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An Integrated Approach to
Improving Quality and Efficiency
CHAPTER 7. USING DATA AND STATISTICAL
TOOLS FOR OPERATIONS IMPROVEMENT
Chapter 7
Using Data and Statistical Tools for
Operations Management
Using Data and Statistical Tools for
Operations Management
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•
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•
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•
•
Data collection
Graphical tools
Mathematical descriptions
Probability and probability distributions
Confidence intervals, hypothesis tests
ANOVA/MANOVA/MANCOVA
Regression
Copyright 2012 Health Administration Press
Data Collection
• Validity: A valid study has no logic, sampling,
or measurement errors.
— Logic
— Selection or sampling
— Measurement
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Data Collection
Diagram created in Inspiration by Inspiration Software, Inc.
Copyright 2012 Health Administration Press
Data Collection: Logic
• Why are the data needed?
• What will the data be used for?
• What questions are going to be asked of the
data?
• Are the patterns of the past going to be repeated
in the future?
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Data Collection: Selection or Sampling
•
•
•
•
•
•
•
Census versus sample
Nonrandom methods
Simple random sampling
Stratified sampling
Systematic or sequential sampling
Cluster or area sampling
Sample size
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Data Collection: Measurement
• Accuracy
• Precision
— How precise should the
measurements be?
— Does the measurement
measure what we want it
to measure (i.e., say = do)?
• Reliability
— Would the
measurement
be the same
if we
repeated it?
Reliable, but
not accurate
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Reliable and
accurate
Not reliable,
but accurate
Graphical Tools
•
•
•
•
•
•
Mapping
Visual representations of data
Histograms and Pareto charts
Stem plots, dot plots
Box (and whisker) plots
Normal probability plots
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Graphical Tools: Histograms and Pareto Charts
Length of Hospital Stay
Diagnosis Category
14
12
10
Frequency
12
6
4
0
H
ea
rt
D
2
0
1-2
3-4
5-6
7-8
9-10 11-12 13-14 15-16 17-18
Length of Hospital Stay (days)
M
2
el
iv
er
6
y
Pn
u
al
em
ig
on
na
ia
nt
N
eo
pl
as
m
s
Ps
yc
ho
se
s
Fr
ac
tu
re
s
4
D
8
8
ise
as
e
Frequency
10
Diagnosis
Microsoft Excel screen shots reprinted with permission from Microsoft Corporation.
Copyright 2012 Health Administration Press
Graphical Tools: Dot Plots
Dotplot of C1
Length of Hospital Stay
3
6
9
12
Days
15
Produced with Minitab statistical software
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18
Graphical Tools: Turnip Graph
Percentage of diabetic Medicare enrollees receiving eye exams
among 306 hospital referral regions (2001)
Source: Wennberg, J. E. 2005. Data from the Dartmouth Atlas Project. Figure copyrighted by the Trustees of
Dartmouth College. Used with permission.
Copyright 2012 Health Administration Press
Graphical Tools: Normal Probability Plots
Length of Hospital Stay
1.00
.75
.50
.25
0.00
0.00
.25
.50
.75
Observed Cumulative Probability
Produced with SPSS for Windows
Copyright 2012 Health Administration Press
1.00
Graphical Tools: Scatter Plots
Strong Positive Correlation
Strong Negative Correlation
Y
Y
r = -0.86
X
r = 0.91
Positive Correlation
X
No Correlation
Y
Y
r = 0.70
X
r = 0.06
Microsoft Excel screen shots reprinted with permission from Microsoft Corporation.
Copyright 2012 Health Administration Press
X
Mathematical Descriptions: Mean
• The mean is the arithmetic average of the
population:
Population mean  μ 
x
, where x  individual values and
N
N  number of values in the population.
• The population mean can be estimated from a
sample:
x

