Title Incremental Cost-Effectiveness (ICE) Statistical Inference from Two Unbiased Samples
Author Bob Obenchain <[email protected]>
Maintainer Bob Obenchain <[email protected]>
Description Given two unbiased samples of patient level data on cost and effectiveness
for a pair of treatments, make head-to-head treatment comparisons by (i) generating thebivariate bootstrap resampling distribution of ICE uncertainty for a specified value ofthe shadow price of health, lambda, (ii) form the wedge-shaped ICE confidence region withspecified confidence fraction within [0.50, 0.99] that is equivariant with respect tochanges in lambda, (iii) color the bootstrap outcomes within the above confidence wedgewith economic preferences from an ICE map with specified values of lambda, beta and gammaparameters, (iv) display VAGR and ALICE acceptability curves, and (v) illustrate variationin ICE preferences by displaying potentially non-linear indifference(iso-preference) curvesfrom an ICE map with specified values of lambda, beta and gamma or eta parameters.
ICEinfer-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dpunc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dpwdg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
dulxparx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
fluoxpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
fluoxtca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ICEalice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ICEcolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ICEepmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ICEscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ICEuncrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ICEwedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot.ICEcolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot.ICEepmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot.ICEuncrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . print.ICEuncrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ICE Statistical Inference and Economic Preference Variation
Functions in the ICE Statistical Inference package make head-to-head comparisons between patientsin two treatment cohorts (assumed to be unbiased samples) in two distinct dimensions, cost andeffectiveness.
Bootstrap resampling methods quantify the endogenous Distribution of ICE Uncertainty and de-fine Wedge-Shaped Statistical Confidence Regions equivariant relative to exogenous choice for thenumerical Shadow Price of Health, lambda.
Preference maps with (linear or nonlinear) indiference curves can be viewed or superimposedupon endogenous confidence wedges to illustrate that considerable additional, potentially self-contradictory Economic Preference Uncertainty results from deliberately varying lambda.
GNU GENERAL PUBLIC LICENSE, Version 2, June 1991
Statistical inference using functions from the ICEinfer package usually start with (possibly multi-ple) invocations of ICEscale() to help determine a reasonable value for the Shadow Price of Health,lambda. This is invariably followed by a single call to ICEuncrt to generate the Bootstrap Distri-bution of ICE Uncertainty corresponding to the chosen value of lambda. However, the print() andplot() functions for objects of type ICEuncrt do have optional arguments, lfact and swu, to help theuser quantify and visualize the consequences of changing lambda and switching between cost andeffe units.
Next, a single call to ICEwedge() yields the equivariant, wedge-shaped region of specified statisticalconfidence within [.50, .99] .by computing ICE Angle Order Statistics around a circle centered atthe ICE Origin, (DeltaEffe, DeltaCost) = (0, 0).
Researchers wishing to view alternative ICE Acceptability Curves would then envoke ICEalice().
Finally, multiple calls to ICEcolor for different values of lambda and/or different forms of (linearor nonlinear) ICE Preference Maps are typically used to illustrate the considerable additional Eco-nomic Preference Uncertainty that can be introduced. This Economic Uncertainty is superimposedon top of the inherent Statistical Uncertainty contained in unbiased, patient level data on the relativecost and effectiveness of two treatments for the same disease / condition.
Black WC. The CE plane: a graphic representation of cost-effectiveness. Med Decis Making 1990;10: 212-214.
Laupacis A, Feeny D, Detsky AS, Tugwell PX. How attractive does a new technology have to be towarrant adoption and utilization? Tentative guidelines for using clinical and economic evaluations. Can Med Assoc J 1992; 146(4): 473-81.
Stinnett AA, Mullahy J. Net health benefits: a new framework for the analysis of uncertainty incost-effectiveness analysis. Medical Decision Making, Special Issue on Pharmacoeconomics 1998;18: S68-S80.
O’Brien B, Gersten K, Willan A, Faulkner L. Is there a kink in consumers’ threshold value forcost-effectiveness in health care? Health Econ 2002; 11: 175-180.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
Output list object of class ICEuncrt for the High Uncertainty numeri-cal example in the ICEinfer package, data(dulxparx).
dpunc is the output list object of class ICEuncrt resulting from the following time consuming com-putation: dpunc <- ICEuncrt(dulxparx, dulx, idb, ru, lambda=0.26)
Output list object of class ICEuncrt.
df Saved value of the name of the data.frame input to ICEuncrt.
lambda Saved positive value of lambda input to ICEuncrt.
unit Saved value of unit, cost or effe, input to ICEuncrt.
R Saved integer value for number of bootstrap replications input to ICEuncrt.
trtm Saved name of the treatment indicator within the df data.frame.
xeffe Saved name of the treatment effectiveness variable within the df data.frame.
ycost Saved name of the treatment cost variable within the df data.frame.
effcst Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
t1 Observed value of (DeltaEffe, DeltaCost) when each patient is included exactly once.
t R x 2 matrix of values of (DeltaEffe, DeltaCost) computed from bootstrap resamples.
seed Saved value of the seed used to start pseudo random number generation.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# Intermediate ICEinfer Output List for the dulxparx dataset.
Output list object of class ICEwedge for the High Uncertainty exam-ple, data(dulxparx)
dpwdg is the output list object of class ICEwdg resulting from the following time consuming com-putation: dpwdg <- ICEwedge(dpunc)
Output list object of class ICEwedge.
ICEinp Name of the ICEuncrt object input to ICEwedge().
lambda Positive value of lfact * ICEu\$lambda
lfact Positive Multiplier for the ICEu\$lambda value input to ICEwedge().
unit Saved value of unit, cost or effe, input to ICEuncrt.
conf Statistical Confidence Level within [0.50, 0.99] input to ICEwedge.
