Power Analysis/Sample Size of Field-Based Mosquito Repellency Studies
EPA Office of Pesticide Programs
July 2017
Objective
To determine the sample size N such that mosquito repellency studies have sufficient power to obtain a
given degree of precision in the estimate of median Complete Protection Time (mCPT). This precision -
designated as "K" - will be expressed as the ratio: 95% LCLmCpi/estimated mCPT
The simulation used to estimate varying sample sizes will require that that 95% LCLmCpi/estimated
mCPTcK; such true variation of the Complete Protection Time (CPT) distribution will be expressed by the
Weibull distribution family and a parameter, P5MR, defined as the 5th percentile/mCPT.
The model simulation was prepared in anticipation of receipt of a proposed mosquito repellency
protocol concerning IR3535 to be reviewed by the HSRB at a future date. EPA anticipates that most, if
not all, future mosquito repellency studies with similar study designs can likely use this approach - and
the simulation results described here - to determine the required sample size to achieve the stated
objectives.
In order to develop estimates of a required sample size for a mosquito repellency study to achieve
certain stated efficacy criteria and estimate a complete protection time (CPT)1, it is necessary to
determine the distribution of mosquito repellent failure times (generally considered to be time to first
landing with intent to bite). However, the underlying distribution of the CPT of a product being tested in
a mosquito repellency study is not known prior to the testing phase. What is known about the
distribution is that CPT values are (necessarily) non-negative and are (generally) right censored after 10
(or 12 hours) in most mosquito repellency studies. From past submitted studies that have been
examined, the EPA has found CPTs display a left skewed distribution in some of the datasets.
On this basis, EPA assumed for this sample size determination exercise that a distribution of mosquito
repellent failure times follows a Weibull distribution. A Weibull distribution is commonly used in
reliability engineering and failure analysis, in survival analysis, in predicting delivery times, in weather
forecasting and hydrology, and in extreme value prediction. Its utility in a wide variety of applications is
due in part to its flexibility to take on a variety of shapes depending on the parameters selected to
describe the distribution. Oftentimes, the Weibull plot is described by two parameters: k (the "shape"
parameter and sometimes referred to in some parameterizations as "a") and X (the scale parameter and
sometimes referred to as "b").2 The PDF (probability density function) and CDF (cumulative distribution
1 The Complete Protection Time (CPT) is defined as the time from initial application of the repellent by the test subject to the
time of first confirmed landing with intent to bite (FCLIB). The FCLIB is considered to be when one landing is followed by
another landing within 30 minutes. The first landing is confirmed by the second landing.
2 A Weibull distribution can sometimes be described by 3 parameters, with a "location" parameter added as a third parameter
to the "scale" and "shape" parameter of the 2-parameter Weibull distribution.
Page 1 of 48
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function) of the aforementioned two-parameter Weibull distribution are defined, respectively, as
follows:
nc ^k-1p-(x/X)k
f(x, K,X) = UV
10
x > 0,
x < 0
F(x,K,X) = (l-e-(*/A''
(.0 x < 0
and are illustrated in the associated plots in Figures 1 and 2 for some illustrative K and X values.
Parameterizing the Weibull distribution in terms of k and X is, however, not necessarily intuitive with
respect to studying - and judging - the efficacy of skin-applied mosquito repellents as measured by CPT
for individuals using the repellent in the field. Instead, it is more natural and desirable to be able to
express the efficacy of the repellent in terms of both the expected precision of the estimated median
CPT (mCPT) and in terms of the estimated variability of mCPT in (or across) the population. More
specifically: the testing of a given repellent in the field should be able to generate a reasonably precise
estimate of the mCPT that is expected to be generally close to what a sizable fraction of the population
would be expected to experience (or, more accurately, a mCPT that only a small fraction of the
population would ideally experience to be much shorter).
Following the above logic, we define the precision of the CPT estimate — designated as "K" - as follows:
K = 95% LCLmcpi/estimated mCPT
where: mCPT= median complete protection time
95% LCLmCpT= 95% lower confidence limit on the estimated mCPT
Similarly, the degree of variation of the CPT distribution in the population will be defined as the P5MR
which we define here as the ratio between the mCPT of the 5th percentile of the population to the mCPT
of the population:
P5MR = CPTsth%iie/mCPT
where:
mCPT= median complete protection time
CPTsth %iie - 5th percentile of the distribution of CPT
Re-parameterization of Standard Weibull Equation
While the above mCPT and P5MR parameterizations of the Weibull distribution are intuitively appealing
for judging and evaluating repellent efficacy, they are non-standard parameterizations and it is
necessary - for comparison and simulation purposes - to convert these to the more standard k (shape)
Page 2 of 48
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and X (scale) values. To do this, EPA developed an equation such that interconversion between the
standard (ic (shape) and X (scale)) parameterization of the Weibull to this alternate version (with the
Weibull distribution instead expressed in terms of P5MR and mCPT). Briefly, the cumulative probability
function of CPT is assumed to be a 2- parameter Weibull distribution:
P(CPT, k, X) = i-e-(CPT/A)K
Given that a value of the mCPT represents the median or 50th percentile of the CPT and the value of
P5MR represents the ratio of the 5%-tile of the CPT distribution to the mCPT, we can develop the
following two equations to represent the cumulative distribution functions at the median CPT and the
5th percentile CPT:
fmCPT\K
P(mCPT,K,X) = 1 — e v a J =0.5 (median)
(P5MRxmCPT\K
P(PSMR x mCPT,K,X) = 1 — e v a ; = 0.05 (5th percentile)
Algebraically solving the equations above (see Appendix 2 for full derivation), we develop expressions
for k and X:
K = In
ln(0.95)
ln(0.5)
/ ln(P5M/0
1 ;;in[ mCPTK]
A = eK I ln(0.5) J
Table 1 below compares these two parameterizations for the example PDF and CDF distributions shown
in Figures 1 and 2, respectively, for the k and X parameterizations shown there, illustrating the
conversion to this new parameterization:
Page 3 of 48
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Table 1. Re-parameterization of Weibull Distribution Parameters from Traditional (k, K) to Revised
(P5MR, mCPT) for Example Weibull Distributions Appearing in Figures 1 and 2.
Parameterization Scheme
Description/Comments
Traditional
Revised
Scale (X)a
Shape (k)
mCPTb
P5MRcd
1
0.5
0.480453
0.005476
- k values of less than 1 indicate a failure
rate decreases overtime, and defective
items fail early or are otherwise removed
from the population.
1
1
0.693147
0.074001
- k values equal to 1 indicate a constant
failure rate over time possibly suggesting
failure is due to random external events.
- Here, the Weibull distribution reduces to
the "exponential" distribution;
- Note that mCPT here = 0.693 = ln(2)
1
1.5
0.78322
0.176261
- k values greater than 1 suggests that the
failure rate increases over time, as when
there is an "aging" process or components
are more likely to fail over time.
1
5
0.92932
0.594083
3 The Weibull scale parameter is the 63.2 percentile of the distribution. If the scale parameter is 1, then this means that 63.2% of
the observed values will be smaller than 1. Note in the CDF in Figure 2, as a consequence, that all \=1 distributions intersect at the
63.2 percentile.
bmCPT = [ln(2)*exp(K *ln(A))](1/lc)
c P5MR = exp(ln(ln(0.95)/ln(0.5))/K)
d Note that as k increases, the P5MR value becomes larger, indicating that the values at the 5th percentile
approaches the values present at the 50th percentile, and the PDF becomes tighter and more peaked. K values of
between 3 and 4 often lead to distributions that appear normal.
An example of the (varied) kinds of distributional "shapes" associated with various parameterizations is
shown in Figure 2 as histograms of the CPT. More specifically, Figure 2 presents the CPT distributions
with different medians and values of P5MR (ratio 5%-tile/mCPT). These present the CPT distributions
with different mCPTs (2-, 4-, 6-, and 8- hrs) and values of the P5MR ratio (P5MR= 0.2, 0.3, 0.4, and 0.6)
for the (assumed) Weibull Distribution3. As seen in Figure 2, larger mCPTs are associated with a shift in
the distribution toward the right. In addition - and importantly - smaller P5MR values in this range are
associated with "flatter" distributions and larger P5MRs are associated with more "peaked"
distributions, with these more peaked distributions showing a greater percentage of the distribution
3 Other simulations were performed for the lognormal, normal, and uniform distributions, with the latter one (particularly)
done as a form of sensitivity analysis but these are not discussed in this report; the simulation outputs, however, are provided
in Appendix 4. Note that the power estimates for a given sample size from the Weibull and Lognormal distributions are similar.
Page 4 of 48
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centered around the median. From a regulatory perspective, a CPT distribution with a larger P5MR is
more desirable than a CPT distribution with smaller P5MR since this means that a greater percentage of
the user population experiences an actual CPT closer to the (advertised) mCPT. Further, it could be
argued from a public policy perspective that a large variability in CPT in the population for a given
repellent is not a desirable characteristic, and does not accurately portray or indicate any "expected"
mCPT on the part of the consumer.
