pilot-study.Rmd 54.8 KB
Newer Older
1
---
linushof's avatar
linushof committed
2
title: "Sampling Strategies in DfE - Pilot"
3
date: "2021"
4
5
6
7
8
9
10
11
bibliography: sampling-strategies-in-dfe.bib
csl: apa.csl
output:
  html_document:
    code_folding: hide
    toc: yes
    toc_float: yes
    number_sections: yes
12
13
---

linushof's avatar
linushof committed
14
15
Some of the `R code` is folded but can be unfolded by clicking the `Code` buttons.

16
17
```{r}
# load packages
18
pacman::p_load(tidyverse,
19
20
21
               knitr)
```

linushof's avatar
linushof committed
22
# Description
23

linushof's avatar
linushof committed
24
Choice data will be generated by applying the *comprehensive-* and *piecewise sampling strategy* to a series of 2-prospect gambles. The simulated data will be explored for characteristic patterns of sampling strategies under varying structures of the choice environment, i.e., features of a gamble's prospects, and aspects of the sampling- and decision behavior (model parameters).
25

linushof's avatar
linushof committed
26
# Generating Choice Data 
27

28
## Method
29

30
### Agents 
31

32
Under each condition, i.e., strategy-parameter combinations, all gambles are played by 100 synthetic agents. 
33
34
35
36

```{r}
n_agents <- 100
```
37

38
### Gambles
39

linushof's avatar
linushof committed
40
A set of gambles, in which one of the prospects contains a safe outcome and the other two risky outcomes (*safe-risky gambles*) will be tested. Therefore, 60 gambles from an initial set of 10,000 are sampled. Both, outcomes and probabilities are drawn from uniform distributions, ranging from 0 to 20 for outcomes and from .01 to .99 for probabilities of the lower risky outcomes $p_L$. The probabilities of the higher risky outcomes are $1-p_L$, respectively. To omit dominant prospects, safe outcomes fall between both risky outcomes. The table below contains the test set of 60 gambles. Sampling of gambles was stratified, randomly drawing an equal number of 20 gambles with no, an attractive, and an unattractive rare outcome. Risky outcomes are considered *"rare"* if their probability is $p < .2$ and *"attractive"* (*"unattractive"*) if they are higher (lower) than the safe outcome. 
41

42
```{r eval = FALSE}
linushof's avatar
linushof committed
43
source("./R/functions/fun_gambles.R") # call generate_gambles() function
44

45
46
47
48
49
50
# generate and select subset of safe-risky gambles
set.seed(3211)
sr_gambles <- generate_gambles(n = 10000, safe = TRUE, lower = 0, upper = 20)
sr_gambles <- sr_gambles %>% mutate(rare = case_when(a_p1 >= .2 & a_p1 <= .8 ~ "None",
                                                     a_p1 < .2 ~ "Unattractive", # a_o1  < a_o2
                                                     a_p1 > .8 ~ "Attractive"))
51
write_rds(sr_gambles, "./R/data/sr_gambles.rds")
52
53
54
55
56
57
58

sr_subset <- tibble()
for(i in unique(sr_gambles$rare)) {
  type <- sr_gambles %>% filter(rare == i)
  smpl <- sample(seq_len(nrow(type)), size = 20)
  sr_subset <- bind_rows(sr_subset, type[smpl, ])
}
59
write_rds(sr_subset, "./R/data/sr_subset.rds")
60
61
62
```

```{r}
63
sr_subset <- read_rds(file = "./R/data/sr_subset.rds")
64
65
66
kable(sr_subset)
```

linushof's avatar
linushof committed
67
### Model Parameters 
68

linushof's avatar
linushof committed
69
**Switching probability:** $s$ is the probability increment added to the unbiased probability $p = .5$ with which agents draw the succesive single sample from the same prospect they get their most recent single sample from. $s$ is varied between -.5 to .4 in increments of .1. To ease interpretation during data analysis, $s$ is transformed by $s_{rec}= 1-(p + s)$ after the simulation such that it ranges from .1 to 1.   
70

linushof's avatar
linushof committed
71
**Boundary type**: Is either the minimum value *any* prospect's sample statistic must reach (absolute) or the minimum value for the difference of these statistics (relative). Sample statistics are sums over outcomes (comprehensive strategy) and sums over wins (piecewise strategy), respectively.
72

linushof's avatar
linushof committed
73
**Boundary value:** For comprehensive sampling, $a$ is varied between 15 to 35 in increments of 5, for piecewise sampling $a$ is varied between 1 to 7 in increments of 2. 
74

linushof's avatar
linushof committed
75
**Noise parameter:** Representations of the sampled outcomes are assumed to be stochastical. Therefore, Gaussian noise $\epsilon \sim N(0, \sigma)$ in units of outcomes is added. To reduce computational load, $\sigma$ is fixed to .5.
76

linushof's avatar
linushof committed
77
78
79
80
81
82
83
84
```{r eval = FALSE, class.source = "fold-show"}
# parameters
parameters <- expand_grid(s = seq(-.5, .4, .1), # probability increment added to unbiased sampling probability of p = .5
                          sigma = .5, # noise 
                          boundary = c("absolute", "relative")) # boundary type
theta_c <- expand_grid(parameters, a = c(15, 20, 25, 30, 35)) # boundaries comprehensive 
theta_p <- expand_grid(parameters, a = c(1, 3, 5, 7)) # boundaries piecewise
```
85

