STA 517 3.0 Programming and Statistical Computing with RAn Introduction to the Bootstrap MethodDr Thiyanga Talagala1 / 23

What is a bootstrap?

Bootstrap is a method of inference.
Bradley Efron first introduced this approach in 1979.
- Link to his original paper is here.

Bootstrapping is a computer-intensive procedure.

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Applications

Estimate the standard error of any statistic and to obtain a confidence interval (CI) for it.
This is useful when CI doesn't have a closed form, or it has a very complicated one.
Hypothesis testing
Bootstrap sampling in regression, etc.

Three forms of bootstrapping

Nonparametric (resampling)
Semiparametric (adding noise)
Parametric (simulation)

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Bootstrapping Resamples

The bootstrap sample is the same size as the original sample data.
As a result, some observations will be represented multiple times in the bootstrap sample while others will not be selected at all.

Example of Bootstrap Samples

Bootstrapping does not take new observations from the population.
It treats the original sample as a proxy for the real population and then draws random samples from it.
Consequently, the central assumption for bootstrapping is that the original sample accurately represents the actual population.

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Steps

Take a sample from population. This is called the original sample. Suppose the sample size is $n$ .
Draw a sample from the original sample data with replacement with size $n$ and repeat this step $N$ times.
Compute the statistic of $θ$ for each Bootstrap Sample, and there will be totally $N$ estimates of $θ$ .
Construct a sampling distribution with these $N$ Bootstrap statistics and use it to make further statistical inference.

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Differences between Bootstrapping and Traditional Inference Methods

how they estimate sampling distributions
"The traditional approach also uses theory to tell what the sampling distribution should look like, but the results fall apart if the assumptions of the theory are not met. The bootstrapping method, on the other hand, takes the original sample data and then resamples it to create many bootstrap samples."

source: towards data science.

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Example 1

Height of randomly selected individuals

height <- c(5.2, 6.1, 5.5, 5.4, 5.3)
height

[1] 5.2 6.1 5.5 5.4 5.3

Construct 1000 bootstrap samples.
Calculate the sample mean for each of the resamples.
Make a histogram of the means of the 1000 bootstrap samples. This is the bootstrap distribution.
Calculate the bootstrap standard error.

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Example 1 (cont.)

Help:

$\bar{X *} = \frac{1}{N} \sum_{i = 1}^{N} {\bar{x}}_{i}^{b o o t s t r a p}$

$s e (\bar{X *}) = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} ({\bar{x}}_{i}^{b o o t s t r a p} - \bar{X *})^{2}}$

The bootstrap standard error, of a statistic is the standard deviation of the bootstrap distribution of that statistic.

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Classical confidence interval

$[\bar{x} - t_{α / 2, n - 1} s e (\bar{x}), \bar{x} - t_{α / 2, n - 1} s e (\bar{x})]$

Confidence intervals via bootstrap

boot package can generate bootstrap samples.
Step 1: To use the boot function for drawing samples, you need a function to compute the statistic of interest.

samplemean <- function(data, indices) {
  return(mean(data[indices]))
}

It should have at least two arguments:

i) data: the original data

ii) a vector containing indices: frequencies or weights which define the bootstrap sample.

data[indices] Creates the bootstrap sample (i.e., subset the provided data by the “indices” parameter). “indices” is automatically provided by the “boot” function; this is the sampling with replacement portion of bootstrapping.

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Confidence intervals via bootstrap (cont.)

Step 2: Conduct the bootstrapping

library(boot)
result <-  boot(data = height, statistic = samplemean, R = 1000)
result


ORDINARY NONPARAMETRIC BOOTSTRAP
Call:
boot(data = height, statistic = samplemean, R = 1000)
Bootstrap Statistics :
    original  bias    std. error
t1*      5.5 0.00528   0.1402647

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View some calculated statistics of boot object

