Power

Randomization-Based Inference

The Null Hypothesis

From data to schedule

library(tidyverse)
library(gt)
my_data <- tibble(i = 1:10,
                  d_i = c(1, 0, 1, 0, 0, 0, 0, 1, 1, 1),
                  y_i = c(.9, -.5, 2.2, -1.4, .1, -1, 1.1, 1, 1.02, 1.5))

my_data |>
  gt() |>
  tab_header(title = "My Data")
My Data
i d_i y_i
1 1 0.90
2 0 -0.50
3 1 2.20
4 0 -1.40
5 0 0.10
6 0 -1.00
7 0 1.10
8 1 1.00
9 1 1.02
10 1 1.50
my_obs_sched <- my_data |>
  mutate(Y_0 = ifelse(d_i == 0, y_i, NA),
         Y_1 = ifelse(d_i == 1, y_i, NA)) |>
  select(i, Y_0, Y_1)
my_obs_sched |>
  gt() |>
  tab_header(title = "My Observed Schedule")
My Observed Schedule
i Y_0 Y_1
1 NA 0.90
2 -0.5 NA
3 NA 2.20
4 -1.4 NA
5 0.1 NA
6 -1.0 NA
7 1.1 NA
8 NA 1.00
9 NA 1.02
10 NA 1.50

From data to schedule

my_obs_sched <- my_data |>
  mutate(Y_0 = ifelse(d_i == 0, y_i, NA),
         Y_1 = ifelse(d_i == 1, y_i, NA)) |>
  select(i, Y_0, Y_1)

my_obs_sched |>
  gt() |>
  tab_header(title = "My Observed Schedule")
My Observed Schedule
i Y_0 Y_1
1 NA 0.90
2 -0.5 NA
3 NA 2.20
4 -1.4 NA
5 0.1 NA
6 -1.0 NA
7 1.1 NA
8 NA 1.00
9 NA 1.02
10 NA 1.50
my_null_sched <- my_obs_sched |>
  mutate(Y_0 = ifelse(is.na(Y_0), Y_1, Y_0),
         Y_1 = ifelse(is.na(Y_1), Y_0, Y_1))

my_null_sched |>
  gt() |>
  tab_header(title = "My Null Schedule")
My Null Schedule
i Y_0 Y_1
1 0.90 0.90
2 -0.50 -0.50
3 2.20 2.20
4 -1.40 -1.40
5 0.10 0.10
6 -1.00 -1.00
7 1.10 1.10
8 1.00 1.00
9 1.02 1.02
10 1.50 1.50

Estimate Dist under H0

rand_stats <- function(schedule, d_i, stat = "diff in means", reps = 1000) {
  # store treatment vector separately
  d_i_vec <- d_i

  # replicate the schedule reps times and stack them on top of one another
  randomized_exp_df <- map_dfr(1:reps, ~ schedule, .id = "experiment") |>
    mutate(experiment = factor(experiment, levels = as.character(1:reps), ordered = TRUE),
           d_i = c(replicate(n = reps, sample(d_i_vec))),  # create random assignments
           y_i = Y_1 * d_i +  Y_0 * (1 - d_i)) |>    # find observed responses
    arrange(experiment)

  # calculate test statistic for every random assignment
  if (stat == "diff in means") {
    stats <- randomized_exp_df |>
      group_by(experiment, d_i) |>
      summarize(ybar = mean(y_i),     # average within each group within each experiment
                .groups = "drop_last") |> 
      summarize(ATE_hat = diff(ybar),
                .groups = "drop") |>   # take the difference between the two groups mean 
      pull()
  } else {
    stop("Statistic not implemented")
  }

