0.009
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.605
                                                                          0.910
                                                                          0.997
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.149
                                                                          0.561
                                                                          0.938
                                                                          0.956

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.027
                                                                          0.167
                                                                          0.534
                                                                          0.554

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.005
                                                                          0.005
                                                                             20
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.999
                                                                          1.000
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.966
                                                                          0.992
                                                                          0.993
                                                                          0.993

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.514
                                                                          0.545
                                                                          0.548
                                                                          0.548

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000

Comment:
 If the true efficacy is 92.5%, the recommended sample size is 8

Simulations of Study Design 5: Narthropods/animal/day = 100+-10 and blood-fed proportion of arthropod on each animal >= 60%
 The study design:
 The results of this study design apply to studies that test a single species of mosquitoes, fleas, sand flies, black flies, or biting midges per case per day.
 Number of arthropod/species/animal/day = 100+-10
 4 different days
 Inclusion criterion: blood-fed proportion of arthropod on each animal >= 60%

 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 >= 60%, the true percent blood-fed arthropods CP in control animal used in the simulation are reasonably assumed to be 0.75.
 N arthropods/animal/day = 100+-10
 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 = 6, 7, ..., 10.

      Table A1.5-1
N per Day = 100+-10, CP = 0.75, 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
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.814
                                                                          0.933
                                                                          0.987
                                                                          0.998

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.522
                                                                          0.801
                                                                          0.942
                                                                          0.963

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.190
                                                                          0.367
                                                                          0.479
                                                                          0.497

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.900
                                                                          0.970
                                                                          0.999
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.691
                                                                          0.903
                                                                          0.969
                                                                          0.976

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.265
                                                                          0.425
                                                                          0.494
                                                                          0.502

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.963
                                                                          0.992
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.853
                                                                          0.960
                                                                          0.987
                                                                          0.989

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.377
                                                                          0.489
                                                                          0.513
                                                                          0.516

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.985
                                                                          0.998
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.925
                                                                          0.981
                                                                          0.991
                                                                          0.992

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.426
                                                                          0.470
                                                                          0.482
                                                                          0.482

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.995
                                                                          1.000
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.958
                                                                          0.987
                                                                          0.991
                                                                          0.991

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.481
                                                                          0.509
                                                                          0.515
                                                                          0.515

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000

      Table A1.5-2
Animal random effects and day random effects assumed to be normal distributions
           N per Day = 100+-10, CP = 0.75, SD_sub = 0, SD_day=0.166
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.624
                                                                          0.871
                                                                          0.988
                                                                          0.999

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.296
                                                                          0.636
                                                                          0.891
                                                                          0.942

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.079
                                                                          0.249
                                                                          0.424
                                                                          0.454

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.002
                                                                          0.002
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.811
                                                                          0.944
                                                                          0.995
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.453
                                                                          0.780
                                                                          0.946
                                                                          0.963

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.147
                                                                          0.357
                                                                          0.500
                                                                          0.513

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.002
                                                                          0.002
                                                                          0.002
                                                                          0.002
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.915
                                                                          0.983
                                                                          0.999
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.651
                                                                          0.894
                                                                          0.961
                                                                          0.967

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.234
                                                                          0.429
                                                                          0.483
                                                                          0.486

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.003
                                                                          0.003
                                                                          0.003
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.959
                                                                          0.996
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.794
                                                                          0.945
                                                                          0.976
                                                                          0.978

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.313
                                                                          0.446
                                                                          0.473
                                                                          0.475

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.987
                                                                          0.998
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.900
                                                                          0.977
                                                                          0.987
                                                                          0.987

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.381
                                                                          0.463
                                                                          0.474
                                                                          0.475

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000

      Table A1.5-3
Animal random effects and day random effects assumed to be Weibull distributions
             N per Day =100+-10, CP = 0.75, SubVar = 0, DayVar=0.5
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.491
                                                                          0.792
                                                                          0.974
                                                                          0.996

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.184
                                                                          0.496
                                                                          0.890
                                                                          0.957

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.052
                                                                          0.184
                                                                          0.491
                                                                          0.551

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.002
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.695
                                                                          0.922
                                                                          0.994
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.331
                                                                          0.697
                                                                          0.962
                                                                          0.983