Bootstrap sample means

result$t

        [,1]
   [1,] 5.38
   [2,] 5.52
   [3,] 5.46
   [4,] 5.50
   [5,] 5.64
   [6,] 5.60
   [7,] 5.40
   [8,] 5.66
   [9,] 5.54
  [10,] 5.50
  [11,] 5.50
  [12,] 5.70
  [13,] 5.32
  [14,] 5.36
  [15,] 5.52
  [16,] 5.60
  [17,] 5.46
  [18,] 5.48
  [19,] 5.40
  [20,] 5.54
  [21,] 5.50
  [22,] 5.48
  [23,] 5.56
  [24,] 5.38
  [25,] 5.40
  [26,] 5.32
  [27,] 5.46
  [28,] 5.60
  [29,] 5.64
  [30,] 5.54
  [31,] 5.72
  [32,] 5.32
  [33,] 5.48
  [34,] 5.64
  [35,] 5.52
  [36,] 5.32
  [37,] 5.32
  [38,] 5.34
  [39,] 5.60
  [40,] 5.36
  [41,] 5.52
  [42,] 5.68
  [43,] 5.68
  [44,] 5.32
  [45,] 5.50
  [46,] 5.46
  [47,] 5.50
  [48,] 5.48
  [49,] 5.38
  [50,] 5.64
  [51,] 5.36
  [52,] 5.54
  [53,] 5.64
  [54,] 5.38
  [55,] 5.70
  [56,] 5.66
  [57,] 5.50
  [58,] 5.54
  [59,] 5.62
  [60,] 5.50
  [61,] 5.50
  [62,] 5.54
  [63,] 5.30
  [64,] 5.62
  [65,] 5.64
  [66,] 5.36
  [67,] 5.34
  [68,] 5.36
  [69,] 5.26
  [70,] 5.70
  [71,] 5.54
  [72,] 5.54
  [73,] 5.68
  [74,] 5.38
  [75,] 5.62
  [76,] 5.78
  [77,] 5.52
  [78,] 5.52
  [79,] 5.24
  [80,] 5.54
  [81,] 5.44
  [82,] 5.68
  [83,] 5.82
  [84,] 5.78
  [85,] 5.58
  [86,] 5.86
  [87,] 5.54
  [88,] 5.46
  [89,] 5.60
  [90,] 5.36
  [91,] 5.42
  [92,] 5.58
  [93,] 5.40
  [94,] 5.26
  [95,] 5.48
  [96,] 5.58
  [97,] 5.44
  [98,] 5.48
  [99,] 5.50
 [100,] 5.50
 [101,] 5.46
 [102,] 5.28
 [103,] 5.26
 [104,] 5.28
 [105,] 5.48
 [106,] 5.36
 [107,] 5.66
 [108,] 5.62
 [109,] 5.80
 [110,] 5.72
 [111,] 5.42
 [112,] 5.36
 [113,] 5.80
 [114,] 5.68
 [115,] 5.44
 [116,] 5.56
 [117,] 5.64
 [118,] 5.28
 [119,] 5.46
 [120,] 5.28
 [121,] 5.46
 [122,] 5.64
 [123,] 5.62
 [124,] 5.50
 [125,] 5.32
 [126,] 5.52
 [127,] 5.26
 [128,] 5.66
 [129,] 5.40
 [130,] 5.78
 [131,] 5.50
 [132,] 5.30
 [133,] 5.66
 [134,] 5.50
 [135,] 5.60
 [136,] 5.26
 [137,] 5.66
 [138,] 5.52
 [139,] 5.56
 [140,] 5.48
 [141,] 5.30
 [142,] 5.32
 [143,] 5.34
 [144,] 5.56
 [145,] 5.56
 [146,] 5.86
 [147,] 5.38
 [148,] 5.40
 [149,] 5.72
 [150,] 5.48
 [151,] 5.54
 [152,] 5.58
 [153,] 5.62
 [154,] 5.50
 [155,] 5.50
 [156,] 5.52
 [157,] 5.52
 [158,] 5.82
 [159,] 5.68
 [160,] 5.50
 [161,] 5.46
 [162,] 5.48
