Statistics question residual risk

Kohn, Michael Michael.Kohn at UCSF.EDU
Tue Sep 8 22:36:36 UTC 2015


I already replied before I saw this email from John Ely.

Using John's notation:

Post-test probability of disease after a negative dichotomous test is

PP*(1-Se) / [PP*(1-Se) + (1-PP)*Sp]

Everything he wrote in his email is correct.  A little algebra will show that the above expression is equivalent.

For tests with 3 possible results, you should use 3 likelihood ratios: LR(+), LR(I), LR(-), where I = indeterminate.  Then, you should use the odds form of Bayes's Rule as outlined in my previous email.

I will be giving a workshop on this material called the Evidence-Based Diagnosis Workshop at the SMDM Annual Meeting in St. Louis on Sunday 10/18.  Please register!  I have not created the new workshop website, but the website from the same workshop given at a difference conference is available at:

http://ebdworkshop.studysites.net/sites/ShowPage.php?site=ebdworkshop

--Michael


Michael A. Kohn, MD, MPP

Associate Professor

Epidemiology and Biostatistics

________________________________
From: Ely, John [john-ely at UIOWA.EDU]
Sent: Tuesday, September 08, 2015 9:53 AM
To: IMPROVEDX at LIST.IMPROVEDIAGNOSIS.ORG
Subject: Re: [IMPROVEDX] Statistics question residual risk

I agree with Mark but I think you can also calculate the residual risk if you know the sensitivity, specificity, and pretest probability.  For example, let’s say you have 1000 people in a study of a new test with a sensitivity of 0.9, specificity of 0.8 and pretest probability of 0.3.  Then you know that 300 people will have the disease and 700 won’t.

true positives = TP
false positives=FP
true negatives=TN
false negatives=FN
pretest probability = PP
total = 1000
those with disease = TP + FN = total*PP = 1000*0.3=300
those without disease = TN + FP = total(1-PP) = total(1 - 0.3) = 1000 * 0.7 = 700
sensitivity=Se
specificity=Sp

Se = TP/(TP+FN)
TP = Se(TP+FN)
= 0.9(300)
=270

Sp = TN/(TN+FP)
TN = Sp(TN+FP)
=0.8(700)
= 560

NPV = TN/(TN+FN)

We know the TN.  We need FN

FN = those with disease minus true positives
FN = (TP+FN) – TP
= (TP + FN) – Se(TP+FN)
= 300 – 0.9(300)
= 300-270
=30

So,

NPV = 0.8(700) / [0.8(700) + 300 – 0.9(300)]
= 560 / [560 + 300 – 270]
= 560/590 = 0.9492

So the residual risk = 1 – NPV = 1 – 0.9492 = 0.0508 (or about 5% residual risk)

But you don’t need to know the total number of people in the study population (in this case 1000) because it cancels out.

If we go back to

NPV = 0.8(700) / [0.8(700) + 300 – 0.9(300)]

More generally,

NPV = Sp(total(1-PP)) / [Sp(total(1-PP)) + total(PP) – Se(total(PP))]

Now, “total” cancels out of the numerator and denominator and we are left with

NPV = Sp(1-PP) / [Sp(1-PP) + PP – Se(PP)]

You can also write it as

NPV = [Sp – Sp(PP)] / [Sp – Sp(PP) + PP – Se(PP)]

Now divide numerator and denominator by PP and we get

NPV = [Sp/PP – Sp] / [Sp/PP – Sp + 1 – Se]

This gives you the NPV in terms of the sensitivity, specificity, and pretest probability, which is what you wanted.

So, in the example above, we have

NPV = [0.8/0.3 – 0.8] / [0.8/0.3 – 0.8 + 1 – 0.9]
= 0.9492

So residual risk is 1 – 0.9492 = 0.0508, which is what we got before.

I wouldn’t do anything important with this unless you confirm it with a statistician (which I’m not) or with Mark Ebell.

You had a second question:  “How would you modify the RR calculation if the screening test could potentially give three results positive or negative or uninformative?”

I would probably lump the uninformatives with the negatives because these are the ones you will miss with your screening test, but it may depend on the specific situation.

By the way, we should probably find a different abbreviation for residual risk because RR usually means relative risk, not residual risk.

John Ely


From: Mark H Ebell [mailto:ebell at UGA.EDU]
Sent: Monday, September 07, 2015 3:31 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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