  return(stats)
}

Estimate Dist under H0

H_0 <- rand_stats(schedule = my_null_sched, d_i = my_data$d_i, reps = 5000)
H_0
   [1]  0.024  0.584  0.664 -0.616 -0.584  0.344  1.136  0.896  1.304 -1.504
  [11]  0.136  0.264  0.304 -0.544  0.776 -1.144 -0.864  1.144  0.496  0.256
  [21] -0.496  1.104 -1.144 -0.696  1.184 -0.864 -0.376  0.664 -1.224  1.304
  [31]  0.096 -0.336 -0.264 -1.744  0.704  0.416 -0.096  0.224 -0.336 -0.544
  [41] -0.336 -0.064  0.584 -0.296  0.376 -1.344  0.896 -0.504  0.584 -0.704
  [51] -0.096 -0.416 -1.224  0.216 -0.864 -0.304 -1.344  1.384 -0.544  0.416
  [61]  0.336 -0.336  0.864  0.776 -0.584  0.904  0.744  0.136  0.536 -1.504
  [71] -0.536  0.224  0.376 -0.704 -0.576 -0.584  0.664  0.224  0.776  0.144
  [81]  0.064  0.776 -0.144 -0.856 -1.136 -0.104  1.104  0.224 -0.496 -0.504
  [91] -0.264 -0.296 -0.016 -0.096  0.296 -0.144 -0.904 -1.144 -0.104  0.304
 [101] -0.496  0.504 -0.184 -1.744 -1.224  0.096  1.504  0.864 -0.064 -0.096
 [111]  0.856  0.056  0.296 -1.384 -1.344 -0.256 -1.104 -0.296 -0.536  0.104
 [121]  0.664 -1.744 -0.256 -1.144  0.904 -0.176 -0.064 -0.304  0.304  0.944
 [131]  0.296  0.584  0.544 -0.664  0.496  0.576  0.584 -1.696 -0.096 -0.344
 [141] -0.104  0.384 -0.584 -0.696  0.056  0.376  0.576 -0.696 -1.184 -0.776
 [151] -0.736  1.504  0.424 -0.384  0.256 -1.664 -0.504  0.664 -0.536 -0.096
 [161] -0.096  0.704  0.856  0.336  0.136  0.096 -1.224 -0.536  0.424 -0.536
 [171] -0.544  0.616  0.856 -1.096  0.376 -0.536  0.224 -1.136  0.904 -0.416
 [181] -1.384 -0.616  0.024  0.744  0.424  1.664 -1.296 -0.144 -0.536  0.824
 [191] -0.704 -1.744  0.224 -0.864  0.424  0.584 -1.504 -0.096 -1.224  0.376
 [201] -0.864 -1.136  0.024  1.144 -1.376  0.104 -0.536  0.056  0.136  0.536
 [211] -0.136  1.384 -0.096  0.384  0.904  0.256 -0.384 -0.256  0.176  0.224
 [221] -0.176 -0.344 -0.784 -1.104 -1.104  0.264 -0.736 -0.104  0.176  0.864
 [231] -0.504 -0.576  0.696 -1.064  0.264 -1.376  0.304 -0.704 -0.704 -0.864
 [241]  0.616 -0.336 -0.944 -0.864  1.664  0.904 -0.536  0.664  0.696  0.376
 [251] -0.496  0.376  0.144 -0.336  0.536 -0.064 -0.664 -1.384  0.056  0.136
 [261] -1.336 -1.664  0.696 -0.104  0.304  0.104 -0.696  0.144  0.824  0.736
 [271]  0.864 -1.184  0.064  0.744 -0.016  0.584  0.576  0.104  0.064 -0.704
 [281]  0.856  0.104 -0.944  0.576 -1.296  0.696  0.784  0.136  0.104 -0.096
 [291] -0.536 -0.224  0.056  0.896 -0.584  0.576 -0.016  0.296 -0.744  0.664
 [301] -1.184 -1.296  0.264 -0.224  0.896 -0.704  0.384  0.304  0.144 -1.696
 [311]  0.216 -0.904  0.144 -0.784 -0.064  0.064  0.856  0.864 -0.296 -0.224
 [321]  0.544  0.664  1.744 -0.744 -0.024  1.104 -0.784  0.704 -0.296 -0.424
 [331] -0.144 -0.424 -0.664  0.496  0.056 -0.136  0.536 -0.696 -0.176 -0.376
 [341] -0.744 -0.896 -0.096  0.104  0.384 -0.944  0.096  1.056 -0.584  0.824
 [351] -0.744  0.776 -0.904  0.896  0.576  0.224 -0.896  0.376  0.344 -0.664
 [361]  0.856 -0.864  0.776  0.864  0.024  0.944 -0.256  0.544 -0.544  0.296
 [371] -0.736 -1.224  0.056 -0.536 -1.104 -0.104  1.064  0.256  0.896  1.696
 [381] -0.144 -0.624  0.144 -0.224 -0.696  0.504 -0.304  0.536  0.864 -1.144
 [391] -1.104 -0.416  0.056  0.544 -0.744  0.304 -0.336  0.256  0.224 -0.176
 [401]  0.744 -0.296 -0.576 -0.104  0.936  0.096  0.104 -1.096  0.336 -0.064
 [411] -0.144 -0.544  0.696 -0.336 -0.144 -0.464 -0.736  0.304  0.384 -0.384
 [421]  1.696  1.696  0.496 -0.584 -0.544  0.904 -0.864 -1.336 -0.344  0.256
 [431]  1.224 -0.336 -0.296 -0.336  0.056  0.904  0.496 -0.496 -0.064  0.424
 [441] -0.384 -0.296  0.696  1.336 -0.576 -0.864  0.224  0.136  0.096  1.136
 [451]  0.664  0.304  0.576  0.064  1.504  0.696  0.304 -0.016  0.536  0.664
 [461]  0.376  0.864 -0.584  0.144  0.664 -0.144  0.904  1.104  0.056  1.136
 [471] -0.784 -0.296  1.296  0.224  0.784  0.896  0.584  0.544  0.224 -0.224
 [481] -1.336 -0.944  0.296 -0.584  1.184  0.224 -0.336  1.504 -0.944 -0.744
 [491]  0.064 -0.536  0.224  1.744 -0.696 -0.144  0.056 -0.376  0.576  0.904
 [501] -0.896 -1.136 -0.304  0.576 -0.696 -1.136 -0.224  0.696  0.024  1.144
 [511] -1.296  0.896  0.696  1.704  1.344 -0.424  1.224 -0.104 -0.496 -0.576
 [521]  0.136 -0.256  0.144  1.336  0.904 -0.136  0.544  0.544  0.376 -0.064
 [531] -1.104 -0.544  0.704 -1.104 -0.136 -1.704 -0.384  0.864 -0.576 -1.664
 [541] -0.304 -0.416 -0.024 -0.016  0.224 -1.704  0.664 -0.136  0.304  0.584
 [551] -0.584  0.224 -0.264 -1.136 -1.136  0.336  0.584  0.264  0.104 -0.784
 [561]  0.296 -0.296  0.864  0.256 -0.224  1.096  0.056 -0.224  1.056  0.216
 [571]  0.104 -1.336 -0.304  0.664 -0.296  1.304  0.056  0.584  0.064  0.376
 [581] -1.296 -0.736  1.296  0.416 -0.296 -1.504  0.024  0.896 -0.304  1.704
 [591] -0.896  0.584  0.824 -0.096 -0.544 -0.536 -0.624 -0.184 -0.896  0.864
 [601]  0.304 -0.416 -1.136 -0.256 -0.144  1.136 -0.216  0.056 -0.736  1.144
 [611]  0.696 -0.104 -0.584  0.904  0.256  1.336 -0.576 -0.056 -0.704 -0.224
 [621] -0.104  0.096  1.384 -1.224  0.544 -0.136 -1.064  0.664  0.776 -0.856
 [631]  0.144  1.184 -0.104  1.344  0.536 -0.344 -0.744 -0.384 -1.136  1.504
 [641]  0.224 -0.776 -0.216  0.584  0.496  0.256 -1.504  1.104  0.544  0.144
 [651] -0.064 -0.544 -1.104  0.024  0.904  0.824 -0.704 -1.336 -1.136  0.704
 [661]  0.744  0.304  0.016  0.224  0.256  1.296 -0.224  0.296 -0.264  1.056
 [671] -0.336  1.104 -0.424  1.064  0.256  0.904 -0.496  0.904 -0.096 -0.416
 [681]  0.776 -0.096 -0.616 -0.736  0.736 -0.264 -1.184 -0.216 -0.296 -0.576
 [691] -0.576 -0.216 -0.176  0.776  0.504 -0.576 -0.696 -0.824  1.184 -0.224
 [701] -0.904  1.136  0.704 -0.064 -0.336 -1.104  0.896 -1.704 -0.744  0.896
 [711] -0.304 -0.496  0.304  0.944 -0.544  1.056  0.024 -0.384  0.216  0.336
 [721]  0.384 -1.224  0.104  1.504  1.336  0.216 -0.536  0.376 -0.464  0.896
 [731]  0.264  0.024  0.224  1.384 -0.824  1.376  0.224  0.104  0.056 -0.296
 [741] -0.096 -0.056  0.024  0.176 -0.744  1.136 -0.584  1.096  0.664  0.224
 [751]  0.584 -1.096 -0.624  0.944  0.096  0.256 -0.504  0.024  0.056  1.304
 [761] -0.744  1.104 -1.056  0.376  0.696  0.344  0.904  0.664  0.024 -0.904
 [771]  0.584  0.664 -0.496 -0.304 -0.536  1.104  0.016  0.304  0.256 -0.864
 [781]  1.064 -0.504 -1.376 -0.384 -1.336 -0.384  0.216 -0.064 -1.744 -0.736
 [791]  0.904  0.136 -1.696  0.584  0.744  0.064  1.344 -0.304 -0.256 -0.024
 [801]  0.744 -0.904  1.504 -0.384  0.664  1.744  0.904  0.064 -0.624  0.224
 [811] -0.744 -0.704 -0.616 -1.136  0.904 -1.376 -1.744 -0.056  1.304 -1.296
 [821]  0.536 -0.704 -0.424 -0.304  0.856  1.144  0.896  0.216  0.904 -0.224
 [831] -0.064 -0.176  1.136 -1.144  0.464  0.944 -0.064  0.776 -0.664  1.344
 [841]  0.016  0.096  0.736  1.696  0.416  1.376  0.304  0.264  0.264 -0.296
 [851] -0.256  0.256  0.864 -0.056  0.064 -0.744  1.144  0.296  0.336 -0.904
 [861]  0.216  0.544  0.096 -0.104  0.896  0.256  0.536  0.544  1.504 -0.224
 [871]  0.144  0.056 -1.136  0.664 -1.136  0.176 -0.616  0.256  1.184  0.856
 [881]  1.144  0.424  0.024 -1.744 -0.264  0.464  0.384 -1.744 -0.136  0.264
 [891] -0.056  1.696 -0.376 -0.544 -0.216  0.264  0.704 -0.544 -0.304 -0.384
 [901]  0.856 -0.576 -0.176  0.056 -0.376  0.224 -0.224 -0.776 -1.296 -0.336
 [911]  0.064  0.376  0.376  0.496 -1.696  0.056  1.064 -1.296  0.296 -0.224
 [921]  1.376 -1.664 -0.176  0.016  0.696 -0.136  0.744 -0.424 -0.504  0.864
 [931] -0.464  0.304 -0.584 -0.024 -0.744  0.064 -0.576 -0.336 -0.056 -0.064
 [941] -0.064  1.136 -0.016  1.096 -0.864  0.704 -0.584  1.224  0.224 -0.064
 [951]  1.304 -0.896  0.136  0.416  0.224 -0.776  0.904 -0.856  0.144 -0.224
 [961]  0.104  0.736  1.664 -0.744 -1.384  0.736 -0.704 -0.896 -0.096 -1.664
 [971]  0.856 -1.064  0.376  0.384  0.224 -0.704 -0.056 -0.064 -0.744 -0.224
 [981] -1.104 -0.576  0.696  0.744  0.896 -1.184 -0.944  1.504  0.424  0.736
 [991]  0.056 -0.504  0.024  0.336 -0.424 -0.384  0.776 -0.936  0.536 -0.304
[1001]  0.256 -0.504 -0.544 -0.576 -0.736  0.424  0.696 -0.304 -0.224 -0.856
[1011]  1.504  0.176 -0.496 -0.216  0.544  0.424  0.336  0.384  0.936  1.704
[1021] -0.944 -0.944  0.704  0.224 -1.376 -0.224 -0.096  0.336  0.144  0.704
[1031] -0.704 -0.056  0.056 -0.264  0.016 -0.584 -1.504 -0.856 -1.344 -1.504
[1041]  0.704 -0.104 -0.536  0.296 -0.576  0.104  0.904  0.184 -0.216  1.184
[1051] -1.664  0.856  0.824 -0.496  1.704  0.144  1.184  0.704  1.136 -1.744
[1061]  1.096  1.344  1.104  0.424  0.216  1.144 -0.696 -0.376 -1.136  0.376
[1071] -1.144 -1.504 -0.864 -0.664 -0.296 -0.744  0.416  0.264 -0.296  0.496
[1081]  1.224 -0.864  0.224 -0.704  0.096  0.336 -0.336  1.344 -1.344 -0.736
[1091] -0.176  0.144 -0.864  0.024  0.864  1.104  1.384  0.304  0.584 -1.344
[1101]  0.776 -0.136  0.056 -0.424 -1.336 -0.664 -0.336  0.104  0.096  0.336
[1111] -1.104 -0.176 -0.496  0.864  0.744  1.104 -0.336  0.056  1.184 -1.384
[1121]  0.664  0.696 -0.104  1.296  0.864  0.784 -0.176 -1.344 -0.736 -1.344
[1131] -0.744  0.064  0.056  0.536  1.184  0.864 -1.696  0.176 -0.944 -0.896
[1141] -0.064  0.544  1.376  0.544 -0.224 -0.304  0.696 -0.624  0.336  0.496
[1151]  0.144  1.304 -0.096  0.496 -0.664  1.056 -0.056  0.064 -0.024  0.696
[1161] -0.224 -0.376  1.664 -0.296 -0.424 -0.304  0.584 -0.104 -0.136  0.056
[1171]  0.544  0.264  0.304  0.776 -0.744 -0.536 -0.296  0.536 -0.304  0.264
[1181] -0.296  0.264 -0.104  0.344  0.856 -0.704 -0.536 -0.376 -0.536  0.424
[1191] -0.296 -0.264  0.096  0.064  0.904 -0.896 -0.336 -0.336  0.736 -0.224
[1201]  0.136 -0.704  1.744 -0.384  0.664  1.184 -0.904  0.064 -0.464 -0.104
[1211] -1.504 -1.296  0.224 -1.136 -0.696  0.384 -0.104 -0.584  0.544 -1.376
[1221]  1.696 -0.664  0.544  0.144 -0.104  0.896  0.384 -0.744 -0.864  0.536
[1231] -0.424  0.544  0.144  0.216 -0.176  0.064  0.544 -1.056 -0.056  0.864
[1241] -1.064  0.856 -0.944 -1.136  1.064  0.544  0.904  0.944  0.016 -0.664
[1251]  0.864 -0.344  0.304  0.056 -0.696 -0.224 -0.136 -0.416 -0.776 -0.136
[1261] -0.776  0.544  0.144 -0.296  0.256  0.944 -0.744 -0.224 -0.136 -0.896
[1271] -0.696 -0.256 -1.296 -0.744  0.584 -0.336 -0.576  0.224  0.504 -0.536
[1281] -0.184 -0.376 -0.704 -0.936 -1.136 -0.304  0.864  0.624 -0.264 -1.336
[1291] -0.376 -0.064 -0.224  0.704 -0.896  0.096  0.136  0.744  0.176 -0.576
[1301] -0.064 -0.704  0.584  0.856 -1.144 -0.304  1.104 -0.744  0.856  0.304
[1311] -1.104  0.496  0.536 -0.056  0.896  0.384 -0.744 -1.104 -0.264 -0.416
[1321]  0.744  0.424 -0.376  1.136  0.304 -0.784  0.184 -0.496  1.504 -0.104
[1331] -0.176  0.616  0.056 -0.896  1.336 -0.464  0.424 -1.104  0.776 -0.584
[1341] -0.664  0.496 -0.344 -0.136  0.144  0.144  1.344  0.544  0.376 -0.264
[1351] -0.944 -0.904  0.104  0.384 -1.136 -0.104  0.864 -0.304 -0.024 -1.144
[1361] -0.336 -0.544 -0.296 -0.464  0.904 -0.384  0.536  0.176  0.184  0.304
[1371] -0.064  0.136  0.104  0.376 -0.304  0.384  0.704  1.384  0.064 -0.536
[1381]  0.264 -1.664 -0.536 -1.136 -1.056  0.104  0.264  0.376  0.896 -0.024
[1391] -0.264 -1.224 -0.176  0.704  0.944 -0.016  0.216 -0.544 -0.104  0.864
[1401]  0.696 -0.424 -0.536 -0.536 -0.144 -0.304  1.144 -0.224 -0.856  0.744
[1411] -1.304 -0.496  1.224  0.056  0.536 -0.336 -0.104  0.376 -0.576  1.384
[1421] -0.104  0.376 -0.216 -1.224 -0.696 -0.376  0.384 -0.144 -0.576 -0.136
[1431] -0.544  0.936  0.064  1.384 -0.136  0.576  1.696  0.064 -0.736 -0.944
[1441]  1.664  0.504 -0.856 -0.904 -1.664  0.744 -0.256  1.104 -0.344  0.784
[1451]  0.664  1.344 -0.944 -1.344  0.856  0.424 -1.056  0.864  0.024 -0.424
[1461]  0.416  0.504 -1.096 -0.024 -1.136 -0.624 -0.464  0.056 -1.144  0.744
[1471]  1.296 -0.376 -0.296  1.704 -0.856  1.184 -0.144  0.864 -0.336 -0.864
[1481]  0.736 -0.424 -0.224 -0.056  0.024 -0.424 -0.904 -0.576  1.384 -0.144
[1491] -1.104  0.144 -0.136 -0.104  0.664 -0.904 -0.584  0.216 -0.896  0.696
[1501] -0.464 -0.744 -1.144 -0.384 -0.216 -0.864  1.144 -0.664 -1.304  0.184
[1511]  0.176  1.504  0.384 -0.064  0.264 -0.304 -0.296 -0.664  0.424 -0.144
[1521] -0.216 -0.256  0.024 -0.504  1.664 -0.304 -0.616  1.504  0.056  0.424
[1531]  0.336 -0.376 -1.104 -0.824  0.296  0.056 -0.304 -1.224 -0.104  1.696
[1541]  0.944 -0.176  1.104  0.744  1.064  0.384 -0.584  0.856 -1.136 -0.304
[1551] -0.224  0.064 -0.384 -0.304  0.496  0.056 -0.104  0.144  0.856  0.744
[1561] -0.056 -0.264  0.304  0.744  0.496 -0.304  1.336 -0.024  0.536 -0.776
[1571] -0.424 -1.184  0.544 -0.104 -0.736  0.016  0.504  0.496  1.344  0.784
[1581] -0.304  1.384 -0.504 -0.864 -0.336  1.136  0.096  0.376 -0.936 -0.704
[1591] -0.136  0.616  0.904 -1.336 -0.144 -0.824  0.904  0.616  0.144  0.424
[1601]  0.576 -0.776 -0.584 -0.944  0.544  1.144  1.136  0.176 -1.104  0.896
[1611]  0.224 -0.256  0.544 -0.224 -1.744  0.904  1.136 -0.144 -0.216  0.376
[1621] -0.136 -0.584 -0.176 -0.504  1.664  0.664 -1.184 -0.056  1.304  0.256
[1631] -0.744 -0.736  0.896 -0.384  0.504  0.664 -1.224 -0.864  0.824 -1.704
[1641] -0.824 -0.696 -0.744 -0.416 -0.224  0.544 -1.064 -0.256  0.744  0.496
[1651]  1.184 -0.896  0.296  0.744 -0.256 -0.264  0.384 -0.216 -0.264 -0.904