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.093
                                                                          0.292
                                                                          0.521
                                                                          0.547

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.003
                                                                          0.003
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.859
                                                                          0.972
                                                                          0.999
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.479
                                                                          0.843
                                                                          0.979
                                                                          0.987

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.145
                                                                          0.405
                                                                          0.544
                                                                          0.554

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.001
                                                                          0.001
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.926
                                                                          0.992
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.668
                                                                          0.940
                                                                          0.995
                                                                          0.996

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.229
                                                                          0.482
                                                                          0.555
                                                                          0.559

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.975
                                                                          0.998
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.793
                                                                          0.960
                                                                          0.991
                                                                          0.991

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.321
                                                                          0.540
                                                                          0.580
                                                                          0.581

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000

      Table A1.5-4
Animal random effects and day random effects assumed to be normal distributions
            N per Day = 100+-10, CP = 0.75, SD_sub = 0.33, SD_day=0
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.451
                                                                          0.743
                                                                          0.967
                                                                          0.997

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.126
                                                                          0.349
                                                                          0.739
                                                                          0.862

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.030
                                                                          0.110
                                                                          0.320
                                                                          0.417

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.007
                                                                          0.009
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.626
                                                                          0.872
                                                                          0.993
                                                                          0.998

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.234
                                                                          0.521
                                                                          0.854
                                                                          0.923

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.051
                                                                          0.172
                                                                          0.400
                                                                          0.460

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.003
                                                                          0.006
                                                                          0.008
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.778
                                                                          0.950
                                                                          0.997
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.338
                                                                          0.665
                                                                          0.903
                                                                          0.932

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.084
                                                                          0.231
                                                                          0.408
                                                                          0.443

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.003
                                                                          0.005
                                                                          0.005
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.892
                                                                          0.984
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.469
                                                                          0.774
                                                                          0.938
                                                                          0.946

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.125
                                                                          0.290
                                                                          0.411
                                                                          0.427

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.004
                                                                          0.004
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.939
                                                                          0.995
                                                                          0.999
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.592
                                                                          0.867
                                                                          0.955
                                                                          0.958

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.179
                                                                          0.338
                                                                          0.439
                                                                          0.443

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.002
                                                                          0.002
                                                                          0.002
                                                                          0.002

      Table A1.5-5
Animal random effects and day random effects assumed to be Weibull distributions
              N per Day =100+-10, CP = 0.75, SubVar = 1, DayVar=0
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.646
                                                                          0.896
                                                                          0.997
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.216
                                                                          0.533
                                                                          0.920
                                                                          0.951

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.052
                                                                          0.189
                                                                          0.484
                                                                          0.533

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.002
                                                                          0.004
                                                                          0.005
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.805
                                                                          0.970
                                                                          0.997
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.328
                                                                          0.707
                                                                          0.960
                                                                          0.966

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.071
                                                                          0.255
                                                                          0.519
                                                                          0.534

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.003
                                                                          0.003
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.919
                                                                          0.996
                                                                          0.999
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.514
                                                                          0.871
                                                                          0.973
                                                                          0.973

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.136
                                                                          0.375
                                                                          0.522
                                                                          0.524

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.002
                                                                          0.002
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.978
                                                                          0.997
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.681
                                                                          0.947
                                                                          0.982
                                                                          0.983

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.203
                                                                          0.451
                                                                          0.505
                                                                          0.506

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                          0.001
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.989
                                                                          0.998
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.827
                                                                          0.975
                                                                          0.985
                                                                          0.986

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.307
                                                                          0.505
                                                                          0.526
                                                                          0.526

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.002
                                                                          0.002

      Table A1.5-6
Animal random effects and day random effects assumed to be normal distributions
          N per Day = 100+-10, CP = 0.75, 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
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.272
                                                                          0.598
                                                                          0.929
                                                                          0.986

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.057
                                                                          0.221
                                                                          0.659
                                                                          0.834

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.016
                                                                          0.063
                                                                          0.277
                                                                          0.409

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.007
                                                                          0.010
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.445
                                                                          0.778
                                                                          0.978
                                                                          0.995

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.119
                                                                          0.361
                                                                          0.785
                                                                          0.870

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.024
                                                                          0.107
                                                                          0.340
                                                                          0.413