86
## Simulation
87

88
```{r}
89
90
# dataset
gambles <- sr_subset
91
```
92

93
```{r}
94
source("./R/functions/fun_moving_stats.R") # call cumsum2() and cummean2() function
95
96
```

linushof's avatar
linushof committed
97
### Comprehensive sampling
98

linushof's avatar
linushof committed
99
```{r eval = FALSE, class.source = "fold-show"}
100
101
102

# simulation

103
theta <- theta_c
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
set.seed(765)
param_list <- vector("list", length(nrow(theta)))
for (set in seq_len(nrow(theta))) {
  gamble_list <- vector("list", length(nrow(gambles)))
  for (gamble in seq_len(nrow(gambles))) {
    agents_list <- vector("list", n_agents)
    for (agent in seq_along(1:n_agents)){ 
      
      ## initial values of an agent's sampling process
      
      fd <- tibble() # state of ignorance
      p <- .5  # no attention bias  
      s <- 0  # no switching at process initiation
      init <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s)) # prospect attended first
      attend <- init
      boundary_reached <- FALSE
      
      ## agent's sampling process 
      
      while(boundary_reached == FALSE) {
        
        #### draw single sample
        
        if(attend == "a") {
          single_smpl <- gambles[gamble, ] %>% 
            mutate(attended = attend,
                   A = sample(x = c(a_o1, a_o2), size = 1, prob = c(a_p1, 1-a_p1)) + 
                     round(rnorm(n = 1, mean = 0, sd = theta[[set, "sigma"]]), 2), # gaussian noise 
                   B = NA)
          s <- theta[[set, "s"]] # get switching probability
          } else {
            single_smpl <- gambles[gamble, ] %>%
              mutate(attended = attend,
                     A = NA,
                     B = b +
                       round(rnorm(n = 1, mean = 0, theta[[set, "sigma"]]), 2))
            s <- -1*theta[[set, "s"]]
          }
        
        #### integrate single sample into frequency distribution
        
        fd <- bind_rows(fd, single_smpl) %>%
          mutate(A_sum = cumsum2(A, na.rm = TRUE),
                 B_sum = cumsum2(B, na.rm = TRUE))
        
        #### evaluate accumulated evidence
        
        if(theta[[set, "boundary"]] == "absolute") {
          fd <- fd %>%
            mutate(choice = case_when(A_sum >= theta[[set, "a"]] ~ "A",
                                      B_sum >= theta[[set, "a"]] ~ "B"))
          } else {
            fd <- fd %>%
              mutate(diff = round(A_sum - B_sum, 2),
                     choice = case_when(diff >= theta[[set, "a"]] ~ "A",
                                        diff <= -1*theta[[set, "a"]] ~ "B"))
          }
        if(is.na(fd[[nrow(fd), "choice"]]) == FALSE) {
          boundary_reached <- TRUE
          } else {
            attend <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s))
          }
      }
      agents_list[[agent]] <- expand_grid(agent, fd)
    }
    all_agents <- agents_list %>% map_dfr(as.list)
    gamble_list[[gamble]] <- expand_grid(gamble, all_agents)
  }
  all_gambles <- gamble_list %>% map_dfr(as.list)
  param_list[[set]] <- expand_grid(theta[set, ], all_gambles)
}
175
sim_comprehensive <- param_list %>% map_dfr(as.list)
176

linushof's avatar
linushof committed
177
# store full simulation 
178

179
write_rds(sim_comprehensive, "./R/data/sim_comprehensive.rds")
180

linushof's avatar
linushof committed
181
# summarize unique sampling processes 
182
183
184
185
186
187

summary_comprehensive <- sim_comprehensive %>%
  group_by(s, sigma, boundary, a, gamble, agent) %>% # group by unique sampling process
  mutate(n_sample = n(), # number of single samples 
         switch = case_when(attended != lag(attended) ~ 1, 
                            attended == lag(attended) ~ 0),
188
189
190
         n_switch = sum(switch, na.rm = TRUE), # number of switches 
         a_ev_exp = round(mean(A, na.rm = TRUE), 2), # experienced expected value
         b_ev_exp = round(mean(B, na.rm = TRUE), 2)) %>% 
191
192
193
  filter(!is.na(choice)) %>% # only return choice data (last obs of unique sampling process)
  select(!c(attended, A, B, switch)) %>% 
  ungroup() %>% 
194
195
196
  mutate(strategy = "comprehensive",
         s = 1-(s+.5)) %>% 
  select(strategy, s:gamble, rare, a_p1:ev_ratio, agent, n_sample, n_switch, A_sum, B_sum, diff, a_ev_exp, b_ev_exp, choice)
197
write_rds(summary_comprehensive, "./R/data/summary_comprehensive.rds")
198
199
```

200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
#### Expansion of boundary range

Because results of the initial analysis indicate scaling issues for the boundary values, the simulation of the comprehensive strategy is rerun. Specifically, the initial set of boundary values $\{15, 20, 25, 30, 35\}$ is relatively small compared to the range of possible outcomes $[0;20]$. As a consequence, boundaries are reached for rather small sample sizes. This may cause potential influences of the boundary value e.g. on decision quality to be masked by a ceiling effect. Furthermore, the effects of switching probability on decision behavior may be confounded with the effect of boundary values. I.e., small switching probabilities cause comparisons of sums over unequal sample sizes - the discrepance in sample size increases with decreasing switching probabilities - immediately after switching. If boundaries are low, decisions may terminate early, however, not so because of real differences in EV but because of the comparison of very different sample sizes. 