[1661]  0.624 -0.056  0.856  0.256  0.104 -0.376 -0.056 -0.064 -0.704 -0.584
[1671] -1.296 -0.536  0.576 -0.304 -0.264  1.304 -1.144 -0.616 -0.544  0.064
[1681]  0.176  0.024  0.696 -0.144 -1.504  0.696 -1.744  0.576  0.304  0.376
[1691]  0.416  0.536  1.704 -0.136 -0.664  0.024 -0.384 -0.944  0.536 -0.864
[1701] -0.416 -0.336  0.336 -0.064  1.064 -0.704  0.824  0.504  0.584 -0.736
[1711] -0.784  0.416  0.264 -1.096 -0.744 -0.856  0.336 -0.864  0.576  0.704
[1721]  0.504 -0.944  0.744 -0.304 -0.304 -1.104  0.064  0.256  0.776  0.904
[1731]  0.856  1.376  0.336 -0.184  0.464 -0.856  0.096  0.256 -0.216 -0.176
[1741]  0.856 -0.784  0.144  0.744  0.864  0.096 -0.576 -0.864 -0.384 -0.016
[1751]  0.384  0.304 -0.384  0.664 -0.224 -0.104 -0.384 -0.944 -1.304 -0.696
[1761] -0.584  0.304 -0.264  1.384  0.384  0.384 -0.776  0.536  0.736  1.224
[1771] -1.384  0.664 -0.136  0.664 -0.536  0.016  0.944  0.416 -0.416  0.264
[1781]  0.184  0.704  1.296  1.096  0.504 -0.296  0.784 -0.216 -0.176 -0.256
[1791]  1.104  0.896  0.664  0.776 -1.136 -0.056 -0.056 -0.536  0.856 -0.296
[1801]  0.744 -0.096  0.144 -0.616 -0.136  0.144  0.096  0.224 -0.064  0.416
[1811]  0.584  0.424 -0.776  0.576  0.176 -0.744  0.696 -0.216  0.104  0.904
[1821]  0.744  1.096  1.336 -0.464  0.576 -0.336  0.864 -0.336  0.064 -0.584
[1831] -1.104 -0.544  0.944 -1.744  0.904 -0.744 -0.304 -0.136 -0.136 -1.104
[1841] -0.144 -0.416  0.544 -1.104 -0.376 -0.144  0.864  0.024  0.064  0.224
[1851] -0.776 -0.944 -0.584 -0.904  0.904  0.664 -1.344 -0.696  0.696 -0.384
[1861]  0.176 -0.224 -1.136  0.696 -0.536 -1.296  0.016 -0.416 -1.064  0.096
[1871] -1.296 -0.176 -0.024  0.336  0.384  1.384  1.504 -0.736  0.384 -0.056
[1881]  0.104 -0.064  0.904 -1.344 -0.504  1.104  0.544 -0.264  0.736  0.384
[1891]  1.096 -0.384  0.536  0.256  0.104 -0.064  0.136  0.584  0.696 -0.056
[1901]  0.344 -0.744 -0.504 -0.776 -0.264 -0.584 -0.224 -0.136  0.864 -0.064
[1911] -0.216 -0.224 -0.144  1.144  0.176 -1.384 -0.584 -0.376  0.576 -0.056
[1921]  0.576 -0.736  1.504 -0.224  0.344 -0.864  0.896  0.304  0.584 -1.664
[1931]  0.144 -0.936 -1.056 -0.416 -1.144 -0.056 -0.064 -0.864 -0.224 -0.224
[1941] -0.776 -0.504  0.584 -0.864 -0.224 -0.384  1.664  0.776 -0.536  0.584
[1951] -0.664 -0.064 -0.064 -0.256 -0.496  0.024  0.496  1.384 -0.936  0.736
[1961] -0.536  1.184 -0.056  0.904 -0.256  0.224  0.664 -0.256  0.256 -1.504
[1971]  0.056 -0.496 -1.704  0.584 -1.384  0.896  0.064 -0.896 -0.304 -0.064
[1981] -0.256 -0.376  1.096  0.136 -0.784  0.296 -0.856  0.696 -0.736 -0.904
[1991] -0.216  1.064  0.384  0.136  0.144 -0.504  0.304  1.144  0.904 -0.664
[2001]  0.944  0.416  1.304 -0.136  0.416 -0.384 -1.704 -1.384  0.664  1.104
[2011]  0.544 -0.216  0.256  0.584  0.664 -0.136 -0.744  0.856 -0.376  0.896
[2021]  0.944  0.576 -0.504 -0.416 -0.216  0.064  0.136  0.744 -0.296 -0.064
[2031] -0.944  0.256  0.064  0.616 -1.664  0.104 -0.504  0.016 -1.344 -1.744
[2041]  1.504  0.416  0.744  1.104  0.064 -0.696  0.904  0.144 -1.744 -0.016
[2051]  0.576  0.896  0.896  0.584 -0.384  0.384 -1.384 -0.496 -0.784 -0.224
[2061] -0.944  0.776 -0.056 -0.384 -0.584  0.904 -0.224 -1.744  0.264 -0.024
[2071]  0.536 -1.184 -0.424 -1.096 -1.376 -0.864 -0.704  0.424  0.224  0.096
[2081]  0.536 -0.776  0.224  1.136  0.424  0.216  0.776 -0.304  0.416  0.776
[2091]  1.056  0.584 -0.024  1.104 -0.336  0.064  0.824  0.864 -0.944 -0.664
[2101] -0.536  0.376 -0.264  0.504  0.136  0.136  0.416  0.696 -1.104  1.704
[2111]  0.304 -1.504 -0.056  0.176 -0.864 -0.056  0.664 -0.136 -0.664  0.744
[2121] -0.416  1.744 -0.264  0.216  0.904 -0.584  0.896  0.704 -1.096  0.864
[2131] -0.384 -0.696 -1.104 -0.344 -0.056 -0.336 -0.336 -0.504  1.384  0.696
[2141] -0.504 -0.224 -0.576  0.776 -0.544  1.744 -0.496 -0.376  0.504 -0.064
[2151]  0.616 -0.576 -0.064 -1.136 -0.216 -0.424  0.624  0.176 -0.064  0.704
[2161] -0.336  0.664  1.144 -1.064  0.224  0.256 -0.744  0.576 -0.424  0.304
[2171]  1.664 -1.336 -0.536 -1.184 -0.504 -0.704  0.224 -0.504 -0.176  1.664
[2181]  0.376  0.104  0.904  1.296 -0.496 -0.144  0.944  0.544  0.136 -0.224
[2191]  1.696  0.464  0.424  0.776  0.016 -0.384 -0.344  1.144  0.696  0.616
[2201]  0.144  1.376  0.904 -0.864  0.144  0.336  0.256  0.576  0.056 -0.256
[2211] -1.704 -0.176 -0.776  0.344 -1.136  1.336  0.416  0.336 -0.784  0.936
[2221]  0.536  0.136  0.736 -0.024  0.696  0.936 -0.944 -1.104 -0.856  0.024
[2231] -0.296 -1.384 -0.304 -0.256 -0.664 -0.776 -0.736  0.096 -0.544 -0.384
[2241] -0.384  1.376  0.064 -0.936  0.336  0.304  0.136 -0.176 -1.184  0.216
[2251]  0.224  0.864  0.104 -0.536 -0.176  0.936 -0.416  1.744 -0.504  1.096
[2261] -0.304 -0.016 -1.136  1.704 -0.416  0.064 -0.304 -0.144 -0.096  0.104
[2271] -0.664 -0.144  0.736 -0.704 -1.056  0.296 -0.336  0.544  0.584 -0.256
[2281]  0.776 -1.224  0.264 -0.224  0.856  0.216  0.904 -0.936 -0.064  0.104
[2291]  0.944  1.344  0.336  0.336  0.696  0.016 -0.936 -0.264 -0.136 -0.384
[2301]  0.544 -0.096 -0.144  1.744 -0.856  0.256  0.056  0.864  0.536  1.344
[2311] -0.304  0.664 -0.904 -0.064 -0.856  1.136  0.296 -1.144 -0.136  1.336
[2321]  0.264 -0.136  1.224  0.504  1.336  1.384 -0.696  0.304  1.144  0.584
[2331] -1.664 -0.424 -0.776  0.304 -0.696 -0.376  1.056  0.784  0.424 -0.336
[2341] -1.296 -0.136 -0.736  1.336  0.296 -0.096  0.024  1.744  0.176  0.136
[2351]  0.584  0.336  0.136  0.344 -0.496  1.064 -0.216  0.096 -0.336 -1.296
[2361]  0.064 -0.296 -0.296 -0.024  1.696  0.896 -0.736  1.296  1.664 -0.264
[2371]  1.064 -0.264  0.544 -0.536  1.376  0.104 -0.576 -1.304 -0.336  0.104
[2381] -0.144  0.064  1.064  0.904  0.576 -0.304 -0.904  0.584  0.496  0.536
[2391] -0.304 -0.216 -0.464  0.136  0.096 -0.104  0.136  0.744  0.904 -0.296
[2401]  0.224 -1.144  0.904  0.896  0.536 -0.416 -1.704 -0.096 -0.856 -0.304
[2411] -0.664 -0.256  0.536 -0.504  0.176  1.096  0.016 -0.016 -0.096  0.264
[2421]  0.064 -1.344 -0.184 -0.904 -1.136 -0.504  0.544 -0.336 -0.264 -0.224
[2431] -0.464 -0.544 -0.264  0.096  0.904  0.416 -1.744 -0.384  0.864  1.384
[2441]  0.184 -0.584  1.664  0.424  1.136 -1.104  0.784 -0.936  0.896  0.424
[2451] -1.384  0.904 -0.024 -0.384 -0.576  0.744 -0.824 -0.856 -0.216  1.336
[2461] -0.256  1.336 -1.504 -1.064 -0.176  0.064 -0.664  1.136  0.424  0.776
[2471] -0.664  0.496 -0.864  1.064  1.336  0.176 -1.056 -0.104  0.224 -1.184
[2481] -0.224  0.904  0.696  0.536  0.536  0.136 -0.384 -0.016 -0.864 -0.784
[2491]  0.064  0.696 -0.176  1.296  0.256 -0.424 -0.056 -0.504  0.336 -0.224
[2501] -0.864  0.544  0.056 -0.744 -0.856 -0.064 -0.144 -0.624 -0.136 -0.496
[2511] -0.336 -0.856  0.904  0.824  0.536 -0.496 -0.776 -0.016 -0.784 -1.104
[2521] -0.376 -0.664  0.896 -0.784 -0.064  0.704  1.064 -0.744 -0.376 -1.744
[2531]  1.344  0.856  0.864  0.256  1.224 -0.384 -0.296  0.024 -0.384 -0.096
[2541]  0.016 -1.136 -0.864  0.224 -0.856 -0.344 -0.536 -0.144  0.544  0.616
[2551]  0.064 -0.264 -0.224 -0.104  0.504  0.944  0.224 -1.136  1.696 -0.936
[2561]  0.496  1.696  0.136  1.704 -0.384 -0.944  0.584  0.336  0.584 -1.744
[2571]  0.264 -0.376 -0.104 -0.296  0.504 -1.064 -0.176 -0.336  0.736 -1.096
[2581] -0.856  0.424 -0.144 -0.664 -0.096 -1.056 -0.304  0.384  0.904  0.096
[2591] -0.336 -0.304 -0.584 -0.864 -0.584  0.704 -0.376  0.176 -0.296  0.416
[2601] -0.264  0.624 -0.136  0.224  1.744  0.864 -0.136 -0.576  0.376 -0.064
[2611] -0.864  0.904 -1.296 -0.144  0.056  1.136 -0.824 -0.424 -0.864 -0.064
[2621] -0.864  0.256 -0.696 -1.144  1.104  0.224  0.384 -0.944  0.664  0.096
[2631]  0.904  1.376 -0.896  0.096  1.184  0.544 -0.576 -0.096 -0.704 -0.504
[2641] -0.776  1.376 -0.424  0.176 -0.216 -0.264 -0.384 -0.264  1.376 -1.144
[2651] -1.304  1.064 -0.744  0.584 -1.184  0.464 -0.096 -0.264  0.496  0.176
[2661]  0.544 -0.096 -1.096 -0.736 -0.376  0.696 -0.224 -0.736  0.536 -0.824
[2671]  0.736  0.904  0.576 -0.736  0.104  0.296  0.256 -1.744 -0.536 -0.384
[2681]  0.296  0.464  0.576  0.696  0.736  0.896  0.144  0.016 -0.944 -0.224
[2691] -1.224 -0.696 -0.704 -0.216 -0.096 -1.056  1.056  0.144 -0.256 -0.544
[2701] -0.496  0.224  0.696 -0.544  0.224  0.544 -0.144  1.384  0.496  0.064
[2711]  0.544  0.576 -0.376 -0.136 -0.144 -1.136  0.104  0.304  1.344  0.056
[2721] -0.304  0.904  0.736 -0.424  0.704 -0.304  0.576 -1.144 -1.376 -0.376
[2731] -0.256  0.384  0.256  0.016  0.696 -0.936  0.264  0.136 -0.416 -1.744
[2741] -0.064 -0.384 -0.376  0.144 -0.216  1.384  1.224  0.904  0.176  0.136
[2751] -0.824 -0.376  1.384 -1.344  0.216  0.056  0.864 -0.664 -0.096 -0.416
[2761] -0.016 -0.776 -0.256  1.104 -0.416 -0.224 -0.504  1.104 -0.176 -0.896
[2771]  0.016 -0.696 -0.504  0.384  0.136 -0.944 -0.936 -1.224  0.664 -1.304
[2781]  0.704 -0.904 -0.496 -0.096  0.224  0.896  0.776 -0.776  0.216  1.144
[2791] -1.096  0.544 -0.544  0.496 -0.576 -0.216 -0.336  0.696 -1.224  0.176
[2801]  0.064  0.696 -1.336 -0.696  0.464  0.544  1.224 -0.024  0.544  0.736
[2811] -0.016  0.864 -0.536 -0.184 -0.136  0.096 -1.336 -0.824  0.736 -0.664
[2821] -1.384  0.944  0.936  0.776  1.344 -0.024 -0.896 -1.336  1.744 -0.096
[2831]  0.336 -0.664 -0.256 -0.584 -0.744 -1.504  0.176  0.736  0.416 -0.064
[2841] -0.504  0.264  0.096 -0.664  0.504 -0.664  0.896 -0.544  0.096 -0.304
[2851]  0.424 -0.296  0.744  0.736 -1.664  0.056  0.496  0.896  1.704 -0.344
[2861] -0.664 -0.376  1.664 -0.256 -1.296  0.856 -0.664  0.056  1.344  0.584
[2871]  0.864  1.184 -1.704  0.224  0.064 -0.496 -0.744 -0.536  0.624 -0.144
[2881]  1.304 -0.864  0.704 -0.464  0.296  0.944 -0.664 -0.056 -0.744 -0.616
[2891] -0.296 -0.824 -0.536 -1.696 -0.296  0.904 -0.296  0.704 -1.104  0.704
[2901]  0.024 -0.776 -0.144 -0.856 -0.056  0.016  0.224  0.024 -1.144  0.664
[2911]  0.384  1.096  1.104  0.296 -0.144  0.376  1.504  0.416 -0.544 -0.944
[2921] -0.336  0.384  0.584  1.136  0.664 -0.904 -0.224 -0.224 -0.584  1.144
[2931]  0.696  0.224 -0.864  0.864  1.064  0.496  0.536 -0.096 -0.064  0.904
[2941] -0.384 -0.544  0.464  0.216  0.176  0.096  0.264 -0.904 -0.304  0.096
[2951] -0.224  1.696  0.616 -0.744  0.096  1.296  0.336  1.744 -1.144  0.064
[2961]  0.064  0.864  1.504  0.256  0.144 -0.464 -0.016  0.056  0.096  0.224
[2971] -0.264  0.104  1.064 -1.224  0.504 -0.136 -0.256 -1.696 -0.864  1.136
[2981] -0.944 -0.096 -0.104  0.344  0.664  0.304  0.664 -0.864  0.784 -0.696
[2991] -0.536  0.664  1.064 -0.864  0.896  0.296 -1.336  0.256 -0.496 -1.336
[3001] -0.016  0.664  0.296 -0.416 -0.424  0.336  0.176  0.664  1.096  0.416
[3011] -0.176 -0.424  0.256  1.224 -0.944 -0.464 -1.504 -0.736  0.296  0.864
[3021]  0.664  1.056 -0.864 -1.664 -0.296  0.704 -1.304 -0.304  0.536  1.304
[3031] -0.336 -0.584  1.664 -0.024 -0.664  0.744 -0.224  0.856 -1.056 -0.536
[3041]  0.216  1.384 -1.104  0.336 -0.304 -0.904 -0.296  1.144 -0.904 -0.144
[3051]  0.224 -0.336 -0.136  0.096 -0.664  0.464  0.584  1.744 -0.384  0.536
[3061]  0.504  0.864  0.576  0.296  0.856  0.704 -0.096  0.344 -0.904 -1.064
[3071] -0.056  1.304 -1.376 -0.664 -0.064  1.704 -0.416 -0.664  0.416  0.304
[3081]  1.136 -0.136  0.096  0.864  0.904  0.696  0.264 -1.104 -0.944 -0.864
[3091] -0.384 -0.016 -0.696 -0.864  0.744  0.536  0.064  1.056 -0.216  0.016
[3101] -1.064  0.864  0.304 -1.704  0.704 -0.856 -0.224 -0.136 -0.424  0.904
[3111] -0.024 -0.904  0.864 -0.256  0.056 -0.296  0.584 -0.136  0.096 -0.096
[3121]  0.184  0.576 -0.264 -0.776  0.336 -1.704  0.536  0.344 -0.864 -0.736
[3131] -0.424 -0.296 -1.144 -0.864  0.384 -0.064  0.416  1.344  0.096  0.136
[3141]  0.016  0.744  0.256  0.544 -0.144 -0.544 -0.256  0.624 -0.896  0.264
[3151] -0.704 -0.216  0.384  0.136 -0.744  0.096 -0.864 -0.736 -0.136 -1.664
[3161]  0.744 -0.776 -0.904 -0.144 -0.056  0.064  1.304  0.064 -0.784 -0.936
[3171]  0.896 -0.224  0.776  1.144  0.104 -0.856  0.376  1.384 -0.616 -1.504
[3181] -0.056 -0.256  1.064  0.904 -1.096 -0.096 -0.496 -0.064  0.736 -1.344
[3191] -0.736  0.544  0.144  0.784  0.624 -1.136 -0.424 -0.016 -0.256 -0.584
[3201] -0.136 -0.176 -0.536  0.536 -0.776 -0.544 -0.696 -0.064 -0.144  0.224
[3211] -0.344  0.336  1.144  0.696 -0.104  0.184  0.256  0.336 -0.896 -0.664
[3221] -0.744  0.704 -0.096 -1.104 -0.344  0.256 -1.296  1.136  0.416 -0.424
[3231]  0.096  0.496  0.056 -0.184  0.064 -1.144 -0.176 -0.424  0.024  0.616
[3241] -1.384  1.144 -0.064  0.296 -0.496  0.064  0.584  0.056  0.144 -0.216
[3251]  0.744 -0.736 -0.224  0.104 -1.144  0.144  0.144  0.224 -0.136  0.096
[3261] -0.576 -1.056  0.336 -0.064 -1.056 -0.184  0.296 -0.704 -0.096  1.344
[3271] -1.344 -0.264  0.216  0.304  0.416 -0.304 -0.336 -0.664 -1.136 -0.536
[3281] -0.296 -0.104 -0.944  0.096  0.384  0.024  0.904 -1.704  0.944  0.944
[3291] -1.064  0.256  0.336 -1.744 -0.624 -0.696  0.536 -0.136 -1.104 -0.096
[3301] -0.864 -0.296 -0.096 -0.496 -0.416  0.424  0.544  1.056  0.056  0.584
[3311] -1.096  1.304  0.224  0.744 -0.056 -0.024  0.904  0.416  0.496 -0.544
[3321] -0.304 -1.696  1.104  0.304 -0.504  0.344  0.064 -0.496  1.664 -0.944
