Decisional Analysis and Statistics
Performance measure Incidence Rate Prevalence Sensitivity Specificity
False-Negative Rate False-Positive Rate Positive Predictive Value Negative
PredictiveValue Overall Accuracy/Diagnostic Efficiency Deriving Missing
Performance Measures Youden's Index Bayes's Theorem and Modifications Odds
and Likelihood Ratios Odds & Likelihood Ratios for Seq.Testing Risk
Sensitivity and Risk Specificity Statistics for Use in Quality Control Mean
Standard Deviation (SD) Coefficient of Variation (CV) Standard Deviation
Interval (SDI) Coefficient of Variation Interval (CVI) Total Allowable Error
Chi Square Outcome Comparison of Two Groups Comparison of Two Observers
Receiver Oper.Characteristics Plots Z-score Westgard Control Rules Series of
Control Rules Evaluating the Medical Literature Assessing the Method.Qual.of
Clin.Stud. Measures of the Consequence of Treatment Number Needed to Treat
Corr.Risk Ratio & Estimating Relat. Risk Confidence Intervals Confidence
Interval for a Single Mean Confid. Interv. for Diff. between Two Means
Confidence Interv. for a Single Proport. Confidence Interv. Observations is
0 or 1 Confidence Interv. btwn Odds Ratio Confidence Interv Difference Btwn
etc. Odds and Percentages Benefit, Risk and Threshold for an Action Testing
and Test Treatment Thresholds Performance Measures Overview: The usefulness
of a test is often judged in how well it makes the diagnosis for the
presence or absence of a disease. A person with the disease who has a
"positive" test is termed a true positive, whereas a person with the disease
but a "negative" test result is termed a false negative. A person without
disease who has a "positive" result is termed a false positive, while a
person without disease having a "negative" result is termed a true negative.
In real life things are not always clear cut; the distinction between
positive and negative in a test result is sometimes artificial while it is
not always possible to say if a person does or does not have a disease.
Positive for Disease (+) Negative for Disease (-) Result Positive (+) a =
true positive b = false positive Result Negative (-) c = false negative d =
true negative TOP References Braunwald E, Isselbacher KJ, et al (editors).
Harrison's Principles of Internal Medicine, 11th edition. McGraw-Hill Book
Publishers. 1987. page 7 Goldman L. Chapter 10: Quantitative aspects of
clinical reasoning. pages 43-48. IN: Isselbacher KJ, Braunwald E, et al.
Harrison's Principles of Internal Medicine, Thirteenth Edition. McGraw-Hill.
1994. Panzer RJ, Black ER, Griner PF. Interpretation of diagnostic tests and
strategies for their use in quantitative decision making. pages 17-28. IN:
Panzer RJ, Black ER, et al. Diagnostic Strategies for Common Medical
Problems. American College of Physicians. 1991. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Incidence Rate Overview: The incidence rate is the number of new cases of a
disease in the total population per unit time. incidence rate = ((A) / (a +
b + c + d)) / T) where: A = number of new cases of a disease for given time
period, which is a subset of all the true positives (a + c) ; (a + b + c +
d) = sum of (true positives, false positives, false negatives, true
negatives) = total population T = unit of time TOP References: Goldman L.
Chapter 10: Quantitative aspects of clinical reasoning. pages 43-48. IN:
Isselbacher KJ, Braunwald E, et al. Harrison's Principles of Internal
Medicine, Thirteenth Edition. McGraw-Hill. 1994. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983., page 51 Prevalence
Overview: Prevalence is all patients with disease divided by all patients
tested. This is also termed the "prior probability." prevalence = (a + c) /
(a + b + c + d) where: a + c = true positives + false negatives = all people
with disease (a + b + c + d) = sum of (true positives, false positives,
false negatives, true negatives) = total population TOP References Braunwald
E, Isselbacher KJ, et al (editors). Harrison's Principles of Internal
Medicine, 11th edition. McGraw-Hill Book Publishers. 1987. page 7 Goldman L.
Chapter 10: Quantitative aspects of clinical reasoning. pages 43-48. IN:
Isselbacher KJ, Braunwald E, et al. Harrison's Principles of Internal
Medicine, Thirteenth Edition. McGraw-Hill. 1994. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Sensitivity Overview: Sensitivity is the true-positive test results divided
by all patients with the disease. sensitivity = (a / (a + c)) where: a =
true positives a + c = true positives+ false negatives = all people with
disease Comments The better the seNsitivity of the test, the fewer the false
Negatives. TOP References Braunwald E, Isselbacher KJ, et al (editors).
Harrison's Principles of Internal Medicine, 11th edition. McGraw-Hill Book
Publishers. 1987. page 7 Goldman L. Chapter 10: Quantitative aspects of
clinical reasoning. pages 43-48. IN: Isselbacher KJ, Braunwald E, et al.
Harrison's Principles of Internal Medicine, Thirteenth Edition. McGraw-Hill.
1994. Panzer RJ, Black ER, Griner PF. Interpretation of diagnostic tests and
strategies for their use in quantitative decision making. pages 17-28. IN:
Panzer RJ, Black ER, et al. Diagnostic Strategies for Common Medical
Problems. American College of Physicians. 1991. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Specificity Overview: The specificity of a test is the true-negative test
results divided by all patients without the disease. specificity = (d / (b +
d)) where: d = true negatives (b + d) = sum of ( false positives, true
negatives) = all people without disease Comments The better the sPecificity
of the test, the fewer the false Positives. TOP References Braunwald E,
Isselbacher KJ, et al (editors). Harrison's Principles of Internal Medicine,
11th edition. McGraw-Hill Book Publishers. 1987. page 7 Goldman L. Chapter
10: Quantitative aspects of clinical reasoning. pages 43-48. IN: Isselbacher
KJ, Braunwald E, et al. Harrison's Principles of Internal Medicine,
Thirteenth Edition. McGraw-Hill. 1994. Panzer RJ, Black ER, Griner PF.
Interpretation of diagnostic tests and strategies for their use in
quantitative decision making. pages 17-28. IN: Panzer RJ, Black ER, et al.
Diagnostic Strategies for Common Medical Problems. American College of
Physicians. 1991. Speicher C, Smith JW Jr.. Choosing Effective Laboratory
Tests. WB Saunders. 1983. pages 50-51, and 210 False-Negative Rate Overview:
The false negative rate for a test is the false-negative test results
divided by all patients with the disease. false-negative rate = (c / (a +
c)) where: c = false negatives (a + c ) = sum of (true positives, false
negatives) = all people with disease TOP References Braunwald E, Isselbacher
KJ, et al (editors). Harrison's Principles of Internal Medicine, 11th
edition. McGraw-Hill Book Publishers. 1987. page 7 Goldman L. Chapter 10:
Quantitative aspects of clinical reasoning. pages 43-48. IN: Isselbacher KJ,
Braunwald E, et al. Harrison's Principles of Internal Medicine, Thirteenth
Edition. McGraw-Hill. 1994. Panzer RJ, Black ER, Griner PF. Interpretation
of diagnostic tests and strategies for their use in quantitative decision
making. pages 17-28. IN: Panzer RJ, Black ER, et al. Diagnostic Strategies
for Common Medical Problems. American College of Physicians. 1991. Speicher
C, Smith JW Jr.. Choosing Effective Laboratory Tests. WB Saunders. 1983.
pages 50-51, and 210 False-Positive Rate Overview: The false positive rate
for a test is the false-positive test results divided by all patients
without the disease. false-positive rate = (b / (b + d)) where: b = false
positives (b + d) = sum of (false positives, true negatives) = all people
without disease TOP References Braunwald E, Isselbacher KJ, et al (editors).
