Practical Applications of
Statistical Methods in the
Clinical Laboratory
Roger L. Bertholf, Ph.D., DABCC
Associate Professor of Pathology
Director of Clinical Chemistry & Toxicology
UF Health Science Center/Jacksonville
“[Statistics are] the only tools by
which an opening can be cut
through the formidable thicket of
difficulties that bars the path of
those who pursue the Science of
Man.”
[Sir] Francis Galton (1822-1911)
“There are three kinds of lies:
Lies, damned lies, and
statistics”
Benjamin Disraeli (1804-1881)
What are statistics, and what are
they used for?
• Descriptive statistics are used to
characterize data
• Statistical analysis is used to distinguish
between random and meaningful variations
• In the laboratory, we use statistics to
monitor and verify method performance,
and interpret the results of clinical
laboratory tests
“Do not worry about your
difficulties in mathematics, I assure
you that mine are greater”
Albert Einstein (1879-1955)
“I don't believe in
mathematics”
Albert Einstein
Summation function
N
 x x x
i 1
i
1
2
 x3  x N
Product function
N
x  x x
i
i 1
1
2
 x3  x N
The Mean (average)
The mean is a measure of the centrality of a
set of data.
Mean (arithmetical)
N
1
x   xi
N i 1
Mean (geometric)
N
x g  N x1  x2  x3 x N  N  xi
i 1
Use of the Geometric mean:
The geometric mean is primarily used to
average ratios or rates of change.
Mean (harmonic)
Example of the use of Harmonic
mean:
Suppose you spend $6 on pills costing 30
cents per dozen, and $6 on pills costing 20
cents per dozen. What was the average
price of the pills you bought?
Example of the use of Harmonic
mean:
You spent $12 on 50 dozen pills, so the
average cost is 12/50=0.24, or 24 cents.
This also happens to be the harmonic mean of
20 and 30:
2
1 1

30 20
 24
Root mean square (RMS)
xrms
x  x  x  x
1
2


xi

N
N i 1
2
1
2
2
2
3
2
N
N
For the data set:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10:
Arithmetic mean
5.50
Geometric mean
4.53
Harmonic mean
3.41
Root mean square
6.20
The Weighted Mean
N
xw 
x w
i
i
i 1
N
w
i
i 1
Other measures of centrality
• Mode
The Mode
The mode is the value that occurs most often
Other measures of centrality
• Mode
• Midrange
The Midrange
The midrange is the mean of the highest and
lowest values
Other measures of centrality
• Mode
• Midrange
• Median
The Median
The median is the value for which half of the
remaining values are above and half are
below it. I.e., in an ordered array of 15
values, the 8th value is the median. If the
array has 16 values, the median is the mean
of the 8th and 9th values.
Example of the use of median vs.
mean:
Suppose you’re thinking about building a house in
a certain neighborhood, and the real estate agent
tells you that the average (mean) size house in
that area is 2,500 sq. ft. Astutely, you ask
“What’s the median size?” The agent replies
“1,800 sq. ft.”
What does this tell you about the sizes of the
houses in the neighborhood?
Measuring variance
Two sets of data may have similar means, but
otherwise be very dissimilar. For example,
males and females have similar baseline LH
concentrations, but there is much wider
variation in females.
How do we express quantitatively the amount
of variation in a data set?
N
1
Mean difference   (xi  x )
N i 1
N
N
1
1
  xi   x
N i 1
N i 1
 xx
0
The Variance
N
1
2
V   ( xi  x )
N i 1
The Variance
The variance is the mean of the squared differences
between individual data points and the mean of the
array.
Or, after simplifying, the mean of the squares minus
the squared mean.
The Variance
1
V 
N
N

i 1
( xi  x ) 2
1

N
1
x  N
1

N
1
1 2
 x  N 2 x  xi  N x  (1)
2
i
1
 2 xi x  N
2
i
 x 2  2x 2  x 2
 x2  x 2
2
x

The Variance
In what units is the variance?
Is that a problem?
The Standard Deviation
 V
N
1
2
( xi  x )