[3331]  0.664 -0.496  0.784  0.136  0.224  0.024  0.784  0.064  0.176 -0.616
[3341] -1.144  0.896  0.904  1.696 -0.776  0.504  0.224  0.664 -0.064  0.544
[3351] -0.904  0.696 -1.704  0.216 -0.496 -0.384 -0.104 -0.864 -0.104  1.096
[3361] -0.064 -0.704  0.184 -0.496 -1.696  0.384 -0.104  0.344 -1.064  0.184
[3371] -0.336  0.304  0.104 -0.104  0.944 -0.224 -0.784 -0.496 -1.704  1.504
[3381]  0.664  0.496 -0.896  0.904 -0.376 -0.576  0.856  0.336 -0.584  0.504
[3391] -0.856  0.384 -0.864  1.296 -1.664  0.384  0.216 -0.736  0.056  1.384
[3401]  0.384 -0.096 -0.664  0.536  0.736  0.664 -0.144  0.416  0.304  0.536
[3411]  0.096  0.136 -0.224  0.416 -0.344  1.136 -0.536 -1.504 -0.904 -0.296
[3421] -0.296 -0.936 -0.056 -0.256 -0.296 -1.344  0.104 -0.104  0.224  0.136
[3431] -0.064 -0.104 -0.904  1.056  0.104  0.704 -1.096  0.864  0.696 -0.536
[3441] -1.376 -0.864  1.136  0.424  1.664  0.784 -0.936 -0.776  0.584  1.704
[3451] -0.016  0.536 -0.304  0.496  0.064  0.944  0.856  0.776 -0.064 -1.336
[3461] -1.296 -1.184  1.104 -0.616  0.224  1.336  0.096  0.864 -0.744  0.904
[3471] -0.424  0.144 -1.184 -0.584 -0.104 -0.776  0.704  0.336  0.344 -0.896
[3481]  1.504 -0.536 -0.336  0.416 -0.224  0.744  0.336  0.224 -0.136 -0.464
[3491]  0.896  1.304  0.216 -0.336  0.256 -0.744  0.096  0.384 -1.144 -0.224
[3501]  0.376  0.304  0.896 -0.584  0.336  0.176 -0.304 -0.536  0.384  1.144
[3511]  0.576  0.504  0.384  1.296 -0.104  0.216  0.104  0.104 -1.744 -0.544
[3521]  0.376 -0.544 -0.504 -0.896 -0.704 -0.824 -0.744 -0.336  0.696  0.856
[3531] -1.104 -0.376  0.104  0.064  0.664 -0.304  0.064 -1.184  0.184  0.064
[3541] -0.296 -0.016 -0.576  0.016  0.696  0.744  1.344 -0.144  0.576  0.064
[3551]  0.864 -0.536  0.024 -0.064 -0.536 -0.664 -0.376  0.144  0.064  0.904
[3561] -0.416 -1.184  0.744  0.544 -0.384  0.624 -0.856 -0.304 -1.104 -0.864
[3571]  0.544  0.104  0.256 -0.096 -0.904 -0.416  0.496 -0.696  1.376 -0.896
[3581] -1.344 -0.936  0.064 -0.736 -0.864  0.856 -0.064  0.696 -0.136  1.296
[3591]  1.336  0.944 -1.664 -1.224 -0.336 -0.264  0.296  0.056  1.296  0.056
[3601]  0.544  0.064  0.936  1.384  0.864  0.864  0.384 -0.064 -0.416 -0.584
[3611] -0.304  0.264  0.064 -0.584 -0.464 -0.336  0.024  0.696  1.304  0.744
[3621] -0.696 -0.064 -0.776 -0.664 -0.744 -0.864  0.736  1.064 -0.144  1.336
[3631]  0.176  0.104  0.144 -0.664  0.704 -0.664  1.136  0.776 -0.384  0.304
[3641] -0.496  1.064  0.336 -0.576 -0.536  0.944  0.864  0.544 -1.296  0.584
[3651] -0.384 -0.304  0.104  0.104 -0.296  0.664 -0.616  0.904  0.736  0.136
[3661]  0.096 -0.024 -0.096  0.256  0.416  0.176  1.136 -0.024 -1.384  0.704
[3671] -0.136  0.696  0.296  1.744  0.936  0.784 -0.544  0.776  0.944 -0.264
[3681] -0.024  0.256 -0.424  1.056  1.144 -0.416 -0.384  0.224 -0.936  1.384
[3691]  0.416 -0.864 -0.944  1.376  0.296  0.104  0.736 -0.896  0.856 -0.336
[3701]  1.064 -0.856  0.944  0.376  0.136 -0.864 -0.536  0.416  0.064  1.384
[3711]  0.664 -0.224  0.136  0.176 -0.696  0.936  0.536 -0.216 -0.664  1.224
[3721] -0.664 -0.296 -0.224  0.224  0.096 -0.264  0.936  0.704  0.704  0.584
[3731]  1.136 -0.304 -0.616  0.384 -1.744  0.584 -0.296  0.664 -0.464 -0.384
[3741]  0.744  0.576  1.104 -0.584 -1.696 -0.096  0.824  1.344  0.296 -0.216
[3751] -0.696 -0.216 -0.024 -0.256  0.544 -0.616  0.544 -0.264 -1.104 -0.096
[3761] -0.624 -0.696  0.176 -0.496 -0.304 -1.504 -0.464 -0.176 -0.384 -0.144
[3771] -0.224  0.944  0.256  0.576 -0.136  0.536 -0.336 -0.016 -0.064  0.376
[3781]  0.144  1.296  0.384  1.096  0.584 -1.104 -0.264 -0.496  1.144 -0.104
[3791]  0.216  0.544 -0.064 -0.304  0.296  0.504 -0.424 -1.104 -0.856  0.336
[3801]  0.256 -0.704 -0.416 -1.504 -0.264  0.104 -0.064  0.664 -0.264 -0.336
[3811]  0.504  0.224 -0.544  0.016  1.136 -0.384  0.696 -1.136  0.416 -0.136
[3821] -0.664  0.304  0.104  0.104  1.144 -0.744 -1.184  0.624 -0.224 -1.696
[3831] -1.344  1.096 -0.144  0.864 -0.096 -0.256 -0.696  0.224 -0.256 -0.096
[3841] -0.224  1.296  0.904 -0.464 -0.784  0.616 -0.296  0.296 -0.104 -0.144
[3851] -0.416 -0.584 -0.064  0.536 -0.904  0.096 -1.344  1.384  1.144 -0.304
[3861]  0.936 -0.144 -0.264  0.224  0.696  0.024 -0.424 -0.224  0.864 -1.096
[3871] -0.544  0.944  1.144 -0.536 -0.824  0.704 -0.056 -1.096  0.704 -0.904
[3881]  0.696  1.376 -0.344  1.744  0.296 -1.096 -0.224 -0.096  1.664  1.344
[3891] -0.176  0.744  0.624 -0.296 -0.136  1.384 -0.544 -0.016 -0.464  0.304
[3901] -0.496  1.136 -0.744 -0.904  0.696  0.384  0.696  0.144 -0.896  1.336
[3911]  0.336  0.424  1.104  0.176  0.056  1.096  0.504 -0.024  1.744  0.096
[3921]  0.296  0.904 -0.384  0.304  0.856 -0.024 -0.104 -1.744 -0.024  1.104
[3931]  0.264 -0.584 -0.296  0.256  0.104 -0.784 -0.824 -0.904 -1.296 -0.584
[3941] -0.576 -1.704  0.104 -1.096 -0.696 -0.296 -0.424  0.904 -0.896  0.224
[3951]  0.864 -1.104  0.384  0.664  1.144 -0.624 -0.576 -1.064 -0.056 -1.096
[3961] -0.464  0.104  1.096 -0.096  0.224  0.384 -0.704 -0.864 -0.264 -0.064
[3971]  1.136  0.904 -1.224 -0.096  1.136  0.736  1.376  0.336 -0.496  0.544
[3981]  0.336  1.136  0.544  0.864 -0.864 -0.344 -0.704  0.896  0.304 -0.256
[3991] -0.896 -0.744 -1.336 -0.896 -0.856  0.336  1.144 -0.096 -0.144  0.144
[4001]  0.824  0.504 -0.224 -0.536 -0.296 -1.504  0.736 -0.296  1.144 -1.096
[4011] -1.064  0.536  0.376 -0.344 -0.304 -0.776  0.304 -0.176  0.704  0.296
[4021] -0.856  0.424 -0.624  0.064  0.064 -0.376 -0.336  1.064 -0.144 -0.944
[4031]  1.104 -0.744  0.496 -0.024 -1.056  0.616 -0.056  0.864  1.064  0.104
[4041]  0.136  0.664 -0.136  0.096  0.384  0.384 -0.584  0.056  0.856 -0.576
[4051] -0.176  0.896 -0.904  0.304 -0.944  1.344 -0.256 -0.864 -0.296 -1.184
[4061]  0.576 -0.664  0.936  0.744  0.896 -0.536 -0.744 -0.416  0.696  0.544
[4071]  0.584  0.664 -0.304 -0.376 -0.096 -0.056 -0.696  1.304 -0.304 -0.664
[4081]  0.176  0.736 -0.464  0.584  0.744  0.616 -1.144 -1.744 -0.144  0.576
[4091]  0.376 -0.064 -1.064 -1.136 -0.864 -0.144  0.096 -0.216 -0.824 -0.584
[4101] -1.664  0.776 -0.664  0.664 -0.056 -1.144 -1.744  0.264 -0.264  0.064
[4111] -0.664 -0.944 -0.224 -0.296  0.864 -0.504 -0.096  0.904 -0.864  0.296
[4121]  0.064 -0.864  0.496  0.336 -1.696 -0.144  1.056  1.304  1.184 -1.744
[4131]  0.696  0.056  0.496  1.184  0.096  0.416 -0.784 -1.296  0.096  0.424
[4141]  0.224 -1.744  0.576  0.824 -0.384 -1.664  0.904 -0.704  0.336  0.104
[4151]  1.344  0.864 -0.384 -0.024 -0.224  0.144  0.864  0.944 -0.136 -1.144
[4161]  0.144 -0.584 -0.776  0.536  1.384 -0.376  1.064  1.144 -0.536 -0.064
[4171]  1.384  0.144  0.536  0.064 -1.144  0.584 -0.336  0.584 -0.104 -1.224
[4181] -0.576  0.544  0.544 -0.304 -0.664 -0.304  0.864  0.904  0.744 -1.096
[4191]  0.896 -1.144 -0.336 -0.264 -0.296  0.416  0.264  1.504  1.704 -0.224
[4201] -0.096 -1.744  0.224 -0.384  0.016  0.536 -0.776 -0.336  0.624  0.144
[4211]  0.536 -0.376 -0.216 -1.096  0.696 -0.056  0.664 -1.064 -0.224  1.504
[4221] -0.096  0.416 -0.224 -0.384 -1.064 -1.224  0.584 -0.464  0.304  1.144
[4231] -0.896 -0.944  0.704  0.136 -0.256 -0.544  0.856  0.136 -0.056  1.376
[4241]  1.344 -1.184  0.136  0.584 -0.904  0.376 -0.024  0.296 -0.216  0.584
[4251]  1.704 -0.224 -1.056  0.224 -0.376  0.064 -0.216 -1.336  1.144 -0.256
[4261]  0.584  0.696  0.664 -0.904 -1.376  0.064  0.896  0.856  0.376 -0.216
[4271] -0.336 -1.064 -1.096  0.376  0.416  0.304  0.384  0.144  0.216 -0.776
[4281] -0.136 -0.136 -1.296 -0.256 -0.536 -0.824 -0.224  0.056  0.944 -0.784
[4291]  0.864 -0.664 -0.064 -0.584 -0.744  0.504 -0.776  0.896 -0.896  0.624
[4301] -0.336  0.384  1.296 -0.776 -1.704 -0.704 -0.296 -0.544 -1.096 -0.384
[4311] -0.296 -0.104 -0.304 -0.704  0.304  0.344  0.544 -0.696  0.336  0.584
[4321]  0.064 -0.416  0.344  0.024  1.696  0.504  0.704 -0.104  0.304 -0.104
[4331] -0.416  0.144 -1.104  0.624 -0.224  0.064 -1.296  0.304  0.224  0.696
[4341] -0.696 -0.664 -1.136  0.216 -0.664 -0.496 -0.904  0.104  1.376 -0.744
[4351]  1.104  0.056  0.584 -0.576  0.384  0.336 -0.096  1.296 -1.744 -0.424
[4361] -1.144  1.136  1.304  1.304  0.064 -0.736  1.304 -0.424  0.536  0.416
[4371] -0.904 -0.264  0.904  1.504  0.576  1.104  0.696  0.536 -0.224 -0.896
[4381]  0.744  0.144 -0.336 -0.064 -0.024 -1.144 -0.064 -0.264 -0.616  0.384
[4391]  1.136 -1.664  0.224 -0.736 -0.136 -0.136  0.744  0.104 -0.536  1.344
[4401]  0.144 -0.384 -1.376 -0.176 -0.104  0.024 -0.096  0.024  0.744 -0.024
[4411] -0.384 -0.824  0.384 -0.064  0.616  1.376 -1.184  0.704  0.176 -0.664
[4421] -0.576 -1.096  0.496 -0.064 -0.216 -0.064  1.136 -0.576  0.384 -0.736
[4431] -0.336  0.144 -0.664  0.304  0.296  0.536 -0.176  0.296  0.064  0.144
[4441] -0.224 -0.216  0.104  1.136  0.904 -0.776  0.664  0.136 -0.744 -0.696
[4451] -0.304 -0.904  0.696 -0.496  0.664 -0.096  0.224 -0.224 -0.584 -0.936
[4461] -0.136  0.144 -0.544 -0.904  0.856 -1.056 -0.736 -0.576 -0.064  0.864
[4471]  1.144  0.744  0.584  0.296  0.176  0.544  0.664  0.576  1.144  0.864
[4481]  0.584 -0.896 -0.584  1.184 -0.864 -0.384  0.096 -0.904 -0.536 -0.136
[4491] -1.704 -0.864  0.224  0.896  1.144 -0.416  0.176  0.776 -0.944 -0.304
[4501] -0.536  0.296  0.296 -1.096 -1.304 -0.256 -0.624 -0.784  0.376 -0.256
[4511] -1.096 -1.144 -0.424  0.264 -0.104  0.256  0.584 -0.056  0.296  0.016
[4521] -1.504 -0.464 -1.104 -1.336  0.384 -1.504  1.384  1.336 -0.504  0.096
[4531]  0.136  1.064  0.064  0.736 -0.104 -0.296 -0.056  0.824 -1.704 -0.224
[4541]  0.744  0.104  0.664 -0.864  0.024 -0.384 -1.504 -1.056  0.576  0.864
[4551]  1.136 -0.064 -1.104 -1.096 -0.536 -0.376 -0.944 -0.344  0.704  0.536
[4561] -0.096  0.464  0.384  0.704 -0.136 -0.016  0.544 -0.384  0.496 -1.184
[4571] -0.064  0.824  0.144  0.384 -0.544  0.944 -0.336  0.944  0.216  0.176
[4581] -0.384  0.936 -0.104  0.576 -0.936  0.536 -1.344  0.416 -1.664  0.424
[4591]  0.744  0.384 -0.584 -0.744  0.304 -0.664 -0.024  0.616 -1.064 -0.024
[4601] -1.664 -1.144  0.136 -0.104  0.936 -0.424  1.144 -1.664  0.704 -0.256
[4611]  0.264  0.544  0.136  1.664 -0.536  0.336  1.696 -0.504 -1.304  1.696
[4621] -1.304  0.256 -0.064  0.896 -0.384 -0.224 -1.664  0.104  1.056 -0.536
[4631] -0.336  0.064 -0.584  0.136 -1.376 -0.176  1.336  0.704  0.664  0.336
[4641] -0.704  1.096  1.096  0.864 -0.024 -0.096  0.024 -0.064  0.064 -1.136
[4651]  0.824  0.424 -1.504 -0.256  0.184  0.504 -1.384 -0.504  1.296  1.144
[4661]  0.736 -0.384  0.864 -0.856 -0.096 -1.136  1.504  0.016 -0.624 -0.336
[4671] -1.696  0.136  1.304 -0.584  0.304 -1.336 -0.096 -0.496 -0.544 -0.736
[4681]  1.344 -0.864  1.504 -0.376  0.176 -0.304 -0.584 -0.576 -0.056 -0.584
[4691] -0.544 -1.136  0.304  0.376 -0.864  0.864 -0.584  0.584  0.104  1.056
[4701]  0.696  0.256  0.736  1.304  0.136  0.824  0.544 -1.104  1.504  1.136
[4711] -0.536  0.096  0.336  0.944 -0.136  1.064  0.576  1.664  0.136 -1.136
[4721] -0.224 -0.904  0.696 -0.576 -0.584 -0.336  0.104  0.544  0.704  0.496
[4731] -0.584 -0.264  0.424  0.096 -0.056  0.896  0.544 -0.544 -0.256  0.696
[4741] -0.696  0.736  0.424  0.336 -0.224  1.376 -0.896  0.576 -0.464 -0.104
[4751]  0.544  0.664 -0.576 -0.064 -0.896  0.104  0.824  0.624 -0.416 -0.024
[4761] -1.104  0.176  0.416 -0.016  0.064  0.864  0.256 -0.176  0.624 -0.296
[4771] -1.056  0.504  0.336 -0.584 -1.104  0.504  0.056 -0.104 -1.664  0.664
[4781]  0.736 -0.536  0.224  0.104 -0.416  0.896 -0.864 -0.064  0.864  0.096
[4791] -0.224 -0.176  0.096 -0.064  0.664 -1.664  1.144  0.536  0.056  0.096
[4801] -0.824  0.704  0.096 -0.464 -0.824  0.256  0.216  0.464  0.776  0.744
[4811]  0.016 -0.416  0.576  1.376  0.424 -0.696  0.944 -0.736 -0.704 -0.064
[4821]  0.536  1.504 -0.904 -1.184  0.264 -0.176 -0.144 -0.056  0.224 -0.336
[4831]  1.336  0.144  1.376  1.704  0.856 -0.256 -0.224  0.944  0.944 -0.424
[4841] -0.584 -0.824  0.056  0.744  0.064 -1.696  1.696  0.096 -0.304 -0.904
[4851]  0.224 -1.504 -0.896 -0.024  0.504 -0.576  1.344  0.224  0.384  0.336
[4861]  0.584  0.416  0.136 -0.856 -0.296  0.696  0.704  1.096 -1.104 -0.264
[4871] -0.944 -0.896 -0.536  0.336 -0.064  0.496  1.704  0.536 -0.504  0.944
[4881]  0.256  0.144  0.944  0.136 -0.664  0.664  0.616  0.376 -1.376  0.616
[4891] -0.736 -0.136  0.336  0.304  0.384  1.144  0.064  0.904 -1.376 -0.736
[4901]  1.104  0.304 -0.136 -1.344 -0.536 -0.496  0.176 -1.344 -0.904  1.104
[4911]  0.584  0.944  0.416 -0.944  0.064 -0.896  0.336 -0.384 -0.216  0.104
[4921]  0.776 -0.384 -0.064  1.224  0.616  0.384  0.624 -0.376  0.384 -0.544
[4931]  0.024  0.496  0.696 -1.744  0.904 -0.376 -1.104  0.304  1.104  1.336
[4941] -1.304  0.704 -0.096  0.704 -0.344  1.296 -0.936 -0.304  0.744  1.136
[4951]  0.856  0.696  1.504  0.216  0.544  0.936 -0.744 -0.744  0.744 -0.584
[4961]  0.136 -0.896  0.864 -0.904  0.104  0.904  0.896  0.544 -0.064  0.664
[4971]  0.696  0.704 -0.296  0.584 -0.224 -0.344  0.544  1.704  1.136  0.064
[4981] -0.096 -0.216 -0.104  0.224 -0.264 -0.944 -1.664 -0.576 -0.584  0.496
[4991]  0.544  0.104  0.296 -0.296 -0.336  0.304 -0.944 -0.176  0.704 -0.536

