Title: Practical Applications of Statistical Methods in the Clinical Laboratory
1Practical 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
2Statistics 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 (18221911)
3There are three kinds of lies Lies, damned
lies, and statistics
 Benjamin Disraeli (18041881)
4What 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
5Do not worry about your difficulties in
mathematics, I assure you that mine are greater
 Albert Einstein (18791955)
6I don't believe in mathematics
7Summation function
8Product function
9The Mean (average)
 The mean is a measure of the centrality of a set
of data.
10Mean (arithmetical)
11Mean (geometric)
12Use of the Geometric mean
 The geometric mean is primarily used to average
ratios or rates of change.
13Mean (harmonic)
14Example 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?
15Example of the use of Harmonic mean
 You spent 12 on 50 dozen pills, so the average
cost is 12/500.24, or 24 cents.  This also happens to be the harmonic mean of 20
and 30
16Root mean square (RMS)
17For the data set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
18The Weighted Mean
19Other measures of centrality
20The Mode
 The mode is the value that occurs most often
21Other measures of centrality
22The Midrange
 The midrange is the mean of the highest and
lowest values
23Other measures of centrality
24The 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.
25Example of the use of median vs. mean
 Suppose youre 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
Whats the median size? The agent replies
1,800 sq. ft.  What does this tell you about the sizes of the
houses in the neighborhood?
26Measuring 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?
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28The Variance
29The 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.
30The Variance
31The Variance
 In what units is the variance?
 Is that a problem?
32The Standard Deviation
33The 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.
34The Standard Deviation
In what units is the standard deviation? Is that
a problem?
35The Coefficient of Variation
 Sometimes called the Relative Standard Deviation
(RSD or RSD)
36Standard Deviation (or Error) of the Mean
 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.
37Exercises
 How many measurements must we average to improve
our precision by a factor of 2?
38Answer
 To improve precision by a factor of 2
39Exercises
 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?
40Answer
 To improve precision by a factor of 10
41Exercises
 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?
42Answer
 Improvement in CV by running duplicates
43Population vs. Sample standard deviation
 When we speak of a population, were referring to
the entire data set, which will have a mean ?
44Population vs. Sample standard deviation
 When we speak of a population, were referring to
the entire data set, which will have a mean ?  When we speak of a sample, were referring to a
subset of the population, customarily designated
xbar  Which is used to calculate the standard deviation?
45Sir, I have found you an argument. I am not
obliged to find you an understanding.
 Samuel Johnson (17091784)
46Population vs. Sample standard deviation
47Distributions
48Statistical (probability) Distribution
 A statistical distribution is a
mathematicallyderived 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
49Distributions
50Binomial 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
51Example
 What is the probability that 10 of the 12 babies
born one busy evening in your hospital will be
girls?
52Solution
53Distributions
 Definition
 Examples
 Binomial
54God does arithmetic
 Karl Friedrich Gauss (17771855)
55The Gaussian Distribution
 What is the Gaussian distribution?
5663 81 36 12 28 7 79 52 96 17 22 4 61 85
etc.
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5863 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
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60. . . etc.
61Probability
x
62The Gaussian Probability Function
 The probability of x in a Gaussian distribution
with mean ? and standard deviation ? is given by
63The Gaussian Distribution
 What is the Gaussian distribution?
 What types of data fit a Gaussian distribution?
64Like 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 )
65Are these Gaussian?
 Human height
 Outside temperature
 Raindrop size
 Blood glucose concentration
 Serum CK activity
 QC results
 Proficiency results
66The Gaussian Distribution
 What is the Gaussian distribution?
 What types of data fit a Gaussian distribution?
 What is the advantage of using a Gaussian
distribution?
67Gaussian probability distribution
Probability
.67
.95
µ
µ?
µ2?
µ3?
µ?
µ2?
µ3?
68What 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
69Some useful Gaussian probabilities
Range
Probability
Odds
/ 1.00 ?
68.3
1 in 3
/ 1.64 ?
90.0
1 in 10
/ 1.96 ?
95.0
1 in 20
/ 2.58 ?
99.0
1 in 100
70Example
That
This
71On the Gaussian curve Experimentalists think
that it is a mathematical theorem while the
mathematicians believe it to be an experimental
fact.
 Gabriel Lippman (18451921)
72Distributions
 Definition
 Examples
 Binomial
 Gaussian
73"Life is good for only two things, discovering
mathematics and teaching mathematics"
 Siméon Poisson (17811840)
74The Poisson Distribution
 The Poisson distribution predicts the frequency
of r events occurring randomly in time, when the
expected frequency is ?
75Examples of events described by a Poisson
distribution
?
 Lightning
 Accidents
 Laboratory?
76A very useful property of the Poisson distribution
77Using the Poisson distribution
 How many counts must be collected in an RIA in
order to ensure an analytical CV of 5 or less?
78Answer
79Distributions
 Definition
 Examples
 Binomial
 Gaussian
 Poisson
80The Students 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
81Questions 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 dont know whether our sample
has the same distribution as the population from
which it was drawn.