Harrison's Principles of Internal Medicine, 11th edition. McGraw-Hill Book
Publishers. 1987. page 7 Goldman L. Chapter 10: Quantitative aspects of
clinical reasoning. pages 43-48. IN: Isselbacher KJ, Braunwald E, et al.
Harrison's Principles of Internal Medicine, Thirteenth Edition. McGraw-Hill.
1994. Panzer RJ, Black ER, Griner PF. Interpretation of diagnostic tests and
strategies for their use in quantitative decision making. pages 17-28. IN:
Panzer RJ, Black ER, et al. Diagnostic Strategies for Common Medical
Problems. American College of Physicians. 1991. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Positive Predictive Value Overview: The positive predictive value is
true-positive test results divided by all positive test results. This is
also referred to as the predictive value of a positive test. This is
equivelent to Bayes's formula for post-test probability given a positive
result. positive predictive value = (a / (a + b)) where: a = true positives
(a + b ) = sum of (true positives, false positives) = all positive test
results TOP References Braunwald E, Isselbacher KJ, et al (editors).
Harrison's Principles of Internal Medicine, 11th edition. McGraw-Hill Book
Publishers. 1987. page 7 Goldman L. Chapter 10: Quantitative aspects of
clinical reasoning. pages 43-48. IN: Isselbacher KJ, Braunwald E, et al.
Harrison's Principles of Internal Medicine, Thirteenth Edition. McGraw-Hill.
1994. Panzer RJ, Black ER, Griner PF. Interpretation of diagnostic tests and
strategies for their use in quantitative decision making. pages 17-28. IN:
Panzer RJ, Black ER, et al. Diagnostic Strategies for Common Medical
Problems. American College of Physicians. 1991. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Negative Predictive Value Overview: The negative predictive value is the
true-negative test results divided by all patients with negative results.
This is also referred to as the predictive value of a negative test. This is
equivelent to Bayes's formula for post-test probability given a negative
result. negative predictive value = (d / (c + d)) where: d = true negatives
(c + d) = sum of ( false negatives, true negatives) = all negative test
results TOP References Braunwald E, Isselbacher KJ, et al (editors).
Harrison's Principles of Internal Medicine, 11th edition. McGraw-Hill Book
Publishers. 1987. page 7 Goldman L. Chapter 10: Quantitative aspects of
clinical reasoning. pages 43-48. IN: Isselbacher KJ, Braunwald E, et al.
Harrison's Principles of Internal Medicine, Thirteenth Edition. McGraw-Hill.
1994. Panzer RJ, Black ER, Griner PF. Interpretation of diagnostic tests and
strategies for their use in quantitative decision making. pages 17-28. IN:
Panzer RJ, Black ER, et al. Diagnostic Strategies for Common Medical
Problems. American College of Physicians. 1991. Speicher C, Smith JW Jr..
Choosing Effective Laboratory Tests. WB Saunders. 1983. pages 50-51, and 210
Overall Accuracy, or Diagnostic Efficiency Overview: The overall accuracy of
a test is the measure of "true" findings (true-positive + true-negative
results) divided by all test results. This is also termed "the efficiency"
of the test. overall accuracy = ((a+d) / (a + b + c + d)) where: a + d =
true positives + true negatives = all people correctly classified by testing
(a + b + c + d) = sum of (true positives, false positives, false negatives,
true negatives) = total population TOP References Braunwald E, Isselbacher
KJ, et al (editors). Harrison's Principles of Internal Medicine, 11th
edition. McGraw-Hill Book Publishers. 1987. page 7 Goldman L. Chapter 10:
Quantitative aspects of clinical reasoning. pages 43-48. IN: Isselbacher KJ,
Braunwald E, et al. Harrison's Principles of Internal Medicine, Thirteenth
Edition. McGraw-Hill. 1994. Clave P, Guillaumes S, Blanco I, et al. Amylase,
Lipase, Pancreatic Isoamylase, and Phospholipase A in Diagnosis of Acute
Pancreatitis. Clin Chem. 1995; 41:1129-1134.
Speicher C, Smith JW Jr.. Choosing Effective Laboratory Tests.
WB Saunders. 1983. pages 50-51, and 210 Deriving Missing Performance
Measures When Only Some Are Known Overview: If some performance measures for
a test are known but others are not, it is often possible to calculate the
missing values from those that are known. Key for equations SE = sensitivity
SP = specificity PPV = positive predictive value NPV = negative predictive
value ACC = accuracy (1)sensitivity = (( 1 + (((NPV^(-1)) - 1) * (((SP^(-1))
- 1)^(-1)) * ((PPV^(-1)) - 1)))^(-1)) (2) specificity = (( 1 + (((PPV^(-1))
- 1) * (((SE^(-1)) - 1)^(-1)) * ((NPV^(-1)) - 1)))^(-1)) (3) positive
predictive value = (( 1 + (((SP^(-1)) - 1) * (((NPV^(-1)) - 1)^(-1)) *
((SE^(-1)) - 1)))^(-1)) (4) negative predictive value = (( 1 + (((SE^(-1)) -
1) * (((PPV^(-1)) - 1)^(-1)) * ((SP^(-1)) - 1)))^(-1)) (5) accuracy = ((1 +
(((((PPV ^ (-1)) - 1) ^ (-1) ) + (((SP ^ (-1)) - 1) ^ (-1)))^(-1)) + (((((SE
^ (-1)) - 1) ^ (-1) ) + (((NPV ^ (-1) ) - 1) ^ (-1)))^(-1))) ^ (-1)) (6)
positive predictive value = ((1 + (((SE ^ (-1)) - (ACC ^ (-1))) * (((((ACC ^
(-1)) - 1) * (((SP ^ (-1)) - 1) ^ (-1))) - 1) ^(-1)))) ^ (-1)) where: The
equation does not apply if SE = SP = ACC (7) sensitivity = ((1 + (((PPV ^
(-1)) - (ACC ^ (-1))) * (((((ACC ^ (-1)) - 1) * (((NPV ^ (-1)) - 1) ^ (-1)))
- 1) ^(-1)))) ^ (-1)) where: The equation does not apply if PPV = NPV = ACC
(8) specificity = (( 1 + ((((SE ^ (-1)) + (PPV ^ (-1)) - (ACC ^ (-1)) - 1) ^
(-1)) * ((ACC ^ (-1)) - 1) * ((PPV ^ (-1)) -1))) ^ (-1)) (9) positive
predictive value = (( 1 + ((((SP ^ (-1)) + (NPV ^ (-1)) - (ACC ^ (-1)) - 1)
^ (-1)) * ((ACC ^ (-1)) - 1) * ((SP ^ (-1)) -1))) ^ (-1)) (10) specificity =
((((((ACC ^ (-1)) - 1) * (((SE ^ (-1)) - 1) ^ (-1)) * ((NPV ^ (-1)) - 1)) +
(ACC ^ (-1)) - (NPV ^ (-1)) + 1)) ^ (-1)) (11) sensitivity = ((((((ACC ^
(-1)) - 1) * (((SP ^ (-1)) - 1) ^ (-1)) * ((PPV ^ (-1)) - 1)) + (ACC ^ (-1))
- (PPV ^ (-1)) + 1)) ^ (-1)) Parameter Known? Equations to SE SP PPV NPV ACC
Apply Y Y Y N N 4 (NPV) 5 (ACC) Y Y N Y N 3 (PPV) 5 (ACC) Y Y N N Y 6 (PPV)
4 (NPV) Y N Y Y N 2 (SP) 5 (ACC) Y N Y N Y 8 (SP) 4 (NPV) Y N N Y Y 10 (SP)
3 (PPV) N Y Y Y N 1 (SE) 5 (ACC) N Y Y N Y 11 (SE) 4 (NPV) N Y N Y Y 9 (PPV)
1 (SE) N N Y Y Y 7 (SE) 2 (SP) Implementation Notes Some of the equation
numbers differ from that in Einstein et al (1997). Equations with similar
structure are grouped together Substituting variables for some of the more
complex structures makes implementing the equations somewhat easier. TOP
References: Einstein AJ, Bodian CA, Gil J. The relationship among
performance measures in the selection of diagnostic tests. Arch Pathol Lab
Med. 1997; 121: 110-117. Youden's Index Overview: Youden's index is one way
to attempt summarizing test accuracy into a single numeric value. Youden's
index = 1 - ((false positive rate) + (false negative rate)) = 1 - ((1 -
(sensitivity)) + (1 - (specificity))) = (sensitivity) + (specificity) - 1 It
may also be expressed as: Youden's index = ( a / (a + b)) + (d / (c + d)) -
1 = ((a * d) - (b * c)) / ((a + b) * (c + d)) where: � a + b = people with
disease � c + d = people without disease � a = people with disease
identified by test (true positive) � b = people with disease not identified
by test (false negatives) � c = people without disease identified by test
(false positives) � d = people without disease not identified by test (true
negatives) Interpretation � minimum index: -1 � maximum index: +1 � A
perfect test would have a Youden index of +1. Limitation � The index by
itself would not identify problems in sensitivity or specificity. TOP
References: Hausen H. Caries prediction - state of the art. Community
Dentistry and Oral Epidemiology. 1997; 25: 87-96. Hilden J, Glasziou P.
