Psych 251 PS4: Simulation + Analysis
Mike Frank
2019
This is problem set #4, in which we want you to integrate your knowledge of data wrangling with some basic simulation skills. It’s a short problem set to help consolidate your ggplot2 skills and then help you get your feet wet in testing statistical concepts through “making up data” rather than consulting a textbook or doing math.
For ease of reading, please separate your answers from our text by marking our text with the > character (indicating quotes).
Part 1: ggplot practice
This part is a warmup, it should be relatively straightforward ggplot2 practice.
Load data from Frank, Vul, Saxe (2011, Infancy), a study in which we measured infants’ looking to hands in moving scenes. There were infants from 3 months all the way to about two years, and there were two movie conditions (Faces_Medium, in which kids played on a white background, and Faces_Plus, in which the backgrounds were more complex and the people in the videos were both kids and adults). An eye-tracker measured children’s attention to faces. This version of the dataset only gives two conditions and only shows the amount of looking at hands (other variables were measured as well).
library(tidyverse)## ── Attaching packages ────────────────────── tidyverse 1.3.0 ──## ✓ ggplot2 3.3.2 ✓ purrr 0.3.4
## ✓ tibble 3.0.3 ✓ dplyr 1.0.2
## ✓ tidyr 1.1.2 ✓ stringr 1.4.0
## ✓ readr 1.3.1 ✓ forcats 0.5.0## ── Conflicts ───────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(ggplot2)
fvs <- read_csv("data/FVS2011-hands.csv")## Parsed with column specification:
## cols(
## subid = col_double(),
## age = col_double(),
## condition = col_character(),
## hand.look = col_double()
## )First, use ggplot to plot a histogram of the ages of children in the study. NOTE: this is a repeated measures design, so you can’t just take a histogram of every measurement.
ages_hist = ggplot(fvs,
aes(x=age)) +
geom_histogram(binwidth=1, aes(fill=condition)) +
facet_grid(. ~ condition) +
labs(x="Age (in months)", y="Count" )
ages_hist
Second, make a scatter plot showing hand looking as a function of age and condition. Add appropriate smoothing lines. Take the time to fix the axis labels and make the plot look nice.
handlooking_scatter = ggplot(fvs,
aes(x=age, y=hand.look, shape=condition, color=condition)) +
geom_point() +
geom_smooth(method=lm) +
labs(x="Age (in months)", y="Hand Looking" )
handlooking_scatter## `geom_smooth()` using formula 'y ~ x'
What do you conclude from this pattern of data?
It appears that as children get older, they tend to look at hands more often in both conditions of this study. While this is true, the effect appears to be stronger in the Faces_Plus condition as it has a steeper positive slope compared to the Faces_Medium condition.
What statistical analyses would you perform here to quantify these differences?
I would want to compare younger children with older children using t-tests to see if there is actually a significant difference between their handlooking behaviors within each condition. I would want to compare the two conditions against each other using an ANOVA to see if their is actually a difference between the two. I would also want to run a linear regression to determine if condition and age are both statistically significant predictors of handlooking.
Part 2: Simulation
library(tidyverse)Let’s start by convincing ourselves that t-tests have the appropriate false positive rate. Run 10,000 t-tests with standard, normally-distributed data from a made up 30-person, single-measurement experiment (the command for sampling from a normal distribution is rnorm).
The goal of these t-tests are to determine, based on 30 observations, whether the underlying distribution (in this case a normal distribution with mean 0 and standard deviation 1) has a mean that is different from 0. In reality, the mean is not different from 0 (we sampled it using rnorm), but sometimes the 30 observations we get in our experiment will suggest that the mean is higher or lower. In this case, we’ll get a “significant” result and incorrectly reject the null hypothesis of mean 0.
What’s the proportion of “significant” results (\(p < .05\)) that you see?
First do this using a for loop.
# set seed to make sure the result is reproducible
set.seed(1)
# define how many observations
nobservations = 30
#define how many samples
nsamples = 10000
# set number of significant results to zero
sigresults = 0
# draw samples and calculate p-values
for (i in 1:nsamples){
random_numbers = rnorm(n = nobservations, mean = 0, sd = 1)
ttest_summary = t.test(random_numbers)
ttest_summary$p.value
# determine if this draw was significant and if so, keep track
if (ttest_summary$p.value < 0.05) {
sigresults = sigresults + 1
}
}
# number of significant results
sigresults## [1] 502# proportion of significant results
proportion_sig = (sigresults / nsamples)
proportion_sig## [1] 0.0502Next, do this using the replicate function:
# make this example reproducible
set.seed(1)
# define how many samples to draw
nobservations = 30
# set number of significant results to zero
sigresults = 0
#define how many samples
nsamples = 10000
# I was not able to figure out how to do this using replicate. I spent quite a bit of time working on it, but I'm unsure how to run the t.tests.
random_numbers = tibble(results = replicate (n = nsamples,
rnorm(n = nobservations, mean = 0, sd = 1)))How does this compare to the intended false-positive rate of \(\alpha=0.05\)?
The values that I got using the “for” loop are comparable to the alpha=0.05 range. They tend to hover right around the 0.05 mark. I haven’t yet been able to figure out how to do this using replicate.
Ok, that was a bit boring. Let’s try something more interesting - let’s implement a p-value sniffing simulation, in the style of Simons, Nelson, & Simonsohn (2011).
Consider this scenario: you have done an experiment, again with 30 participants (one observation each, just for simplicity). The question is whether the true mean is different from 0. You aren’t going to check the p-value every trial, but let’s say you run 30 - then if the p-value is within the range p < .25 and p > .05, you optionally run 30 more and add those data, then test again. But if the original p value is < .05, you call it a day, and if the original is > .25, you also stop.
