Simulating

September 23 + 25, 2024

Jo Hardin

Agenda 9/23/24

Why simulate?
What makes a good simulation?
Simulating to calculate probabilities

Goals of Simulating Complicated Models

The goal of simulating a complicated model is not only to create a program which will provide the desired results. We also hope to be able to write code such that:

The problem is broken down into small pieces.
The problem has checks in it to see what works (run the lines inside the functions!).
uses simple code (as often as possible).

Aside: a few R functions (`ifelse()`)

data.frame(value = c(-2:2)) |>
  mutate(abs_value = ifelse(value >=0, value, -value))  # abs val

  value abs_value
1    -2         2
2    -1         1
3     0         0
4     1         1
5     2         2

Aside: a few R functions (`case_when()`)

set.seed(4747)
diamonds |> select(carat, cut, color, price) |>
  sample_n(20) |>
  mutate(price_cat = case_when(
    price > 10000 ~ "expensive",
    price > 1500 ~ "medium", 
    TRUE ~ "inexpensive"))

# A tibble: 20 × 5
   carat cut       color price price_cat  
   <dbl> <ord>     <ord> <int> <chr>      
 1  1.23 Very Good F     10276 expensive  
 2  0.35 Premium   H       706 inexpensive
 3  0.7  Good      E      2782 medium     
 4  0.4  Ideal     D      1637 medium     
 5  0.53 Ideal     G      1255 inexpensive
 6  2.22 Ideal     G     14637 expensive  
 7  0.3  Ideal     G       878 inexpensive
 8  1.05 Ideal     H      4223 medium     
 9  0.53 Premium   E      1654 medium     
10  1.7  Ideal     H      7068 medium     
11  0.31 Good      E       698 inexpensive
12  0.31 Ideal     F       840 inexpensive
13  1.03 Ideal     H      4900 medium     
14  0.31 Premium   G       698 inexpensive
15  1.56 Premium   G      8858 medium     
16  1.71 Premium   G     11032 expensive  
17  1    Good      E      5345 medium     
18  1.86 Ideal     J     10312 expensive  
19  1.08 Very Good E      3726 medium     
20  0.31 Premium   E       698 inexpensive

Aside: a few R functions (`sample()`)

sampling, shuffling, and resampling: sample()

set.seed(47)
alph <- letters[1:10]
alph

 [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"

sample(alph, 5, replace = FALSE) # sample (from a population)

[1] "i" "b" "g" "d" "a"

sample(alph, 5, replace = TRUE) # sample (from a population)

[1] "j" "g" "f" "i" "f"

sample(alph, 10, replace = FALSE)  # shuffle

 [1] "f" "h" "i" "e" "g" "d" "c" "j" "b" "a"

sample(alph, 10, replace = TRUE)  # resample

 [1] "e" "j" "e" "b" "e" "c" "f" "a" "e" "a"

Why?

Three simulating methods are used for different purposes:

Monte Carlo methods - use repeated sampling from a population with known characteristics.
Randomization / Permutation methods - use shuffling (sampling without replacement from a sample) to test hypotheses of “no effect”.
Bootstrap methods - use resampling (sampling with replacement from a sample) to establish confidence intervals.

Aside: a few R functions (`set.seed()`)

What if we want to be able to generate the same random numbers (here on the interval from 0 to 1) over and over?

set.seed(4747)
runif(4, 0, 1) # random uniform numbers

[1] 0.1949071 0.3390270 0.5147919 0.4516470

set.seed(123)
runif(4, 0, 1) # random uniform numbers

[1] 0.2875775 0.7883051 0.4089769 0.8830174

set.seed(4747)
runif(4, 0, 1) # random uniform numbers

[1] 0.1949071 0.3390270 0.5147919 0.4516470

Small simulation example

Simulation as easier than calculus for estimating probabilities.
Simulation to understand complicated models.
Simulation to consider statistical methods with fewer assumptions.

Sally & Joan

Consider a situation where Sally and Joan plan to meet to study in their college campus center (Mosteller 1987; Baumer, Kaplan, and Horton 2021). They are both impatient people who will wait only 10 minutes for the other before leaving.

But their planning was incomplete. Sally said, “Meet me between 7 and 8 tonight at the student center.” When should Joan plan to arrive at the campus center? And what is the probability that they actually meet?

Simulate their meeting

Assume that Sally and Joan are both equally likely to arrive at the campus center anywhere between 7pm and 8pm.

vectorized
function
map()
for() loop

library(tictoc)
n <- 10000

tic()
meet_vect <- tibble(
  sally = runif(n, 0, max = 60),
  joan = runif(n, min = 0, max = 60),
  result = ifelse(
    abs(sally - joan) <= 10, "They meet", "They do not"
  )
)
toc()

0.002 sec elapsed

meet_func <- function(nada){
  tibble(
    sally = runif(1, min = 0, max = 60),
    joan = runif(1, min = 0, max = 60),
    result = ifelse(
    abs(sally - joan) <= 10, "They meet", "They do not"
  )
  )
}

tic()
meet_map <- 1:n |> 
  map(meet_func) |> 
  list_rbind()
toc()

2.291 sec elapsed

tic()
meet_for <- data.frame()
for(i in 1:n){
  meet_for <- meet_for |> rbind(
    tibble(
      sally = runif(1, min = 0, max = 60),
      joan = runif(1, min = 0, max = 60),
      result = ifelse(
    abs(sally - joan) <= 10, "They meet", "They do not"
  )))
}
toc()

3.837 sec elapsed

Results

The results themselves are equivalent. Differing values due to randomness in the simulation.

vectorized
map()
for() loop

meet_vect |> 
  group_by(result) |> 
  summarize(number = n()) |> 
  mutate(proprotion = number / sum(number))

# A tibble: 2 × 3
  result      number proprotion
  <chr>        <int>      <dbl>
1 They do not   6966      0.697
2 They meet     3034      0.303

meet_map |> 
  group_by(result) |> 
  summarize(number = n())|> 
  mutate(proprotion = number / sum(number))

# A tibble: 2 × 3
  result      number proprotion
  <chr>        <int>      <dbl>
1 They do not   7087      0.709
2 They meet     2913      0.291

meet_for |> 
  group_by(result) |> 
  summarize(number = n())|> 
  mutate(proprotion = number / sum(number))

# A tibble: 2 × 3
  result      number proprotion
  <chr>        <int>      <dbl>
1 They do not   6942      0.694
2 They meet     3058      0.306

Visualizing the meet up

meet_map |> 
  ggplot(aes(x = joan, y = sally, color = result)) +
  geom_point(alpha = 0.3) + 
  geom_abline(intercept = 10, slope = 1) + 
  geom_abline(intercept = -10, slope = 1) + 
  scale_color_brewer(palette = "Paired")

Simulation best practices

no magic numbers
comment your code
use informative names
set a seed for reproducibility

Bigger simulation example

Thoughts on simulating

break down the problem into small steps
check to see what works
simple is better

Roulette

A roulette wheel has 38 slots numbered 00, 0, and 1–36. Two are green, 18 are red, and 18 are black.
If a gambler bets based on color, the return on a $1 bet is $2
A gambler has $20, and will continuously bet $1 on red until they double their money (have $40) or lose the money they came with
What is the probability the gambler doubles their money?

Question: Without calculating probabilities, how could you design an experiment to estimate this probability?

The example is taken directly from Stat 279, Fall 2023 with Ciaran Evans.

Design the simulation

Step 1: Need a Roulette wheel! … and money
Step 2: Will spin the wheel
Step 3: Depending on the wheel, update the money.
* if spin is red: money + 1
* if spin is not red: money - 1
Step 4: Keep spinning until money = 40 or money = 0
Step 5: Repeat many times

Step 1: create a roulette wheel

wheel <- c(rep("green", 2), rep("black", 18), rep("red", 18))

wheel

 [1] "green" "green" "black" "black" "black" "black" "black" "black" "black"
[10] "black" "black" "black" "black" "black" "black" "black" "black" "black"
[19] "black" "black" "red"   "red"   "red"   "red"   "red"   "red"   "red"  
[28] "red"   "red"   "red"   "red"   "red"   "red"   "red"   "red"   "red"  
[37] "red"   "red"

rep() repeats a value the specified number of times
c() combines vectors (or scalars) into a single vector

Step 2: spin the wheel

spin <- sample(wheel, size = 1)

spin

[1] "red"

Step 3: update the money

money <- 20

spin <- sample(wheel, size = 1)
money <- ifelse(spin == "red", money + 1, money  - 1)

spin

[1] "red"

money

[1] 21

Step 4: keep spinning (function first)

spin_func <- function(money = 20, i){
  wheel <- c(rep("green", 2), rep("black", 18), rep("red", 18))
  spin <- sample(wheel, size = 1)
  money <- ifelse(spin == "red", money + 1, money  - 1)
  money
}

spin_func(10)

[1] 9

spin_func(spin_func(10))

[1] 10

Aside: the `accumulate()` function

In R, the accumulate() function will run f(f(f(f(x)))) as many times as asked.

1:10 |> 
  accumulate(`+`)

 [1]  1  3  6 10 15 21 28 36 45 55

1:10 |> 
  accumulate(`+`, .init = 47)

 [1]  47  48  50  53  57  62  68  75  83  92 102

Step 4: run multiple times

set.seed(4747)
1:20 |> accumulate(spin_func, .init = 20)

 [1] 20 21 20 19 20 19 20 19 18 19 18 17 16 15 16 17 18 19 18 19 18

Step 4: another function

game_func <- function(rep, money, iter){
  data.frame(interim_money = 1:iter |> accumulate(spin_func, .init = money),
             game = 0:iter,
             rep = rep)
}

game_func(1, 20, 3)

  interim_money game rep
1            20    0   1
2            19    1   1
3            18    2   1
4            17    3   1

Step 5: Repeat many times

Note that map() is specifically designed to work in vectorized ways, so it doesn’t have an easy while() companion. Which means that we need to make sure to iterate long enough to either lose or double our money.

