
(x  x )  t
1
2
2
s
s

n
n
1
.025
1

2
2
2
Software provides the t-score with righttail probability of 0.025
Agresti/Franklin Statistics, 49 of 111
Confidence Interval for the
Difference between Two Population
Means

This method assumes:
• Independent random samples from the two
groups
• An approximately normal population
distribution for each group
• this is mainly important for small sample sizes,
and even then the method is robust to
violations of this assumption
Agresti/Franklin Statistics, 50 of 111
Example: Nicotine – How Much More
Addicted Are Smokers than
Ex-Smokers?

Data as summarized by HONC scores
for the two groups:

Smokers: x1 = 5.9, s1 = 3.3, n1 = 75

Ex-smokers:x2 = 1.0, s2 = 2.3, n2 = 257
Agresti/Franklin Statistics, 51 of 111
Example: Nicotine – How Much More
Addicted Are Smokers than
Ex-Smokers?



Were the sample data for the two
groups approximately normal?
Most likely not for Group 2 (based on
the sample statistics): x2 = 1.0, s2 =
2.3)
Since the sample sizes are large, this
lack of normality is not a problem
Agresti/Franklin Statistics, 52 of 111
Example: Nicotine – How Much More
Addicted Are Smokers than
Ex-Smokers?

95% CI for (µ1- µ2):
2
2
3.3 2.3
(5.9  1)  1.985


75 257
4.9  0.8, or (4.1, 5.7)

We can infer that the population mean for the
smokers is between 4.1 higher and 5.7 higher
than for the ex-smokers
Agresti/Franklin Statistics, 53 of 111
How Can We Interpret a Confidence
Interval for a Difference of Means?


Check whether 0 falls in the interval
When it does, 0 is a plausible value for
(µ1 – µ2), meaning that it is possible that
µ1 = µ2

A confidence interval for (µ1 – µ2) that
contains only positive numbers suggests
that (µ1 – µ2) is positive

We then infer that µ1 is larger than µ2
Agresti/Franklin Statistics, 54 of 111
How Can We Interpret a Confidence
Interval for a Difference of Means?

A confidence interval for (µ1 – µ2) that
contains only negative numbers suggests
that (µ1 – µ2) is negative

We then infer that µ1 is smaller than µ2

Which group is labeled ‘1’ and which is
labeled ‘2’ is arbitrary
Agresti/Franklin Statistics, 55 of 111
Significance Tests Comparing
Population Means
1. Assumptions:
• Quantitative response variable for two
groups
• Independent random samples
Agresti/Franklin Statistics, 56 of 111
Significance Tests Comparing
Population Means
Assumptions (continued):

Approximately normal population
distributions for each group
•
This is mainly important for small sample sizes,
and even then the two-sided test is robust to
violations of this assumption
Agresti/Franklin Statistics, 57 of 111
Significance Tests Comparing
Population Means
2. Hypotheses:
The null hypothesis is the hypothesis of
no difference or no effect:
H0: (µ1- µ2) =0
Agresti/Franklin Statistics, 58 of 111
Significance Tests Comparing
Population Proportions
2. Hypotheses (continued):
The alternative hypothesis:
Ha: (µ1- µ2) ≠ 0 (two-sided test)
Ha: (µ1- µ2) < 0 (one-sided test)
Ha: (µ1- µ2) > 0 (one-sided test)
Agresti/Franklin Statistics, 59 of 111
Significance Tests Comparing
Population Means
3. The test statistic is:
(x  x )  0
z
s s

n n
1
2
2
2
1
2
1
2
Agresti/Franklin Statistics, 60 of 111
Significance Tests Comparing
Population Means
4. P-value: Probability obtained from
the standard normal table
5. Conclusion: Smaller P-values give
stronger evidence against H0 and
supporting Ha
Agresti/Franklin Statistics, 61 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Experiment:
• 64 college students
• 32 were randomly assigned to the cell
phone group
• 32 to the control group
Agresti/Franklin Statistics, 62 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Experiment (continued):
• Students used a machine that simulated
•
•
driving situations
At irregular periods a target flashed red or
green
Participants were instructed to press a
“brake button” as soon as possible when
they detected a red light
Agresti/Franklin Statistics, 63 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?