 [163,] 5.26
 [164,] 5.56
 [165,] 5.54
 [166,] 5.42
 [167,] 5.52
 [168,] 5.44
 [169,] 5.48
 [170,] 5.64
 [171,] 5.64
 [172,] 5.58
 [173,] 5.36
 [174,] 5.48
 [175,] 5.34
 [176,] 5.38
 [177,] 5.44
 [178,] 5.32
 [179,] 5.48
 [180,] 5.58
 [181,] 5.44
 [182,] 5.38
 [183,] 5.74
 [184,] 5.32
 [185,] 5.48
 [186,] 5.34
 [187,] 5.72
 [188,] 5.56
 [189,] 5.52
 [190,] 5.50
 [191,] 5.40
 [192,] 5.60
 [193,] 5.28
 [194,] 5.36
 [195,] 5.68
 [196,] 5.64
 [197,] 5.52
 [198,] 5.54
 [199,] 5.34
 [200,] 5.28
 [201,] 5.76
 [202,] 5.48
 [203,] 5.60
 [204,] 5.52
 [205,] 5.52
 [206,] 5.54
 [207,] 5.52
 [208,] 5.56
 [209,] 5.54
 [210,] 5.40
 [211,] 5.46
 [212,] 5.84
 [213,] 5.64
 [214,] 5.30
 [215,] 5.48
 [216,] 5.54
 [217,] 5.50
 [218,] 5.34
 [219,] 5.36
 [220,] 5.38
 [221,] 5.44
 [222,] 5.44
 [223,] 5.52
 [224,] 5.26
 [225,] 5.82
 [226,] 5.38
 [227,] 5.80
 [228,] 5.32
 [229,] 5.22
 [230,] 5.32
 [231,] 5.52
 [232,] 5.30
 [233,] 5.54
 [234,] 5.80
 [235,] 5.34
 [236,] 5.62
 [237,] 5.66
 [238,] 5.34
 [239,] 5.62
 [240,] 5.66
 [241,] 5.32
 [242,] 5.62
 [243,] 5.62
 [244,] 5.52
 [245,] 5.36
 [246,] 5.54
 [247,] 5.50
 [248,] 5.50
 [249,] 5.66
 [250,] 5.52
 [251,] 5.54
 [252,] 5.44
 [253,] 5.50
 [254,] 5.50
 [255,] 5.70
 [256,] 5.72
 [257,] 5.64
 [258,] 5.78
 [259,] 5.54
 [260,] 5.42
 [261,] 5.38
 [262,] 5.38
 [263,] 5.48
 [264,] 5.50
 [265,] 5.28
 [266,] 5.32
 [267,] 5.48
 [268,] 5.48
 [269,] 5.66
 [270,] 5.54
 [271,] 5.70
 [272,] 5.80
 [273,] 5.78
 [274,] 5.50
 [275,] 5.86
 [276,] 5.42
 [277,] 5.66
 [278,] 5.38
 [279,] 5.48
 [280,] 5.66
 [281,] 5.44
 [282,] 5.68
 [283,] 5.64
 [284,] 5.66
 [285,] 5.50
 [286,] 5.56
 [287,] 5.60
 [288,] 5.58
 [289,] 5.48
 [290,] 5.36
 [291,] 5.30
 [292,] 5.54
 [293,] 5.52
 [294,] 5.34
 [295,] 5.52
 [296,] 5.54
 [297,] 5.66
 [298,] 5.42
 [299,] 5.38
 [300,] 5.44
 [301,] 5.50
 [302,] 5.36
 [303,] 5.56
 [304,] 5.52
 [305,] 5.34
 [306,] 5.34
 [307,] 5.74
 [308,] 5.64
 [309,] 5.76
 [310,] 5.40
 [311,] 5.50
 [312,] 5.40
 [313,] 5.42
 [314,] 5.46
 [315,] 5.70
 [316,] 5.52
 [317,] 5.64
 [318,] 5.68
 [319,] 5.32
 [320,] 5.40
 [321,] 5.42
 [322,] 5.50
 [323,] 5.44
 [324,] 5.50
 [325,] 5.46
 [326,] 5.68
 [327,] 5.32