Estimate Dist under H0

p_H0 <- data.frame(stat = H_0) |>
  ggplot(aes(x = stat)) +
  geom_histogram() +
  #geom_vline(xintercept = c(left_threshold, right_threshold), col = "tomato") +
  theme_bw()
p_H0

Find Reject Region

left_threshold <- quantile(H_0, .025)
right_threshold <- quantile(H_0, .975)

Estimate Dist under H0

p_H0 +
  geom_vline(xintercept = c(left_threshold, right_threshold), col = "tomato")

An Alternative Hypothesis

An Alternative Hypothesis

Consider a constant shift of 1 for all units.

tau <- 1

From data to schedule

my_obs_sched <- my_data |>
  mutate(Y_0 = ifelse(d_i == 0, y_i, NA),
         Y_1 = ifelse(d_i == 1, y_i, NA)) |>
  select(i, Y_0, Y_1)

my_obs_sched |>
  gt() |>
  tab_header(title = "My Observed Schedule")
My Observed Schedule
i Y_0 Y_1
1 NA 0.90
2 -0.5 NA
3 NA 2.20
4 -1.4 NA
5 0.1 NA
6 -1.0 NA
7 1.1 NA
8 NA 1.00
9 NA 1.02
10 NA 1.50
my_alt_sched <- my_obs_sched |>
  mutate(Y_0 = ifelse(is.na(Y_0), Y_1 - tau, Y_0),
         Y_1 = ifelse(is.na(Y_1), Y_0 + tau, Y_1))

my_alt_sched |>
  gt() |>
  tab_header(title = "My Alt Schedule")
My Alt Schedule
i Y_0 Y_1
1 -0.10 0.90
2 -0.50 0.50
3 1.20 2.20
4 -1.40 -0.40
5 0.10 1.10
6 -1.00 0.00
7 1.10 2.10
8 0.00 1.00
9 0.02 1.02
10 0.50 1.50

