Estimation of post-test probability Pre- and post-test probability

1 estimation of post-test probability

1.1 predictive values
1.2 likelihood ratio

1.2.1 example
1.2.2 specific sources of inaccuracy

1.2.2.1 interference test
1.2.2.2 overlap of tests
1.2.2.3 methods overcome inaccuracy




1.3 relative risk

1.3.1 1 risk factor
1.3.2 multiple risk factors


1.4 diagnostic criteria , clinical prediction rules

estimation of post-test probability

in clinical practice, post-test probabilities estimated or guessed. acceptable in finding of pathognomonic sign or symptom, in case target condition present; or in absence of finding sine qua non sign or symptom, in case target condition absent.

in reality, however, subjective probability of presence of condition never 0 or 100%. yet, there several systematic methods estimate probability. such methods based on having performed test on reference group in presence or absence on condition known (or @ least estimated test considered highly accurate, such gold standard ), in order establish data of test performance. these data subsequently used interpret test result of individual tested method. alternative or complement reference group-based methods comparing test result previous test on same individual, more common in tests monitoring.

the important systematic reference group-based methods estimate post-test probability includes ones summarized , compared in following table, , further described in individual sections below.

by predictive values

predictive values can used estimate post-test probability of individual if pre-test probability of individual can assumed equal prevalence in reference group on both test results , knowledge on presence or absence of condition (for example disease, such may determined gold standard ) available.

if test result of binary classification either positive or negative tests, following table can made:

pre-test probability can calculated diagram follows:

pretest probability = (true positive + false negative) / total sample

also, in case, positive post-test probability (the probability of having target condition if test falls out positive), numerically equal positive predictive value, , negative post-test probability (the probability of having target condition if test falls out negative) numerically complementary negative predictive value ([negative post-test probability] = 1 - [negative predictive value]), again assuming individual being tested not have other risk factors result in individual having different pre-test probability reference group used establish positive , negative predictive values of test.

in diagram above, positive post-test probability, is, posttest probability of target condition given positive test result, calculated as:

positive posttest probability = true positives / (true positives + false positives)

similarly:

the post-test probability of disease given negative result calculated as:

negative posttest probability = false negatives / (false negatives + true negatives)

the validity of equations above depend on sample population not have substantial sampling bias make groups of have condition , not substantially disproportionate corresponding prevalence , non-prevalence in population. in effect, equations above not valid merely case-control study separately collects 1 group condition , 1 group without it.

by likelihood ratio

the above methods inappropriate use if pretest probability differs prevalence in reference group used establish, among others, positive predictive value of test. such difference can occur if test preceded, or person involved in diagnostics considers pretest probability must used because of knowledge of, example, specific complaints, other elements of medical history, signs in physical examination, either calculating on each finding test in own sensitivity , specificity, or @ least making rough estimation of individual pre-test probability.

in these cases, prevalence in reference group not accurate in representing pre-test probability of individual, and, consequently, predictive value (whether positive or negative) not accurate in representing post-test probability of individual of having target condition.

in these cases, posttest probability can estimated more accurately using likelihood ratio test. likelihood ratio calculated sensitivity , specificity of test, , thereby not depend on prevalence in reference group, and, likewise, not change changed pre-test probability, in contrast positive or negative predictive values (which change). also, in effect, validity of post-test probability determined likelihood ratio not vulnerable sampling bias in regard , without condition in population sample, , can done case-control study separately gathers , without condition.

estimation of post-test probability pre-test probability , likelihood ratio goes follows:

pretest odds = (pretest probability / (1 - pretest probability)
posttest odds = pretest odds * likelihood ratio

in equation above, positive post-test probability calculated using likelihood ratio positive, , negative post-test probability calculated using likelihood ratio negative.

posttest probability = posttest odds / (posttest odds + 1)


fagan nomogram

the relation can estimated so-called fagan nomogram (shown @ right) making straight line point of given pre-test probability given likelihood ratio in scales, which, in turn, estimates post-test probability @ point straight line crosses scale.

the post-test probability can, in turn, used pre-test probability additional tests if continues calculated in same manner.

