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Estimating Incremental Cost-Effectiveness Ratios from Cluster Randomized Intervention Trials
M. Ashraf Chaudhary & M. Shoukri
Sep 28, 2005 2CREATE Biostat Core Meeting, Cape Town
Incremental Cost-Effectiveness Ratio
E2E1
C2C1
R
21
21ˆEECC
R
Assuming numerator and denominator positive, R is the cost per additional outcome achieved by the treatment.
Sep 28, 2005 3CREATE Biostat Core Meeting, Cape Town
Statistical Properties
• Biased• Consistent• Positively skewed• Limiting distribution is normal• Very sensitive to changes in the denominator• No exact method of estimating variance
Sep 28, 2005 4CREATE Biostat Core Meeting, Cape Town
Methods
Parametric:
• 1) Taylor Series Expansion
• 2) Fieller's Method
Non-Parametric Bootstrap:
• 3) Percentile Method
• 4) BCa
Sep 28, 2005 5CREATE Biostat Core Meeting, Cape Town
Cluster Randomized Trials
i
ii
m
CmC
1
111
1111
212
1 11 CC
C mmk
2)(1
2)(1
2)(1
1eCbC
bCC
11111 ),cov( ECEC
For fixed cluster size , so that .11 mm i 111 kCC i
Sep 28, 2005 6CREATE Biostat Core Meeting, Cape Town
Coefficients of Variation
2
21
2C2
2C1
CC
sscnn
221
2E2
2E1
EE
sscdd
2121
E2C22111
EECC
ssrssrc EC
nd
Sep 28, 2005 7CREATE Biostat Core Meeting, Cape Town
Taylor Series Method
Large sample normal approximation yields,
2/1)ˆ(ˆ RvzRR
• Inaccurate if is far from normal or the sample is not large enough.• Affected by extreme values• Interval is symmetrical even if is not.
R̂
R̂
ndddnn2 2ˆ)ˆ( cccRRv
Sep 28, 2005 8CREATE Biostat Core Meeting, Cape Town
Fieller's Method
dd2
2/12ndddnn
2ndddnnnd
2
1)()2()1(ˆ
czccczccczcz
RR
» Not symmetrical if is not » Assumes bivariate normality» Assumes unbiasedness of » Imaginary roots for certain samples» Volatility of making a negative quantity leading in correct intervals.
R̂
R̂
ddcddcz21
Sep 28, 2005 9CREATE Biostat Core Meeting, Cape Town
Non-Parametric Bootstrap
1. a- resample observations or b- clusters or c- two stage bootstrap?a – not appropriate as observations within cluster are correlatedb – theoretically appropriate c – mathematically preferable but assumes no correlation within clusters
2. We use approach b and resample clusters retaining all observations in resampled clusters
3. A cluster level summary data is prepared with mean cost and mean effect in each cluster
4. boot package in R was used to implement bootstrap stratified by study arm and estimate intervals
5. bootstrap replications =1999
Sep 28, 2005 10CREATE Biostat Core Meeting, Cape Town
Percentile Method
• If is normal, agrees with delta method
• Interval not symmetrical if is not
• Transformation respecting
• Range preserving
• Robust to extreme replication
• No adjustment for bias due to for asymmetry
R̂
R̂
Sep 28, 2005 11CREATE Biostat Core Meeting, Cape Town
BCa All the advantages of percentile method
Adjustment for bias due to asymmetry
More accurate in terms of coverage
Sep 28, 2005 12CREATE Biostat Core Meeting, Cape Town
Simulation (1)– Balanced CRT with two treatment groups
• clusters of fixed size
• varying but equal ICC in cost in two groups and zero for the effectiveness
– Cost - Random effects model framework used separately for each arm, between and within cluster effects assumed normal
– Effect - A correlated normal variable generated within each cluster and dichotomized – 0.20 in control and 0.40 in treatment and cost-effect correlation 0.30 in each group
– Mean cost in control and treatment set at $20.00 and $30.00
– R = $50.00
– Box-Cox transformation of normal for positively skewed cost data
Sep 28, 2005 13CREATE Biostat Core Meeting, Cape Town
Simulation (2)
– 54 Scenarios:• number of clusters (12, 24, 48)• cluster sizes (25, 50, 100)• ICC (0.25, 0.10, 0.01)• Normal/positively skewed cost data
– Sum of between and within components of variance in cost data constrained to be 100
– 2000 simulation replications of data under each scenario– All four types of intervals computed for each replication– The programming for simulation and analysis in R
Sep 28, 2005 14CREATE Biostat Core Meeting, Cape Town
Skewness & VariabilityRho Clusters Size Skewness nnc
ddc ndc
0.25 12 25 0.40 0.47 0.44 0.31 0.25 12 100 0.08 0.46 0.30 0.25 0.25 48 25 0.10 0.33 0.30 0.21 0.25 48 100 0.17 0.32 0.21 0.18
0.01 12 25 0.63 0.31 0.43 0.26 0.01 12 100 0.26 0.24 0.30 0.19 0.01 48 25 0.24 0.21 0.30 0.18 0.01 48 100 0.04 0.17 0.21 0.13
Sep 28, 2005 15CREATE Biostat Core Meeting, Cape Town
Skewness & Variability
– The coefficient of skewness estimated as 3(mean-median)/sd
– Distributions of the simulation replications of are generally positively skewed.