For each subject, the experiment
analyzed their mean response time
over all the trials
Averaged over all trials and subjects,
the mean response time for the cellphone group was 585.2 milliseconds
The mean response time for the
control group was 533.7 milliseconds
Agresti/Franklin Statistics, 64 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Data:
Agresti/Franklin Statistics, 65 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Test the hypotheses:
H0: (µ1- µ2) =0
vs.
Ha: (µ1- µ2) ≠ 0
•
using a significance level of 0.05
Agresti/Franklin Statistics, 66 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?
Agresti/Franklin Statistics, 67 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Conclusion:
• The P-value is less than 0.05, so we can
•
•
reject H0
There is enough evidence to conclude that
the population mean response times differ
between the cell phone and control groups
The sample means suggest that the
population mean is higher for the cell
phone group
Agresti/Franklin Statistics, 68 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

What do the box plots tell us?
• There is an extreme outlier for the cell
•
phone group
It is a good idea to make sure the results
of the analysis aren’t affected too strongly
by that single observation
• Delete the extreme outlier and redo the
analysis
• In this example, the t-statistic changes only
slightly
Agresti/Franklin Statistics, 69 of 111
Example: Does Cell Phone Use
While Driving Impair Reaction
Times?

Insight:
• In practice, you should not delete outliers
•
•
from a data set without sufficient cause
(i.e., if it seems the observation was
incorrectly recorded)
It is however, a good idea to check for
sensitivity of an analysis to an outlier
If the results change much, it means that
the inference including the outlier is on
shaky ground
Agresti/Franklin Statistics, 70 of 111
How much more time do women spend on
housework than men? Data is Hours per Week.
Sample Size Mean
Gender:
St. Dev.
Women
Men
6764
4252
32.6
18.1
What is a point estimate of µ1- µ2?
a. 18.2 – 12.9
b. 32.6 – 18.1
c. 6764 - 4252
d. 32.6/18.2 – 18.1/12.9
Agresti/Franklin Statistics, 71 of 111
18.2
12.9
How much more time do women spend on
housework than men? Data is Hours per Week.
Sample Size Mean
Gender:
St. Dev.
Women
Men
6764
4252
32.6
18.1
18.2
12.9
What is the standard error for comparing the
means?
a. 5.3
b. .076
c. .297
d. .088
Agresti/Franklin Statistics, 72 of 111
How much more time do women spend on
housework than men? Data is Hours per Week.
Sample Size Mean
Gender:
St. Dev.
Women
Men
6764
4252
32.6
18.1
18.2
12.9
What factor causes the standard error to be
small compared to the sample standard
deviations for the two groups?
a. sample means
b. sample standard deviations
c. sample sizes
d. genders
Agresti/Franklin Statistics, 73 of 111
Section 9.3
Other Ways of Comparing Means
and Comparing Proportions
Agresti/Franklin Statistics, 74 of 111
Alternative Method for
Comparing Means


An alternative t- method can be used
when, under the null hypothesis, it is
reasonable to expect the variability as
well as the mean to be the same
This method requires the assumption
that the population standard
deviations be equal
Agresti/Franklin Statistics, 75 of 111
The Pooled Standard Deviation

This alternative method estimates the
common value σ of σ1 and σ1 by:
(n  1) s  (n  1) s
s
n n 2
2
1
1
1
2
2
Agresti/Franklin Statistics, 76 of 111
2
2
Comparing Population Means,
Assuming Equal Population Standard
Deviations