Sample mean  x 
, where n  number of values in the sample.
n
For our simple data set, x 
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3 6853
 5.
5
Mathematical Descriptions: Median and Mode
• The median is the middle value of the sample or
population. If the data are arranged into an array (an
ordered data set):
3, 3, 5, 6, 8
5 would be the middle value or median.
• The mode is the most frequently occurring value. In
the above example, the value 3 occurs more often
(two times) than any other value, so 3 would be the
mode.
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Mathematical Descriptions
Range and Mean Absolute Deviation
• The range is the difference between the high
and low values in a data set.
Range  xhigh  xlow  8  3  5
• The mean absolute deviation (MAD) is the
average of the absolute value of the differences
from the mean.
xx

MAD 
n
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2  2  0 1 3 8

  1.6
5
5
Mathematical Descriptions
Variance, Standard Deviation
• The variance is the average square difference from
the mean.
(x  μ)
4  4  0  1 9 18

Population variance  σ 


 3.6
2
2
Sample variance  s 2 
N
2
(x

x
)

n-1
5

5
4  4  0  1 9 18

 4.5
5 1
4
• This standard deviation is the square root of the
variance.
 (x  μ)
2
Population standard deviation  σ 
2
Sample standard deviation  s 
2
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N
 (x  x)
n
2


4  4  0  1 9
18

 3.6  1.9
5
5
4  4  0  1 9
18

 4.5  2.1
5 1
4
Mathematical Descriptions
Coefficient of Variation
The coefficient of variation (CV) is a measure of
the relative variation in the data. It is the standard
deviation divided by the mean.
σ
s 1.9
CV  or 
 0.4
μ
x
5
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Probability and Probability Distributions
•
•
•
•
•
Determination of probabilities
Properties of probabilities
Probability distributions
Discrete probability distributions
Continuous probability distributions
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Determination of Probabilities
Observed Probability
Observed probability is the relative frequency of
an event—the number of times the event occurred
divided by the total number of trials.
P(A) 
Number of times A occured
r

Total number of observations, trials, or experiments n
P(drug is effective) 
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Number of times patients are cured
r

Total number of patients given the drug n
Determination of Probabilities
Theoretical Probability
Theoretical probability is the theoretical relative
frequency of an event; the theoretical number of
times an event will occur divided by the total
number of possible outcomes.
Number of times A could occur
r
P(A) 

Total number of possible outcomes n
P(card is a spade) 
Number of spades in the deck
13

 0.25
Total number of cards in the deck 52
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Determination of Probabilities
Opinion Probability
Opinion probability is a subjective determination
of the number of times an event will occur
divided by the imaginary total number of
possible outcomes or trials.
Opinion of number of times an event will occur r
P(A) 

Theoretical total
n
P(Secretariat winning the Belmont Stakes) 
Opinion on the number of times Secretariat would win the Belmont r

Imaginary total number of times the Belmont would be run
n
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Properties of Probabilities
Bounds on Probability
• Probabilities always must be 0, and an event that cannot
occur has a probability of 0.
Least number of times A could occur
0
P(A) 

0
Total number of possible outcomes
Any number
• Probabilities must always be 1.
P(A) 
Greatest number of times A could occur n
 1
Total number of possible outcomes
n
0  P(A)  1
• P(A) + P(A') = 1 and 1 − P(A') = P(A), where A' is not A.
Copyright 2012 Health Administration Press
Properties of Probabilities
Multiplicative Property
For two independent events, the probability of
both A and B occurring, or the intersection () of
A and B, is the probability of A occurring times
the probability of B occurring.
P(A and B occurring) = P(A  B) = P(A) × P(B)
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Properties of Probabilities: Multiplicative Property
Coin Toss
H
Die Toss
Probability
1
1/12
2
1/12
3
1/12
4
1/12
5
1/12
6
1/12
1
1/12
2
1/12
3
1/12
4
1/12
5
1/12
6
1/12
P(3) = 1/6
P(H) × P(3) =
P(H  3) = 1/12
Start
T
P(H) = 1/2
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1/2 × 1/6 = 1/12
Properties of Probabilities
Additive Property
• For two events, the probability of A or B
occurring, or the union () of A with B, is the
probability of A occurring plus the probability of
B occurring, minus the probability of both A and
B occurring.
P(A or B occurring) = P(A  B) = P(A) + P(B) + P(A  B)
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Properties of Probabilities: Additive Property
Coin Toss
H
Die Toss
Probability
1
1/12
2
1/12
3
1/12
4
1/12
5
1/12
6
1/12
P(H  3) = 7/12
Start
T
P(H) = 1/2
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1
1/12
2
1/12
3
1/12
4
1/12
5
1/12
6
1/12
P(3) = 1/6
P(H) + P(3) − P(H  3) = 7/12
Properties of Probabilities: Conditional Probability
The probability of an event occurring if more information
is obtained:
P( A B) 
P( A  B)
P( B)
Contingency Table for ER Wait Times
30 minute wait
>30 minute wait
Friday night
20
30
50
Other times
40
10
50
60
40
100
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Properties of Probabilities: Conditional Probability
• Note that:
P( A  B)  P( A B)  P( B)  P( B A)  P( A)
and if one event has no effect on the other event
(the events are independent), then
.
P( A B)  P( A) and P( A  B)  P( A)  P( B)
• Bayes’ theorem
P( B A)  P( A)
P( A  B) P( B A)  P( A)
P( A B) 


