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)

(expected clinical benefit) Δ((expected medical effort or cost)

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 failureL: mean future life-spani: incidence rate of episodes of illness

s: start cost m: annual maintenance cost

”Δ” 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 life-years 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 THEORYEND 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.

CriticismCriticism• 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.• Unverifiable assumptions ~ ! sensitivity analysis.

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 EXAMPLEEND OF EXAMPLE

DISCOUNTINGIn 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-yrsNo. of interventions needed to gain one expected discounted year of life

= 1 / ( LA - LO )

(2) Life-long regimen to add life-yearsDiscounted 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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