R Saved integer value for number of bootstrap replications input to ICEuncrt.
axys R x 4 data.frame with ICE Angle in column 1, bootstrap resampled values of (DeltaEffe,
DeltaCost) in columns 2 and 3, and the binary flag with 0 => outcome outSide the ConfidenceWedge and 1 => outcome inSide the Confidence Wedge in column 4.
t1 Observed value of (DeltaEffe, DeltaCost) when each patient is sampled exactly once.
ia1 The ICE Angle corresponding to the Objerved ICE Ratio.
center The largest value of j such that axys[j, 1] < ia1 <= axys[j+1, 1].
jlo Number of the ICE Angle Order Statistic defining the Clockwise or lower ICE Ray boundary
kup Number of the ICE Angle Order Statistic defining the Counter-Clockwise or upper ICE Ray
subangle Subtended Polar ICE Angle between Order Statistics jlo and kup.
xmax Alias plots of ICEwedge have horizontal range [-xmax, +xmax].
ymax Alias plots of ICEwedge have vertical range [-ymax, +ymax].
ab ICE angle computation perspective of alibi or alias.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# Intermediate ICEinfer Output List for the dulxparx dataset.
Data for the High Uncertainty numerical example of Obenchain et al. (2005)
The data are from two arms of a double-blind clinical trial in which 91 patients were randomizedto the SNRI duloxetine 80 mg/d (40 mg BID) and 87 patients were randomized to the SSRI parox-etine 20 mg/d for treatment of major depressive disorder (MDD). Missing-data- imputation andsensitivity-analyses were needed to make meaningful cost-effectiveness comparisons in this study.
A data frame of 3 variables on 178 patients; no NAs.
idb This measure of overall effectiveness is integrated decrease in HAMD-17 score from baseline
to endpoint, Hamilton (1967). This is a (signed) area-under-the-curve measure with largervalues more favorable. Missing values were imputed via the MMRM models reported inGoldstein et al. (2004).
ru Patient self-reported health-care resource utilization above and beyond that provided within
study protocol was collected using the Resource Utilization Survey, Copley-Merriman et al. (1992), with published 1998 dollars-per-unit costs, Schoenbaum et al. (2001), rounded to thenearest 50 dollars. Dollars/week were then calculated by multiplying (total accumulated cost)for a patient by 7 and dividing by the (total days of cost accumulation) for that patient. Forpatients who discontinued early, this is Average-Value-Carried-Forward imputation.
dulx Treatment indicator variable. dulx = 1 implies receipt of duloxetine 80 mg/d (40 mg BID).
dulx = 0 implies receipt of paroxetine 20 mg/d.
Copley-Merriman C, Egbuonu-Davis L, Kotsanos JG, Conforti P, Franson T, Gordon G. Clinicaleconomics: a method for prospective health resource data collection. Pharmacoeconomics 1992;1(5): 370–376.
Goldstein DJ, Lu Y, Detke MJ, Wiltse C, Mallincrodt C, Demitrack MA. Duloxetine in the treatmentof depression - A double-blind, placebo-controlled comparison with paroxetine. J Clin Psychophar-macol 2004; 24: 389–399.
Hamilton M. Development of a rating scale for primary depressive illness. British Journal of Socialand Clinical Psychology 1967; 6: 278–296.
Obenchain RL, Robinson RL, Swindle RW. Cost-effectiveness inferences from bootstrap quadrantconfidence levels: three degrees of dominance. J Biopharm Stat 2005; 15(3): 419–436.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
Schoenbaum M, Unutzer J, Sherbourne C, Duan N, Rubenstein LV, Miranda J, Meredith LS, CarneyMF, Wells K. Cost-effectiveness of practice-initiated quality improvement for depression: resultsof a randomized controlled trial. JAMA 2001; 286(11): 1325–1330.
# Demo of ICEinfer functionality on the dulxparx dataset.
Data from a double-blind clinical trial comparing fluoxetine plus pin-dolol with fluoxetine alone
These data are from a Spanish double-blind clinical trial in which 55 patients were randomized tofluoxetine (an SSRI) plus pindolol (a Beta Blocker) and 56 patients were randomized to fluoxetineplus placebo for treatment of major depressive disorder (MDD), Sacristan et al. (2000).
A data frame of 3 variables on 111 patients; no NAs.
respond Patients are considered to have responded to treatment when a 50% or greater decrease
in HAMD-17 total score occurred between baseline and end-point (at day 42), with no morethan 10% additional variation between intermediate visits.
cost Resource utilization was prospectively collected alongside the clinical trial. Patients and care-
givers were interviewed by the researcher concerning all resources consumed during the studyperiod. Resources dictated by the protocol were not counted. Costs are expressed in Pesetas(Pts.) at 1996 prices (1 Dollar = 145 Pts.) Observed differences in average direct medicalcosts were mainly due to hospitalizations within the FlxPin = 0 group.
flxpin Treatment indicator variable. FlxPin = 1 implies receipt of fluoxetine 20 mg/day plus pin-
dolol 7.5 mg/day (2.5 mg tid). FlxPin = 0 implies receipt of fluoxetine 20 mg/day plus placebo(tid).
Since both samples are rather small (55 and 56 patients) here and the Effectiveness variable, re-spond, is binary, this example illustrates how the Law of Large Numbers can fail to apply to ICEinferences. Specifically, the bootstrap distribution of sample differences between AVERAGES ap-pears to be quite different from bivariate normal in three ways: (i) The Bootstrap Distribution ofICE Uncertainty appears to consist of vertical stripes because the horizontal variable is discrete herewhile the vertical variable is continuous. (ii) The Bootstrap Distribution of cost differences appears
to end somewhat abruptly near the horizontal axis at DeltaCost = 0, rather than have a long upwardstail like its downwards tail. (iii) The equal density contours of the bivariate Bootstrap Distributionappear to NOT be elliptical. This third point can be dramaticaly illustrated by computing the OwenEmpirical Likelihood contour that passes through the origin of the ICE plane.