output <- 1:20 |> map(game_func, money = 20, iter = 100000) |> 
  list_rbind()

output

      interim_money  game rep
1                20     0   1
2                19     1   1
3                18     2   1
4                19     3   1
5                18     4   1
6                17     5   1
7                16     6   1
8                15     7   1
9                14     8   1
10               13     9   1
11               12    10   1
12               13    11   1
13               12    12   1
14               13    13   1
15               12    14   1
16               13    15   1
17               12    16   1
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19               14    18   1
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12314          -699 12313   1
12315          -700 12314   1
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12317          -700 12316   1
12318          -701 12317   1
12319          -702 12318   1
12320          -703 12319   1
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12531          -720 12530   1
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13021          -738 13020   1
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13027          -736 13026   1
13028          -735 13027   1
13029          -734 13028   1
13030          -733 13029   1
13031          -732 13030   1
13032          -733 13031   1
13033          -732 13032   1
13034          -731 13033   1
13035          -730 13034   1
13036          -731 13035   1
13037          -732 13036   1
13038          -733 13037   1
13039          -732 13038   1
13040          -731 13039   1
13041          -732 13040   1
13042          -733 13041   1
13043          -732 13042   1
13044          -731 13043   1
13045          -732 13044   1
13046          -733 13045   1
13047          -732 13046   1
13048          -731 13047   1
13049          -732 13048   1
13050          -733 13049   1
13051          -732 13050   1
13052          -731 13051   1
13053          -732 13052   1
13054          -733 13053   1
13055          -732 13054   1
13056          -731 13055   1
13057          -730 13056   1
13058          -731 13057   1
13059          -730 13058   1
13060          -731 13059   1
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13063          -732 13062   1
13064          -731 13063   1
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13066          -733 13065   1
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13068          -735 13067   1
13069          -734 13068   1
13070          -735 13069   1
13071          -734 13070   1
13072          -735 13071   1
13073          -736 13072   1
13074          -737 13073   1
13075          -736 13074   1
13076          -737 13075   1
13077          -736 13076   1
13078          -735 13077   1
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13081          -734 13080   1
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13091          -738 13090   1
13092          -739 13091   1
13093          -738 13092   1
13094          -737 13093   1
13095          -736 13094   1
13096          -737 13095   1
13097          -738 13096   1
13098          -737 13097   1
13099          -738 13098   1
13100          -739 13099   1
13101          -740 13100   1
13102          -741 13101   1
13103          -740 13102   1
13104          -741 13103   1
13105          -742 13104   1
13106          -743 13105   1
13107          -744 13106   1
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13111          -748 13110   1
13112          -747 13111   1
13113          -746 13112   1
13114          -747 13113   1
13115          -748 13114   1
13116          -747 13115   1
13117          -746 13116   1
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13119          -746 13118   1
13120          -745 13119   1
13121          -744 13120   1
13122          -745 13121   1
13123          -746 13122   1
13124          -745 13123   1
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13145          -744 13144   1
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13161          -748 13160   1
13162          -747 13161   1
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13169          -744 13168   1
13170          -745 13169   1
13171          -746 13170   1
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13183          -744 13182   1
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13191          -744 13190   1
13192          -743 13191   1
13193          -744 13192   1
13194          -743 13193   1
13195          -742 13194   1
13196          -743 13195   1
13197          -742 13196   1
13198          -741 13197   1
13199          -740 13198   1
13200          -741 13199   1
13201          -742 13200   1
13202          -741 13201   1
13203          -740 13202   1
13204          -739 13203   1
13205          -740 13204   1
13206          -741 13205   1
13207          -742 13206   1
13208          -743 13207   1
13209          -744 13208   1
13210          -745 13209   1
13211          -744 13210   1
13212          -743 13211   1
13213          -742 13212   1
13214          -741 13213   1
13215          -742 13214   1
13216          -743 13215   1
13217          -742 13216   1
13218          -741 13217   1
13219          -740 13218   1
13220          -739 13219   1
13221          -740 13220   1
13222          -741 13221   1
13223          -742 13222   1
13224          -743 13223   1
13225          -742 13224   1
13226          -743 13225   1
13227          -742 13226   1
13228          -741 13227   1
13229          -742 13228   1
13230          -741 13229   1
13231          -742 13230   1
13232          -741 13231   1
13233          -742 13232   1
13234          -741 13233   1
13235          -742 13234   1
13236          -743 13235   1
13237          -744 13236   1
13238          -743 13237   1
13239          -744 13238   1
13240          -743 13239   1
13241          -744 13240   1
13242          -745 13241   1
13243          -744 13242   1
13244          -745 13243   1
13245          -746 13244   1
13246          -745 13245   1
13247          -746 13246   1
13248          -747 13247   1
13249          -746 13248   1
13250          -747 13249   1
13251          -748 13250   1
13252          -747 13251   1
13253          -748 13252   1
13254          -747 13253   1
13255          -748 13254   1
13256          -749 13255   1
13257          -750 13256   1
13258          -751 13257   1
13259          -752 13258   1
13260          -753 13259   1
13261          -752 13260   1
13262          -753 13261   1
13263          -752 13262   1
13264          -753 13263   1
13265          -754 13264   1
13266          -755 13265   1
13267          -754 13266   1
13268          -755 13267   1
13269          -756 13268   1
13270          -755 13269   1
13271          -754 13270   1
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13273          -756 13272   1
13274          -757 13273   1
13275          -758 13274   1
13276          -759 13275   1
13277          -760 13276   1
13278          -759 13277   1
13279          -760 13278   1
13280          -761 13279   1
13281          -760 13280   1
13282          -761 13281   1
13283          -762 13282   1
13284          -761 13283   1
13285          -760 13284   1
13286          -761 13285   1
13287          -762 13286   1
13288          -761 13287   1
13289          -762 13288   1
13290          -761 13289   1
13291          -762 13290   1
13292          -763 13291   1
13293          -764 13292   1
13294          -763 13293   1
13295          -764 13294   1
13296          -765 13295   1
13297          -764 13296   1
13298          -765 13297   1
13299          -764 13298   1
13300          -763 13299   1
13301          -762 13300   1
13302          -763 13301   1
13303          -762 13302   1
13304          -763 13303   1
13305          -762 13304   1
13306          -763 13305   1
13307          -762 13306   1
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13309          -764 13308   1
13310          -765 13309   1
13311          -764 13310   1
13312          -765 13311   1
13313          -766 13312   1
13314          -765 13313   1
13315          -764 13314   1
13316          -763 13315   1
13317          -762 13316   1
13318          -761 13317   1
13319          -760 13318   1
13320          -761 13319   1
13321          -760 13320   1
13322          -761 13321   1
13323          -760 13322   1
13324          -761 13323   1
13325          -762 13324   1
13326          -763 13325   1
13327          -764 13326   1
13328          -765 13327   1
13329          -766 13328   1
13330          -765 13329   1
13331          -764 13330   1
13332          -763 13331   1
13333          -764 13332   1
13334          -763 13333   1
13335          -764 13334   1
13336          -763 13335   1
13337          -764 13336   1
13338          -765 13337   1
13339          -764 13338   1
13340          -763 13339   1
13341          -764 13340   1
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13343          -764 13342   1
13344          -765 13343   1
13345          -764 13344   1
13346          -765 13345   1
13347          -764 13346   1
13348          -763 13347   1
13349          -764 13348   1
13350          -765 13349   1
13351          -766 13350   1
13352          -767 13351   1
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13354          -767 13353   1
13355          -768 13354   1
13356          -767 13355   1
13357          -766 13356   1
13358          -767 13357   1
13359          -768 13358   1
13360          -769 13359   1
13361          -768 13360   1
13362          -769 13361   1
13363          -768 13362   1
13364          -769 13363   1
13365          -770 13364   1
13366          -771 13365   1
13367          -772 13366   1
13368          -771 13367   1
13369          -770 13368   1
13370          -771 13369   1
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13372          -771 13371   1
13373          -770 13372   1
13374          -769 13373   1
13375          -768 13374   1
13376          -767 13375   1
13377          -766 13376   1
13378          -767 13377   1
13379          -766 13378   1
13380          -767 13379   1
13381          -768 13380   1
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13384          -767 13383   1
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13389          -768 13388   1
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13393          -766 13392   1
13394          -765 13393   1
13395          -764 13394   1
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13397          -766 13396   1
13398          -767 13397   1
13399          -766 13398   1
13400          -767 13399   1
13401          -766 13400   1
13402          -765 13401   1
13403          -764 13402   1
13404          -763 13403   1
13405          -762 13404   1
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13410          -763 13409   1
13411          -764 13410   1
13412          -765 13411   1
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13414          -765 13413   1
13415          -766 13414   1
13416          -767 13415   1
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13419          -768 13418   1
13420          -767 13419   1
13421          -768 13420   1
13422          -767 13421   1
13423          -766 13422   1
13424          -767 13423   1
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13427          -768 13426   1
13428          -767 13427   1
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13432          -767 13431   1
13433          -766 13432   1
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13435          -766 13434   1
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13437          -766 13436   1
13438          -767 13437   1
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13441          -766 13440   1
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13445          -770 13444   1
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13450          -769 13449   1
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13460          -767 13459   1
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13464          -767 13463   1
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13468          -767 13467   1
13469          -766 13468   1
13470          -765 13469   1
13471          -766 13470   1
13472          -765 13471   1
13473          -766 13472   1
13474          -765 13473   1
13475          -764 13474   1
13476          -765 13475   1
13477          -764 13476   1
13478          -765 13477   1
13479          -764 13478   1
13480          -763 13479   1
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13500          -759 13499   1
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13531          -764 13530   1
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13533          -766 13532   1
13534          -767 13533   1
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13538          -769 13537   1
13539          -770 13538   1
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13575          -776 13574   1
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13591          -776 13590   1
13592          -777 13591   1
13593          -778 13592   1
13594          -777 13593   1
13595          -778 13594   1
13596          -779 13595   1
13597          -780 13596   1
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13599          -778 13598   1
13600          -777 13599   1
13601          -776 13600   1
13602          -775 13601   1
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13606          -773 13605   1
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13610          -769 13609   1
13611          -768 13610   1
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13619          -766 13618   1
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13622          -765 13621   1
13623          -766 13622   1
13624          -767 13623   1
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13626          -767 13625   1
13627          -766 13626   1
13628          -765 13627   1