Regret graphs, diagnostic uncertainty and Youden's index. Statistics in
Medicine. 1996; 15: 969-986. Youden WJ. Index for rating diagnostic tests.
Cancer. 1950; 3: 32-35. Bayes's Theorem and Modifications Bayes's Theorem
Overview: Bayes's theorem gives the probability of disease in a patient
being tested based on disease prevalence and test performance. post-test
probability disease present given a positive test result = = ((pretest
probability that disease present) * (probability test positive if disease
present)) / (((pretest probability that disease present) * (probability test
positive if disease present)) + ((pretest probability that disease absent) *
(probability test positive if disease absent))) post-test probability
disease present given a negative test result = = ((pretest probability that
disease present) * (probability test negative if disease present)) / (((pretest
probability that disease absent) * (probability disease absent when test
negative)) + ((pretest probability that disease present) * (probability test
negative if disease present))) Variable Alternative Statement pretest
probability that disease present prevalence probability test positive if
disease present sensitivity pretest probability that disease absent (1 -
(prevalence)) probability test positive if disease absent false positive
rate = (1 - (specificity)) probability test negative if disease present
false negative rate = (1 - (sensitivity)) probability disease absent when
test negative specificity Bayes's formula can also be expressed in the
positive and negative predictive values: post-test probability given a
positive result = = positive predictive value = = (true positives) / (all
positives) = = (true positives) / ((true positives) + (false positives))
post-test probability given a negative result = = negative predictive value
= = (false negatives) / (all negatives) = = (false negatives) / ((true
negatives) + (false negatives)) Limitations of Bayes's theorem � Bayes's
theorem assumes test independence, which may not occur if multiple tests are
used for diagnosis TOP References Einstein AJ, Bodian CA, Gil J. The
relationship among performance measures in the selection of diagnostic
tests. Arch Pathol Lab Med. 1997; 121: 110-117. Nicoll D, Detmer WM. Chapter
1: Basic principles and diagnostic test use and interpretation. pages 1 -
16. IN: Nicoll D, McPhee SJ, et al. Pocket Guide to Diagnostic Tests, Second
Edition. Appleton & Lange. 1997. Noe DA. Chapter 3: Diagnostic
Classification. pages 27-43. IN: Noe DA, Rock RC (Editors). Laboratory
Medicine. Williams and Wilkins. 1994. Schultz EK. Chapter 14: Analytical
goals and clinical interpretation of laboratory proceedures, pages 485-507.
IN: Burtis C, Ashwood E. Tietz Textbook of Clinical Chemistry, Second
edition. W.B. Saunders Company. 1994. Scott TE. Chapter 2: Decision making
in pediatric trauma. pages 20-40. IN: Ford EG, Andrassy RJ. Pediatric Trauma
- Initial Assessment and Management. W.B. Saunders. 1994 Suchman AL, Dolan
JG. Odds and likelihood ratios. pages 29-34. IN: Panzer RJ, Black ER, et al.
Diagnostic Strategies for Common Medical Problems. American College of
Physicians. 1991. Weissler AM. Chapter 11: Assessment and use of
cardovascular tests in clinical prediction. pages 400-421. IN: Giuliani ER,
Gersh BJ, et al. Mayo Clinic Practice of Cardiology, Third Edition. Mosby.
1996 Odds and Likelihood Ratios Overview: One form of Bayes's theorem is to
calculate the post-test odds for a disorder from the pre-test odds and
performance characteristics for the test. odds ratio = (probability of
disease) / (1 - (probability of disease)) likelihood ratio = (probability of
a test result in a person with the disease) / (probability of a test result
in a person without the disease) post-test odds = (pre-test odds) *
(likelihood ratio) where: � disease prevalence in the population can be used
as the pretest odds � likelihood ratios can be expressed in terms of the
sensitivity and specificity of the test for the diagnosis � positive
likelihood ratio is the likelihood ratio for a positive test result; it is
the true-positive rate divided by the false-positive rate, or (sensitivity)
/ (1 - (specificity)) � negative likelihood ratio is the likelihood ratio
for a negative test result; it is the false negative rate divided by the
true negative rate, or (1 - (sensitivity)) / (specificity) post-test odds
that the person has the disease if there is a positive test result =
(pre-test odds) * (positive likelihood ratio) post-test odds that the person
has the disease if there is a negative test result = (pre-test odds) *
(negative likelihood ratio) TOP Calculating Post-Test Odds Step 1: Calculate
the positive and negative likelihood ratios for the test � positive
likelihood ratio = = (sensitivity) / (1 - (specificity)) � negative
likelihood ratio = = (1 - (sensitivity)) / (specificity) Step 2: Convert the
prior probability to prior odds: ((prior probability) * 10) : ((1 - (prior
probability)) * 10) Step 3: Multiply the prior odds by the likelihood ratios
to obtain the post-test odds � ((positive likelihood ratio) * (prior
probability) * 10) : ((1 - (prior probability)) * 10) � ((negative
likelihood ratio) * (prior probability) * 10) : ((1 - (prior probability)) *
10) Step 4: Convert the post-test odds to post-test probabilities � positive
post-test probability = = ((positive likelihood ratio) * (prior probability)
* 10) / (((positive likelihood ratio) * (prior probability) * 10) + ((1 -
(prior probability)) * 10)) � negative post-test probability = = ((negative
likelihood ratio) * (prior probability) * 10) / (((negative likelihood
ratio) * (prior probability) * 10) + ((1 - (prior probability)) * 10)) TOP
References: Einstein AJ, Bodian CA, Gil J. The relationship among
performance measures in the selection of diagnostic tests. Arch Pathol Lab
Med. 1997; 121: 110-117. Noe DA. Chapter 3: Diagnostic Classification. pages
27-43. IN: Noe DA, Rock RC (Editors). Laboratory Medicine. Williams and
Wilkins. 1994. Scott TE. Chapter 2: Decision making in pediatric trauma.
pages 20-40. IN: Ford EG, Andrassy RJ. Pediatric Trauma - Initial Assessment
and Management. W.B. Saunders. 1994 Suchman AL, Dolan JG. Odds and
likelihood ratios. pages 29-34. IN: Panzer RJ, Black ER, et al. Diagnostic
Strategies for Common Medical Problems. American College of Physicians.