In the following, the extended boundary range of $\{40, 45, 50, 55, 60, 65, 70, 75, 80\}$ will be tested. 

```{r eval = FALSE, class.source = "fold-show"}
# parameters
parameters <- expand_grid(s = seq(-.5, .4, .1), # probability increment added to unbiased sampling probability of p = .5
                          sigma = .5, # noise 
                          boundary = c("absolute", "relative")) # boundary type
theta_c <- expand_grid(parameters, a = c(40, 45, 50, 55, 60, 65, 70, 75, 80)) # boundaries comprehensive 
```

```{r eval = FALSE, class.source = "fold-show"}

# param combinations 1:122
theta <- theta_c
set.seed(76754)
param_list <- vector("list", length(nrow(theta)))
for (set in 1:122) {
  gamble_list <- vector("list", length(nrow(gambles)))
  for (gamble in seq_len(nrow(gambles))) {
    agents_list <- vector("list", n_agents)
    for (agent in seq_along(1:n_agents)){ 
      
      ## initial values of an agent's sampling process
      
      fd <- tibble() # state of ignorance
      p <- .5  # no attention bias  
      s <- 0  # no switching at process initiation
      init <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s)) # prospect attended first
      attend <- init
      boundary_reached <- FALSE
      
      ## agent's sampling process 
      
      while(boundary_reached == FALSE) {
        
        #### draw single sample
        
        if(attend == "a") {
          single_smpl <- gambles[gamble, ] %>% 
            mutate(attended = attend,
                   A = sample(x = c(a_o1, a_o2), size = 1, prob = c(a_p1, 1-a_p1)) + 
                     round(rnorm(n = 1, mean = 0, sd = theta[[set, "sigma"]]), 2), # gaussian noise 
                   B = NA)
          s <- theta[[set, "s"]] # get switching probability
          } else {
            single_smpl <- gambles[gamble, ] %>%
              mutate(attended = attend,
                     A = NA,
                     B = b +
                       round(rnorm(n = 1, mean = 0, theta[[set, "sigma"]]), 2))
            s <- -1*theta[[set, "s"]]
          }
        
        #### integrate single sample into frequency distribution
        
        fd <- bind_rows(fd, single_smpl) %>%
          mutate(A_sum = cumsum2(A, na.rm = TRUE),
                 B_sum = cumsum2(B, na.rm = TRUE))
        
        #### evaluate accumulated evidence
        
        if(theta[[set, "boundary"]] == "absolute") {
          fd <- fd %>%
            mutate(choice = case_when(A_sum >= theta[[set, "a"]] ~ "A",
                                      B_sum >= theta[[set, "a"]] ~ "B"))
          } else {
            fd <- fd %>%
              mutate(diff = round(A_sum - B_sum, 2),
                     choice = case_when(diff >= theta[[set, "a"]] ~ "A",
                                        diff <= -1*theta[[set, "a"]] ~ "B"))
          }
        if(is.na(fd[[nrow(fd), "choice"]]) == FALSE) {
          boundary_reached <- TRUE
          } else {
            attend <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s))
          }
      }
      agents_list[[agent]] <- expand_grid(agent, fd)
    }
    all_agents <- agents_list %>% map_dfr(as.list)
    gamble_list[[gamble]] <- expand_grid(gamble, all_agents)
  }
  all_gambles <- gamble_list %>% map_dfr(as.list)
  param_list[[set]] <- expand_grid(theta[set, ], all_gambles)
}
sim_comprehensive_ext_122 <- param_list %>% map_dfr(as.list)


summary_comprehensive_ext_l122 <- l122 %>%
  group_by(s, sigma, boundary, a, gamble, agent) %>% # group by unique sampling process
  mutate(n_sample = n(), # number of single samples 
         switch = case_when(attended != lag(attended) ~ 1, 
                            attended == lag(attended) ~ 0),
         n_switch = sum(switch, na.rm = TRUE), # number of switches 
         a_ev_exp = round(mean(A, na.rm = TRUE), 2), # experienced expected value
         b_ev_exp = round(mean(B, na.rm = TRUE), 2)) %>% 
  filter(!is.na(choice)) %>% # only return choice data (last obs of unique sampling process)
  select(!c(attended, A, B, switch)) %>% 
  ungroup() %>% 
  mutate(strategy = "comprehensive",
         s = 1-(s+.5)) %>% 
  select(strategy, s:gamble, rare, a_p1:ev_ratio, agent, n_sample, n_switch, A_sum, B_sum, diff, a_ev_exp, b_ev_exp, choice)
write_rds(summary_comprehensive_ext_122, "./R/data/summary_comprehensive_ext_122.rds")
```

```{r}

# # param combinations 1:122

theta <- theta_c
set.seed(659)
param_list <- vector("list", length(nrow(theta)))
for (set in 123:nrow(theta)) {
  gamble_list <- vector("list", length(nrow(gambles)))
  for (gamble in seq_len(nrow(gambles))) {
    agents_list <- vector("list", n_agents)
    for (agent in seq_along(1:n_agents)){ 
      
      ## initial values of an agent's sampling process
      
      fd <- tibble() # state of ignorance
      p <- .5  # no attention bias  
      s <- 0  # no switching at process initiation
      init <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s)) # prospect attended first
      attend <- init
      boundary_reached <- FALSE
      
      ## agent's sampling process 
      
      while(boundary_reached == FALSE) {
        
        #### draw single sample
        
        if(attend == "a") {
          single_smpl <- gambles[gamble, ] %>% 
            mutate(attended = attend,
                   A = sample(x = c(a_o1, a_o2), size = 1, prob = c(a_p1, 1-a_p1)) + 
                     round(rnorm(n = 1, mean = 0, sd = theta[[set, "sigma"]]), 2), # gaussian noise 
                   B = NA)
          s <- theta[[set, "s"]] # get switching probability
          } else {
            single_smpl <- gambles[gamble, ] %>%
              mutate(attended = attend,
                     A = NA,
                     B = b +
                       round(rnorm(n = 1, mean = 0, theta[[set, "sigma"]]), 2))
            s <- -1*theta[[set, "s"]]
          }
        
        #### integrate single sample into frequency distribution
        
        fd <- bind_rows(fd, single_smpl) %>%
          mutate(A_sum = cumsum2(A, na.rm = TRUE),
                 B_sum = cumsum2(B, na.rm = TRUE))
        
        #### evaluate accumulated evidence
        
        if(theta[[set, "boundary"]] == "absolute") {
          fd <- fd %>%
            mutate(choice = case_when(A_sum >= theta[[set, "a"]] ~ "A",
                                      B_sum >= theta[[set, "a"]] ~ "B"))
          } else {
            fd <- fd %>%
              mutate(diff = round(A_sum - B_sum, 2),
                     choice = case_when(diff >= theta[[set, "a"]] ~ "A",
                                        diff <= -1*theta[[set, "a"]] ~ "B"))
          }
        if(is.na(fd[[nrow(fd), "choice"]]) == FALSE) {
          boundary_reached <- TRUE
          } else {
            attend <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s))
          }
      }
      agents_list[[agent]] <- expand_grid(agent, fd)
    }
    all_agents <- agents_list %>% map_dfr(as.list)
    gamble_list[[gamble]] <- expand_grid(gamble, all_agents)
  }
  all_gambles <- gamble_list %>% map_dfr(as.list)
  param_list[[set]] <- expand_grid(theta[set, ], all_gambles)
  svMisc::progress(set, 58, progress.bar = TRUE)
}
sim_comprehensive_ext_123 <- param_list %>% map_dfr(as.list)

summary_comprehensive_ext_123 <- sim_comprehensive_ext_123 %>%
  group_by(s, sigma, boundary, a, gamble, agent) %>% # group by unique sampling process
  mutate(n_sample = n(), # number of single samples 
         switch = case_when(attended != lag(attended) ~ 1, 
                            attended == lag(attended) ~ 0),
         n_switch = sum(switch, na.rm = TRUE), # number of switches 
         a_ev_exp = round(mean(A, na.rm = TRUE), 2), # experienced expected value
         b_ev_exp = round(mean(B, na.rm = TRUE), 2)) %>% 
  filter(!is.na(choice)) %>% # only return choice data (last obs of unique sampling process)
  select(!c(attended, A, B, switch)) %>% 
  ungroup() %>% 
  mutate(strategy = "comprehensive",
         s = 1-(s+.5)) %>% 
  select(strategy, s:gamble, rare, a_p1:ev_ratio, agent, n_sample, n_switch, A_sum, B_sum, diff, a_ev_exp, b_ev_exp, choice)

write_rds(summary_comprehensive_123, "./R/data/summary_comprehensive_123.rds")


