Sample Size for Pet Product Studies

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Table of Contents

Executive Summary	2
1.	Background	3
2.	Simulations and Assumptions	3
3.	Conclusions and Suggested sample sizes based on the results of simulations	6
Simulations of Study Design 1: Narthropods/animal/day = 50+-5 and blood-fed proportion of arthropod on each animal >= 25%	8
Simulations of Study Design 2: Narthropods/animal/day = 50+-5 and blood-fed proportion of arthropod on each animal >= 40%	15
Simulations of Study Design 3: Narthropods/animal/day = 100+-10 and blood-fed proportion of arthropod on each animal >= 40%	21
Simulations of Study Design 4: Narthropods/animal/day = 50+-5 and blood-fed proportion of arthropod on each animal >= 60%	27
Simulations of Study Design 5: Narthropods/animal/day = 100+-10 and blood-fed proportion of arthropod on each animal >= 60%	32
Appendix II: Evaluation of random effects of logit generated from normal distributions and Weibull distribution	37
Appendix III: True blood-fed proportion vs. proportions of animals/datasets meet the required blood-fed proportion	42

SAS Programs of Simulations

 Executive Summary
This document is directly related to the guideline for Efficacy Testing of Topically Applied Pesticides Used Against Certain Ectoparasitic Pests on Pets, Charge Question #9.

Briefly, the revision of guideline for testing topically applied pet products provides an opportunity to assess sample sizes from a statistical standpoint. Historically, EPA has recommended 6 animals per treatment, but no statistical simulation or calculation had been conducted to evaluate the statistical power of this experimental design to detect a treatment-related effect. In practice, the agency has often received studies which utilized between 2 and 12 animals per treatment.

Here we provide statistical simulations based on sections (j)  -  (l) of the draft guideline, which illustrate the potential power of several different experimental designs to detect efficacy of 90% or greater based on the behavior of different types of arthropods. Adequate power is typically considered 80% power to detect a given true effect at a given precision. For the studies in the guideline, EPA determined the sample size which results in 80% power to detect the true efficacy of >=90% with a precision of <= 5% using a normal distribution. The Agency will be soliciting comments of this analysis and how it relates to the draft guideline.

Specifically:
 For studies assessing efficacy against ticks (study design 1), 14 animal subjects would be appropriate per treatment.
 For studies assessing efficacy against fleas, 7 animal subjects per treatment would be appropriate for tests on cats (study design 3), and 6 animal subjects are appropriate for tests on dogs (study design 5).
 For studies assessing efficacy against mosquitoes, 6 animal subjects per treatment would be appropriate if testing against one species (study design 5), and 8 animal subjects per treatment if testing against three mosquito species simultaneously (study design 4).
 For studies assessing efficacy against stable flies, horse flies, and deer flies, 8 animal subjects per treatment would be appropriate (study design 4).
 For studies assessing efficacy against sand flies, black flies, or biting midges, 6 animal subjects would be appropriate (study design 5).

Background
   Purpose
   Historically, the Agency has recommended using a minimum of 6 animals per study group for efficacy testing of topically applied products on pets, however studies have been submitted with 2-12 animals per group. Previously no statistical simulation or calculation had been conducted to evaluate the statistical power associated with the sample size of these experimental designs in detecting a treatment-related effect. Therefore, this document presents the simulations that have been conducted to determine the sample sizes and associated power of the study designs in the proposed guideline.
   Study design
 2 groups (1 control and 1 treatment) with X animals per group
 Data are collected at 4 different days over the study duration
 Each day, Y arthropods (+-10%) per animal
   Objective of study
 Evaluate the efficacy or effectiveness of the treatment where efficacy or effectiveness of the treatment is defined as ABBOTT's formula:
 (retained or blood-fed proportion of control group  -  retained or blood-fed proportion of treated group)/retained or blood-fed proportion of control group for repellency evaluation.
 (Survival proportion of control group  -  survival proportion of treated group)/survival proportion of control group for mortality evaluation.
   Objective of Simulations
   Determine the sample size and the power to obtain an observed efficacy >= 90% with a given (suitable) pre-stated level of precision in studies to evaluate repellency of pet products.