it possible calculation of likelihood ratios tests continuous values or more 2 outcomes similar calculation dichotomous outcomes. purpose, separate likelihood ratio calculated every level of test result , called interval or stratum specific likelihood ratios.

example

an individual screened test of fecal occult blood (fob) estimate probability person having target condition of bowel cancer, , fell out positive (blood detected in stool). before test, individual had pre-test probability of having bowel cancer of, example, 3% (0.03), have been estimated evaluation of, example, medical history, examination , previous tests of individual.

the sensitivity, specificity etc. of fob test established population sample of 203 people (without such heredity), , fell out follows:

from this, likelihood ratios of test can established:

pretest probability (in example) = 0.03
pretest odds = 0.03 / (1 - 0.03) = 0.0309
positive posttest odds = 0.0309 * 7.4 = 0.229
positive posttest probability = 0.229 / (0.229 + 1) = 0.186 or 18.6%

thus, individual has post-test probability (or post-test risk ) of 18.6% of having bowel cancer.

the prevalence in population sample calculated be:

prevalence = (2 + 1) / 203 = 0.0148 or 1.48%

the individual s pre-test probability more twice 1 of population sample, although individual s post-test probability less twice 1 of population sample (which estimated positive predictive value of test of 10%), opposite result less accurate method of multiplying relative risks.

specific sources of inaccuracy

specific sources of inaccuracy when using likelihood ratio determine post-test probability include interference determinants or previous tests or overlap of test targets, explained below:

interference test

post-test probability, estimated pre-test probability likelihood ratio, should handled caution in individuals other determinants (such risk factors) general population, in individuals have undergone previous tests, because such determinants or tests may influence test in unpredictive ways, still causing inaccurate results. example risk factor of obesity additional abdominal fat can make difficult palpate abdominal organs , decrease resolution of abdominal ultrasonography, , similarly, remnant barium contrast previous radiography can interfere subsequent abdominal examinations, in effect decreasing sensitivities , specificities of such subsequent tests. on other hand, effect of interference can potentially improve efficacy of subsequent tests compared usage in reference group, such abdominal examinations being easier when performed on underweight people.

overlap of tests

furthermore, validity of calculations upon pre-test probability derived previous test depend on 2 tests not overlap in regard target parameter being tested, such blood tests of substances belonging 1 , same deranged metabolic pathway. example of extreme of such overlap sensitivity , specificity has been established blood test detecting substance x , , likewise 1 detecting substance y . if, in fact, substance x , substance y 1 , same substance, then, making 2 consecutive tests of 1 , same substance may not have diagnostic value @ all, although calculation appears show difference. in contrast interference described above, increasing overlap of tests decreases efficacy. in medical setting, diagnostic validity increased combining tests of different modalities avoid substantial overlap, example in making combination of blood test, biopsy , radiograph.

methods overcome inaccuracy

to avoid such sources of inaccuracy using likelihood ratios, optimal method gather large reference group of equivalent individuals, in order establish separate predictive values use of test in such individuals. however, more knowledge of individual s medical history, physical examination , previous test etc. individual becomes more differentiated, increasing difficulty find reference group establish tailored predictive values, making estimation of post-test probability predictive values invalid.

another method overcome such inaccuracies evaluating test result in context of diagnostic criteria, described in next section.

by relative risk

post-test probability can estimated multiplying pre-test probability relative risk given test. in clinical practice, applied in evaluation of medical history of individual, test question (or assumption) regarding various risk factors, example, sex, tobacco smoking or weight, can potentially substantial test such putting individual on weighing scale. when using relative risks, resultant probability rather related individual developing condition on period of time (similarly incidence in population), instead of being probability of individual of having condition in present, can indirectly estimation of latter.

usage of hazard ratio can used relative risk.

one risk factor

to establish relative risk, risk in exposed group divided risk in unexposed group.

if 1 risk factor of individual taken account, post-test probability can estimated multiplying relative risk risk in control group. control group represents unexposed population, if low fraction of population exposed, prevalence in general population can assumed equal prevalence in control group. in such cases, post-test probability can estimated multiplying relative risk risk in general population.