– Increasing the number of clusters and the cluster sizes leads to more symmetrical distribution of
– Distribution of is more skewed with smaller ICC – with more within cluster variability
– The order of the numerator and denominator coefficients of skewness is about the same as observed in most cost-effectiveness studies.
R̂R̂
Sep 28, 2005 16CREATE Biostat Core Meeting, Cape Town
Histograms
k=12 m=25
Histogram of R2
R2
Fre
qu
ency
20 40 60 80 100 120
050
10
015
020
025
0
Histogram of R2
R2
Fre
qu
en
cy
40 60 80 100
05
01
00
15
02
00
25
0
k = 12, m = 100
Histogram of R2
R2
Fre
qu
en
cy
20 40 60 80
05
01
00
15
0
Histogram of R2
R2
Fre
qu
en
cy
40 50 60 70
01
00
20
03
00
k = 48 m = 25
Histogram of R2
R2
Fre
qu
en
cy
30 40 50 60 70 80
050
10
015
020
025
0
Histogram of R2
R2
Fre
qu
en
cy
40 50 60 70
01
00
20
03
00
40
0
k = 48 m = 100
Histogram of R2
R2F
req
uency
30 40 50 60 70
050
10
015
020
025
030
0
Histogram of R2
R2
Fre
qu
en
cy
45 50 55
05
01
00
15
0
Rho = 0.25 Rho = 0.01
Sep 28, 2005 17CREATE Biostat Core Meeting, Cape Town
95% Confidence Intervals
Rho Clus- Cluster Skew- Methods
ters Size ness Delta Fieller’s Percentile BCa
in R̂ Cov. Width Shape Cov. Width Shape Cov. Width Shape Cov. Width Shape
25 0.41 93.8 53.2 1.0 91.4 62.5 1.6 92.5 63.6 1.7 92.3 64.7 1.7 50 0.07 93.4 44.7 1.0 92.6 46.8 1.3 92.3 48.8 1.3 92.7 49.2 1.3 12
100 0.10 92.0 40.6 1.0 91.7 41.4 1.1 91.9 42.8 1.1 92.1 43.0 1.2
25 0.22 94.6 36.1 1.0 93.2 38.0 1.3 94.2 40.1 1.4 94.5 40.2 1.4 50 0.12 93.2 31.1 1.0 93.0 31.7 1.2 94.1 33.6 1.2 93.9 33.7 1.2 24
100 0.11 93.2 28.7 1.0 93.1 29.0 1.1 94.3 30.6 1.1 94.3 30.7 1.1
25 0.18 94.2 24.9 1.0 93.6 25.5 1.2 94.6 27.3 1.3 94.3 27.4 1.3 50 0.21 93.2 21.9 1.0 92.6 22.1 1.1 94.3 23.7 1.1 94.3 23.8 1.1
0.25
48
100 0.00 92.6 20.3 1.0 92.5 20.4 1.0 94.1 21.8 1.1 93.6 21.8 1.1
25 0.43 94.3 44.6 1.0 93.4 52.9 1.8 92.7 52.3 1.8 92.4 53.5 1.9 50 0.11 94.6 34.3 1.0 93.2 36.2 1.4 93.8 37.3 1.4 93.5 37.7 1.4 12
100 0.17 92.6 28.5 1.0 92.1 29.1 1.2 92.9 30.3 1.2 93.1 30.5 1.2
25 0.28 93.8 30.1 1.0 93.6 32.0 1.5 93.5 33.2 1.5 93.9 33.3 1.5 50 0.10 93.2 23.5 1.0 92.7 24.1 1.2 94.2 25.6 1.3 93.9 25.6 1.3 24
100 0.11 92.7 20.1 1.0 92.6 20.3 1.1 94.4 21.7 1.1 94.0 21.7 1.1
25 0.27 94.4 20.8 1.0 93.4 21.3 1.3 94.5 22.5 1.3 94.6 22.6 1.3 50 0.15 92.3 16.6 1.0 92.4 16.7 1.2 94.3 17.9 1.2 94.2 18.0 1.2
0.10
48
100 0.04 93.3 14.2 1.0 93.0 14.2 1.1 94.9 15.4 1.1 94.6 15.4 1.1
25 0.49 94.1 40.3 1.0 94.2 51.0 2.0 92.0 47.0 2.0 91.7 47.6 2.1 50 0.28 94.7 27.0 1.0 94.5 28.9 1.6 93.2 28.0 1.6 92.9 28.3 1.6 12
100 0.21 93.6 19.1 1.0 93.6 19.7 1.3 92.7 19.4 1.3 92.5 19.6 1.4
25 0.29 94.2 26.6 1.0 95.0 28.5 1.6 93.9 28.0 1.6 93.6 28.2 1.6 50 0.23 94.3 18.5 1.0 94.6 19.0 1.4 94.2 18.9 1.4 94.1 19.0 1.4 24
100 0.12 94.7 13.3 1.0 94.5 13.5 1.2 94.5 13.7 1.2 94.1 13.7 1.2
25 0.19 94.8 18.4 1.0 94.2 18.9 1.4 93.8 18.9 1.4 93.5 18.9 1.4 50 0.10 95.3 13.0 1.0 95.8 13.2 1.2 95.4 13.3 1.2 95.4 13.3 1.3
0.01
48
100 0.10 94.7 9.4 1.0 94.9 9.4 1.2 94.9 9.7 1.2 94.8 9.7 1.2
Overall 93.7 26.7 1.0 93.4 28.4 1.3 93.8 29.1 1.3 93.6 29.3 1.4