OPP staff have judged what might be considered reasonable values for input parameters (precision of
the estimated mCPT and variability in CPT in (or among) users of the tested product) in order to
estimate required number of test subjects for the field exercise to achieve a desired set of aims
regarding precision around the estimate of the mCPT. These judgments are based in part on available
data and past experiences4 and in part on general thoughts regarding consumer expectations with
respect to product efficacy. Specifically, EPA has estimated the power associated with various sample
sizes where power - as defined here - is the probability that the ratio of the (95% LCLmCpi)/(estimated
mCPT) is greater than a given acceptable K (a scalar which measures the precision of the estimates in
estimating the mCPT). Such mosquito repellency study design power depends on:
• Number of test subjects
o The larger the number of test subjects, the greater the power
(The required) precision (K) for estimated mCPT
o The precision of an estimated mCPT from a study is expressed by the value of the ratio 95%
LCLmcpi/estirnated mCPT. The value of ratio is in the interval (0,1).
o K is the smallest acceptable value of the ratio 95% LCLmcpi/estimated mCPT for a given trial
to be considered a "success", and conceptually represents an inverse of precision
("tightness") in the estimate of the mCPT: a larger K represents a greater "tightness" around
the estimated mCPT. As K is chosen to be smaller, there is a greater probability that ratio
95% LCLmcpi/estimated mCPT > K (and the trial is considered to be a "success" in the power
calculation)
P5MR
o P5MR = ratio of the 5th percentile/mCPT
o As the variation (dispersion or spread) of the distribution of CPT in the population becomes
smaller, the 95% confidence interval of the estimated mCPT also becomes narrower (i.e. the
95% LCLmcpT is closer to the estimated mCPT and the mCPT is better estimated, certeris
paribus). Therefore, a smaller variation in the distribution of CPT will result in a larger P5MR
and a higher probability that the ratio 95% LCLmcpi/estimated mCPT > K. A CPT distribution
with greater P5MR is generally more desirable than a CPT distribution with smaller P5MR
Ideally, a mosquito repellency study will be designed to have a sufficient number of test subjects
such that one can have reasonable assurance that there is adequate power (defined here as a high
probability that the ratio 95% LCL/estimated mCPT > K) given a shape and spread of the CPT
4 See Appendix 2 for Weibull parameters fit to previous mosquito efficacy field data that the EPA has evaluated for a similar
design and experimental set-up. In general, the values found in these (prior) studies support the values selected here to be
used for the simulation
Page 5 of 48
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distribution in the population. This shape/spread of the CPT in the population is defined by the
P5MR.
Brief Description of the Conduct of a Field Mosquitoes Repellent Study
In mosquito field repellency studies, test subjects are exposed in the field for 5-minute intervals
immediately following product application and then for 5 minutes every 30 minutes until a "first
confirmed landing" occurs. For subjects who receive confirmed landings, the CPTs are set as 0 if the
first confirmed landing occurs during the first 5 minutes after application of the repellent;
otherwise, the CPTs are rounded down to the nearest half hour (i.e., the starting time of the
exposure period in which the first confirmed landing occurs). For those subjects for which there are
no confirmed landings through the end of the testing day, CPTs are considered to be right censored
at a time that is rounded down to the nearest half hour.
Description of (Computer) Simulation Procedure:
To simulate the field study trials, 4000 datasets were created with each dataset consisting of 10 data
points (representing CPTs of 10 subjects) that were generated randomly from a Weibull distribution
with a median CPT=2 and ratio of the 5%-tile/median P5MR= 0.2. If the randomly generated CPTs
for the 10 subjects are < 5, 6-35, 36-65, 66-95, ... 576-605 minutes, the CPTs are set to be 0-, 0.5-,
1-, 1.5- hours...10 hours, respectively, to simulate the study design in which each study participant is
exposed for 5 of every 30 minutes until the first confirmed mosquito landing. If the randomly
generated CPTs are greater than 10 hours (or 605 minutes), they are considered in the calculation to
be (right) censored at 10 hours.
After generating the CPTs as described in the previous paragraph, the Kaplan Meier Estimator is
used to estimate the mCPT and its 95% CI for each of the 4000 (10-person) datasets. The proportion
of datasets in which the ratio of 95% LCLmCpi/mCPT > K as 0.6 is considered to be the "power" of the
study design. More specifically: if the value of 95% LCL/mCPT >0.6 is considered a "success", the
power is calculated as the proportion of successes in the 4000 datasets consisting of 10 data points
each.
The process described in previous paragraph is then repeated for each combination of different
mCPT = 2, 4, 6, and 8 hours; P5MR = 0.2, 0.4, 0.5, 0.6, 0.7, and 0.8; sample size per dataset = 10, 11,
12 ... 20; and the lowest acceptable K = 0.6, 0.7, and 0.8; all assuming that CPT follows a Weibull
distribution5.
5 Such calculations were similarly done for the lognormal distribution, normal distribution, and uniform
distribution, but are not discussed further in this report. The SAS output from these calculations and various
associated tables and graphs, however, is shown in Appendix 4 for completeness.
Page 6 of 48
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Results of Simulation
Tables 1, 2, and 3 present the power estimates from simulations in which the data were randomly
generated from Weibull distributions for K = 0.6, 0.7, and 0.8, respectively. These are shown for
various values of mCPT (ranging from 2 to 8 hours), P5MR (ranging from 0.2 to 0.8), and Sample Size
(ranging from 10 to 20). As described earlier, K reflects a measure the precision of the estimate of
mCPT with larger K values representing tighter estimates. For example, the K value of 0.6 requires
that the 95% LCL on a median protection of 10 hours be no less than 6 hours (for a "success") while
a K value of 0.8 requires that the 95% LCL on that same median protection time be no less than 8
hours. A required precision of a K of 0.8, then, requires a more precise estimate of the mCPT than a
K of 0.6 for this trial to be considered a "success" in the power calculation.
Figures 4, 5, and 6 present visually the same results in Tables 1, 2, and 3 (as power curves rather
than tables).
As can be seen within each Table or Figure, the power of a study to achieve a given acceptable ratio
K value (e.g., 0.6, 0.7, or 0.8 representing 95% LCLmCpi/mCPT) value increases as the assumed P5MR
value of the distribution increases (for example, from 0.2 to 0.8) or as the sample size increases
(from 10 to 20). This is expected since a tighter (or more "peaked") distributions (as evidenced by a
larger P5MR value) will require fewer random "draws" to accurately estimate the mCPT. Across the
Figures or Tables, we also see that as the acceptable K value increases from 0.6 to 0.8, the power of
a study to achieve "95% LCLmCpi/mCPT > K" decreases since stricter requirements for a "success" are
being levied.
The SAS Code used to generate the simulated data and the associated tables and graphs are
presented in Appendix 3. Note - as described earlier - that simulations were also performed for the
lognormal, normal, and uniform distributions, in part to serve as a sensitivity analysis and these are
presented in the Appendix 4 for completeness, but are not discussed further here.
Page 7 of 48
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Figure 1. Probability Density Function (PDF) for Weibull Plot with A (scale) =1 and k (shape) ranging from 0.5 to 5
0*0 o's 10 1.6 20 2*5
Figure 2. Cumulative Distribution Function (CDF) for Above Weibull PDF
Weibull(2-Parameter) Cumulative distribution function
o!o o'5 l'o 1*5 20 25
Page 8 of 48
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Figure 3: Histograms of CPT distributions for various CPTs and P5MRs (assume CPTs are Weibull distributions)
weibull distribution: median=2
CPT
weibull distribution: median=6
-
P5MR= 0.2
Ti ITTTTlTlTTrrTTTTTTT-^
TTITTTtti-^—
—rrrrTT
"TTTtt-^-,—
t-ttT I
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
CPT
weibull distribution: median=4
8-
6-
4-
2-
0-
P5MR= 0.2
—rffT 1T
111 rrrmnTiTn 11 i-rm-n-
8-
6-
4-
2-
0-
TirrriTTTTriTm-rr^™—
Percent
O W -b (J) O]
—-r-rt-rfrnTT
8-
6-
4-
2-
0-
8-
6-
4-
2-
^rrrf
u I I I I
0 12 3
1 ryi-, j j j j
4 5 6 7 8 9 10
CPT
weibull distribution: median=8
-
P5MR= 0.2
Ti iTlTTlTlTTrrTTTTTTT-m-
- i i
r —¦-t — "W
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
CPT
Page 9 of 48
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Table 1: Results of power analysis when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.6
(Weibull distribution)
Median
(hours)
P5MR
Sample size
_10
_11
_12
_13
_14
_15
_16
_17
_18
_19
_20
2
0.2
0.071
0.291
0.207
0.473
0.362
0.369
0.502
0.494
0.637
0.626
0.521
0.4
0.297
0.691
0.594
0.841
0.804
0.779
0.893
0.898
0.939
0.945
0.932
0.5
0.498
0.850
0.802
0.942
0.938
0.921
0.968
0.977
0.964
0.982
0.986
0.6
0.733
0.949
0.943
0.962
0.971
0.955
0.954
0.979
0.915
0.951
0.971
0.7
0.893
0.945
0.955
0.875
0.918
0.852
0.855
0.893
0.810
0.859
0.886
0.8
0.819
0.786
0.826
0.666
0.734
0.591
0.637
0.708
0.558
0.632
0.689
4
0.2
0.043
0.208
0.146
0.356
0.289
0.254
0.432
0.380
0.567
0.516
0.435
0.4
0.241
0.595
0.521
0.783
0.737
0.709
0.849
0.842
0.930
0.920
0.884
0.5
0.412
0.795
0.730
0.921
0.901
0.888
0.956
0.964
0.986
0.988
0.973
0.6
0.648
0.938
0.899
0.987
0.980
0.976
0.995
0.997
0.997
0.999
0.998
0.7
0.869
0.988
0.986
0.993
0.995
0.994
0.992
0.998
0.977
0.992
0.996
0.8
0.975
0.982
0.987
0.948
0.970
0.949
0.934
0.968
0.887
0.932
0.954
6
0.2
0.075
0.204
0.153
0.339
0.280
0.252
0.426
0.369
0.557
0.490
0.424
0.4
0.227
0.572
0.504
0.759
0.743
0.689
0.851
0.826
0.929
0.916
0.885
0.5
0.408
0.779
0.729
0.914
0.905
0.873
0.963
0.958
0.987
0.981
0.978
0.6
0.645
0.925
0.906
0.984
0.980
0.977
0.997
0.997
1.000
0.999
0.999
0.7
0.874
0.990
0.988
0.998
0.999
0.999
1.000
1.000
0.999
1.000
1.000
0.8
0.986
0.998
0.999
0.993
0.995
0.994
0.990
0.997
0.975
0.989
0.995
8
0.2
0.323
0.346
0.362
0.457
0.443
0.361
0.537
0.453
0.636
0.564
0.522
0.4
0.314
0.586
0.552
0.769
0.753
0.700
0.858
0.836
0.934
0.919
0.891
0.5
0.421
0.779
0.732
0.914
0.904
0.875
0.960
0.956
0.989
0.985
0.979
0.6
0.638
0.927
0.906
0.983
0.979
0.974
0.997
0.997
1.000
0.999
1.000
0.7
0.874
0.990
0.989
0.999
1.000
0.999
1.000
1.000
1.000
1.000
1.000
0.8
0.985
0.999
1.000
0.997
1.000
0.999
0.998
1.000
0.994
0.998
0.999
NOTE: Yellow indicates power > 0.8; orange indicates power > 0.9; blue indicates unusual power when median complete
protection time = 2 hours and P5MR = 0.8.