13629          -766 13628   1
13630          -767 13629   1
13631          -768 13630   1
13632          -767 13631   1
13633          -766 13632   1
13634          -767 13633   1
13635          -766 13634   1
13636          -767 13635   1
13637          -768 13636   1
13638          -769 13637   1
13639          -768 13638   1
13640          -769 13639   1
13641          -770 13640   1
13642          -771 13641   1
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13644          -771 13643   1
13645          -770 13644   1
13646          -771 13645   1
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13648          -771 13647   1
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13650          -769 13649   1
13651          -770 13650   1
13652          -769 13651   1
13653          -768 13652   1
13654          -769 13653   1
13655          -768 13654   1
13656          -767 13655   1
13657          -766 13656   1
13658          -765 13657   1
13659          -764 13658   1
13660          -765 13659   1
13661          -764 13660   1
13662          -765 13661   1
13663          -766 13662   1
13664          -765 13663   1
13665          -766 13664   1
13666          -765 13665   1
13667          -764 13666   1
13668          -763 13667   1
13669          -762 13668   1
13670          -761 13669   1
13671          -762 13670   1
13672          -763 13671   1
13673          -764 13672   1
13674          -763 13673   1
13675          -764 13674   1
13676          -763 13675   1
13677          -764 13676   1
13678          -763 13677   1
13679          -762 13678   1
13680          -763 13679   1
13681          -762 13680   1
13682          -761 13681   1
13683          -760 13682   1
13684          -761 13683   1
13685          -762 13684   1
13686          -761 13685   1
13687          -760 13686   1
13688          -759 13687   1
13689          -758 13688   1
13690          -759 13689   1
13691          -760 13690   1
13692          -759 13691   1
13693          -758 13692   1
13694          -757 13693   1
13695          -758 13694   1
13696          -757 13695   1
13697          -756 13696   1
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13699          -756 13698   1
13700          -755 13699   1
13701          -754 13700   1
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13704          -751 13703   1
13705          -752 13704   1
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13707          -750 13706   1
13708          -749 13707   1
13709          -750 13708   1
13710          -751 13709   1
13711          -750 13710   1
13712          -751 13711   1
13713          -752 13712   1
13714          -751 13713   1
13715          -750 13714   1
13716          -749 13715   1
13717          -750 13716   1
13718          -751 13717   1
13719          -752 13718   1
13720          -753 13719   1
13721          -754 13720   1
13722          -753 13721   1
13723          -754 13722   1
13724          -755 13723   1
13725          -754 13724   1
13726          -755 13725   1
13727          -754 13726   1
13728          -755 13727   1
13729          -754 13728   1
13730          -753 13729   1
13731          -752 13730   1
13732          -751 13731   1
13733          -752 13732   1
13734          -751 13733   1
13735          -752 13734   1
13736          -751 13735   1
13737          -750 13736   1
13738          -751 13737   1
13739          -752 13738   1
13740          -753 13739   1
13741          -752 13740   1
13742          -753 13741   1
13743          -754 13742   1
13744          -753 13743   1
13745          -752 13744   1
13746          -753 13745   1
13747          -754 13746   1
13748          -755 13747   1
13749          -756 13748   1
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17353          -936 17352   1
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17355          -938 17354   1
17356          -937 17355   1
17357          -936 17356   1
17358          -937 17357   1
17359          -936 17358   1
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17361          -936 17360   1
17362          -935 17361   1
17363          -936 17362   1
17364          -935 17363   1
17365          -934 17364   1
17366          -935 17365   1
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17377          -936 17376   1
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17394          -947 17393   1
17395          -948 17394   1
17396          -947 17395   1
17397          -946 17396   1
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17401          -946 17400   1
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17410          -949 17409   1
17411          -950 17410   1
17412          -951 17411   1
17413          -950 17412   1
17414          -949 17413   1
17415          -948 17414   1
17416          -949 17415   1
17417          -948 17416   1
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17433          -946 17432   1
17434          -947 17433   1
17435          -946 17434   1
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17437          -944 17436   1
17438          -945 17437   1
17439          -946 17438   1
17440          -947 17439   1
17441          -948 17440   1
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17443          -950 17442   1
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17445          -950 17444   1
17446          -949 17445   1
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17465          -950 17464   1
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17503          -954 17502   1
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17511          -958 17510   1
17512          -957 17511   1
17513          -958 17512   1
17514          -959 17513   1
17515          -960 17514   1
17516          -961 17515   1
17517          -960 17516   1
17518          -961 17517   1
17519          -962 17518   1
17520          -961 17519   1
17521          -962 17520   1
17522          -961 17521   1
17523          -960 17522   1
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17528          -959 17527   1
17529          -960 17528   1
17530          -961 17529   1
17531          -962 17530   1
17532          -961 17531   1
17533          -960 17532   1
17534          -961 17533   1
17535          -962 17534   1
17536          -963 17535   1
17537          -964 17536   1
17538          -963 17537   1
17539          -964 17538   1
17540          -965 17539   1
17541          -964 17540   1
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17555          -966 17554   1
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17560          -965 17559   1
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17562          -967 17561   1
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17564          -967 17563   1
17565          -966 17564   1
17566          -965 17565   1
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17569          -964 17568   1
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17576          -965 17575   1
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18050         -1037 18049   1
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18052         -1037 18051   1
18053         -1036 18052   1
18054         -1037 18053   1
18055         -1036 18054   1
18056         -1037 18055   1
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18139         -1036 18138   1
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18298         -1057 18297   1
18299         -1058 18298   1
18300         -1059 18299   1
18301         -1060 18300   1
18302         -1061 18301   1
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18304         -1061 18303   1
18305         -1062 18304   1
18306         -1061 18305   1
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18311         -1062 18310   1
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18313         -1062 18312   1
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18315         -1064 18314   1
18316         -1063 18315   1
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18320         -1065 18319   1
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18364         -1063 18363   1
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18399         -1070 18398   1
18400         -1071 18399   1
18401         -1072 18400   1
18402         -1073 18401   1
18403         -1074 18402   1
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18405         -1076 18404   1
18406         -1075 18405   1
18407         -1076 18406   1
18408         -1075 18407   1
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18410         -1077 18409   1
18411         -1078 18410   1
18412         -1077 18411   1
18413         -1076 18412   1
18414         -1077 18413   1
18415         -1078 18414   1
18416         -1079 18415   1
18417         -1078 18416   1
18418         -1077 18417   1
18419         -1076 18418   1
18420         -1077 18419   1
18421         -1078 18420   1
18422         -1079 18421   1
18423         -1078 18422   1
18424         -1079 18423   1
18425         -1078 18424   1
18426         -1077 18425   1
18427         -1076 18426   1
18428         -1077 18427   1
18429         -1078 18428   1
18430         -1079 18429   1
18431         -1080 18430   1
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18434         -1077 18433   1
18435         -1076 18434   1
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18438         -1073 18437   1
18439         -1074 18438   1
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18460         -1075 18459   1
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18465         -1076 18464   1
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18468         -1077 18467   1
18469         -1078 18468   1
18470         -1079 18469   1
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18472         -1081 18471   1
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18474         -1083 18473   1
18475         -1084 18474   1
18476         -1085 18475   1
18477         -1086 18476   1
18478         -1087 18477   1
18479         -1086 18478   1
18480         -1087 18479   1
18481         -1088 18480   1
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18484         -1089 18483   1
18485         -1088 18484   1
18486         -1089 18485   1
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18488         -1091 18487   1
18489         -1092 18488   1
18490         -1091 18489   1
18491         -1092 18490   1
18492         -1091 18491   1
18493         -1092 18492   1
18494         -1093 18493   1
18495         -1094 18494   1
18496         -1095 18495   1
18497         -1096 18496   1
18498         -1095 18497   1
18499         -1096 18498   1
18500         -1095 18499   1
18501         -1096 18500   1
18502         -1097 18501   1
18503         -1096 18502   1
18504         -1097 18503   1
18505         -1098 18504   1
18506         -1097 18505   1
18507         -1096 18506   1
18508         -1097 18507   1
18509         -1096 18508   1
18510         -1097 18509   1
18511         -1096 18510   1
18512         -1097 18511   1
18513         -1096 18512   1
18514         -1097 18513   1
18515         -1096 18514   1
18516         -1095 18515   1
18517         -1096 18516   1
18518         -1095 18517   1
18519         -1096 18518   1
18520         -1097 18519   1
18521         -1096 18520   1
18522         -1097 18521   1
18523         -1098 18522   1
18524         -1099 18523   1
18525         -1098 18524   1
18526         -1099 18525   1
18527         -1100 18526   1
18528         -1099 18527   1
18529         -1098 18528   1
18530         -1099 18529   1
18531         -1098 18530   1
18532         -1097 18531   1
18533         -1098 18532   1
18534         -1097 18533   1
18535         -1096 18534   1
18536         -1097 18535   1
18537         -1098 18536   1
18538         -1097 18537   1
18539         -1098 18538   1
18540         -1099 18539   1
18541         -1100 18540   1
18542         -1101 18541   1
18543         -1102 18542   1
18544         -1101 18543   1
18545         -1100 18544   1
18546         -1099 18545   1
18547         -1098 18546   1
18548         -1097 18547   1
18549         -1098 18548   1
18550         -1097 18549   1
18551         -1098 18550   1
18552         -1097 18551   1
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18554         -1097 18553   1
18555         -1098 18554   1
18556         -1099 18555   1
18557         -1098 18556   1
18558         -1099 18557   1
18559         -1098 18558   1
18560         -1097 18559   1
18561         -1096 18560   1
18562         -1097 18561   1
18563         -1096 18562   1
18564         -1097 18563   1
18565         -1096 18564   1
18566         -1097 18565   1
18567         -1098 18566   1
18568         -1097 18567   1
18569         -1096 18568   1
18570         -1097 18569   1
18571         -1096 18570   1
18572         -1095 18571   1
18573         -1096 18572   1
18574         -1095 18573   1
18575         -1096 18574   1
18576         -1097 18575   1
18577         -1096 18576   1
18578         -1095 18577   1
18579         -1096 18578   1
18580         -1095 18579   1
18581         -1096 18580   1
18582         -1097 18581   1
18583         -1098 18582   1
18584         -1097 18583   1
18585         -1096 18584   1
18586         -1095 18585   1
18587         -1096 18586   1
18588         -1095 18587   1
18589         -1094 18588   1
18590         -1095 18589   1
18591         -1096 18590   1
18592         -1095 18591   1
18593         -1096 18592   1
18594         -1095 18593   1
18595         -1094 18594   1
18596         -1095 18595   1
18597         -1096 18596   1
18598         -1095 18597   1
18599         -1096 18598   1
18600         -1097 18599   1
18601         -1098 18600   1
18602         -1097 18601   1
18603         -1096 18602   1
18604         -1097 18603   1
18605         -1098 18604   1
18606         -1099 18605   1
18607         -1100 18606   1
18608         -1099 18607   1
18609         -1098 18608   1
18610         -1099 18609   1
18611         -1100 18610   1
