likelihood ratiosclinepi.cmcvellore.org/.../10/likelihood-ratios-dr... · •sensitivity (sn) and...
TRANSCRIPT
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Likelihood RatiosPrathap Tharyan MD, MRCPsych
Adjunct Professor, Clinical Epidemiology Unit
Prof BV Moses Centre for Evidence-Informed Healthcare
& Health Policy
Christian Medical College, Vellore
9 December 2019
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DOCTOR, WHAT DO I HAVE? AND WHAT
SHOULD I DO?
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Test accuracy alone is insufficient for accurate
diagnosis
• Sensitivity (Sn) and specificity (Sp) describe the accuracy of performance
of a test but:
• There are often trade-offs that one needs to make regarding sensitivity and specificity.
• These are amplified if the threshold / cut off for a positive or a negative test are not
absolute but vary.
• They are two separate measures; clinicians need a measure that combines true and
false results into one
• Sn and Sp are derived from studies on populations that are often different from ones
that a clinician would see.
• Clinicians want to know if their patients have or do not have the target
condition, given the test‟s results.
• A clinician should be able to use a diagnostic test result to make a clinical
decision about whether to investigate further, treat immediately, or tell a
patient to come back later, or that he/she does not have a problem
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Consequence of Diagnostic Errors
• False negative errors, i.e., missing disease that is present.
-can result in people foregoing needed treatment for the disease
- the consequence can be as serious as death
Important to reduce false negative rates in tests (A test should be
sensitive to pick up all with disease)
• False positive errors, i.e., falsely indicating disease
- disease-free are subjected to unnecessary work-up procedures or
even treatment.
- negative impact include personal inconvenience and/or unnecessary
stress, anxiety, etc.
Important to reduce false positive rates in tests (Should be specific in
picking up only those with disease)
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Overdiagnosis and Overtreatment
• Overdiagnosis is the diagnosis of an
abnormality that bears no
substantial health hazard and no
benefit for patients.
• Mainly due to the use of increasingly
sensitive screening and diagnostic
tests, as well as broadened definitions
of conditions requiring an intervention,
overdiagnosis is a growing but still
largely misunderstood public health
issue..
The main consequence of
overdiagnosis is overtreatment.
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Moynihan et al: BMJ2012;344:e3502doi:10.1136/bmj.e350
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Drivers of overdiagnosis
• Technological changes detecting ever smaller “abnormalities”
• Commercial and professional vested interests
• Conflicted guideline panels producing expanded disease definitions and writing
guidelines
• Legal incentives that punish underdiagnosis but not overdiagnosis
• Health system incentives favouring more tests and treatments
• Cultural beliefs that more is better; faith in early detection unmodified by its risks
• Confusion between risk and disease
• Physician‟s fear of missing a disease or not meeting their patients‟ expectations
• Lack of access to or understanding about evidence of lack of benefit of
overdiagnosisMoynihan et al: BMJ2012;344:e3502doi:10.1136/bmj.e350
Bulliard and Chiolero Public Health Reviews (2015) 36:8
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BALANCING SENSITIVITY AND SPECIFICITY:
PERSPECTIVES ON RELATIVE IMPORTANCE
• Patients may prefer not to miss a cancer diagnosis and hence may not
mind over-investigation
• Health-administrators may prefer not to spend on un-necessary
investigations or may welcome it (depends on who pays for it)
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The utility of a diagnostic test
• Thresholds for testing and for treating
Low pre-test
probability
Uncertain High pre-test
probability
Diagnostic tests should increase the post- test probability of an
accurate diagnosis over the pre-test probability of the diagnosis
(prevalence) and help with triggering treatment decisions
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Predictive values help in interpretation of test results for the presence
of disease
Disease
(Reference test)
Present Absent
Index
test
+ TP FP TP+FP
- FN TN FN+TN
TP+FN FP+TNTP+FP+
FN+TN
• Positive Predictive
Value (PPV) is the
probability of disease in
a patient with a positive
(abnormal) test result
• TP / TP+FP• Negative Predictive
Value (NPV) is the
probability of not having
the disease when the
result is negative (normal)
• TN / FN +TN
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Predictive values are dependant on prevalence
• Prevalence is an important determinant of the interpretation of the result
of a diagnostic test
• When the prevalence of disease in the population tested is relatively
high – the test performs well
• At lower prevalence, the PPV drops to nearly zero, and the test is
virtually useless
• As Sn and Sp fall, the influence of prevalence on PV becomes more
pronounced!
• PPV derived from hospital populations where the prevalence of disease
is high will over-estimate probability if applied to a community setting
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LIKELIHOOD RATIOS
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Likelihood Ratios
• A useful single measure of accuracy of a diagnostic test is the likelihood
ratio (LR).
