MOVER-R confidence intervals for ratios and products of two independently estimated quantities

Abstract

The purpose of this note is to show how the MOVER algorithm for a ratio of two independently estimated quantities, recently published by Donner and Zou,¹ requires modification to cope with the general case in which the lower and upper confidence limits for the numerator can take either sign.

The MOVER² (Method Of Variance Estimates Recovery) algorithm as originally formulated relates to a difference (or sum) of two independently estimated parameters. Suppose that for i = 1 and 2, θ_i has maximum likelihood estimate ${\overset{\land}{θ}}_{i}$ and 1 − α confidence limits L_i and U_i. Then the MOVER interval for the difference θ₁ − θ₂, called MOVER-D is delimited by

L = \overset{\land}{θ} 1 - \overset{\land}{θ} 2 - \sqrt{(\overset{\land}{θ} 1 - L 1) 2 + (U 2 - \overset{\land}{θ} 2) 2}, U = \overset{\land}{θ} 1 - \overset{\land}{θ} 2 + \sqrt{(U 1 - \overset{\land}{θ} 1) 2 + (\overset{\land}{θ} 2 - L 2) 2}

The MOVER algorithm does not explicitly take into account how the limits (L_i, U_i) were calculated, but obviously for validity we require that for each parameter,

L_{i} \leq {\overset{\land}{θ}}_{i} \leq U_{i}

. It has been used extensively to derive intervals for various quantities, notably a difference of independent proportions.³ It is particularly useful for quantities derived from proportions as it preserves boundary-respecting properties. Donner and Zou^4,5 provided a theoretical justification. L_i and U_i are used to derive local variance estimates for θ_i, separately for θ_i ranging from L_i to

{\overset{\land}{θ}}_{i}

and from

{\overset{\land}{θ}}_{i}

to U_i. For the lower tail of the interval for θ_i, the variance estimate recovered from L_i is simply (

{\overset{\land}{θ}}_{i}

− L_i)²/z². Similarly, the variance estimate appropriate for the upper tail is (U_i −

{\overset{\land}{θ}}_{i}

)²/z². These recovered local variance estimates are then used to construct lower and upper limits for θ₁ − θ₂ in an obvious manner. Obviously these algorithms require great care with regard to signs and which limit is which.

MOVER may be extended to ratios and products by log transformation, but this process fails when it would involve taking the log of zero or a negative number. A purpose-built CI for the ratio R = θ₁/θ₂ was formulated by Zou and co-workers.^1,6–8 We refer to this as MOVER-R (MOVER for a ratio)⁹ – a distinctive label is needed because the formulae bear little resemblance to the original ‘square-and-add’ formulae above. MOVER-R was conceived as a generalisation of Fieller’s theorem, using local variance estimates reconstructed as described above. Earlier, Dube et al.¹⁰ gave equivalent formulae, specifically designed for asymmetrical intervals for θ_i, i = 1, 2, in a context where θ₁ and θ₂ are necessarily strictly positive. Consider f(r) = θ₁ − rθ₂ for arbitrary r ∈ (−∞, ∞), where θ₁ and θ₂ are estimated independently. Equating the MOVER limits for f(r) to 0 leads to quadratic equations in r. Choosing appropriate roots of these quadratics leads to the required interval (R_L, R_U) for R. For wider applicability, we let θ₁ take either sign, but always require θ₂ to be positive.

Sometimes, including in the context of the GRADE¹¹ (Grading of Recommendations Assessment, Development and Evaluation) system, an absolute risk difference (RD) is derived from independent estimates of relative risk (RR) (often from a meta-analysis) and baseline risk (BR): RD = BR × (RR − 1). The interval for the RD should reflect the uncertainty of both the BR and the RR, but an interval derived in the obvious manner from lower and upper limits for the two parameters is much too wide. Because the RD is derived as a product here, we identify θ₁ = RR − 1 and θ₂ = 1/BR, then R = θ₁/θ₂ represents the RD. Newcombe and Bender¹² give illustrative examples, both with RR > 1 and RR < 1. The latter is the usual case of a beneficial intervention.