N i 1
The Standard Deviation
The standard deviation is the square root of
the variance. Standard deviation is not the
mean difference between individual data
points and the mean of the array.
1
1
2
xx 
(x  x)


N
N
The Standard Deviation
In what units is the standard deviation?
Is that a problem?
The Coefficient of
*
Variation

CV   100
x
*Sometimes
called the Relative Standard
Deviation (RSD or %RSD)
Standard Deviation (or Error) of
the Mean
x 
x
N
The standard deviation of an average
decreases by the reciprocal of the square
root of the number of data points used to
calculate the average.
Exercises
How many measurements must we average to
improve our precision by a factor of 2?
Answer
To improve precision by a factor of 2:
1
1
 05
. 
2
N
1
N
2
05
.
2
N  2  4 (quadruplicate)
Exercises
• How many measurements must we average
to improve our precision by a factor of 2?
• How many to improve our precision by a
factor of 10?
Answer
To improve precision by a factor of 10:
1
1
 01
. 
10
N
1
N
 10
01
.
2
N  10  100 times!
Exercises
• How many measurements must we average
to improve our precision by a factor of 2?
• How many to improve our precision by a
factor of 10?
• If an assay has a CV of 7%, and we decide
run samples in duplicate and average the
measurements, what should the resulting
CV be?
Answer
Improvement in CV by running duplicates:
CVdup
CV
7


 4.9%
2 141
.
Population vs. Sample standard
deviation
• When we speak of a population, we’re
referring to the entire data set, which will
have a mean :
1
Population mean   xi  
N i
Population vs. Sample standard
deviation
• When we speak of a population, we’re
referring to the entire data set, which will
have a mean 
• When we speak of a sample, we’re referring
to a subset of the population, customarily
designated “x-bar”
• Which is used to calculate the standard
deviation?
“Sir, I have found you an
argument. I am not obliged to
find you an understanding.”
Samuel Johnson (1709-1784)
Population vs. Sample standard
deviation

1
2
(xi   )

N i
s
1
2
(xi  x )

N 1 i
Distributions
• Definition
Statistical (probability)
Distribution
• A statistical distribution is a
mathematically-derived probability function
that can be used to predict the
characteristics of certain applicable real
populations
• Statistical methods based on probability
distributions are parametric, since certain
assumptions are made about the data
Distributions
• Definition
• Examples
Binomial distribution
The binomial distribution applies to events
that have two possible outcomes. The
probability of r successes in n attempts,
when the probability of success in any
individual attempt is p, is given by:
P(r; p, n)  p  (1  p)
r
n r
n!

r !(n  r )!
Example
What is the probability that 10 of the 12
babies born one busy evening in your
hospital will be girls?
Solution
P(10;05
. ,12)  05
.  (1  05
.)
10
12  10
 0016
.
or 16%
.
12!

10!(12  10)!
Distributions
• Definition
• Examples
– Binomial
“God does arithmetic”
Karl Friedrich Gauss (1777-1855)
The Gaussian Distribution
What is the Gaussian distribution?
63
81
36
12
28
7
79
52
96
17
22
4
61
85 etc.
F
1
number
100
63
81
36
12
28
7
79
52
96
17
22
4
61
85
+
22
73
54
33
99
5
61
28
58
24
16
77
43
8
=
85
152
90
45
127
12
140
70
154
41
38
81
104
93
F
2
number
200
. . . etc.
x
Probability
The Gaussian Probability
Function
The probability of x in a Gaussian distribution
with mean  and standard deviation  is
given by:
1
 ( x   ) 2 / 2 2
P ( x;  ,  ) 
e
 2
The Gaussian Distribution
• What is the Gaussian distribution?
• What types of data fit a Gaussian
distribution?
“Like the ski resort full of girls
hunting for husbands and husbands
hunting for girls, the situation is
not as symmetrical as it might
seem.”
Alan Lindsay Mackay (1926- )
Are these Gaussian?
•
•
•
•
•
•
•
Human height
Outside temperature
Raindrop size
Blood glucose concentration
Serum CK activity
QC results
Proficiency results
The Gaussian Distribution
• What is the Gaussian distribution?
• What types of data fit a Gaussian
distribution?
• What is the advantage of using a Gaussian
distribution?
Probability
Gaussian probability distribution
.67
.95
µ-3 µ-2 µ- µ µ+ µ+2 µ+3
What are the odds of an
observation . . .
• more than 1 from the mean (+/-)
• more than 2  greater than the mean
• more than 3  from the mean
Some useful Gaussian
probabilities
Range Probability Odds
68.3%
1 in 3
+/- 1.00 
90.0%
1 in 10
+/- 1.64 
95.0%
1 in 20
+/- 1.96 
99.0%
1 in 100
+/- 2.58 
That
Example
This
[On the Gaussian curve]
“Experimentalists think that it is a
mathematical theorem while the
mathematicians believe it to be
an experimental fact.”
Gabriel Lippman (1845-1921)
Distributions
• Definition
• Examples
– Binomial
– Gaussian
"Life is good for only two
things, discovering
mathematics and teaching
mathematics"
Siméon Poisson (1781-1840)
The Poisson Distribution
The Poisson distribution predicts the
frequency of r events occurring randomly in
time, when the expected frequency is 