Hamilton M. Development of a rating scale for primary depressive illness. British Journal of Socialand Clinical Psychology 1967; 6: 278–296.
Sacristan JA, Obenchain RL. Reporting cost-effectiveness analyses with confidence. JAMA 1997;277: 375.
Obenchain RL, Sacristan JA. In reply to: The negative side of cost-effectiveness ratios. JAMA 1997;277: 1931–1933.
Sacristan JA, Gilaberte I, Boto B, Buesching DP, Obenchain RL, Demitrack M, Perez Sola V, Al-varez E, and Artigas F. Cost-effectiveness of fluoxetine plus pindolol in patients with major de-pressive disorder: results from a randomized, double blind clinical trial. Int Clin Psychopharmacol2000; 15: 107–113.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
Owen AB. Empirical Likelihood New York: Chapman and Hall/CRC. 2001.
# Demo of ICEinfer functionality on the fluoxpin dataset.
Cost-Effectiveness data for 1242 MDD patients from Marketscan(SM)claims database
In 1990-1992, the Marketscan(SM) database included medical and pharmacy claims for approxi-mately 700,000 individuals whose health insurance was provided by large corporations throughoutthe United States. Outcomes for 1242 patients treated with either fluoxetine (SSRI) or with a TCA /HCA for major depressive disorder (MDD) were discussed in Croghan et al. (1996) and Obenchainet al. (1997). All 1242 patients were continuously enrolled for at least 4 months prior to their initialantidepressant prescription and for the following 12 months.
A data frame of 3 variables on 1242 patients; no NAs.
stable stable = 1 indicates that the patient remained on his/her initial antidepressant medication for
cost cost is the Marketscan(SM) 12 month total annual charge for a patient.
fluox Treatment indicator variable; fluox = 1 indicates receipt of fluoxetine 20 mg/d by 799 pa-
tients. fluox = 0 implies receipt of either a tricyclic (TCA) or a heterocyclic (HCA) by 443patients.
This dataset contains measures of cost and efffectiveness for 799 patients treated with fluoxetine(a Selective Serotonin Reuptake Inhibitor or SSRI), 104 patients treated with a first generationtricyclic, TCA (amitriptyline or imipramine), 250 patients treated with a second generation TCA(desipramine or nortriptyline), and 89 patients treated with trazodone (a heterocyclic, HCA).
Croghan TW, Lair TJ, Engelhart L, et al. Effect of antidepressant therapy on health care utilizationand costs in primary care. Working paper, Eli Lilly and Company, 1996. (Presented in part at theAssociation for Health Services Research meeting, Chicago, June 9, 1995.)
Obenchain RL, Melfi CA, Croghan TW, Buesching DP. Bootstrap analyses of cost-effectiveness inantidepressant pharmacotherapy. PharmacoEconomics 1997; 17: 1200–1206.
Obenchain RL. ICEplane: a windows application for incremental cost-effectiveness (ICE) statis-tical inference. Copyright(c) Pharmaceutical Research and Manufacturers of America (PhRMA.)http://members.iquest.net/~softrx/ Revised 1997–2007.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
Sclar DA, Robinson LM, Skaer TL, Legg RF, Nemec NL, Galin RS, Hugher TE, Buesching DP. Antidepressant pharmacotherapy: economic outcomes in a health maintenance organization. ClinTher 1994; 16: 715–730.
# Demo of ICEinfer functionality on the fluoxtca dataset.
Functions to compute and display ICE Acceptability Curves
ICEalice() computes statistics for the VAGR Acceptability Curve and for the Buckingham ALICEcurve. Plots for the resulting ICEalice object are of two types: [1] a VAGR curve where the hor-izontal axis is the Willingness to Pay (WTP) ICE Ratio, and [2] a monotone ALICE curve wherethe horizontal axis is the Absolute Value of the ICE Polar Angle, which varies from +45 degreesto +135 degrees. Printing an ICEalice object yields a 13 x 5 table (matrix) of numerical values forAbsolute ICEangle, WTP, VAGR Acceptability, WTA and ALICE acceptability, respectively.
The VAGR Acceptability Curve displays the fraction of outcomes within the Bootstrap distributionof ICE Uncertainty that lie below and/or to the right of a rotating straight line through the origin ofthe ICE plane. This straight line starts out horizontal, representing lambda = WTP = 0, and rotatescounter-clockwise until it becomes vertical, representing lambda = WTP = +Inf.
The Buckingham ALICE Curve assumes that lambra is held fixed. It displays the fraction of out-comes within the Bootstrap distribution of ICE Uncertainty that lie on or between a pair of rotat-ing ICE rays (eminating from the ICE origin) with slopes representing KINKed values of WTP< WTA that always satisfy Obenchain’s LINK function, lambda = sqrt(WTP*WTA), with lambdaheld fixed. The right-hand ray for WTP starts out horizontal and pointing to the right, then rotatescounter-clockwise until it is vertical, as in a VAGR curve. The left-hand ray for WTA starts outvertical and pointing downwards, then rotates clockwise until it is horizontal. Since lambda is heldfixed, the slopes of the rotating rays corresponding to decreasing WTA as WTP increases. The start-ing point of an ALICE curve at an Absolute ICE Angle of 45 degrees always represents the fractionof outcomes in the Bootstrap Distribution of ICE Uncertainty for which the new treatment is bothless costly AND more effective than the std treatment. The ending point of an ALICE curve at anAbsolute ICE Angle of 135 degrees always represents the fraction of outcomes in the BootstrapDistribution of ICE Uncertainty for which the new treatment is either less costly OR more effectivethan the std treatment. The middle point of an ALICE curve at an Absolute ICE Angle of 90 degreesrepresents the fraction of outcomes in the Bootstrap Distribution of ICE Uncertainty falling belowand/or to the right of the straight line through the ICE origin of slope lambda = WTP = WTA.