1991. Weissler AM. Chapter 11: Assessment and use of cardovascular tests in
clinical prediction. pages 400-421. IN: Giuliani ER, Gersh BJ, et al. Mayo
Clinic Practice of Cardiology, Third Edition. Mosby. 1996 Odds and
Likelihood Ratios for Sequential Testing Overview: If more than one test or
finding is used for diagnosis, the final post-test probability can be
calculated by combining the likelihood ratio for each test. post-test odds =
(pre-test odds) * (likelihood ratio for test 1) * (likelihood ratio for test
2) * .... * (likelihood ratio for test n) Limitation � For valid results,
tests must be conditionally independent of each other, where conditionally
independent indicates that the results of the tests are not associated with
each other. � If conditionally dependent tests are used, then the calculated
post-test probability will be over-estimated. TOP References: Nicoll D,
Detmer WM. Chapter 1: Basic principles and diagnostic test use and
interpretation. pages 1 - 16. IN: Nicoll D, McPhee SJ, et al. Pocket Guide
to Diagnostic Tests, Second Edition. Appleton & Lange. 1997. Schultz EK.
Chapter 14: Analytical goals and clinical interpretation of laboratory
proceedures, pages 485-507. IN: Burtis C, Ashwood E. Tietz Textbook of
Clinical Chemistry, Second edition. W.B. Saunders Company. 1994. Suchman AL,
Dolan JG. Odds and likelihood ratios. pages 29-34. IN: Panzer RJ, Black ER,
et al. Diagnostic Strategies for Common Medical Problems. American College
of Physicians. 1991. Weissler AM. Chapter 11: Assessment and use of
cardovascular tests in clinical prediction. pages 400-421. IN: Giuliani ER,
Gersh BJ, et al. Mayo Clinic Practice of Cardiology, Third Edition. Mosby.
1996 Risk Sensitivity and Risk Specificity Overview: Risk sensitivity and
specificity can be used to evaluate how good a risk factor is for predicting
mortality in the population. � Risk sensitivity is the proportion of people
who die during the follow-up period who were identified as high risk. � Risk
specificity is the proportion of people who survive during the follow-up
period who were identified as low risk. Patient subgroups � high risk
fraction = those with risk factor � low risk fraction = those without risk
factor risk sensitivity in percent = (mortality for high risk subgroup in
percent) * (percent of population identified as high risk) / (cumulative
mortality in percent for the whole population) risk specificity in percent =
(survival for low risk subgroup in percent) * (percent of population
identified as low risk) / (cumulative survival in percent for the whole
population) percent of population in high risk group = 100 - (percent of
population in low risk group) cumulative survival of high risk group = 100 -
(cumulative mortality of high risk group) cumulative survival of low risk
group = 100 - (cumulative mortality of low risk group) cumulative survival
of population = 100 - (cumulative mortality of population) TOP References:
Weissler AM. Chapter 11: Assessment and use of cardovascular tests in
clinical prediction. pages 400-421. IN: Giuliani ER, Gersh BJ, et al. Mayo
Clinic Practice of Cardiology, Third Edition. Mosby. 1996 Statistics for the
Normal Distribution and Use in Quality Control Mean of Values in a Normal
Distribution Overview: If data follows a normal Gaussian distribution, then
the mean of the data can be calculated. mean of values = (sum of all values)
/ (number of values) TOP References: Woo J, Henry JB. Chapter 6: Quality
management. pages 125-136 (128). IN: Henry JB (editor-in-chief). Clinical
Diagnosis and Management by Laboratory Methods, 19th edition. WB
Saunders.1996. Standard Deviation (SD) Overview: The standard deviation is a
measure of the dispersion of data about the mean. standard deviation =
absolute value [square root of the variance] where � variance = ((sum of
((each value) - (mean of values))^2) / ((number of values) - 1)) TOP
References: Barnett RN. Clinical Laboratory Statistics, Second Edition.
Little, Brown and Company. 1979. page 4 Woo J, Henry JB. Chapter 6: Quality
management. pages 125-136 (128). IN: Henry JB (editor-in-chief). Clinical
Diagnosis and Management by Laboratory Methods, 19th edition. WB
Saunders.1996. Coefficient of Variation (CV) Overview: The coefficient of
variation for a test (CV) gives a true picture of deviation regardless of
the nature of the measurement or the methodology. CV (expressed as a
percent) = ((standard deviation) * 100 / (mean)) TOP References: Dharan,
Murali. Total Quality Control in the Clinical Laboratory. C.V. Mosby Co.
1977. page 22 Standard Deviation Interval (SDI) Overview: The Standard
Deviation Interval gives information about how a given laboratory's mean
differs from the mean of a group of comparable laboratories, taking into
account the variation among the laboratories. This is also called the
Standard Deviation Index. This is a measure of accuracy. SDI = (((mean) -
(average of all means)) / (standard deviation of all means)) where: �
average of all means = ((sum of all means) / (number of means))
Interpretation � Values > +2.0 or < -2.0 need to be investigated. TOP
References: College of American Pathologists QAS Program Coefficient of
Variation Interval (CVI) Overview: The CVI is a measure of precision., but
it is difficult to get a good definition of. It also may be called the
Coefficient of Variation Index, or the CVR. CVI = (CV for laboratory) /
(pool CV) or (CV for laboratory for time period) / (peer group CV for time
period) Interpretation: � Values > +2.0 or < -2.0 need to be investigated.