e 
P ( r;  ) 
r!
r
Examples of events described by
a Poisson distribution
?
• Lightning
• Accidents
• Laboratory?
A very useful property of the
Poisson distribution
V( r )  
 
Using the Poisson distribution
How many counts must be collected in an RIA
in order to ensure an analytical CV of 5% or
less?
Answer


Since CV  (100)  (100)
x

and   


0.05  
 
  400 counts
Distributions
• Definition
• Examples
– Binomial
– Gaussian
– Poisson
The Student’s t Distribution
When a small sample is selected from a large
population, we sometimes have to make
certain assumptions in order to apply
statistical methods
Questions about our sample
• Is the mean of our sample, x bar, the same as the
mean of the population, ?
• Is the standard deviation of our sample, s, the
same as the standard deviation for the population,
?
• Unless we can answer both of these questions
affirmatively, we don’t know whether our sample
has the same distribution as the population from
which it was drawn.
Recall that the Gaussian distribution is defined by
the probability function:
1
 ( x   ) 2 / 2 2
P ( x;  ,  ) 
e
 2
Note that the exponential factor contains both
and , both population parameters. The factor
is often simplified by making the substitution:
z
(x   )

The variable z in the equation:
z
(x   )

is distributed according to a unit
gaussian, since it has a mean of zero
and a standard deviation of 1
Probability
Gaussian probability distribution
.67
.95
-3
-2
-1
0
z
1
2
3
But if we use the sample mean and
standard deviation instead, we get:
(x  x )
t
s
and we’ve defined a new quantity, t,
which is not distributed according to
the unit Gaussian. It is distributed
according to the Student’s t
distribution.
Important features of the
Student’s t distribution
• Use of the t statistic assumes that the parent
distribution is Gaussian
• The degree to which the t distribution
approximates a gaussian distribution
depends on N (the degrees of freedom)
• As N gets larger (above 30 or so), the
differences between t and z become
negligible
Application of Student’s t
distribution to a sample mean
The Student’s t statistic can also be used
to analyze differences between the
sample mean and the population mean:
(x   )
t
 s 


 N
Comparison of Student’s t and
Gaussian distributions
Note that, for a sufficiently large N (>30), t
can be replaced with z, and a Gaussian
distribution can be assumed
Exercise
The mean age of the 20 participants in one
workshop is 27 years, with a standard
deviation of 4 years. Next door, another
workshop has 16 participants with a mean age
of 29 years and standard deviation of 6 years.
Is the second workshop attracting older
technologists?
Preliminary analysis
• Is the population Gaussian?
• Can we use a Gaussian distribution for our
sample?
• What statistic should we calculate?
Solution
First, calculate the t statistic for the two
means:
t