First, write a function that implements this sampling regime.
double.sample <- function () {
random_numbers = rnorm(n = nobservations, mean = 0, sd = 1)
ttest_summary = t.test(random_numbers)
while (ttest_summary$p.value > 0.05 & ttest_summary$p.value < 0.25){
random_numbers = random_numbers + (rnorm(n = nobservations, mean = 0, sd = 1))
ttest_summary = t.test(random_numbers)
}
print(ttest_summary$p.value)
}Now call this function 10k times and find out what happens.
values = tibble(result = replicate(n = nsamples,
double.sample()))## [1] 0.4890051
## [1] 0.9163058
## [1] 0.4589652
## [1] 0.273501
## [1] 0.422761
## [1] 0.006808159
## [1] 0.4167755
## [1] 0.9607634
## [1] 0.4955661
## [1] 0.560613
## [1] 0.2591467
## [1] 0.2569887
## [1] 0.004333556
## [1] 0.8541639
## [1] 0.6444015
## [1] 0.272929
## [1] 0.4816703
## [1] 0.7103748
## [1] 0.4104597
## [1] 0.5659474
## [1] 0.9420142
## [1] 0.7655076
## [1] 0.441554
## [1] 0.9612409
## [1] 0.3695909
## [1] 0.4425443
## [1] 0.6712761
## [1] 0.6871076
## [1] 0.8376753
## [1] 0.7667599
## [1] 0.2673588
## [1] 0.4487867
## [1] 0.3292278
## [1] 0.5687781
## [1] 0.3272819
## [1] 0.4917471
## [1] 0.7893293
## [1] 0.9484404
## [1] 0.3015309
## [1] 0.03967249
## [1] 0.6167427
## [1] 0.9933041
## [1] 0.3141151
## [1] 0.9864266
## [1] 0.5405613
## [1] 0.4693691
## [1] 0.7434421
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## [1] 0.4415097
## [1] 0.5396973
## [1] 0.770864
## [1] 0.3438575
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## [1] 0.04062805
## [1] 0.6408117
## [1] 0.8966006
## [1] 0.9477693
## [1] 0.6785956
## [1] 0.3962696
## [1] 0.4054167
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## [1] 0.00361867
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## [1] 0.6655312
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## [1] 0.3982337
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## [1] 0.04524597
## [1] 0.01829155
## [1] 0.9169704
## [1] 0.2516379
## [1] 0.4157927
## [1] 0.8991188
## [1] 0.9520454
## [1] 0.5331687
## [1] 0.932855
## [1] 0.8026517
## [1] 0.785334
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## [1] 0.267737
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## [1] 0.3176784
## [1] 0.6779429
## [1] 0.5370841
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## [1] 0.3977158
## [1] 0.5726408
## [1] 0.8498758
## [1] 0.3336698
## [1] 0.2672334
## [1] 0.7181442
## [1] 0.9316717
## [1] 0.3094899
## [1] 0.3770941
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## [1] 0.7385999
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## [1] 0.6080524
## [1] 0.8510859
## [1] 0.5185916
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## [1] 0.8126917
## [1] 0.6863468
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## [1] 0.04453269
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## [1] 0.9559671
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## [1] 0.5533807
## [1] 0.477323
## [1] 0.5825148
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## [1] 0.04294104
## [1] 0.5741169
## [1] 0.846201
## [1] 0.01233911
## [1] 0.2517155
## [1] 0.725133
## [1] 0.687367
## [1] 0.9212177
## [1] 0.9384585
## [1] 0.638615
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## [1] 0.6021446
## [1] 0.01278445
## [1] 0.5890745
## [1] 0.4947288
## [1] 0.7531011
## [1] 0.4023199
## [1] 0.7417204
## [1] 0.3534487
## [1] 0.0385077
## [1] 0.4675759
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## [1] 0.3651046
## [1] 0.4717161
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## [1] 0.03912388
## [1] 0.9870988
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## [1] 0.02749414
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## [1] 0.6308084
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## [1] 0.5902948
## [1] 0.3219852
## [1] 0.3896973
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## [1] 0.5244631
## [1] 0.01457161
## [1] 0.7773451
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## [1] 0.9958072
## [1] 0.4409623
## [1] 0.3367279
## [1] 0.83755
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## [1] 0.4975953