TOP References: The Interlaboratory Quality Assurance Program. Coulter
Diagnostics. 1988. Total Allowable Error Overview: Analysis of the total
allowable error (TEa) can help a laboratory meet its goals for precision
performance. Variables laboratory mean = mean at laboratory for period of
stability in reagents & controls "true" mean = mean for all methods &
laboratories laboratory standard deviation = standard deviation noted at
laboratory method standard deviation = standard deviation reported by vendor
CLIA limit � given as a range, either using a percent or an absolute value �
if both specified, use whichever is greater Calculations calculated bias =
laboratory's deviation (based on site and method) from mean of all sites =
((laboratory mean) - (true mean)) laboratory imprecision = (factor) *
(laboratory standard deviation) where � factor is 1.96 for 95% � factor is
2.50 for 99% Total allowable error = TEa = ((CLIA limit) * (true mean)) bias
as percent of CLIA limit = ((calculated bias) / ((CLIA limit) * (true
mean))) = (((laboratory mean) - (true mean)) / (total allowable error)) = (
((laboratory mean) - (true mean)) / ((CLIA limit) * (true mean))) total
error = calculated bias + imprecision = (((laboratory mean) - (true mean)) +
(laboratory imprecision)) ( ((laboratory mean) - (true mean)) + ((factor) *
(laboratory standard deviation))) assessment of performance = (total error)
/ (total allowable error) * 100 = (((laboratory mean) - (true mean))+
(laboratory imprecision)) / (((CLIA limit) * (true mean)))* 100 =
(((laboratory mean) - (true mean))+ (1.96 * (laboratory standard
deviation))) / (((CLIA limit) * (true mean))) * 100 systemic error
(critical) = SEc = ( ( ( (total allowable error) - (calculated bias) ) /
(laboratory standard deviation) ) - 1.65) = ( ( ( ((CLIA limit) * (true
mean)) - ((laboratory mean) - (true mean))) / (laboratory standard
deviation)) - 1.65) Use the systemic error for selection of QC control rules
to use � standard deviation to use = ((calculated standard deviation) *
((denominator of primary rule) / 2)) Example: If using 1:3s rule, where the
denominator = 3 standard deviation to use = (laboratory standard deviation)
* (3 / 2) = 1.5 * (laboratory standard deviation) TOP References Blanchard
J-M, O'Grady M. Applicationof the Westgard quality control selection grids (QCSG)
to the Kodak Ektachem 700 analyzer. Abstract presented athte 43rd AACC
National Meeting, Washington, DC. July 30-August 1, 1991. Westgard JO, Bawa
N, et al. Laboratory precision performance. Arch Pathol Lab Med. 1996; 120:
621-625. Westgard JO. Error budgets for quality management: Practical tools
for planning and assuring the analytical quality of laboratory testing
processes. Clinical Laboratory Management Review. July/August, 1996. pages
377-403. Westgard JO. Chapter 150: Planning statistical quality control
procedures. pages 1191-1200. IN: Rose NR, de Macario EC, et al (editors).
Manual of Clinical Laboratory Immunology, Fifth Edition. ASM Press. 1997.
Chi Square Outcome Comparison of Two Groups Overview: When 2 different
groups receive different treatment, the number of each group improved and
not improved can be compared as follows: group A improved group B improved
total improved group A not improved group B not improved total not improved
total group A total group B total patients This data shows 1 degree of
freedom. chi square value using Yates correction for 1 degree of freedom =
((total number) * ((ABS(((number of group A improved) * (number group B not
improved)) - ((number of group A not improved) * (number of group B
improved))) - ((total number) / 2))^2)) / (((number of group A improved) +
(number of group B improved)) * ((number of group A not improved) + (number
of group B not improved)) * (total number of group A) * (total number of
group B)) From the chi-square value, it is the probability that a difference
is due to chance can be calculated. The Excel function CHIDIST will give the
probability of the difference being due to chance for the chi-square value.
The probability that the difference is not due to chance is then (1 -
(probability due to chance)). TOP References: Barnett RN. Clinical
Laboratory Statistics, 2nd edition. Little, Brown and Company. 1979. pages
26-29 Beyer WH. CRC Standard Mathematical Tables, 25th edition. CRC Press.
1978. page 537 Keeping ES. Introduction to Statistical Inference. Dover
Publications. 1995 printing of 1962 work. pages 314-322 Comparison of Two
Observers Overview: When 2 observers tally data from the same material, it
is useful to see whether the differences in their tabulations is due to
chance or due to observer variation. For more than 2 observations: chi
square = (summation from 1 to number of observations ( (((observer A value)
- (observer B value)) ^ 2) / ((observer A value) + (observer B value)) ) )
From this value, the probability that the differences between the 2
observers is due to chance can be calculated. The equation can be simplified
from an integral depending on whether there is an even or odd degree of
freedom. Even Degrees of Freedom For even degrees of freedom, this is
relatively simple. probability due to chance (chisquare, degree of freedom)
= ((e) ^ ((-1) * (chisquare) / 2)) * (summation of i from 0 to I ( (((chisquare)
/ 2) ^ (i)) / (factorial (i))) where: � I = (1/2 * ((degree of freedom) -
2)) 2 degrees of freedom probability = e^((-1) * (chisquare) / 2) 4 degrees
of freedom probability = (e^((-1) * (chisquare) / 2)) * (1 + ((chisquare) /
2) ) 6 degrees of freedom probability = (e^((-1) * (chisquare) / 2)) * (1 +
((chisquare) / 2))+ (((chisquare) ^2) / 8)) 8 degrees of freedom probability
= (e^((-1) * (chisquare) /2 )) * (1 + ((chisquare) / 2))+ (((chisquare) ^2)
/ 8) + + (((chisquare) ^3) / 48)) 10 degrees of freedom probability =
(e^((-1) * (chisquare) / 2)) * (1 + ((chisquare) / 2))+ (((chisquare) ^2) /
8)+ (((chisquare) ^3) / 48) + (((chisquare) ^4) / 384)) Odd Degrees of
Freedom For odd degrees of freedom, this is quite complex, and it is easier
to use the Excel function CHIDIST. probability due to chance (chisquare,
degree of freedom) = 1 - ( (1 / (gamma function (I + 1))) * (summation from
0 to infinity (((-1)^i) * (((chisquare) / 2) ^ (I + i + 1)) / ((factorial (i))
* (I + i + 1))) where � I = (1/2 * ((degree of freedom) - 2)) TOP
References: Barnett RN. Clinical Laboratory Statistics, 2nd edition. Little,
Brown and Company. 1979. pages 26-29 Beyer WH. CRC Standard Mathematical
Tables, 25th edition. CRC Press. 1978. page 537 Keeping ES. Introduction to
Statistical Inference. Dover Publications. 1995 printing of 1962 work. pages
314-322 Test Comparison Using Receiver Operating Characteristics (ROC) Plots
Overview: The Receiver Operating Curve (ROC) originated during World War II
with the use of radar in signal detection. This was extended to the use of
diagnostic tests for identifying disease states, using plots of sensitivity
versus specificity for different test results. The area under a ROC curve
serves as a measure of the diagnostic accuracy (discrimination performance)
for a test. Receiver Operating Curve To generate a receiver operating curve
it is first necessary to determine the sensitivity and specificity for each
test result in the diagnosis of the disorder in question. The x axis ranges
from 0 to 1, or 0% to 100%, and can be either the � false positive rate (1 -
(specificity)), or � true negative rate (specificity) The false positive
rate is the one typically used. The y axis ranges from 0 to 1, or 0% to 100%
� true positive rate (sensitivity), with range 0 to 1 (or 0 to 100%) When
the x-axis is the false positive rate (1 - (specificity)), the curve starts
at (0,0) and increases towards (1,1). When the x-axis is the true negative
rate (specificity), the curve starts at (0, 1) and drops towards (1, 0). The
endpoints for the curve will run to these points. Area under Curve One way
of measuring the area under a curve is by measuring subcomponent trapezoids.