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.006
                                                                          0.008
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.612
                                                                          0.892
                                                                          0.995
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.178
                                                                          0.504
                                                                          0.868
                                                                          0.909

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.038
                                                                          0.162
                                                                          0.382
                                                                          0.421

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.005
                                                                          0.005
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.748
                                                                          0.947
                                                                          0.998
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.306
                                                                          0.645
                                                                          0.904
                                                                          0.929

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.066
                                                                          0.214
                                                                          0.367
                                                                          0.393

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.002
                                                                          0.003
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.876
                                                                          0.984
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.400
                                                                          0.773
                                                                          0.935
                                                                          0.939

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.099
                                                                          0.263
                                                                          0.398
                                                                          0.409

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.002
                                                                          0.002
                                                                          0.002

      Table A1.5-7
Animal random effects and day random effects assumed to be Weibull distributions
            N per Day = 100+-10, CP = 0.75, SubVar = 1, DayVar=0.5
                                       N
                                      Day
                                    true_TE
                                     ReqTE
                             Precision <= 0.025
                             Precision <= 0.030
                             Precision <= 0.040
                             Precision <= 0.050
                                                                              6
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.385
                                                                          0.722
                                                                          0.960
                                                                          0.987

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.097
                                                                          0.313
                                                                          0.787
                                                                          0.930

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.016
                                                                          0.094
                                                                          0.407
                                                                          0.539

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.002
                                                                          0.006
                                                                          0.008
                                                                              7
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.547
                                                                          0.845
                                                                          0.977
                                                                          0.991

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.163
                                                                          0.461
                                                                          0.917
                                                                          0.962

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.036
                                                                          0.136
                                                                          0.491
                                                                          0.549

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.002
                                                                          0.005
                                                                          0.006
                                                                              8
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.728
                                                                          0.935
                                                                          0.991
                                                                          0.998

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.256
                                                                          0.629
                                                                          0.951
                                                                          0.973

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.052
                                                                          0.245
                                                                          0.548
                                                                          0.564

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.001
                                                                          0.002
                                                                          0.003
                                                                              9
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.851
                                                                          0.966
                                                                          0.992
                                                                          0.999

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.389
                                                                          0.794
                                                                          0.971
                                                                          0.982

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.101
                                                                          0.343
                                                                          0.551
                                                                          0.560

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.001
                                                                          0.003
                                                                          0.003
                                                                          0.003
                                                                             10
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          0.923
                                                                          0.979
                                                                          0.997
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.552
                                                                          0.887
                                                                          0.974
                                                                          0.982

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.131
                                                                          0.435
                                                                          0.565
                                                                          0.570

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                             20
                                                                              1
                                                                           0.95
                                                                            0.9
                                                                          1.000
                                                                          1.000
                                                                          1.000
                                                                          1.000

                                                                              2
                                                                          0.925
                                                                            0.9
                                                                          0.988
                                                                          0.997
                                                                          0.999
                                                                          0.999

                                                                              3
                                                                            0.9
                                                                            0.9
                                                                          0.575
                                                                          0.582
                                                                          0.584
                                                                          0.584

                                                                              4
                                                                           0.85
                                                                            0.9
                                                                          0.000
                                                                          0.000
                                                                          0.000
                                                                          0.000