```


409
410
### Piecewise sampling

linushof's avatar
linushof committed
411
```{r eval = FALSE, class.source = "fold-show"}
412
413
414

# simulation

415
theta <- theta_p
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
set.seed(8739)
param_list <- vector("list", length(nrow(theta)))
for (set in seq_len(nrow(theta))) {
  gamble_list <- vector("list", length(nrow(gambles)))
  for (gamble in seq_len(nrow(gambles))) {
    agents_list <- vector("list", n_agents)
    for (agent in seq_along(1:n_agents)){ 
      
      ## initial values of an agent's sampling process
      
      fd <- tibble() # state of ignorance
      p <- .5  # no attention bias
      s <- 0  # no switching at process initiation
      init <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s)) # prospect attended first
      attend <- init
      round <- 1
      boundary_reached <- FALSE
      
      ## agent's sampling process 
      
      while(boundary_reached == FALSE) {
        
        #### sampling round
        
        smpl_round <- tibble()
        while(attend == init) {
          
          ##### draw single sample from prospect attended first
          
          if(attend == "a") {
            single_smpl <- gambles[gamble, ] %>%
              mutate(round = round,
                     attended = attend,
                     A = sample(x = c(a_o1, a_o2), size = 1, prob = c(a_p1, 1-a_p1)) +
                       round(rnorm(1, mean = 0, sd = theta[[set, "sigma"]]), 2),
                     B = NA)
            s <- theta[[set, "s"]]
            } else {
              single_smpl <- gambles[gamble, ] %>%
                mutate(round = round,
                       attended = attend,
                       A = NA,
                       B = b + 
                         round(rnorm(1, mean = 0, sd = theta[[set, "sigma"]]), 2))
              s <- -1*theta[[set, "s"]]
            }
          smpl_round <- bind_rows(smpl_round, single_smpl)
          attend <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s))
        }
        
        while(attend != init) {
          
          ##### draw single sample from prospect attended second
          
          if(attend == "a") {
            single_smpl <- gambles[gamble, ] %>%
              mutate(round = round,
                     attended = attend, 
                     A = sample(x = c(a_o1, a_o2), size = 1, prob = c(a_p1, 1-a_p1)) + 
                       round(rnorm(1, mean = 0, sd = theta[[set, "sigma"]]), 2),
                     B = NA)
            s <- theta[[set, "s"]]
            } else {
              single_smpl <- gambles[gamble, ] %>%
                mutate(round = round,
                       attended = attend,
                       A = NA,
                       B = b + 
                         round(rnorm(1, mean = 0, sd = theta[[set, "sigma"]]), 2))
              s <- -1*theta[[set, "s"]]
            }
          smpl_round <- bind_rows(smpl_round, single_smpl)
          attend <- sample(c("a", "b"), size = 1, prob = c(p + s, p - s))
        }
        
        ##### compare mean outcomes 
        
        smpl_round <- smpl_round %>%
          mutate(A_rmean = cummean2(A, na.rm = TRUE),
                 B_rmean = cummean2(B, na.rm = TRUE),
                 rdiff = A_rmean - B_rmean)
        smpl_round[[nrow(smpl_round), "A_win"]] <- case_when(smpl_round[[nrow(smpl_round), "rdiff"]] > 0 ~ 1,
                                                             smpl_round[[nrow(smpl_round), "rdiff"]] <= 0 ~ 0)
        smpl_round[[nrow(smpl_round), "B_win"]] <-  case_when(smpl_round[[nrow(smpl_round), "rdiff"]] >= 0 ~ 0,
                                                              smpl_round[[nrow(smpl_round), "rdiff"]] < 0 ~ 1)
        
        ##### integrate sampling round into frequency distribution
        
        fd <- bind_rows(fd, smpl_round)
        fd[[nrow(fd), "A_sum"]] <- sum(fd[["A_win"]], na.rm = TRUE)
        fd[[nrow(fd), "B_sum"]] <- sum(fd[["B_win"]], na.rm = TRUE)
        
        #### evaluate accumulated evidence
        
        if(theta[[set, "boundary"]] == "absolute") {
          fd <- fd %>%
            mutate(choice = case_when(A_sum >= theta[[set, "a"]] ~ "A",
                                      B_sum >= theta[[set, "a"]] ~ "B"))
          } else {
            fd[[nrow(fd), "wdiff"]] <- fd[[nrow(fd), "A_sum"]] - fd[[nrow(fd), "B_sum"]]
            fd <- fd %>%
              mutate(choice = case_when(wdiff >= theta[[set, "a"]] ~ "A",
                                        wdiff <= -1*theta[[set, "a"]] ~ "B"))
          }
        if(is.na(fd[[nrow(fd), "choice"]]) == FALSE) {
          boundary_reached <- TRUE
          } else {
            round <- round + 1
          }
      }
      agents_list[[agent]] <- expand_grid(agent, fd)
    }
    all_agents <- agents_list %>% map_dfr(as.list)
    gamble_list[[gamble]] <- expand_grid(gamble, all_agents)
  }
  all_gambles <- gamble_list %>% map_dfr(as.list)
  param_list[[set]] <- expand_grid(theta[set, ], all_gambles)
}
sim_piecewise <- param_list %>% map_dfr(as.list)
535
536
537
538
539
540
541
542
543
544

# store simulation 

write_rds(sim_piecewise, "./R/data/sim_piecewise.rds")

# summarize unique sampling processes

summary_piecewise <- sim_piecewise %>%
  group_by(s, sigma, boundary, a, gamble, agent) %>% 
  mutate(n_sample = n(),  
545
546
         a_ev_exp = mean(A, na.rm = TRUE), 
         b_ev_exp = mean(B, na.rm = TRUE)) %>%
547
548
549
  ungroup() %>%
  filter(!is.na(choice)) %>% 
  mutate(strategy = "piecewise",
550
551
552
553
554
         s = 1-(s+.5),
         diff = wdiff,
         n_switch = (round*2)-1) %>%
  select(!c(attended, A, B, A_rmean, B_rmean, rdiff, A_win, B_win, wdiff)) %>% 
  select(strategy, s:gamble, rare, a_p1:ev_ratio, agent, n_sample, n_switch, A_sum, B_sum, diff, a_ev_exp, b_ev_exp, choice)
555
write_rds(summary_piecewise, "./R/data/summary_piecewise.rds")
556
557
```