Using the pooled standard deviation
estimate, a 95% CI for (µ1 - µ2) is:
1 1
(x  x )  t s

n n
1
2
.025
1

This method has df =n1+ n2- 2
Agresti/Franklin Statistics, 77 of 111
2
Comparing Population Means,
Assuming Equal Population Standard
Deviations

The test statistic for H0: µ1=µ2 is:
(x  x )
t
1
1
s

n n
1
2
1

2
This method has df =n1+ n2- 2
Agresti/Franklin Statistics, 78 of 111
Comparing Population Means,
Assuming Equal Population Standard
Deviations

These methods assume:
• Independent random samples from the two
•
groups
An approximately normal population
distribution for each group
• This is mainly important for small sample sizes,
and even then, the CI and the two-sided test are
usually robust to violations of this assumption
• σ1=σ2
Agresti/Franklin Statistics, 79 of 111
The Ratio of Proportions: The
Relative Risk

The ratio of proportions for two groups is:
pˆ
1

pˆ
2
In medical applications for which the
proportion refers to a category that is an
undesirable outcome, such as death or having
a heart attack, this ratio is called the relative
risk
Agresti/Franklin Statistics, 80 of 111
 Section 9.4
How Can We Analyze Dependent
Samples?
Agresti/Franklin Statistics, 81 of 111
Dependent Samples

Each observation in one sample has a
matched observation in the other
sample

The observations are called matched
pairs
Agresti/Franklin Statistics, 82 of 111
Example: Matched Pairs Design for
Cell Phones and Driving Study

The cell phone analysis presented
earlier in this text used independent
samples:
• One group used cell phones
• A separate control group did not use cell
phones
Agresti/Franklin Statistics, 83 of 111
Example: Matched Pairs Design for
Cell Phones and Driving Study

An alternative design used the same
subjects for both groups
• Reaction times are measured when
subjects performed the driving task
without using cell phones and then again
while using cell phones
Agresti/Franklin Statistics, 84 of 111
Example: Matched Pairs Design for
Cell Phones and Driving Study
Data:
Agresti/Franklin Statistics, 85 of 111
Example: Matched Pairs Design for
Cell Phones and Driving Study

Benefits of using dependent samples
(matched pairs):
• Many sources of potential bias are
•
•
controlled so we can make a more
accurate comparison
Using matched pairs keeps many other
factors fixed that could affect the analysis
Often this results in the benefit of smaller
standard errors
Agresti/Franklin Statistics, 86 of 111
Example: Matched Pairs Design for
Cell Phones and Driving Study

To Compare Means with Matched
Pairs, Use Paired Differences:
• For each matched pair, construct a
•
•
difference score
d = (reaction time using cell phone) –
(reaction time without cell phone)
Calculate the sample mean of these
differences: xd
Agresti/Franklin Statistics, 87 of 111
For Dependent Samples
(Matched Pairs)
Mean of Differences
=
Difference of Means
Agresti/Franklin Statistics, 88 of 111
For Dependent Samples
(Matched Pairs)


The difference (x1 – x2) between the
means of the two samples equals the
mean xd of the difference scores for
the matched pairs
The difference (µ1 – µ2) between the
population means is identical to the
parameter µd that is the population
mean of the difference scores
Agresti/Franklin Statistics, 89 of 111
For Dependent Samples
(Matched Pairs)



Let n denote the number of observations in each
sample
This equals the number of difference scores
The 95 % CI for the population mean difference
is:
sd
xd  t.025
n
xd is the sample mean of the difference s
s d is their standard deviation
Agresti/Franklin Statistics, 90 of 111
For Dependent Samples
(Matched Pairs)


To test the hypothesis H0: µ1 = µ2 of equal
means, we can conduct the single-sample test of
H0: µd = 0 with the difference scores
The test statistic is:
x 0
t
with df  n  1
s
n
d
d
Agresti/Franklin Statistics, 91 of 111
For Dependent Samples
(Matched Pairs)