Objects of class ICEalice contain the following output list:
Positive numerical value for the Shadow Price of Health, lambda
Common unit of measurement of either cost or effe.
R x 1 Vector of Sorted ICE Angles. Default value of R = 25000.
13 x 5 Matrix of Absolute ICEangle, WTP, VAGR Acceptability, WTA and AL-ICE statistics.
Van Hout BA, Al MJ, Gordon GS, Rutten FFH. Costs, effects and C/E ratios alongside a clinicaltrial. (VAGR curve) Health Economics 1994; 3: 309-319.
Buckingham K. Personal communications including a draft manuscript entitled: Representing thecumulative probability of Acceptability Levels In Cost Effectiveness. (ALICE curve) 2003.
Fenwick E, O’Brien BJ, Briggs AH. Cost-effectiveness acceptability curves - facts, fallacies andfrequently asked questions. Health Economics 2004; 13: 405-415.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# Read in previously computed ICEwedge output list.
# Display VAGR and ALICE acceptability curves.
Compute Preference Colors for Outcomes in a Bootstrap ICE Scatterwithin a Confidence Wedge
Assuming ICEw is an object of class ICEwedge, ICEcolor uses the value of lambda given by lfact* (ICEw item lambda) and the ICE Preference Map with parameters beta and gamma to computethe Economic Preference value for only the points in a Bootstrap Distribution of ICE Uncertaintythat also happen to fall within the ICE confidence wedge. When the overall level of confidence(statistical size of the wedge) is held fixed, the points to be colored are always the very same pointsfor all choices of lambda. However, the numerical value of preference (and thus the color) for eachsuch point as well as potential overall asymmetry in the resulting ICE map do depend greatly uponchoice of lambda.
ICEcolor(ICEw, lfact = 1, beta = 1, gamma = 3+2*sqrt(2))
Strictly positive multiplier for ICEw item lambda.
Strictly positive Returns-to-Scale power parameter for the ICE Preference Map. beta = 1 implies linear (constant) Returns to Scale. beta > 0 and < 1 impliesdiminishing Returns to Scale. beta > 1 implies increasing Returns to Scale.
Strictly positive Directional power parameter. The smallest reasonable valuefor gamma is usually gamma = beta, which yields a (generalized) linear map. The largest reasonable value for gamma is usually gamma = beta*(3+2*sgrt(2)),which yields a map that satisfies the Cartesian Monotonicity Axiom and alsoadmits all possible finite values for WTP and WTA, i.e. all values greater thanor equal to 0 but less than +Inf.
Multiple calls to ICEcolor() are usually made for different lfact multipliers of the lambda itemwithin ICEw as well as different choices for the ICE Preference power parameters, beta and gamma. Calls to plot(x, alibi) for these alternative ICEcolor x-objects can be used to illustrate that exogenousEconomic Uncertainty can literally SWAMP the Statistical Uncertainty endogenous to patient leveldata on the relative cost and effectiveness of two treatments.
Object of class ICEcolor containing an output list with the following items:
Saved value of the name of the data.frame input to ICEcolor.
Saved positive value of lambda input to ICEcolor.
Saved value of unit, cost or effe, input to ICEcolor.
Saved integer value for number of bootstrap replications input to ICEcolor.
Saved name of the treatment indicator within the df data.frame.
Saved name of the treatment effectiveness variable within the df data.frame.
Saved name of the treatment cost variable within the df data.frame.
Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
Observed value of (DeltaEffe, DeltaCost) when each patient is sampled exactlyonce.
R x 2 matrix of values of (DeltaEffe, DeltaCost) computed from bootstrap re-samples.
Saved value of the seed used to start pseudo random number generation.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# Read in previously computed ICEwedge output list.
# Display preference coloring for the stored value of lambda.
# Display preference coloring when lambda is increased by a factor of 10.
Set Parameter Values defining ICE Economic Preference Maps
ICEepmap() and ICEomega() set numerical values for lambda (the full, fair shadow price of health)and for the two so-called power-parameters of a parametric ICE Preference Map. These functionsreturn a value, epm, that is an output list object of class ICEepmap for display using print(epm) orplot(epm, xygrid). The primary purpose of such plots is to allow the user to more easily visualizethe profound effects that changing numerical values for lambda, beta and either gamma or eta =gamma / beta can have on the iso-preference contours (level curves) of an ICE map.
From the statistical prospective championed here, lambda is little more than a nusiance parameter. For example, the wedge-shaped ICE confidence regions formed by ICEwedge() are equivariantunder changes in lambda. Unfortunately, the resulting economic preferences that can be visualizedusing ICEcolor() can change drastically with changes in lambda.
A stardardized ICE map results when the specified value of lambda is used to assure that the x effedifference and the y cost difference are both expressed in the same units (i.e. both in cost unitsor else both in effe units.) Unfortunately, the only way to assure display of this particular sort ofrescaling in ICE plane depictions is to use alibi = TRUE in plot.ICEuncrt(). Both plot.ICEwedge()and plot.ICEcolor() always use alias axis rescaling so that the equivariance property of the wedge-shaped ICE confidence region is depicted as if it were invariant under changes in lambda.