linushof's avatar
linushof committed
558
## Summary
559

linushof's avatar
linushof committed
560
```{r eval = FALSE, class.source = "fold-show"}
561
562
563
564
565
sr_data <- bind_rows(summary_comprehensive, summary_piecewise) %>% 
  mutate(across(c(strategy, boundary, a, gamble, agent, rare, choice), as.factor)) # convert to factor
write_rds(sr_data, "./R/data/sr_data.rds")
```

linushof's avatar
linushof committed
566
# Descriptive Analyses
567

linushof's avatar
linushof committed
568
Analyses of the generated choice data focuses on the interplay of sampling strategies, different gamble features (e.g. existence and valence of rare outcomes), and parameter combinations in their effects on sample sizes and choice behavior. 
569

570
```{r}
linushof's avatar
linushof committed
571
572
573
574
# read summary data set
data <- read_rds("./data/sr_data.rds")
data <- data %>% 
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp))) # use only choices in which all prospects were seen
575
576
```

linushof's avatar
linushof committed
577
578
579
## Sample Size

Below, median sample sizes of all strategy-parameter combinations (circles) are plotted, ranging from $2 \leq \tilde{x} \leq 162$ for piecewise sampling and $3 \leq \tilde{x} \leq 18$ for comprehensive sampling.
580
581

```{r}
linushof's avatar
linushof committed
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, a, s) %>% 
  summarise(group = as.factor(cur_group_id()),
            med = round(median(n_sample), 0)) %>%
  ggplot(.) + 
  geom_point(aes(x = reorder(group, med), y = med, color = s, size = a), alpha = .8) +
  facet_wrap(~boundary) + 
  scale_color_gradient(low = "blue", high = "red") + 
  scale_y_continuous(breaks = seq(0, 170, 10)) +
  scale_x_discrete(breaks = NULL, expand = expansion(add = 3)) + 
  theme_minimal() +
  labs(title = "Piecewise Sampling",
       x ="Strategy-Parameter Combination", 
       y="Sample Size", size="Boundary Value", 
       col="Switching Probability")
598
599
600
601
```

```{r}
data %>%
linushof's avatar
linushof committed
602
603
604
605
  filter(strategy == "comprehensive") %>% 
  group_by(boundary, a, s) %>% 
  summarise(group = as.factor(cur_group_id()),
            med = round(median(n_sample), 0)) %>%
606
  ggplot(.) + 
linushof's avatar
linushof committed
607
608
609
610
611
612
613
614
615
616
  geom_point(aes(x = reorder(group, med), y = med, color = s, size = a), alpha = .7) +
  facet_wrap(~boundary) + 
  scale_color_gradient(low = "blue", high = "red") + 
  scale_y_continuous(breaks = seq(0, 25, 1)) +
  scale_x_discrete(breaks = NULL, expand = expansion(add = 3)) + 
  theme_minimal() +
  labs(title = "Comprehensive Sampling",
       x ="Strategy-Parameter Combination", 
       y="Sample Size", size="Boundary Value", 
       col="Switching Probability")
617
618
619
620
```

### Boundary value 

linushof's avatar
linushof committed
621
622
623
624
625
626
```{r}
col_a_p = c("#0302FC", "#A1015D", "#63009E", "#FE0002")
col_a_c = c("#0302FC", "#A1015D", "#63009E", "#2A00D5", "#FE0002")
```

