Handbook for Health Care Research, Second Edition
Chapter 13
CHAPTER 13
Statistical Methods for Continuous Measures
© 2010 Jones and Bartlett Publishers, LLC
Handbook for Health Care Research, Second Edition
Chapter 13
Testing for Normality
 A key assumption of test is the sample data come from a
population that is normally distributed
• Kolmogorov-Smirnov Test - tests whether the distribution of a
continuous variable is the same for two groups
-Example: You want to test the effect of prone positioning on
the Pao2 of patients in the ICU. Data are collected before and
after positioning the patient and you intend to use a paired t
test.
-Null: Two distributions are the same
• See Figure 13.1 in book
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Handbook for Health Care Research, Second Edition
Chapter 13
Testing for Equal Variances
Key assumption of the tests is the data in two
or more samples have equal variances
• F Ratio Test- is calculated as the ratio of two
sample variances and shows whether the
variance of one group is smaller, larger, or
equal to the variance of the other group
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Handbook for Health Care Research, Second Edition
Chapter 13
Testing for Equal Variances
Example: One of the pulmonary physicians in your hospital
has questioned the accuracy of your lab’s pulmonary
function test results. Specifically, she questions whether
the new lab technician you have hired (Tech A) can
produce as consistent results as the other technician, who
has years of experience (Tech B). You gather two sets of
FEV1 measurements on the same patient by the two techs.
Since you are using only one patient, most of the variance
in measurements will be due to the two technicians.
-Null Hypothesis: The variances of the two groups of data
are the same.
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Handbook for Health Care Research, Second Edition
Chapter 13
F Ratio Test
Results of F Ratio Test
Tech A
Tech B
30
30
1.206
1.173
0.143
0.033
0.378
0.182
0.069
0.033
Report from the Statistics Program
The p value is less than 0.05, so we reject the null hypothesis
And conclude that the variances are not equal. The variance
of measurements made by Tech A (0.143) is much greater
than the variance of measurements made by Tech B (0.033).
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Handbook for Health Care Research, Second Edition
Chapter 13
Correlation and Regression
Basic assumption of this section is that the
association between the two variables is linear
• Pearson Product-Moment Correlation Coefficientcorrelation coefficient with a continuous variable
measurable on an interval level
The Pearson r statistic ranges:1.0 (perfect negative
correlation) through 0 (no correlation) to 1.0 (perfect
positive correlation).
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Handbook for Health Care Research, Second Edition
Chapter 13
Correlation and Regression
Example: You decide to evaluate a device called the
EzPAP for lung expansion therapy. It generates a
continuous airway pressure proportional to the flow
introduced at its inlet port. However, the user’s
manual does not say how the set flow is related to
resultant airway pressure, and we generally use the
device without a pressure gauge attached. You
connect a flowmeter and pressure gauge to the
device and record the pressures as you adjust the
gas flow over a wide range.
-Null Hypothesis: The two variables have no
significant linear association.
© 2010 Jones and Bartlett Publishers, LLC
Handbook for Health Care Research, Second Edition
Chapter 13
Pearson Product-Moment Correlation
Coefficient
Data Entry for Calculating the Correlation Coeffi cient
Flow
3
4
5
6
etc.
Pressure
5
5
6
7
etc.
Report from Statistics Program
Correlation coefficient: 0.942.
The p value is less than 0.001.
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Handbook for Health Care Research, Second Edition
Chapter 13
Correlation and Regression
• Simple Linear Regression-A simple regression uses
the values of one independent variable (x) to predict
the value of a dependent variable (y). Regression
analysis fits a straight line to a plot of the data.
-Example: Using the data table from the EzPAP, we
perform a simple linear regression. The purpose of
the experiment is to allow us to predict the amount
of flow required for a desired level of pressure.
Therefore, we designate pressure as the (known)
independent variable and fl ow as the dependent
variable.
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Handbook for Health Care Research, Second Edition
Chapter 13
Simple Linear Regression
Results of linear regression analysis.
Pressure (cm H2O)
Report from Statistics Program
Normality test: Passed (p 0.511)
Constant variance test: Passed (p 0.214)
Coeffi cient p
Y-intercept
1.045 0.021
Pressure
0.653 0.001
R2: 0.89
Standard error of estimate: 1.482
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Handbook for Health Care Research, Second Edition
Chapter 13
Correlation and Regression
• Multiple Linear Regression- simple linear regression
can be extended to cases where more than one
independent (predictor) variable is present
- Multiple linear regression assumes an association
between one dependent variable and an arbitrary
number (symbolized by k) of independent variables.