Simulations and Assumptions
   Assumptions in the simulations
 Given a total number of arthropods tested on an s[th] animal on a specific i[th] day, the number of blood-fed arthropods follows a binomial distribution with p= Psi, where Psi is the true proportion of blood-fed arthropods on the s[th] animal on i[th] day.

 Additional to the variation of the arthropods (described by the variation of binomial distribution), animal random effects and day random effects are permitted in the simulation.  These last two sources of variation typically cause the data to be over dispersed.

 The results from simulations in Appendix II indicate that the random effects of animal or day can be reasonably assumed to follow normal distributions or Weibull distributions.  Estimated proportions of blood-fed and estimated treatment efficacy of simulated datasets where random animal effects and random day effects generated from normal distributions or Weibull distributions are unbiased estimates of the true values.
 For the purpose of simulation, efficacy was defined as "blood-fed proportion." However, for application in the guideline, the words "mortality" and "blood-fed proportion" can be used interchangeably for each model when determining sample size.

 The true logit of the s[th] animal on a given i[th] day is Qsi = Q + rands + randi, where rands is the random effect of the s[th] animal (on i[th] day), and randi is the random effect of the i[th] day on the s[th] animal.

 The random animal effects were generated as below:
 P is the true blood-fed proportion of a group (on a specific day)
 Logit Q = log(P/(1-P))
 Logit Qs= Q + rands is the logit of s[th] animal
 When the animal random effects are assumed to be normal distribution, rands = random number from normal distribution with mean = 0 and SD = 0.33.
 When the animal random effects are assumed to follow a Weibull distribution, rands = (random number from Weibull distribution shape a and location b)  -  SubVar, where a = 1/P and b = exp((1/a)xlog(-(SubVar[a])/log(0.5))).

                  Note that the Weibull distribution shape a and location b has a median value SubVar.  If SubVar = 0, then rands is set = 0.

                  Since the value of blood-fed proportion P can vary by day (due to the change in efficacy of the treatment, for example), the shape a = 1/P of the Weibull distributions of random effects can differ by day.  Therefore, a same value rands cannot be used for the same s[th] animal on different days.  However, the quantile value of random effect of an animal in the distribution of random effects is assumed to be consistent on different days (for example, one animal has a random effect located at the 75[th] percentile of the random effect distribution on one day; its random effect on any other day is also located at the 75[th] percentile).

                  A quantile value U that describes the location of random effects of an s[th] animal is randomly generated from the uniform distribution [0,1], and the associated rands of the s[th] animal on each specific i[th] day is then calculated based on the quantile value U and the specific animal random effect Weibull distribution (a, b) on i[th] day.

 Similarly, the random day effects were generated as below
 Ps is the blood-fed proportion of arthropods on an s[th] animal (on a specific i[th] day)
 Logit Qs = log(Ps/(1-Ps))
 Logit Qsi = Qs + randi is the logit of animal s[th] on specific i[th] day
 When the day random effects are assumed to be normal distribution, randi = random number from normal distribution with mean = 0 and SD = 0.166.
 When the day random effects are assumed to be Weibull distribution, randi = (random number from Weibull distribution shape ai and location bi)  -  DayVar, where ai = 1/Ps and bi = exp((1/ai) xlog(-(DayVar[a][i])/log(0.5))).
                     Note that the Weibull distribution shape ai and location bi has a median value DayVar.  Also, on a given day, different animals have different value of ai, so they have different Weibull distributions of day effects.  If DayVar = 0, then randi is set = 0.
 Therefore, the logit of the s[th] animal on a given i[th] day is Qsi = Q + rands + randi.  The blood-fed proportion of arthropods on s[th] animal on a specific i[th] day is Psi = exp(Qsi)/(1+exp(Qsi)).

 Number of arthropods used on s[th] animal on i[th] day is randomly generated from Uniform distribution Nsi =  U [ NPDayx(1  -  Rndiff), NPDayx (1  -  Rndiff) ]

 Number of blood-fed arthropods on s[th] animal on i[th] day is randomly generated from binomial distribution of Nsi trials and success proportion =  Psi.