The easy way to visualize a standardized ICE map is to always use the default value of lambda = 1in ICEepmap() and ICEomega(). A standardized ICE map always has the following two character-istics: [i] it always assigns a zero overall preference to all (x, y) outcomes everywhere along the x= y ICE diagonal, and [ii] its iso-preference contours are always exactly symmetric about the x = -y(upper-left to lower-right) ICE diagonal.
ICEepmap(lambda = 1, beta = 1, gamma = 3+2*sqrt(2))
ICEomega(lambda = 1, beta = 1, eta = 3+2*sqrt(2))
Positive value for the fair, full-retail Shadow Price of Health.
Positive Returns-to-Scale Power parameter for the ICE Preference Map. beta =1 implies linear (constant) Returns-to-Scale. A beta > 0 and < 1 implies dimin-ishing Returns-to-Scale. A beta > 1 implies increasing Returns-to-Scale.
Positive Directional Power parameter for ICEepmap(). The smallest reasonablevalue for gamma is usually gamma = beta, which yields a (generalized) linearmap. The largest reasonable value for gamma is usually gamma = beta*(3+2*sgrt(2)),which yields a map that satisfies Cartesian Monotonicity and also yields WTPand WTA values within [0, +Inf).
Positive Power Parameter Ratio for ICEomega(). Generalized linear maps re-sult when eta = 1. The eta for the more realistic Nonlinear maps is greater thanone, but not greater than the ICE Omega limit of (3+2*sgrt(2)), which is ap-proximately 5.828. This upper limit on eta is required to assure that CartesianMonotonicity of preferences holds.
The ICEepmap() and ICEomega() functions specify numerical values for the Shadow Price ofHealth Parameter, lambda, for the Returns to Scale Power Parameter, beta, and for either the Direc-tional Power Parameter, gamma, or else the Power Parameter Ratio, eta = gamma / beta.
Object of class ICEepmap containing an output list with the following items:
Saved positive value of Shadow Price of Health, lambda, read by the print andplot methods for objects of class ICEepmap.
Saved Positive Returns-to-Scale Power parameter, beta, read by the print andplot methods for objects of class ICEepmap.
Saved Positive Directional Power parameter, gamma, read by the print and plotmethods for objects of class ICEepmap.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Obenchain RL. Incremental Cost-Effectiveness (ICE) Preference Maps. 2001 JSM Proceedings(Biopharmaceutical Section) on CD-ROM. (10 pages.) Alexandria, VA: American Statistical Asso-ciation. 2002.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
ICEscale() functions compute or print ICE Statistical Inference Sum-mary Statistics relative to choice for the numerical value of theShadow Price of Health, lambda
ICEscale() computes Summary Statistics for 2-sample, 2-variable inference where one variable is ameasure of effectiveness (higher values are better) and the other variable is a measure of cost (lowervalues are better). The 2 samples are of patients receiving only 1 of the 2 possible treatments. The treatment called new is the one with the higher numerical level for the specified treatmentindicator variable, while the treatment called std corresponds to the lower numerical level. Thepivotal statistic for inference is (DeltaEffe, DeltaCost), which are the head-to-head mean differencesfor new treatment minus std treatment. Each sample is assumed to provide unbiased estimates ofthe overall expected effectiveness and cost for that treatment.
ICEscale(df, trtm, xeffe, ycost, lambda = 1, unit = cost)
Required; Existing data.frame object containing the trtm, xeffe and ycost vari-ables.
Required; Name of the treatment indicator variable contained within the dfdata.frame that assumes one of only two different numerical values for eachpatient.
Required; Name of the treatment effectiveness variable within the df data.frame.
Required; Name of the treatment cost variable within the df data.frame.
Optional; lambda strictly positive value for the Shadow Price of Health.
Optional; unit character string containing either cost (default) or effe.
After an initial call with the default value of lambda = 1, multiple additional calls to ICEscale() withdifferent numerical values for lambda are usually made at the very beginning of analyses using otherfunctions from the ICEinfer package. For example, the statistical choice for lambda assures thatthe DeltaEffe and DeltaCost mean treatment differences (new minus std) will have approximatelyequal variability when expressed in either cost or effe units. The power of ten value of lambda thatis closest to the statistical value for lambda assures use of units that, except for the position of thedecimal point, are identical to the cost/effectiveness ratio implied by the scales in which data valuesare stored within the input data.frame.
Object of class ICEscale containing an output list with the following items:
Saved name of the treatment indicator within the input data.frame.
Saved name of the treatment effectiveness variable within the input data.frame.
Saved name of the treatment cost variable within the input data.frame.
Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
Value for the Shadow Price of Health, lambda, input to ICEscals().
Observed values of (DeltaEffe, DeltaCost) when each distinct patient is sampledexactly once.
Observed values for the standard deviations of (DeltaEffe, DeltaCost) when eachdistinct patient is sampled exactly once.
Statistical Shadow Price computed as s1[2]/s1[1] and rounded to digits = 3.
Power-of-Ten Shadow Price computed as 10\^(as.integer(log10(slam))).
Obenchain RL. Issues and algorithms in cost-effectiveness inference. Biopharmaceutical Reports1997; 5(2): 1-7. Washington, DC: American Statistical Association.
Obenchain RL. ICEplane: a windows application for incremental cost-effectiveness (ICE) statisticalinference. Copyright (c) Pharmaceutical Research and Manufacturers of America (PhRMA.)http://www.math.iupui.edu/~indyasa/bobodown.htm 1997–2007.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Compute Bootstrap Distribution of ICE Uncertainty for given ShadowPrice of Health, lambda
ICEuncrt() uses bootstrap resampling (with replacement) to compute the distribution of uncertaintyfor 2-sample, 2-variable statistical inference. The 2 variables must be measures of effectiveness(higher values are better) and cost (lower values are better). The 2 samples are of patients receivingonly 1 of the 2 possible treatments. The treatment called new is the one with the higher numericallevel for the specified treatment indicator variable, while the treatment called std corresponds to thelower numerical level. The pivotal statistic for inference is (DeltaEffe, DeltaCost), which are thehead-to-head mean differences for new treatment minus std treatment. Each sample is assumed toprovide unbiased estimates of the overall expected effectiveness and cost for that treatment.