 [328,] 5.30
 [329,] 5.32
 [330,] 5.50
 [331,] 5.66
 [332,] 5.66
 [333,] 5.32
 [334,] 5.58
 [335,] 5.50
 [336,] 5.32
 [337,] 5.68
 [338,] 5.48
 [339,] 5.50
 [340,] 5.50
 [341,] 5.48
 [342,] 5.36
 [343,] 5.60
 [344,] 5.34
 [345,] 5.48
 [346,] 5.38
 [347,] 5.42
 [348,] 5.80
 [349,] 5.48
 [350,] 5.48
 [351,] 5.26
 [352,] 5.54
 [353,] 5.44
 [354,] 5.50
 [355,] 5.80
 [356,] 5.26
 [357,] 5.50
 [358,] 5.64
 [359,] 5.30
 [360,] 5.48
 [361,] 5.34
 [362,] 5.62
 [363,] 5.50
 [364,] 5.82
 [365,] 5.68
 [366,] 5.52
 [367,] 5.44
 [368,] 5.78
 [369,] 5.50
 [370,] 5.52
 [371,] 5.36
 [372,] 5.68
 [373,] 5.50
 [374,] 5.36
 [375,] 5.42
 [376,] 5.38
 [377,] 5.68
 [378,] 5.46
 [379,] 5.40
 [380,] 5.80
 [381,] 5.66
 [382,] 5.46
 [383,] 5.54
 [384,] 5.56
 [385,] 5.32
 [386,] 5.60
 [387,] 5.48
 [388,] 5.40
 [389,] 5.54
 [390,] 5.28
 [391,] 5.50
 [392,] 5.34
 [393,] 5.54
 [394,] 5.68
 [395,] 5.32
 [396,] 5.36
 [397,] 5.50
 [398,] 5.44
 [399,] 5.54
 [400,] 5.42
 [401,] 5.66
 [402,] 5.56
 [403,] 5.32
 [404,] 5.82
 [405,] 5.56
 [406,] 5.44
 [407,] 5.72
 [408,] 5.62
 [409,] 5.72
 [410,] 5.66
 [411,] 5.60
 [412,] 5.52
 [413,] 5.48
 [414,] 5.58
 [415,] 5.30
 [416,] 5.46
 [417,] 5.26
 [418,] 5.38
 [419,] 5.34
 [420,] 5.56
 [421,] 5.44
 [422,] 5.46
 [423,] 5.66
 [424,] 5.62
 [425,] 5.52
 [426,] 5.32
 [427,] 5.70
 [428,] 5.66
 [429,] 5.50
 [430,] 5.62
 [431,] 5.36
 [432,] 5.64
 [433,] 5.66
 [434,] 5.40
 [435,] 5.26
 [436,] 5.58
 [437,] 5.48
 [438,] 5.70
 [439,] 5.48
 [440,] 5.74
 [441,] 5.52
 [442,] 5.42
 [443,] 5.52
 [444,] 5.64
 [445,] 5.78
 [446,] 5.30
 [447,] 5.52
 [448,] 5.38
 [449,] 5.46
 [450,] 5.36
 [451,] 5.30
 [452,] 5.34
 [453,] 5.62
 [454,] 5.50
 [455,] 5.52
 [456,] 5.66
 [457,] 5.60
 [458,] 5.48
 [459,] 5.50
 [460,] 5.66
 [461,] 5.48
 [462,] 5.40
 [463,] 5.32
 [464,] 5.52
 [465,] 5.34
 [466,] 5.68
 [467,] 5.72
 [468,] 5.60
 [469,] 5.40
 [470,] 5.58
 [471,] 5.50
 [472,] 5.46
 [473,] 5.62
 [474,] 5.44
 [475,] 5.52
 [476,] 5.54
 [477,] 5.40
 [478,] 5.74
 [479,] 5.34
 [480,] 5.46
 [481,] 5.52
 [482,] 5.30
 [483,] 5.50
 [484,] 5.60
 [485,] 5.76
 [486,] 5.52
 [487,] 5.50
 [488,] 5.70
 [489,] 5.62
 [490,] 5.40
 [491,] 5.46
 [492,] 5.48
 [493,] 5.34