Comment:
 if the true efficacy is 92.5%, the recommended sample size is 6
Appendix II: Evaluation of random effects of logit generated from normal distributions and Weibull distribution
Due to the variation between animals, the blood-fed proportion of an arthropod species may be different between different animals; and due to the variation of unknown factors, the blood-fed proportion of an arthropod species on a specific animal can be different between days.  To create the random effects due to the variation between animals or variation between days given the overall blood-fed proportion of population P, an animal random effect rands and a day random effect randi are added to the population logit.
Specifically, if P is the blood-fed proportion of the animal population, the population logit is
      Logit Q = log(P/(1-P)), and
the logit of an s[th] animal on i[th] day is:
      Qs= Q + rands + randi
Since we don't know the true values of rands and randi, we assume that the range of lower half distribution (from minimum value to median) of rands distribution is 1 and the range of lower half distribution of randi is 0.5.  With these assumed values, if the population proportion is 0.95, the population logit is = ln(0.95/(1-0.95)) = 2.9444, the logit of true proportion of an animal on a day at low range can be = 2.9444  -  1  -  0.5 = 1.444, and the true proportion of the animal on the day is = exp(1.444)/(1+exp(1.444) = 0.8091.   Similarly, if the population proportion is 0.4, the true proportion of an animal on a day can be as low as 0.1294.
Since we don't exactly know the underlying distributions of rands and randi of these effects, we evaluate and determine whether normal distributions or Weibull distributions is a better assumption for these random effects.  Basically, we would evaluate the estimated blood-fed proportions and the estimated treatment efficacy from the simulated datasets where rands and randi are generated from normal distributions or Weibull distribution, given the population blood-fed proportions and the treatment efficacy.
Details simulations
The study design for each dataset includes 2 groups: control and treatment.  There are 4 different testing days and the blood-fed proportions in the control are P = 0.95, 0.60, 0.40, and 0.20.  The treatment effect (efficacy) are TE = 0.95, 0.925, 0.90, and 0.85.  The blood-fed proportions of treated group are calculated as P*(1-TE) and equal 0.0475, 0.045, 0.04, and 0.3.  Within each dataset, the numbers of animals in control group and treatment group are the same.
When the animal random effects and day random effects are assumed to follow normal distributions, the logit of the s[th] animal on i[th] day is Qsi= Q + rands + randi, where the animal random effect rands is randomly generated from normal distribution with mean = 0 and standard deviation SD_sub =0.33 and day random effect is randomly generated from normal distribution mean =0 and standard deviation SD_day = 0.166 (the standard deviation of the distribution is estimated as the half range of distribution divided by 3).  Each animal would have same value of rands for different days and different values of randi for different days.
When the animal random effects and day random effects are assumed to follow Weibull distributions, the logit of the s[th] animal on i[th] day is Qsi= Q + rands + randi, where the animal random effect rands is randomly generated as
 Randomly generated u from uniform distribution U(0,1).  Each animal has same value u for all days.
 Derive the temps associated with the quantile value u of the Weibull distribution (a, b), where a = 1/P and b = exp((1/a)xlog(-(SubVar[a])/log(0.5))), with SubVar = 1
 the animal random effect of s[th] animal on i[th] day is rands = temps  -  SubVar.  Since value of P is different on different day, value of a is different on different days and value of rands of same s[th] animal is different on different days.
 For each i[th] day, compute Ps = exp(Qs)/(1 + exp(Qs)), where Qs = Q + rands
 Randomly generated tempi from Weibull distribution shape ai and location bi, where ai = 1/Ps and bi = exp((1/ai) xlog(-(DayVar[ai])/log(0.5))), with DayVar = 0.5.
 The random day effect of s[th] animal on i[th] day is randi = tempi  -  DayVar,
From the logit of the s[th] animal on i[th] day is Qsi= Q + rands + randi, we can compute the blood-fed proportion of the s[th] animal on i[th] day as Psi = exp(Qsi)/(1 + exp(Qsi)).
The total arthropod used for each animal on a day is randomly generated from a uniform distribution Nsi = U[0.9*Narthropod, 1.1*Narthropod), where Narthropod = 100 is target number of arthropod/animal/day.
The number of blood-fed arthropods for the s[th] animal on i[th] day is randomly generated from binomial distribution B(Nsi, Psi).
The number of animals per group is set to be 10 (expected sample size for actual studies) and 100 (large but unpractical sample size).
The simulations will include 2000 datasets for each scenario.  Generalized linear mixed effects model (PROC GLIMMIX) for overdispersion Poisson distribution data with animal and day as random effects (RANDOM Day/subject = animal type = cs; RANDOM _residual_;) is used to analyze each dataset.  For those datasets that the generalized linear mixed effects model has a computational (convergence) issue, generalized linear model for Poisson distribution data is used to separately analyze the data by Day.  The point estimated blood-fed proportions and blood-fed proportions ratio (i.e. adjusted blood-fed proportion of treatment group) for each day are obtained.  The efficacy of the treatment group = 1  -  adjusted blood-fed proportion.  Medians of observed blood-fed proportions and treatment efficacy will be compared to the true value to determine whether normal distributions or Weibull distributions for the random animal effect and random day effect would have less bias in the estimates.