 Generalized linear mixed effects model (PROC GLIMMIX) for Poisson distribution data with animal and day as random effects for over-dispersed data (RANDOM Day/subject = animal type = cs; RANDOM _residual_;) is used to analyze each dataset.  For those datasets that have a computational (convergence) issue in the generalized linear mixed effects model, generalized linear model for Poisson distribution data is used to separately analyze the data by Day.  The blood-fed proportion ratio (i.e. adjusted blood-fed proportion of treatment group) and its 95% CI are estimated is by each Day.  The efficacy of the treatment group = 1  -  adjusted blood-fed proportion.  The precision of the estimated efficacy = estimated efficacy  -  lower limit of 95% CI.

 The power of a study design is the proportion of datasets that have observed treatment efficacy >= 0.90 and observed precision <= ReqPrecision.
   Important Parameters in simulations
 NminSub = the lowest number of ANIMALS/group
 NmaxSub = the highest number of ANIMALS/group
 NPDay = number of arthropods/animal/day
 RNdiff = allowable proportion of deviation in number arthropods/animal
 CP = true blood-fed proportion of control group
 TE = true efficacy of treatment, (1 - TE is the adjusted blood-fed proportion of treated group)
 ReqTE = required observed efficacy of treatment
 ReqPrecision = required maximum precision value
 SubVar  or SD_sub = degree of variation between ANIMAL
 DayVar or SD_day = degree of variation between DAY
 Nsim = number of simulated datasets per scenario (set to be 2000)
 seed = seed to regenerate the results of simulations

Conclusions and Suggested sample sizes based on the results of simulations
   Conclusions
   The results of power in Appendix I indicate that when the true efficacy is 0.85, the power (i.e. false discovery rate to claim the efficacy >= 0.90) is very low even when the required precision is 0.05.  Therefore, a required precision of 0.05 is suggested for use to optimize the power to detect the products with true efficacy >= 0.90.
   The statistical model used to analyze the simulated data is an acceptable technique for evaluating the efficacy of the products. As mentioned above, when the true efficacy of a product is 0.85 (i.e., less than the required efficacy of 0.90), there is a very low probability (< 0.02) that the product would be incorrectly accepted (i.e., low probability that the estimated efficacy from a study of this product >= 0.90 and the precision <=0.05). When the true efficacy of a product is 0.925 (slightly greater than the required efficacy of 0.90), there is high probability that the product would be correctly accepted based on the results of an adequate sample size study (i.e., there is at least 0.80 power that the study would result an estimated efficacy >= 0.90 and a precision <=0.05).
   Suggested sample sizes
   Below is the summary table including simulation results of all the scenarios analyzed where both animal random effects and day random effects were randomly generated from normal distributions or Weibull distributions (i.e. both animal random effects and day random effects exist in the dataset). From the simulations based on sections (j)-(l) of the draft guideline, EPA determined the sample size which results in 80% power to detect the true efficacy of >=90% with a precision of <= 5% using a normal distribution for the studies in the guideline. For studies assessing efficacy against ticks (study design 1), 14 animal subjects would be appropriate per treatment. For studies assessing efficacy against fleas, 7 animal subjects per treatment would be appropriate for tests on cats (study design 3), and 6 animal subjects are appropriate for tests on dogs (study design 5). For studies assessing efficacy against mosquitoes, 6 animal subjects per treatment would be appropriate if testing against one species (study design 5), and 8 animal subjects per treatment if testing against three mosquito species simultaneously (study design 4).  For studies assessing efficacy against stable flies, horse flies, and deer flies, 8 animal subjects per treatment would be appropriate (study design 4). For studies assessing efficacy against sand flies, black flies, or biting midges, 6 animal subjects would be appropriate (study design 5). EPA will be soliciting comments on this analysis and how it relates to the draft guideline.

                                 Study design
                         Relevant section of guideline
     Required lowest observed blood-fed proportion on each control animal
                  True blood fed proportion in control group
                       No. arthropods per animal per day
                                    True TE
                          Assumed distribution types
                                  Sample size
                                     Power
               to observe TE >= 0.90 and precision <= 0.050
                                       1
                         Ticks on cats or dogs, (k)(2)

                                     0.25
                                     0.45
                                     50+-5
                                     0.925
                                    Normal
                                      14
                                     0.811

                                    Weibull
                                      13
                                     0.828
                                       2
                           Not included in guideline
                                     0.40
                                     0.60
                                     50+-5
                                     0.925
                                    Normal
                                      11
                                     0.834