Page 10 of 48
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Table 2: Results of power analysis when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.7
(Weibull distribution)
Median
(hours)
P5MR
Sample size
_10
_11
_12
_13
_14
_15
_16
_17
_18
_19
_20
2
0.2
0.022
0.096
0.066
0.199
0.121
0.128
0.217
0.211
0.304
0.299
0.222
0.4
0.132
0.415
0.299
0.607
0.484
0.522
0.634
0.677
0.751
0.767
0.668
0.5
0.267
0.633
0.516
0.789
0.697
0.748
0.803
0.851
0.868
0.895
0.838
0.6
0.476
0.813
0.732
0.881
0.845
0.864
0.885
0.919
0.893
0.926
0.918
0.7
0.694
0.895
0.876
0.861
0.889
0.847
0.850
0.888
0.799
0.837
0.888
0.8
0.768
0.780
0.821
0.673
0.750
0.591
0.652
0.699
0.566
0.622
0.694
4
0.2
0.016
0.075
0.053
0.166
0.109
0.088
0.190
0.171
0.276
0.245
0.177
0.4
0.103
0.332
0.267
0.517
0.452
0.402
0.620
0.555
0.752
0.681
0.638
0.5
0.210
0.525
0.468
0.715
0.685
0.624
0.830
0.776
0.914
0.866
0.848
0.6
0.402
0.736
0.714
0.886
0.880
0.833
0.955
0.923
0.979
0.966
0.969
0.7
0.673
0.914
0.917
0.971
0.975
0.962
0.987
0.986
0.982
0.988
0.995
0.8
0.927
0.970
0.987
0.945
0.971
0.946
0.931
0.955
0.892
0.922
0.958
6
0.2
0.047
0.083
0.066
0.158
0.105
0.083
0.174
0.150
0.247
0.218
0.149
0.4
0.079
0.294
0.225
0.473
0.387
0.356
0.556
0.507
0.690
0.636
0.566
0.5
0.172
0.483
0.406
0.679
0.622
0.573
0.779
0.735
0.887
0.841
0.806
0.6
0.335
0.697
0.649
0.861
0.851
0.804
0.938
0.909
0.977
0.963
0.956
0.7
0.607
0.894
0.885
0.975
0.970
0.958
0.994
0.989
0.997
0.996
0.997
0.8
0.897
0.987
0.992
0.988
0.995
0.993
0.989
0.993
0.978
0.987
0.995
8
0.2
0.309
0.234
0.297
0.289
0.320
0.210
0.347
0.251
0.392
0.306
0.306
0.4
0.180
0.321
0.294
0.497
0.435
0.379
0.598
0.521
0.726
0.654
0.592
0.5
0.206
0.499
0.439
0.692
0.645
0.603
0.804
0.762
0.904
0.867
0.830
0.6
0.357
0.731
0.684
0.892
0.872
0.831
0.957
0.933
0.983
0.978
0.966
0.7
0.634
0.922
0.907
0.985
0.981
0.976
0.998
0.994
0.998
0.998
0.999
0.8
0.913
0.996
0.995
0.999
0.999
0.998
0.997
1.000
0.994
0.997
0.999
NOTE: Yellow indicates power > 0.8; orange indicates power > 0.9; blue indicates unusual power when median complete
protection time = 2 hours and P5MR = 0.8.
Page 11 of 48
-------
Table 3: Results of power analysis when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.8
(Weibull distribution)
Median
(hours)
P5MR
Sample size
_10
_11
_12
_13
_14
_15
_16
_17
_18
_19
_20
2
0.2
0.007
0.032
0.027
0.064
0.044
0.027
0.079
0.061
0.115
0.088
0.071
0.4
0.035
0.119
0.103
0.219
0.188
0.140
0.275
0.227
0.355
0.287
0.258
0.5
0.068
0.210
0.185
0.339
0.297
0.251
0.418
0.364
0.506
0.460
0.425
0.6
0.133
0.341
0.306
0.496
0.473
0.407
0.594
0.560
0.678
0.670
0.641
0.7
0.251
0.561
0.532
0.679
0.692
0.598
0.753
0.753
0.749
0.775
0.809
0.8
0.455
0.696
0.728
0.654
0.728
0.562
0.648
0.694
0.565
0.620
0.692
4
0.2
0.004
0.026
0.012
0.053
0.027
0.022
0.054
0.045
0.085
0.065
0.042
0.4
0.026
0.093
0.080
0.186
0.151
0.103
0.254
0.182
0.346
0.245
0.228
0.5
0.060
0.170
0.165
0.315
0.297
0.202
0.436
0.315
0.546
0.413
0.414
0.6
0.135
0.317
0.320
0.494
0.499
0.374
0.650
0.529
0.760
0.643
0.639
0.7
0.295
0.548
0.565
0.726
0.754
0.651
0.863
0.784
0.914
0.854
0.873
0.8
0.619
0.828
0.867
0.884
0.923
0.864
0.913
0.922
0.886
0.909
0.947
6
0.2
0.038
0.033
0.027
0.055
0.037
0.027
0.053
0.033
0.076
0.058
0.039
0.4
0.022
0.098
0.072
0.206
0.135
0.115
0.234
0.196
0.341
0.289
0.214
0.5
0.054
0.205
0.154
0.364
0.281
0.248
0.438
0.382
0.567
0.493
0.425
0.6
0.133
0.383
0.335
0.572
0.525
0.473
0.694
0.626
0.812
0.748
0.716
0.7
0.316
0.646
0.614
0.819
0.818
0.750
0.918
0.874
0.965
0.943
0.938
0.8
0.670
0.916
0.917
0.967
0.974
0.962
0.986
0.984
0.977
0.985
0.993
8
0.2
0.301
0.193
0.270
0.198
0.264
0.155
0.250
0.157
0.244
0.171
0.206
0.4
0.122
0.136
0.141
0.227
0.182
0.142
0.267
0.208
0.340
0.292
0.229
0.5
0.082
0.209
0.165
0.368
0.282
0.256
0.434
0.392
0.561
0.505
0.424
0.6
0.124
0.390
0.321
0.588
0.514
0.490
0.688
0.655
0.823
0.779
0.710
0.7
0.299
0.683
0.610
0.857
0.808
0.794
0.915
0.909
0.966
0.963
0.939
0.8
0.644
0.940
0.909
0.989
0.978
0.981
0.994
0.995
0.993
0.995
0.998
NOTE: Yellow indicates power > 0.8; orange indicates power > 0.9.
Page 12 of 48
-------
Figure 4; Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.6 (Weibull distributions)
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
Weibull median = 4 hours, K = 0.6
1.0-
Weibull median = 2 hours, K = 0.6
Weibull median = 6 hours, K = 0.6
Weibull median = 8 hours, K = 0.6
Page 13 of 48
-------
re 5: Power curves of study design when the lowest acceptable ratio 95% LCLmCpi/mCPT = 0.7 (Weibull distributions)
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
Weibull median = 2 hours, K = 0.7
1.0 -
Weibull median = 8 hours, K = 0.7
1.0-
Weibull median = 4 hours, K = 0.7
Weibull median = 6 hours, K = 0.7
Page 14 of 48
-------
re 6: Power curves of study design when the lowest acceptable ratio 95% LCLmCpi/mCPT = 0.8 (Weibull distributions)
Weibull median = 2 hours, K = 0.8
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
Weibull median = 4 hours, K = 0.8
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 + 0.8 |
Weibull median = 6 hours, K = 0.8
Weibull median = 8 hours, K = 0.8
.^Sr=r.:
/
„ /
/ ' — -El
/
/
- A-
V / * / ~
/
/ V
/ /¦
s /
/ / ^
14 16
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
Page 15 of 48
-------
APPENDIX 1
Re-oarameterization of Standard Weibull Equation:
Given the definition of PDF and CDF from first principles:
fmCPT\K
P(mCPT,K,X) = 1 — e v A J =0.5 (median)
(PSMRxmCPT\K
P(PSMR x mCPT,K,X) = 1 — e v a / = 0.05 (5th percentile)
rmCPT\K
e v A ) =0.5 (median)
fP5MRxmCPT\K
e v A J = 0.95 (5th percentile)
and
Then:
(1)
(2)
Divide (2) by (1), we have:
[P5MR x mCPTf
A
ln(0.95)
ln(0.5)
mCPT
~~r~
fln(0.95)l .
k = ln /'n(p5MR)
(3)
From (1):
16
-------
fmCPT\
k x Iny—-—J = ln[—ln(0.5)]
(mCPT\ 1
In ^—-—J = - ln[-ln(0.5)]
1
In (mCPT) — ln(2) = — ln[—ln(0.5)]
K
1
ln(2) = ln(mCPT) — — ln[—ln(0.5)]
K
l
= —[Kln(mCPT)— ln[—ln(0.5)]]
K
l
= - [In(mCPTK)~ ln[— ln(0.5)]]
K
l
= - In
K
mCPTh
ln(0.5)J
1 , I" mCPTK 1
I = [ ln(o.s) J
So.
k = In
ln(0.95)
ln(0.5)
/ ln(P5Mi?)