18612         -1099 18611   1
18613         -1100 18612   1
18614         -1099 18613   1
18615         -1098 18614   1
18616         -1097 18615   1
18617         -1098 18616   1
18618         -1099 18617   1
18619         -1098 18618   1
18620         -1097 18619   1
18621         -1098 18620   1
18622         -1099 18621   1
18623         -1100 18622   1
18624         -1101 18623   1
18625         -1100 18624   1
18626         -1099 18625   1
18627         -1100 18626   1
18628         -1101 18627   1
18629         -1100 18628   1
18630         -1099 18629   1
18631         -1100 18630   1
18632         -1099 18631   1
18633         -1098 18632   1
18634         -1097 18633   1
18635         -1098 18634   1
18636         -1097 18635   1
18637         -1098 18636   1
18638         -1099 18637   1
18639         -1100 18638   1
18640         -1099 18639   1
18641         -1100 18640   1
18642         -1099 18641   1
18643         -1100 18642   1
18644         -1101 18643   1
18645         -1102 18644   1
18646         -1101 18645   1
18647         -1100 18646   1
18648         -1101 18647   1
18649         -1100 18648   1
18650         -1101 18649   1
18651         -1102 18650   1
18652         -1101 18651   1
18653         -1100 18652   1
18654         -1101 18653   1
18655         -1102 18654   1
18656         -1101 18655   1
18657         -1102 18656   1
18658         -1101 18657   1
18659         -1102 18658   1
18660         -1103 18659   1
18661         -1104 18660   1
18662         -1105 18661   1
18663         -1106 18662   1
18664         -1107 18663   1
18665         -1106 18664   1
18666         -1107 18665   1
18667         -1108 18666   1
18668         -1109 18667   1
18669         -1110 18668   1
18670         -1111 18669   1
18671         -1112 18670   1
18672         -1111 18671   1
18673         -1110 18672   1
18674         -1111 18673   1
18675         -1110 18674   1
18676         -1111 18675   1
18677         -1110 18676   1
18678         -1111 18677   1
18679         -1112 18678   1
18680         -1113 18679   1
18681         -1114 18680   1
18682         -1113 18681   1
18683         -1112 18682   1
18684         -1113 18683   1
18685         -1112 18684   1
18686         -1111 18685   1
18687         -1112 18686   1
18688         -1113 18687   1
18689         -1112 18688   1
18690         -1113 18689   1
18691         -1112 18690   1
18692         -1113 18691   1
18693         -1112 18692   1
18694         -1113 18693   1
18695         -1112 18694   1
18696         -1113 18695   1
18697         -1114 18696   1
18698         -1115 18697   1
18699         -1116 18698   1
18700         -1117 18699   1
18701         -1118 18700   1
18702         -1119 18701   1
18703         -1120 18702   1
18704         -1121 18703   1
18705         -1122 18704   1
18706         -1121 18705   1
18707         -1122 18706   1
18708         -1123 18707   1
18709         -1122 18708   1
18710         -1123 18709   1
18711         -1124 18710   1
18712         -1125 18711   1
18713         -1126 18712   1
18714         -1125 18713   1
18715         -1126 18714   1
18716         -1125 18715   1
18717         -1126 18716   1
18718         -1127 18717   1
18719         -1126 18718   1
18720         -1127 18719   1
18721         -1126 18720   1
18722         -1127 18721   1
18723         -1126 18722   1
18724         -1125 18723   1
18725         -1124 18724   1
18726         -1125 18725   1
18727         -1124 18726   1
18728         -1125 18727   1
18729         -1124 18728   1
18730         -1123 18729   1
18731         -1122 18730   1
18732         -1123 18731   1
18733         -1124 18732   1
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31032         -1991 31031   1
31033         -1990 31032   1
31034         -1989 31033   1
31035         -1988 31034   1
31036         -1989 31035   1
31037         -1988 31036   1
31038         -1989 31037   1
31039         -1990 31038   1
31040         -1991 31039   1
31041         -1992 31040   1
31042         -1993 31041   1
31043         -1992 31042   1
31044         -1993 31043   1
31045         -1992 31044   1
31046         -1993 31045   1
31047         -1992 31046   1
31048         -1993 31047   1
31049         -1994 31048   1
31050         -1995 31049   1
31051         -1996 31050   1
31052         -1997 31051   1
31053         -1998 31052   1
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31055         -1996 31054   1
31056         -1997 31055   1
31057         -1998 31056   1
31058         -1999 31057   1
31059         -2000 31058   1
31060         -2001 31059   1
31061         -2002 31060   1
31062         -2001 31061   1
31063         -2000 31062   1
31064         -2001 31063   1
31065         -2002 31064   1
31066         -2001 31065   1
31067         -2002 31066   1
31068         -2003 31067   1
31069         -2004 31068   1
31070         -2003 31069   1
31071         -2004 31070   1
31072         -2003 31071   1
31073         -2002 31072   1
31074         -2003 31073   1
31075         -2002 31074   1
31076         -2003 31075   1
31077         -2002 31076   1
31078         -2003 31077   1
31079         -2002 31078   1
31080         -2001 31079   1
31081         -2000 31080   1
31082         -2001 31081   1
31083         -2002 31082   1
31084         -2001 31083   1
31085         -2000 31084   1
31086         -2001 31085   1
31087         -2002 31086   1
31088         -2003 31087   1
31089         -2002 31088   1
31090         -2001 31089   1
31091         -2002 31090   1
31092         -2001 31091   1
31093         -2000 31092   1
31094         -1999 31093   1
31095         -2000 31094   1
31096         -2001 31095   1
31097         -2002 31096   1
31098         -2003 31097   1
31099         -2004 31098   1
31100         -2005 31099   1
31101         -2006 31100   1
31102         -2007 31101   1
31103         -2006 31102   1
31104         -2007 31103   1
31105         -2008 31104   1
31106         -2009 31105   1
31107         -2008 31106   1
31108         -2007 31107   1
31109         -2008 31108   1
31110         -2009 31109   1
31111         -2010 31110   1
31112         -2009 31111   1
31113         -2008 31112   1
31114         -2009 31113   1
31115         -2008 31114   1
31116         -2009 31115   1
31117         -2008 31116   1
31118         -2009 31117   1
31119         -2010 31118   1
31120         -2011 31119   1
31121         -2012 31120   1
31122         -2013 31121   1
31123         -2014 31122   1
31124         -2013 31123   1
31125         -2012 31124   1
31126         -2011 31125   1
31127         -2012 31126   1
31128         -2011 31127   1
31129         -2010 31128   1
31130         -2011 31129   1
31131         -2012 31130   1
31132         -2013 31131   1
31133         -2012 31132   1
31134         -2013 31133   1
31135         -2014 31134   1
31136         -2015 31135   1
31137         -2014 31136   1
31138         -2013 31137   1
31139         -2012 31138   1
31140         -2011 31139   1
31141         -2010 31140   1
31142         -2009 31141   1
31143         -2010 31142   1
31144         -2009 31143   1
31145         -2010 31144   1
31146         -2011 31145   1
31147         -2012 31146   1
31148         -2011 31147   1
31149         -2012 31148   1
31150         -2013 31149   1
31151         -2014 31150   1
31152         -2013 31151   1
31153         -2014 31152   1
31154         -2013 31153   1
31155         -2014 31154   1
31156         -2013 31155   1
31157         -2012 31156   1
31158         -2011 31157   1
31159         -2010 31158   1
31160         -2009 31159   1
31161         -2008 31160   1
31162         -2007 31161   1
31163         -2006 31162   1
31164         -2005 31163   1
31165         -2004 31164   1
31166         -2003 31165   1
31167         -2004 31166   1
31168         -2003 31167   1
31169         -2004 31168   1
31170         -2005 31169   1
31171         -2006 31170   1
31172         -2007 31171   1
31173         -2008 31172   1
31174         -2007 31173   1
31175         -2008 31174   1
31176         -2007 31175   1
31177         -2008 31176   1
31178         -2007 31177   1
31179         -2008 31178   1
31180         -2009 31179   1
31181         -2008 31180   1
31182         -2009 31181   1
31183         -2008 31182   1
31184         -2007 31183   1
31185         -2006 31184   1
31186         -2007 31185   1
31187         -2008 31186   1
31188         -2007 31187   1
31189         -2008 31188   1
31190         -2007 31189   1
31191         -2008 31190   1
31192         -2007 31191   1
31193         -2006 31192   1
31194         -2005 31193   1
31195         -2004 31194   1
31196         -2005 31195   1
31197         -2004 31196   1
31198         -2005 31197   1
31199         -2006 31198   1
31200         -2007 31199   1
31201         -2008 31200   1
31202         -2009 31201   1
31203         -2008 31202   1
31204         -2009 31203   1
31205         -2008 31204   1
31206         -2009 31205   1
31207         -2008 31206   1
31208         -2007 31207   1
31209         -2008 31208   1
31210         -2007 31209   1
31211         -2008 31210   1
31212         -2009 31211   1
31213         -2010 31212   1
31214         -2009 31213   1
31215         -2008 31214   1
31216         -2007 31215   1
31217         -2006 31216   1
31218         -2005 31217   1
31219         -2006 31218   1
31220         -2005 31219   1
31221         -2004 31220   1
31222         -2005 31221   1
31223         -2004 31222   1
31224         -2005 31223   1
31225         -2004 31224   1
31226         -2003 31225   1
31227         -2002 31226   1
31228         -2001 31227   1
31229         -2002 31228   1
31230         -2003 31229   1
31231         -2004 31230   1
31232         -2005 31231   1
31233         -2006 31232   1
31234         -2005 31233   1
31235         -2006 31234   1
31236         -2007 31235   1
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31240         -2007 31239   1
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31250         -2007 31249   1
31251         -2008 31250   1
31252         -2009 31251   1
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31254         -2009 31253   1
31255         -2010 31254   1
31256         -2011 31255   1
31257         -2010 31256   1
31258         -2011 31257   1
31259         -2010 31258   1
31260         -2009 31259   1
31261         -2010 31260   1
31262         -2009 31261   1
31263         -2008 31262   1
31264         -2007 31263   1
31265         -2008 31264   1
31266         -2009 31265   1
31267         -2010 31266   1
31268         -2009 31267   1
31269         -2010 31268   1
31270         -2011 31269   1
31271         -2012 31270   1
31272         -2011 31271   1
31273         -2010 31272   1
31274         -2011 31273   1
31275         -2010 31274   1
31276         -2011 31275   1
31277         -2010 31276   1
31278         -2011 31277   1
31279         -2012 31278   1
31280         -2013 31279   1
31281         -2012 31280   1
31282         -2013 31281   1
31283         -2012 31282   1
31284         -2013 31283   1
31285         -2014 31284   1
31286         -2013 31285   1
31287         -2012 31286   1
31288         -2013 31287   1
31289         -2014 31288   1
31290         -2015 31289   1
31291         -2016 31290   1
31292         -2015 31291   1
31293         -2014 31292   1
31294         -2013 31293   1
31295         -2014 31294   1
31296         -2013 31295   1
31297         -2012 31296   1
31298         -2013 31297   1
31299         -2012 31298   1
31300         -2011 31299   1
31301         -2010 31300   1
31302         -2011 31301   1
31303         -2010 31302   1
31304         -2009 31303   1
31305         -2008 31304   1
31306         -2007 31305   1
31307         -2006 31306   1
31308         -2005 31307   1
31309         -2004 31308   1
31310         -2005 31309   1
31311         -2006 31310   1
31312         -2005 31311   1
31313         -2004 31312   1
31314         -2003 31313   1
31315         -2002 31314   1
31316         -2001 31315   1
31317         -2002 31316   1
31318         -2001 31317   1
31319         -2000 31318   1
31320         -1999 31319   1
31321         -2000 31320   1
31322         -2001 31321   1
31323         -2002 31322   1
31324         -2001 31323   1
31325         -2000 31324   1
31326         -2001 31325   1
31327         -2002 31326   1
31328         -2003 31327   1
31329         -2002 31328   1
31330         -2001 31329   1
31331         -2000 31330   1
31332         -2001 31331   1
31333         -2000 31332   1
31334         -1999 31333   1
31335         -2000 31334   1
31336         -1999 31335   1
31337         -2000 31336   1
31338         -2001 31337   1
31339         -2000 31338   1
31340         -1999 31339   1
31341         -1998 31340   1
31342         -1997 31341   1
31343         -1998 31342   1
31344         -1999 31343   1
31345         -1998 31344   1
31346         -1999 31345   1
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31350         -1999 31349   1
31351         -1998 31350   1
31352         -1999 31351   1
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31354         -1999 31353   1
31355         -2000 31354   1
31356         -1999 31355   1
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31358         -1999 31357   1
31359         -1998 31358   1
31360         -1997 31359   1
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31362         -1997 31361   1
31363         -1996 31362   1
31364         -1997 31363   1
31365         -1998 31364   1
31366         -1997 31365   1
31367         -1998 31366   1
31368         -1999 31367   1
31369         -2000 31368   1
31370         -1999 31369   1
31371         -1998 31370   1
31372         -1997 31371   1
31373         -1998 31372   1
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31375         -1998 31374   1
31376         -1997 31375   1
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31380         -1997 31379   1
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31390         -1995 31389   1
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31398         -1995 31397   1
31399         -1994 31398   1