P( B)
P( B)
P( B A)  P( A)  P( B A)  P( A)
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Confidence Intervals, Hypothesis Testing
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•
•
•
•
•
Central limit theorem
Hypothesis testing
Type I () and Type II () errors
T-tests
Proportions
Practical significance versus statistical
significance
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Confidence Intervals, Hypothesis Testing
Central Limit Theorem
• As the sample size becomes large, the
sampling distribution of the mean approaches
normality, no matter what the distribution the
original variable, and
 x   and  x  
n
Sampling Distribution Simulation
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Confidence Intervals
Confidence interval for the true value of the population
mean:
x  z *    x  z *
 /2
x  z / 2 *
 /2
x

n
.    x  z / 2 *
x

n
95%
P(X)
0.4
0.2
2.5%
2.5%
0
-3
-2
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-1
0
Z
1
2
3
Hypothesis Testing
• Belief or null hypothesis, Ho:  = b
• Alternate belief or hypothesis, Ha:   b
• Decision rule: If z  z*, reject the null
x
:
hypothesis. Where z 
x
-Z*< Z < Z* (95% confidence)
P(X)
0.4
0.2
Z<-Z*
Z>Z*
0
-3
-2
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-1
0
Z
1
2
3
Hypothesis Testing: Type I () and Type II () Errors
Ho: 1=2
Ha: 12
Type I and Type II Error—Clinic Wait Time Example
Reality
Wait times at Wait times at the
the two clinics two clinics are
are the same
NOT the same
1=2
Assessment or
guess
Wait times at the
two clinics are the
same
1=2
Wait times at the
two clinics are
NOT the same
12
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12
Type II or
 error
Type I or
 error
Equal Variance t-Test
• t-tests are used to test hypotheses about two
means.
• Ho: 1=2 Ha: 12
• Decision rule: If t  t*, reject Ho
(x  x )  (μ1  μ2 )
t 1 2
1 1
sp

n1 n2
(n1  1) s12  (n2  1) s22
where s p 
n1  n2  2
• Confidence interval


1 1 
1 1 
*
( x1  x2 )  t *  s p
   1   2  ( x1  x2 )  t *  s p
 
n1 n2 
n1 n2 


*
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Proportions
Ho: 1= 2
Ha: 12
Decision rule: If z  z*, reject Ho
( p1  p2 )  (1   2 )
z
where
p (1  p ) p (1  p )

n1
n2
n1 p1  n2 p2
p
n1  n2
Confidence interval
( p1  p2 )  z
*
p (1  p ) p (1  p )
p (1  p ) p (1  p )
*