1 x In mCPT
A = ex L In(0.5)
(As shown in the main text)
-------
APPENDIX 2
Product
Location
Sample
size
Est. mCPT
(95% CI)
Ratio of
95% LCiyest. mCPT
Est. Weibull
(shape K; scale A)
Est.
P5MR
A
1
10
7.5 (4.0, 8.0)
0.53
6.602; 7.777
0.674
2
10
8.5 (4.5, 10.0)
0.53
5.855; 8.624
0.641
B
1
10
12.0 (6.0, 12.0)
0.50
4.311; 10.669
0.547
2
10
12.0 (8.5, 12.0)
0.71
10.424; 11.516
0.779
C
1
10
7.5 (4.0, 9.0)
0.53
4.430; 8.146
0.556
2
8
5.0 (2.5, 5.5)
0.50
5.318; 4.915
0.613
D
1
10
2.0 (1.5, 2.0)
0.75
7.004; 2.135
0.690
2
10
2.5 (1.0, 3.5)
0.40
3.557; 2.8970
0.481
E
1
10
8.25 (6.0, 10.0)
0.73
7.609; 8.733
0.710
2
10
8.0 (3.5, 8.5)
0.44
4.009; 7.442
0.522
18
-------
SAS codes
APPENDIX 3
Programmer: James Nguyen, USEPA
Project: Mosquito Repellenov Studies
Purpose: Power Analysis/sample size calculation
Descrii. >i i < >11::
r N( > i in. ii. r 11F Uniform
"ie ( i I : : I I II II.I I I C >| 1; :
[ FE' .CLIFETEST
options formdlim=,,=" ps=90 ls=90 nonumber nodate;
libname MOS "C:\Users\JNguyen\Desktop\MOS";
%Macro distParam;
if upcase(Distribution) = "WEIBULL" then do;
* "Weibull :::::: f (x, a , b) ;
a = log (log(0.95)/log(0.5))/log(P5MR) ; b = exp ( (1/a)*log(-
(MED**a)/log(0.5)));
end;
if upcase(Distribution) = "UNIFORM" then do;
* uni form U [ a F b] ;
a = MED* (0. 5*P5MR - 0.05) / 0.45; b = MED"4" 2 - a;
end;
if upcase(Distribution) = "NORMAL" then do;
* norm a 1 :::::: N (a , b) ;
a = MED; b = MED* (1-P5MR)/1.645;
end;
if upcase(Distribution) = "LOGNORMAL" then do;
* log normal :::::: exp(N(a,b));
a = log(MED); b = (log(MED)-log(MED*P5MR))/l.645;
end;
%Mend;title;
%Macro generate',
if upcase
if upcase
if upcase
if upcase
%Mend;
%Macro Histogram(MED=, P5MRS=, dist=, seed=);
%let N=l;
%let P5MR&N = %nrbquote(%scan(&P5MRS,&N, %str( )));
%do %while (&&P5MR&N A=);
%let N=%eval(&N+1);
%let P5MR&N = %nrbquote(%scan(&P5MRS,&N, %str ( )));
%end;
%let N=%eval(&N-1);
(Distribution) = "WEIBULL" then CPT = rand ("Weibull", a, b) ;
(Distribution) = "LOGNORMAL" then CPT = exp (rand ( "Normal", a, b) ) ;
(Distribution) = "NORMAL" then CPT = rand("Normal", a, b);
(Distribution) = "UNIFORM" then CPT = a + (b-a)*rand("Uniform") ,
Data Parameters;
MED = &MED;
%do i = 1 %to &N;
P5MR = &&P5MR&i;
P5 = MED*P5MR;
output;
19
-------
run;
%end;
label MED = "P50";
Data Parameters;
set Parameters;
Distribution = "&dist";
%di stParam;
run;
data simmer;
call streaminit(&seed) ;
set parameters;
do i = 1 to 50000;
%generate;
output;
end; *i;
drop i a b;
title "&dist distribution: median=&MED";
Proc SGPANEL data = Simmer;
panelby P5MR/rows=&N;
Histogram CPT/binwidth=%sysevalf(2.5*&MED/50);
refline P5 /axis=x lineattrs=(pattern=l thickness=l color=blue);
refline MED/axis=x lineattrs=(pattern=l thickness=l color=blue);
colaxis values = (0 to %sysevalf(2.5*&MED) by 1);
run;
Proc datasets nolist; save sasmacr; run;quit;
%Mend;title;
iHistogram(MED=2, P5MRS=
& Histogram(ME D= 4, P 5MRS=
iHistogram (ME D= S, P5MRS=
&Histogram (MED=8, P5MRS=
& Histogram (ME D=10, P 5MRS=
0 . 2 0 . 3
0 „ 2 0 „ 3
0 . 2 0 . 3
0.5
0 „ 5
0 „ 5
0 „ 5
0.5
dist=weibull,
dist=weibull,
dist=weibull,
dist=weibull,
dist=weibull,
seed=279420) ,
seed=279420) ,
seed=279420) ,
seed=279420) ,
seed=279420) ,
%Macro Histograml(MED=, P5MR=, seed=);
Data Parameters;
MED = &MED;
P5MR = &P5MR;
P5 = MED^P5MR;
do i = 1 to 4;
if i = 1 then Distribution = "Lognormal";
if i = 2 then Distribution = "Normal";
if i = 3 then Distribution = "Uniform";
if i = 4 then Distribution = "Weibull";
%distParam;
output;
end;
label MED = "P50" P5MR="5%-tile/median"; drop i;
run;
data simmer;
call streaminit(&seed) ;
set parameters;
do i = 1 to 50000;
%generate;
output;
end; "J;'i;
drop i a b;
title "median=&MED P5MR=&P5MR" ;
Proc SGPANEL data = Simmer;
panelby Distribution/rows=4;
20
-------
Histogram CPT/binwidth=%sysevalf(2.5*&MED/50);
refline P5 /axis=x lineattrs=(pattern=l thickness=l color=blue);
refline MED/axis=x lineattrs=(pattern=l thickness=l color=blue);
colaxis values = (0 to %sysevalf(2.5*&MED) by 1);
run;
Proc datasets nolist; save sasmacr; run;quit;
%Mend;title;
% Histograml(MED=2, P5MR=0.2, seed=279420),
iHistograml(ME D=2, P5MR=0.4, seed=279420) ,
iHistograml(ME D=2, P5MR=0.5, seed=279420) ,
iHistograml(ME D=2, P5MR=0.6, seed=279420) ,
iHistograml(ME D=2, P5MR=0.7, seed=279420) ,
%Macro CPT;
CPT=CPT*60;
if CPT <= 5 then do;
LT =0; RT = 0; CPT= 0; censor = 0;
end;
else if CPT >= &maxT*S0 then do;
LT = &maxT*SO; RT=.; CPT=&maxT*80; censor = 1;
end;
else do;
LT = 30*floor( (CPT-5)/30)+5; RT = 30*ceil( (CPT-5)/30); CPT = RT;
censor = 0;
end;
CPT = CPT/60;
LT = LT/60;
RT = RT/60;
%Mend;title;
%Macro power;
ods select none;
%if &censor=right %then %do;
ods output Quartiles=MPT;
Proc lifetest data = Simmer(keep=MED P5MR N Sim CPT Censor);
by MED P5MR N Sim;
time CPT*Censor(1);
run;
%end;
%if &censor=interval %then %do;
ods output quartiles=MPT;
Proc iclifetest data = simmer(keep=MED P5MR N Sim LT RT) method=turnbull
impute(seed=1234);
by MED P5MR N Sim;
time (LT, RT);
run;
%end;
ods select default;
Proc datasets nolist; delete simmer; run;quit;
Data MPT;
set MPT;
if percent = 50;
power = (LowerLimit >= &K*Estimate);
%if &censor=right %then %do; Censor = "right";%end;
%if &censor=interval %then %do; Censor = "interval"; %end;
Proc SQL;
create table &dist&MED as
select Censor, MED, P5MR, N, avg(Power) as Power
from MPT
group by Censor, MED, P5MR, N;
quit;
21
-------
%Mend;title;
%Macro Mosquito(med=, P5MRS=, nmin=,nmax=,maxT=,K=,dist=,censor=,NSim=, seed=);
%let N=l;
%let P5MR&N = %nrbquote(%scan(&P5MRS,&N, %str( )));
%do %while (&&P5MR&N A=);
%let N=%eval(&N+1);
%let P5MR&N = %nrbquote(%scan(&P5MRS,&N, %str ( )));
%end;
%let N=%eval(&N-1);
%if &i = 1 %then %do; data All_&dist&MED; set _NULL_; run; %end;
Data Parameters;
MED = &MED;
P5MR = &&P5MR&i;
P5 = MED*P5MR;
label MED = "median" P5MR="5%-tile/median ratio";
run;
Data Parameters;
set Parameters;
Distribution = "&dist";
%dlstParam;
data simmer;
call streaminit(&seed) ;
set Parameters;
do N = &Nmin to &Nmax;
do Sim = 1 to ScNSim;
do ID = 1 to N;
%generate;
output;
end; *ID;
end; *Sirn;
end; *N;
drop a b;
Data Simmer;
set Simmer;
%CPT'f
run;
%pover;
Data All_&dist&MED;
set All_&dist&MED &dist&MED;
run;
Proc datasets nolist; delete Parameters simmer MPT &dist&MED; quit;
%end;
Data MOS.&distcensor._MED&MED._K%sysevalf(100*&K);
set All_&dist&MED;
run;
Proc datasets nolist; save sasmacr; run;quit;
%Mend;
dm log 'clear';%Mosqaito(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8, nmin=10, nmax=20, rnaxT=10,
K = 0.8, dist= weibull, censor=right, NSim=4000, seed=561);
22
-------
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.8
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.6
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
dm log
K = 0.7
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
clear
dist
%Mosc[uito (med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=561);
%Mosqu±to(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=561);
%Mosc[u.ito (med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=561);
%Mosqu±to(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosc[u.ito (med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
weibull, censor=right, NSim=4000, seed=352);
%Mosqu±to(med=2, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=4, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=6, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=8, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=2, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=4, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=6, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=8, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=2, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=4, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=6, P5MRS=0.