Estimate Dist under HA

H_A <- rand_stats(schedule = my_alt_sched, d_i = my_data$d_i, reps = 5000)
H_A
   [1]  0.736  1.624  1.376  1.776  0.536  1.096  1.264  1.664  0.896  1.744
  [11]  1.216  0.984  0.696  1.104  1.304  0.544  0.456  1.696  2.096  0.944
  [21]  1.296  0.944  0.944  1.304  0.896  0.504  0.784  0.744  1.264  1.424
  [31]  0.856  1.496  1.904  1.096  0.864  0.976  1.424  0.856  1.704  1.464
  [41]  0.984  0.616  0.584  0.896  1.464  0.664  0.536  0.944  0.896  0.536
  [51]  0.496  0.224  0.056  1.376  1.744  0.864  1.184  1.784  1.064  0.784
  [61]  1.584  1.384  0.064  1.024  0.544  1.224  1.064  0.696  0.656  0.816
  [71]  1.336  1.144  1.456  0.456  1.544  1.224  0.064  0.224  1.256  1.224
  [81]  1.056  1.136  1.744  2.136  1.104  0.424  1.504  1.304  0.736  0.896
  [91]  0.944  1.304  1.464  1.064  0.776  0.664  1.376  0.936  0.224  1.576
 [101]  0.936  1.184  0.456  1.704  0.624  0.584  0.904  0.864  0.984  1.504
 [111]  0.816  2.096  1.416  1.944  0.264  0.936  1.056  0.216  1.056  0.624
 [121]  0.424  2.096  1.184  1.296  0.336  0.496  0.856  0.456  0.896  0.936
 [131]  1.144  1.976  1.696  1.064  1.656  2.136  2.176  0.864  1.264  1.896
 [141]  0.944  0.824  1.664  0.304  1.064  0.064  0.336  2.096  0.656  1.544
 [151]  1.504  0.224  0.504  0.656 -0.144  0.696  0.776  0.816  0.936  0.856
 [161]  1.304  1.584  0.704  1.496  1.016  1.744  0.104  1.024  1.696  0.824
 [171]  1.576  1.256  0.616  0.344  1.104  0.984  2.144  0.656  0.944  0.936
 [181]  0.856  0.976  0.464  0.504  0.656  0.816  0.304  0.224  0.336  1.136
 [191]  1.776  1.304  0.704  1.496  1.184  0.304  1.904  0.776  0.656  1.344
 [201]  1.256  1.104  1.344  0.216  1.976  1.224  0.584  1.624  0.984  0.936
 [211]  0.496  1.336  1.376  1.376  1.176  1.056  1.576  0.336  1.536  0.944
 [221]  0.416  1.616  0.464  1.096  0.936  1.504  1.696  0.856  1.016  0.104
 [231]  1.264  0.544  1.304  0.944  1.576  0.024  1.104  1.096  0.384  0.656
 [241]  0.416  0.896  0.656  1.536  0.096  0.224  0.984  1.336  1.224  0.424
 [251]  1.136  0.736  1.096  1.016  0.456  1.136  0.624  0.704  0.864  0.256
 [261]  1.704  0.784  0.664  0.936  1.296  1.256  0.056  1.096  1.104  1.944
 [271]  0.384  1.104  0.464  1.096  0.296  1.576 -0.136  0.744  0.744  0.336
 [281]  0.944  1.664  0.016  0.464  1.464  0.896  0.784  0.504  0.856  1.424
 [291]  0.904  0.464  1.304  1.264  1.064  1.304  0.536  0.864  0.984  1.024
 [301]  0.384  1.056  1.944  1.096  1.664  1.016  0.904  1.024  1.784  1.576
 [311]  1.096  1.144  0.296  0.264  0.984  1.424  0.944  0.936  1.176  0.384
 [321]  0.344  1.384  1.944  1.896  0.576  0.696  0.896  1.496  0.864  1.304
 [331]  0.904  1.504  0.624  1.176  1.624  1.264  0.704  1.224  1.184  0.016
 [341]  0.104  1.104  0.936  0.824  1.576  1.256  1.144  1.776  0.024  0.824
 [351]  0.936  0.544  0.096  1.624  1.464  1.616  1.056  2.104  0.984  1.064
 [361]  1.544  1.224  1.344  0.664  1.144  1.304  0.704  0.456  0.744  1.136
 [371]  0.336  0.744  0.696  0.776  0.536  1.104  1.944  1.664  0.856  1.104
 [381]  0.704  0.904  1.256  1.704  1.336  1.064  1.936  0.664  0.256  0.944
 [391]  1.504  1.304  0.936  1.104  0.936  0.504  0.864  0.624  0.464  2.104
 [401]  1.184  0.944  1.136  2.096  0.776  1.584  1.704  1.296  1.264  0.296
 [411]  1.184  0.384  0.376  1.264  0.496  0.896  1.264  0.704  0.656  0.584
 [421]  0.824  0.704  0.536  0.504  2.176  1.376  0.704  1.544  1.344  1.496
 [431] -0.104  0.504  0.464  1.016  0.656  0.416  0.296  0.496  0.624  0.944
 [441]  1.544  2.144  0.904  0.896  1.016  1.264  0.464  0.416  0.656  0.456
 [451]  0.464  0.944  1.344  1.136  1.376  1.296 -0.096  0.696  1.336  0.504
 [461]  0.856  1.704 -0.096  1.064  0.224  0.024  1.176  1.944  1.904 -0.136
 [471]  0.536  1.016  1.056  1.016  0.744  1.176  1.344  0.824  1.944  1.256
 [481]  1.696  0.464  1.336  2.144  2.144  1.064  1.016  1.216  0.696  1.584
 [491]  0.864  0.496  0.704  1.304  2.176  0.664  1.376  1.704  1.096  1.064
 [501]  1.896  1.304  1.696  1.496  0.864  0.056  1.064  0.624  1.016  0.664
 [511]  1.216  0.816  1.024  1.344  0.224  1.056  1.696  0.616  0.056  1.264
 [521]  1.304  0.944  0.496  1.456  1.344  0.904  1.056  0.704  0.416  1.536
 [531]  0.456  0.256  1.424  0.416  0.904  1.544  0.896 -0.176  0.936  1.056
 [541]  1.216  0.944  0.056  1.536  1.536  0.056  1.584 -0.184  0.456  1.696
 [551]  1.224  1.416  1.416  0.496  0.696  2.136  0.744  0.896  1.016  0.784
 [561]  1.056  1.384  0.264  0.896  0.496  0.464  1.056  1.544  1.376  1.456
 [571]  1.056  1.896  0.896  0.944  0.696  0.264  0.536  0.904  0.776  0.896
 [581]  1.536  0.584  1.616  1.704  0.296  0.896  0.936  1.336  1.984  1.104
 [591]  1.096  1.296  1.136  1.096  0.376  0.896  1.336  0.736  1.096  0.544
 [601]  1.936  1.136  0.656  1.544  1.936  0.376  1.264  0.816  0.464  1.344
 [611]  0.896  0.656  1.936  1.496  1.976  0.384  1.496  0.576  1.584  1.936
 [621]  1.624  0.776  1.016  0.456  0.296  1.184  1.496  0.904  0.776  0.904
 [631]  1.104  1.536  1.096  0.736  0.576  0.064  0.976  1.336  1.344  1.376
 [641]  1.784  0.656  1.216  0.096  0.776  0.336  1.376  0.984  1.944  0.304
 [651]  0.656  0.776 -0.184  1.696  1.016  1.184  0.816  0.664  1.704  1.336
 [661]  0.744  0.056  0.656  0.864  0.496  1.184  1.336  1.016  1.304  1.056
 [671]  0.504  0.864  1.224  1.136  1.904  1.344  0.544  0.864  0.896  1.736
 [681]  0.416  0.496  1.456  1.536  0.704  0.416 -0.144  1.904  0.576  1.336
 [691]  0.104  1.656  0.744  0.224  0.664  1.016  1.976  1.416  1.944  0.344
 [701]  1.144  1.024  1.184  0.736  1.016  0.496  1.136  0.704  1.216  2.176
 [711]  1.056 -0.176  0.864  0.656  0.496  1.096  0.696  0.656  1.944  1.304
 [721]  0.024  0.776  1.736  1.504  1.064  1.264  0.864  0.824 -0.184  0.976
 [731]  1.144  1.344  1.256  0.056  0.656  1.224  1.584 -0.144  0.024  0.936
 [741]  1.224  1.664  1.176  1.376  1.744  1.736  0.504  0.984  1.184  0.504
 [751]  1.344  2.184  1.696  1.304  0.304  0.704  0.904  0.776  0.464  1.104
 [761]  0.224  1.096  1.456  0.856  1.576  0.936  0.344  1.136  0.504  0.984
 [771]  1.096  0.456  0.696  1.096  0.416  1.024  0.944  1.504  1.096  1.304
 [781]  1.616  0.936  1.424  1.104  0.344  0.656  1.304  0.464  0.536  0.776
 [791]  1.504  0.456  1.136  1.704  1.096  1.696  1.016  0.896  1.064  1.296
 [801]  0.784  0.496  0.224  1.096  1.744  0.776  0.616  0.856  2.136  1.096
 [811]  1.056  0.784  0.624  2.176  0.584  0.816  1.504  1.616  0.304  0.536
 [821]  0.424  1.784  0.304  1.696  1.576  0.936  1.984  1.024  1.304  1.144
 [831]  0.976  0.784  0.696  0.464  1.544  1.984  1.504  0.504  1.024  0.576
 [841]  0.776  1.704  1.504  0.744  1.336  0.976  0.224  1.144  0.784  0.416
 [851]  1.104  0.464  0.496  0.104  2.144  0.264  2.176  0.056  0.656  2.144
 [861]  0.536  0.336  1.944  0.376  0.744  0.624  0.704  0.824  1.304  1.304
 [871]  0.664  0.784  0.304  1.624  1.384  1.976  1.056  1.296  1.536  1.264
 [881]  0.504  1.096  0.264  1.784  0.504  0.896  1.504  1.496  0.704  0.336
 [891]  1.096  1.784  0.296  1.704  1.536  0.944  1.784  1.104  0.656  0.104
 [901]  0.464  0.256  1.456  0.696  1.496  1.096  1.456  1.336  0.304  0.584
 [911]  1.224  0.656  1.064  1.264  2.096  1.264  0.576  0.656  0.696  1.984
 [921]  1.384  0.656  1.144  1.744  0.864  1.136  1.776  0.784  1.616  0.296
 [931]  0.656  1.344  0.096  2.136  0.656  1.104  1.224 -0.136  1.264 -0.136
 [941]  1.776  1.496  1.664  1.136  0.464  1.104  0.864  0.304  1.136  1.096
 [951]  0.816  1.664 -0.144  1.904  1.496  1.344  1.096  1.536  0.264  0.576
 [961]  0.704  1.504  0.864  0.256  1.016  1.056  0.664  0.536  1.696  1.696
 [971]  0.904  0.616  0.424  1.736  0.624  0.704  0.936  1.136  1.104  1.456
 [981]  0.464  1.704  0.864  0.336  1.304  1.224  1.656  0.864  0.896  0.936
 [991]  1.016  0.736  1.024  1.224  1.264  1.256  0.736  0.736  0.104  0.944
[1001]  1.696  0.816  0.464  0.256  0.856  0.624  0.456  0.704  0.464  1.584
[1011]  0.304  0.784  1.704  0.696  0.816  1.224  0.936  1.504  1.096  0.536
[1021]  1.584  0.696  1.784  0.896  1.776  2.184  1.664  0.544  1.256  1.104
[1031]  1.544  1.136  0.064  1.296  0.904  0.936  1.224  0.496  0.816  1.016
[1041]  0.496  0.064  1.304  1.504  0.104  0.056  0.256  1.576  0.336  1.464
[1051]  1.944  0.936  0.696  0.896  0.696  0.344  1.136  0.864  2.144  0.984
[1061]  1.216  0.616  1.144  1.504  1.104  0.696  1.536  0.544  1.536  1.016
[1071]  2.136  1.736  0.984  1.344  0.376  0.624  1.784  0.536  1.584  1.056
[1081]  0.096  1.544  0.384  0.464  0.624  0.936  1.736  0.296  0.976  0.256
[1091]  1.136  0.464  0.256  0.464  0.864  0.696  0.936  1.464  0.424  0.784
[1101]  1.304  1.144  1.624  0.624  1.344  1.736  1.424  1.616  0.456  0.056
[1111]  1.936  0.904  1.544  0.744  0.496  1.744  0.944  1.216  1.424 -0.136
[1121]  0.496  0.936  1.176  1.464  0.576  0.304  1.336  1.616  0.856  1.056
[1131]  1.736  1.176  0.736  1.776  1.704  1.544  0.664  0.464  0.936  0.264
[1141]  1.704  1.176  1.504  0.904  1.176  1.024  1.296  0.936  0.936  1.104
[1151]  0.416  0.584  2.144  0.824  0.544  0.584  1.136  1.184  1.136  0.576
[1161]  0.056  1.496  0.464  1.696  1.256  0.736  1.696  0.656  1.296  1.096
[1171]  1.464  1.584  0.896  0.816  0.304  0.264  1.736  0.456  1.144 -0.096
[1181]  0.704  1.504  1.704  1.376  0.664  0.776  0.824  1.176  1.496  1.744
[1191]  1.456  1.024  0.976  1.064  0.784  1.184  1.136  0.736  1.744  0.944
[1201]  0.496  1.376  0.264  0.664  1.544  0.456  0.104  0.864  0.296  0.016
[1211] -0.176  1.104  1.144  0.824  0.704  0.656  0.256  0.664  0.424  1.264
[1221]  0.984  2.104  0.544  1.064  0.544  0.576  0.064  0.736  0.656  0.384
[1231]  1.096  1.296  0.264  0.944  1.704  0.664  0.744  0.816  0.096  1.664
[1241]  0.976  1.176  1.704  1.064  0.536  0.016  1.216  0.424  0.664  0.424
[1251]  1.696  1.344  1.744  0.376  0.744 -0.136  0.344  1.256  0.384  0.976
[1261]  1.696  1.064  0.864  1.056  1.976 -0.144  1.144  1.896  1.584  0.056
[1271]  0.944  1.496  0.896  0.104  1.336  0.464  0.944  1.984  0.776  0.304
[1281]  0.864  1.376  1.096  1.584  0.816  0.896  0.296  0.736  1.736  1.344
[1291]  0.936  0.904  0.704  0.936  0.496  0.776  0.784  1.016  0.944  0.704
[1301]  0.976  1.336  0.216  1.144  1.416  0.856  0.744  1.696  0.616  0.456
[1311]  1.944  1.664  1.536  1.296  1.264  0.776  0.336 -0.144  0.336  0.904
[1321]  1.536  1.584  1.264  0.424  0.544  1.696  1.304  0.504  0.536  1.696
[1331]  2.144  1.696  1.184  1.136  0.696  0.024  0.736  0.536  1.224  0.704
[1341]  0.944  0.864  1.256  0.864  1.256  1.264  0.984  0.216  0.696  1.144
[1351]  1.544  0.224  1.416  1.264  1.224  0.496  1.664  1.224  1.056  1.304
[1361]  0.624  1.944  0.024  0.704 -0.104  1.216  1.304  0.936  1.664  1.544
[1371]  1.704  1.136  1.304  0.744  0.056  0.496  2.096  0.744  0.656  1.944
[1381]  0.904  1.744  1.064  0.984  0.264  0.256  2.184  0.696  0.376  1.424
[1391]  0.216  0.504  1.336  0.696  1.784  0.904  0.824  0.536  0.056  0.304
[1401]  1.096  0.544  0.536  0.976  2.176  1.216 -0.104  0.104  1.504  1.456
[1411]  1.296  1.704  1.464  2.104  0.656  1.384  1.976  1.584  1.936  1.984
[1421]  0.624  0.656  0.816  1.104  0.216  1.776  0.464  1.296  1.224  1.696
[1431]  1.264  0.696 -0.184  0.456  0.264  1.384  0.584  0.496  1.544  0.904
[1441]  1.304  0.936  1.096  0.224  1.624  1.064  2.176  1.496  1.944  2.184
[1451]  0.296  0.584  0.464  0.056  1.536  1.384  1.464  1.296  1.096  0.496
[1461]  0.064  1.336  0.744 -0.144  0.616  0.864  1.064  0.944  1.056  0.544
[1471]  1.056  0.424  1.056  1.216  0.656  0.736  0.864  1.456  0.936  0.104
[1481]  0.256  1.184  0.944  1.296  0.424  0.344  0.104  0.336  1.504  1.304
[1491]  0.416  1.896  0.944  0.664  1.544  0.944 -0.096  0.256  1.136  0.056
[1501]  1.184  1.576  1.536  1.424  1.736  0.544  0.256  1.456  0.376  0.624
[1511]  2.096  2.184  0.704  1.136  0.464  1.664  0.816  1.704  1.736  1.104
[1521]  1.176  0.944  1.504  0.776  1.216  0.904  1.304  0.896  1.064  0.576
[1531]  0.936  1.064  1.176  0.536  0.536  1.464  0.064  0.576  0.744  0.264
[1541]  0.256  0.424  0.904  1.584  2.096  0.776  1.504  1.224  0.544  1.424
[1551]  1.744  0.576  1.424  0.016  1.344  1.224  0.656  0.536  0.704  0.784
[1561]  1.224  1.016 -0.144  0.304  0.904  1.056  0.464  0.056  1.104  0.864
[1571]  0.496  1.344  1.696  0.376  1.376  0.856  1.256  0.776  1.144  1.064
[1581]  0.736  1.464  0.256  1.064  0.936 -0.184  0.776  0.336  0.856  1.224
[1591]  0.016  2.136  2.176  0.904  0.496  0.296  0.544  1.144  0.984  1.264
[1601]  0.664  0.056  0.704  1.496  1.744  0.496  0.224  1.256  1.136  1.344
[1611]  0.264  0.416  0.304  0.104  1.544  1.256  2.184  1.744  1.696  1.176
[1621]  1.704  0.896  1.664  0.624  0.824  0.656  0.504  0.736  0.264  1.744
[1631]  1.904  0.856  0.896  1.496  1.064  1.944  0.736  1.744  0.976  0.704
[1641]  0.944  1.576  1.616  0.104  0.744  0.864  0.936  0.864  0.296  1.216
[1651]  0.024  1.704  1.304  0.784  0.944  0.696  0.296  0.824  0.664  1.696
[1661]  0.904  1.064  0.056  1.304  1.944  1.216  1.344  1.904  1.144  1.704
[1671]  1.304  0.216  0.616  0.664  0.696  1.904  0.576  1.456  0.664  1.464
[1681]  0.424  1.104  0.656  1.496  2.176  0.696  1.096  1.264  0.304  0.496
[1691]  0.824 -0.096  1.656  1.264  2.184  0.536  1.344  0.224  0.896  0.696
[1701]  0.896  0.896  0.616  0.456  0.464  1.544  0.504  1.056  1.096  0.056
[1711]  0.904  0.096  0.864  0.936  0.304  1.064  1.176  1.776  0.744  1.416
[1721]  1.456  0.624  0.704  0.464  1.016  1.744 -0.104  1.424  1.944  1.096
[1731]  1.104  1.224  1.544  1.504  1.104  1.744  1.984  1.144  1.936  0.256
[1741]  1.656  0.776  2.144  0.256  0.744  1.976  1.304  0.304  0.504  0.104
[1751]  0.336  1.016  1.496  1.184  1.064  1.496  0.784  1.304  0.936  0.064
[1761]  0.496  0.896  2.096  1.016  0.856  0.864  1.704  1.056  1.056  0.864
[1771]  1.536  1.344 -0.136  0.296  1.536  0.904  0.624  0.856  0.624  1.224
[1781]  0.224  1.944  1.944  0.696  0.256  0.536  0.064  0.936  0.256 -0.176
[1791]  0.944  1.976  0.856  1.536  1.064  0.696  0.256  0.056  1.184  1.024
[1801] -0.144  0.536  1.336  0.976  1.104  0.656  1.224  1.496  1.656 -0.184
[1811]  0.664  0.736  0.296  0.296  0.296  1.456  0.896  1.424  0.696  1.344
[1821]  1.296  1.056  0.984  1.416  1.576  1.544  1.296  0.576  1.424  0.744
[1831]  0.336  1.224  0.256  0.256  1.696  1.304  1.496  2.104  1.136  0.816
[1841]  2.136  0.736  1.344  1.264  1.176  0.736  1.664  1.136  1.304  1.944
[1851]  0.504  1.664  0.896  1.424  0.344  1.016  1.256  1.304  1.896  0.736
[1861]  1.104  0.064  1.664  1.504  1.656  1.256  1.336  1.096  1.544  1.416
[1871]  0.416  1.456  1.096  1.496  1.224  0.736  0.416  0.456  0.944  1.264
[1881]  1.256  0.824  0.816  1.536  0.536  0.224  1.304  1.504  0.856  1.216
[1891]  1.296  1.536  0.096  0.864  0.056  0.304  1.096  2.144  1.784  1.144
[1901]  0.016  0.824  0.056  1.416  1.096  1.064  0.664  1.544  0.744  0.624
[1911]  1.136  1.136  0.416  1.376  1.056  1.744  0.576  0.984  0.984  0.744
[1921]  0.976  1.184  2.104  0.016  0.664  1.696  0.864  0.664  0.656  0.976
[1931]  1.056  1.064  1.504  1.464  0.664  0.384 -0.136  0.944  0.016  1.896
[1941]  0.096  1.224  1.144  0.696  0.504  0.976  0.904  1.376  0.656  1.184
[1951]  0.696  1.544  0.824  1.936  0.656  0.296  0.864  1.064  1.504  1.216
[1961]  1.056  0.104  0.736  1.576  0.936  0.944  1.096  1.496  1.936  1.344
[1971]  0.944  0.736  0.416  0.696  0.584  1.464  1.776  1.376  0.704  1.456
[1981]  1.104  0.936  1.096  1.504  0.576  0.744  0.376  0.304  0.856  0.424
[1991]  1.664  0.704  1.536  1.024  1.376  0.936  1.296  0.984  1.464  0.736
[2001]  1.984  1.024  0.856  0.344  0.904  0.424  1.256  0.896  1.584  1.976
[2011]  0.896  0.064  0.896  1.464  1.056  0.704  1.944  0.104  1.696  1.104
[2021]  1.024  0.624  0.456 -0.104  2.096  0.864  1.504  0.824  1.224  0.816
[2031]  1.344  0.416  0.624  0.664  1.576  0.664  0.504  2.136  1.464  0.416
[2041]  0.624  0.304  0.776  0.664  0.296 -0.176  1.184  1.384  0.544  1.696
[2051]  1.136  0.776  2.096  1.984 -0.104  0.936  1.136  0.496  0.504  1.776
[2061]  1.176  1.704  0.944 -0.136  1.944  0.936  1.576  0.616  1.104  1.416
[2071]  0.936  1.304  0.664  1.544  1.344  1.424  0.056  1.736  1.024  1.064
[2081]  1.064  1.344  1.304  1.376  1.256  0.304  1.304  0.536  1.736  1.256
[2091]  0.496  0.296  1.176  0.856  1.664  0.824  0.896  1.464  0.904  1.336
[2101]  1.424  0.904  1.464  1.544  0.584  0.104  1.136  1.696  1.096  0.304
[2111]  1.104  0.256  1.144  0.224  1.776  1.104  1.984  0.656  0.936  1.944
[2121]  0.976  1.496  1.624  2.144  1.744  1.344  1.136  0.296  1.536  0.936
[2131]  1.904  1.376  1.496  1.504  1.064  1.736  0.584  1.064  1.456  0.936
[2141]  0.456  0.104  0.696  1.416  0.704  0.656  0.864  0.776  1.936  1.184
[2151]  0.864  0.496  0.736  1.336  0.856  0.536  0.384  1.104  0.464  0.456
[2161]  0.744  1.024  0.296  1.344  0.856  1.744  0.856  1.376  0.784  1.264
[2171]  1.536  0.296  1.336  1.496  1.096  0.256  1.416  0.536  1.296  1.064
[2181]  1.536  0.424  0.536  1.656  1.264  2.176  0.976  0.536  0.696  0.656
[2191]  0.576  1.376  0.944  1.664  1.024  1.696  1.536  1.504  1.264  1.056
[2201]  0.744  1.776  1.736  1.544  0.504  0.856  0.496  1.336  0.224  1.384
[2211]  1.704  0.896  1.064  1.104  1.736  1.896  0.496  0.504  1.016  1.376
[2221]  0.304  0.984  0.664  1.936  1.024  0.696  0.624  1.576  1.776  1.056
[2231]  1.904  1.344  1.736  1.296  0.736  0.304  1.344  0.744  0.264  1.104
[2241]  0.016  0.664  0.976  0.624  1.496  1.136  0.464  1.696  0.776  1.136
[2251]  1.376  0.664  1.784  1.504  0.856  0.696  0.224  1.456  0.936  1.376
[2261]  1.184  0.664  0.824  1.416  0.464 -0.176  1.216  1.504  1.416  1.136
[2271]  1.264  0.856  1.536  0.696  0.304  1.136  0.904  1.624  0.704  0.544
[2281]  1.024  1.176  1.016  0.064  1.264  1.344  1.184  0.064  0.656  1.944
[2291]  1.096  1.264  1.024  0.896  0.896  0.256  1.776  0.496  0.304  1.064
[2301]  1.944  0.096  1.224  1.304  0.784  1.304  1.064  1.616  0.584  1.536
[2311]  1.056  0.224  0.664  0.984  0.696  0.936  0.856  0.384  1.304  0.424
[2321]  1.224  0.216  1.104  0.224  1.176  0.696  1.064  0.936  0.536  1.344
[2331]  0.216  0.656  0.696  0.624  0.696  0.496  0.736  0.896  0.304  1.176
[2341]  1.704  1.256  2.104  1.096  0.976  1.104  1.024  0.864  1.576  0.256
[2351]  0.056  0.976  0.656  0.936  1.344  1.264  1.616  1.464  0.464  0.624
[2361]  0.704  1.416  1.536  1.696  2.176  1.504  1.184  1.536  0.464  0.984
[2371]  0.064  0.936  0.224  1.976  2.096  0.616  1.016  0.744  1.536  0.936
[2381]  1.504  0.336  1.736  1.656 -0.136  1.136  1.344  1.136  1.296  1.016
[2391]  1.536  0.504  1.296  0.944  0.744  2.144  0.464  0.496  0.736  1.064
[2401]  1.296  0.504  0.984  0.744 -0.136  0.304  0.776  1.344  1.536  0.816
[2411]  1.304  1.496  0.416  0.296  0.944  0.776  0.344  1.504  1.464  1.184
[2421]  1.744  1.696  1.944  0.264  0.856  0.624  1.144  0.656  1.464  2.184
[2431]  1.424  1.544  1.304  0.296  1.304  0.256  1.424  1.264  1.104  1.424
[2441]  1.696  1.064  1.024  0.656  0.096  0.424  0.976  0.896  0.736  1.496
[2451]  1.376  0.056  0.656  1.704  0.104  0.696  0.696  1.464  0.896  0.336
[2461]  1.504  0.784  1.776  1.096  0.736  1.536  0.704  1.264  0.544  0.496
[2471]  1.704  1.104  1.064  1.264  0.544  0.416  1.136  1.344  1.304  0.984
[2481]  0.896  0.824  0.784  0.296  1.416  1.544  0.496  0.984  0.696  1.944