                                    Weibull
                                      10
                                     0.860
                                       3
                             Fleas on cats, (j)(2)

                                     0.40
                                     0.60
                                    100+-10
                                     0.925
                                    Normal
                                       7
                                     0.798

                                    Weibull
                                       7
                                     0.875
                                       4
          Mosquitoes or flies on cats or dogs using 50 pests, (l)(2)
                                     0.60
                                     0.80
                                     50+-5
                                     0.925
                                    Normal
                                       8
                                     0.839

                                    Weibull
                                       6
                                     0.801
                                       5
Fleas on dogs, (j)(2) Mosquitoes or flies for cats and dogs using 100 pests, (l)(2)
                                     0.60
                                     0.75
                                    100+-10
                                     0.925
                                    Normal
                                       6
                                     0.834

                                    Weibull
                                       6
                                     0.930

Appendix I: Details of study designs and input parameters
     Simulations of Study Design 1: Narthropods/animal/day = 50+-5 and blood-fed proportion of arthropod on each animal >= 25%
 The study design:
 The results of this study design apply to studies that test 1 tick species/animal/day
 Number of arthropods/animal/day = 50+-5
 4 different days
 Inclusion criterion: blood-fed proportion of arthropod on each animal >= 25%
 Input parameters for the simulations:
 From the simulation in SAS program "percent animals or datasets have at least  a certain percent arthropod non-blood-fed.sas" using the criterion that the blood-fed proportion of arthropod on each animal >= 25%, a true blood-fed proportion CP of arthropods in control animals can reasonably be as low as 0.40.
 NPDay = 50 arthropods/animal/day
 RNdiff = 0.1
 True efficacy of treatment is set = 0.95, 0.925, 0.90, and 0.85 for the 1[st], 2[nd], 3[rd], and 4[th] days.
 Required observed treatment efficacy is ReqTE = 0.90
 Required precision of efficacy is evaluated as ReqPrecision = 0.025, 0.03, 0.04, and 0.05.
 The simulations are set to have 2000 datasets per each sample size = 8, 9, 10, 11, and 15.

      Table A1.1-1
 N per Day = 50+-5, CP = 0.45, no animal random effect, no day random effects
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.010
                                                                          0.078
                                                                          0.433
                                                                          0.825

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.004
                                                                          0.016
                                                                          0.177
                                                                          0.565

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.003
                                                                          0.061
                                                                          0.267

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.009
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.026
                                                                          0.135
                                                                          0.632
                                                                          0.923

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.004
                                                                          0.025
                                                                          0.297
                                                                          0.700

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.002
                                                                          0.005
                                                                          0.097
                                                                          0.373

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.006
                                                                          0.023
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.040
                                                                          0.186
                                                                          0.746
                                                                          0.967

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.006
                                                                          0.038
                                                                          0.412
                                                                          0.826

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.002
                                                                          0.007
                                                                          0.134
                                                                          0.439

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.017
                                                                             11
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.065
                                                                          0.292
                                                                          0.848
                                                                          0.986

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.010
                                                                          0.069
                                                                          0.553
                                                                          0.883

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.012
                                                                          0.208
                                                                          0.472

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.013
                                                                             12
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.094
                                                                          0.388
                                                                          0.922
                                                                          0.996

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.012
                                                                          0.090
                                                                          0.661
                                                                          0.893

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.014
                                                                          0.303
                                                                          0.500

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.008
                                                                          0.017
                                                                             13
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.141
                                                                          0.518
                                                                          0.964
                                                                          0.999

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.017
                                                                          0.133
                                                                          0.774
                                                                          0.925

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.003
                                                                          0.025
                                                                          0.353
                                                                          0.494

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.007
                                                                             14
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.201
                                                                          0.634
                                                                          0.987
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.023
                                                                          0.186
                                                                          0.840
                                                                          0.926

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.004
                                                                          0.040
                                                                          0.429
                                                                          0.520

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.007
                                                                             15
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.274
                                                                          0.751
                                                                          0.995
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.051
                                                                          0.293
                                                                          0.908
                                                                          0.944

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.005
                                                                          0.060
                                                                          0.452
                                                                          0.491

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.007
                                                                          0.007