Lognormal, censor=right,
%Mosquito(med=8, P5MRS=0.
Lognormal, censor=right,
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=561);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=561);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=561);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=561);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
2 0.4 0.5 0.6 0.7 0.8
NSim=4000, seed=352);
%Mosqu±to{med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=561);
%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=561);
%Mosqu±to(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=561);
%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=561);
%Mosquito(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=352);
%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=352);
%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=352);
%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
Normal, censor=right, NSim=4000, seed=352);
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmin=10
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
nmax=20
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
23
-------
dm log 'clear%Mbsqu±to(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Normal, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Normal, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Normal, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Normal, censor=right, NSim=4000, seed=352);
dm log 'clear%Mbsqu±to(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.6, dist= Uniform, censor=right, NSim=4000, seed=561);
dm log 'clear%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.6, dist= Uniform, censor=right, NSim=4000, seed=561);
dm log 'clear%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.6, dist= Uniform, censor=right, NSim=4000, seed=561);
dm log 'clear%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.6, dist= Uniform, censor=right, NSim=4000, seed=561);
dm log 'clear%Mbsquito(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.7, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.7, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.7, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.7, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mbsquito(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Uniform, censor=right, NSim=4000, seed=352);
dm log 'clear%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.
K = 0.8, dist= Uniform, censor=right, NSim=4000, seed=352);
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10,
nmin=10,
nmin=10,
nmin=10,
nmax=20,
nmax=20,
nmax=20,
nmax=20,
maxT=10,
maxT=10,
maxT=10,
maxT=10,
/*
dm
K =
dm
K =
dm
K =
dm
K =
*/
log 'clear%Mosquito(med=2, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
0.7, dist= Weibull, censor=interval, NSim=4000, seed=352)
log 'clear%Mosquito(med=4, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
0.7, dist= Weibull, censor=interval, NSim=4000, seed=352)
log 'clear%Mosquito(med=6, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
0.7, dist= Weibull, censor=interval, NSim=4000, seed=352)
log 'clear%Mosquito(med=8, P5MRS=0.2 0.4 0.5 0.6 0.7 0.8
0.7, dist= Weibull, censor=interval, NSim=4000, seed=352)
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
nmin=10, nmax=20, maxT=10,
Create Figures and Print Results;
libname MOS "C:\Users\JNguyen\Desktop\MOS";
%let folder=C:\Users\JNguyen\Desktop\MOS;
%Macro SGPLOT(distribution=, K=);
title "Scdistribution median = 2 hours, K = O.&K";
Proc SGPLOT data = MOS.^distribution._right_med2_k&k.0;
scatter x = N y = Power/group = P5MR;
series x = N y = Power/group = P5MR;
refline 0.8 0.9/axis=y;
yaxis min=0 max=l;
run;
24
-------
title "Scdistribut
Proc SGPLOT data
scatter x
series
refline
x
yaxis min
run;
title "Scdistribut
Proc SGPLOT data
scatter x
series x
refline 0
yaxis min
run;
title "Scdistribut
Proc SGPLOT data
scatter x
series x
refline 0
yaxis min
run;
ion median = 4 hours,
MOS.^distribution._
N y = Power/group
N y = Power/group =
0.9/axis=y;
max=l;
ion median = 6 hours,
MOS.^distribution._
N y = Power/group
N y = Power/group =
0.9/axis=y;
max=l;
ion median = 8 hours,
= MOS.^distribution._
= N y = Power/group
N y = Power/group =
8 0.9/axis=y;
0 rnax=l;
K = 0.&K";
right_med4_k&
= P5MR;
P5MR;
K = 0.ScK";
right_med6_k&
= P5MR;
P5MR;
K = 0.ScK";
right_med8_k&
= P5MR;
P5MR;
%Mend;
%Macro print(dis
data &di;
run;
Proc transpose data
by MED P5MR;
ID N;
var Power;
run;
title
'-) ;
K;
i. but ion. _right_med2_k&
istribution._right_med4_k&
istribution._right_med6_k&
istribution._right_med8_k&
&distribution._K&K out = ^distribution._K&K(drop=_NAME_) ,
'^distribution K=0.&K.0'
Proc print data = &distribution._K&K noobs label; format
%mend;
%5GPX0T(distribution=Weibull, K= 6);
%S,GPI,OT(distribution=Weibull, K=7) ;
%S,GPI,OT(distribution=Weibull, K=8) ;
%5GPX0T(distribution=Lognormal, K= 6)
%5GPX0T(distribution=Lognormal, K=7)
%5GPX0T(distribution=Lognormal, K=8)
%5GPX0T(distribution=Normal, K= 6);
%S,GPI,OT(distribution=Normal, K=7) ;
%S,GPI,OT(distribution=Normal, K=8) ;
%S'GPXOT(distribution=Uniform, K= 6) ;
%S,GPI,OT(distribution=Uniform, K=7) ;
%S,GPI,OT(distribution=Uniform, K=8) ;
ods rtf file = "&folder\&dist Median=&MED K=&K..rtf" bodytitle;
%print(distribution=Weibull, K= 6);
%print(distribution=Weibull, K=7);
%print(distribution=Weibull, K=8);
%print(distribution=Lognormal, K= 6);
%print(distribution=Lognormal, K=7);
%print(distribution=Lognormal, K=8);
%print(distribution=Normal, K= 6);
%print(distribution=Normal, K=7);
%print(distribution=Normal, K=8);
25
-------
%print(distribution=Uniform, K= 6);
%print(distribution=Uniform, K=7);
%print(distribution=Uniform, K=8);
ods rtf close;
26
-------
APPENDIX 4
27
-------
Table 4-1. Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.6
(Lognormal distribution)
Median
Sample size
(hours)
2
rjlVIK
0.2
_10
0.045
_11
0.223
_12
0.126
_13
0.344
_14
0.239
_15
0.257
_16
0.355
_17
0.362
_18
0.440
_19
0.467
_20
0.353
0.4
0.236
0.580
0.476
0.776
0.666
0.678
0.815
0.812
0.900
0.903
0.869
0.5
0.449
0.820
0.770
0.937
0.902
0.899
0.958
0.962
0.982
0.985
0.984
0.6
0.768
0.969
0.955
0.979
0.988
0.980
0.980
0.991
0.963
0.981
0.988
0.7
0.964
0.971
0.985
0.933
0.961
0.931
0.926
0.948
0.886
0.915
0.940
0.8
0.894
0.867
0.914
0.801
0.851
0.761
0.797
0.839
0.743
0.792
0.827
4
0.2
0.048
0.169
0.103
0.259
0.188
0.162
0.295
0.254
0.381
0.343
0.279
0.4
0.176
0.472
0.411
0.682
0.595
0.578
0.749
0.735
0.851
0.844
0.796
0.5
0.367
0.729
0.662
0.895
0.834
0.828
0.924
0.927
0.973
0.972
0.960
0.6
0.638
0.932
0.896
0.984
0.977
0.972
0.989
0.992
0.998
0.999
0.997
0.7
0.919
0.996
0.991
0.999
0.999
0.998
0.997
0.999
0.995
0.998
0.999
0.8
0.994
0.992
0.997
0.976
0.990
0.981
0.971
0.984
0.949