( x1  x 2 )
s1   s 2

N1   N 2
(29  27)
2
2
6
4

20 16




 1.19

( x1  x 2 )
2
1
2
2
s
s

N1 N 2
Solution, cont.
Next, determine the degrees of freedom:
N df  N 1  N 2  2
 16  20  2
 34
Statistical Tables
df
t0.050
t0.025
t0.010
-
-
-
-
34
1.645
1.960
2.326
-
-
-
-
Conclusion
Since 1.16 is less than 1.64 (the t value
corresponding to 90% confidence limit), the
difference between the mean ages for the
participants in the two workshops is not
significant
The Paired t Test
Suppose we are comparing two sets of data in
which each value in one set has a
corresponding value in the other. Instead of
calculating the difference between the
means of the two sets, we can calculate the
mean difference between data pairs.
Instead of:
(x1  x2 )
N
we use:
to calculate t:
1
( x1  x2 )   ( x1i  x2i )
N i 1
(x1  x2 )
t
2
sd
N
Advantage of the Paired t
If the type of data permit paired analysis, the
paired t test is much more sensitive than the
unpaired t.
Why?
Applications of the Paired t
• Method correlation
• Comparison of therapies
Distributions
• Definition
• Examples
–
–
–
–
Binomial
Gaussian
Poisson
Student’s t
The 2 (Chi-square) Distribution
There is a general formula that relates actual
measurements to their predicted values
N
 
2
i 1
[ yi  f (xi )]
2

2
i
The 2 (Chi-square) Distribution
A special (and very useful) application of the
2 distribution is to frequency data
( ni  f i )
 
fi
i 1
N
2
2
Exercise
In your hospital, you have had 83 cases of
iatrogenic strep infection in your last 725
patients. St. Elsewhere, across town,
reports 35 cases of strep in their last 416
patients.
Do you need to review your infection control
policies?
Analysis
If your infection control policy is roughly as
effective as St. Elsewhere’s, we would
expect that the rates of strep infection for
the two hospitals would be similar. The
expected frequency, then would be the
average
83  35
118

 01034
.
725  416 1141
Calculating 2
First, calculate the expected frequencies at your
hospital (f1) and St. Elsewhere (f2)
f 1  725  01034
.
 75 cases
f 2  416  01034
.
 43 cases
Calculating 2
Next, we sum the squared differences between
actual and expected frequencies
(ni  f i )
 
fi
i
2
2
(83  75) 2 (35  43) 2


75
43
 2.34
Degrees of freedom
In general, when comparing k sample
proportions, the degrees of freedom for 2
analysis are k - 1. Hence, for our problem,
there is 1 degree of freedom.
Conclusion
A table of 2 values lists 3.841 as the 2
corresponding to a probability of 0.05.
So the variation (2between strep
infection rates at the two hospitals is within
statistically-predicted limits, and therefore
is not significant.
Distributions
• Definition
• Examples
–
–
–
–
–
Binomial
Gaussian
Poisson
Student’s t
2
The F distribution
• The F distribution predicts the expected
differences between the variances of two
samples
• This distribution has also been called
Snedecor’s F distribution, Fisher
distribution, and variance ratio distribution
The F distribution
The F statistic is simply the ratio of two
variances
V1
F
V2
(by convention, the larger V is the numerator)
Applications of the F distribution
There are several ways the F distribution can
be used. Applications of the F statistic are
part of a more general type of statistical
analysis called analysis of variance
(ANOVA). We’ll see more about ANOVA
later.
Example
You’re asked to do a “quick and dirty”
correlation between three whole blood
glucose analyzers. You prick your finger
and measure your blood glucose four times
on each of the analyzers.
Are the results equivalent?
Data
Analyzer 1
Analyzer 2
Analyzer 3
71
90
72
75
80
77
65
86
76
69
84
79
Analysis
The mean glucose concentrations for the three
analyzers are 70, 85, and 76.
If the three analyzers are equivalent, then we
can assume that all of the results are drawn
from a overall population with mean  and
variance 2.
Analysis, cont.
Approximate  by calculating the mean of the
means:
70  85  76
 77
3
Analysis, cont.
Calculate the variance of the means:
(70  77)  (85  77)  (76  77)
Vx 
3
 38
2
2
2
Analysis, cont.
But what we really want is the variance of the
population. Recall that:
x 