Both sampling strategies show a similiar effect of boundary value on sample size. I.e., large boundary values lead to larger sample sizes.
627
628

```{r}
linushof's avatar
linushof committed
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673

# piecewise

med_a <- data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, a) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, a, s) %>%
  summarise(med = round(median(n_sample), 0)) %>%
  ggplot(., aes(x = a, y = med, color = a)) + 
  facet_wrap(~boundary) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = med_a, size = 4) +
  scale_color_manual(values = col_a_p) +
  labs(title = "Piecewise Sampling",
       x ="Boundary value", 
       y="Sample Size", 
       col="Boundary Value") + 
  theme_minimal()


# comprehensive

med_a <- data %>%
  filter(strategy == "comprehensive") %>% 
  group_by(boundary, a) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "comprehensive") %>% 
  group_by(boundary, a, s) %>%
  summarise(med = round(median(n_sample), 0)) %>%
  ggplot(., aes(x = a, y = med, color = a)) + 
  facet_wrap(~boundary) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = med_a, size = 4) +
  scale_color_manual(values = col_a_c) +
  labs(title = "Comprehensive Sampling",
       x ="Boundary value", 
       y="Sample Size", 
       col="Boundary Value") + 
  theme_minimal()
674
675
676
677
```

### Boundary type 

linushof's avatar
linushof committed
678
For both sampling strategies, relative (as compared to absolute) boundaries lead to larger sample sizes as sequential sampling can either stabilize or reduce a prospects' distance to absolute boundaries while the distance to relative boundaries can also increase. 
679
680

```{r}
linushof's avatar
linushof committed
681
682
683
684
685
686
687
688

# piecewise

med_a <- data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, a) %>% 
  summarise(med = round(median(n_sample), 0))

689
data %>%
linushof's avatar
linushof committed
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
  filter(strategy == "piecewise") %>% 
  group_by(boundary, a, s) %>%
  summarise(med = round(median(n_sample), 0)) %>%
  ggplot(., aes(x = a, y = med, color = boundary)) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = med_a, size = 4) +
  scale_color_manual(values = col_a_p) +
  labs(title = "Piecewise Sampling",
       x ="Boundary value", 
       y="Sample Size", 
       col="Boundary Type") + 
  theme_minimal()


# comprehensive

med_a <- data %>%
  filter(strategy == "comprehensive") %>% 
  group_by(boundary, a) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "comprehensive") %>% 
  group_by(boundary, a, s) %>%
  summarise(med = round(median(n_sample), 0)) %>%
  ggplot(., aes(x = a, y = med, color = boundary)) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = med_a, size = 4) +
  scale_color_manual(values = col_a_p) +
  labs(title = "Comprehensive Sampling",
       x ="Boundary value", 
       y="Sample Size", 
       col="Boundary Type") + 
  theme_minimal()
724
725
726
727
```

### Switching probability 

linushof's avatar
linushof committed
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
For piecewise sampling, there is an inverse relationship between switching probability and sample size. I.e., the lower the switching probability, the the less frequent prospects are compared and thus, boundaries are only approached with larger sample sizes. This effect is particularly pronounced for low probabilities such that the increase in sample size accelerates as switching probability decreases. Consequentially, the magnitude of the effect of the boundary value increases.

```{r}

med_s <- data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, s) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "piecewise") %>% 
  group_by(boundary, s, a) %>% 
  summarise(med = round(median(n_sample), 0)) %>% 
  ggplot(., aes(x = s, y = med, color = s, shape = boundary)) + 
  geom_jitter(size = 2, alpha = .5) + 
  geom_point(data = med_s, size = 4) +
  geom_line(data = med_s) +
  scale_color_gradient(low = "blue", high = "red") + 
  labs(title = "Piecewise Sampling",
       x ="Switching Probability", 
       y="Sample Size", 
       col="Switching Probability",
       shape = "Boundary Type") + 
  theme_minimal()
```

For comprehensive sampling, boundary types differ in the effects of switching probability. Regarding absolute boundaries, switching probability has no apparent effect on sample size as the distance of a given prospect to its absolute boundary is not changed by switching to (and sampling from) the other prospect. 

```{r}
med_s <- data %>%
  filter(strategy == "comprehensive", boundary == "absolute") %>% 
  group_by(boundary, s) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "comprehensive", boundary == "absolute") %>% 
  group_by(s, a) %>% 
  summarise(med = round(median(n_sample), 0)) %>% 
  ggplot(., aes(x = s, y = med, color = s)) + 
  geom_jitter(size = 2, alpha = .5) + 
  geom_point(data = med_s, size = 4) +
  geom_line(data = med_s) +
  scale_color_gradient(low = "blue", high = "red") + 
  labs(title = "Comprehensive Sampling - Absolute Boundary",
       x ="Switching Probability", 
       y="Sample Size", 
       col="Switching Probability",
       shape = "Boundary Type") + 
  theme_minimal()
```

For relative boundaries, switching probability has a more nuanced effect on sample size. Particularly, plotting sample size as a function of switching probability across all gambles produces a somewhat unexpected U-shaped pattern of an inverse relationship for small probabilities and a positive relationship for larger probabilities. 

```{r}
med_s <- data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  group_by(s) %>% 
  summarise(med = round(median(n_sample), 0))

data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  group_by(s, a) %>% 
  summarise(med = round(median(n_sample), 0)) %>% 
  ggplot(., aes(x = s, y = med, color = s)) + 
  geom_jitter(size = 2, alpha = .5) + 
  geom_point(data = med_s, size = 4) +
  geom_line(data = med_s) +
  scale_color_gradient(low = "blue", high = "red") + 
  labs(title = "Comprehensive Sampling - Relative Boundary",
       x ="Switching Probability", 
       y="Sample Size", 
       col="Switching Probability",
       shape = "Boundary Type") + 
  theme_minimal()
```

However, inspecting all gambles separately, one does not observe an U-shaped relation but rather two gamble clusters, one of which shows a positive and the other an inverse relationship. Inspecting the qualitative differences between the gamble features, one finds that the "positive" cluster is indicated by only small differences in the expected value (EV) of prospects ($0.5 \leq \frac{EV_{safe}}{EV_{risky}} \leq 1.5$), whereas the negative" cluster is indicated by larger EV-differences ($0.5 < \frac{EV_{safe}}{EV_{risky}} > 1.5$). 