0.971
0.979
6
0.2
0.175
0.207
0.202
0.283
0.240
0.199
0.343
0.266
0.417
0.355
0.304
0.4
0.180
0.474
0.400
0.677
0.600
0.561
0.751
0.706
0.845
0.827
0.794
0.5
0.360
0.703
0.665
0.876
0.826
0.804
0.930
0.916
0.971
0.963
0.956
0.6
0.635
0.917
0.900
0.982
0.976
0.964
0.992
0.993
0.999
0.998
0.999
0.7
0.922
0.994
0.993
1.000
0.999
0.999
1.000
1.000
0.999
1.000
1.000
0.8
0.999
0.999
1.000
0.998
0.999
0.998
0.996
0.999
0.993
0.998
0.999
8
0.2
0.408
0.389
0.438
0.449
0.470
0.371
0.535
0.418
0.594
0.501
0.487
0.4
0.378
0.567
0.551
0.739
0.697
0.635
0.813
0.766
0.886
0.864
0.842
0.5
0.469
0.742
0.731
0.898
0.868
0.831
0.942
0.924
0.979
0.967
0.963
0.6
0.680
0.923
0.919
0.987
0.983
0.966
0.994
0.992
0.998
0.999
0.998
0.7
0.929
0.994
0.994
1.000
1.000
0.999
1.000
1.000
1.000
1.000
1.000
0.8
0.999
1.000
1.000
1.000
1.000
0.999
1.000
1.000
0.999
1.000
1.000
-------
Table 4-2: Results of power analysis when the lowest acceptable ratio 95% LCL
0.7 (Lognormal distribution)
/mCPT =
Median
P5M
Sample size
(hours)
R
_10
_11
_12
_13
_14
_15
_16
_17
_18
_19
_20
0.2
0.008
0.048
0.034
0.110
0.070
0.063
0.117
0.106
0.172
0.159
0.106
0.4
0.072
0.285
0.197
0.465
0.355
0.390
0.489
0.530
0.608
0.619
0.519
2
0.5
0.197
0.529
0.419
0.697
0.591
0.641
0.699
0.750
0.783
0.811
0.720
0.6
0.440
0.749
0.648
0.856
0.775
0.816
0.844
0.893
0.877
0.907
0.874
0.7
0.677
0.878
0.831
0.896
0.895
0.870
0.886
0.933
0.855
0.909
0.926
0.8
0.800
0.866
0.905
0.796
0.849
0.744
0.795
0.843
0.737
0.786
0.837
0.2
0.024
0.056
0.037
0.110
0.064
0.058
0.107
0.093
0.160
0.139
0.090
0.4
0.057
0.227
0.173
0.411
0.312
0.297
0.462
0.423
0.593
0.541
0.469
0.5
0.157
0.430
0.368
0.627
0.561
0.521
0.714
0.669
0.819
0.765
0.745
4
0.6
0.353
0.666
0.643
0.838
0.827
0.769
0.909
0.890
0.958
0.928
0.935
0.7
0.692
0.894
0.906
0.966
0.970
0.945
0.988
0.983
0.992
0.993
0.996
0.8
0.954
0.986
0.993
0.977
0.991
0.979
0.970
0.985
0.945
0.965
0.982
0.2
0.161
0.128
0.142
0.153
0.149
0.099
0.165
0.118
0.200
0.163
0.121
0.4
0.068
0.210
0.157
0.368
0.279
0.258
0.402
0.361
0.542
0.487
0.403
0.5
0.127
0.379
0.300
0.591
0.487
0.466
0.646
0.614
0.775
0.727
0.669
6
0.6
0.284
0.623
0.559
0.809
0.773
0.735
0.880
0.867
0.941
0.912
0.914
0.7
0.608
0.879
0.873
0.964
0.959
0.944
0.988
0.981
0.997
0.995
0.997
0.8
0.941
0.993
0.996
0.996
0.999
0.998
0.998
0.999
0.995
0.997
0.999
0.2
0.394
0.311
0.381
0.331
0.382
0.274
0.390
0.296
0.407
0.320
0.333
0.4
0.273
0.317
0.326
0.461
0.415
0.331
0.525
0.432
0.628
0.540
0.500
0.5
0.260
0.439
0.406
0.621
0.577
0.504
0.712
0.650
0.824
0.771
0.731
8
0.6
0.362
0.663
0.617
0.844
0.822
0.775
0.904
0.891
0.959
0.941
0.933
0.7
0.653
0.905
0.898
0.979
0.975
0.972
0.993
0.992
0.999
0.999
0.998
0.8
0.959
0.999
0.998
0.999
1.000
1.000
1.000
1.000
0.999
1.000
1.000
-------
Table 4-3 Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.8
(Lognormal distribution)
Median
P5MR
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.005
0.022
0.020
0.043
0.031
0.019
0.050
0.032
0.074
0.044
0.033
0.4
0.021
0.067
0.062
0.144
0.117
0.087
0.178
0.134
0.252
0.193
0.164
0.5
0.043
0.142
0.125
0.253
0.212
0.157
0.288
0.246
0.378
0.334
0.283
0.6
0.094
0.253
0.219
0.414
0.364
0.301
0.487
0.448
0.571
0.549
0.510
0.7
0.210
0.491
0.465
0.661
0.649
0.571
0.734
0.750
0.771
0.797
0.809
0.8
0.534
0.794
0.824
0.780
0.833
0.723
0.792
0.839
0.736
0.785
0.836
4
0.2
0.019
0.022
0.014
0.042
0.025
0.021
0.029
0.028
0.048
0.046
0.019
0.4
0.014
0.073
0.044
0.146
0.101
0.075
0.155
0.119
0.228
0.173
0.140
0.5
0.036
0.129
0.109
0.237
0.207
0.151
0.311
0.226
0.412
0.300
0.303
0.6
0.095
0.253
0.256
0.399
0.404
0.297
0.540
0.409
0.654
0.513
0.532
0.7
0.258
0.462
0.496
0.649
0.666
0.558
0.782
0.688
0.861
0.770
0.801
0.8
0.623
0.779
0.834
0.880
0.914
0.845
0.926
0.914
0.923
0.916
0.951
6
0.2
0.157
0.098
0.119
0.088
0.108
0.061
0.095
0.054
0.096
0.066
0.061
0.4
0.028
0.068
0.046
0.138
0.092
0.073
0.144
0.125
0.203
0.184
0.121
0.5
0.033
0.141
0.094
0.266
0.187
0.176
0.292
0.257
0.408
0.355
0.285
0.6
0.078
0.292
0.228
0.469
0.397
0.358
0.541
0.482
0.680
0.606
0.560
0.7
0.253
0.552
0.513
0.737
0.721
0.655
0.839
0.799
0.917
0.862
0.877
0.8
0.680
0.882
0.904
0.962
0.971
0.946
0.989
0.982
0.992
0.992
0.996
8
0.2
0.392
0.279
0.366
0.270
0.348
0.244
0.332
0.245
0.315
0.234
0.289
0.4
0.240
0.194
0.230
0.235
0.245
0.169
0.267
0.193
0.305
0.237
0.214
0.5
0.173
0.203
0.193
0.301
0.266
0.195
0.337
0.275
0.435
0.368
0.306
0.6
0.145
0.301
0.251
0.495
0.414
0.369
0.538
0.516
0.680
0.639
0.558
0.7
0.250
0.588
0.502
0.773
0.714
0.707
0.837
0.841
0.916
0.888
0.876
0.8
0.660
0.910
0.901
0.978
0.973
0.971
0.993
0.990
0.998
0.998
0.997
30
-------
Table 4-4: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.6
(Normal distribution)
Median
P5MR
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.139
0.454
0.358
0.655
0.556
0.540
0.722
0.701
0.824
0.820
0.770
0.4
0.326
0.711
0.653
0.877
0.825
0.818
0.916
0.912
0.960
0.962
0.952
0.5
0.506
0.858
0.833
0.946
0.934
0.922
0.966
0.974
0.967
0.981
0.985
0.6
0.730
0.951
0.949
0.963
0.978
0.962
0.958
0.979
0.934
0.957
0.969
0.7
0.925
0.957
0.976
0.908
0.940
0.901
0.900
0.923
0.842
0.882
0.915
0.8
0.875
0.846
0.899
0.768
0.821
0.723
0.760
0.812
0.695
0.753
0.791
4
0.2
0.098
0.360
0.291
0.569
0.476
0.448
0.646
0.618
0.775
0.757
0.692
0.4
0.253
0.635
0.562
0.838
0.764
0.749
0.880
0.874
0.950
0.949
0.926
0.5
0.415
0.800
0.747
0.938
0.901
0.899
0.965
0.960
0.989
0.990
0.982
0.6
0.638
0.936
0.913
0.991
0.983
0.975
0.992
0.994
0.996
0.998
0.999
0.7
0.888
0.993
0.987
0.996
0.998
0.995
0.994
0.998
0.988
0.996
0.997
0.8
0.991
0.988
0.995
0.967
0.983
0.973
0.959
0.980
0.934
0.958
0.970
6
0.2
0.088
0.344
0.272
0.552
0.461
0.426
0.648
0.594
0.777
0.743
0.693
0.4
0.246
0.607
0.558
0.820
0.761
0.719
0.889
0.864
0.956
0.936
0.932
0.5
0.408
0.780
0.745
0.930
0.905
0.876
0.966
0.955
0.990
0.986
0.984
0.6
0.638
0.927
0.918
0.987
0.982
0.973
0.996
0.995
1.000
0.998
1.000
0.7
0.893
0.993
0.992
1.000
0.999
0.998
1.000
1.000
0.999
1.000
1.000
0.8
0.997
0.999
0.999
0.996
0.998
0.997
0.994
0.998
0.988
0.996
0.997
8
0.2
0.231
0.395
0.362
0.574
0.504
0.459
0.667
0.605
0.780
0.756
0.705
0.4
0.303
0.618
0.578
0.821
0.765
0.723
0.891
0.860
0.959
0.941
0.934
0.5
0.422
0.785
0.759
0.933
0.910
0.879
0.967
0.956
0.991
0.986
0.987
0.6
0.646
0.927
0.920
0.989
0.984
0.972
0.997
0.994
1.000
0.999
0.999
0.7
0.896
0.992
0.991
1.000
1.000
0.998
1.000
1.000
1.000
1.000
1.000
0.8
0.997
1.000
1.000
1.000
1.000
0.999
0.999
1.000
0.998
0.999
1.000
31
-------
Table 4-5: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.7
(Normal distribution)
Media" DCIWID
P5MR
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.044
0.216
0.143
0.390
0.287
0.293
0.429
0.441
0.566
0.559
0.450
0.4
0.136
0.465
0.347
0.662
0.549
0.592
0.684