N
Analysis, cont.
Since we just calculated
Vx    38
2
x
we can solve for 
  

Vx    
 
 N
N
2
2
2
x
  N    4  38  152
2
2
x
Analysis, cont.
So we now have an estimate of the population
variance, which we’d like to compare to the
real variance to see whether they differ. But
what is the real variance?
We don’t know, but we can calculate the
variance based on our individual
measurements.
Analysis, cont.
If all the data were drawn from a larger
population, we can assume that the variances
are the same, and we can simply average the
variances for the three data sets.
V1  V2  V3
 14.4
3
Analysis, cont.
Now calculate the F statistic:
152
F
 10.6
14.4
Conclusion
A table of F values indicates that 4.26 is the limit
for the F statistic at a 95% confidence level
(when the appropriate degrees of freedom are
selected). Our value of 10.6 exceeds that, so
we conclude that there is significant variation
between the analyzers.
Distributions
• Definition
• Examples
–
–
–
–
–
–
Binomial
Gaussian
Poisson
Student’s t
2
F
Unknown or irregular distribution
• Transform
Probability
Probability
Log transform
x
log x
Unknown or irregular distribution
• Transform
• Non-parametric methods
Non-parametric methods
• Non-parametric methods make no
assumptions about the distribution of the
data
• There are non-parametric methods for
characterizing data, as well as for
comparing data sets
• These methods are also called distributionfree, robust, or sometimes non-metric tests
Application to Reference Ranges
The concentrations of most clinical analytes
are not usually distributed in a Gaussian
manner. Why?
How do we determine the reference range
(limits of expected values) for these
analytes?
Application to Reference
Ranges
• Reference ranges for normal, healthy populations
are customarily defined as the “central 95%”.
• An entirely non-parametric way of expressing this
is to eliminate the upper and lower 2.5% of data,
and use the remaining upper and lower values to
define the range.
• NCCLS recommends 120 values, dropping the two
highest and two lowest.
Application to Reference Ranges
What happens when we want to compare one
reference range with another? This is
precisely what CLIA ‘88 requires us to do.
How do we do this?
“Everything should be made as
simple as possible, but not
simpler.”
Albert Einstein
Solution #1: Simple comparison
Suppose we just do a small internal reference
range study, and compare our results to the
manufacturer’s range.
How do we compare them?
Is this a valid approach?
NCCLS recommendations
• Inspection Method: Verify reference
populations are equivalent
• Limited Validation: Collect 20 reference
specimens
– No more than 2 exceed range
– Repeat if failed
• Extended Validation: Collect 60 reference
specimens; compare ranges.
Solution #2:
*
Mann-Whitney
Rank normal values (x1,x2,x3...xn) and the reference
population (y1,y2,y3...yn):
x1, y1, x2, x3, y2, y3 ... xn, yn
Count the number of y values that follow each x, and call
the sum Ux. Calculate Uy also.
*Also
called the U test, rank sum test, or Wilcoxen’s test.
Mann-Whitney, cont.
It should be obvious that: Ux + Uy = NxNy
If the two distributions are the same, then:
Ux = Uy = 1/2NxNy
Large differences between Ux and Uy indicate
that the distributions are not equivalent
“‘Obvious’ is the most dangerous
word in mathematics.”
Eric Temple Bell (1883-1960)
Solution #3: Run test
In the run test, order the values in the two
distributions as before:
x1, y1, x2, x3, y2, y3 ... xn, yn
Add up the number of runs (consecutive values from
the same distribution). If the two data sets are
randomly selected from one population, there will
be few runs.
Solution #4: The Monte Carlo
method
Sometimes, when we don’t know anything
about a distribution, the best thing to do is
independently test its characteristics.
The Monte Carlo method
Asq  xy  x
2
x

