NNT Analogues When Time Does Matter

1. 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

2. Numbers Needed to Treat are geared to one-shot interventions such as cures for an acute infection. They do not capture the time aspectof long-term therapeutic effortsor treatments aimed at long-termgoals. Both the numerator and the denominatorof 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.

3. 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)

4. Let’s list some NNT-like (”reciprocal”) measures of clinical profitability:effort [or cost] per unit expected clinical gain when oneswitches 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.

5. NotationA: the new regimen, O: reference regimenP: probability of treatment failureL: mean future life-spani: incidence rate of episodes of illnesss: start cost m: annual maintenance cost ”Δ” means AO difference

6. The classical NNT Acute disease: case treatments needed per averted treatment failure, NNT = 1 / ( PO - PA ), (dimensionless, >1)P: prob. of treatment failure

7. Performing an operation to gain life-yearsNo. of interventions needed to gain one expected year of life = 1 / ( LA - LO ) (yrs-1, > 0) L: mean post-intervention lifespan

8. 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

9. 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

10. 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.

11. 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. • Unverifiable assumptions ~ ! sensitivity analysis.

12. 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

13. 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 onthe annual discount rate, a. **but the necessary changes may be simple:

14. The two formulae below remain valid when survival is exponential withmortality 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 )

15. Long-term treatment aimed at reducing attack incidenceDiscounted 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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