      Table A1.1-2
Animal random effects and day random effects assumed to be normal distributions
             N per Day = 50+-5, CP = 0.45, SD_sub = 0.33, SD_day=0
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.004
                                                                          0.039
                                                                          0.292
                                                                          0.669

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.000
                                                                          0.005
                                                                          0.077
                                                                          0.326

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.014
                                                                          0.121

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.007
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.008
                                                                          0.069
                                                                          0.401
                                                                          0.805

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.001
                                                                          0.008
                                                                          0.139
                                                                          0.480

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.032
                                                                          0.194

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.008
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.027
                                                                          0.113
                                                                          0.557
                                                                          0.889

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.003
                                                                          0.014
                                                                          0.205
                                                                          0.595

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.050
                                                                          0.260

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.011
                                                                             11
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.033
                                                                          0.165
                                                                          0.671
                                                                          0.949

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.001
                                                                          0.020
                                                                          0.292
                                                                          0.703

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.073
                                                                          0.317

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.016
                                                                             12
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.048
                                                                          0.234
                                                                          0.809
                                                                          0.975

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.004
                                                                          0.028
                                                                          0.377
                                                                          0.792

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.003
                                                                          0.130
                                                                          0.391

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.017
                                                                             13
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.077
                                                                          0.340
                                                                          0.875
                                                                          0.988

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.007
                                                                          0.056
                                                                          0.523
                                                                          0.844

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.004
                                                                          0.176
                                                                          0.407

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.009
                                                                          0.018
                                                                             14
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.105
                                                                          0.394
                                                                          0.919
                                                                          0.991

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.008
                                                                          0.078
                                                                          0.606
                                                                          0.862

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.010
                                                                          0.213
                                                                          0.410

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.007
                                                                             15
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.161
                                                                          0.521
                                                                          0.958
                                                                          0.997

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.014
                                                                          0.114
                                                                          0.722
                                                                          0.894

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.020
                                                                          0.281
                                                                          0.409

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.003
                                                                          0.006

      Table A1.1-3
Animal random effects and day random effects assumed to be Weibull distributions
              N per Day = 50+-5, CP = 0.45, SubVar = 1, DayVar=0
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.017
                                                                          0.078
                                                                          0.368
                                                                          0.727

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.001
                                                                          0.008
                                                                          0.110
                                                                          0.397

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.030
                                                                          0.168

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.013
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.024
                                                                          0.112
                                                                          0.502
                                                                          0.836

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.002
                                                                          0.012
                                                                          0.181
                                                                          0.541

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.048
                                                                          0.235

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.017
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.043
                                                                          0.156
                                                                          0.619
                                                                          0.905

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.004
                                                                          0.023
                                                                          0.242
                                                                          0.658

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.003
                                                                          0.066
                                                                          0.316

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.003
                                                                          0.019
                                                                             11
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.054
                                                                          0.223
                                                                          0.734
                                                                          0.946

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.006
                                                                          0.034
                                                                          0.354
                                                                          0.766

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.005
                                                                          0.109
                                                                          0.415

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.020
                                                                             12
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.083
                                                                          0.321
                                                                          0.841
                                                                          0.977

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.009
                                                                          0.057
                                                                          0.478
                                                                          0.861

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.007
                                                                          0.146
                                                                          0.476

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.019
                                                                             13
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.123
                                                                          0.409
                                                                          0.907
                                                                          0.985

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.007
                                                                          0.079
                                                                          0.586
                                                                          0.896

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.008
                                                                          0.210
                                                                          0.507

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.003
                                                                          0.012
                                                                             14
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.158
                                                                          0.498
                                                                          0.929
                                                                          0.987

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.010
                                                                          0.112
                                                                          0.676
                                                                          0.907

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.013
                                                                          0.291
                                                                          0.522

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.011
                                                                             15
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.214
                                                                          0.609
                                                                          0.958
                                                                          0.992

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.019
                                                                          0.152
                                                                          0.762
                                                                          0.917

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.015
                                                                          0.359
                                                                          0.532

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.008
                                                                          0.011

      Table A1.1-4
Animal random effects and day random effects assumed to be normal distributions
            N per Day = 50+-5, CP = 0.45, SD_sub = 0, SD_day=0.166
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.007
                                                                          0.042
                                                                          0.306
                                                                          0.722

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.001
                                                                          0.006
                                                                          0.092
                                                                          0.392