Acir  r    
 2
y
2
Acir
  4
Asq
x
2
The Monte Carlo method
N
mean, SD
N
mean, SD
N
mean, SD
N
mean, SD
Reference population
The Monte Carlo method
With the Monte Carlo method, we have
simulated the test we wish to apply--that is,
we have randomly selected samples from
the parent distribution, and determined
whether our in-house data are in agreement
with the randomly-selected samples.
Analysis of paired data
• For certain types of laboratory studies, the
data we gather is paired
• We typically want to know how closely the
paired data agree
• We need quantitative measures of the extent
to which the data agree or disagree
• Examples?
Examples of paired data
•
•
•
•
Method correlation data
Pharmacodynamic effects
Risk analysis
Pathophysiology
Correlation
50
45
40
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10
5
0
0
5
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Linear regression (least squares)
Linear regression analysis generates an equation
for a straight line
y = mx + b
where m is the slope of the line and b is the
value of y when x = 0 (the y-intercept).
The calculated equation minimizes the differences
between actual y values and the linear
regression line.
Correlation
50
45
40
y = 1.031x - 0.024
35
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5
0
0
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Covariance
Do x and y values vary in concert, or
randomly?
1
cov( x , y )   ( yi  y )( xi  x )
N i
1
cov( x , y )   ( yi  y )( xi  x )
N i
• What if y increases when x increases?
• What if y decreases when x increases?
• What if y and x vary independently?
Covariance
It is clear that the greater the covariance, the
stronger the relationship between x and y.
But . . . what about units?
e.g., if you measure glucose in mg/dL, and I
measure it in mmol/L, who’s likely to have
the highest covariance?
The Correlation Coefficient
1
( yi  y )( xi  x )

cov( x , y ) N i


 x y
 y x
1    1
The Correlation Coefficient
• The correlation coefficient is a unitless
quantity that roughly indicates the degree to
which x and y vary in the same direction.
•  is useful for detecting relationships
between parameters, but it is not a very
sensitive measure of the spread.
Correlation
50
45
40
y = 1.031x - 0.024
 = 0.9986
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
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50
Correlation
50
45
40
y = 1.031x - 0.024
 = 0.9894
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
Standard Error of the Estimate
The linear regression equation gives us a way
to calculate an “estimated” y for any given x
value, given the symbol ŷ (y-hat):
y  mx  b
Standard Error of the Estimate
Now what we are interested in is the average
difference between the measured y and its
estimate, ŷ :
sy / x
1
2

( yi  yi )