0.727
0.790
0.813
0.728
0.5
0.262
0.627
0.521
0.803
0.710
0.744
0.801
0.857
0.857
0.892
0.840
0.6
0.460
0.796
0.712
0.891
0.836
0.854
0.877
0.925
0.883
0.920
0.916
0.7
0.690
0.885
0.860
0.882
0.895
0.855
0.867
0.915
0.822
0.879
0.908
0.8
0.795
0.843
0.886
0.761
0.819
0.706
0.759
0.811
0.689
0.743
0.801
4
0.2
0.033
0.175
0.130
0.340
0.257
0.216
0.405
0.346
0.535
0.469
0.413
0.4
0.111
0.369
0.317
0.581
0.516
0.466
0.683
0.622
0.811
0.746
0.709
0.5
0.214
0.526
0.475
0.742
0.704
0.643
0.833
0.797
0.915
0.871
0.870
0.6
0.391
0.718
0.697
0.882
0.881
0.821
0.944
0.925
0.976
0.957
0.964
0.7
0.683
0.905
0.915
0.971
0.979
0.957
0.990
0.988
0.990
0.993
0.996
0.8
0.940
0.983
0.993
0.966
0.984
0.968
0.961
0.978
0.926
0.957
0.973
6
0.2
0.028
0.153
0.105
0.300
0.215
0.191
0.345
0.301
0.483
0.426
0.347
0.4
0.085
0.328
0.259
0.550
0.455
0.416
0.619
0.580
0.754
0.708
0.646
0.5
0.174
0.486
0.416
0.710
0.648
0.600
0.794
0.764
0.886
0.851
0.827
0.6
0.334
0.681
0.635
0.863
0.837
0.798
0.927
0.914
0.970
0.951
0.952
0.7
0.607
0.892
0.885
0.971
0.969
0.958
0.991
0.989
0.997
0.996
0.998
0.8
0.925
0.991
0.995
0.994
0.999
0.996
0.997
0.998
0.992
0.996
0.998
8
0.2
0.185
0.222
0.214
0.351
0.296
0.231
0.416
0.335
0.522
0.448
0.390
0.4
0.162
0.350
0.308
0.560
0.492
0.428
0.653
0.603
0.784
0.730
0.680
0.5
0.205
0.494
0.439
0.726
0.678
0.621
0.809
0.785
0.909
0.875
0.851
0.6
0.358
0.713
0.662
0.890
0.870
0.835
0.945
0.932
0.981
0.971
0.970
0.7
0.642
0.916
0.902
0.985
0.980
0.980
0.996
0.996
0.999
0.999
0.998
0.8
0.943
0.998
0.998
0.999
1.000
0.999
1.000
1.000
0.998
0.999
1.000
32
-------
Table 4-6: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.8
(Normal distribution)
Media" DCIWID
P5MR
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.013
0.069
0.049
0.142
0.109
0.081
0.176
0.136
0.250
0.193
0.154
0.4
0.032
0.142
0.120
0.259
0.214
0.167
0.311
0.259
0.401
0.352
0.305
0.5
0.067
0.210
0.183
0.355
0.314
0.245
0.426
0.382
0.504
0.469
0.433
0.6
0.116
0.328
0.291
0.502
0.464
0.387
0.578
0.559
0.661
0.650
0.628
0.7
0.239
0.537
0.508
0.683
0.685
0.601
0.741
0.776
0.760
0.799
0.824
0.8
0.512
0.768
0.801
0.745
0.802
0.685
0.756
0.807
0.689
0.741
0.800
4
0.2
0.008
0.051
0.035
0.112
0.084
0.055
0.140
0.098
0.208
0.138
0.123
0.4
0.029
0.114
0.093
0.224
0.192
0.133
0.304
0.214
0.420
0.293
0.295
0.5
0.055
0.182
0.169
0.320
0.305
0.221
0.451
0.323
0.572
0.432
0.443
0.6
0.118
0.289
0.310
0.479
0.484
0.363
0.633
0.503
0.750
0.620
0.632
0.7
0.271
0.512
0.528
0.695
0.720
0.616
0.827
0.744
0.892
0.824
0.853
0.8
0.618
0.800
0.844
0.892
0.923
0.854
0.927
0.925
0.911
0.920
0.952
6
0.2
0.005
0.051
0.031
0.115
0.070
0.054
0.129
0.104
0.196
0.155
0.105
0.4
0.022
0.120
0.088
0.247
0.179
0.159
0.297
0.247
0.410
0.347
0.287
0.5
0.044
0.206
0.163
0.377
0.297
0.265
0.453
0.389
0.584
0.509
0.457
0.6
0.111
0.357
0.307
0.558
0.502
0.444
0.664
0.600
0.799
0.721
0.691
0.7
0.285
0.605
0.573
0.796
0.786
0.716
0.885
0.861
0.949
0.907
0.921
0.8
0.680
0.898
0.913
0.968
0.977
0.955
0.990
0.986
0.990
0.993
0.996
8
0.2
0.164
0.131
0.143
0.164
0.163
0.101
0.190
0.135
0.245
0.177
0.146
0.4
0.097
0.152
0.128
0.260
0.204
0.158
0.298
0.248
0.411
0.353
0.283
0.5
0.082
0.213
0.171
0.389
0.303
0.265
0.440
0.401
0.589
0.536
0.445
0.6
0.119
0.362
0.290
0.591
0.496
0.465
0.660
0.635
0.789
0.759
0.695
0.7
0.273
0.638
0.564
0.827
0.773
0.760
0.887
0.891
0.945
0.933
0.918
0.8
0.658
0.923
0.909
0.983
0.976
0.978
0.995
0.995
0.998
0.998
0.998
33
-------
Table 4-7: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.6
(Uniform distribution)
ra: p™*
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.059
0.266
0.177
0.434
0.319
0.331
0.471
0.450
0.576
0.586
0.491
0.4
0.242
0.561
0.468
0.727
0.657
0.664
0.792
0.772
0.876
0.870
0.824
0.5
0.429
0.788
0.736
0.923
0.880
0.889
0.955
0.949
0.981
0.982
0.974
0.6
0.987
0.995
0.994
0.995
0.995
0.993
0.990
0.998
0.981
0.990
0.996
0.7
0.996
0.990
0.994
0.970
0.981
0.968
0.960
0.977
0.931
0.961
0.975
0.8
0.900
0.904
0.934
0.851
0.886
0.793
0.849
0.875
0.813
0.851
0.883
4
0.2
0.038
0.183
0.126
0.336
0.253
0.226
0.385
0.340
0.499
0.467
0.397
0.4
0.175
0.446
0.372
0.641
0.577
0.547
0.723
0.675
0.814
0.807
0.743
0.5
0.357
0.681
0.627
0.851
0.796
0.802
0.906
0.883
0.947
0.948
0.917
0.6
0.693
0.919
0.876
0.975
0.956
0.960
0.986
0.984
0.996
0.996
0.991
0.7
1.000
1.000
1.000
1.000
1.000
0.999
0.999
1.000
0.999
0.999
1.000
0.8
0.999
0.997
1.000
0.996
0.996
0.994
0.990
0.998
0.981
0.990
0.996
6
0.2
0.037
0.161
0.114
0.313
0.228
0.198
0.368
0.312
0.485
0.442
0.384
0.4
0.159
0.437
0.359
0.626
0.559
0.541
0.722
0.660
0.819
0.784
0.740
0.5
0.371
0.653
0.620
0.823
0.797
0.766
0.895
0.853
0.946
0.930
0.910
0.6
0.674
0.899
0.879
0.971
0.960
0.951
0.987
0.985
0.996
0.996
0.992
0.7
0.970
0.998
0.999
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.8
1.000
1.000
1.000
1.000
1.000
0.999
0.999
1.000
0.999
0.999
1.000
8
0.2
0.291
0.302
0.323
0.423
0.390
0.299
0.499
0.375
0.564
0.491
0.458
0.4
0.314
0.493
0.463
0.665
0.625
0.561
0.747
0.668
0.834
0.796
0.751
0.5
0.452
0.687
0.668
0.835
0.813
0.784
0.905
0.872
0.949
0.934
0.918
0.6
0.744
0.903
0.902
0.968
0.965
0.952
0.991
0.983
0.995
0.995
0.992
0.7
0.977
0.998
0.997
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.8
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
34
-------
Table 4-8: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.7
(Uniform distribution)
ra: p™*
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.016
0.079
0.052
0.168
0.117
0.098
0.184
0.186
0.277
0.267
0.197
0.4
0.073
0.242
0.172
0.417
0.310
0.335
0.445
0.489
0.570
0.601
0.491
0.5
0.180
0.470
0.371
0.645
0.540
0.584
0.662
0.711
0.725
0.780
0.683
0.6
0.543
0.731
0.636
0.803
0.711
0.767
0.783
0.839
0.826
0.874
0.808
0.7
0.624
0.847
0.757
0.890
0.853
0.855
0.885
0.914
0.896
0.931
0.918
0.8
0.820
0.904
0.914
0.845
0.885
0.803
0.841
0.882
0.813
0.856
0.882
4
0.2
0.014
0.066
0.041
0.145
0.100
0.073
0.162
0.145
0.266
0.220
0.155
0.4
0.055
0.203
0.148
0.351
0.285
0.251
0.425
0.379
0.557
0.494
0.432
0.5
0.136
0.365
0.305
0.551
0.499
0.442
0.648
0.594
0.746
0.690
0.655
0.6
0.354
0.600
0.572
0.750
0.761
0.670
0.854
0.807
0.908
0.872
0.874
0.7
0.782
0.850
0.876
0.934
0.947
0.896
0.973
0.954
0.989
0.976
0.980
0.8
1.000
0.999
1.000
0.994
0.995
0.995
0.992
0.995
0.984
0.991
0.996
6
0.2
0.016
0.055
0.035
0.129
0.079
0.062
0.141
0.120
0.224
0.190
0.125
0.4
0.045
0.180
0.118
0.313
0.244
0.209
0.367
0.334
0.493
0.450
0.371
0.5
0.116
0.330
0.252
0.496
0.421
0.384
0.572
0.525
0.672
0.638
0.581
0.6
0.280
0.549
0.493
0.713
0.683
0.616
0.799
0.764
0.875
0.848
0.828
0.7
0.665
0.839
0.846
0.931
0.933
0.896
0.970
0.956
0.992
0.979
0.982
0.8
1.000
1.000
1.000
1.000
1.000
1.000
0.999
1.000
0.999
1.000
1.000
8
0.2
0.268
0.210
0.267
0.257
0.268