The published MOVER-R algorithm works directly for the case of a statistically significant increase in risk. Arzola and Wieczorek¹³ evaluated the use of low-dose bupivacaine (≤8 mg) in spinal anaesthesia for elective caesarean section. Whilst use of a low dose instead of the conventional dose (>8 mg) may help prevent hypotension resulting from spinal anaesthesia, it may compromise anaesthetic efficacy. In a meta-analysis, the need for analgesic supplementation during surgery was higher (RR = 3.76, 95% CI 2.38 to 5.92) in women receiving the low dose compared to the conventional dose. The BR of needing analgesic supplementation during surgery is 175/1610 = 0.109 (95% CI 0.094 to 0.125).¹⁴ The resulting RD is 0.109 × 2.76 = 0.301. MOVER-R leads to the interval (0.149, 0.546): using low-dose bupivacaine leads to the need for analgesic supplementation in an additional 30% of women compared to the conventional dose, with 95% CI 15 to 55%.

Figure 1 shows how the limits for the RD are derived from limits for the RR and BR. Here, ${\overset{\land}{θ}}_{1}$ = RR – 1 = 2.76 (95% CI 1.38 to 4.92) and ${\overset{\land}{θ}}_{2}$ = 1/BR = 9.20 (95% CI 8.01 to 10.59). Upper and lower 95% MOVER-D confidence limits for f(r) = θ₁ − rθ₂ are plotted against r across the valid range 0 < r < 1. For any value of r, the plausible range of values for f can be read off the graph. For example, if r = 0.2, the 95% interval for f is (−0.49, 3.09). If r = 0.1, the 95% interval for f is (0.45, 4.00). However, when r = R = θ₁/θ₂, f(r) = θ₁ − rθ₂ ≡ 0. The value r = 0.2 is plausible for the data, because the 95% interval for f includes 0; the value r = 0.1 is not plausible for the data, because the 95% interval excludes 0. Consequently, the range of values for r for which the interval for f includes zero, 0.149 ≤ r < 0.546, may be regarded as a confidence interval for the absolute RD, R.

Figure 1.

Derivation of the confidence interval for the absolute risk difference calculated from the relative risk (assumed >1 here) and baseline risk. Source data from Arzola and Wieczorek.¹³

The calculations for the example above use formulae for the lower and upper limits published by Donner and Zou¹:

r 1 = \frac{\overset{\land}{θ} 1 \overset{\land}{θ} 2 - \sqrt{(\overset{\land}{θ} 1 \overset{\land}{θ} 2) 2 - L 1 U 2 (2 \overset{\land}{θ} 1 - L 1) (2 \overset{\land}{θ} 2 - U 2)}}{U 2 (2 \overset{\land}{θ} 2 - U 2)}, r 2 = \frac{\overset{\land}{θ} 1 \overset{\land}{θ} 2 + \sqrt{(\overset{\land}{θ} 1 \overset{\land}{θ} 2) 2 - U 1 L 2 (2 \overset{\land}{θ} 1 - U 1) (2 \overset{\land}{θ} 2 - L 2)}}{L 2 (2 \overset{\land}{θ} 2 - L 2)}

However, if θ₁ can take either sign, these formulae can produce incorrect limits, which can sometimes be seriously misleading. For r > 0, the relevant roots are r₁ and r₂ as above. For r < 0, the relevant roots are

r_{3} = \frac{{\overset{\land}{θ}}_{1} {\overset{\land}{θ}}_{2} - \sqrt{({\overset{\land}{θ}}_{1} {\overset{\land}{θ}}_{2}) 2 - L_{1} L_{2} (2 {\overset{\land}{θ}}_{1} - L_{1}) (2 {\overset{\land}{θ}}_{2} - L_{2})}}{L_{2} (2 {\overset{\land}{θ}}_{2} - L_{2})}, r_{4} = \frac{{\overset{\land}{θ}}_{1} {\overset{\land}{θ}}_{2} + \sqrt{({\overset{\land}{θ}}_{1} {\overset{\land}{θ}}_{2}) 2 - U_{1} U_{2} (2 {\overset{\land}{θ}}_{1} - U_{1}) (2 {\overset{\land}{θ}}_{2} - U_{2})}}{U_{2} (2 {\overset{\land}{θ}}_{2} - U_{2})}

However, in the event that

U_{2} = 2 {\overset{\land}{θ}}_{2}

, the respective roots need to be replaced by

r_{1} = L_{1} (1 - L_{1} / (2 {\overset{\land}{θ}}_{1})) / {\overset{\land}{θ}}_{2} and r_{4} = U_{1} (1 - U_{1} / (2 {\overset{\land}{θ}}_{1})) / {\overset{\land}{θ}}_{2} .