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.003
                                                                          0.031
                                                                          0.184

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.010
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.013
                                                                          0.060
                                                                          0.434
                                                                          0.839

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.002
                                                                          0.009
                                                                          0.156
                                                                          0.553

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.038
                                                                          0.247

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.004
                                                                          0.016
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.017
                                                                          0.113
                                                                          0.594
                                                                          0.918

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.003
                                                                          0.018
                                                                          0.259
                                                                          0.684

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.005
                                                                          0.081
                                                                          0.338

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.011
                                                                             11
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.031
                                                                          0.162
                                                                          0.739
                                                                          0.967

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.003
                                                                          0.023
                                                                          0.344
                                                                          0.784

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.004
                                                                          0.115
                                                                          0.408

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.003
                                                                          0.012
                                                                             12
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.042
                                                                          0.235
                                                                          0.831
                                                                          0.983

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.006
                                                                          0.041
                                                                          0.478
                                                                          0.857

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.006
                                                                          0.182
                                                                          0.456

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.013
                                                                             13
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.073
                                                                          0.335
                                                                          0.907
                                                                          0.995

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.009
                                                                          0.074
                                                                          0.614
                                                                          0.890

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.007
                                                                          0.228
                                                                          0.470

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.009
                                                                             14
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.100
                                                                          0.423
                                                                          0.957
                                                                          0.999

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.007
                                                                          0.087
                                                                          0.713
                                                                          0.916

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.016
                                                                          0.300
                                                                          0.476

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.006
                                                                             15
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.149
                                                                          0.550
                                                                          0.976
                                                                          0.998

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.015
                                                                          0.137
                                                                          0.797
                                                                          0.920

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.022
                                                                          0.360
                                                                          0.484

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.004

      Table A1.1-5
Animal random effects and day random effects assumed to be Weibull distributions
             N per Day = 50+-5, CP = 0.45, SubVar = 0, DayVar=0.5
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.009
                                                                          0.038
                                                                          0.305
                                                                          0.681

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.001
                                                                          0.005
                                                                          0.092
                                                                          0.405

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.024
                                                                          0.166

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.002
                                                                          0.011
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.011
                                                                          0.062
                                                                          0.442
                                                                          0.804

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.002
                                                                          0.011
                                                                          0.160
                                                                          0.542

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.036
                                                                          0.243

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.010
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.014
                                                                          0.107
                                                                          0.563
                                                                          0.904

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.002
                                                                          0.022
                                                                          0.251
                                                                          0.688

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.057
                                                                          0.326

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.016
                                                                             11
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.028
                                                                          0.160
                                                                          0.706
                                                                          0.954

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.003
                                                                          0.028
                                                                          0.364
                                                                          0.788

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.002
                                                                          0.103
                                                                          0.406

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.019
                                                                             12
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.047
                                                                          0.234
                                                                          0.792
                                                                          0.979

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.003
                                                                          0.040
                                                                          0.461
                                                                          0.859

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.001
                                                                          0.004
                                                                          0.162
                                                                          0.479

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.002
                                                                          0.011
                                                                             13
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.068
                                                                          0.320
                                                                          0.898
                                                                          0.991

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.008
                                                                          0.063
                                                                          0.596
                                                                          0.894

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.008
                                                                          0.239
                                                                          0.524

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                          0.014
                                                                             14
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.100
                                                                          0.432
                                                                          0.938
                                                                          0.994

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.008
                                                                          0.090
                                                                          0.714
                                                                          0.939

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.013
                                                                          0.307
                                                                          0.531

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.005
                                                                          0.009
                                                                             15
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.131
                                                                          0.535
                                                                          0.971
                                                                          0.999

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.012
                                                                          0.141
                                                                          0.798
                                                                          0.945

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.003
                                                                          0.025
                                                                          0.388
                                                                          0.542

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.007
                                                                          0.010

      Table A1.1-6
Animal random effects and day random effects assumed to be normal distributions
           N per Day = 50+-5, CP = 0.45, SD_sub = 0.33, SD_day=0.166
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.003
                                                                          0.017
                                                                          0.196
                                                                          0.540

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.000
                                                                          0.002
                                                                          0.042
                                                                          0.224

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.012
                                                                          0.078

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.004
                                                                              9