 [494,] 5.54
 [495,] 5.40
 [496,] 5.38
 [497,] 5.52
 [498,] 5.54
 [499,] 5.50
 [500,] 5.42
 [501,] 5.54
 [502,] 5.70
 [503,] 5.36
 [504,] 5.34
 [505,] 5.36
 [506,] 5.46
 [507,] 5.38
 [508,] 5.34
 [509,] 5.40
 [510,] 5.44
 [511,] 5.48
 [512,] 5.74
 [513,] 5.46
 [514,] 5.40
 [515,] 5.26
 [516,] 5.44
 [517,] 5.50
 [518,] 5.52
 [519,] 5.78
 [520,] 5.50
 [521,] 5.30
 [522,] 5.30
 [523,] 5.34
 [524,] 5.42
 [525,] 5.28
 [526,] 5.56
 [527,] 5.50
 [528,] 5.66
 [529,] 5.66
 [530,] 5.62
 [531,] 5.30
 [532,] 5.62
 [533,] 5.50
 [534,] 5.82
 [535,] 5.58
 [536,] 5.66
 [537,] 5.30
 [538,] 5.84
 [539,] 5.56
 [540,] 5.40
 [541,] 5.30
 [542,] 5.84
 [543,] 5.42
 [544,] 5.24
 [545,] 5.60
 [546,] 5.58
 [547,] 5.60
 [548,] 5.56
 [549,] 5.82
 [550,] 5.34
 [551,] 5.70
 [552,] 5.64
 [553,] 5.48
 [554,] 5.36
 [555,] 5.32
 [556,] 5.48
 [557,] 5.48
 [558,] 5.48
 [559,] 5.64
 [560,] 5.48
 [561,] 5.38
 [562,] 5.62
 [563,] 5.38
 [564,] 5.52
 [565,] 5.44
 [566,] 5.78
 [567,] 5.58
 [568,] 5.66
 [569,] 5.40
 [570,] 5.46
 [571,] 5.30
 [572,] 5.48
 [573,] 5.48
 [574,] 5.64
 [575,] 5.34
 [576,] 5.48
 [577,] 5.72
 [578,] 5.40
 [579,] 5.58
 [580,] 5.48
 [581,] 5.44
 [582,] 5.68
 [583,] 5.66
 [584,] 5.64
 [585,] 5.66
 [586,] 5.34
 [587,] 5.50
 [588,] 5.38
 [589,] 5.50
 [590,] 5.96
 [591,] 5.66
 [592,] 5.54
 [593,] 5.48
 [594,] 5.36
 [595,] 5.64
 [596,] 5.42
 [597,] 5.42
 [598,] 5.54
 [599,] 5.34
 [600,] 5.48
 [601,] 5.68
 [602,] 5.32
 [603,] 5.30
 [604,] 5.42
 [605,] 5.48
 [606,] 5.52
 [607,] 5.58
 [608,] 5.36
 [609,] 5.38
 [610,] 5.66
 [611,] 5.64
 [612,] 5.46
 [613,] 5.34
 [614,] 5.46
 [615,] 5.48
 [616,] 5.56
 [617,] 5.52
 [618,] 5.40
 [619,] 5.32
 [620,] 5.38
 [621,] 5.26
 [622,] 5.44
 [623,] 5.50
 [624,] 5.76
 [625,] 5.66
 [626,] 5.30
 [627,] 5.30
 [628,] 5.38
 [629,] 5.56
 [630,] 5.48
 [631,] 5.40
 [632,] 5.30
 [633,] 5.28
 [634,] 5.50
 [635,] 5.48
 [636,] 5.40
 [637,] 5.54
 [638,] 5.50
 [639,] 5.54
 [640,] 5.68
 [641,] 5.30
 [642,] 5.34
 [643,] 5.20
 [644,] 5.80
 [645,] 5.34
 [646,] 5.50
 [647,] 5.34
 [648,] 5.50
 [649,] 5.48
 [650,] 5.42
 [651,] 5.52
 [652,] 5.62
 [653,] 5.50
 [654,] 5.62
 [655,] 5.50
 [656,] 5.74
 [657,] 5.62
 [658,] 5.54