N i
Correlation
50
45
40
y = 1.031x - 0.024
 = 0.9986
sy/x=1.83
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
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Correlation
50
45
40
y = 1.031x - 0.024
 = 0.9894
sy/x = 5.32
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
Standard Error of the Estimate
If we assume that the errors in the y
measurements are Gaussian (is that a safe
assumption?), then the standard error of the
estimate gives us the boundaries within
which 67% of the y values will fall.
2sy/x defines the 95% boundaries..
Limitations of linear regression
• Assumes no error in x measurement
• Assumes that variance in y is constant
throughout concentration range
Alternative approaches
• Weighted linear regression analysis can
compensate for non-constant variance
among y measurements
• Deming regression analysis takes into
account variance in the x measurements
• Weighted Deming regression analysis allows
for both
Evaluating method performance
• Precision
Method Precision
• Within-run: 10 or 20 replicates
– What types of errors does within-run precision
reflect?
• Day-to-day: NCCLS recommends
evaluation over 20 days
– What types of errors does day-to-day precision
reflect?
Evaluating method performance
• Precision
• Sensitivity
Method Sensitivity
• The analytical sensitivity of a method refers
to the lowest concentration of analyte that
can be reliably detected.
• The most common definition of sensitivity
is the analyte concentration that will result
in a signal two or three standard deviations
above background.
Signal
Signal/Noise threshold
time
Other measures of sensitivity
• Limit of Detection (LOD) is sometimes defined as the
concentration producing an S/N > 3.
– In drug testing, LOD is customarily defined as the lowest
concentration that meets all identification criteria.
• Limit of Quantitation (LOQ) is sometimes defined as
the concentration producing an S/N >5.
– In drug testing, LOQ is customarily defined as the lowest
concentration that can be measured within ±20%.
Question
At an S/N ratio of 5, what is the minimum CV
of the measurement?
If the S/N is 5, 20% of the measured signal is
noise, which is random. Therefore, the CV
must be at least 20%.
Evaluating method performance
• Precision
• Sensitivity
• Linearity
Method Linearity
• A linear relationship between concentration and
signal is not absolutely necessary, but it is
highly desirable. Why?
• CLIA ‘88 requires that the linearity of analytical
methods is verified on a periodic basis.
Ways to evaluate linearity
• Visual/linear regression
Signal
Concentration
Outliers
We can eliminate any point that differs from
the next highest value by more than 0.765
(p=0.05) times the spread between the
highest and lowest values (Dixon test).
Example: 4, 5, 6, 13
(13 - 4) x 0.765 = 6.89
Limitation of linear regression
method
If the analytical method has a high variance
(CV), it is likely that small deviations from
linearity will not be detected due to the high
standard error of the estimate
Signal
Concentration
Ways to evaluate linearity
• Visual/linear regression
• Quadratic regression
Quadratic regression
Recall that, for linear data, the relationship
between x and y can be expressed as
y = f(x) = a + bx
Quadratic regression
A curve is described by the quadratic
equation:
y = f(x) = a + bx + cx2
which is identical to the linear equation except
for the addition of the cx2 term.
Quadratic regression
It should be clear that the smaller the x2
coefficient, c, the closer the data are to
linear (since the equation reduces to the
linear form when c approaches 0).
What is the drawback to this approach?
Ways to evaluate linearity
• Visual/linear regression
• Quadratic regression
• Lack-of-fit analysis
Lack-of-fit analysis
• There are two components of the variation
from the regression line
– Intrinsic variability of the method
– Variability due to deviations from linearity
• The problem is to distinguish between these
two sources of variability
• What statistical test do you think is
appropriate?
Signal
Concentration
Lack-of-fit analysis
The ANOVA technique requires that method
variance is constant at all concentrations.
Cochran’s test is used to test whether this is
the case.
VL
 0.5981 ( p  0.05)
Vi
i
Lack-of-fit method calculations
• Total sum of the squares: the variance
calculated from all of the y values
• Linear regression sum of the squares: the
variance of y values from the regression line
• Residual sum of the squares: difference
between TSS and LSS
• Lack of fit sum of the squares: the RSS
minus the pure error (sum of variances)
Lack-of-fit analysis
• The LOF is compared to the pure error to give
the “G” statistic (which is actually F)
• If the LOF is small compared to the pure
error, G is small and the method is linear
• If the LOF is large compared to the pure error,
G will be large, indicating significant
deviation from linearity
Significance limits for G
• 90% confidence = 2.49
• 95% confidence = 3.29
• 99% confidence = 5.42
“If your experiment needs
statistics, you ought to have done
a better experiment.”
Ernest Rutherford (1871-1937)
Evaluating Clinical Performance
of laboratory tests
• The clinical performance of a laboratory
test defines how well it predicts disease
• The sensitivity of a test indicates the
likelihood that it will be positive when
disease is present
Clinical Sensitivity
If TP as the number of “true positives”, and FN
is the number of “false negatives”, the
sensitivity is defined as:
TP
Sensitivity 
100
TP  FN
Example
Of 25 admitted cocaine abusers, 23 tested
positive for urinary benzoylecgonine and 2
tested negative. What is the sensitivity of the
urine screen?
23
 100  92%
23  2
Evaluating Clinical Performance
of laboratory tests
• The clinical performance of a laboratory test
defines how well it predicts disease
• The sensitivity of a test indicates the
likelihood that it will be positive when disease
is present
• The specificity of a test indicates the
likelihood that it will be negative when
disease is absent
Clinical Specificity
If TN is the number of “true negative” results,
and FP is the number of falsely positive
results, the specificity is defined as:
TN
Specificity 
100
TN  FP
Example
What would you guess is the specificity of
any particular clinical laboratory test?
(Choose any one you want)
Answer
Since reference ranges are customarily set to
include the central 95% of values in healthy
subjects, we expect 5% of values from
healthy people to be “abnormal”--this is the
false positive rate.
Hence, the specificity of most clinical tests is
no better than 95%.
Sensitivity vs. Specificity
• Sensitivity and specificity are inversely
related.
Disease
+
Marker concentration
Sensitivity vs. Specificity
• Sensitivity and specificity are inversely
related.
• How do we determine the best compromise
between sensitivity and specificity?
True positive rate
(sensitivity)
Receiver Operating
Characteristic
False positive rate
1-specificity
Evaluating Clinical Performance
of laboratory tests
• The sensitivity of a test indicates the likelihood that it
will be positive when disease is present
• The specificity of a test indicates the likelihood that it
will be negative when disease is absent
• The predictive value of a test indicates the
probability that the test result correctly classifies a
patient
Predictive Value
The predictive value of a clinical laboratory test
takes into account the prevalence of a certain
disease, to quantify the probability that a
positive test is associated with the disease in a
randomly-selected individual, or alternatively,
that a negative test is associated with health.
Illustration
• Suppose you have invented a new screening
test for Addison disease.
• The test correctly identified 98 of 100 patients
with confirmed Addison disease (What is the
sensitivity?)
• The test was positive in only 2 of 1000 patients
with no evidence of Addison disease (What is
the specificity?)
Test performance
• The sensitivity is 98.0%
• The specificity is 99.8%
• But Addison disease is a rare disorder-incidence = 1:10,000
• What happens if we screen 1 million
people?
Analysis
• In 1 million people, there will be 100 cases of
Addison disease.
• Our test will identify 98 of these cases (TP)
• Of the 999,900 non-Addison subjects, the test
will be positive in 0.2%, or about 2,000 (FP).
Predictive value of the positive
test
The predictive value is the % of all positives that
are true positives:
TP
PV 
 100
TP  FP
98