Results
Table A2.1
Observed blood-fed proportions
Random animal effect and day effect randomly generated from Normal distribution, N = 100 animal/group
Group
Day
TrueP
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
0
1
0.95
2000
0.9328454
0.9402728
0.9421845
0.9443784
0.9466070
0.9484965
0.9553941

2
0.6
2000
0.5651210
0.5817944
0.5879972
0.5945176
0.6013941
0.6078380
0.6315579

3
0.4
2000
0.3700825
0.3882950
0.3938229
0.4009357
0.4074620
0.4137459
0.4370733

4
0.2
2000
0.1812036
0.1958686
0.2000952
0.2050870
0.2099574
0.2145404
0.2324107
1
1
0.0475
2000
0.0415714
0.0464748
0.0481624
0.0500982
0.0519519
0.0537152
0.0596553

2
0.045
2000
0.0374002
0.0442125
0.0457950
0.0475080
0.0493431
0.0511296
0.0591684

3
0.04
2000
0.0337842
0.0390285
0.0404892
0.0422281
0.0439873
0.0455478
0.0518940

4
0.03
2000
0.0248943
0.0292099
0.0305113
0.0318165
0.0330973
0.0344033
0.0385632

Table A2.2
Observed blood-fed proportions
Random animal effect and day effect randomly generated from Weibull distribution, N = 100 animal/group

Group
Day
TrueP
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
0
1
0.95
2000
0.9303659
0.9393636
0.9423884
0.9461275
0.9492810
0.9521507
0.9632905

2
0.6
2000
0.5585443
0.5907012
0.6004746
0.6119333
0.6231649
0.6331901
0.6720473

3
0.4
2000
0.3611335
0.3928335
0.4004552
0.4091731
0.4180852
0.4263132
0.4574537

4
0.2
2000
0.1778586
0.1912323
0.1945159
0.1986015
0.2026500
0.2062123
0.2223280
1
1
0.0475
2000
0.0402851
0.0440981
0.0455001
0.0469498
0.0484082
0.0497623
0.0537063

2
0.045
2000
0.0369101
0.0417713
0.0430978
0.0445859
0.0460477
0.0472637
0.0519586

3
0.04
2000
0.0330633
0.0371344
0.0382563
0.0395118
0.0407533
0.0419070
0.0495017

4
0.03
2000
0.0236409
0.0276042
0.0285592
0.0296799
0.0308359
0.0318953
0.0349423

Table A2.3
Observed Treatment Efficacy TE
Random animal effect and day effect randomly generated from Normal distribution, N = 100 animal/group
Day
TrueTE
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
1
0.95
2000
0.9368674
0.9431031
0.9449707
0.9469610
0.9490180
0.9507364
0.9558333
2
0.925
2000
0.9006239
0.9137789
0.9168844
0.9200212
0.9231180
0.9257359
0.9381344
3
0.9
2000
0.8662430
0.8855087
0.8898869
0.8946578
0.8992225
0.9033674
0.9165963
4
0.85
2000
0.8044660
0.8301122
0.8374219
0.8448203
0.8521580
0.8592251
0.8822268

Table A2.4
Observed Treatment Efficacy TE
Random animal effect and day effect randomly generated from Weibull distribution, N = 100 animal/group
Day
TrueTE
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
1
0.95
2000
0.9431504
0.9473898
0.9487798
0.9503326
0.9518947
0.9533259
0.9576572
2
0.925
2000
0.9125166
0.9219001
0.9245417
0.9271314
0.9298943
0.9323712
0.9397698
3
0.9
2000
0.8820273
0.8963840
0.8995518
0.9034971
0.9073198
0.9106829
0.9197308
4
0.85
2000
0.8200519
0.8374681
0.8434998
0.8502472
0.8571217
0.8627663
0.8798294

Table A2.5
Observed blood-fed proportions
Random animal effect and day effect randomly generated from Normal distribution, N = 10 animal/group
Group
Day
TrueP
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
0
1
0.95
2000
0.9088040
0.9327243
0.9395808
0.9461810
0.9522739
0.9572993
0.9703017

2
0.6
2000
0.4765415
0.5533382
0.5738590
0.5958000
0.6160173
0.6351628
0.7020218

3
0.4
2000
0.3031381
0.3611132
0.3816398
0.4020775
0.4221412
0.4423258
0.5187859

4
0.2
2000
0.1319066
0.1779262
0.1905663
0.2045801
0.2202250
0.2341645
0.3004032
1
1
0.0475
2000
0.0261650
0.0389766
0.0439661
0.0496388
0.0551633
0.0610804
0.0796229

