Delta-method confidence interval for an ICER

Linearises the ratio of the two estimated differences around their point estimates to obtain a large-sample standard error for the ICER, then forms a symmetric Wald interval around the ratio. The function abs returns the absolute value, so the standard error stays positive when the ICER is negative. The method is also called the Taylor series method.

Signature

R = Delta_C / Delta_E; SE_R = abs(R) * sqrt(SE_C^2 / Delta_C^2 + SE_E^2 / Delta_E^2 - 2 * Cov_CE / (Delta_C * Delta_E)); R_L = R - z * SE_R; R_U = R + z * SE_R
Inputs
InputsDefinitionUnit
Delta_CEstimated difference in mean cost, intervention minus comparatorcurrency, for example pounds
Delta_EEstimated difference in mean effect, intervention minus comparatoreffect, for example QALYs
SE_CStandard error of Delta_Ccurrency
SE_EStandard error of Delta_Eeffect
Cov_CEEstimated covariance between Delta_C and Delta_E, zero when the two differences are uncorrelatedcurrency times effect
zStandard normal critical value, 1.96 for a two-sided 95% intervalnone
Output
REstimated ICER, the difference in mean cost divided by the difference in mean effectcurrency per unit of effect, for example pounds per QALY
SE_RLarge-sample standard error of R from the delta methodsame as R
R_LLower limit of the delta-method intervalsame as R
R_UUpper limit of the delta-method intervalsame as R

Function

Large-sample normal confidence interval function

Maps an estimate and its estimated sampling variance to an approximate confidence interval by treating the estimator as normally distributed, which asymptotic normality justifies in large samples even when patient-level data are skewed. In trial-based cost-effectiveness analysis the same operation is applied to a difference in mean cost or effect, to the ICER through the delta method, to the pair of differences through Fieller's method, and to incremental net benefit, which is linear in the two differences.

Try this function

Implementations

  • Excel

    Delta-method ICER limits in two cells

    With named cells DeltaCost, DeltaEffect, SeCost, SeEffect, CovCostEffect and ZValue, the two formulas return the lower and upper limits.

    =DeltaCost/DeltaEffect-ZValue*ABS(DeltaCost/DeltaEffect)*SQRT(SeCost^2/DeltaCost^2+SeEffect^2/DeltaEffect^2-2*CovCostEffect/(DeltaCost*DeltaEffect)); =DeltaCost/DeltaEffect+ZValue*ABS(DeltaCost/DeltaEffect)*SQRT(SeCost^2/DeltaCost^2+SeEffect^2/DeltaEffect^2-2*CovCostEffect/(DeltaCost*DeltaEffect))

Assumptions

  • Effect difference well away from zero for the delta method

    The incremental effect is large relative to its standard error. When its sampling distribution has positive density at zero, the ratio estimator has no finite mean or variance in any finite sample, so the delta-method variance describes only the limiting distribution and the symmetric interval can be a poor guide in trials of realistic size.

  • Non-zero cost and effect differences in the relative variance form

    Both estimated differences are non-zero, because the relative form of the variance divides by each of them.

Worked examples

  • Delta-method interval for an ICER of £20,000 per QALY

    In the article's trial, a cost difference of £2,000 and a QALY gain of 0.10, with standard errors of £600 and 0.04 and no correlation, give an ICER of £20,000 per QALY with a standard error of £10,000. The approximate 95% interval runs from £400 to £39,600 per QALY.

    Delta_C = 2000; Delta_E = 0.10; SE_C = 600; SE_E = 0.04; Cov_CE = 0; z = 1.96; R = 20000; SE_R = 10000; R_L = 400; R_U = 39600

  • Delta-method interval with a QALY gain that is not significant

    With the QALY standard error raised to 0.06 the effect difference is no longer significant at the 5% level, yet the delta method still returns a bounded interval of about -£6,296 to £46,296 per QALY. Fieller's method gives an unbounded set for the same data.

    Delta_C = 2000; Delta_E = 0.10; SE_C = 600; SE_E = 0.06; Cov_CE = 0; z = 1.96; R = 20000; SE_R = 13416.41; R_L = -6296.16; R_U = 46296.16

Common errors

  • Trusting the delta-method interval when the effect difference is not significant

    With a QALY standard error of 0.06 the delta method returns about -£6,296 to £46,296 per QALY, which looks informative, while Fieller's confidence set is every value below about -£110,622 together with every value above about £6,177. The data cannot place an upper limit on the ICER, and the symmetric interval hides that.

  • Reading a negative delta-method limit as a cost saving

    A negative limit such as -£6,296 per QALY does not identify a cheaper and more effective option. A negative ICER arises both for a cheaper, more effective option and for a dearer, less effective one, in opposite quadrants of the cost-effectiveness plane.

  • Taking the delta-method upper limit as the plausible maximum ICER

    In the article's trial the delta method puts the upper limit at £39,600 per QALY whereas Fieller's interval reaches about £96,786, because a much smaller QALY gain remains plausible and would push the ratio up sharply. Polsky and colleagues found that the Taylor series method underestimated the upper limit.

Sources

  • Simulation comparison of Taylor series, Fieller, bootstrap and box intervals

    Polsky D, Glick HA, Willke R, Schulman K. Confidence intervals for cost-effectiveness ratios: a comparison of four methods. Health Economics. 1997;6(3):243-252. Abstract: in a Monte Carlo comparison of the box, Taylor series, non-parametric bootstrap and Fieller methods, the Taylor series method gave intervals that underestimated the upper limit, and the bootstrap and Fieller intervals had miscoverage closer to the nominal level.

    View source →

Canonical Identity

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