  1   2  ( p1  p2 )  z

n1
n2
n1
n2
Copyright 2012 Health Administration Press
Practical Significance Versus Statistical Significance
• Basic confidence interval
statistic – [(z*) * (s.e. statistic)]  parameter 
statistic + [(z*) * (s.e. statistic)]
• As n increases, s.e. decreases and the
confidence interval gets larger.
• Large samples may give statistically
significant results that are not practically
significant.
Copyright 2012 Health Administration Press
ANOVA/MANOVA/MANCOVA
• One-way ANalysis Of VAariance (ANOVA) is used
to test hypotheses about three or more levels of
treatment. A t-test will give the same information as
an ANOVA when there are only two treatment levels
of interest.
• Two-way and higher ANOVAs are used when there is
more than one type of treatment variable of interest.
• MANOVA/MANCOVA are used when there is more
than one outcome or dependent variable of interest.
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Regression
• Simple linear regression—used to describe the
relationship between two variables
• Multiple regression—used to describe the
relationship between multiple predictor
variables and a single dependent variable
• General linear model
• Artificial neural networks
• Design of experiments
Copyright 2012 Health Administration Press
What Is the Equation of a Line?
Algebra: y  mx  b
Statistics: Ŷ  bX  a
Where
rise Δy
b  slope 

run Δx
Copyright 2012 Health Administration Press
a  y intercept
 y, when x  0
Problem
Student A owns a health insurance firm and
wants us to determine the cost (price would be a
more difficult problem) of providing healthcare
to insured individuals.
Copyright 2012 Health Administration Press
Seeing the Future
Data
Experiences
are relevant
Judgment: To what degree are
these experiences still relevant?
Experiences
are irrelevant
Deductive reasoning versus inductive reasoning
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What Is the Cost of Healthcare Related To?
Quantitative
______________
______________
______________
______________
______________
______________
Copyright 2012 Health Administration Press
Qualitative
_____________
_____________
_____________
_____________
_____________
_____________
Selection
•
•
•
•
Define population
Census or sample
Type of sample
Measurement—accurate, reliable, precise?
X = number of dependents; Y = annual healthcare
expense ($1,000)
• Is the study valid?
• How do we create knowledge from data?
Copyright 2012 Health Administration Press
Data
Copyright 2012 Health Administration Press
Number of
Dependents
Annual
Healthcare
Expense
($1,000)
0
3
1
2
2
6
3
7
4
7
Scatterplot
Y—Annual Healthcare Cost $1,000
10
y = 1.3x + 2.4
9
8
7
6
y=x+3
5
y=5
y = 1.2x + 2
4
3
2
1
0
0
1
2
3
X—Number of Dependents
Copyright 2012 Health Administration Press
4
5
6
Scatterplot Questions
• Which is the “best” line on the scatterplot?
• How would you define “best” (e.g., must be
quantifiable)?
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Professor’s Model
Ŷ  bX  a
Ŷ  cost estimate ($1,000)
a  Y intercept  3
Y
b  slope 
1
X
Ŷ  1X  3 knowledge
Copyright 2012 Health Administration Press
Model Comparison
X
Y
Yhat =
X+3
Prof’s
e=
Y − Yhat
Yˆ  1.2( X ) Yˆ  1.3( X )
 2.4
2
Student 1
e
Student 2
e
0
3
3
0
−1
−0.6
1
2
4
-2
1.2
1.7
2
6
5
1
−1.6
−1
3
7
6
1
−1.4
−0.7
4
7
7
0
−0.2
0.6
0
−3
0
 (sum)
Copyright 2012 Health Administration Press
Good Model
• A good model must be unbiased.
e = 0
• Is that enough? What else? Does this remind
you of 2?
• How do we get rid of signs?
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Model Comparison
X
Y
Yhat =
X+3
e=
Y − Yhat
e2
Student 1
e2
0
3
3
0
0
1
1
2
2
−2
4
1.44
2
6
6
1
1
2.56
3
7
7
1
1
1.96
4
7
7
0
0
0.04
 (sum)
25
25
0
6
7
Copyright 2012 Health Administration Press
Least Squares Technique
Gauss proved that if you use:
(Y  Y)(X  X)
b
and a  Y  bX
2
(X  X)
You are guaranteed that
e = 0 and e2 is a minimum.
Yhat = 1.3X + 2.4, e = 0, and e2 = 5.1.
Copyright 2012 Health Administration Press
Coefficient of Determination
Are we better off making estimates by using
information (X = number of dependents) and
having created knowledge (Yhat = 1.3X + 2.1)
than using no information or knowledge (i.e.,
is the model “better”)?
How would you estimate without using our
knowledge (our model)?
Copyright 2012 Health Administration Press
Sum of Squares Total
X
Y
Yhat = Ybar
e=Y−
Ybar
SSTO
(Y −
Ybar)2
0
3
5
−2
4
1
2
5
−3
9
2
6
5
1
1
3
7
5
2
4
4
7
5
2
4
 (sum)
25
25
0
22
Note that this method is unbiased.
Copyright 2012 Health Administration Press
Graph
10
Y—Annual Healthcare Cost $1,000
9
8
7
6
5
y=5
4
3
2
1
0
0
1
2
3
4
X—Number of Dependents
Copyright 2012 Health Administration Press
5
6
Y—Annual Healthcare Costs $1,000
Errors
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
X—Number of Dependents
Copyright 2012 Health Administration Press
3.5
4
4.5
Sum of Squares Error
e=
Y−
Yhat
SSE
e2
= (Y −
Yhat)2
X
Y
Yhat =
1.3X +
2.4
0
3
2.4
0.6
0.36
5
−2
4
1
2
3.7
−1.7
2.89
5
−3
9
2
6
5
1.0
1.00
5
1
1
3
7
6.3
0.7
0.49
5
2
4
4
7
7.6
−0.6
0.36
5
2
4
 (sum)
25
25
0
5.1
25
0
22
Copyright 2012 Health Administration Press
Ybar
Y−
Ybar
SSTO
(Y −
Ybar)2
Coefficient of Determination
What is the percentage of improvement when we
use knowledge gained from our model?
New error level  old error level
% improvement 
Old error level
5.1  22  16.9