ICEuncrt(df, trtm, xeffe, ycost, lambda = 1, unit = cost, R = 25000, seed = 0)
Required; Existing data.frame object containing the trtm, xeffe and ycost vari-ables.
Required; Name of the treatment indicator variable contained within the dfdata.frame that assumes one of only two different numerical values for eachpatient.
Required; Name of the treatment effectiveness variable within the df data.frame.
Required; Name of the treatment cost variable within the df data.frame.
Optional; lambda strictly positive value for the Shadow Price of Health.
Optional; unit character string containing either cost (default) or effe.
Optional; R positive integer value for the number of Bootstrap Replications de-sired. Minimum allowed value is 50; default value is 25000.
Optional; seed is an integer between 0 and 25000. A seed value of 0 causes arandom integer seed between 1 and 25000 to be generated. To reproduce resultsfrom a previous invocation of ICEuncrt(), use the seed value saved in its outputlist object.
A single call to ICEuncrt() is usually made for a particular value of the Shadow Price of Health,lambda. Alternative statistical choices for lambda can be suggested by making calls to ICEscale()with different values for lambda. Because the bootstrap distribution of ICE uncertainty is equivari-ant under changes in lambda, it is much faster to transform an existing bootstrap distribution than togenerate a new one for a different value of lambda. The print.ICEuncrt() and plot.ICEuncrt() func-tions thus have 2 special parameters, lfact and swa, that can change lambda and switch the unitsof measurement, respectively, without actually generating a new bootstrap distribution via a call toICEuncrt().
Object of class ICEuncrt containing an output list with the following items:
Saved value of the name of the data.frame input to ICEuncrt.
Saved positive value of lambda input to ICEuncrt.
Saved value of unit, cost or effe, input to ICEuncrt.
Saved integer value for number of bootstrap replications input to ICEuncrt.
Saved name of the treatment indicator within the df data.frame.
Saved name of the treatment effectiveness variable within the df data.frame.
Saved name of the treatment cost variable within the df data.frame.
Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
Observed value of (DeltaEffe, DeltaCost) when each patient is included exactlyonce.
R x 2 matrix of values of (DeltaEffe, DeltaCost) computed from bootstrap re-samples.
Saved value of the seed used to start pseudo random number generation.
Obenchain RL. ICEplane: a windows application for incremental cost-effectiveness (ICE) statisticalinference. Copyright (c) Pharmaceutical Research and Manufacturers of America (PhRMA.)http://members.iquest.net/~softrx/ 1997-2007.
Obenchain RL, Melfi CA, Croghan TW, Buesching DP. Bootstrap analyses of cost-effectiveness inantidepressant pharmacotherapy. PharmacoEconomics 1997; 17: 1200-1206.
Obenchain RL. Resampling and multiplicity in cost-effectiveness inference. Journal of Biophar-maceutical Statistics 1999; 9(4): 563-582.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# Generating a bootstrap ICE uncertainty distribution is time consuming.
dpunc <- ICEuncrt(dulxparx, dulx, idb, ru, lambda=0.26)
# Transforming an existing bootstrap ICE uncertainty distribution is fast.
Equivariant Wedge-Shaped ICE Region with Confidence Level from0.50 to 0.99
ICEwedge() uses the Bootstrap Distribution of ICE Uncertainty generated by ICEuncrt() to cal-culate and sort ICE Angle Order Statistics around a circle. ICEwedge() then counts outwards thesame number of ICE Angle Order Statistics, floor(R*conf/2), both Counter-Clockwise and Clock-wise from the so-called center Order Statistic (the one nearest to the Observed ICE Ratio) to definea pair of ICE Ray Endpoints at ICE Angle Order Statistics (reported as numbers jlo and kup, re-spectively) that subtend an ICE Polar Angle reported as being of subangle in degrees.
Output list object of class ICEuncrt.
Either a strictly positive multiplicative factor for ICEu item lambda or else 0 tocause ICEwedge to compute the positive lfact and lambda values which trans-forms the alibi display so that it has an alias interpretation.
Statistical Confidence Level within [0.50, 0.99].
The plot() of an object of class ICEwedge displays the Bootstrap Distribution of ICE Uncertaintywith a small, circular, colored dot (pch = 20). Outcomes outside the Wedge are displayed in black,while outcomes inside the Wedge are displayed in cyan. Upper and lower ICE Ray Limits aredisplayed as solid black lines, and the ICE Ray through the center ICE Angle Order Statistic isshown as a dashed black line.
An object of class ICEwedge with the following output list:
Name of the ICEuncrt object input to ICEwedge().
Positive value of lfact * ICEu item lambda
Positive Multiplier for the ICEu item lambda value input to ICEwedge().
Saved value of unit, cost or effe, input to ICEuncrt.
Statistical Confidence Level within [0.50, 0.99] input to ICEwedge.
Saved integer value for number of bootstrap replications input to ICEuncrt.
R x 4 data.frame with ICE Angle in column 1, bootstrap resampled values of(DeltaEffe, DeltaCost) in columns 2 and 3, and the binary flag with 0 => out-come outSide the Confidence Wedge and 1 => outcome inSide the ConfidenceWedge in column 4.
Observed value of (DeltaEffe, DeltaCost) when each patient is sampled exactlyonce.