## [1] 0.6704613
## [1] 0.7831458
## [1] 0.3640663
## [1] 0.953537
## [1] 0.2710857
## [1] 0.3972706
## [1] 0.2838599
## [1] 0.4889828
## [1] 0.6692048
## [1] 0.3612271
## [1] 0.5030328
## [1] 0.8446526
## [1] 0.3022184
## [1] 0.9953376
## [1] 0.2756861
## [1] 0.4039842
## [1] 0.7791321
## [1] 0.9581294
## [1] 0.9723685
## [1] 0.5781411
## [1] 0.5420294
## [1] 0.2506497
## [1] 0.4840296
## [1] 0.3330564
## [1] 0.4993753
## [1] 0.4804418
## [1] 0.4730326
## [1] 0.2697477
## [1] 0.4021052
## [1] 0.3940756
## [1] 0.4490217
## [1] 0.5692382
## [1] 0.02373574
## [1] 0.01715138
## [1] 0.7110826
## [1] 0.4754208
## [1] 0.3070303
## [1] 0.296367
## [1] 0.03401587
## [1] 0.3724406
## [1] 0.4603473
## [1] 0.7684978
## [1] 0.8297717
## [1] 0.3069705
## [1] 0.7130154
## [1] 0.872117
## [1] 0.2907996
## [1] 0.7540971
## [1] 0.9960914
## [1] 0.5716934
## [1] 0.5020211
## [1] 0.4684509
## [1] 0.9288899
## [1] 0.6060069
## [1] 0.6845895
## [1] 0.962694
## [1] 0.01059879
## [1] 0.81426
## [1] 0.3517107
## [1] 0.5185506
## [1] 0.9776091
## [1] 0.7595703
## [1] 0.3201196
## [1] 0.9869001
## [1] 0.8248234
## [1] 0.8588568
## [1] 0.7057129
## [1] 0.9708804
## [1] 0.8310175
## [1] 0.5296459
## [1] 0.4759869
## [1] 0.9874587
## [1] 0.6646653
## [1] 0.276227
## [1] 0.5581668
## [1] 0.3727491
## [1] 0.386949
## [1] 0.4413231
## [1] 0.4526956
## [1] 0.2545366
## [1] 0.4422355
## [1] 0.8638905
## [1] 0.2965511
## [1] 0.04353299
## [1] 0.4724639
## [1] 0.9881038
## [1] 0.3779903
## [1] 0.7256884
## [1] 0.7079268
## [1] 0.5789237
## [1] 0.8238193
## [1] 0.4759169
## [1] 0.3983964
## [1] 0.2823739
## [1] 0.3746808
## [1] 0.3405687
## [1] 0.988157
## [1] 0.02940058
## [1] 0.3299062
## [1] 0.5306077
## [1] 0.8776695
## [1] 0.005772545
## [1] 0.4267687
## [1] 0.007818426
## [1] 0.9646814
## [1] 0.8788776
## [1] 0.5245791
## [1] 0.6628959
## [1] 0.6467873
## [1] 0.8357113
## [1] 0.6697111
## [1] 0.4128631
## [1] 0.6032833
## [1] 0.5397706
## [1] 0.890389
## [1] 0.5525595
## [1] 0.5974846
## [1] 0.3577967
## [1] 0.3902639
## [1] 0.9834604
## [1] 0.3843886
## [1] 0.7663993
## [1] 0.395744
## [1] 0.4719238
## [1] 0.529776
## [1] 0.7229671
## [1] 0.4116702
## [1] 0.8715305
## [1] 0.5760593
## [1] 0.3088852
## [1] 0.4467221
## [1] 0.4823709
## [1] 0.4161673
## [1] 0.009521325
## [1] 0.2715002
## [1] 0.9719208
## [1] 0.5728649
## [1] 0.8252347
## [1] 0.8942096
## [1] 0.3356954
## [1] 0.8509099
## [1] 0.7941876
## [1] 0.8157297
## [1] 0.8862778
## [1] 0.2681752
## [1] 0.8789502
## [1] 0.8779779
## [1] 0.555125
## [1] 0.0143018
## [1] 0.01471447
## [1] 0.2779553
## [1] 0.3045481
## [1] 0.4413146
## [1] 0.2545054
## [1] 0.8683174
## [1] 0.01570066
## [1] 0.7398696
## [1] 0.6160517
## [1] 0.2617995
## [1] 0.33543
## [1] 0.8190634
## [1] 0.3424843
## [1] 0.8659163
## [1] 0.9011108
## [1] 0.2933964
## [1] 0.332472
## [1] 0.009342972
## [1] 0.8156746
## [1] 0.6180039
## [1] 0.01389664
## [1] 0.02453588
## [1] 0.7120773
## [1] 0.4679921
## [1] 0.7111275
## [1] 0.8282183
## [1] 0.8184395
## [1] 0.335728
## [1] 0.99253
## [1] 0.8531251
## [1] 0.5385747
## [1] 0.9021604
## [1] 0.5367117
## [1] 0.4026544
## [1] 0.3345605
## [1] 0.6010134
## [1] 0.5812019
## [1] 0.4863109
## [1] 0.7097842
## [1] 0.739487
## [1] 0.9162009
## [1] 0.4005491
## [1] 0.5027998
## [1] 0.8974283
## [1] 0.9169745
## [1] 0.7845569
## [1] 0.7523788
## [1] 0.7711421
## [1] 0.4732595
## [1] 0.3945917
## [1] 0.353771
## [1] 0.5994878
## [1] 0.9034391
## [1] 0.8093076
## [1] 0.5888852
## [1] 0.5023966
## [1] 0.4367446
## [1] 0.3122074
## [1] 0.3558686
## [1] 0.8926867
## [1] 0.9541854
## [1] 0.6278448
## [1] 0.9100764
## [1] 0.006504916
## [1] 0.9564332
## [1] 0.5045893
## [1] 0.9840261
## [1] 0.8384912
## [1] 0.7834068
## [1] 0.6840138
## [1] 0.4939393
## [1] 0.3480538
## [1] 0.7311872
## [1] 0.03655649
## [1] 0.9204633
## [1] 0.8710073
## [1] 0.9354911
## [1] 0.309225
## [1] 0.8564847
## [1] 0.948754
## [1] 0.3423926
## [1] 0.3942583
## [1] 0.5572514
## [1] 0.6832815
## [1] 0.6464587
## [1] 0.6336227
## [1] 0.5653734
## [1] 0.7646608
## [1] 0.7508209
## [1] 0.7548812
## [1] 0.3394169
## [1] 0.290384
## [1] 0.5090683
## [1] 0.7196674
## [1] 0.286956
## [1] 0.9423503
## [1] 0.01049803
## [1] 0.6607733
## [1] 0.4858743
## [1] 0.9779673
## [1] 0.9293136
## [1] 0.0481848
## [1] 0.7456951
## [1] 0.9714922
## [1] 0.7100252
## [1] 0.04398012
## [1] 0.7420285
## [1] 0.836348
## [1] 0.3793992
## [1] 0.278554
## [1] 0.6565815