2
0.045
2000
0.0267567
0.0367934
0.0412758
0.0470691
0.0531056
0.0588088
0.0853381

3
0.04
2000
0.0216858
0.0322226
0.0368126
0.0416141
0.0467908
0.0529292
0.0727324

4
0.03
2000
0.0157180
0.0236748
0.0272710
0.0315263
0.0359851
0.0402597
0.0593183

Table A2.6
Observed blood-fed proportions
Random animal effect and day effect randomly generated from Weibull distribution, N = 10 animal/group

Group
Day
TrueP
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
0
1
0.95
2000
0.8837114
0.9256713
0.9363142
0.9475478
0.9574173
0.9665922
0.9899296

2
0.6
2000
0.4474323
0.5449825
0.5762213
0.6104993
0.6453740
0.6790745
0.7876322

3
0.4
2000
0.2898584
0.3605328
0.3823695
0.4089170
0.4390268
0.4638160
0.5571623

4
0.2
2000
0.1364718
0.1775665
0.1863037
0.1985275
0.2101781
0.2220527
0.2583755
1
1
0.0475
2000
0.0281432
0.0386449
0.0425075
0.0467563
0.0516444
0.0559265
0.0725646

2
0.045
2000
0.0230692
0.0361864
0.0399485
0.0443134
0.0488494
0.0527465
0.0660680

3
0.04
2000
0.0182927
0.0316249
0.0353893
0.0395499
0.0436574
0.0475643
0.0632315

4
0.03
2000
0.0132069
0.0230203
0.0258402
0.0294418
0.0333539
0.0366646
0.0519890

Table A2.7
Observed Treatment Efficacy TE
Random animal effect and day effect randomly generated from Normal distribution, N = 10 animal/group
Day
TrueTE
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
1
0.95
2000
0.9164632
0.9356072
0.9416649
0.9474740
0.9534573
0.9587734
0.9717145
2
0.925
2000
0.8614778
0.9000384
0.9096871
0.9211580
0.9307886
0.9388335
0.9571959
3
0.9
2000
0.7878601
0.8658581
0.8813963
0.8961819
0.9092991
0.9213917
0.9514290
4
0.85
2000
0.6470266
0.7950162
0.8212458
0.8461488
0.8692246
0.8876879
0.9301326

Table A2.8
Observed Treatment Efficacy TE
Random animal effect and day effect randomly generated from Weibull distribution, N = 10 animal/group
Day
TrueTE
N Obs
Minimum
10th Pctl
25th Pctl
50th Pctl
75th Pctl
90th Pctl
Maximum
1
0.95
2000
0.9248519
0.9407501
0.9454142
0.9506733
0.9551289
0.9593058
0.9704267
2
0.925
2000
0.8735442
0.9107129
0.9186265
0.9273257
0.9357941
0.9422458
0.9637692
3
0.9
2000
0.8063440
0.8789001
0.8914272
0.9034103
0.9150339
0.9252431
0.9523135
4
0.85
2000
0.7311387
0.8102500
0.8302189
0.8514160
0.8710503
0.8881022
0.9405302

Conclusions
 The medians of estimated blood-fed proportions from simulated datasets are very close to the true values, when the animal random effects and day random effects are assumed to follow normal distributions or Weibull distributions.

 When the random animal effects and random day effects are randomly generated from Weibull distributions, the observed efficacy is slightly over-estimated.  When the random animal effects and random day effects are randomly generated from normal distributions, the observed efficacy is slightly under-estimated.

 Both assumptions that random animal effects and random day effects follow normal distributions and Weibull distributions are reasonable for the simulation of sample size/power of Pet Product studies.