31400         -1995 31399   1
31401         -1996 31400   1
31402         -1995 31401   1
31403         -1996 31402   1
31404         -1995 31403   1
31405         -1994 31404   1
31406         -1995 31405   1
31407         -1994 31406   1
31408         -1995 31407   1
31409         -1994 31408   1
31410         -1995 31409   1
31411         -1994 31410   1
31412         -1993 31411   1
31413         -1994 31412   1
31414         -1993 31413   1
31415         -1994 31414   1
31416         -1993 31415   1
31417         -1992 31416   1
31418         -1991 31417   1
31419         -1990 31418   1
31420         -1989 31419   1
31421         -1988 31420   1
31422         -1989 31421   1
31423         -1988 31422   1
31424         -1989 31423   1
31425         -1988 31424   1
31426         -1987 31425   1
31427         -1988 31426   1
31428         -1989 31427   1
31429         -1990 31428   1
31430         -1989 31429   1
31431         -1988 31430   1
31432         -1987 31431   1
31433         -1986 31432   1
31434         -1985 31433   1
31435         -1984 31434   1
31436         -1983 31435   1
31437         -1982 31436   1
31438         -1983 31437   1
31439         -1982 31438   1
31440         -1983 31439   1
31441         -1982 31440   1
31442         -1983 31441   1
31443         -1982 31442   1
31444         -1981 31443   1
31445         -1982 31444   1
31446         -1981 31445   1
31447         -1980 31446   1
31448         -1979 31447   1
31449         -1980 31448   1
31450         -1981 31449   1
31451         -1980 31450   1
31452         -1981 31451   1
31453         -1980 31452   1
31454         -1979 31453   1
31455         -1980 31454   1
31456         -1979 31455   1
31457         -1980 31456   1
31458         -1979 31457   1
31459         -1978 31458   1
31460         -1977 31459   1
31461         -1978 31460   1
31462         -1977 31461   1
31463         -1976 31462   1
31464         -1977 31463   1
31465         -1976 31464   1
31466         -1975 31465   1
31467         -1976 31466   1
31468         -1977 31467   1
31469         -1976 31468   1
31470         -1975 31469   1
31471         -1974 31470   1
31472         -1973 31471   1
31473         -1974 31472   1
31474         -1975 31473   1
31475         -1974 31474   1
31476         -1973 31475   1
31477         -1972 31476   1
31478         -1973 31477   1
31479         -1972 31478   1
31480         -1971 31479   1
31481         -1970 31480   1
31482         -1969 31481   1
31483         -1968 31482   1
31484         -1969 31483   1
31485         -1970 31484   1
31486         -1971 31485   1
31487         -1970 31486   1
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31489         -1970 31488   1
31490         -1971 31489   1
31491         -1970 31490   1
31492         -1971 31491   1
31493         -1972 31492   1
31494         -1973 31493   1
31495         -1974 31494   1
31496         -1973 31495   1
31497         -1974 31496   1
31498         -1973 31497   1
31499         -1972 31498   1
31500         -1973 31499   1
31501         -1972 31500   1
31502         -1973 31501   1
31503         -1972 31502   1
31504         -1971 31503   1
31505         -1972 31504   1
31506         -1973 31505   1
31507         -1972 31506   1
31508         -1973 31507   1
31509         -1974 31508   1
31510         -1973 31509   1
31511         -1974 31510   1
31512         -1975 31511   1
31513         -1976 31512   1
31514         -1977 31513   1
31515         -1978 31514   1
31516         -1979 31515   1
31517         -1978 31516   1
31518         -1977 31517   1
31519         -1976 31518   1
31520         -1975 31519   1
31521         -1974 31520   1
31522         -1975 31521   1
31523         -1974 31522   1
31524         -1975 31523   1
31525         -1976 31524   1
31526         -1975 31525   1
31527         -1976 31526   1
31528         -1977 31527   1
31529         -1978 31528   1
31530         -1977 31529   1
31531         -1978 31530   1
31532         -1979 31531   1
31533         -1978 31532   1
31534         -1977 31533   1
31535         -1978 31534   1
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31538         -1977 31537   1
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31540         -1979 31539   1
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31544         -1979 31543   1
31545         -1978 31544   1
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31550         -1977 31549   1
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31554         -1979 31553   1
31555         -1978 31554   1
31556         -1977 31555   1
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31560         -1977 31559   1
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31563         -1978 31562   1
31564         -1977 31563   1
31565         -1978 31564   1
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31570         -1979 31569   1
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31588         -1981 31587   1
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31590         -1981 31589   1
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31592         -1981 31591   1
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31594         -1981 31593   1
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31598         -1983 31597   1
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31600         -1985 31599   1
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31602         -1985 31601   1
31603         -1986 31602   1
31604         -1987 31603   1
31605         -1988 31604   1
31606         -1989 31605   1
31607         -1990 31606   1
31608         -1991 31607   1
31609         -1990 31608   1
31610         -1989 31609   1
31611         -1990 31610   1
31612         -1991 31611   1
31613         -1992 31612   1
31614         -1993 31613   1
31615         -1992 31614   1
31616         -1993 31615   1
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31618         -1993 31617   1
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31620         -1993 31619   1
31621         -1994 31620   1
31622         -1995 31621   1
31623         -1996 31622   1
31624         -1995 31623   1
31625         -1996 31624   1
31626         -1995 31625   1
31627         -1994 31626   1
31628         -1993 31627   1
31629         -1992 31628   1
31630         -1991 31629   1
31631         -1992 31630   1
31632         -1991 31631   1
31633         -1990 31632   1
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31635         -1988 31634   1
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31638         -1987 31637   1
31639         -1986 31638   1
31640         -1985 31639   1
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31662         -1989 31661   1
31663         -1990 31662   1
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31727         -1990 31726   1
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31774         -1995 31773   1
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31900         -1983 31899   1
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31902         -1981 31901   1
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31906         -1979 31905   1
31907         -1978 31906   1
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31910         -1975 31909   1
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31912         -1975 31911   1
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31914         -1975 31913   1
31915         -1974 31914   1
31916         -1973 31915   1
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31920         -1975 31919   1
31921         -1974 31920   1
31922         -1973 31921   1
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31925         -1972 31924   1
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31927         -1974 31926   1
31928         -1975 31927   1
31929         -1974 31928   1
31930         -1975 31929   1
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31932         -1975 31931   1
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31934         -1973 31933   1
31935         -1974 31934   1
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31939         -1974 31938   1
31940         -1975 31939   1
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31944         -1975 31943   1
31945         -1976 31944   1
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31950         -1977 31949   1
31951         -1978 31950   1
31952         -1979 31951   1
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31954         -1979 31953   1
31955         -1978 31954   1
31956         -1979 31955   1
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31960         -1977 31959   1
31961         -1976 31960   1
31962         -1977 31961   1
31963         -1978 31962   1
31964         -1979 31963   1
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31966         -1981 31965   1
31967         -1982 31966   1
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31969         -1984 31968   1
31970         -1983 31969   1
31971         -1984 31970   1
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31975         -1982 31974   1
31976         -1983 31975   1
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31978         -1981 31977   1
31979         -1982 31978   1
31980         -1981 31979   1
31981         -1982 31980   1
31982         -1983 31981   1
31983         -1984 31982   1
31984         -1985 31983   1
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31986         -1985 31985   1
31987         -1986 31986   1
31988         -1985 31987   1
31989         -1986 31988   1
31990         -1985 31989   1
31991         -1984 31990   1
31992         -1983 31991   1
31993         -1982 31992   1
31994         -1983 31993   1
31995         -1982 31994   1
31996         -1983 31995   1
31997         -1982 31996   1
31998         -1981 31997   1
31999         -1982 31998   1
32000         -1981 31999   1
32001         -1980 32000   1
32002         -1981 32001   1
32003         -1980 32002   1
32004         -1979 32003   1
32005         -1980 32004   1
32006         -1979 32005   1
32007         -1980 32006   1
32008         -1979 32007   1
32009         -1978 32008   1
32010         -1977 32009   1
32011         -1976 32010   1
32012         -1975 32011   1
32013         -1976 32012   1
32014         -1977 32013   1
32015         -1976 32014   1
32016         -1977 32015   1
32017         -1976 32016   1
32018         -1975 32017   1
32019         -1974 32018   1
32020         -1973 32019   1
32021         -1974 32020   1
32022         -1975 32021   1
32023         -1974 32022   1
32024         -1973 32023   1
32025         -1974 32024   1
32026         -1975 32025   1
32027         -1976 32026   1
32028         -1977 32027   1
32029         -1976 32028   1
32030         -1977 32029   1
32031         -1978 32030   1
32032         -1979 32031   1
32033         -1978 32032   1
32034         -1979 32033   1
32035         -1978 32034   1
32036         -1979 32035   1
32037         -1978 32036   1
32038         -1979 32037   1
32039         -1980 32038   1
32040         -1979 32039   1
32041         -1980 32040   1
32042         -1979 32041   1
32043         -1980 32042   1
32044         -1981 32043   1
32045         -1980 32044   1
32046         -1981 32045   1
32047         -1982 32046   1
32048         -1981 32047   1
32049         -1982 32048   1
32050         -1981 32049   1
32051         -1982 32050   1
32052         -1981 32051   1
32053         -1980 32052   1
32054         -1979 32053   1
32055         -1978 32054   1
32056         -1977 32055   1
32057         -1976 32056   1
32058         -1977 32057   1
32059         -1978 32058   1
32060         -1979 32059   1
32061         -1978 32060   1
32062         -1979 32061   1
32063         -1980 32062   1
32064         -1979 32063   1
32065         -1978 32064   1
32066         -1977 32065   1
32067         -1976 32066   1
32068         -1975 32067   1
32069         -1976 32068   1
32070         -1977 32069   1
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32072         -1975 32071   1
32073         -1976 32072   1
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32078         -1977 32077   1
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32080         -1979 32079   1
32081         -1980 32080   1
32082         -1979 32081   1
32083         -1980 32082   1
32084         -1981 32083   1
32085         -1982 32084   1
32086         -1981 32085   1
32087         -1980 32086   1
32088         -1979 32087   1
32089         -1978 32088   1
32090         -1977 32089   1
32091         -1978 32090   1
32092         -1977 32091   1
32093         -1976 32092   1
32094         -1977 32093   1
32095         -1978 32094   1
32096         -1977 32095   1
32097         -1976 32096   1
32098         -1977 32097   1
32099         -1978 32098   1
32100         -1979 32099   1
32101         -1978 32100   1
32102         -1979 32101   1
32103         -1978 32102   1
32104         -1977 32103   1
32105         -1976 32104   1
32106         -1975 32105   1
32107         -1974 32106   1
32108         -1973 32107   1
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32110         -1975 32109   1
32111         -1976 32110   1
32112         -1977 32111   1
32113         -1976 32112   1
32114         -1977 32113   1
32115         -1976 32114   1
32116         -1977 32115   1
32117         -1976 32116   1
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32119         -1976 32118   1
32120         -1977 32119   1
32121         -1978 32120   1
32122         -1979 32121   1
32123         -1980 32122   1
32124         -1981 32123   1
32125         -1982 32124   1
32126         -1983 32125   1
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32128         -1985 32127   1
32129         -1986 32128   1
32130         -1987 32129   1
32131         -1986 32130   1
32132         -1985 32131   1
32133         -1986 32132   1
32134         -1987 32133   1
32135         -1988 32134   1
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32137         -1986 32136   1