82 Recall that the Gaussian distribution is defined
by the probability function  Note that the exponential factor contains both
??and ?, both population parameters. The factor
is often simplified by making the substitution
83 The variable z in the equation
 is distributed according to a unit gaussian,
since it has a mean of zero and a standard
deviation of 1
84Gaussian probability distribution
Probability
.67
.95
0
1
2
3
1
2
3
z
85 But if we use the sample mean and standard
deviation instead, we get  and weve defined a new quantity, t, which is not
distributed according to the unit Gaussian. It
is distributed according to the Students t
distribution.
86Important features of the Students 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
87Application of Students t distribution to a
sample mean
 The Students t statistic can also be used to
analyze differences between the sample mean and
the population mean
88Comparison of Students t and Gaussian
distributions
 Note that, for a sufficiently large N (gt30), t
can be replaced with z, and a Gaussian
distribution can be assumed
89Exercise
 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?
90Preliminary analysis
 Is the population Gaussian?
 Can we use a Gaussian distribution for our
sample?  What statistic should we calculate?
91Solution
 First, calculate the t statistic for the two
means
92Solution, cont.
 Next, determine the degrees of freedom
93Statistical Tables
94Conclusion
 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
95The 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.
96 Instead of
 we use
 to calculate t
97Advantage 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?
98Applications of the Paired t
 Method correlation
 Comparison of therapies
99Distributions
 Definition
 Examples
 Binomial
 Gaussian
 Poisson
 Students t
100The ?2 (Chisquare) Distribution
 There is a general formula that relates actual
measurements to their predicted values
101The ?2 (Chisquare) Distribution
 A special (and very useful) application of the ?2
distribution is to frequency data
102Exercise
 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?
103Analysis
 If your infection control policy is roughly as
effective as St. Elsewheres, we would expect
that the rates of strep infection for the two
hospitals would be similar. The expected
frequency, then would be the average
104Calculating ?2
 First, calculate the expected frequencies at your
hospital (f1) and St. Elsewhere (f2)
105Calculating ?2
 Next, we sum the squared differences between
actual and expected frequencies
106Degrees 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.
107Conclusion
 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
statisticallypredicted limits, and therefore is
not significant.
108Distributions
 Definition
 Examples
 Binomial
 Gaussian
 Poisson
 Students t
 ?2
109The F distribution
 The F distribution predicts the expected
differences between the variances of two samples  This distribution has also been called Snedecors
F distribution, Fisher distribution, and variance
ratio distribution
110The F distribution
 The F statistic is simply the ratio of two
variances  (by convention, the larger V is the numerator)
111Applications 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). Well see
more about ANOVA later.
112Example
 Youre 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?
113Data
114Analysis
 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.
115Analysis, cont.
 Approximate ? by calculating the mean of the
means
116Analysis, cont.
 Calculate the variance of the means
117Analysis, cont.
 But what we really want is the variance of the
population. Recall that
118Analysis, cont.
 Since we just calculated
 we can solve for ??
119Analysis, cont.
 So we now have an estimate of the population
variance, which wed like to compare to the real
variance to see whether they differ. But what is
the real variance?  We dont know, but we can calculate the variance
based on our individual measurements.
120Analysis, 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.
121Analysis, cont.
 Now calculate the F statistic
122Conclusion
 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.
123Distributions
 Definition
 Examples
 Binomial
 Gaussian
 Poisson
 Students t
 ?2
 F
124Unknown or irregular distribution
125Log transform
Probability
Probability
log x
x
126Unknown or irregular distribution
 Transform
 Nonparametric methods
127Nonparametric methods
 Nonparametric methods make no assumptions about
the distribution of the data  There are nonparametric methods for
characterizing data, as well as for comparing
data sets  These methods are also called distributionfree,
robust, or sometimes nonmetric tests
128Application 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?
129Application to Reference Ranges
 Reference ranges for normal, healthy populations
are customarily defined as the central 95.  An entirely nonparametric 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.
130Application 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?
131Everything should be made as simple as possible,
but not simpler.
132Solution 1 Simple comparison
 Suppose we just do a small internal reference
range study, and compare our results to the
manufacturers range.  How do we compare them?
 Is this a valid approach?
133NCCLS 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.
134Solution 2 MannWhitney
 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
Wilcoxens test.
135MannWhitney, 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
136Obvious is the most dangerous word in
mathematics.
 Eric Temple Bell (18831960)
137Solution 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.
138Solution 4 The Monte Carlo method
 Sometimes, when we dont know anything about a
distribution, the best thing to do is
independently test its characteristics.
139The Monte Carlo method
y
x
140The Monte Carlo method
Reference population
141The Monte Carlo method
 With the Monte Carlo method, we have simulated
the test we wish to applythat is, we have
randomly selected samples from the parent
distribution, and determined whether our inhouse
data are in agreement with the randomlyselected
samples.
142Analysis 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?
143Examples of paired data
 Method correlation data
 Pharmacodynamic effects
 Risk analysis
 Pathophysiology
144Correlation
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145Linear 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 yintercept).  The calculated equation minimizes the differences
between actual y values and the linear regression
line.