100  77%
22
22
r2 = coefficient of determination = 77%
r2 = 0.77
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Another Viewpoint
Variation in cost of removal is either explained by
knowledge (the model) or not explained.
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Explained and Unexplained Error
Y—Annual Healthcare Costs $1,000
8
7
6
5
4
3
----- Explained
2
___ Unexplained
1
0
0
0.5
1
1.5
2
2.5
3
X—Number of Dependents
Copyright 2012 Health Administration Press
3.5
4
4.5
Sum of Squares Regression
e=
Y−
Yhat
SSE
e2
= (Y −
Yhat)2
SSTO
(Y −
Ybar)2
Yhat
–
Ybar
SSR
(Yhat
−
Ybar)2
X
Y
Yhat =
1.3X +
2.4
0
3
2.4
0.6
0.36
5
−2
4
−2.6
6.76
1
2
3.7
−1.7
2.89
5
−3
9
−1.3
1.69
2
6
5
1.0
1.00
5
1
1
0
0
3
7
6.3
0.7
0.49
5
2
4
1.3
1.69
4
7
7.6
−0.6
0.36
5
2
4
2.6
6.76

(sum)
35
25
0
5.1
25
0
22
0
16.9
Y
Y−
bar Ybar
Coefficient of Determination
Explained SSR
16.9
r 