Specifically, for gambles (in the gain range), the distance of a given prospect to its relative boundary is probabilistically reduced by switching and sampling from the other prospect. If the EV-difference is low, then frequent switching will lead to a more oscillating behavior of prospects approaching and moving away from the relative border. I.e., for small EV-differences frequent switching reduces the probability that the relative boundary can be reached with the subsequent sample(s) of a given prospect.

```{r}
med_s <- data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  filter(ev_ratio >= .5 & ev_ratio <= 1.5) %>% 
  group_by(gamble, s) %>% 
  summarise(med = round(median(n_sample), 0))


data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  filter(ev_ratio >= .5 & ev_ratio <= 1.5) %>% 
  group_by(gamble, s, a) %>% 
  summarise(med = round(median(n_sample), 0)) %>% 
  ggplot(., aes(x = s, y = med, color = s)) + 
  geom_jitter(size = 1, alpha = .5) + 
  geom_point(data = med_s, size = 2) +
  geom_line(data = med_s) +
  facet_wrap(~gamble) +
  scale_color_gradient(low = "blue", high = "red") + 
  scale_y_continuous(limits = c(0, 50)) +
  labs(title = "Comprehensive Sampling - Relative Boundary - Small EV-Difference",
       x ="Switching Probability", 
       y="Sample Size", 
       col="Switching Probability",
       shape = "Boundary Type") + 
  theme_minimal()

```

In contrast, if the EV-difference is high, small samples are already diagnostic for the difference of prospects, i.e., we do not expect stark oscillation but rather a continuous drift to the boundary of the higher EV-prospect. Consequentially, small switching probabilities may lead to larger samples than is actually required. 

```{r}
med_s <- data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  filter(ev_ratio < .5 | ev_ratio > 1.5) %>% 
  group_by(gamble, s) %>% 
  summarise(med = round(median(n_sample), 0))


data %>%
  filter(strategy == "comprehensive", boundary == "relative") %>% 
  filter(ev_ratio < .5 | ev_ratio > 1.5) %>% 
  group_by(gamble, s, a) %>% 
  summarise(med = round(median(n_sample), 0)) %>% 
  ggplot(., aes(x = s, y = med, color = s)) + 
  geom_jitter(size = 1, alpha = .5) + 
  geom_point(data = med_s, size = 2) +
  geom_line(data = med_s) +
  facet_wrap(~gamble) +
  scale_color_gradient(low = "blue", high = "red") + 
  labs(title = "Comprehensive Sampling - Relative Boundary - Large EV-difference",
       x ="Switching Probability", 
       y="Sample Size", 
       col="Switching Probability",
       shape = "Boundary Type") + 
  theme_minimal()
```

## Choice Behavior and Underweighting

Below, in extension to Hills and Hertwig [-@hillsInformationSearchDecisions2010], the interplay of sampling strategies, gambles' features, and model parameters in their effects on choice behavior in general and their contribution to underweighting of rare events in particular is investigated. The working definition of underweighting of rare events is as follows: The rarity of an attractive (unattractive) outcome leads to choose the safe (risky) prospect although the risky (safe) prospect has a higher expected value.  

### Interpretation of heatmaps

The heatmaps below show the proportions of normatively (i.e., according to the EV-difference) false choices with blueish cells indicating low proportions and redish cells indicating high proportions. For each combination of sampling strategy and boundary type, 6 heatmaps are plotted (3 x 2 grid). The upper panel of each grid shows the proportions of false safe choices and the lower panel of false risky choices. The horizontal grid dimension separates gambles in which the rare outcome ($p <= .2$) of the risky option is either larger (attractive) or smaller (unattractive) than the safe outcome. Accordingly, underweighting of rare events is indicated in the Attractive Rare-False Safe (top left) and the Unattractive Rare-False Risky heatplots (bottom right). Within each heatplot, false response proportions are plotted as a function of the model parameters $s$ (switching probability) and $a$ (boundary value). 

### Piecewise sampling

As can be seen below, the differences between both boundary types (absolute vs. relative) are rather minor and of magnitude than of qualitative pattern. Therefore, the remaining analyses are aggregated over both boundary types.