• It‟s a ratio of the likelihood of a positive test result being a true positive
rather than a false positive result; or a negative test result being a true
negative test rather than a false negative result
• The LR is equivalent to a relative risk in other epidemiological studies and
is calculated in the same way
• 95% CI for the LR can be calculated as is done for relative risks
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Increasing the utility of a diagnostic test result: Likelihood Ratios
• LR is more stable than predictive values (depends on the ratio of Sn
and Sp not prevalence)
• It is possible to calculate LRs for different test results (e.g. for a
positive or a negative test result) and for different thresholds of test
results
• LRs can be estimated for binary (positive or negative), ordinal (more
than two categories) or continuous (number scale) diagnostic test
outcomes.
• However, ordinal and continuous outcomes are often dichotomized using a cut-
off value to help with the decision-making process.
• LR can be used to determine if the application of a diagnostic test
increases the probability of a target disorder compared to the pre-test
probability of the disorder in a given patient
• LRs can be used to combine the results of multiple diagnostic tests
and can be used to increase the post-test probability for a target
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Likelihood Ratio for a positive test (LR+)
Disease
(Reference test)
Present Absent
Index
test
+ TP FP TP+FP
- FN TN FN+TN
TP+FN FP+TNTP+FP+
FN+TN
• How much more likely is a positive
test to be found in a person with the
disease than in a person without it?
• The probability of having a true
positive test result rather than a
false positive test result
• LR+ = sensitivity /(1-specificity)
• (TP/TP+FN) / [1- (TN / FP+TN)]
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Likelihood Ratio for a negative test (LR-)
Disease
(Reference test)
Present Absent
Index
test
+ TP FP TP+FP
- FN TN FN+TN
TP+FN FP+TNTP+FP+
FN+TN
• How much more likely is a
negative test to be found in a
person without the disease than
in a person with it?
• The probability that the patient
has a true negative test and not
a false negative test result
• LR - = (1-sens)/spec
• [FN / (TP+FN] / [TN / FP+TN]
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LRs with more than two test results
• LR for high probability result=(a/x)/(b/y).
• LR is likelihood of a high probability test result when disease is present
divided by likelihood of a high probability test result when no disease is
present.
• LR for intermediate probability result=(c/x)/(d/y).
• LR for low probability result=(e/x)/(f/y).
• * n=a+b+c+d+e+f.Hayden SR, Brown, MD. Likelihood ratio: A powerful tool for incorporating the results of a
diagnostic test into clinical decision making. Ann Emerg Med 1999; 33:575–80.
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Likelihood ratios for levels of serum ferritin in Iron Deficiency
Anaemia
Serum Ferritin (mcg/l) LR for Iron Deficiency
Anaemia
>100 0.08
45 to 99 0.54
35 to 44 1.83
25 to 34 2.54
15 to 24 8.83
<15 51.85
Modified from: Guyatt G et al. Laboratory diagnosis of iron deficiency anaemia. J Gen Intern Med. 1992 Mar–Apr;
7(2):145–53.
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The Bayesian Approach to using the Likelihood Ratios in
diagnosis
• In a Bayesian approach, one starts with an initial probability estimate
that is based on one‟s knowledge of disease prevalence or from
one‟s previous experiences.
• This initial probability estimate, termed the prior probability , is then
sequentially modified on the basis of each piece of additional
evidence encountered to form new probabilities, termed posterior
probabilities .
• Bayes‟ Theorem is basically a mathematical recognition of context as
an important factor in decision making.
• In other words no diagnostic test is perfect, and because every test
will be wrong sometimes the likelihood that a test is right will depend
heavily upon its context.
Goodman SN. Toward evidence-based medical statistics. II. The Bayes factor. Ann Intern
Med 1999;130:1005-13.
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The Bayesian Approach to using the Likelihood Ratios in
diagnosis
• This approach requires an estimate of the probability of a disease
before any test is ordered (i.e. the „pre-test probability‟)
• Bayes‟s theorem of conditional probability states that the pre-test
odds of a hypothesis being true multiplied by the weight of new
evidence (likelihood ratio) generates post-test odds of the hypothesis
being true.
• Conditional probability is the probability of an event occurring given in
the context of some other event or events
• When used for diagnosis of disease, this refers to the odds of having
a certain disease versus not having that disease
Goodman SN. Toward evidence-based medical statistics. II. The Bayes factor. Ann Intern
Med 1999;130:1005-13.
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Bayes Theorem
• Bayes Theorem: P(x/A) = [P(A|x)* P(x)]/P(A)
• Where:
• P(x) = the probability of condition × being present. (prior or pre-test
probability)
• P(A) = the probability of A being present. (test result)
• P(x|A) = the probability of condition × being present given the
presence of A (Posterior or post-test probability).