for example, incidence of breast cancer in woman in united kingdom @ age 55 59 estimated @ approximately 280 cases per 100.000 per year, , risk factor of having been exposed high-dose ionizing radiation chest (for example, treatments other cancers) confers relative risk of breast cancer between 2.1 , 4.0, compared unexposed. because low fraction of population exposed, prevalence in unexposed population can assumed equal prevalence in general population. subsequently, can estimated woman in united kingdom aged between 55 , 59 , has been exposed high-dose ionizing radiation should have risk of developing breast cancer on period of 1 year of between 588 , 1.120 in 100.000 (that is, between 0,6% , 1.1%).

multiple risk factors

theoretically, total risk in presence of multiple risk factors can estimated multiplying each relative risk, less accurate using likelihood ratios, , done because easier perform when relative risks given, compared to, example, converting source data sensitivities , specificities , calculate likelihood ratios. likewise, relative risks given instead of likelihood ratios in literature because former more intuitive. sources of inaccuracy of multiplying relative risks include:

relative risks affected prevalence of condition in reference group (in contrast likelihood ratios, not), , issue results in validity of post-test probabilities become less valid increasing difference between prevalence in reference group , pre-test probability individual. known risk factor or previous test of individual confers such difference, decreasing validity of using relative risks in estimating total effect of multiple risk factors or tests. physicians not appropriately take such differences in prevalence account when interpreting test results, may cause unnecessary testing , diagnostic errors.
a separate source of inaccuracy of multiplying several relative risks, considering positive tests, tends overestimate total risk compared using likelihood ratios. overestimation can explained inability of method compensate fact total risk cannot more 100%. overestimation rather small small risks, becomes higher higher values. example, risk of developing breast cancer @ age younger 40 years in women in united kingdom can estimated @ approximately 2%. also, studies on ashkenazi jews has indicated mutation in brca1 confers relative risk of 21.6 of developing breast cancer in women under 40 years of age, , mutation in brca2 confers relative risk of 3.3 of developing breast cancer in women under 40 years of age. these data, may estimated woman brca1 mutation have risk of approximately 40% of developing breast cancer @ age younger 40 years, , woman brca2 mutation have risk of approximately 6%. however, in rather improbable situation of having both brca1 , brca2 mutation, multiplying both relative risks result in risk of on 140% of developing breast cancer before 40 years of age, can not possibly accurate in reality.

the (latter mentioned) effect of overestimation can compensated converting risks odds, , relative risks odds ratios. however, not compensate (former mentioned) effect of difference between pre-test probability of individual , prevalence in reference group.

a method compensate both sources of inaccuracy above establish relative risks multivariate regression analysis. however, retain validity, relative risks established such must multiplied other risk factors in same regression analysis, , without addition of other factors beyond regression analysis.

in addition, multiplying multiple relative risks has same risk of missing important overlaps of included risk factors, when using likelihood ratios. also, different risk factors can act in synergy, result that, example, 2 factors both individually have relative risk of 2 have total relative risk of 6 when both present, or can inhibit each other, interference described using likelihood ratios.

by diagnostic criteria , clinical prediction rules

most major diseases have established diagnostic criteria and/or clinical prediction rules. establishment of diagnostic criteria or clinical prediction rules consists of comprehensive evaluation of many tests considered important in estimating probability of condition of interest, including how divide subgroups, , when , how treat condition. such establishment can include usage of predictive values, likelihood ratios relative risks.

for example, acr criteria systemic lupus erythematosus defines diagnosis presence of @ least 4 out of 11 findings, each of can regarded target value of test own sensitivity , specificity. in case, there has been evaluation of tests these target parameters when used in combination in regard to, example, interference between them , overlap of target parameters, thereby striving avoid inaccuracies otherwise arise if attempting calculate probability of disease using likelihood ratios of individual tests. therefore, if diagnostic criteria have been established condition, appropriate interpret post-test probability condition in context of these criteria.

also, there risk assessment tools estimating combined risk of several risk factors, such online tool [1] framingham heart study estimating risk coronary heart disease outcomes using multiple risk factors, including age, gender, blood lipids, blood pressure , smoking, being more accurate multiplying individual relative risks of each risk factor.

still, experienced physician may estimate post-test probability (and actions motivates) broad consideration including criteria , rules in addition other methods described previously, including both individual risk factors , performances of tests have been carried out.