These paired-difference inferences
are special cases of single-sample
inferences about a population mean
so they make the same assumptions
Agresti/Franklin Statistics, 92 of 111
Paired-difference Inferences

Assumptions:
• The sample of difference scores is a
•
random sample from a population of such
difference scores
The difference scores have a population
distribution that is approximately normal
•
This is mainly important for small samples
(less than about 30) and for one-sided
inferences
Agresti/Franklin Statistics, 93 of 111
Paired-difference Inferences


Confidence intervals and two-sided
tests are robust: They work quite well
even if the normality assumption is
violated
One-sided tests do not work well
when the sample size is small and the
distribution of differences is highly
skewed
Agresti/Franklin Statistics, 94 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study

Boxplot of the 32 difference scores
Agresti/Franklin Statistics, 95 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study

The box plot shows skew to the right
for the difference scores
• Two-sided inference is robust to violations
of the assumption of normality

The box plot does not show any
severe outliers
Agresti/Franklin Statistics, 96 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study
Agresti/Franklin Statistics, 97 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study

Significance test:
•
•
H0: µd = 0 (and hence equal population means for
the two conditions)
Ha: µd ≠ 0
•
Test statistic:
50.6
t
 5.46
52.5
32
Agresti/Franklin Statistics, 98 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study


The P-value displayed in the output is
0.000
There is extremely strong evidence
that the population mean reaction
times are different
Agresti/Franklin Statistics, 99 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study

95% CI for µd =(µ1 - µ2):
52.5
50.6  2.040(
)  50.6  18.9
32
or (31.7, 69.5)
Agresti/Franklin Statistics, 100 of 111
Example: Matched Pairs Analysis for
Cell Phones and Driving Study


We infer that the population mean when
using cell phones is between about 32
and 70 milliseconds higher than when not
using cell phones
The confidence interval is more
informative than the significance test,
since it predicts just how large the
difference must be
Agresti/Franklin Statistics, 101 of 111
Section 9.5
How Can We Adjust for Effects of
Other Variables?
Agresti/Franklin Statistics, 102 of 111
A Practically Significant
Difference

When we find a practically significant
difference between two groups, can
we identify a reason for the
difference?

Warning: An association may be due
to a lurking variable not measured in
the study
Agresti/Franklin Statistics, 103 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?

In a previous example, we saw that
teenagers who watch more TV have a
tendency later in life to commit more
aggressive acts

Could there be a lurking variable that
influences this association?
Agresti/Franklin Statistics, 104 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?

Perhaps teenagers who watch more
TV tend to attain lower educational
levels and perhaps lower education
tends to be associated with higher
levels of aggression
Agresti/Franklin Statistics, 105 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?


We need to measure potential lurking
variables and use them in the
statistical analysis
If we thought that education was a
potential lurking variable we would
what to measure it
Agresti/Franklin Statistics, 106 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?
Agresti/Franklin Statistics, 107 of 111
Example: Is TV Watching Associated
with Aggressive Behavior?

This analysis uses three variables:
• Response variable:
Whether the
subject has committed aggressive acts
• Explanatory variable:
Level of TV
watching
• Control variable:
Educational level
Agresti/Franklin Statistics, 108 of 111
Control Variable

A control variable is a variable that is
held constant in a multivariate
analysis (more than two variables)
Agresti/Franklin Statistics, 109 of 111
Can An Association Be
Explained by a Third Variable?



Treat the third variable as a control
variable
Conduct the ordinary bivariate
analysis while holding that control
variable constant at fixed values
Whatever association occurs cannot
be due to effect of the control variable
Agresti/Franklin Statistics, 110 of 111
Example: Is TV Watching
Associated with Aggressive
Behavior?


At each educational level, the
percentage committing an aggressive
act is higher for those who watched
more TV
For this hypothetical data, the
association observed between TV
watching and aggressive acts was not
because of education
Agresti/Franklin Statistics, 111 of 111