 100
98  2000
 4.7%
What about the negative
predictive value?
• TN = 999,900 - 2000 = 997,900
• FN = 100 * 0.002 = 0 (or 1)
TN
PV 
 100
TN  FN
997,900

 100
997,900  1
 100%
Summary of predictive value
Predictive value describes the usefulness of a
clinical laboratory test in the real world.
Or does it?
Lessons about predictive value
• Even when you have a very good test, it is
generally not cost effective to screen for
diseases which have low incidence in the
general population. Exception?
• The higher the clinical suspicion, the better the
predictive value of the test. Why?
Efficiency
We can combine the PV+ and PV- to give a
quantity called the efficiency:
TP  TN
Efficiency 
100
TP  FP  TN  FN
The efficiency is the percentage of all patients
that are classified correctly by the test
result.
Efficiency of our Addison screen
98  997,900
 100  99.8%
98  2000  997,900  2
“To call in the statistician after the
experiment is done may be no more
than asking him to perform
a postmortem examination: he may
be able to say what the experiment
died of.”
Ronald Aylmer Fisher (1890 - 1962)
Application of Statistics to
Quality Control
• We expect quality control to fit a Gaussian
distribution
• We can use Gaussian statistics to predict the
variability in quality control values
• What sort of tolerance will we allow for
variation in quality control values?
• Generally, we will question variations that have
a statistical probability of less than 5%
“He uses statistics as a
drunken man uses lamp posts -for support rather than
illumination.”
Andrew Lang (1844-1912)
Westgard’s rules
•
•
•
•
•
•
12s
13s
22s
R4s
41s
10x
1 in 20
1 in 300
1 in 400
1 in 800
1 in 600
1 in 1000
Some examples
+3sd
+2sd
+1sd
mean
-1sd
-2sd
-3sd
Some examples
+3sd
+2sd
+1sd
mean
-1sd
-2sd
-3sd
Some examples
+3sd
+2sd
+1sd
mean
-1sd
-2sd
-3sd
Some examples
+3sd
+2sd
+1sd
mean
-1sd
-2sd
-3sd
“In science one tries to tell
people, in such a way as to be
understood by everyone,
something that
no one ever knew before. But in
poetry, it's the exact opposite.”
Paul Adrien Maurice Dirac (1902- 1984)