The general equation is:
Y= b0 +b1x1+ b2x2 + b3x3 + … + bkxk
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Handbook for Health Care Research, Second Edition
Chapter 13
Correlation and Regression
• Logistic Regression -designed for predicting a
qualitative dependent variable from observations of
one or more independent variables.
-The qualitative dependent variable must be nominal
and dichotomous (take only two possible values such
as lived or died, presence or absence, etc.),
represented by values of 0 and 1.
-The general logistic regression: P=ey
1+eY
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing One Sample to a Known Value
• One-Sample t Test - compares a sample mean to
a hypothesized population mean and determines
the probability that the observed difference
between sample and hypothesized mean
occurred by chance.
• Probability of chance occurrence is the p value:
– p close to 1.0 implies that the hypothesized and
sample means are the same
– A small p value (less than 0.05) suggests that such a
difference is unlikely (only 1 in 20) to occur by chance
if the sample came from a population with the
hypothesized mean.
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Two Samples, Unmatched
Data
• Unpaired t Test - compares the means of two
groups and determines the probability that the
observed difference occurred by chance.
• Probability of chance occurrence is the p value:
-A p value close to 1.0 implies that the two
sample means are the same
-A small p value (less than 0.05) suggests that
such a difference is unlikely (only 1 in 20) to occur
by chance if the sample came from a population
with the hypothesized mean.
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Two Samples, Matched
Data
• Paired t Test- used for comparing two measurements from
the same individual or experimental unit.
-The two measurements can be made at different times or
under different conditions.
• The paired t test is used to evaluate the hypothesis that the
mean of the differences between pairs of experimental units
is equal to some hypothesized value, usually zero.
• The paired t test compares the two samples and determines
the probability of the observed difference occurring by
chance. The chance is reported as the p value.
-A small p value (less than 0.05) suggests that such a
difference is unlikely (only 1 in 20) to occur by chance if the
sample came from a population with the hypothesized mean.
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Three or More Samples,
Unmatched Data
 With several independent comparisons, the probability of
getting at least one falsely significant comparison when there
are actually no significant differences
 The proper way to compare more than two mean values is to
use the analysis of variance (ANOVA)
• One-Way ANOVA- assesses values collected at one point in
time for more than two different groups of subjects. It is used
when you want to determine if the means of two or more
different groups are affected by a single experimental factor.
-ANOVA tests the null hypothesis that the mean values of all
the groups are the same versus the hypothesis that at least
one of the mean values is different.
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Three or More Samples,
Unmatched Data
• Two-Way ANOVA - compares values within groups as well as
between groups.
-Appropriate for looking at comparisons of groups at different
times as well as the differences within each group over the
course of the study.
• In a two-factor ANOVA, there are two experimental factors,
which are varied for each experimental group.
• A two-factor ANOVA tests three hypotheses:
-There is no difference among the levels of the first factor.
-There is no difference among the levels of the second factor.
-There is no interaction between factors. That is, if there is
any difference among levels of one factor, the differences are
the same regardless of the second factor level.
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Three or More Samples,
Matched Data
• One-Way Repeated Measures ANOVA - examines
values collected at more than one point in time for a
single group of subjects.
-It is used when you want to determine if a single
group of individuals was affected by a series of
experimental treatments or conditions.
• ANOVA tests the null hypothesis that all the mean
values of all the groups are the same versus the
hypothesis that at least one of the mean values is
different.
© 2010 Jones and Bartlett Publishers, LLC
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Handbook for Health Care Research, Second Edition
Chapter 13
Comparing Three or More Samples,
Matched Data
• Two-Way Repeated Measures ANOVA - compares values
within groups as well as between groups
- This analysis would be appropriate for looking at
comparisons of groups at different times as well as the
differences within each group over the course of the study
• The test is for differences between the different levels of
each factor and for interactions between the factors.
-A two-factor ANOVA tests three hypotheses:
-There is no difference among the levels of the first factor.
-There is no difference among the levels of the second factor.
-There is no interaction between factors.
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