[2491]  0.576  0.336  0.784  0.464  1.664  1.616  1.056  1.104  0.304  0.856
[2501]  0.464  1.104  1.296  0.504  0.856  1.064  0.936  0.464  0.024  1.296
[2511]  0.024  0.936  1.336  0.776  0.464  0.624  1.344  1.216  0.696  0.944
[2521]  1.104  1.176  1.096  2.136  1.104  0.584  0.464  0.976  1.304  2.104
[2531]  2.136  0.904  1.704  0.904  1.976  0.336  0.864  1.224  0.496  1.344
[2541]  1.064  2.104  0.576  0.936  1.704  0.304  0.496  1.344  0.544  0.816
[2551]  0.736  1.344  0.904  1.144  0.656  1.336  0.904  1.376  0.856  1.144
[2561]  0.576  0.856  1.144 -0.136  0.576 -0.136  0.656  0.936  0.336  0.776
[2571]  1.536  0.696  1.224 -0.096  0.456  1.064  1.024  1.936  1.944 -0.184
[2581]  0.264  0.656  1.336  0.824 -0.096  1.336  0.824  0.496  0.736  0.544
[2591]  1.664  1.136  1.256  0.824  1.184  0.424  0.664  0.456  1.056  0.944
[2601]  1.984  0.856  1.096  0.904  1.304  1.936  0.064  0.536  0.896  0.656
[2611]  1.096  1.656  0.704  0.296  1.344  0.896  0.904  1.344  1.736  0.336
[2621]  0.936  1.264  1.416  1.144  0.696  1.896  0.856  1.096  2.176  0.944
[2631]  0.936  0.664  0.576  1.496  0.456  0.336  1.536  1.184  1.216  0.584
[2641]  0.736  1.096  1.624  0.776  1.704  1.416  0.744  0.744  1.936  1.984
[2651]  1.224  0.896  1.064  1.016  1.616  2.104  0.984  0.856  0.304  0.704
[2661]  0.944  0.064  0.736  1.896  1.776  1.224  0.704  0.896  0.704  1.496
[2671]  1.584  0.904  0.744  2.096  0.496  1.936  0.216  0.336  1.576  1.216
[2681]  0.736  0.056  0.656  1.296  1.016  0.984  1.264  0.896  0.256  0.056
[2691]  1.224  1.504  0.424  0.256  1.704  0.936  0.504  1.104  0.504  1.736
[2701]  1.016  0.656  1.456  1.496  0.944  1.136  1.136  1.344  0.816  0.904
[2711]  0.816  1.296  0.856  0.496  0.896  0.984  1.616  0.896  0.024  0.664
[2721]  1.936  0.216  1.056  1.176  0.224  1.144  1.144  1.096  0.936  1.184
[2731]  1.296  1.176  1.104  0.824  1.496  0.864  1.536  0.864  1.344  1.504
[2741]  0.256  1.736  0.944  0.944  0.704  1.256  0.256  1.336  0.816  0.336
[2751]  0.544  0.936  1.336  1.776  1.224  1.336  1.024  1.144  0.704  0.696
[2761]  1.296  0.296  1.144  1.624  0.504  1.224  1.136  1.096  0.896  1.256
[2771]  0.344  1.056  1.256  0.656  1.264  0.696  0.216  1.264  1.264  1.984
[2781]  1.744  1.144  0.984  0.496  1.304  1.304  0.664  2.096  1.504  0.656
[2791]  0.816  0.104  0.824  1.176  0.904  0.544  1.344  1.144  1.304  0.936
[2801]  0.856  1.216  1.336  0.856  1.984  0.504  1.304  0.056  0.904  0.824
[2811] -0.096  0.344  1.416  0.504  0.904  1.416  0.856  1.144  0.776  1.536
[2821]  1.584  1.104  0.256  1.224  0.976  1.264  1.304  0.736  1.296  0.704
[2831]  1.296  1.176  1.744  1.056  0.944  0.456 -0.184  1.504  0.944  0.936
[2841]  0.464  1.904  0.944  0.896  1.704  0.496  1.056  1.424  0.656  0.744
[2851]  2.184  0.024  1.096  1.176  0.704  1.104  1.544  1.104  0.536  1.344
[2861]  0.064  0.096  1.424  1.144  0.864  0.656  1.344  0.704  0.784  0.856
[2871]  1.264  0.856  2.184  0.776  0.696  0.736  0.216  1.704  0.776  0.936
[2881]  1.624  0.416  0.296  1.896  0.416  1.584  2.184  0.624  0.696  1.744
[2891]  1.944  1.096  0.904  1.304  1.744  0.464  0.384  1.616  1.016  1.336
[2901]  0.664  0.304  1.264  0.864  0.336  0.896  1.024  1.104  0.784  0.584
[2911]  0.624  0.664  1.376  0.304  1.176  0.504  0.896  1.064  1.016  0.424
[2921]  0.736  0.696  0.576  0.624  1.344  1.064  0.496  0.624  0.224  2.176
[2931]  1.704  2.176  0.296  1.496  1.176  1.984  0.296  0.304  1.184  1.576
[2941]  0.696  1.024  0.816  0.296  1.224  0.936  1.464  1.416  1.336  1.224
[2951] -0.136  0.376  1.176  0.224  1.304  1.536  1.376  1.296  0.504  1.096
[2961]  0.296  0.216  0.736  0.584  0.656  0.336  0.816  1.064  0.856  0.744
[2971]  0.824  1.064  0.544  0.664  1.304  1.264  1.784  0.816  1.144 -0.176
[2981]  1.016  0.456  1.344  0.824  1.144  0.256  0.224  2.096  0.696  1.344
[2991]  0.864  1.696  0.224  1.296  2.184  1.056  0.504  0.296  0.216  0.704
[3001]  1.144  0.776  1.744  0.936  1.744  0.704  0.416  0.464  0.536  0.936
[3011]  0.336  0.856  0.696  1.144  1.264  1.584  1.384  1.016  1.144  1.416
[3021]  1.656  0.264  0.696  0.664  0.416  0.464  0.304  1.016  0.576  0.304
[3031]  0.984  0.896  1.104  0.264  0.264  1.464  0.904  1.384  0.784  0.904
[3041]  0.984  0.496  0.944  1.224  0.336  1.776  0.704  1.416  1.136  2.184
[3051]  1.024  0.656  0.656  0.304  1.056  0.344  0.776  1.504  1.696  0.744
[3061]  0.216  1.416  1.784  0.504  1.304  0.896  0.456  1.024  0.896  1.376
[3071] -0.096  1.416  1.256  0.256  0.704  1.696  1.896  1.504  0.464  1.104
[3081]  1.464  0.944  0.696  0.256  1.576  0.456  0.464  0.024  1.784  0.024
[3091]  1.184  0.424  1.104  1.064  0.896  0.696  0.864  1.304 -0.096  0.936
[3101]  0.944  1.344  0.984  1.936  1.176  1.336  0.656  1.456  0.496  1.656
[3111]  0.704  1.976  1.104  1.536  0.904  0.064  2.144  0.296  1.544  1.424
[3121]  0.704  0.344  0.296  0.984  1.064  1.104  0.584  0.024  1.424  1.344
[3131]  1.296  0.664  0.784  0.696  0.696  0.224  0.496  0.024  1.744  0.704
[3141]  1.496  0.656  0.584  1.904  1.064  0.984  1.136  1.736  0.336  0.544
[3151]  0.696  0.304  0.416  0.456  0.224  0.896  0.816  0.336  0.024  0.824
[3161]  1.504  1.056  0.536  1.104  0.704  1.936  1.504  1.944  0.416  0.696
[3171]  0.304  1.056  0.856  1.136  1.096 -0.136  0.944  0.704  1.736  0.336
[3181]  0.904  1.096  1.576  0.904  0.016  0.744  1.736  0.936  1.264  2.184
[3191]  1.784  0.304  0.496  0.904  1.664  1.224  0.616  1.424  1.416  0.736
[3201]  0.896  0.936  1.296  0.904  1.376  0.336  0.824  0.744  1.336  0.504
[3211]  0.216  1.216  1.176  1.456  1.104  0.936  1.536  0.256  0.816  1.064
[3221]  2.096  1.336  0.904  1.496  1.216  0.944  0.496  0.024  0.064  0.784
[3231]  1.616  0.864  1.576  0.296  0.216  0.576  0.496  1.184  1.464  0.736
[3241]  1.584  0.456  1.776  1.096  1.264  0.264  1.136  0.864  0.616  1.496
[3251]  1.144  0.936  0.984  1.696  0.016  1.584 -0.104  1.264  1.056  0.016
[3261]  1.744  1.536  0.336  1.016  0.944  0.376  1.584  0.296  1.024  0.536
[3271]  1.944  0.936  1.256  1.304  0.504  1.104  1.376  0.576  0.456  0.096
[3281]  1.744  2.096  1.784  1.664  1.344  1.664  1.256  1.144  0.584  0.824
[3291]  2.104  0.944  0.944  0.696  0.944  0.064  1.296  0.896  1.504  0.584
[3301]  1.056  1.944  0.016  1.096  1.536  1.176  1.736  1.504  1.944  1.056
[3311]  0.464  1.064  1.376  0.296  0.784  0.816  0.584  1.064  0.464  1.776
[3321]  1.496  0.496  0.656  0.784  0.584  1.064  1.304  1.344  0.104  1.296
[3331]  0.456  1.296  1.504  0.936  1.536  1.064  0.936  0.096  0.856  0.656
[3341]  0.696  1.624  0.024  0.984  0.984  0.496  0.536  0.296 -0.136  0.304
[3351]  0.816  1.424  0.864 -0.136  0.824  1.496  1.024  0.696  0.224 -0.104
[3361]  0.504  1.064  0.256  0.864  2.136  1.504 -0.184  0.904  0.424  0.536
[3371]  1.304  0.896  1.104  0.024  0.704  1.264  0.936  0.504  1.256  1.144
[3381]  1.296  0.256  1.336  1.256  1.384  1.064  1.224  0.736  0.496  1.336
[3391]  0.936  0.576  1.504  0.736  1.744  1.984  0.936  0.256  1.776  1.144
[3401]  0.976  0.664  1.696  1.624  1.376  1.416  1.544  0.664  1.984  1.984
[3411]  0.536  1.776  1.024  0.904  0.456  0.624  1.704  1.584  0.704  0.056
[3421]  0.704  0.976  0.864  0.656  0.744  0.464  1.224  0.944  0.704  0.536
[3431]  1.136  0.376  1.496  0.984 -0.144 -0.176  1.024  0.016  0.904  1.304
[3441]  0.496  1.504  1.704  0.224 -0.184  0.984  1.144  0.736  0.656  2.104
[3451]  1.264  0.904  1.096  1.536  2.136  0.944  1.496  0.504  0.976  1.544
[3461]  0.944  2.176  1.656  1.256  0.704  0.496  1.616  0.696  0.344  1.736
[3471]  0.656  0.416  1.104  1.376  0.344  0.944  1.464  0.424  0.464  1.264
[3481]  0.256  0.336  0.936  1.264  1.336  0.944  1.104  2.096  1.504  0.904
[3491]  0.576 -0.136  0.064  0.464  0.624  1.104  1.704  0.264  1.776  1.456
[3501]  0.984  1.264  1.144  0.936  0.424  0.664  0.624  1.504  0.856  1.704
[3511]  1.224  1.496  1.656  0.624  1.576  0.664  1.144  1.264  1.576  0.976
[3521]  0.416  1.256  2.136  0.584  0.704  1.424  0.064  0.656  1.176  1.744
[3531]  0.944  1.544  0.064  1.944  0.656  1.104  0.504  1.184  1.496  1.144
[3541]  0.864  1.376  1.984  0.576 -0.104  1.096  1.704  1.024  1.096  1.704
[3551]  0.824  1.136 -0.136  1.056  1.064  1.304  0.936  1.696  1.224  0.664
[3561]  1.304  2.136  1.296  0.336  1.296  1.904  1.096  0.584 -0.136  1.104
[3571]  0.056  0.824  0.664  1.424  0.496  1.056  1.464  1.576  2.096  1.936
[3581]  0.376  1.496  0.864 -0.096  0.936  0.384  0.944  1.344  0.744  1.536
[3591]  1.144  0.416  0.824  0.096  0.904  1.344  1.536  1.224  0.536  1.376
[3601]  0.816  1.464  0.424  1.256  0.496  0.744  0.024  1.944  0.864  0.944
[3611]  0.864  1.336  0.544  0.464  1.776  1.544  0.464  0.096  1.224 -0.136
[3621]  1.056  0.696  1.496  1.096  1.776  1.544  1.976  0.624  1.544  1.064
[3631]  1.704  1.504  0.424  0.936  1.936  0.504  0.464  1.104 -0.144  2.096
[3641]  0.976  1.896  1.656  1.136  0.696  1.456  1.336  1.696  1.304  0.624
[3651]  1.536  1.296  1.504  1.704  0.456  1.496  0.464  1.016  0.904  0.096
[3661]  0.464  0.536  1.104  1.136  0.336  0.384  0.296  0.936 -0.096  0.944
[3671]  0.936  0.416  1.744  0.936  1.096  0.504  0.504  1.376  0.496  1.296
[3681]  1.096  0.944  0.536  0.656  1.224  1.216  0.904  1.096  0.504  0.536
[3691]  1.176  0.864  2.176  1.496  0.696  1.096  1.904  0.464  1.896  1.296
[3701]  0.664  1.504  1.064  2.136  0.464  0.856  1.464  1.456  1.024  0.304
[3711]  0.296  0.864  1.496  0.784  0.464  1.144  1.104  0.384  1.296  1.016
[3721]  0.616  1.424  1.104  1.024  0.696  1.016  0.216  1.584  1.536  0.504
[3731]  1.304  1.064  0.544  1.416  0.504  0.856  1.296  0.896  1.504  1.336
[3741]  0.224  1.776  1.656  1.424  1.216  0.896  1.264  1.656  1.264  0.296
[3751]  1.976  1.744  1.944  0.576  1.296  1.136  0.096  1.376  0.704  1.704
[3761]  0.944  0.864  0.496  1.704  0.704  1.104  2.136  0.304  1.696  0.496
[3771]  0.496  1.496  0.656  0.696  1.344  1.184  0.344  1.336  0.536  2.104
[3781]  1.944  1.536  1.056  1.944  1.096  0.856  1.504  0.664  1.536  0.504
[3791]  0.336  0.864  0.424  1.536  0.904  1.256  0.896  0.536  2.096  1.784
[3801] -0.136  1.104  0.824  1.296  1.056  1.184  1.384  0.464  1.936  0.864
[3811]  1.264  1.064  0.936  1.184  1.304  1.336  0.704  0.904  0.496  0.536
[3821]  0.656  0.696  1.176  0.776  0.936  0.424  0.304  0.824  0.264  0.976
[3831]  0.104  0.936  1.384  1.024  1.136  0.344  0.736  0.936  0.416  2.136
[3841]  0.104  1.296  0.256  1.584  0.936  0.504  2.104  0.424  1.464  0.296
[3851]  0.824  1.376  1.064  1.216  1.656  0.584  0.856  0.296  1.616  1.176
[3861]  0.864  0.824  1.264  1.496  0.936  1.544  1.064  0.496  0.496  1.456
[3871]  1.696  0.944  0.904  0.936  0.096  1.296  0.056  2.144  0.416  1.016
[3881]  1.504  0.704  0.336  0.704  0.936  0.584  1.664  1.536  0.896  0.384
[3891]  0.376  0.304  1.096  0.464  0.656  0.376  0.624  0.896  1.936  1.104
[3901]  1.504  0.416  1.696  1.896  1.976  1.616  1.704  1.336  1.264  1.536
[3911]  0.576  1.376  0.896  0.496  1.216  0.656  0.704  1.136  0.064  1.376
[3921]  1.424  1.136  1.344  0.424  1.344  0.776  1.704  0.856  0.736  0.056
[3931]  0.896  1.704  1.064  0.256  0.104  0.256  0.344  1.496  0.936  1.104
[3941]  0.496  1.056  0.496  1.536  0.264  0.704  0.064  1.464  1.616  1.536
[3951]  0.536  0.856  0.656  1.176  0.536  2.136 -0.104  0.976  0.936  0.104
[3961]  0.896  1.336  0.704  1.016  1.136  0.064  0.696  1.136  0.224  0.256
[3971]  0.904  0.744  0.704  0.656  1.704  1.464  1.144  1.304  0.296  1.264
[3981]  1.504 -0.176  0.464 -0.136  0.744  0.944  1.736  1.664  1.296  1.176
[3991]  1.264  2.096  0.864  0.496  0.984  0.704  1.616  0.624  1.056  0.696
[4001]  1.064  1.304  0.496  1.544  1.016  1.656  1.536  2.136  1.544  1.256
[4011]  1.016  1.344  2.096  1.296  1.096  1.536  1.504  0.224  1.176  0.024
[4021]  0.504  1.016  0.944  0.696  1.456  1.024  1.584  0.736  1.264  1.944
[4031]  2.144  0.296  0.536  0.544  1.104  1.064  0.624  0.256  0.976  1.744
[4041]  1.264  0.296  0.464  1.544  1.616  0.824  1.416  1.376  1.416  1.664
[4051]  1.104  1.984  2.136  0.656  0.744  0.336  1.536  0.944  0.864  0.896
[4061]  1.184  0.896  0.536  0.856  1.776  1.056  1.104  0.264  0.736  1.976
[4071]  1.496  1.264  0.256  0.824  1.696  0.576  0.376  1.776  1.136  1.104
[4081]  0.384  1.576  0.856  0.864  0.864  1.096  1.296  0.696  1.096  1.096
[4091]  1.176  0.976  0.904  0.976  0.216  0.664  2.096  0.464  1.256  0.544
[4101]  1.576  1.296  1.904  0.064  2.136  0.656  1.136  1.584  0.264  1.496
[4111]  1.096  1.016  2.176  1.944  1.296  1.456  0.304  0.296  0.744  1.416
[4121]  0.896  0.064  0.656  0.976  1.056  1.496 -0.176  0.576  0.704  0.584
[4131]  0.744  0.536  1.096  0.736  1.504  1.464  0.904  1.184  0.736  0.656
[4141]  0.864  1.344  1.504  0.744  0.656  1.384  0.696  0.824  1.096  0.264
[4151]  0.736  0.784  1.344 -0.144  0.496  0.744  0.384  0.944  1.304  1.224
[4161]  1.264  1.056  0.664  2.104  0.776 -0.144  1.136  1.016  1.416  1.064
[4171]  1.136  0.864  0.984  1.304  0.256  0.296  1.136  1.336  1.344  1.064
[4181]  0.424  0.536  1.424  1.104  1.264  0.016  1.664  0.704  0.656  0.616
[4191]  0.296  0.896  0.376  1.096  0.056  2.144  1.424  1.696  0.016  1.984
[4201]  1.096  1.496  0.456  0.984  0.864  1.096  1.104  0.704  0.056  0.056
[4211]  0.784  1.224  1.096  1.736  1.544  0.736  1.064  0.784  1.424  0.936
[4221]  1.096  2.096  1.536  1.536  1.064  0.864  1.176  0.864  1.144  0.416
[4231]  1.496  0.256  0.696  1.216  1.344  1.624  0.864 -0.096  1.536  1.304
[4241]  1.144  1.064  0.936  1.904  1.504  1.104  1.064  0.936  1.744  0.736
[4251]  1.256  1.464  1.104  0.104  0.736  0.904  0.744  0.496  1.144  1.456
[4261]  1.584  1.464  0.064  0.304  0.424  0.864  0.856  1.096  0.424  1.496
[4271]  1.064  1.576  1.024  0.544  0.864  1.056  1.344  1.264  0.896  0.696
[4281]  0.256  1.584  2.136  1.376  1.184  1.504  1.696  1.376  1.536  1.304
[4291]  1.336  0.536  1.456  0.864  0.264  2.184  1.144  0.496  1.336  1.264
[4301]  0.984  0.584  0.944  0.624  1.336  0.904  1.744  1.776  0.656  1.224
[4311]  1.104  1.144  1.576  0.656  0.696  1.416  0.656  0.944 -0.136  1.936
[4321]  1.144  0.576  0.216  1.056  2.144  0.904  0.336  1.496  0.656  0.296
[4331]  1.104  1.624  0.256  1.704  0.936  0.944  1.576  0.296  0.824  0.896
[4341]  0.224  2.176  1.304  0.464  1.704  0.904  1.376  0.456  1.144  0.744
[4351]  1.704  1.696  2.144  2.144  2.184  0.736  1.064  1.496 -0.136  1.616
[4361]  2.136  1.056  1.096  0.296  0.656  1.696  1.296  1.464  1.384  0.704
[4371]  0.064  1.144  1.216  1.136  0.936  0.536  1.344  1.696  0.064  0.304
[4381]  1.776  1.696  1.376  1.024  1.344  1.784  1.744  0.664  0.736  0.896
[4391]  1.304  0.704  1.136  0.864  1.064  1.184  0.936  1.464  0.736  1.296
[4401]  0.944  0.624  1.336  1.184  1.464  0.864  0.864  1.136  1.024  0.224
[4411]  1.304  0.224  1.416  0.624  0.736  0.216  0.256  0.576  1.304  0.584
[4421]  0.624  0.464  0.744  0.824  0.424  1.064  1.016  0.904  0.904  0.256
[4431]  1.464  0.304  1.136  0.704  0.824  0.016  0.536  0.784  0.984  1.064
[4441]  0.744  0.944  0.864  0.464  0.784  0.904  0.696  1.496  1.064  1.496
[4451]  1.496  0.056  1.736  1.504  1.056  1.136  0.016  2.136  1.296  0.936
[4461]  0.896  0.384  1.176  0.944  0.296  1.776  0.904  0.856  0.736  1.584
[4471]  0.784  1.504  1.344  1.736  1.704  0.544  0.504  0.744  1.256  0.864
[4481]  0.904  0.376  0.544  0.504  0.736  1.536  1.536  0.904  1.616  1.376
[4491]  0.544  1.344  0.336  0.104  1.416  0.704  0.504  0.784  0.464  0.464
[4501]  0.624  1.096  0.856  1.976  1.344  0.464  0.944  1.504  1.216  1.296
[4511]  1.424  1.056  0.584  1.264  1.784 -0.144  0.704  0.664  1.016  1.376
[4521]  1.216  0.976  0.936  1.064  0.256  1.424  0.976  0.936  0.504  1.784
[4531]  0.536  1.016  0.744  0.304  2.104  0.376  1.176  1.016  1.576  2.136
[4541]  1.176  0.576  1.536  0.976  0.064  0.624  0.904  0.496  1.096  1.224
[4551]  1.304  0.704  0.896  1.216 -0.136  0.736  1.184  1.496  0.264  1.936
[4561]  0.584  1.544  1.744  1.344  1.496  0.704  0.944  1.336  0.416  1.496
[4571]  1.536  0.056  1.224  1.136  0.696  0.984  0.016  1.616  1.264  0.304
[4581]  0.936  1.264  0.464  0.376  0.464  0.816  0.616 -0.144  0.056  1.376
[4591]  0.864  0.896  0.056  0.824  1.216  1.016  0.776  0.944  1.504  0.704
[4601]  0.336  2.096  0.896  1.224  0.896  1.016  0.464  1.096  1.344  0.584
[4611]  0.904  1.784  0.496  1.064  0.456  1.536  1.696  0.424  1.104  1.104
[4621]  1.456  1.064  1.256  1.504  1.104 -0.104  1.264 -0.144  1.136  0.944
[4631]  0.696  1.304  1.184  0.664  0.336  1.056  1.416  1.736  0.544  0.736
[4641]  0.664  0.624  0.464  1.056  1.496  0.856  0.696  0.304  0.856  0.736
[4651]  0.304  0.896  0.224  1.216  0.984  0.056  1.936  0.616  1.064  1.104
[4661]  0.496  1.464  1.104  0.896  0.664  0.904  0.704  1.376 -0.104  1.136
[4671]  2.096  0.736  0.736  0.336  1.296  0.864  1.256  1.304  0.544  0.296
[4681]  1.464  0.304  1.344  2.136  1.584  1.144  0.696  0.304  0.304  1.496
[4691]  0.464  0.504  1.944  1.456  1.024  0.016  1.224  1.384  1.136  0.464
[4701]  1.016  0.824  0.816  0.736  1.936  1.256  1.024  1.056  0.424  0.904
[4711]  0.976  1.696  0.056  1.104  2.184  1.776  1.024  0.336  1.984  0.504
[4721]  1.376  0.544  0.904  1.376  1.656  0.944  1.576  1.104  1.016  0.464
[4731]  0.904  1.544  0.096  1.176  0.264  1.584  0.256  0.864  0.576  0.744
[4741]  0.056  0.824  2.136  1.024  0.016  0.256  0.024  1.456  0.464  0.536
[4751]  1.256  0.336  1.744  1.456  0.856  0.576  0.464  1.696  1.744  1.664
[4761]  0.344 -0.184  0.464  1.304  0.336  1.136  1.704  0.456  1.984 -0.096
[4771]  0.336  0.624  1.304  1.216  1.096  0.304 -0.104  1.576  0.784  1.096
[4781]  0.696  0.984  1.504  1.104  0.816  2.104  1.416  1.384  0.224  1.976
[4791]  0.744  1.104  1.224  1.304  1.136  0.656  1.104  1.096  1.104  1.136
[4801]  1.536  0.824  0.704  1.144  1.504  0.904  0.944  0.344  0.736  0.656
[4811]  0.536  1.224  1.304  1.696  0.944  0.504  1.504  0.856  1.536  1.096
[4821]  1.416  1.304  1.944  1.224  0.736  1.504  1.944  1.576  1.016  1.104
[4831]  0.656  1.256  1.184  0.896  0.904  1.096  1.704  0.864  0.976  1.224
[4841]  1.024  1.704  0.464  0.664  1.104  0.784  0.736  1.704  1.904  0.704
[4851]  0.096  0.064  0.936  0.056  1.744  1.264  1.696  0.064  0.064  1.336
[4861]  1.176  0.904  0.536  1.936 -0.184  0.784  1.264  1.576  0.656  1.976
[4871]  1.296 -0.176  1.136  0.336  0.056  0.344  0.784  1.416  0.784  0.536
[4881]  0.056  0.024  1.176  0.344  0.456  0.616  1.136  1.096  1.064  0.424
[4891]  1.896  1.736  0.464  1.024  1.776  0.936  0.064  1.456  1.304  0.096
[4901]  0.496  1.224  0.424  0.664  1.104  1.016 -0.096  1.904  2.136  0.816
[4911]  1.896  0.944  0.704  0.064  0.104  1.264  1.224  0.896  0.896  0.896
[4921]  1.496  1.944  1.696  1.696  0.744  0.096  0.336  0.064  0.416  0.416
[4931]  1.536  0.416  0.864  1.336  1.376  1.336  1.136  0.736  0.056  0.296
[4941]  0.424  1.264  0.584  0.656  0.264  1.944  1.744  0.256  1.664  0.464
[4951]  1.096  0.504  0.424  0.464  0.056  1.136  0.296  0.496  0.824  1.744
[4961]  1.056  0.904  1.584  0.304  1.096  0.224  1.624  0.384  1.136  1.456
[4971]  2.096  1.264  1.296  1.896  0.896 -0.096  0.504  0.896 -0.184  1.544
[4981]  1.344 -0.176  1.224  1.696  0.344  1.544  0.816  1.144  1.576  1.504
[4991]  1.584  0.704  0.464 -0.184  0.824  0.496  0.736  0.424  0.256  0.944