However, false response proportions across the cell configurations are systematically different. Apparently, the piecewise sampling strategy produces extreme response proportions in either direction, for some configurations generating almost no EV-incoherent  decisions (blue areas) and for some almost only EV-incoherent decisions (red areas). Below, possible determinants of these distinct patterns of false response rates are identified.

```{r}

## Absolute 

data %>% 
  filter(strategy == "piecewise", boundary == "absolute") %>% 
  mutate(optimal = case_when(ev_ratio > 1 ~ "A",
                             ev_ratio < 1 ~ "B")) %>% 
  group_by(optimal, rare, s, a, choice) %>% 
  summarise(n = n()) %>%
  mutate(prop = round(n/sum(n), 2)) %>% 
  mutate(kind = case_when(optimal == choice ~ "norm",
                          optimal == "A" & choice == "B" ~ "False Safe",
                          optimal == "B" & choice == "A" ~ "False Risky")) %>% 
  mutate(kind = as.factor(kind)) %>% 
  filter(kind != "norm") %>%
  ggplot(., aes(x = a, y = s, fill = prop)) + 
  facet_grid(kind~rare, switch = "y") +
  geom_tile(colour="white",size=0.25) +
  scale_x_discrete(expand=c(0,0), name = "Boundary Value")+
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1), name = "Switching Probability")+
  scale_fill_gradient(low="blue", high="red") + 
  theme_minimal() + 
  labs(title = "Piecewise Sampling - Absolute Boundary",
       x = "Boundary Value", 
       y= "Switching Probability", 
       fill = "% False Responses") 

## Relative 

data %>% 
  filter(strategy == "piecewise", boundary == "relative") %>% 
  mutate(optimal = case_when(ev_ratio > 1 ~ "A",
                             ev_ratio < 1 ~ "B")) %>% 
  group_by(optimal, rare, s, a, choice) %>% 
  summarise(n = n()) %>%
  mutate(prop = round(n/sum(n), 2)) %>% 
  mutate(kind = case_when(optimal == choice ~ "norm",
                          optimal == "A" & choice == "B" ~ "False Safe",
                          optimal == "B" & choice == "A" ~ "False Risky")) %>% 
  mutate(kind = as.factor(kind)) %>% 
  filter(kind != "norm") %>%
  ggplot(., aes(x = a, y = s, fill = prop)) + 
  facet_grid(kind~rare, switch = "y") +
  geom_tile(colour="white",size=0.25) +
  scale_x_discrete(expand=c(0,0), name = "Boundary Value")+
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1), name = "Switching Probability")+
  scale_fill_gradient(low="blue", high="red") + 
  theme_minimal() + 
  labs(title = "Piecewise Sampling - Relative Boundary",
       x = "Boundary Value", 
       y= "Switching Probability", 
       fill = "% False Responses") 
```

#### Existence and Attractiveness of Rare Events

The inversed color gradients from left to right panels indicate that the presence and attractiveness of rare events is a large determinant of false response rates with the direction of the effect dependent on whether the risky or the safe prospect has a higher EV. I.e., consistent with the notion of underweighting, the rarity of an attractive outcome leads to choose the safe prospect although the risky prospect has a higher expected value (top panel). Conversely, the rarity of an unattractive outcome leads to choose the risky prospect although the safe prospect has a higher expected value (bottom panel). 

Below, this relation is emphasized by plotting the false response proportions against the probability of the unattractive outcome. If the latter increases, the piecewise strategy is more likely to falsely choose the safe option (top panel) but less likely to falsely choose the risky option (bottom panel) - cf. signal-detection-theory.

```{r}
data %>% filter(strategy == "piecewise") %>% 
  mutate(optimal = case_when(ev_ratio > 1 ~ "A",
                             ev_ratio < 1 ~ "B")) %>% 
  group_by(optimal, gamble, a_p1, s, a, choice) %>% 
  summarise(n = n()) %>% 
  mutate(prop = round(n/sum(n), 2)) %>% 
  mutate(kind = case_when(optimal == choice ~ "norm",
                          optimal == "A" & choice == "B" ~ "False Safe",
                          optimal == "B" & choice == "A" ~ "False Risky")) %>% 
  mutate(kind = as.factor(kind)) %>% 
  filter(kind != "norm") %>%
  ggplot(., aes(x = a_p1, y = prop, color = prop)) + 
  geom_jitter(alpha = .5, size = 2) + 
  scale_color_gradient(low="blue", high="red")+
  facet_wrap(~kind, nrow = 2) + 
  labs(title = "Piecewise Sampling",
       x = "Propability of Unattractive Event", 
       y= "% False responses", 
       color = "% False Responses")  + 
  theme_minimal()
```
963

linushof's avatar
linushof committed
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
#### Switching Probability and Boundary Value

The heatplots and scatterplots indicate that above and beyond the interplay of the rarity and attractiveness of outcomes, there are additional sources of variation of the choice pattern. I.e., within some of the panels of the 3 x 2 grids, especially in those indicative for underweighting, we observe a color gradient from low to high switching probabilities. Precisely, the plots below indicate that rates of false responses in general and underweighting in particular increase if switching probability increases. This is because round-wise comparisons are based on smaller sample sizes for high switching probabilities, pronouncing the effect of rare events described above.

```{r}
data %>% 
  filter(strategy == "piecewise") %>% 
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp))) %>%
  mutate(optimal = case_when(ev_ratio > 1 ~ "A",
                             ev_ratio < 1 ~ "B")) %>% 
  group_by(optimal, boundary, rare, s, a, choice) %>% 
  summarise(n = n()) %>%
  mutate(prop = round(n/sum(n), 2)) %>% 
  mutate(kind = case_when(optimal == choice ~ "norm",
                          optimal == "A" & choice == "B" ~ "False Safe",
                          optimal == "B" & choice == "A" ~ "False Risky")) %>% 
  mutate(kind = as.factor(kind)) %>% 
  filter(kind != "norm") %>%
  ggplot(., aes(x = s, y = prop, color = prop)) + 
  facet_grid(kind~rare, switch = "y") +
  geom_jitter(size = 3) +
  scale_color_gradient(low="blue", high="red") + 
  labs(title = "Piecewise Sampling",
       x = "Switching Probability", 
       y= "% False responses", 
       color = "% False Responses")  + 
  theme_minimal()
```

The effects of different boundary values are less independent, and therefore relatively nuanced, but in the expexted directions. I.e., in the absence of rare events, larger boundary values (indicating a larger number of rounds that must be won) lead to lower false response proportions - in line with the law of large numbers. In contrast, in the presence of rare events, the influence of the boundary value appears to be rather low, indicating that a larger number of necessary wins does not reduce the underweighting of rare events. I.e., also a larger number of rounds increases the likelihood of a rare event being sampled, the latters magnitude is largely ignored because all rounds are weighted equally.    
994
995
996
997

```{r}
data %>% 
  filter(strategy == "piecewise") %>% 
linushof's avatar
linushof committed
998
999
1000
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp))) %>%
  mutate(optimal = case_when(ev_ratio > 1 ~ "A",
                             ev_ratio < 1 ~ "B")) %>%