## [1] 0.7419391
## [1] 0.8399357
## [1] 0.859593
## [1] 0.008888441
## [1] 0.6577199
## [1] 0.9892708
## [1] 0.7297388
## [1] 0.03510716
## [1] 0.02925133
## [1] 0.9024859
## [1] 0.8404651
## [1] 0.03516443
## [1] 0.4336204
## [1] 0.5153792
## [1] 0.2532633
## [1] 0.5044091
## [1] 0.2706173
## [1] 0.472625
## [1] 0.02081029
## [1] 0.801092
## [1] 0.3600648
## [1] 0.8711601
## [1] 0.3201659
## [1] 0.8602648
## [1] 0.3414984
## [1] 0.3513247
## [1] 0.8283698
## [1] 0.8760676
## [1] 0.8294038
## [1] 0.7918983
## [1] 0.8742268
## [1] 0.7742309
## [1] 0.3229941
## [1] 0.5707003
## [1] 0.7726644
## [1] 0.04223799
## [1] 0.9024229
## [1] 0.7285934
## [1] 0.5967349
## [1] 0.8277165
## [1] 0.2704769
## [1] 0.9119937
## [1] 0.4867073
## [1] 0.4437935
## [1] 0.03888682
## [1] 0.3730445
## [1] 0.4521857
## [1] 0.357691
## [1] 0.6165764
## [1] 0.2632913
## [1] 0.9439644
## [1] 0.6897476
## [1] 0.882658
## [1] 0.9926255
## [1] 0.9453402
## [1] 0.9097079
## [1] 0.5047794
## [1] 0.7572031
## [1] 0.9707356
## [1] 0.739612
## [1] 0.8373518
## [1] 0.04213875
## [1] 0.8459266
## [1] 0.5062385
## [1] 0.8480154
## [1] 0.3455199
## [1] 0.4352851
## [1] 0.6771904
## [1] 0.9239703
## [1] 0.7307465
## [1] 0.7824039
## [1] 0.3869615
## [1] 0.9993826
## [1] 0.5607056
## [1] 0.3604616
## [1] 0.02770932
## [1] 0.283118
## [1] 0.3273147
## [1] 0.2645126
## [1] 0.3105025
## [1] 0.4604656
## [1] 0.04000018
## [1] 0.6525607
## [1] 0.570801
## [1] 0.6187017
## [1] 0.5496892
## [1] 0.5917716
## [1] 0.4219662
## [1] 0.02011335
## [1] 0.2691935
## [1] 0.474783
## [1] 0.2891931
## [1] 0.3810313
## [1] 0.3074388
## [1] 0.845636
## [1] 0.4341878
## [1] 0.4460966
## [1] 0.7052041
## [1] 0.2710504
## [1] 0.971439
## [1] 0.7733289
## [1] 0.01737456
## [1] 0.3653483
## [1] 0.2825045
## [1] 0.8306339
## [1] 0.3291119
## [1] 0.9100002
## [1] 0.9449547
## [1] 0.4768116
## [1] 0.5403943
## [1] 0.5024295
## [1] 0.6927261
## [1] 0.986866
## [1] 0.03277077
## [1] 0.5763761
## [1] 0.7029138
## [1] 0.2666318
## [1] 0.9622979
## [1] 0.888798
## [1] 0.5673053
## [1] 0.9343244
## [1] 0.4739326
## [1] 0.02183928
## [1] 0.9533112
## [1] 0.6676377
## [1] 0.358073
## [1] 0.999075sig_values = sum(values$result < 0.05)
sig_values## [1] 831proportion = (sig_values/10000)
proportion## [1] 0.0831Is there an inflation of false positives? How bad is it?
There is an inflation of false positives. Approximately 8% of the values are false positives, indicating that the assumption of an alpha of 0.05 is not applicable in this case. While some may argue that the difference between thw two is small, if we know that this procedure will cause an inflation in the false positives, we still need to account for that in our calculations of significance (or not use the procedure).
Now modify this code so that you can investigate this “double the sample” rule in a bit more depth. In the previous question, the researcher doubles the sample only when they think they got “close” to a significant result, i.e. when their not-significant p is less than 0.25. What if the researcher was more optimistic? See what happens in these 3 other scenarios:
- The researcher doubles the sample whenever their pvalue is not significant, but it’s less than 0.5.
- The researcher doubles the sample whenever their pvalue is not significant, but it’s less than 0.75.
- The research doubles their sample whenever they get ANY pvalue that is not significant.
How do these choices affect the false positive rate?
HINT: Try to do this by making the function double.sample take the upper p value as an argument, so that you can pass this through dplyr.
HINT 2: You may need more samples. Find out by looking at how the results change from run to run.
# Altered using 0.5 as the upper limit
double.sample <- function () {
random_numbers = rnorm(n = nobservations, mean = 0, sd = 1)
ttest_summary = t.test(random_numbers)
while (ttest_summary$p.value > 0.05 & ttest_summary$p.value < 0.5){
random_numbers = random_numbers + (rnorm(n = nobservations, mean = 0, sd = 1))
ttest_summary = t.test(random_numbers)
}
print(ttest_summary$p.value)
}
values = tibble(result = replicate(n = 10000,
double.sample()))## [1] 0.5749599