Estimate Dist under HA

p_HA <- data.frame(stat = H_A) |>
  ggplot(aes(x = stat)) +
  geom_histogram() +
  theme_bw()
p_HA

Power

Power

power <- mean(H_A > right_threshold | H_A < left_threshold)
power
[1] 0.2274

n_0 <- 5
n_1 <- 5

data.frame(stat = c(H_0, H_A),
           type = factor(c(rep("Null", length(H_0)),
                           rep("Alt", length(H_A))))) |>
  mutate(in_reject_region = ifelse(stat > right_threshold | stat < left_threshold, "Reject Null", NA)) |>
  ggplot(aes(x = stat, fill = type, color = in_reject_region)) +
  geom_histogram(position = "identity", alpha = .3) +
  geom_vline(xintercept = c(left_threshold, right_threshold), col = "tomato") +
  scale_color_manual(
    values = c("Reject Null" = "black"),
    na.value = NA,
    guide = "none") +
  labs(title = paste0("Statistical Power: ", round(power, 2)),
       subtitle = paste0("n0 = ", n_0, ", n1 = ", n_1, ", tau = ", tau)) +
  theme_bw()

Functions

rejection_region <- function(df, alpha = .05, reps = 1000, seed = 4892) {
  # create observed schedule (with missing data)
  obs_sched <- df |>
    mutate(Y_0 = ifelse(d_i == 0, y_i, NA),
           Y_1 = ifelse(d_i == 1, y_i, NA))
  
  # impute the missing potential outcomes with fisher sharp null
  null_sched <- obs_sched |>
    mutate(Y_0 = ifelse(is.na(Y_0), Y_1, Y_0),
           Y_1 = ifelse(is.na(Y_1), Y_0, Y_1))
  
  # simulate from sampling distribution under null
  set.seed(seed)
  H_0 <- rand_stats(schedule = null_sched,
                    d_i = df$d_i,
                    stat = "diff in means",
                    reps = reps)
  
  # find rejection region
  left_threshold <- quantile(H_0, alpha/2)
  right_threshold <- quantile(H_0, 1 - alpha/2)
  
  return(list(LT = left_threshold, RT = right_threshold))
}

HA_tau <- function(df, tau, reps = 1000, seed = 4892) {
  # create observed schedule (with missing data)
  obs_sched <- df |>
    mutate(Y_0 = ifelse(d_i == 0, y_i, NA),
           Y_1 = ifelse(d_i == 1, y_i, NA))
  
  # impute the missing potential outcomes under particular HA
  alt_sched <- obs_sched |>
    mutate(Y_0 = ifelse(is.na(Y_0), Y_1 - tau, Y_0),
           Y_1 = ifelse(is.na(Y_1), Y_0 + tau, Y_1))
  
  H_A <- rand_stats(schedule = alt_sched,
                    d_i = df$d_i,
                    stat = "diff in means",
                    reps = reps)

  return(H_A)
}