# Statistics question residual risk

Kohn, Michael Michael.Kohn at UCSF.EDU
Tue Sep 8 22:10:03 UTC 2015

```The post-test probability of disease after a negative test is given by

[P(D+) x (1-Sensitivity)] / [P(D+) x (1-Sensitivity) + (1-P(D+)) x Specificity]

Where P(D+) = Pre-test probability of disease.

Substituting "*" for "x" to denote multiplication and using conditional probability where P(A|B) is read "Probability of A given B",

P(-|D+) = Probability of a negative test given disease = 1 - sensitivity
P(-|D-) = Probability of a negative test given NO disease = specificity

Post test probability after a negative test is

P(D+)*P(-|D+) /[P(D+)*P(-|D+) + (1-P(D+))*(P(-|D-)]

If you are willing to use Odds instead of probability, post-test odds after a negative test is

Odds(D+|-) = Odds(D+) * LR(-)

Where LR(-) = P(-|D+) / P(-|D-)

To convert between probability and odds:

Odds(D+) = P(D+)/(1 - P(D+))

To convert between odds and probability:

P(D+|-) = Odds(D+|-) / (1 + Odds(D+|-)

The reason you might like to use the Odds Form of Bayes's Rule is that it generalizes to all test results including results of non-dichotomous tests.

Post-test Odds = Pre-Test Odds x Likelihood Ratio of Test Result

Odds(D+|r) = Odds(D+) * LR(r)

Where "r" is an arbitrary test result and LR(r) = P(r|D+) / P(r|D-)

Best,

Michael

Michael A. Kohn, MD, MPP

Associate Professor

Epidemiology and Biostatistics

________________________________
From: Mark H Ebell [ebell at UGA.EDU]
Sent: Monday, September 07, 2015 1:31 PM
To: IMPROVEDX at LIST.IMPROVEDIAGNOSIS.ORG
Subject: Re: [IMPROVEDX] Statistics question residual risk

Yes, as you define it, residual risk is the converse of negative predictive value. It is also post test probability of disease given a negative test.

Mark Ebell

Sent from Outlook<http://aka.ms/Ox5hz3>

On Mon, Sep 7, 2015 at 1:00 PM -0700, "Twest54973" <000000040134e744-dmarc-request at LIST.IMPROVEDIAGNOSIS.ORG<mailto:000000040134e744-dmarc-request at LIST.IMPROVEDIAGNOSIS.ORG>> wrote:

Can this question be posted on the list serve?

Residual risk (used in the context of a screening test which gives dichotomous results ie, positive or negative)  is usually defined as the risk of disease after a negative result on the screening test

I am looking for a formula to calculate residual risk using sensitivity, specificity, and pretest probability (akin to the formulas that exist for PPV and NPV)

I have seen some authors state that residual risk = 1- NPV

Is there another way to calculate RR?

How would you modify the RR calculation if the screening test could potentially give three results positive or negative or uninformative?

Thank you
Tom Westover MD

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