Data points can be connected by straight lines defined by: y = ((slope) * x)
+ intercept The area under each line can be determined by integration of (y
* dx) over the interval of x1 to x2: area = (((slope) / 2) * ((x2 ^ 2) - (x1
^2))) + ((intercept) * (x2 - x1)) By summating the areas under each segment,
an approximation of the area under the entire curve can be reached. However,
the trapezoidal method tends to underestimate areas (Hanley, 1983), so that
other techniques for measuring area should be used if greater accuracy is
required. The maximum area under ROC curve is 1 and is seen with the ideal
test. The closer the area under the ROC curve is to 1, the better (more
accurate) the test. Comparison of Two Methods Two methods can be compared by
the area under their respective ROC curves. The method with the larger area
under the ROC curve is preferable over one with a smaller area, allowing for
variability, as being more accurate. TOP References: Bamber D. The area
above the ordinal dominance graph and the area below the receiver operating
characteristic graph. J Math Psych. 1975; 12: 387-415. Beck JR, Shultz EK.
The use of relative operating characteristic (ROC) curves in test
performance evaluation. Arch Pathol Lab Med. 1986; 110: 13-20. Dorfman DD.
Maximum-likelihood estimation of parameters of signal-detection theory and
determination of confidence intervals - rating method data. J Math Psychol.
1969; 6: 487-496. Hanley JA, McNeil BJ. The meaning and use of the area
under a receiver operating characteristic (ROC) curve. Radiology. 1982; 143:
29-36. Hanley JA, McNeil BJ. A method of comparing the areas under receiver
operating characteristics curves derived from the same cases. Radiology.
1983; 148: 839-843. Henderson AR. Assessing test accuracy and its clinical
consequences: a primer for receiver operating characteristic curve analysis.
Ann Clin Biochem. 1993; 30: 521-539. Henderson AR, Bhayana V. A modest
proposal for the consistent presentation of ROC plots in Clinical Chemistry
(Letter to the Editor). Clin Chem. 1995; 41: 1205-1206. Lett RR, Hanley JA,
Smith JS. The comparison of injury severity instrument performance using
likelihood ratio and ROC curve analysis. J Trauma. 1995; 38: 142-148. Pellar
TG, Leung FY, Henderson AR. A computer program for rapid generation of
receiver operating characteristic curves and likelihood ratios in the
evaluation of diagnostic tests. Ann Clin Biochem. 1988; 25: 411-416.
Pritchard ML, Woosley JT. Comparison of two prognostic models predicting
survival in patients with malignant melanoma. Hum Pathol. 1995; 26:
1028-1031. Raab SS, Thomas PA, et al. Pathology and probability: LIkelihood
ratios and receiver operating characteristic curves in the interpretation of
bronchial brush specimens. Am J Clin Pathol. 1995; 103: 588-593. Schoonjans
F, Depuydt C, Comhaire F. Presentation of receiver-operating characteristic
(ROC) plots (Letter to the Editor). Clin Chem. 1996; 42: 986-987. Shultz EK.
Multivariate receiver-operating characteristic curve analysis: Prostate
cancer screening as an example. Clin Chem. 1995; 41: 1248-1255. Vida S. A
computer program for non-parametric receiver operating characteristic
analysis. Comput Meth Prog Biomed. 1993; 40: 95-101. Zweig MH. Evaluation of
the clinical accuracy of laboratory tests. Arch Pathol Lab Med. 1988; 112:
383-386. Zweig MH, Campbell G. Receiver-operating characteristic (ROC)
plots: A fundamental evaluation tool in clinical medicine. Clin Chem. 1993;
39: 561-577. Zweig MH, Ashwood ER, et al. Assessment of the clinical
accuracy of laboratory tests using receiver operating characteristics (ROC)
plots: Approved guideline. NCCLS. 1995; 15 (19). Z-score Overview: The
Z-score can be used to put a patient result in perspective with reference
values from a control population. It basically gives the number of standard
deviations that a given value is from the reference population mean. Z-score
= ((patient value) - (mean for reference population)) / (standard deviation
for reference population) This appears to be share features with the
Standard Deviation Interval (SDI). TOP References Withold W, Schulte U,
Reinauer H. Methods for determination of bone alkaline phosphatase activity:
analytical performance and clinical usefulness in patients with metabolic
and malignant bone diseases. Clin Chem. 1996; 42: 210-217. Westgard Rules
and the Multirule Shewhart Procedure Westgard Control Rules Overview:
Westgard et al have proposed a series of multiple rules (multirules) for
interpreting quality control data. The rules are sensitive to random and
systemic errors, and they are selected to keep the probability of false
rejection low. Procedure 1. Starting with a stable testing system and stable
control material, a control material is analyzed for at least 20 different
days. This data is used to calculate a mean and standard deviation for the
control material. 2. Usually 2 control materials are analyzed (one with a
low value, one with a higher value in the analytical range). Sometimes 3 or
more control materials may be used, and rarely only 1. 3. The controls are
included with each analytical run of the test system. 4. A Levey-Jennings
control chart is prepared to graphically represent the data for each control
relative to the mean and multiples of the standard deviation. 5. With each
analytical run, the pattern of the current and previous control results are
analyzed using all of the selected Westgard control rules. 6. If none of the
rules fail, then the run is accepted. If one or more rules fail, then
different responses may occur. This may include rejecting the run, adjusting
the stated mean, and/or recalibrating the test. Westgard Control Rule
Definition 1:2S control result is outside +/- 2 standard deviations of the
mean 1:3S control result is outside +/- 3 standard deviations of the mean
2:2S 2 consecutive control results are more than 2 standard deviations from
the mean R:4S either (a) one control is more than 2 SD above mean and other
is more than 2 SD below the mean; or (b) the range between 2 controls
exceeds 4 SD 4:1S the last 4 consecutive control results are all either 1 SD
above or below the mean 10:X the last 10 consecutive control results all lie
on the same side of the mean Rule Failure Systemic Error Random Error 1:3S
yes yes 2:2S yes R:4S yes 4:1S yes 10:X yes TOP References: Lott JA. Chapter
18: Process control and method evaluation. pages 293-325 (Figure 18-4, page
302). IN: Snyder JR, Wilkinson DS. Management in Laboratory Medicine, Third
Edition. Lippincott. 1998. Westgard JO. Chapter 150: Planning statistical
quality control procedures. pages 1191-1200. IN: Rose NR, de Macario EC, et
al (editors). Manual of Clinical Laboratory Immunology, Fifth Edition. ASM
Press. 1997. Westgard JO, Klee GG. Chapter 17: Quality management. pages
384-418. IN: Burtis CA, Ashwood ER. Tietz Textbook of Clinical Chemistry,
Third Edition. WB Saunders Company. 1999 (1998). Using a Series of Control
Rules in the Multirule Shewhart Procedure Overview: Westgard et al have
developed a series of rules for evaluating controls which can be used to
judge if the data from an analysis is acceptable. The results of the rule
analysis can be employed sequentially in a multirule Shewhart procedure to
determine whether to accept or reject an analytic run. Westgard Rule Failed?
Yes No 1:2S go to next rule in control, accept run 1:3S out of control,
reject run go to next rule 2:2S out of control, reject run go to next rule
R:4S out of control, reject run go to next rule 4:1S out of control, reject
run go to next rule 10:X out of control, reject run in control, accept run
TOP References: Lott JA. Chapter 18: Process control and method evaluation.
pages 293-325 (Figure 18-4, page 302). IN: Snyder JR, Wilkinson DS.