## [1] 0.6097946
## [1] 0.6295649
## [1] 0.04503786
## [1] 0.5179949
## [1] 0.5835702
## [1] 0.8187902
## [1] 0.9859687
## [1] 0.02978119
## [1] 0.8198756
## [1] 0.8819271
## [1] 0.559847
## [1] 0.5484903
## [1] 0.6758651
## [1] 0.8596373
## [1] 0.8111728
## [1] 0.8610195
## [1] 0.5901644
## [1] 0.9693614
## [1] 0.6082196
## [1] 0.9208829
## [1] 0.9753879
## [1] 0.7513814
## [1] 0.5262931
## [1] 0.8250463
## [1] 0.006470294
## [1] 0.6294834
## [1] 0.8657381
## [1] 0.5858503
## [1] 0.7784282
## [1] 0.9344797
## [1] 0.7705448
## [1] 0.6348126
## [1] 0.5801729
## [1] 0.5424415
## [1] 0.590257
## [1] 0.593912
## [1] 0.6167834
## [1] 0.0334702
## [1] 0.6438709
## [1] 0.8904109
## [1] 0.9139396
## [1] 0.7880908
## [1] 0.6111004
## [1] 0.5741862
## [1] 0.5948444
## [1] 0.9426461
## [1] 0.8698497
## [1] 0.8733527
## [1] 0.01594924
## [1] 0.6261885
## [1] 0.5506267
## [1] 0.7415528
## [1] 0.8540124
## [1] 0.6474456
## [1] 0.6973335
## [1] 0.8806629
## [1] 0.8286031
## [1] 0.5308488
## [1] 0.7949364
## [1] 0.5852748
## [1] 0.9808684
## [1] 0.7326971
## [1] 0.8800427
## [1] 0.6132274
## [1] 0.01393262
## [1] 0.6837001
## [1] 0.8071749
## [1] 0.936915
## [1] 0.8944502
## [1] 0.7855554
## [1] 0.7832039
## [1] 0.8171744
## [1] 0.9287321
## [1] 0.6100138
## [1] 0.5374961
## [1] 0.5985036
## [1] 0.8175004
## [1] 0.5124655
## [1] 0.9226697
## [1] 0.6656714
## [1] 0.6766924
## [1] 0.04959809
## [1] 0.9850836
## [1] 0.00731754
## [1] 0.7010912
## [1] 0.5834277
## [1] 0.9886152
## [1] 0.9381228
## [1] 0.5961799
## [1] 0.8625363
## [1] 0.5281865
## [1] 0.7166101
## [1] 0.8800446
## [1] 0.7675254
## [1] 0.02736946
## [1] 0.6207256
## [1] 0.02037595
## [1] 0.8850055
## [1] 0.6081811
## [1] 0.700665
## [1] 0.8264124
## [1] 0.5629889
## [1] 0.6832271
## [1] 0.02935046
## [1] 0.6572859
## [1] 0.5164243
## [1] 0.538846
## [1] 0.7242431
## [1] 0.0125758
## [1] 0.8713412
## [1] 0.04282286
## [1] 0.855866
## [1] 0.9749589
## [1] 0.6127041
## [1] 0.9640271
## [1] 0.9306154
## [1] 0.9472787
## [1] 0.8288736
## [1] 0.03144146
## [1] 0.6380211
## [1] 0.6902496
## [1] 0.6316027
## [1] 0.8270458
## [1] 0.6978165
## [1] 0.564708
## [1] 0.9804732
## [1] 0.9607077
## [1] 0.7080387
## [1] 0.7213511
## [1] 0.016134
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## [1] 0.00979848
## [1] 0.72547
## [1] 0.7204962
## [1] 0.8680746
## [1] 0.7857071
## [1] 0.9189443
## [1] 0.6398618
## [1] 0.6915025
## [1] 0.5020998
## [1] 0.6660165
## [1] 0.7519386
## [1] 0.6822088
## [1] 0.7046534
## [1] 0.5424403
## [1] 0.9644865
## [1] 0.6221322
## [1] 0.5279317
## [1] 0.5479066
## [1] 0.6054524
## [1] 0.6301112
## [1] 0.9610689
## [1] 0.006904701
## [1] 0.7798376
## [1] 0.7346779
## [1] 0.8714298
## [1] 0.8756231
## [1] 0.03552609
## [1] 0.6120945
## [1] 0.9405149
## [1] 0.9708329
## [1] 0.7342528
## [1] 0.5814908
## [1] 0.5291961
## [1] 0.9645441
## [1] 0.7038692
## [1] 0.9247516
## [1] 0.5457299
## [1] 0.5595185
## [1] 0.9844145
## [1] 0.03299036
## [1] 0.5976315
## [1] 0.810721
## [1] 0.5398177
## [1] 0.6211562
## [1] 0.6261236
## [1] 0.6000427
## [1] 0.9955812
## [1] 0.002855432
## [1] 0.5387814
## [1] 0.5106429
## [1] 0.04867937
## [1] 0.9549181
## [1] 0.9622338
## [1] 0.6786979
## [1] 0.7073071
## [1] 0.6063827
## [1] 0.8167989
## [1] 0.9063459
## [1] 0.8232307
## [1] 0.0204309
## [1] 0.5725594
## [1] 0.01169393
## [1] 0.553542
## [1] 0.006810485
## [1] 0.6912655
## [1] 0.01966106
## [1] 0.5268715
## [1] 0.04829334
## [1] 0.6141599
## [1] 0.8554675
## [1] 0.6250695
## [1] 0.6895195
## [1] 0.9194548
## [1] 0.6583208
## [1] 0.8808077
## [1] 0.8409311
## [1] 0.56721
## [1] 0.04970294
## [1] 0.6600362
## [1] 0.8822414
## [1] 0.8555398
## [1] 0.5298321
## [1] 0.03248956
## [1] 0.6807281
## [1] 0.8232342
## [1] 0.5312589
## [1] 0.6243744
## [1] 0.781848
## [1] 0.6704037
## [1] 0.8472077
## [1] 0.7207082
## [1] 0.6421304
## [1] 0.7684729
## [1] 0.9205244
## [1] 0.646189
## [1] 0.9037835
## [1] 0.04314785
## [1] 0.02787053
## [1] 0.9951364
## [1] 0.8896143
## [1] 0.505003
## [1] 0.9679008
## [1] 0.5427996
## [1] 0.7327675
## [1] 0.8080164
## [1] 0.7091709
## [1] 0.9687559
## [1] 0.03830743
## [1] 0.9867796
## [1] 0.8209024
## [1] 0.7526665
## [1] 0.6460352
## [1] 0.04625934
## [1] 0.7625524
## [1] 0.6894524
## [1] 0.6404566
## [1] 0.952698
## [1] 0.520854
## [1] 0.7571568
## [1] 0.5010824
## [1] 0.8910073
## [1] 0.8896016
## [1] 0.5804675
## [1] 