Sep 28, 2005 18CREATE Biostat Core Meeting, Cape Town
Results (1) Coverage: Proportion of intervals containing R
Width:
Shape:
ICER is highly unstable with small effect difference. If |R| < 0.0001 then R= +/- 0.0001. Replications of R are bounded by Chebychev’s inequality -
the probability of having a more extreme replication < 1/100. Possibly no effect on bootstrap confidence intervals.
lowup RR ˆˆ )ˆˆ/()ˆˆ( lowup RRRR
Sep 28, 2005 19CREATE Biostat Core Meeting, Cape Town
Results (2)
In terms of coverage all the four methods seem to perform equally well
Generally the coverage falls below the nominal value of 0.95
The coverage does not seem to improve with increase in sample size
The lower levels of ICC seem to be associated with better coverage of the intervals.
The width of the intervals shrinks with the increase in the number of clusters and the cluster size
The shape of the intervals tends to be more even with large cluster and of bigger size.
Sep 28, 2005 20CREATE Biostat Core Meeting, Cape Town
Results (3)
The shapes of the Fieller’s, Percentile and the BCa intervals are similar and reflect the direction of asymmetry in the distribution of ICER.
The distribution of simulation replications of R is more symmetrical as the number of clusters and the size of the clusters increase. This translates to Fieller’s, Percentile and BCa intervals to be more even around the R. On top of this trend, these intervals are more symmetrical with smaller ICC. The same trend is evident in the width of the intervals. Further the width of the intervals shrinks with reduced levels of ICC.
It is evident that confidence intervals ignoring the ICC will be shorter in length and would not provide the desired coverage probability and would be misleading.
Sep 28, 2005 21CREATE Biostat Core Meeting, Cape Town
Results (4) The big question – why all the methods perform equally well in
terms of coverage when the distribution of replications of R is clearly positively skewed? The overall sample is big in each combination? The cost data are assumed normally distributed? The cluster specific means are used in the analysis leading to normality by
CLT?
Sep 28, 2005 22CREATE Biostat Core Meeting, Cape Town
References1. Chaudhary, M.A., and Stearns, S.C., "Estimating Confidence
Intervals for Cost Effectiveness Ratios: An Example from a Randomized Trial", Statistics in Medicine, Vol. 15, 1447-1458 (1996)
2. O'Brien, B.J., M.F. Drummond, R.J. LaBelle, A. Willan 'In Search of Power and Significance: Issues in the Design and Analysis of Stochastic Cost-Effectiveness Studies in Health Care', Medical Care, 32(2):150-163 (1994)
3. Mullahy, J. and W. Manning 'Chapter Eight: Statistical Issues in Cost-Effectiveness Analyses', in Costs, Benefits, and Effectiveness of Pharmaceuticals and Other Medical Technologies, edited by Frank Sloan, Cambridge University Press. (1995)
4. Cochran, W. G. Sampling Techniques, John Wiley and Sons. N.Y. (1977)
5. Efron, B., and Tibshirani, R.J. An Introduction to the Bootstrap, Chapman and Hall, N.Y. (1993)