The center ICE Angle closest to the Objerved ICE Ratio.
The largest value of j such that axys[j, 1] < ia1 <= axys[j+1, 1].
Number of the ICE Angle Order Statistic defining the Clockwise or lower ICERay boundary of the Confidence Wedge.
Number of the ICE Angle Order Statistic defining the Counter-Clockwise orupper ICE Ray boundary of the Confidence Wedge.
Subtended Polar ICE Angle between Order Statistics numbers jlo and kup.
Alias plots of ICEwedge have horizontal range [-xmax, +xmax].
Alias plots of ICEwedge have vertical range [-ymax, +ymax].
ICE angle computation perspective of alibi or alias.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Obenchain RL. Resampling and multiplicity in cost-effectiveness inference. Journal of Biophar-maceutical Statistics 1999; 9(4): 563-582.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Obenchain RL. ICEinR.pdf Vignette-like documentation for ICEinfer stored in the R library/ICEinfer/docfolder. 2009; 30 pages.
# ICEwedge() calculations are rather slow
# ICE Angle computations using the alibi axes with an alias interpretation
Add Economic Preference Colors to Bootstrap Uncertainty Scatterswithin a Confidence Wedge
Assuming x is an object of class ICEcolor, the default invocation of plot(x) recolors the default aliasdisplay of the points within the bootstrap distribution of ICE uncertainty that are within its statisticalconfidence wedge. An invocation of the form plot(x, alibi=TRUE) recolors the alibi display. Whenready, the user should click within this graphics window to display a Histogram of all the EconomicPreference values falling within the ICE Statistical Confidence Wedge.
Required; Output list object of class ICEcolor.
Optional; Logical value of TRUE or FALSE to control scaling of axes. alibi= FALSE produces the default alias graphic in which points in the bootstrapuncertainty scatter are held fixed in space, and changes in lambda change thescaling (tick marks) along either the horizontal axis of a cost unit display orelse along the vertical axis of an effe unit display. alibi = TRUE produces analibi graphic in which the scaling (and range) is the same along both axes, andchanges in lambda cause the points in the bootstrap uncertainty scatter to moveeither left or right in a cost unit display or else up or dowm in an effe unit display.
Optional; Argument(s) passed on to plot().
To illustrate the sensitivity of Economic Preferences to choice of lambda, multiple calls are usuallymade to ICEcolor() for different values of lambda as well as for different choices of the beta andgamma parameters that determine the shape of and spacing between the Indifference Curves of anICE Preference Map.
The plot() of an object of class ICEcolor displays the Bootstrap Distribution of ICE Uncertaintyusing small, circular, colored dots (pch = 20). Outcomes outside the Confidence Wedge are dis-played in black, while outcomes inside the Wedge are displayed in a rainbow of colors (within thered-tan-yellow-green range) that represent Economic Preferences.
Upper and lower ICE Ray Limits are again displayed as solid black lines, while the Straight Linethrough the ICE origin that represents lambda is shown as a dashed black line. In an Alias graphic,the slope of this dashed, black line will always be one; however, this dashed line usually does notappear to bisect the North-East and South-West ICE quadrants because DIFFERENT SCALINGSare being used along the horizontal and vertical axes. In an Alibi graphic where the scaling alongboth axes is the SAME, the slope of this dashed, black line will always be lambda; this dashed linewill thus not bisect the North-East and South-West ICE quadrants unless lambda = 1.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Obenchain RL. Incremental Cost-Effectiveness (ICE) Preference Maps. 2001 JSM Proceedings(Biopharmaceutical Section) on CD-ROM. (10 pages.) Alexandria, VA: American Statistical Asso-ciation. 2002.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Display Indifference Curves on a standardized ICE Economic Prefer-ence Map
Display plots of the Indifference Curves of an ICE Economic Preference Map using the contour-plot() and expand.grid() functions from the lattice R-package.
Output list object from either ICEepmap or ICEomega.
Either FALSE or a grid object for a lattice of (x, y) plotting positions.
Optional argument(s) passed on to contourplot().
If xygrid == FALSE, the default xygrid wiil be a 201 x 201 lattice of equally spaced plottingpositions covering the x=DeltaEffe and y=DeltaCost ranges [-10,+10]. This default is: x <- seq(-10, +10, length = 201); y <- x; xygrid <- expand.grid(x = x, y = y)
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
Obenchain RL. Incremental Cost-Effectiveness (ICE) Preference Maps. 2001 JSM Proceedings(Biopharmaceutical Section) on CD-ROM. (10 pages.) Alexandria, VA: American Statistical Asso-ciation. 2002.
Obenchain RL. ICE Preference Maps: Nonlinear Generalizations of Net Benefit and Acceptabil-ity. Health Serv Outcomes Res Method 2008; 8: 31-56. DOI 10.1007/s10742-007-0027-2. OpenAccess.
Display Scatter for a possibly Transformed Bootstrap Distribution ofICE Uncertainty
Assuming x is an output list object of class ICEuncrt, the default invocation of plot(x) graphicallydisplays the bootstrap distrib of ICE uncertainty currently stored in x. An invocation of the form x10<- plot(x, lfact=10) increases the value of x item lambda by a factor of 10, displays that transformedbootstrap distribution, and stores it in object x10. When the x item unit is cost, an invocation of theform xs <- plot(x, swu=TRUE) displays the bootstrap distribution stored in x using effe units andstores the transformed distribution in object xs.
plot(x, lfact = 1, swu = FALSE, alibi = FALSE, .)
Output list object of class ICEuncrt.
Positive factor multiplying the stored value of x item lambda.
Logical value of TRUE or FALSE to control switching the stored value of x itemunit between the 2 possibilities, cost and effe.