0.7628324
## [1] 0.862065
## [1] 0.7659279
## [1] 0.6731178
## [1] 0.5431645
## [1] 0.9523242
## [1] 0.8066886
## [1] 0.5384176
## [1] 0.8192944
## [1] 0.5000048
## [1] 0.8006216
## [1] 0.9050756
## [1] 0.7495153
## [1] 0.6096518
## [1] 0.5766832
## [1] 0.5245591
## [1] 0.5836914
## [1] 0.5132284
## [1] 0.7574331
## [1] 0.690627
## [1] 0.6475309
## [1] 0.8596703
## [1] 0.8404321
## [1] 0.605983
## [1] 0.6463136
## [1] 0.7729405
## [1] 0.7288345
## [1] 0.7756418
## [1] 0.7279487
## [1] 0.707542
## [1] 0.6610839
## [1] 0.6373966
## [1] 0.8609655
## [1] 0.7238798
## [1] 0.8675812
## [1] 0.9360199
## [1] 0.8995008
## [1] 0.7602579
## [1] 0.01910352
## [1] 0.895465
## [1] 0.889353
## [1] 0.7150955
## [1] 0.6168632
## [1] 0.9673716
## [1] 0.8264102
## [1] 0.582291
## [1] 0.5894786
## [1] 0.5820336
## [1] 0.5068804
## [1] 0.6784534
## [1] 0.8722016
## [1] 0.04604915
## [1] 0.04096189
## [1] 0.5975902
## [1] 0.6952673
## [1] 0.6036189
## [1] 0.5192601
## [1] 0.5903263
## [1] 0.03563641
## [1] 0.8055859
## [1] 0.9262276
## [1] 0.5676831
## [1] 0.04401056
## [1] 0.9818761
## [1] 0.953171
## [1] 0.7376943
## [1] 0.02416441
## [1] 0.8366521
## [1] 0.5908433
## [1] 0.03468598
## [1] 0.5459435
## [1] 0.8063849
## [1] 0.5258392
## [1] 0.6176297
## [1] 0.04497291
## [1] 0.8123025
## [1] 0.04002185
## [1] 0.03460355
## [1] 0.862851
## [1] 0.6348581
## [1] 0.7064931
## [1] 0.03353523
## [1] 0.6065046
## [1] 0.6709532
## [1] 0.5767165
## [1] 0.04979554
## [1] 0.6623695
## [1] 0.9124997
## [1] 0.9701107
## [1] 0.8405851
## [1] 0.03681277
## [1] 0.5921546
## [1] 0.5040695
## [1] 0.6723924
## [1] 0.0313377
## [1] 0.6659669
## [1] 0.8756973
## [1] 0.04457857
## [1] 0.8767938
## [1] 0.5240332
## [1] 0.7701474
## [1] 0.5280777
## [1] 0.944918
## [1] 0.6013414
## [1] 0.0445265
## [1] 0.6134484
## [1] 0.7550937
## [1] 0.8045568
## [1] 0.5876619
## [1] 0.51943
## [1] 0.8122446
## [1] 0.9383772
## [1] 0.5588568
## [1] 0.0416824
## [1] 0.9184342
## [1] 0.728337
## [1] 0.8058403
## [1] 0.8009046
## [1] 0.7639763
## [1] 0.7461059
## [1] 0.7896216
## [1] 0.6404377
## [1] 0.6735846
## [1] 0.7410941
## [1] 0.9361033
## [1] 0.02253074
## [1] 0.6033458
## [1] 0.9122059
## [1] 0.5448733
## [1] 0.7043703
## [1] 0.6683749
## [1] 0.5231318
## [1] 0.5071644
## [1] 0.6752267
## [1] 0.5047878
## [1] 0.8601178
## [1] 0.9335601
## [1] 0.5583709
## [1] 0.02875556
## [1] 0.8506147
## [1] 0.5196592
## [1] 0.7037252
## [1] 0.5420758
## [1] 0.9890728
## [1] 0.6714512
## [1] 0.6950995
## [1] 0.5865626
## [1] 0.5474425
## [1] 0.719676
## [1] 0.9001824
## [1] 0.8473877
## [1] 0.5022167
## [1] 0.6014565
## [1] 0.8235372
## [1] 0.01596121
## [1] 0.5177563
## [1] 0.6216225
## [1] 0.5106313sig_values = sum(values$result < 0.05)
sig_values## [1] 1297proportion = (sig_values/10000)
proportion## [1] 0.1297# Altered using 0.75 as the upper limit
double.sample <- function () {
random_numbers = rnorm(n = nobservations, mean = 0, sd = 1)
ttest_summary = t.test(random_numbers)
while (ttest_summary$p.value > 0.05 & ttest_summary$p.value < 0.75){
random_numbers = random_numbers + (rnorm(n = nobservations, mean = 0, sd = 1))
ttest_summary = t.test(random_numbers)
}
print(ttest_summary$p.value)
}
values = tibble(result = replicate(n = 10000,
double.sample()))## [1] 0.02570015
## [1] 0.03629254
## [1] 0.8094768
## [1] 0.02098587
## [1] 0.9046258
## [1] 0.01228637
## [1] 0.8227006
## [1] 0.03645389
## [1] 0.9436793
## [1] 0.7684706
## [1] 0.8510336
## [1] 0.8706952
## [1] 0.9375576
## [1] 0.8207272
## [1] 0.7776061
## [1] 0.9753037
## [1] 0.9050938
## [1] 0.927448
## [1] 0.04895954
## [1] 0.7963002
## [1] 0.8073066
## [1] 0.8135705
## [1] 0.9971906
## [1] 0.9590927
## [1] 0.8434595
## [1] 0.006867801
## [1] 0.8163759
## [1] 0.04401802
## [1] 0.04504788
## [1] 0.7848403
## [1] 0.7951666
## [1] 0.783132
## [1] 0.9519792
## [1] 0.934784
## [1] 0.7544984
## [1] 0.960324
## [1] 0.9546686
## [1] 0.003133284
## [1] 0.7796701
## [1] 0.8375464
## [1] 0.9572028
## [1] 0.8823262
## [1] 0.9158134
## [1] 0.9421108
## [1] 0.8797951
## [1] 0.994409
## [1] 0.7892714
## [1] 0.9958394
## [1] 0.8328205
## [1] 0.00283173
## [1] 0.9442026
## [1] 0.7945562
## [1] 0.916765
## [1] 0.9618547
## [1] 0.830586
## [1] 0.7720073
## [1] 0.8489895
## [1] 0.7973149
## [1] 0.8754375
## [1] 0.9534864