• P(A|x) = the probability of A being present given the presence of
condition x.
• Bayes‟ Theorem states that the pre-test odds of disease
multiplied by the likelihood ratio yields the post-test odds of
disease.
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Converting Probabilities to Odds
• Bayes Theorem:
Post-test Odds = Pre-test Odds LR
• Pre-test odds =
Prevalence /(1-prevalence)
• Post-test probability = post-test
odds/(post-test odds +1)
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Pre-test Probability is the Prevalence of the
disease
Disease
(Reference test)
Present Absent
Index
test
+ TP FP TP+FP
- FN TN FN+TN
TP+FN FP+TNTP+FP+
FN+TN
• Prevalence or Pre-test
Probability can be
estimated from prior
knowledge or local data or
calculated from results of a
study
• Pre-test Probability =
TP+FN / TP+FP+FN+TN
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Exercise: Serum Ferritin in diagnosis of Iron Deficiency Anaemia
Iron deficiency anemia
(Bone Marrow Iron)Totals
Present Absent
Diagnostic
test result
(serum
ferritin)
Positive
(< 65 mmol/L)
731
TP
a
270
FP
b
1001
a+b
Negative
( 65 mmol/L)
78
FN
c
1500
TN
d
1578
c+d
Totals
809
a+c
1770
b+d
2579
a+b+c+d
What is the post test
probability of Iron deficiency
anemia in a 56 year old man
with a serum ferritin level of
35 who is afebrile and has
pallor?
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Exercise: Serum Ferritin in diagnosis of anaemia
Iron deficiency anemia
(Bone Marrow Iron)Totals
Present Absent
Diagnostic
test result
(serum
ferritin)
Positive
(< 65 mmol/L)731
TP
a
270
FP
b
1001
a+b
Negative
( 65 mmol/L)78
FN
c
1500
TN
d
1578
c+d
Totals809
a+c
1770
b+d
2579
a+b+c+d
• Sensitivity = a/(a+c)
• Specificity = d/(b+d)
• Positive Predictive Value =
a/(a+b)
• Negative Predictive Value =
d/(c+d)
• Likelihood ratio for a positive
test result = LR+ = sens/(1-
spec)
• Likelihood ratio for a negative
test result = LR - = (1-
sens)/spec
• Pre-test probability
(prevalence) = (a+c)/(a+b+c+d)
• Pre-test odds = prevalence/(1-
prevalence)
• Post-test odds = pre-test odds
LR
• Post-test probability = post-test
odds/(post-test odds +1)
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Exercise: Serum Ferritin in diagnosis of anaemia
Iron deficiency anemia
(Bone Marrow Iron)Totals
Present Absent
Diagnosti
c test
result
(serum
ferritin)
Positive
(< 65
mmol/L)
731
TP
a
270
FP
b
1001
a+b
Negative
( 65
mmol/L)
78
FN
c
1500
TN
d
1578
c+d
Totals
809
a+c
1770
b+d
2579
a+b+c+d
• Sensitivity = a/(a+c) = 731/809 = 90%
• Specificity = d/(b+d) = 1500/1770 =
85%
• Positive Predictive Value = a/(a+b) =
731/1001 = 73%
• Negative Predictive Value = d/(c+d) =
1500/1578 = 95%
• Likelihood ratio for a positive test
result = LR+ = sens/(1-spec) =
90%/15% = 6
• Likelihood ratio for a negative test
result = LR - = (1-sens)/spec =
10%/85% = 0.12
• Pre-test probability (prevalence) =
(a+c)/(a+b+c+d) = 809/2579 = 32%
• Pre-test odds = prevalence/(1-
prevalence) = 0.31/0.69 = 0.45
• Post-test odds = pre-test odds LR =
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The Fagan's nomogram.
Charles G B Caraguel, and Raphaël Vanderstichel Evid
Based Med 2013;18:125-128
©2013 by BMJ Publishing Group Ltd
MRI screening test for breast
cancer in high-risk female patients:
Sensitivity = 75%
Specificity = 96%
LR+ (0.75/(1-0.96) = 0.75/0.04
=18.75
LR- (1-0.75)/0.96) = 0.26
A patient from a high-risk population
has an estimated pre-test
probability of 2%
If MR +: post-test probability that
she truly has cancer = ~28% (red
line).