Management in Laboratory Medicine, Third Edition. Lippincott. 1998. Westgard
JO, Barry PL, Hunt MR. A multi-rule Shewhart chart for quality control in
clinical chemistry. Clin Chem. 1981; 27: 493-501. Evaluating Reports in the
Medical Literature Criteria for Assessing the Methodologic Quality of
Clinical Studies Overview: The methodologic quality of a clinical study or
trial can be evaluated by examining its design and implementation. A score
based on the key parameters can be used to evaluate the study and to compare
it with other similar studies. Parameters (1) randomization (2) blinding (3)
analysis (4) patient selection (5) comparability of groups at baseline (6)
extent of follow-up (7) description of treatment protocol (8)
cointerventions (9) description of outcomes Parameter Finding Points
randomization not concealed or not sure 1 concealed randomization 2 blinding
not blinded 0 adjudicators blinded 2 analysis other 0 intention to treat 2
patient selection selected patients or unable to tell 0 consecutive eligible
patients 1 comparability of groups at baseline no or not sure 0 yes 1 extent
of follow-up < 100% 0 100% 1 treatment protocol poorly described 0
reproducibly described 1 cointerventions (extent to which interventions
applied equally across groups) not described 0 described but not equal or
not sure 1 well described and all equal 2 outcomes not described 0 partially
described 1 objectively defined 2 Interpretation � minimum score: 0 �
maximum score: 14 � The higher the score, the higher the quality in the
study design and implementation. TOP References: Heyland DK, Cook D, et al.
Maximizing oxygen delivery in critically ill patients: a methodologic
appraisal of the evidence. Crit Care Med. 1996; 24: 517-524. Heyland DK,
MacDonald S, et al. Total parenteral nutrition in the critically ill
patient. JAMA. 1998; 280: 2013-2019. Measures of the Consequences of
Treatment Number Needed to Treat Overview: The number needed to treat is a
simple method of looking at the benefit of a treatment intervention to
prevent a condition or complication. It is the inverse of the absolute risk
reduction for the treated versus untreated control populations. It can be
used to extrapolate findings in the literature to a given patient at an
arbitrary specified baseline risk when the relative risk reduction
associated with treatment is constant for all levels of risk. Variables: �
number of people in control group � number of people in control group who
develop condition of interest during time interval � number of people in
active treatment group � number of people in active treatment group who
develop condition of interest during time interval event rate in control
group = (number of people in control group with condition) / (number of
people in control group) event rate in active treatment group = (number of
people in active treatment group with condition) / (number of people in
active treatment group) relative risk reduction = ((event rate in control
group) - (event rate in active treatment group)) / (event rate in control
group) absolute risk reduction = (event rate in control group) - (event rate
in active treatment group) number needed to treat = 1 / (absolute risk
reduction) = 1 / ((event rate in control group) - (event rate in active
treatment group)) Interpretation � The number needed to treat indicates the
number of patients who need to be treated to prevent the condition of
interest during the time interval. � The smaller the number needed to treat,
the greater the benefit of the treatment to prevent the condition. � The
number needed to treat should be considered together with other factors such
as the seriousness of the condition to be prevented and the risk of adverse
side effects from the treatment. TOP References: Altman DG. Confidence
intervals for the number needed to treat. BMJ. 1998; 317: 13091312. Cook RJ,
Sackett DL. The number needed to treat: a clinically useful measure of
treatment effect. BMJ. 1995; 310: 452-454. Laupacis A, Sackett DL, Roberts
RS. An assessment of clinically useful measures of the consequences of
treatment. N Engl J Med. 1988; 318: 1728-1733. The Corrected Risk Ratio and
Estimating Relative Risk Overview: The corrected risk ratio can be used to
derive an estimate of an association or treatment effect that better
represents the true relative risk. Odds ratio and relative risk (see Figure
on page 1690 of Zhang and Yu) � If the incidence of an outcome in the study
population is < 10%, then the odds ratio is close to the risk ratio. � As
the incidence of the outcome increases, the odds ratio overestimates the
relative risk if it is more than 1, or underestimates the relative risk is
less than 1. Situations when desirable to perform correction � if the
incidence of the outcome in the nonexposed population is more than 10%, AND
� if the odds ratio is > 2.5 or < 0.5 incidence of outcome in nonexposed
group = N = (number with outcome in nonexposed group) / (number in
nonexposed group) incidence of outcome in exposed group = E = (number with
outcome in exposed group) / (number in exposed group) risk ratio = E / N
odds ratio = (E / (1 - E)) / (N / (1 - N)) E / N = (odds ratio) / [(1 - N) +
(N * (odds ratio))] corrected risk ratio = (odds ratio) / [(1 - N) + (N *
(odds ratio))] This equation can be used to correct the adjusted odds ratio
obtained from logistic regression. TOP References Wacholder S. Binomail
regression in GLIM: Estimating risk ratios and risk differences. Am J
Epidemiol. 1986; 123: 174-184. Zhang J, Yu KF. What's the relative risk? A
method for correcting the odds ratio in cohort studies of common outcomes.
JAMA. 1998; 280: 1690-1691. Confidence Intervals Confidence Interval for a
Single Mean Overview: The confidence interval for a series of findings can
be calculated from the number of values, the mean, standard deviation and
standard statistical tables. Data assumptions: single mean, symmetrical
distribution confidence interval = (mean) +/- ((one-tailed value of
Student's t distribution) * (standard deviation) / ((number of values) ^
(0.5)) where: � for a 95% confidence interval, the one-tailed value is for
2.5% (F 0.975, t 0.025) � degrees of freedom = (number of values) - 1 � as
the number of values increases, the closer the one-tailed value for t=0.025
approaches 1.96; at 120 degrees of freedom it is 1.98 TOP References: Beyer
WH. CRC Standard Mathematical Tables, 25th Edition. CRC Press. 1978.
Section: Probability and Statistics. Percentage points, Student's
t-distribution. page 536. Young KD. Lewis RJ. What is confidence? Part 2:
Detailed definition and determination of confidence intervals. Ann Emerg
Med. 1997; 30: 311-318. Confidence Interval for the Difference Between Two
Means Overview: The confidence interval for the observed difference in the
means for two sets of data can be calculated from standard statistical
tables and data characteristics (number of values, mean, standard deviation)
for the two data sets. Data assumptions: 2 sets of data with symmetrical
distribution confidence interval for the difference in the means between 2
sets of data = ABS((mean first group) - (mean second group)) +/- (factor)
factor = (one-sided value of Student's t-distribution) * (pooled standard
deviation) * (((1 / (number in first set)) + (1 / (number in second set))) ^
(0.5)) degrees of freedom = (number in first set) + (number in second set) -
2 pooled standard deviation = ((A + B) / (degrees of freedom)) ^ (0.5) A =
((number in first set) - 1) * ((standard deviation of first set) ^ 2) B =
((number in second set) - 1) * ((standard deviation of second set) ^ 2)
where: � for a 95% confidence interval, the one-tailed value is for 2.5% (F
0.975, t 0.025) � as the number of values increases, the closer the
one-tailed value for t=0.025 approaches 1.96; at 120 degrees of freedom it
is 1.98 TOP References: Beyer WH. CRC Standard Mathematical Tables, 25th
Edition. CRC Press. 1978. Section: Probability and Statistics. Percentage
points, Student's t-distribution. page 536. Young KD. Lewis RJ. What is
confidence? Part 2: Detailed definition and determination of confidence
intervals. Ann Emerg Med. 1997; 30: 311-318. Confidence Interval for a
Single Proportion Overview: When a certain event occurs several times in a
series of observations, then its proportion and confidence interval can be
calculated. Variables � N observations � X events of interest Distribution
used � F distribution, with F = 0.975 for the 95% confidence interval � uses
m and n as degrees of freedom proportion of events = X / N lower limit for
the 95% confidence interval = X / (X + ((N - X + 1) * (F distribution for m
and n))) where � m = 2 * (N - X + 1) � n = 2 * X upper limit for the 95%
confidence interval = ((X + 1) * (F distribution for m and n)) / (N - X +
((X + 1) * (F distribution for m and n))) where � m = 2 * (X + 1) = n + 2 �
n = 2 * (N - X) = m - 2 TOP References Beyer WH. CRC Standard Mathematical
Tables, 25th Edition. CRC Press. 1978. Section: Probability and Statistics.