## [1] 0.9623009
## [1] 0.9989329
## [1] 0.8983095
## [1] 0.02502
## [1] 0.9043522
## [1] 0.0003442263
## [1] 0.773437
## [1] 0.7876842
## [1] 0.986969
## [1] 0.9633476
## [1] 0.03292521
## [1] 0.8118697
## [1] 0.9308457
## [1] 0.03836528
## [1] 0.9019915
## [1] 0.9835792
## [1] 0.002927839
## [1] 0.01843365
## [1] 0.8088357
## [1] 0.8535636
## [1] 0.03966847
## [1] 0.8460261
## [1] 0.8156386
## [1] 0.9256339
## [1] 0.8702562
## [1] 0.7776669
## [1] 0.007018639
## [1] 0.9343104
## [1] 0.8611446
## [1] 0.9375149
## [1] 0.03084853
## [1] 0.8591727
## [1] 0.8125696
## [1] 0.8622753
## [1] 0.02032547
## [1] 0.9732579
## [1] 0.9456295
## [1] 0.9845395
## [1] 0.7830561
## [1] 0.7678701
## [1] 0.8651914
## [1] 0.8746662
## [1] 0.7659454
## [1] 0.8539597
## [1] 0.9695325
## [1] 0.8666239
## [1] 0.002100953
## [1] 0.8235758
## [1] 0.9270093
## [1] 0.8479309
## [1] 0.8452778
## [1] 0.7860164
## [1] 0.753974
## [1] 0.8119514
## [1] 0.03459539
## [1] 0.86885
## [1] 0.02375469
## [1] 0.804114
## [1] 0.8594702
## [1] 0.9780884
## [1] 0.9873372
## [1] 0.9653267
## [1] 0.7926327
## [1] 0.7699162
## [1] 0.02639097
## [1] 0.8785819
## [1] 0.818886
## [1] 0.7890447
## [1] 0.8280407
## [1] 0.9433486
## [1] 0.9197116
## [1] 0.8688034
## [1] 0.943492
## [1] 0.04051529
## [1] 0.8317247
## [1] 0.7515105
## [1] 0.8179525
## [1] 0.8589586
## [1] 0.9346392
## [1] 0.03856383
## [1] 0.9365306
## [1] 0.8107832
## [1] 0.809772
## [1] 0.7689971
## [1] 0.7783373
## [1] 0.9263294
## [1] 0.9051722
## [1] 0.04601627
## [1] 0.8051712
## [1] 0.7968332
## [1] 0.9254308
## [1] 0.8997907
## [1] 0.8919278
## [1] 0.7806167
## [1] 0.7958458
## [1] 0.8795923
## [1] 0.8684265
## [1] 0.9096648
## [1] 0.9050169
## [1] 0.04259512
## [1] 0.02529947
## [1] 0.03054371
## [1] 0.7628508
## [1] 0.9268033
## [1] 0.7770627
## [1] 0.7659103
## [1] 0.8071857
## [1] 0.76525
## [1] 0.9017868
## [1] 0.03342839
## [1] 0.7501217
## [1] 0.7508473
## [1] 0.8820733
## [1] 0.8245547
## [1] 0.8906816
## [1] 0.7833858
## [1] 0.0020622
## [1] 0.8113891
## [1] 0.7797879
## [1] 0.777152
## [1] 0.9526508
## [1] 0.9820142
## [1] 0.8526622
## [1] 0.7632147
## [1] 0.9747764
## [1] 0.9120753
## [1] 0.8023065
## [1] 0.02186342
## [1] 0.8451622
## [1] 0.8688338
## [1] 0.02673303
## [1] 0.03301371
## [1] 0.9922499
## [1] 0.9293101
## [1] 0.000152152
## [1] 0.8252063
## [1] 0.8664551
## [1] 0.7824751
## [1] 0.04746171
## [1] 0.9199562
## [1] 0.04981981
## [1] 0.7919328
## [1] 0.8068805
## [1] 0.03998162
## [1] 0.9003875
## [1] 0.04202388
## [1] 0.9139858
## [1] 0.7825536
## [1] 0.7879622
## [1] 0.7984531
## [1] 0.8634305
## [1] 0.8074989
## [1] 0.7884682
## [1] 0.01286677
## [1] 0.8229911
## [1] 0.8878228
## [1] 0.9587898
## [1] 0.8612384
## [1] 0.7994853
## [1] 0.9160629
## [1] 0.9621502
## [1] 0.8595355
## [1] 0.04061442
## [1] 0.9270135
## [1] 0.8063627
## [1] 0.7934671
## [1] 0.8331256
## [1] 0.03833307
## [1] 0.03082379
## [1] 0.9487565
## [1] 0.01890317
## [1] 0.9883622
## [1] 0.7611866
## [1] 0.9724231
## [1] 0.8821715
## [1] 0.04486682
## [1] 0.9027407
## [1] 0.7856313
## [1] 0.01934502
## [1] 0.8996579
## [1] 0.7801101
## [1] 0.9103516
## [1] 0.8310454
## [1] 0.8340373
## [1] 0.9134029
## [1] 0.8540683
## [1] 0.04812184
## [1] 0.03834262
## [1] 0.880673
## [1] 0.01061746
## [1] 0.9490403
## [1] 0.7592373
## [1] 0.045883
## [1] 0.8005406
## [1] 0.8487326
## [1] 0.9947288
## [1] 0.9914045
## [1] 0.03958371
## [1] 0.8043322
## [1] 0.9853707
## [1] 0.7743779
## [1] 0.02002083
## [1] 0.8417894
## [1] 0.9184956
## [1] 0.7684657
## [1] 0.03560656
## [1] 0.9825327
## [1] 0.793102
## [1] 0.04334432
## [1] 0.8163527
## [1] 0.02596995
## [1] 0.8284236
## [1] 0.9848944
## [1] 0.04192359
## [1] 0.04516551
## [1] 0.9982604
## [1] 0.8795953
## [1] 0.8991536
## [1] 0.9535829
## [1] 0.01500599
## [1] 0.8246014
## [1] 0.8346798
## [1] 0.8884942
## [1] 0.7558055
## [1] 0.9767722
## [1] 0.7562373
## [1] 0.8575198
## [1] 0.9756215
## [1] 0.8676513
## [1] 0.8440142
## [1] 0.8799658
## [1] 0.7844001
## [1] 0.9255117
## [1] 0.8506194
## [1] 0.9487014
## [1] 0.7555962
## [1] 0.8122134
## [1] 0.8370965
## [1] 0.04319858
## [1] 0.9630783
## [1] 0.8525175
## [1] 0.9437042
## [1] 0.8067675
## [1] 0.9381091
## [1] 0.9733939
## [1] 0.8595529
## [1] 0.9599714
## [1] 0.02864595
## [1] 0.7915434