If MR-: post-test probability that she
truly has cancer = ~0.6% (blue line).Warner E, Messersmith H, Causer P, et
al. Systematic review: using magnetic
resonance imaging to screen women at
high risk for breast cancer. Ann Intern
Med 2008;148:671–9
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Pre-test probability = 32%
LR+ = 6
Post-test probability test += 73%
LR- = 0.12
Post-test probability test- = 5%
Serum Ferritin in
diagnosis of Iron
Deficiency
Anaemia
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Dynamic Fagan Nomogram:
https://cscheid.net/projects/fagan_nomogram/
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Two-Step Fagan Nomogram
Charles G B Caraguel, and Raphaël Vanderstichel Evid Based
Med 2013;18:125-128
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Charles G B Caraguel, and Raphaël
Vanderstichel Evid Based Med
2013;18:125-128
Two-Step Fagan
Nomogram
• MRI screening test for breast
cancer in high-risk female patients
• Sensitivity of 75% and specificity
of 96%.
• A positive result from the MRI
provides a likelihood ratio (LR+) of
~ 19 (red line, I).
• A patient from a high-risk
population has an estimated pre-
test probability of 2%
• If she tested positive, the post-test
probability for this patient to truly
have cancer would be ~28% (red
line, II).
• A negative test result would
produce a likelihood ratio (LR−) of
approximately 0.25 (blue line, I)
• The post-test probability for this
patient to truly have cancer would
be approximately 0.6% (blue line,
II
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Likelihood ratio interpretation at the bedside
McGee, S. Simplifying Likelihood Ratios. J Gen Intern Med. 2002 Aug; 17(8):
647– 650.
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Benchmarking LRs and probability of change in post-test
probability
• Remember 3 specific LRs: 2, 5, 10
• And the first 3 multiples of 15 (15, 30,
45)
• LR 2 increases post test probability by
15%, LR 5 by 30% and LR 10 by 45%
• For LRs between 0 and 1, invert 2, 5,
and 10 (i.e.: ½ = 0.5. 1/5 = 0.2 and
1/10 = 0.1).
• Inverse of LR 2 (0.5) decreases
probability 15%,
• Inverse of LR 5 (0.2) decreases
probability 30%
• Inverse of LR 10 (0.1) decreases
probability 45%
• These benchmark LRs can be used to
deduce the restMcGee, S. Simplifying Likelihood Ratios. J Gen Intern Med. 2002 Aug; 17(8):
647– 650.
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Benchmarking LRs and probability of change in post-test
probability
• Change in probability ~ 0.19 x log LR
• Regardless of a patient‟s pre-test
probability, the change in probability
from a finding is approximated by a
constant (0.19 x log LR).
• The bedside estimates are rounded off
to the nearest 5% for easy recall
• Not accurate for pre-test probabilities
of less than 10% or greater than 90%
but these do not warrant further tests
• Useful when pre-test probability is not
readily known
McGee, S. Simplifying Likelihood Ratios. J Gen Intern
Med. 2002 Aug; 17(8): 647– 650.
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Uses of LR
McGee, S.
Simplifying
Likelihood Ratios. J
Gen Intern Med.
2002 Aug; 17(8):
647– 650.
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Using LRs
•
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Using LRs in sequence
• When one test increases the post-test probability somewhat and
another test is done, the pre-test probability is now the post test
probability after the first test.
• In clinical practice, the history and physical exam serve to increase or
decrease post-test probability and supplemented by additional tests aid
more accurate diagnoses by acting synergistically,
• For example: While numerous elements of the clinical examination are
associated with the diagnosis of COPD, only 3 are significant on
multivariate analysis. Patients having all 3 of these findings have an LR
of 33 (ruling in COPD); those with none have an LR of 0.18 (ruling out
COPD) [Strauss et al, J Gen Intern Med 2002; 17 (9):684-8]
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Limitations of Likelihood Ratios
• The accuracy of a LR depends entirely upon the relevance and
quality of the studies that generated the numbers (sensitivity and
specificity) that inform that LR.
• Clinical decision making occurs by absorbing multiple factors and
generating impressions simultaneously. LRs demand that we
consider one element of diagnosis at a time.
• Some clinicians use one LR to generate a post-test probability, and
then use the new post-test probability as a pre-test probability for
application of the next LR related to a different test. There is no
evidence to support or refute the use of LRs in this fashion
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Decision process in making a diagnosis
Lancet 2005; 365: 1500–05
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Diagnostic odds ratios can be derived from
LRs
FNFP
TNTPORDiagnostic
veLR
veLR
yspecificit
yspecificit
ysensitivit
ysensitivit
DOR
1
1
Ratio of the odds of positivity in the diseased to the odds of positivity in the non-diseased
Also a stable measure and not affected by prevalence
Disease
(Reference test)
Present Absent
Index
test
+ TP FP TP+FP
- FN TN FN+TN
TP+FN FP+TNTP+FP+
FN+TN
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LR CALCULATORS
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http://araw.mede.uic.edu/cgi-bin/testcalc.pl
http://araw.mede.uic.edu/cgi-bin/testcalc.pl?DT=&Dt=&dT=&dt=&2x2=Compute
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https://www.medcalc.net/statisticaltests/diagnostic_test.php