Appendix III: True blood-fed proportion vs. proportions of animals/datasets meet the required blood-fed proportion
One of the criteria for an animal to be selected for a study is that the observed blood-fed proportion of the animal from the screening test is greater than certain required value to ensure the attractiveness of the animal to the arthropods.  It also requires that the observed blood-fed proportion of each control animal is greater than certain required value during the study period.  This simulation is simply to determine the lowest blood-fed proportion such that a low probability for an animal to have an observed blood-fed proportion below the required blood-fed proportion and a low probability for a dataset (single measurement/animal) to have at least 1 animal with observed blood-fed proportion below the required blood-fed proportion.  It is worth mentioning that the lowest true blood-fed proportions obtained from these simulations are used for simulations of sample size and used for the purpose of helping experts to determine whether the required blood-fed proportions are reasonable or not.  In reality, we cannot assign a value for true blood-fed proportion for animals and we don't exactly know these values.  We determine the lowest true blood-fed for simulations because the power of the study will increase if the true blood-fed proportion is greater.
Two Tables A3.1 and A3.2 below present the results of simulations.  From these simulations, we conclude that the lowest blood-fed proportion are assumed to be:
 When the required blood-fed proportion for an animal is 0.25 and the number of arthropods/animal = 50, the lowest true value for blood-fed-proportion that can be used is 0.45

 When the required blood-fed proportion for an animal is 0.40 and the number of arthropods/animal = 50, the lowest true value for blood-fed- proportion that can be used is 0.60

 When the required blood-fed proportion for an animal is 0.40 and the number of arthropods/animal = 100, the lowest true value for blood-fed- proportion that can be used is 0.60

 When the required blood-fed proportion for an animal is 0.60 and the number of arthropods/animal = 50, the lowest true value for blood-fed- proportion that can be used is 0.80

 When the required blood-fed proportion for an animal is 0.60 and the number of arthropods/animal = 100, the lowest true value for blood-fed- proportion that can be used is 0.75
Note that if the true blood-fed proportion is greater than the assumed value in the simulations, the true power of the study design is greater than the power observed from the simulations.

TableA3.1: Proportion of animals with observed blood-fed proportion < require blood fed proportion
                                          True             blood-fed proportion
                                         Required          blood-fed proportion
                                                   Number arthropods per animal
                                                                 Total animals
Proportion of animals with observed blood-fed proportion < require blood fed proportion
                                                                           0.75
                                                                            0.6
                                                                             50
                                                                         460000
                                                                        0.01065
                                                                            0.8
                                                                            0.6
                                                                             50
                                                                         460000
                                                                        0.00073
                                                                            0.7
                                                                            0.6
                                                                            100
                                                                         460000
                                                                        0.01759
                                                                           0.75
                                                                            0.6
                                                                            100
                                                                         460000
                                                                        0.00060
                                                                           0.55
                                                                            0.4
                                                                             50
                                                                         460000
                                                                        0.01845
                                                                            0.6
                                                                            0.4
                                                                             50
                                                                         460000
                                                                        0.00255
                                                                           0.55
                                                                            0.4
                                                                            100
                                                                         460000
                                                                        0.00152
                                                                            0.6
                                                                            0.4
                                                                            100
                                                                         460000
                                                                        0.00005
                                                                            0.4
                                                                           0.25
                                                                             50
                                                                         460000
                                                                        0.01486
                                                                           0.45
                                                                           0.25
                                                                             50
                                                                         460000
                                                                        0.00209

Table A3.2: Proportion of datasets having at least one animal with observed blood-fed proportion < require blood fed proportion
                                          True             blood-fed proportion
                                         Required          blood-fed proportion
                                                   Number arthropods per animal
                                                       Number animals per group
                                                             Number of datasets
Proportion of datasets having  at least one animal with observed blood-fed proportion < require blood fed proportion
                                                                           0.75
                                                                            0.6
                                                                             50
                                                                              8
                                                                           5000
                                                                        0.08320

                                                                              9
                                                                           5000
                                                                        0.08700

                                                                             10
                                                                           5000
                                                                        0.09940

                                                                             11
                                                                           5000
                                                                        0.11240

                                                                             12
                                                                           5000
                                                                        0.12300

                                                                             13
                                                                           5000
                                                                        0.12660

                                                                             14
                                                                           5000
                                                                        0.13780

                                                                             15
                                                                           5000
                                                                        0.15900
                                                                            0.8
                                                                            0.6
                                                                             50
                                                                              8
                                                                           5000
                                                                        0.00600

                                                                              9
                                                                           5000
                                                                        0.00560

                                                                             10
                                                                           5000
                                                                        0.00560