32138         -1985 32137   1
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32140         -1983 32139   1
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32142         -1983 32141   1
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32144         -1983 32143   1
32145         -1982 32144   1
32146         -1981 32145   1
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32150         -1979 32149   1
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32154         -1979 32153   1
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32160         -1979 32159   1
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32166         -1981 32165   1
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32168         -1981 32167   1
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32170         -1981 32169   1
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32180         -1981 32179   1
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32186         -1981 32185   1
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32190         -1979 32189   1
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32194         -1981 32193   1
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32198         -1983 32197   1
32199         -1984 32198   1
32200         -1983 32199   1
32201         -1984 32200   1
32202         -1985 32201   1
32203         -1984 32202   1
32204         -1983 32203   1
32205         -1982 32204   1
32206         -1981 32205   1
32207         -1982 32206   1
32208         -1983 32207   1
32209         -1984 32208   1
32210         -1983 32209   1
32211         -1984 32210   1
32212         -1985 32211   1
32213         -1984 32212   1
32214         -1985 32213   1
32215         -1986 32214   1
32216         -1987 32215   1
32217         -1988 32216   1
32218         -1987 32217   1
32219         -1988 32218   1
32220         -1987 32219   1
32221         -1986 32220   1
32222         -1985 32221   1
32223         -1984 32222   1
32224         -1985 32223   1
32225         -1986 32224   1
32226         -1987 32225   1
32227         -1988 32226   1
32228         -1989 32227   1
32229         -1990 32228   1
32230         -1991 32229   1
32231         -1992 32230   1
32232         -1993 32231   1
32233         -1992 32232   1
32234         -1991 32233   1
32235         -1992 32234   1
32236         -1993 32235   1
32237         -1992 32236   1
32238         -1993 32237   1
32239         -1994 32238   1
32240         -1995 32239   1
32241         -1996 32240   1
32242         -1995 32241   1
32243         -1996 32242   1
32244         -1997 32243   1
32245         -1998 32244   1
32246         -1999 32245   1
32247         -2000 32246   1
32248         -2001 32247   1
32249         -2000 32248   1
32250         -2001 32249   1
32251         -2002 32250   1
32252         -2003 32251   1
32253         -2002 32252   1
32254         -2003 32253   1
32255         -2004 32254   1
32256         -2003 32255   1
32257         -2004 32256   1
32258         -2005 32257   1
32259         -2004 32258   1
32260         -2003 32259   1
32261         -2002 32260   1
32262         -2001 32261   1
32263         -2000 32262   1
32264         -2001 32263   1
32265         -2002 32264   1
32266         -2001 32265   1
32267         -2000 32266   1
32268         -2001 32267   1
32269         -2000 32268   1
32270         -2001 32269   1
32271         -2002 32270   1
32272         -2003 32271   1
32273         -2004 32272   1
32274         -2005 32273   1
32275         -2006 32274   1
32276         -2007 32275   1
32277         -2008 32276   1
32278         -2007 32277   1
32279         -2008 32278   1
32280         -2009 32279   1
32281         -2010 32280   1
32282         -2009 32281   1
32283         -2010 32282   1
32284         -2011 32283   1
32285         -2010 32284   1
32286         -2009 32285   1
32287         -2010 32286   1
32288         -2011 32287   1
32289         -2012 32288   1
32290         -2013 32289   1
32291         -2012 32290   1
32292         -2011 32291   1
32293         -2012 32292   1
32294         -2013 32293   1
32295         -2014 32294   1
32296         -2013 32295   1
32297         -2012 32296   1
32298         -2011 32297   1
32299         -2010 32298   1
32300         -2009 32299   1
32301         -2008 32300   1
32302         -2009 32301   1
32303         -2008 32302   1
32304         -2009 32303   1
32305         -2010 32304   1
32306         -2009 32305   1
32307         -2008 32306   1
32308         -2009 32307   1
32309         -2008 32308   1
32310         -2007 32309   1
32311         -2008 32310   1
32312         -2007 32311   1
32313         -2008 32312   1
32314         -2009 32313   1
32315         -2008 32314   1
32316         -2007 32315   1
32317         -2006 32316   1
32318         -2005 32317   1
32319         -2006 32318   1
32320         -2005 32319   1
32321         -2004 32320   1
32322         -2003 32321   1
32323         -2004 32322   1
32324         -2005 32323   1
32325         -2004 32324   1
32326         -2005 32325   1
32327         -2004 32326   1
32328         -2005 32327   1
32329         -2006 32328   1
32330         -2005 32329   1
32331         -2006 32330   1
32332         -2005 32331   1
32333         -2006 32332   1
32334         -2007 32333   1
32335         -2006 32334   1
32336         -2005 32335   1
32337         -2004 32336   1
32338         -2005 32337   1
32339         -2006 32338   1
32340         -2007 32339   1
32341         -2006 32340   1
32342         -2007 32341   1
32343         -2008 32342   1
32344         -2007 32343   1
32345         -2006 32344   1
32346         -2005 32345   1
32347         -2006 32346   1
32348         -2007 32347   1
32349         -2006 32348   1
32350         -2007 32349   1
32351         -2008 32350   1
32352         -2007 32351   1
32353         -2006 32352   1
32354         -2007 32353   1
32355         -2006 32354   1
32356         -2007 32355   1
32357         -2008 32356   1
32358         -2009 32357   1
32359         -2008 32358   1
32360         -2009 32359   1
32361         -2008 32360   1
32362         -2007 32361   1
32363         -2006 32362   1
32364         -2005 32363   1
32365         -2004 32364   1
32366         -2005 32365   1
32367         -2004 32366   1
32368         -2005 32367   1
32369         -2006 32368   1
32370         -2007 32369   1
32371         -2006 32370   1
32372         -2005 32371   1
32373         -2006 32372   1
32374         -2007 32373   1
32375         -2006 32374   1
32376         -2007 32375   1
32377         -2008 32376   1
32378         -2009 32377   1
32379         -2010 32378   1
32380         -2011 32379   1
32381         -2010 32380   1
32382         -2011 32381   1
32383         -2012 32382   1
32384         -2013 32383   1
32385         -2014 32384   1
32386         -2015 32385   1
32387         -2016 32386   1
32388         -2015 32387   1
32389         -2016 32388   1
32390         -2017 32389   1
32391         -2018 32390   1
32392         -2019 32391   1
32393         -2020 32392   1
32394         -2021 32393   1
32395         -2022 32394   1
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32397         -2024 32396   1
32398         -2023 32397   1
32399         -2024 32398   1
32400         -2025 32399   1
32401         -2026 32400   1
32402         -2027 32401   1
32403         -2028 32402   1
32404         -2027 32403   1
32405         -2028 32404   1
32406         -2029 32405   1
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32410         -2029 32409   1
32411         -2030 32410   1
32412         -2029 32411   1
32413         -2028 32412   1
32414         -2029 32413   1
32415         -2030 32414   1
32416         -2029 32415   1
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32420         -2031 32419   1
32421         -2030 32420   1
32422         -2029 32421   1
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32427         -2032 32426   1
32428         -2033 32427   1
32429         -2034 32428   1
32430         -2035 32429   1
32431         -2034 32430   1
32432         -2033 32431   1
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32435         -2030 32434   1
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32438         -2033 32437   1
32439         -2034 32438   1
32440         -2033 32439   1
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32456         -2037 32455   1
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32460         -2039 32459   1
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32464         -2041 32463   1
32465         -2042 32464   1
32466         -2041 32465   1
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32468         -2041 32467   1
32469         -2042 32468   1
32470         -2043 32469   1
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32475         -2040 32474   1
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32477         -2040 32476   1
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32480         -2039 32479   1
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32488         -2045 32487   1
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32490         -2045 32489   1
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32492         -2043 32491   1
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32495         -2044 32494   1
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32499         -2048 32498   1
32500         -2047 32499   1
32501         -2048 32500   1
32502         -2049 32501   1
32503         -2050 32502   1
32504         -2051 32503   1
32505         -2052 32504   1
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33188         -2125 33187   1
33189         -2124 33188   1
33190         -2125 33189   1
33191         -2126 33190   1
33192         -2127 33191   1
33193         -2128 33192   1
33194         -2127 33193   1
33195         -2126 33194   1
33196         -2125 33195   1
33197         -2124 33196   1
33198         -2125 33197   1
33199         -2126 33198   1
33200         -2125 33199   1
33201         -2126 33200   1
33202         -2125 33201   1
33203         -2124 33202   1
33204         -2125 33203   1
33205         -2124 33204   1
33206         -2123 33205   1
33207         -2124 33206   1
33208         -2123 33207   1
33209         -2124 33208   1
33210         -2125 33209   1
33211         -2124 33210   1
33212         -2125 33211   1
33213         -2124 33212   1
33214         -2123 33213   1
33215         -2122 33214   1
33216         -2123 33215   1
33217         -2122 33216   1
33218         -2123 33217   1
33219         -2122 33218   1
33220         -2121 33219   1
33221         -2122 33220   1
33222         -2123 33221   1
33223         -2122 33222   1
33224         -2123 33223   1
33225         -2122 33224   1
33226         -2123 33225   1
33227         -2122 33226   1
33228         -2123 33227   1
33229         -2122 33228   1
33230         -2123 33229   1
33231         -2124 33230   1
33232         -2125 33231   1
33233         -2126 33232   1
33234         -2125 33233   1
33235         -2124 33234   1
33236         -2125 33235   1
33237         -2124 33236   1
33238         -2123 33237   1
33239         -2124 33238   1
33240         -2125 33239   1
33241         -2124 33240   1
33242         -2123 33241   1
33243         -2124 33242   1
33244         -2123 33243   1
33245         -2124 33244   1
33246         -2123 33245   1
33247         -2124 33246   1
33248         -2123 33247   1
33249         -2124 33248   1
33250         -2123 33249   1
33251         -2124 33250   1
33252         -2125 33251   1
33253         -2124 33252   1
33254         -2125 33253   1
33255         -2126 33254   1
33256         -2127 33255   1
33257         -2126 33256   1
33258         -2127 33257   1
33259         -2126 33258   1
33260         -2127 33259   1
33261         -2126 33260   1
33262         -2125 33261   1
33263         -2124 33262   1
33264         -2125 33263   1
33265         -2124 33264   1
33266         -2125 33265   1
33267         -2126 33266   1
33268         -2127 33267   1
33269         -2128 33268   1
33270         -2127 33269   1
33271         -2126 33270   1
33272         -2127 33271   1
33273         -2126 33272   1
33274         -2127 33273   1
33275         -2128 33274   1
33276         -2129 33275   1
33277         -2130 33276   1
33278         -2129 33277   1
33279         -2128 33278   1
33280         -2127 33279   1
33281         -2126 33280   1
33282         -2125 33281   1
33283         -2124 33282   1
33284         -2123 33283   1
33285         -2122 33284   1
33286         -2121 33285   1
33287         -2120 33286   1
33288         -2121 33287   1
33289         -2120 33288   1
33290         -2121 33289   1
33291         -2120 33290   1
33292         -2119 33291   1
33293         -2118 33292   1
33294         -2117 33293   1
33295         -2116 33294   1
33296         -2117 33295   1
33297         -2116 33296   1
33298         -2117 33297   1
33299         -2118 33298   1
33300         -2117 33299   1
33301         -2116 33300   1
33302         -2117 33301   1
33303         -2118 33302   1
33304         -2117 33303   1
33305         -2118 33304   1
33306         -2117 33305   1
33307         -2116 33306   1
33308         -2115 33307   1
33309         -2114 33308   1
33310         -2115 33309   1
33311         -2116 33310   1
33312         -2117 33311   1
33313         -2118 33312   1
33314         -2117 33313   1
33315         -2118 33314   1
33316         -2119 33315   1
33317         -2120 33316   1
33318         -2119 33317   1
33319         -2120 33318   1
33320         -2119 33319   1
33321         -2118 33320   1
33322         -2117 33321   1
33323         -2116 33322   1
33324         -2117 33323   1
33325         -2116 33324   1
33326         -2117 33325   1
33327         -2116 33326   1
33328         -2117 33327   1
33329         -2116 33328   1
33330         -2115 33329   1
33331         -2116 33330   1
33332         -2115 33331   1
33333         -2114 33332   1
 [ reached 'max' / getOption("max.print") -- omitted 1966687 rows ]