## [1] 0.01545393
## [1] 0.04699579
## [1] 0.8827166
## [1] 0.04997578
## [1] 0.7739644
## [1] 0.9722375
## [1] 0.9704449
## [1] 0.9905917
## [1] 0.791175
## [1] 0.9525742
## [1] 0.910581
## [1] 0.8330674
## [1] 0.01846725
## [1] 0.8049028
## [1] 0.8009224
## [1] 0.8316626
## [1] 0.9896633
## [1] 0.04988315
## [1] 0.9132356
## [1] 0.9924453
## [1] 0.9840641
## [1] 0.04240763
## [1] 0.04414505
## [1] 0.9232818
## [1] 0.8654227
## [1] 0.9313496
## [1] 0.8927233
## [1] 0.9143886
## [1] 0.9303797
## [1] 0.8164392
## [1] 0.8137528
## [1] 0.9849591
## [1] 0.7605404
## [1] 0.844245
## [1] 0.03990386
## [1] 0.7786873
## [1] 0.0270328
## [1] 0.8676843
## [1] 0.8985622
## [1] 0.01483944
## [1] 0.02063555
## [1] 0.01201791
## [1] 0.8585607
## [1] 0.8988697
## [1] 0.7823022
## [1] 0.9816849
## [1] 0.9204484
## [1] 0.8433086
## [1] 0.9143962
## [1] 0.8091275
## [1] 0.953074
## [1] 0.9955194
## [1] 0.8902805
## [1] 0.7567067
## [1] 0.04404848
## [1] 0.01779659
## [1] 0.8889248
## [1] 0.7903471
## [1] 0.8170546
## [1] 0.7885673
## [1] 0.02851005
## [1] 0.9460952
## [1] 0.792283
## [1] 0.991547
## [1] 0.9263642
## [1] 0.7548398
## [1] 0.8611509
## [1] 0.960677
## [1] 0.7624027
## [1] 0.9851369
## [1] 0.9317588
## [1] 0.989687
## [1] 0.8849649
## [1] 0.01396027
## [1] 0.7991569
## [1] 0.04205743
## [1] 0.8199078
## [1] 0.9928571
## [1] 0.7619387
## [1] 0.03862261
## [1] 0.8014119
## [1] 0.7810365
## [1] 0.8962504
## [1] 0.03958821
## [1] 0.9305153
## [1] 0.7643964
## [1] 0.04541692
## [1] 0.9639755
## [1] 0.8605657
## [1] 0.9000414
## [1] 0.9876409
## [1] 0.8051462
## [1] 0.9980732
## [1] 0.9172038
## [1] 0.908614
## [1] 0.8002137
## [1] 0.9966133
## [1] 0.8946077
## [1] 0.9717571
## [1] 0.8059139
## [1] 0.7936021
## [1] 0.8683454
## [1] 0.8757785
## [1] 0.9111193
## [1] 0.9192894
## [1] 0.7728054
## [1] 0.9967194
## [1] 0.8499875
## [1] 0.7541727
## [1] 0.03397889
## [1] 0.0448286
## [1] 0.9608699
## [1] 0.7903115
## [1] 0.9401923
## [1] 0.85613
## [1] 0.8990097
## [1] 0.9471681
## [1] 0.7741319
## [1] 0.7918009
## [1] 0.9047592
## [1] 0.9577041
## [1] 0.950858
## [1] 0.8238881
## [1] 0.9069174
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## [1] 0.75469
## [1] 0.7684407
## [1] 0.9094003
## [1] 0.8342096
## [1] 0.973894
## [1] 0.9383343
## [1] 0.7622349
## [1] 0.7647431
## [1] 0.8104502
## [1] 0.9102307
## [1] 0.7977567
## [1] 0.8719308
## [1] 0.8351858
## [1] 0.8131873
## [1] 0.0279516
## [1] 0.9380256
## [1] 0.836335
## [1] 0.8148104
## [1] 0.9525802
## [1] 0.7909791
## [1] 0.7791768
## [1] 0.8557388
## [1] 0.8878204
## [1] 0.9976154
## [1] 0.8779532
## [1] 0.9659557
## [1] 0.8659588
## [1] 0.8298529
## [1] 0.9422923
## [1] 0.7655662
## [1] 0.03530172
## [1] 0.7584655
## [1] 0.04100955
## [1] 0.9930302
## [1] 0.7653371
## [1] 0.7897604
## [1] 0.9973186
## [1] 0.7851245
## [1] 0.7717393
## [1] 0.8810096
## [1] 0.7698055
## [1] 0.9911927
## [1] 0.8960391sig_values = sum(values$result < 0.05)
sig_values## [1] 1849proportion = (sig_values/10000)
proportion## [1] 0.1849# This code is commented out because it runs indefinitely.
# Altered for any p-value that is insignificant
#double.sample <- function () {
# random_numbers = rnorm(n = nobservations, mean = 0, sd = 1)
# ttest_summary = t.test(random_numbers)
# while (ttest_summary$p.value > 0.05){
# random_numbers = random_numbers + (rnorm(n = nobservations, mean = 0, sd = 1))
# ttest_summary = t.test(random_numbers)
# }
# print(ttest_summary$p.value)
#}
#values = tibble(result = replicate(n = 10000,
# double.sample()))
#sig_values = sum(values$result < 0.05)
#sig_values
#proportion = (sig_values/10000)
#proportionWhat do you conclude on the basis of this simulation? How bad is this kind of data-dependent policy?
What I conclude is that as you increase the “threshold” of the significance value and double your sample to search for a significant value, you greatly increase your chances of finding a false positive each time. This data-dependent policy is pretty bad because significance tests are based on the probabliity of observing a particular result (or more extreme) from a given population. By doubling the sample each time a non-significant value is observed, you are changing that probability which makes it much more likely to observe a false positive.