Logical value of TRUE or FALSE to control scaling of axes. alibi = FALSEproduces the default alias graphic in which points in the bootstrap uncertaintyscatter are held fixed in space, and changes in lambda merely change the scaling(tick marks) along either the horizontal axis of a cost unit display or else alongthe vertical axis of an effe unit display. alibi = TRUE produces an alibi graphicin which the scaling (and range) is the same along both axes, and changes inlambda cause the points in the bootstrap uncertainty scatter to literally moveeither left or right in a cost unit display or else up or dowm in an effe unitdisplay.
Optional argument(s) passed on to plot().
After a single call to ICEuncrt() for an initial value of the Shadow Price of Health, lambda, andan initial choice of display unit (cost or effe), multiple calls to plot.ICEuncrt() are usually made. Alternative economic choices for lambda can be suggested by making calls to ICEscale() withdifferent values for lambda. Because the Bootstrap Distribution of ICE Uncertainty is equivariantunder changes in lambda, it is much faster to transform an existing bootstrap distribution thanto generate a new one for a different value of lambda. The print.ICEuncrt() and plot.ICEuncrt()functions thus have 2 special parameters, lfact and swa, that can change lambda and switch theunits of measurement, respectively, without actually regenerating the bootstrap distribution via acall to ICEuncrt().
Object of class ICEuncrt containing a possibly TRANSFORMED output list with items:
Saved value of the name of the data.frame in the original call to ICEuncrt().
Possibly changed, positive value of lfact * (x item lambda).
Possibly switched value of x item unit, cost or effe.
Saved integer value for number of bootstrap replications input to ICEuncrt.
Saved name of the treatment indicator within the df data.frame.
Saved name of the treatment effectiveness variable within the df data.frame.
Saved name of the treatment cost variable within the df data.frame.
Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
Observed value of (DeltaEffe, DeltaCost) when each patient is included exactlyonce.
R x 2 matrix of values of (DeltaEffe, DeltaCost) computed by transformation.
Saved value of the seed used to start pseudo random number generation.
Obenchain RL. Issues and algorithms in cost-effectiveness inference. Biopharmaceutical Reports1997; 5(2): 1-7. Washington, DC: American Statistical Association.
Obenchain RL. ICEplane: a windows application for incremental cost-effectiveness (ICE) statisticalinference. Copyright (c) Pharmaceutical Research and Manufacturers of America (PhRMA.)http://members.iquest.net/~softrx/ 1997–2007.
Obenchain RL. Resampling and multiplicity in cost-effectiveness inference. Journal of Biophar-maceutical Statistics 1999; 9(4): 563–582.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
# Transformation of a bootstrap distribution is fast.
Summary Statistics for a possibly Transformed Bootstrap Distributionof ICE Uncertainty
Assuming x is an output list object of class ICEuncrt, the default invocations of x or print(x) describethe bootstrap distribution of ICE uncertainty currently stored in x. An invocation of the form x10 <-print(x, lfact=10) increases the value of x item lambda by a factor of 10, describes that transformedbootstrap distribution, and stores it in object x10. When x item unit is cost, an invocation of theform xs <- print(x, swu=TRUE) describes the bootstrap distribution stored in x using effe units andstores the transformed distribution in object xs.
Required; Output list object of class ICEuncrt.
Optional; Positive factor multiplying the stored value of x item lambda.
Optional; Logical value of TRUE or FALSE to control switching the storedvalue of x item unit between the 2 possibilities, cost and effe.
Optional; argument(s) passed on to plot().
After a single call to ICEuncrt() for an initial value of the Shadow Price of Health, lambda, and aninitial choice of common display unit (cost or effe), multiple print() and/or plot() calls are usuallymade. Because the bootstrap distribution of ICE uncertainty is equivariant under changes in lambda,it is much faster to transform an existing Bootstrap ICE Uncertainty Distribution than to generate anew one for a different value of lambda.
The print.ICEuncrt() and plot.ICEuncrt() functions thus have 2 special parameters, lfact and swa,that can change lambda and switch the units of measurement, respectively, without actually regen-erating the bootstrap distribution via a new call to ICEuncrt().
Object of class ICEuncrt containing a possibly TRANSFORMED output list with items:
Saved value of the name of the data.frame in the original call to ICEuncrt().
Possibly changed, positive value of (lfact * x item lambda).
Possibly switched value of x item unit, cost or effe.
Saved integer value for number of bootstrap replications input to ICEuncrt.
Saved name of the treatment indicator within the df data.frame.
Saved name of the treatment effectiveness variable within the df data.frame.
Saved name of the treatment cost variable within the df data.frame.
Saved value of the sorted 3-variable (trtm,effe,cost) data.frame.
Observed value of (DeltaEffe, DeltaCost) when each patient is included exactlyonce.
R x 2 matrix of values of (DeltaEffe, DeltaCost) computed by transformation.
Saved value of the seed used to start pseudo random number generation.
Obenchain RL. Issues and algorithms in cost-effectiveness inference. Biopharmaceutical Reports1997; 5(2): 1-7. Washington, DC: American Statistical Association.
Obenchain RL. ICEplane: a windows application for incremental cost-effectiveness (ICE) statisticalinference. Copyright (c) Pharmaceutical Research and Manufacturers of America (PhRMA.)http://members.iquest.net/~softrx/ 1997–2007.
Obenchain RL. Resampling and multiplicity in cost-effectiveness inference. Journal of Biophar-maceutical Statistics 1999; 9(4): 563–582.
Cook JR, Heyse JF. Use of an angular transformation for ratio estimation in cost-effectivenessanalysis. Statistics in Medicine 2000; 19: 2989-3003.
# Transformation of bootstrap distributions is fast.

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