Step 6: Wrangle the output

end <- output |> 
  group_by(rep) |> 
  filter(interim_money >= 40 | interim_money <= 0) |> 
  group_by(rep) |> 
  slice_min(game) |> 
  ungroup()

end

# A tibble: 20 × 3
   interim_money  game   rep
           <dbl> <int> <int>
 1             0   104     1
 2             0   286     2
 3             0   410     3
 4             0   136     4
 5             0   156     5
 6             0   442     6
 7             0   110     7
 8             0   170     8
 9             0   150     9
10             0   318    10
11             0   210    11
12             0   268    12
13             0   750    13
14             0   116    14
15             0   204    15
16             0    92    16
17             0   354    17
18            40   264    18
19             0   124    19
20            40   472    20

Step 6: Wrangle the output

end |> 
  summarize(prop_double = mean(interim_money == 40))

# A tibble: 1 × 1
  prop_double
        <dbl>
1         0.1

Without magic numbers

set.seed(4747)
money <- 10
games <- 20
iterations <- 10000


1:games |> map(game_func, money = money, iter = iterations) |> 
  list_rbind() |> 
  group_by(rep) |> 
  filter(interim_money >= 2 * money | interim_money <= 0) |> 
  group_by(rep) |> 
  slice_min(game) |> 
  ungroup() |> 
  summarize(prop_double = mean(interim_money == 2 * money))

# A tibble: 1 × 1
  prop_double
        <dbl>
1        0.25

Agenda 9/25/24

Simulating to understand complicated models
Simulating to consider statistical methods with fewer assumptions

Modeling

Simulation as easier than calculus for estimating probabilities.
Simulation to understand complicated models.
Simulation to consider statistical methods with fewer assumptions.

Bias in a model

Population:

talent ~ Normal (100, 15)
grades ~ Normal (talent, 15)
SAT ~ Normal (talent, 15)

The example is taken directly (and mostly verbatim) from a blog by Aaron Roth Algorithmic Unfairness Without Any Bias Baked In.

Goal for admitting students

College wants to admit students with

talent > 115

… but they only have access to grades and SAT which are noisy estimates of talent.

Plan for accepting students

Run a regression on a training dataset (talent is known for existing students)
Find a model which predicts talent based on grades and SAT
Choose students for whom predicted talent is above 115

Flaw in the plan …

there are two populations of students, the Reds and Blues.
Reds are the majority population (99%), and Blues are a small minority population (1%)
the Reds and the Blues are no different when it comes to talent: they both have the same talent distribution, as described above.
there is no bias baked into the grading or the exams: both the Reds and the Blues also have exactly the same grade and exam score distributions

What is really different?

But there is one difference: the Blues have more money than the Reds, so they each take the SAT twice, and report only the highest of the two scores to the college. This results in a small but noticeable bump in their average SAT scores, compared to the Reds.

Key aspect:

The value of SAT means something different for the Reds versus the Blues

(They have different feature distributions.)

Let’s see what happens …

Two models:

Red model (SAT taken once):

# A tibble: 3 × 5
  term        estimate std.error statistic p.value
  <chr>          <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)   33.2     0.152        218.       0
2 SAT            0.334   0.00149      224.       0
3 grades         0.334   0.00149      225.       0

Blue model (SAT is max score of two):

# A tibble: 3 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)   25.3      1.60        15.8 2.04e- 50
2 SAT            0.432    0.0161      26.7 3.35e-119
3 grades         0.277    0.0154      18.0 6.83e- 63

New data

Generate new data, use the two models separately.
Can the models predict if a student has talent > 115?

New data

# A tibble: 2 × 5
  color   tpr    fpr   fnr error
  <chr> <dbl>  <dbl> <dbl> <dbl>
1 blue  0.503 0.0379 0.497 0.109
2 red   0.509 0.0378 0.491 0.109

TWO models doesn’t seem right????

What if we fit only one model to the entire dataset?

After all, there are laws against using protected classes to make decisions (housing, jobs, money loans, college, etc.)

# A tibble: 3 × 5
  term        estimate std.error statistic p.value
  <chr>          <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)   33.1     0.151        219.       0
2 SAT            0.334   0.00148      225.       0
3 grades         0.334   0.00148      226.       0

(The coefficients kinda look like the red model…)

How do the error rates change?

One model:

# A tibble: 2 × 5
  color   tpr    fpr   fnr error
  <chr> <dbl>  <dbl> <dbl> <dbl>
1 blue  0.619 0.0627 0.381 0.112
2 red   0.507 0.0375 0.493 0.109

Two separate models:

# A tibble: 2 × 5
  color   tpr    fpr   fnr error
  <chr> <dbl>  <dbl> <dbl> <dbl>
1 blue  0.503 0.0379 0.497 0.109
2 red   0.509 0.0378 0.491 0.109

What did we learn?

with two populations that have different feature distributions, learning a single classifier (that is prohibited from discriminating based on population) will fit the bigger of the two populations

depending on the nature of the distribution difference, it can be either to the benefit or the detriment of the minority population
no explicit human bias, either on the part of the algorithm designer or the data gathering process
the problem is exacerbated if we artificially force the algorithm to be group blind
well-intentioned “fairness” regulations prohibiting decision makers form taking sensitive attributes into account can actually make things less fair and less accurate at the same time

Simulate?

different varying proportions
effect due to variability
effect due to SAT coefficient
different number of times the blues get to take the test
etc.

Presidential election

Harris is ahead in the national polling but behind in North Carolina, Arizona, and Georgia.

Presidential election

538 simulates many many elections to see how the individual state percentages play out.

Election Data

(Big caveat: the data are old!)

# A tibble: 54 × 5
   date                state                Harris Trump evotes
   <dttm>              <chr>                 <dbl> <dbl>  <dbl>
 1 2020-11-03 00:00:00 Alabama                37.8 57.4       9
 2 2020-11-03 00:00:00 Alaska                 43.6 51.2       3
 3 2024-09-17 00:00:00 Arizona                46.9 47.3      11
 4 2020-11-03 00:00:00 Arkansas               36.2 58.9       6
 5 2024-09-17 00:00:00 California             59.5 35.3      54
 6 2020-11-03 00:00:00 Colorado               53.6 41.1      10
 7 2020-11-03 00:00:00 Connecticut            58.6 32.4       7
 8 2020-11-03 00:00:00 Delaware               58.9 34.6       3
 9 2020-11-03 00:00:00 District of Columbia   90.8  5.78      3
10 2024-09-17 00:00:00 Florida                45.4 49.4      30
# ℹ 44 more rows

Data sources include: 538 and the National Archives.

Election model

each state has a Bernoulli probability of going for either Harris or Trump
the states are independent of one another
the probabilities are an accurate model for how each state will vote

(Do we believe the model conditions?)

Function to determine who wins

Need 270 to win!

election <- function(i, data) {
  rand_number <- runif(nrow(data)) * 100
  data |> 
    cbind(rand_number) |> 
    mutate(state_winner = case_when(
      rand_number < Harris ~ "Harris",
      rand_number < Harris + Trump ~ "Trump",
      rand_number < (Harris + Trump) + (100 - Harris - Trump) /2 ~ "Harris",
      TRUE ~ "Trump")) |> 
    group_by(state_winner) |> 
    summarize(votes = sum(evotes)) |> 
    mutate(simulation = paste0("simulation", i))
}

election(47, poll_data)

# A tibble: 2 × 3
  state_winner votes simulation  
  <chr>        <dbl> <chr>       
1 Harris         304 simulation47
2 Trump          234 simulation47

Running the function many times

set.seed(47)
map(1:1000, election, data = poll_data) |> 
  list_rbind() |> 
  group_by(simulation) |> 
  slice_max(votes) |> 
  ungroup() |> 
  mutate(winner = case_when(
    votes >= 270 ~ state_winner,
    TRUE ~ "no winner"
  )) |> 
  select(winner) |> 
  table()

winner
   Harris no winner     Trump 
      591        10       404

Caveats

The simulation assumes that each state is independent (538’s model is much more sophisticated modeling connections across states and other demographic groups)
The poll numbers are very old
The poll numbers are constantly changing

Sensitivity of assumptions

Simulation as easier than calculus for estimating probabilities.
Simulation to understand complicated models..
Simulation to consider statistical methods with fewer assumptions

t-test with Poisson data

The t-test says that if the null hypothesis is true, we’d expect to reject a true null hypothesis $\alpha \cdot 100$ percent of the time. That is, if $\alpha = 0.05$, we’d reject true null hypotheses 5% of the time.