 0.77
Total
SSTO 22.0
2
Note: r2 is not based on statistics or
probability; it is just a percentage.
Copyright 2012 Health Administration Press
Correlation Coefficient
r =  r2
r = Correlation coefficient
= Measure of the strength of the linear
relationship between two variables
−1  r  1
r = −1
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r = +1
Correlation Coefficient Examples
r = 0.0
r = 0.9
r = −0.5
Copyright 2012 Health Administration Press
Coefficient of Determination
Questions:
• If r2 is low, does that mean there is no relationship
between your variables?
• If r2 is high (close to 1), does that mean you always
get useful predictions from your model?
• If r2 is high, does that mean your model has a
“good” fit?
Copyright 2012 Health Administration Press
r2 and Curves
• Can we fit a straight line to this?
• Yes, and we are guaranteed that the errors sum
to zero and are a minimum.
• However, a curve would be better.
Y
X
Copyright 2012 Health Administration Press
Excel Output
To get this sheet, go to Tools -> Data Analysis -> Regression. If you don't have Data Analysis
listed in your tools, see Excel help "Install and Use the Analysis ToolPak.”
X—Number of
Dependents
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.8765
R Square
0.7682
Adjusted R
Square
0.6909
Standard
Error
0.8790
Observations
5
SS
7.6818
2.3182
10
Coefficients Standard Error
-0.9545
1.0162
0.5909
0.1874
MS
7.6818
0.7727
F
Significance F
9.9412
0.0511
Residual Plot
1.0000
0.5000
t Stat
P-value
Lower 95% Upper 95% Lower 0.0000
90.0% Upper 90.0%
-0.9393 0.4169
-4.1885
2.2794
-3.3460
1.4369
2
-0.5000 0
-1.0000
Residuals
Intercept
Y - $ 1000
Annual
Health Care
Expense
1
3
4
Predicted X
3.1530
RESIDUAL OUTPUT
Predicted X Number of
Standard
Observation Dependents Residuals Residuals
1
0.8182
-0.8182 -1.0747
2
0.2273
0.7727
1.0150
3
2.5909
-0.5909 -0.7762
4
3.1818
-0.1818 -0.2388
5
3.1818
0.8182
1.0747
Copyright 2012 Health Administration Press
0.0511
-0.0055
1.1873
PROBABILITY OUTPUT
X - Number
of
Percentile Dependents
10
0
30
1
50
2
70
3
90
4
4
6
8
Y—$ 1,000 Annual Healthcare Expense
1.0320
0.1499
X—Number
of
Dependents
df
X—Number
of
Dependents
—Number of
0
2
4
6
8
Dependents
Y—$ 1,000 Annual Healthcare Expense
ANOVA
Regression
Residual
Total
Line Fit Plot
5
4
3
2
1
0
Normal Probability Plot
5
0
0
20
40
60
Sample Percentile
80
100
F Test
MSR
SSR / 1

 F*
MSE SSE / n  2
If F* > F(1-;1;n-2), reject H0:  = 0 (in this case)
MSR/MSE  1   = 0
MSR/MSE  big    0
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Assumptions of Linear Regression
Linear regression is based on several
assumptions. If these assumptions are violated,
the resulting model will be misleading. The
principal assumptions are:
• The dependent and independent variables are
linearly related.
• The errors associated with the model are not serially
correlated.
• The errors are normally distributed and have
constant variance.
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Transformations
X
Y
Transform
X ->X2
−3
9
9
−2
4
4
−1
1
1
0
0
0
1
1
1
2
4
4
3
9
9
Copyright 2012 Health Administration Press
Y
If the variables are not linearly related or the assumptions
of regression are violated, the variables can be
transformed to produce a possibly better model.
10
8
6
4
2
0
0
2
4
6
X2
8
10
General Linear Model
• The most general of all linear models
• Multiple predictor variables:
— Metric
— Categorical
— Both
• Multiple dependent variables:
— Metric
— Categorical
— Both
• Can be used to build complex models
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Outline for Analyses
1. Define the problem/question.
2. Determine what data will be needed to address the
problem question.
3. Collect the data.
4. Graph the data.
5. Analyze the data using the appropriate tool.
6. “Fix” the problem.
7. Evaluate the effectiveness of the “fix.”
8. Start again.
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Choice of Statistical Technique
Independent
Variable
Categorical
Dependent
Variable
One
Categorical
Metric
Many
Categorical
Metric
Both
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Mathematical
Graphical
One
2
Many
2 (layered)
One
t-Test
Histogram
type
Many
MANOVA
Box plot
One
2
Many
2 (layered)
One
ANOVA
Many
MANOVA
GLM
Box plots
Choice of Statistical Technique
Independent
Variable
Metric
Dependent
Variable
One
Categorical
Mathematical
One
Graphical
Logit
Many GLM
Metric
One
Simple regression
Many GLM
Both
Many Categorical
MANCOVA
One
Logit
Many GLM
Metric
One
Multiple regression
Many GLM
Both
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GLM; neural net
Scatterplot
Choice of Statistical Technique
Independent
Variable
Dependent
Variable
Both
Categorical
Metric
Both
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Mathematical
One
ANCOVA
Many
MANCOVA
One
Simple regression
Many
Multiple regression
GLM
Neural Net
Graphical
End of Chapter 7
Copyright 2012 Health Administration Press