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Tests for a medical condition can be classified into two categories:

  1. Tests on an asymptomatic (no reason to suspect condition) patient where the result is a circumstantial finding; and
  2. Tests on a symptomatic (prior reason to suspect condition) patient where the result is a diagnostic finding.

In the first case, the emphasis of any test should be on maximum specificity. For the second case, the emphasis of any test should be on maximum sensitivity.

Intuitively, I understand why. However, can somebody give me a formal run-through, preferably from a Bayesian perspective, as to why this is the case?

References:

Slide presentation, "Categorizing variants after whole genome sequencing", J.S. Berg

Lang, E., Naraghi, R. (2005) Neurovascular relationship at the trigeminal root entry zone in persistent idiopathic facial pain: findings from MRI 3D visualisation. J Neurol Neurosurg Psychiatry 76:1506-1509. Full text

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I am not sure I agree with the statements. Whether specificity or sensitivity depends on more than whether the patient is symptomatic. It depends on which error (false positive or false negative) is worse. I am also not sure that the division into "circumstantial" and "diagnostic" findings is useful. – Peter Flom Aug 18 '12 at 17:15
The division into "circumstantial" and "diagnostic" is just for descriptive purposes; not essential. Symptomatic - reason to suspect patient has condition; asymptomatic - no reason to suspect patient has condition. Perhaps the distinct error rates for the two types of patient could be framed as a prior in a Bayesian formula? – oisyutat Aug 18 '12 at 17:23
You could specify different priors, certainly. But I am not really a Bayesian, so I hope you get some more answers from those who are. – Peter Flom Aug 18 '12 at 17:29
@Peter Flom makes a very important point about whether false positives or false negatives are worse. False negatives can be bad if the condition is serious and must be identified early. False positives are bad if follow-up tests are invasive. – tristan Aug 19 '12 at 8:29
@PeterFlom This is a good point and would be good to incorporate into any model but is not what I'm after in the first instance. – oisyutat Aug 19 '12 at 9:11

1 Answer

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The simple way to work it out is that a test with high specificity has a high proportion of disease negative patients resulting in test negatives. Therefore high specificity implies low false positive rate. When you are not very convinced a patient has a condition you want to make sure a test is only likely to give a positive result if the patient has the condition.

From a Bayesian perspective, let $P(D+)$ be your prior belief that the patient has the disease. Let $T+$ and $T-$ denote the outcomes test positive and test negative respectively.

$sensitivity = P(T+\mid D+)$ and $specificity = P(T- \mid D-)$

By Bayes' rule

$P(D+ \mid T+) = \frac{P(T+ \mid D+)P(D+)}{P(T+)} = \frac{sensitivity \times P(D+)}{P(T+)} = \frac{sensitivity \times P(D+)}{P(T+ \mid D+)P(D+)+P(T+ \mid D-)P(D-)} = \frac{sensitivity \times P(D+)}{sensitivity \times P(D+)+(1-specificity)\times P(D-)}$

A similar formula can be constructed for $P(T- \mid D-)$.

You should observe that when the prior belief that the patient has the disease is small the denominator is strongly affected by $specificity$

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Thanks, but would $P(D+)$ not represent the probability the patient actually has the disease in order to get the $sensitivity$ and $specificity$ formulae correct? Perhaps a new variable, $P(S+)$ could represent prior belief of disease? – oisyutat Aug 19 '12 at 9:09
@oisyutat yes I assume that the sensitivity and specificity are not affected by patient selection, which may not be legitimate, eg if the test has only ever been used on people from a certain age group. Incidentally I don't believe there is really a 'probability' that the patient has the disease, it's a case of what you belief on the basis of evidence you've accumulated. – tristan Aug 19 '12 at 12:24
Sorry, what I meant was: is there not a distinction to be made between $P(T+|D+)$ where $D+$ refers to suspicion of disease and $P(T+|D+)$ where $D+$ refers to a true positive? – oisyutat Aug 19 '12 at 20:46
Ah I think I understand where we are confused. Within the framework I have outlined $D+$ is the outcome that the patient actually has the disease, independent of whether we know that. We then have our prior belief that the patient has the disease, which we express as $P(D+)$, which is low if the patient is asymptomatic and high if the patient has clinical symptoms and signs very suggestive. We will have the sensitivity and specificity of our test from experiments, such that $P(T+\mid D+)$ and $P(T-\mid D-)$ are known, and then we use Bayes' rule to update our belief as shown above...[MTF] – tristan Aug 20 '12 at 8:12
In your suggested framework, perhaps you are saying that $S$ is an additional variable with value $S-$ if we do not have a high suspicion of disease and $S+$ if we do have a suspicion. We then want to know $P(D+ \mid S+,T+)$, P(D+ \mid S-,T+)$, P(D+ \mid S+,T-)$ and $P(D+ \mid S-,T-)$. You are unlikely to be able to evaluate these without making assumptions, eg, S is conditionally independent of T given D, and even then you may not have data as different clinicians would have individual $P(S\mid D)$. Have I resolved the confusion? – tristan Aug 20 '12 at 8:18
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