0.179
0.297
0.215
0.358
0.267
0.244
0.4
0.203
0.248
0.240
0.367
0.331
0.245
0.434
0.367
0.542
0.467
0.400
0.5
0.203
0.356
0.318
0.513
0.467
0.394
0.615
0.554
0.713
0.660
0.612
0.6
0.338
0.576
0.550
0.741
0.722
0.661
0.838
0.806
0.902
0.877
0.854
0.7
0.693
0.892
0.882
0.959
0.956
0.942
0.982
0.980
0.995
0.990
0.990
0.8
0.996
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
35
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Table 4-9: Results of power analysis when the lowest acceptable ratio 95% LCL /mCPT = 0.8
(Uniform distribution)
ra: p™*
(hours)
Sample size
_10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20
2
0.2
0.009
0.031
0.023
0.065
0.053
0.031
0.081
0.051
0.120
0.091
0.070
0.4
0.027
0.070
0.057
0.144
0.113
0.084
0.178
0.128
0.253
0.190
0.162
0.5
0.047
0.121
0.119
0.227
0.192
0.142
0.275
0.211
0.344
0.278
0.246
0.6
0.131
0.205
0.200
0.313
0.283
0.221
0.375
0.322
0.476
0.409
0.373
0.7
0.142
0.364
0.315
0.530
0.483
0.418
0.609
0.595
0.703
0.692
0.651
0.8
0.641
0.864
0.871
0.837
0.879
0.795
0.841
0.880
0.813
0.856
0.882
4
0.2
0.003
0.017
0.009
0.048
0.024
0.019
0.050
0.038
0.081
0.063
0.033
0.4
0.013
0.061
0.038
0.121
0.090
0.066
0.138
0.103
0.205
0.160
0.122
0.5
0.031
0.112
0.080
0.195
0.159
0.114
0.249
0.185
0.342
0.251
0.231
0.6
0.092
0.198
0.194
0.309
0.325
0.211
0.428
0.317
0.528
0.403
0.428
0.7
0.303
0.374
0.415
0.523
0.564
0.428
0.673
0.562
0.755
0.654
0.675
0.8
0.766
0.735
0.806
0.805
0.860
0.769
0.902
0.839
0.916
0.874
0.899
6
0.2
0.006
0.015
0.009
0.041
0.019
0.017
0.042
0.029
0.069
0.054
0.029
0.4
0.010
0.049
0.028
0.110
0.069
0.050
0.120
0.101
0.189
0.160
0.106
0.5
0.026
0.101
0.065
0.197
0.142
0.114
0.214
0.201
0.327
0.284
0.219
0.6
0.074
0.238
0.164
0.368
0.298
0.271
0.421
0.382
0.541
0.480
0.426
0.7
0.246
0.461
0.422
0.608
0.611
0.521
0.732
0.656
0.808
0.751
0.748
0.8
0.773
0.824
0.860
0.913
0.939
0.880
0.966
0.940
0.986
0.970
0.978
8
0.2
0.261
0.178
0.247
0.181
0.220
0.141
0.215
0.139
0.214
0.146
0.159
0.4
0.175
0.129
0.162
0.175
0.176
0.100
0.203
0.141
0.245
0.184
0.143
0.5
0.134
0.144
0.135
0.214
0.185
0.128
0.253
0.206
0.338
0.282
0.223
0.6
0.119
0.224
0.182
0.354
0.289
0.254
0.418
0.378
0.529
0.492
0.407
0.7
0.223
0.485
0.419
0.652
0.594
0.564
0.730
0.701
0.803
0.794
0.748
0.8
0.679
0.868
0.851
0.939
0.942
0.916
0.976
0.968
0.991
0.984
0.987
36
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Figure 4-1: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.6 (Lognormal distributions)
Lognormal median = 2 hours, K = 0.6
14 16
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 + 0.8 |
Lognormal median = 4 hours, K = 0.6
14 16
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 <¦ 0.8 |
Lognormal median = 6 hours, K = 0.6
Lognormal median = 8 hours, K = 0.6
15%-tlie/median ratio o 0.2 -i- 0.4 0.5 Ajjj 0.; 0.8 |
| 5%-tlie/'median ratio O 0.2 +0.4 0.5 A 0.0 07 0.8 |
37
-------
Figure 4-2: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.7 (Lognormal distributions)
Lognormal median = 2 hours, K = 0.7
^£3 B- _
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 + 0.8 |
Lognormal median = 4 hours, K = 0.7
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 <¦ 0.8 |
Lognormal median = 6 hours, K = 0.7
Lognormal median = 8 hours, K = 0.7
XT""
a q —
- * s
p ,
X"
/
/
/ -V /
/ .
/ /
A.
/ /
y
14 16
N
15%-tlie/median ratio o 0.2 + 0.4 0.5 A 0.6 0.; O S |
15%-tile/inedian ratio O 0.2 +0.4 0.5 A 0.0 07 0.8 |
38
-------
re 4-3: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.8 (Lognormal distributions)
Lognormal median = 2 hours, K = 0.8
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 + 0.8 |
Lognormal median = 4 hours, K = 0.8
A
/
\
^
A- —V /
/X /
V
>< *
/
/
/ X1--
14 16
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 <¦ 0.8 |
Lognormal median = 6 hours, K = 0.8
Lognormal median = 8 hours, K = 0.8
->H-
* *—
xu.
_ " *
*
/
B- . _
/
/
-•¦a--—-Q
/
/
P~ —
/
/
/
-cf
/
— B. ^
A-
/
y
^ -a
/
/
/
*
\
i A-.
/
X
\
^x
/
V
^ ,+ — ___
-K.
-e o
14 16
N
15%-tlie/median ratio o 0.2 -i- 0.4 0.5 Ajjj 0.; 0.8 |
| 5%-tlie/'median ratio O 0.2 +0.4 0.5 A 0.0 07 0.8 |
39
-------
Figure 4-4: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.6 (Normal distributions)
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 * 0.8 |
Normal median = 2 hours, K = 0.6
Normal median = 4 hours, K = 0.6
Normal median = 6 hours, K = 0.6
Normal median = 8 hours, K = 0.6
40
-------
Figure 4-5: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.7 (Normal distributions)
Normal median = 4 hours, K = 0.7
Normal median = 8 hours, K = 0.7
Normal median = 2 hours, K = 0.7
Normal median = 6 hours, K = 0.7
41
-------
Figure 4-6: Power curves of study design when the lowest acceptable ratio 95% LCLmcpi/mCPT = 0.8 (Normal distributions)
Normal median = 2 hours, K = 0.8
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 + 0.8 |
Normal median = 4 hours, K = 0.8
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 <¦ 0.8 |
Normal median = 6 hours, K = 0.8
Normal median = 8 hours, K = 0.8
15%-tlie/median ratio o 0.2 -i- 0.4 0.5 Ajjj 0.; 0.8 |
| 5%-tlie/'median ratio O 0.2 +0.4 0.5 A 0.0 07 0.8 |
42
-------
Figure 4-7: Power curves of study design when the lowest acceptable ratio 95% LCLmCpi/mCPT = 0.6
(Uniform distributions)
Uniform median = 2 hours, K = 0.6
N
15%-tile/median ratio o 0.2 + 0.4 0.5 A 0.6 07 Q.8 |
Uniform median = 4 hours, K = 0.6
N
15%-tile/median ratio o 0.2 + 0.4 0.5 A 0.0 07 0.8 |
43
-------
Uniform median = 6 hours, K = 0.6
N
15%-tile/median ratio o 0.2 + 0.4 0.5 A 0.6 07 0.8 |
Uniform median = 8 hours, K = 0.6
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 * 0.8 |
Figure 4-8: Power curves of study design when the lowest acceptable ratio 95% LCLmCpi/mCPT = 0.7
(Uniform distributions)
44
-------
Uniform median = 2 hours, K = 0.7
N
15%-tile/median ratio o 0.2 + Q.4 0.5 A 0.6 07 0.8 |
Uniform median = 4 hours, K = 0.7
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
45
-------
Uniform median = 6 hours, K = 0.7
N
15%-tile/median ratio o 0.2 + 0.4 0.5 A 0.6 07 0.8 |
Uniform median = 8 hours, K = 0.7
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 * 0.8 |
Figure 4-9: Power curves of study design when the lowest acceptable ratio 95% LCLmCpi/mCPT = 0.8
(Uniform distributions)
46
-------
Uniform median = 2 hours, K = 0.8
N
15%-tile/median ratio o 0.2 + Q.4 0.5 A 0.6 0.7 0.3 |
Uniform median = 4 hours, K = 0.8
1.0-
10 12 14 16 18 20
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
47
-------
Uniform median = 6 hours, K = 0.8
10 12 14 16 18 20
N
15%-tile/median ratio o 0.2 + Q.4 0.5 A 0.6 0.7 0.3 |
Uniform median = 8 hours, K = 0.8
N
|5%-tile/median ratio o 0.2 + 0.4 x 0.5 a 0.6 ~ 0.7 t 0.8 |
48
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