Transcript Document

NNT Analogues
When Time Does Matter
Numbers Needed to Treat
Rarely Capture the Temporal Aspects
of Therapeutic Efforts
J.Hilden @ biostat.ku.dk
ESMDM Rotterdam
June 2004
Numbers Needed to Treat are geared
to one-shot interventions
such as cures for an acute infection.
They do not capture the time aspect
of long-term therapeutic efforts
or treatments aimed at long-term goals.
Both the numerator and the denominator
of a clinical effort-to-benefit ratio,
or its reciprocal for that matter,
may have several kinds of temporal features.
We examine a few typical examples
and make a critical appraisal of current practice.*
*with its sometimes improper use of the NNT idea.
NNT belongs to the ”reciprocal” measures of
therapeutic superiority.
Direct measures ask:
What do we get for a man-hour or $
by switching to the new treatment?
Reciprocal measures ask:
What is the price of one additional unit of benefit?
- i.e.,
)(expected medical effort or cost)
Δ(expected medical effort or cost)
)(expected clinical benefit)
Δ(expected
clinical benefit)
Let’s list some NNT-like (”reciprocal”)
measures of clinical profitability:
effort [or cost] per unit expected clinical gain
when one
switches from Regimen O to Regimen A
¤¤¤¤¤¤¤
In ( … ) are shown
units of measurement, & also the
interval in which the quantity will fall
when A is superior to O
but is also more expensive.
Notation
A: the new regimen, O: reference regimen
P: probability of treatment failure
L: mean future life-span
i: incidence rate of episodes of illness
s: start cost
”Δ”
m: annual maintenance cost
means AO difference
The classical NNT
Acute disease:
case treatments needed
per averted treatment failure,
NNT = 1 / ( PO - PA ),
(dimensionless, >1)
P: prob. of treatment failure
Performing an operation to gain lifeyears
No. of interventions needed
to gain one expected year of life
= 1 / ( LA - LO )
(yrs-1, > 0)
L: mean post-intervention lifespan
Life-long regimen to add life-years
Years of treatment needed to gain one
expected year of life
= LA / ( LA - LO )
(dimensionless, >1)
L: mean lifespan on regimen
-----------------------------------------------------Cost per life-year gained
= ( Δs + mALA - mOLO ) / ( LA - LO )
($/yr, > mA)
Δs: Δstart cost, m: annual maintenance cost
Long-term treatment aimed at reducing
attack incidence (epilepsy, etc.)
Years on regimen needed
to prevent one episode = 1 / [ iO - iA ]
(yrs, > 0)
i: attack incidence rate
.
Cost per prevented episode
[when treatment duration = D]
= [Δs / D + Δm ] / [ iO - iA ]
(unit price, > 0)
Subtract Cost of treating one episode to obtain the
Net cost per prevented episode (hopefully < 0)
END OF THEORY
Example (common but misleading):
• An RCT compares drug A with drug O
•
in 1000+1000 patients
• for prevention of ”major cardiac events.”
• If A proves advantageous, it is foreseen that
•
the treatment should be life-long in most cases.
• However, trial duration is only 23 months
[all event-free patients do receive drug for 23 months].
• ”Success rates” of 87 % vs. 80 % prompt
•
an NNT of 14 (95%CI: 11-21) to be reported.
Criticism
• Neither the hoped-for effect nor the envisaged
treatment duration is limited to 23 months.
Indeed, the envisaged duration depends on how the patient fares.
• Who wants to know how many will pass the 23-mths
point without events? - Answering the wrong question!
• Instead one ought to report an appropriate version of
•
LA / ( LA - LO ) .
• Extrapolation beyond 23 months is needed!
• That goes against the evidence-based paradigm •
but it is necessary in order to give
•
an approximate answer to the right question.
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One assumption might be this
The event-free fraction decays exponentially.
It implies estimated mean event-free periods of
LA = 13.76 yrs, LO = 8.59 yrs,
LA / ( LA - LO ) = 2.66 treatment months
per event-free month gained (95%CI: 2.00-4.55).
Sensitivity analysis: exponential model not decisive.
Can we do better than that? Yes, we can
(1) study the individual data;
(2) try to include what happens after a cardiac event
- using external evidence.
END OF EXAMPLE
DISCOUNTING
In the classical NNT situation, the formula
(treatment cost per averted treatm. failure)
= (costA - costO) × NNT
is unaffected by discounting due to synchronicity of cost and effect.
Otherwise, the answer depends on
the annual discount rate, a. *
*but the necessary changes may be simple:
The two formulae below remain valid when
survival is exponential with
mortality hazard i [implying L = 1/i],
provided that i is replaced with (i+a)
[L = 1/ (i+a) = discounted life expectation]:
(1) Performing an operation to gain life-yrs
No. of interventions needed
to gain one expected discounted year of life
= 1 / ( LA - LO )
(2) Life-long regimen to add life-years
Discounted cost / discounted life-year gained
= ( Δs + mALA - mOLO ) / ( LA - LO )
Long-term treatment aimed at
reducing attack incidence
Discounted cost per prevented episode
= [Δs / D* + Δm ] / [ iO - iA ]
almost as before; only,
i is replaced with (i+a), and the
actual depreciation period, = planned duration of
treatment, D, has been replaced with
D* = (1 - exp(-aD))/a,
an ”effective depreciation time” (somewhat < D).
As before,
subtract Cost of treating one episode to obtain the
Net discounted cost per prevented episode.
END OF DISCOUNTING
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