146Correlation
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147Covariance
 Do x and y values vary in concert, or randomly?
148 What if y increases when x increases?
 What if y decreases when x increases?
 What if y and x vary independently?
149Covariance
 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, whos likely to have the
highest covariance?
150The Correlation Coefficient
151The 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.
152Correlation
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153Correlation
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154Standard 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 y (yhat)
155Standard Error of the Estimate
 Now what we are interested in is the average
difference between the measured y and its
estimate, y
156Correlation
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157Correlation
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158Standard 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..
159Limitations of linear regression
 Assumes no error in x measurement
 Assumes that variance in y is constant throughout
concentration range
160Alternative approaches
 Weighted linear regression analysis can
compensate for nonconstant variance among y
measurements  Deming regression analysis takes into account
variance in the x measurements  Weighted Deming regression analysis allows for
both
161Evaluating method performance
162Method Precision
 Withinrun 10 or 20 replicates
 What types of errors does withinrun precision
reflect?  Daytoday NCCLS recommends evaluation over 20
days  What types of errors does daytoday precision
reflect?
163Evaluating method performance
164Method 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.
165Signal
time
166Other measures of sensitivity
 Limit of Detection (LOD) is sometimes defined as
the concentration producing an S/N gt 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 gt5.  In drug testing, LOQ is customarily defined as
the lowest concentration that can be measured
within 20.
167Question
 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.
168Evaluating method performance
 Precision
 Sensitivity
 Linearity
169Method 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.
170Ways to evaluate linearity
171Signal
Concentration
172Outliers
 We can eliminate any point that differs from the
next highest value by more than 0.765 (p0.05)
times the spread between the highest and lowest
values (Dixon test).  Example 4, 5, 6, 13
 (13  4) x 0.765 6.89
173Limitation 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
174Signal
Concentration
175Ways to evaluate linearity
 Visual/linear regression
 Quadratic regression
176Quadratic regression
 Recall that, for linear data, the relationship
between x and y can be expressed as  y f(x) a bx
177Quadratic 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.
178Quadratic 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?
179Ways to evaluate linearity
 Visual/linear regression
 Quadratic regression
 Lackoffit analysis
180Lackoffit 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?
181Signal
Concentration
182Lackoffit analysis
 The ANOVA technique requires that method variance
is constant at all concentrations. Cochrans
test is used to test whether this is the case.
183Lackoffit 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)
184Lackoffit 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
185Significance limits for G
 90 confidence 2.49
 95 confidence 3.29
 99 confidence 5.42
186If your experiment needs statistics, you ought
to have done a better experiment.
 Ernest Rutherford (18711937)
187Evaluating 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
188Clinical Sensitivity
 If TP as the number of true positives, and FN
is the number of false negatives, the
sensitivity is defined as
189Example
 Of 25 admitted cocaine abusers, 23 tested
positive for urinary benzoylecgonine and 2 tested
negative. What is the sensitivity of the urine
screen?
190Evaluating 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
191Clinical Specificity
 If TN is the number of true negative results,
and FP is the number of falsely positive results,
the specificity is defined as
192Example
 What would you guess is the specificity of any
particular clinical laboratory test? (Choose any
one you want)
193Answer
 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 abnormalthis is the false
positive rate.  Hence, the specificity of most clinical tests is
no better than 95.
194Sensitivity vs. Specificity
 Sensitivity and specificity are inversely related.
195Marker concentration

Disease
196Sensitivity vs. Specificity
 Sensitivity and specificity are inversely
related.  How do we determine the best compromise between
sensitivity and specificity?
197Receiver Operating Characteristic
198Evaluating 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
199Predictive 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 randomlyselected individual, or alternatively,
that a negative test is associated with health.
200Illustration
 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?)
201Test performance
 The sensitivity is 98.0
 The specificity is 99.8
 But Addison disease is a rare disorderincidence
110,000  What happens if we screen 1 million people?
202Analysis
 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 nonAddison subjects, the test
will be positive in 0.2, or about 2,000 (FP).
203Predictive value of the positive test
 The predictive value is the of all positives
that are true positives
204What about the negative predictive value?
 TN 999,900  2000 997,900
 FN 100 0.002 0 (or 1)
205Summary of predictive value
 Predictive value describes the usefulness of a
clinical laboratory test in the real world.  Or does it?
206Lessons 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?
207Efficiency
 We can combine the PV and PV to give a quantity
called the efficiency  The efficiency is the percentage of all patients
that are classified correctly by the test result.
208Efficiency of our Addison screen
209To 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)
210Application 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
211He uses statistics as a drunken man uses lamp
posts  for support rather than illumination.
212Westgards rules
 1 in 20
 1 in 300
 1 in 400
 1 in 800
 1 in 600
 1 in 1000
213Some examples
3sd
2sd
1sd
mean
1sd
2sd
3sd
214Some examples
3sd
2sd
1sd
mean
1sd
2sd
3sd
215Some examples
3sd
2sd
1sd
mean
1sd
2sd
3sd
216Some examples
3sd
2sd
1sd
mean
1sd
2sd
3sd
217In 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)