F-distribution. page 540. Young KD. Lewis RJ. What is confidence? Part 2:
Detailed definition and determination of confidence intervals. Ann Emerg
Med. 1997; 30: 311-318. Confidence Interval When the Proportion in N
Observations is 0 or 1 Overview: If either 0 or n events occur in n
observations, then the limits of the confidence interval can be calculated
based on the confidence interval and the number of observations. X = 1 -
((confidence interval in percent) / 100) If 0 events occur in n observations
� lower limit for the confidence interval: 0 � upper limit for the
confidence interval: 1 - ((X/2) ^ (1/n)) If n events occur in n observations
� lower limit for the confidence interval: ((X/2)) ^ (1/n)) � upper limit
for the confidence interval: 1 (100%) TOP References Young KD. Lewis RJ.
What is confidence? Part 2: Detailed definition and determination of
confidence intervals. Ann Emerg Med. 1997; 30: 311-318. Confidence Interval
for the Difference Between Two Proportions Based on the Odds Ratio Overview:
When comparing two populations for an event, the odds ratio and 95%
confidence intervals can be calculated from looking at the number in each
group positive and negative for the event. Group 1 Group 2 Negative A B
Positive C D (Table page 316, Young 1997) odds for the event in group 1 = C
/ A odds for the event in group 2 D / B odds ratio for group 2 relative to
group 1 = (odds group 2) / (odds group 1) = (A * D) / (B * C) confidence
interval for 95% = EXP( X +/- Y) X = LN ((A * D) / (B * C)) Y = 1.96 * SQRT((1/A)
+ (1/B) + (1/C) + (1/D)) where: � 1.96 is the value for Z from the standard
normal distribution with F(Z) = 0.975 If the odds ratio is 1.0, then there
is no difference between the two groups. If the 2 groups are comparing an
intervention, then this is equivalent to a null hypothesis of no
intervention difference. Small Sample Sizes If sample sizes are small (less
than 10 or 20), then 0.5 is added to each of the factors. odds ratio = (odds
group 2) / (odds group 1) = ((A+0.5) * (D+0.5)) / ((B+0.5) * (C+0.5))
confidence interval for 95% = EXP( X +/- Y) X = LN (((A+0.5) * (D+0.5)) /
((B+0.5) * (C+0.5))) Y = 1.96 * SQRT((1/ (A+0.5)) + (1/ (B+0.5)) + (1/
(C+0.5)) + (1/ (D+0.5))) NOTE: I am using sample size as (A + B + C + D).
TOP References Beyer WH. CRC Standard Mathematical Tables, 25th Edition. CRC
Press. 1978. Section: Probability and Statistics. F-distribution. page 524.
Young KD. Lewis RJ. What is confidence? Part 2: Detailed definition and
determination of confidence intervals. Ann Emerg Med. 1997; 30: 311-318.
Confidence Interval for the Difference Between Two Proportions Using the
Normal Approximation Overview: If the events for two proportions are
normally distributed, then the confidence interval for the difference
between the two proportions can be calculated using the normal
approximation. Requirements (1) events occur with a normal distributions (2)
populations and events are sufficiently large (3) the proportions for the 2
populations are not too close to 0 or 1 Population 1 Population 2 total
number N1 N2 number showing response R1 R2 proportion responding in
population 1 = P1 = (R1) / (N1) proportion responding in population 2 = P2 =
(R2) / (N2) confidence interval = P1 - P2 +/- ((one tailed value of the
standard normal distribution) * (SQRT (((P1 * (1 - P1)) / N1) + ((P2 * (1 -
P2)) / N2))) where: � The one tailed values for standard normal
distributions with two-tailed confidence intervals, assuming an infinite
degree of freedom: Confidence Intervals one-tailed value 80% 1.282 90% 1.645
95% 1.960 98% 2.326 99% 2.576 99.8% 3.090 Interpretation � If the confidence
interval includes 0, then the data shows no statistically significant
difference between the 2 proportions. TOP References Beyer WH. CRC Standard
Mathematical Tables, 25th Edition. CRC Press. 1978. page 524. Young KD.
Lewis RJ. What is confidence? Part 2: Detailed definition and determination
of confidence intervals. Ann Emerg Med. 1997; 30: 311-318. Odds and
Percentages Overview: The rate of occurrence for a condition can be
expressed as the odds or the percentage of a population involved. total
population = (number affected) + (number unaffected) odds denominator =
(total population) / (number affected) = 1 + ((number unaffected) / (number
affected)) odds = 1 in (odds denominator) (number affected) to (number
unaffected) percent affected = (number affected) / (total population) * 100%
= 1 / (odds denominator) TOP References Harper PS. Practical Genetic
Counselling, Fifth Edition. Butterworth Heinemann. 1999. Table 1.1, page 10.
Benefit, Risk and Threshold for an Action Benefit-to-Risk Ratio and
Treatment Threshold for Using a Treatment Strategy Overview: Each treatment
strategy has potential risks and benefits. The treatment threshold uses a
treatment's benefit and risk when used for a given condition to help decide
if and when to treat. benefit for treatment = (risk of adverse outcome from
the disease in those untreated) - (risk of adverse outcome from the disease
with treatment) risk of treatment = (risk of significant adverse
complication due to treatment) benefit-to-risk ratio = (benefit for
treatment) / (risk for treatment) treatment threshold = 1 /
((benefit-to-risk ratio) +1 ) = (risk) / ((benefit) + (risk)) Interpretation
� Treatment should be given when the risk of having the condition exceeds
the treatment threshold. � Treatment should be withheld if the risk of
having the condition is less than the treatment threshold. TOP References
Beers MH, Berkow R, et al (editors). The Merck Manual of Diagnosis and
Therapy, Seventeenth Edition. Merck Research Laboratories. 1999. Chapter
295. Clinical Decision Making. page 2523. Testing and Test Treatment
Thresholds Overview: If a test is performed to determine whether a treatment
strategy is used, then the testing and test treatment thresholds can help
decide if the test should be done. Test features � performance
characteristics (sensitivity and specificity) for the condition are known �
assume that the test has no direct adverse risk to the patient benefit for
treatment = (risk of adverse outcome from the disease in those untreated) -
(risk of adverse outcome from the disease with treatment) risk of treatment
= (risk of significant adverse complication due to treatment) testing
threshold = ((1 - (specificity of test)) * (risk of treatment)) / (((1 -
(specificity of test)) * (risk of treatment)) + ((sensitivity of test) *
(benefit of test))) test treatment threshold = ((specificity of test) *
(risk of treatment)) / (((specificity of test) * (risk of treatment)) + ((1
- (sensitivity of test)) * (benefit of test))) Interpretation � If the
probability of disease is equal or more than the testing threshold and equal
or less than the test treatment threshold, then the test should be done. �
If the probability TOP |