The above choices are expressed in terms of the sign of r. In the application, the choices depend on the signs of the two limits for θ₁ (not on the sign of

{\overset{\land}{θ}}_{1}

). The lower limit R_L is r₁ if L₁ > 0, r₃ if L₁ ≤ 0. Dually, the upper limit R_U is r₂ if U₁ ≥ 0, r₄ if U₁ < 0. So we choose r₁ or r₃ according to sgn(L₁) and r₂ or r₄ according to sgn(U₁) after substituting for r₁ and r₄ if

U_{2} = 2 {\overset{\land}{θ}}_{2}

. This algorithm yields the correct limits in all cases.

Table 1 gives some examples of the MOVER-R calculations, based on the scenario of deriving an absolute RD from a baseline rate and a RR. As above, θ₁ represents RR − 1 and θ₂ represents 1/BR. Results are shown for RR ranging from 1.3 to 0.7, with confidence limits 0.8RR and 1.25RR, including cases with L₁ = 0 or U₁ = 0, in line with the approximate log scale symmetry of intervals for the RR. The assumed BR is 0.04 (95% CI 0.016 to 0.1) leading to

{\overset{\land}{θ}}_{2}

= 25 (95% CI 10 to 62.5). We also show results for the special case U₂ = 50 =

2 {\overset{\land}{θ}}_{2}

Table 1.

Examples of calculation of MOVER-R intervals for the absolute risk difference R = θ₁/θ₂ derived from the baseline rate and the relative risk.

${\overset{\land}{θ}}_{1}$	L₁	U₁	${\overset{\land}{θ}}_{1} / {\overset{\land}{θ}}_{2}$	With U₂ = 62.5		With U₂ = 50 = 2θ₂
${\overset{\land}{θ}}_{1}$	L₁	U₁	${\overset{\land}{θ}}_{1} / {\overset{\land}{θ}}_{2}$	R_L	R_U	R_L	R_U
0.30	0.04	0.625	0.012	0.00139	0.03851	0.00149	0.03851
0.25	0	0.5625	0.010	0.00000	0.03385	0.00000	0.03385
0.10	−0.12	0.375	0.004	−0.00537	0.02050	−0.00537	0.02050
0	−0.20	0.25	0	−0.01000	0.01250	−0.01000	0.01250
−0.10	−0.28	0.125	−0.004	−0.01600	0.00561	−0.01600	0.00561
−0.20	−0.36	0	−0.008	−0.02347	0.00000	−0.02347	0.00000
−0.30	−0.44	−0.125	−0.012	−0.03200	−0.00337	−0.03200	−0.00396

RR: relative risk; BR: baseline risk. Here, θ₁ = RR − 1 and θ₂ = 1/BR = 25.

An MS Excel spreadsheet MOVER-R.xls to calculate the MOVER-R limits is freely downloadable from http://medicine.cf.ac.uk/primary-care-public-health/resources/. A derived spreadsheet RD from BR and RR.xls uses MOVER-R to calculate limits for the RD from those of the BR and the RR.

Coverage of a MOVER-R interval for R = θ₁/θ₂ can only be as good as the coverage of the intervals used for θ₁ and θ₂. Fagerland and Newcombe¹⁵ showed that the performance of MOVER-R intervals for ratios of proportions derived from Wilson¹⁴ intervals is comparable to that of the score method¹⁶ which requires iteration. This finding provides some evidence that MOVER-R, like MOVER-D, is likely to propagate favourable coverage properties.

A more general approach to combining two or more parameters, Propagating Imprecision (PropImp) is described by Newcombe.^9[chap. ^13],17 If an estimated odds ratio is available instead of a RR, the corresponding formula is RD = BR – 1/[1 + (1/BR – 1)/OR]. However, in this formula, the two parameters, the BR and OR are interlocked in such a way that the MOVER algorithms are inapplicable and only PropImp gives a solution.

Footnotes

Acknowledgements

I am deeply indebted to Ralf Bender, Head of Department of Medical Biometry, Institute for Quality and Efficiency in Health Care (IQWiG), Cologne, and Professor of Medical Statistics and Epidemiology, Faculty of Medicine, University of Cologne, for drawing my attention to the evidence-based healthcare papers considered in this article, and for many helpful comments on this article and accompanying spreadsheet.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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