The guarantee of 5% (across many tests) happens only when the data are normal (or reasonably normal).

Non-normal data

data.frame(pois_data = rpois(n = 100000, lambda = 1)) |> 
  ggplot(aes(x = pois_data)) +
  geom_histogram()

t-test function

t.test.pval <- function(data1, data2){
  t.test(data1, data2) |> 
    broom::tidy() |> 
    select(p.value)
}

t.test.pval(rpois(n = 10, lambda = 1), rpois(n = 7, lambda = 1))

# A tibble: 1 × 1
  p.value
    <dbl>
1   0.696

t-test 10 times

map(1:10, ~t.test.pval(rpois(n = 10, lambda = 1),
                       rpois(n = 7, lambda = 1))) |> 
          list_rbind()

# A tibble: 10 × 1
   p.value
     <dbl>
 1  0.643 
 2  0.304 
 3  0.784 
 4  0.0874
 5  0.0379
 6  0.361 
 7  0.0215
 8  0.532 
 9  0.319 
10  0.539

set.seed(4747)
results <- map(1:10000, 
               ~t.test.pval(rpois(n = 8, lambda = 1),
                            rpois(n = 5, lambda = 1))) |> 
  list_rbind()

results |> 
  ggplot(aes(x = p.value)) + 
  geom_histogram(bins = 30) + 
  labs(x = "p.values")

results |> 
  summarize(type1 = mean(p.value < 0.05))

# A tibble: 1 × 1
   type1
   <dbl>
1 0.0498

Sensitivity of CI to tech conditions

Consider the following set up:

set.seed(474)
beta0 <- -1; beta1 <- 0.5; beta2 <- 1.5
n_obs <- 100; reps <- 1000


x1 <- rep(c(0,1), each=n_obs/2)
x2 <- runif(n_obs, min=-1, max=1)
y <- beta0 + beta1*x1 + beta2*x2 + rnorm(n_obs, mean=0, sd = 1)

Plot of data

Capture the slope parameter?

We know the truth (!!!!!)
We can create a single CI for the parameter given a single data set
Is the parameter in the interval?
Should it always be in the interval? How often should it be?
Repeat the process to see whether the confidence interval captures the parameter at the claimed rate.

See: Rossman Chance regression applet

Running a linear model

CI <- lm(y~x1+x2) |> tidy(conf.int=TRUE) |> data.frame()
CI

         term   estimate std.error statistic      p.value    conf.low
1 (Intercept) -0.9291632 0.1424025 -6.524909 3.098281e-09 -1.21179266
2          x1  0.3304882 0.2016861  1.638627 1.045307e-01 -0.06980279
3          x2  1.4447884 0.1919184  7.528137 2.662556e-11  1.06388341
   conf.high
1 -0.6465337
2  0.7307792
3  1.8256934

CI |>
  filter(term == "x2") |>
  select(term, estimate, conf.low, conf.high) |>
  mutate(inside = between(beta2, conf.low, conf.high))

  term estimate conf.low conf.high inside
1   x2 1.444788 1.063883  1.825693   TRUE

What happens to the CI of coefficients in repeated samples? (eq var)

Datasets
Function
map()

eqvar_data <- data.frame(row_id = seq(1, n_obs, 1)) |>
  slice(rep(row_id, each = reps)) |>
  mutate(
    sim_id = rep(1:reps, n_obs),
    x1 = rep(c(0,1), each = n()/2),
    x2 = runif(n(), min = -1, max = 1),
    y = beta0 + beta1*x1 + beta2*x2 + rnorm(n(), mean = 0, sd = 1)
  ) |>
  arrange(sim_id, row_id) |>
  group_by(sim_id) |>
  nest()

eqvar_data

# A tibble: 1,000 × 2
# Groups:   sim_id [1,000]
   sim_id data              
    <int> <list>            
 1      1 <tibble [100 × 4]>
 2      2 <tibble [100 × 4]>
 3      3 <tibble [100 × 4]>
 4      4 <tibble [100 × 4]>
 5      5 <tibble [100 × 4]>
 6      6 <tibble [100 × 4]>
 7      7 <tibble [100 × 4]>
 8      8 <tibble [100 × 4]>
 9      9 <tibble [100 × 4]>
10     10 <tibble [100 × 4]>
# ℹ 990 more rows

beta_coef <- function(df){
  lm(y ~ x1 + x2, data = df) |>
    tidy(conf.int = TRUE) |>
    filter(term == "x2") |>
    select(estimate, conf.low, conf.high, p.value) 
}

eqvar_data |> 
  mutate(b2_vals = map(data, beta_coef)) |>
  select(b2_vals) |> 
  unnest(b2_vals) |>
  summarize(capture = between(beta2, conf.low, conf.high)) |>
  summarize(capture_rate = sum(capture)/reps)

# A tibble: 1 × 1
  capture_rate
         <dbl>
1        0.953

Sensitivity of CI to tech conditions

Consider the following set up (note the difference in variability):

set.seed(470)
beta0 <- -1; beta1 <- 0.5; beta2 <- 1.5
n_obs <- 100; reps <- 1000

  x1 <- rep(c(0,1), each=n_obs/2)
  x2 <- runif(n_obs, min=-1, max=1)
  y <- beta0 + beta1*x1 + beta2*x2 + 
             rnorm(n_obs, mean=0, sd = 1 + x1 + 10*abs(x2))

Plot of data

What happens to the CI of coefficients in repeated samples? (uneq var)

Datasets
map()

uneqvar_data <- data.frame(row_id = seq(1, n_obs, 1)) |>
  slice(rep(row_id, each = reps)) |>
  mutate(sim_id = rep(1:reps, n_obs), 
         x1 = rep(c(0,1), each = n()/2),
         x2 = runif(n(), min = -1, max = 1),
         y = beta0 + beta1*x1 + beta2*x2 + 
              rnorm(n(), mean = 0, sd = 1 + x1 + 10*abs(x2))  ) |>
  arrange(sim_id, row_id) |>
  group_by(sim_id) |> nest()

uneqvar_data |> 
  mutate(b2_vals = map(data, beta_coef)) |>
  select(b2_vals) |> unnest(b2_vals) |>
  summarize(capture = between(beta2, conf.low, conf.high)) |>
  summarize(capture_rate = sum(capture)/reps)

# A tibble: 1 × 1
  capture_rate
         <dbl>
1        0.877

Equal variance: type I error?

Another method for thinking about type I error rates. With equal variance, the type I error rate is close to what we’d expect, 5%.

function
generate data
summarize p-values

t_test_pval <- function(df){
  t.test(y ~ x1, data = df, var.equal = TRUE) |>
    tidy() |>
    select(estimate, p.value) 
}

set.seed(470)
reps <- 1000
n_obs <- 20
null_data_equal <- 
  data.frame(row_id = seq(1, n_obs, 1)) |>
  slice(rep(row_id, each = reps)) |>
  mutate(
    sim_id = rep(1:reps, n_obs),
    x1 = rep(c("group1", "group2"), each = n()/2),
    y = rnorm(n(), mean = 10, 
               sd = rep(c(1,1), each = n()/2))
  ) |>
  arrange(sim_id, row_id) |>
  group_by(sim_id) |>
  nest()

null_data_equal |> 
  mutate(t_vals = map(data,t_test_pval)) |>
  select(t_vals) |> 
  unnest(t_vals) |>
  ungroup(sim_id) |>
  summarize(type1error_rate = sum(p.value < 0.05)/reps)

# A tibble: 1 × 1
  type1error_rate
            <dbl>
1           0.045

Unequal variance: type I error?

With unequal variance, the type I error rate is much higher than we set. We set the the type I error rate to be 5%, but the simulated rate was 6.36%.

generate data
summarize p-values

set.seed(470)
reps <- 1000
n_obs <- 20
null_data_unequal <- 
  data.frame(row_id = seq(1, n_obs, 1)) |>
  slice(rep(row_id, each = reps)) |>
  mutate(
    sim_id = rep(1:reps, n_obs),
    x1 = rep(c("group1", "group2"), each = n()/2),
    y = rnorm(n(), mean = 10, 
               sd = rep(c(1,100), each = n()/2))
  ) |>
  arrange(sim_id, row_id) |>
  group_by(sim_id) |>
  nest()

null_data_unequal |> 
  mutate(t_vals = map(data,t_test_pval)) |>
  select(t_vals) |> 
  unnest(t_vals) |>
  ungroup(sim_id) |>
  summarize(type1error_rate = sum(p.value < 0.05)/reps)

# A tibble: 1 × 1
  type1error_rate
            <dbl>
1           0.057

References

Baumer, Ben, Daniel Kaplan, and Nicholas Horton. 2021. Modern Data Science with r. CRC Press. https://mdsr-book.github.io/mdsr2e/.

Mosteller, Frederick. 1987. Fifty Challenging Problems in Probability with Solutions. Dover Publications: Mineola, NY.

Simulating

Agenda 9/23/24

Goals of Simulating Complicated Models

Aside: a few R functions (ifelse())

Aside: a few R functions (case_when())

Aside: a few R functions (sample())

Why?

Aside: a few R functions (set.seed())

Small simulation example

Sally & Joan

Simulate their meeting

Results

Visualizing the meet up

Simulation best practices

Bigger simulation example

Thoughts on simulating

Roulette

Design the simulation

Step 1: create a roulette wheel

Step 2: spin the wheel

Step 3: update the money

Step 4: keep spinning (function first)

Aside: the accumulate() function

Step 4: run multiple times

Step 4: another function

Step 5: Repeat many times

Step 6: Wrangle the output

Step 6: Wrangle the output

Without magic numbers

Agenda 9/25/24

Modeling

Bias in a model

Goal for admitting students

Plan for accepting students

Flaw in the plan …

What is really different?

Key aspect:

Let’s see what happens …

Two models:

New data

New data

TWO models doesn’t seem right????

How do the error rates change?

What did we learn?

Simulate?

Presidential election

Presidential election

Election Data

Election model

Function to determine who wins

Running the function many times

Caveats

Sensitivity of assumptions

t-test with Poisson data

Non-normal data

t-test function

t-test 10 times

t-test many times

Sensitivity of CI to tech conditions

Plot of data

Capture the slope parameter?

Running a linear model

What happens to the CI of coefficients in repeated samples? (eq var)

Sensitivity of CI to tech conditions

Plot of data

What happens to the CI of coefficients in repeated samples? (uneq var)

Equal variance: type I error?

Unequal variance: type I error?

References

Aside: a few R functions (`ifelse()`)

Aside: a few R functions (`case_when()`)

Aside: a few R functions (`sample()`)

Aside: a few R functions (`set.seed()`)

Aside: the `accumulate()` function