 [659,] 5.34
 [660,] 5.32
 [661,] 5.42
 [662,] 5.32
 [663,] 5.66
 [664,] 5.60
 [665,] 5.52
 [666,] 5.48
 [667,] 5.36
 [668,] 5.64
 [669,] 5.52
 [670,] 5.66
 [671,] 5.62
 [672,] 5.42
 [673,] 5.34
 [674,] 5.60
 [675,] 5.28
 [676,] 5.46
 [677,] 5.44
 [678,] 5.66
 [679,] 5.70
 [680,] 5.54
 [681,] 5.64
 [682,] 5.54
 [683,] 5.52
 [684,] 5.78
 [685,] 5.38
 [686,] 5.80
 [687,] 5.42
 [688,] 5.38
 [689,] 5.48
 [690,] 5.32
 [691,] 5.56
 [692,] 5.28
 [693,] 5.28
 [694,] 5.52
 [695,] 5.52
 [696,] 5.46
 [697,] 5.50
 [698,] 5.54
 [699,] 5.30
 [700,] 5.50
 [701,] 5.62
 [702,] 5.60
 [703,] 5.44
 [704,] 5.44
 [705,] 5.66
 [706,] 5.36
 [707,] 5.46
 [708,] 5.36
 [709,] 5.46
 [710,] 5.36
 [711,] 5.52
 [712,] 5.52
 [713,] 5.44
 [714,] 5.50
 [715,] 5.64
 [716,] 5.40
 [717,] 5.60
 [718,] 5.50
 [719,] 5.50
 [720,] 5.32
 [721,] 5.40
 [722,] 5.54
 [723,] 5.42
 [724,] 5.30
 [725,] 5.36
 [726,] 5.30
 [727,] 5.40
 [728,] 5.48
 [729,] 5.68
 [730,] 5.40
 [731,] 5.52
 [732,] 5.50
 [733,] 5.46
 [734,] 5.58
 [735,] 5.44
 [736,] 5.52
 [737,] 5.64
 [738,] 5.46
 [739,] 5.50
 [740,] 5.62
 [741,] 5.54
 [742,] 5.68
 [743,] 5.64
 [744,] 5.52
 [745,] 5.50
 [746,] 5.64
 [747,] 5.34
 [748,] 5.80
 [749,] 5.54
 [750,] 5.64
 [751,] 5.48
 [752,] 5.52
 [753,] 5.50
 [754,] 5.72
 [755,] 5.64
 [756,] 5.52
 [757,] 5.62
 [758,] 5.36
 [759,] 5.40
 [760,] 5.50
 [761,] 5.72
 [762,] 5.54
 [763,] 5.48
 [764,] 5.64
 [765,] 5.68
 [766,] 5.72
 [767,] 5.36
 [768,] 5.46
 [769,] 5.56
 [770,] 5.64
 [771,] 5.48
 [772,] 5.60
 [773,] 5.66
 [774,] 5.36
 [775,] 5.36
 [776,] 5.54
 [777,] 5.34
 [778,] 5.78
 [779,] 5.48
 [780,] 5.64
 [781,] 5.40
 [782,] 5.46
 [783,] 5.40
 [784,] 5.44
 [785,] 5.78
 [786,] 5.30
 [787,] 5.64
 [788,] 5.24
 [789,] 5.70
 [790,] 5.44
 [791,] 5.38
 [792,] 5.38
 [793,] 5.72
 [794,] 5.50
 [795,] 5.34
 [796,] 5.64
 [797,] 5.74
 [798,] 5.54
 [799,] 5.46
 [800,] 5.48
 [801,] 5.42
 [802,] 5.48
 [803,] 5.50
 [804,] 5.40
 [805,] 5.64
 [806,] 5.60
 [807,] 5.52
 [808,] 5.42
 [809,] 5.50
 [810,] 5.70
 [811,] 5.52
 [812,] 5.50
 [813,] 5.38
 [814,] 5.36
 [815,] 5.38
 [816,] 5.62
 [817,] 5.24
 [818,] 5.56
 [819,] 5.40
 [820,] 5.44
 [821,] 5.50
 [822,] 5.62
 [823,] 5.36
 [824,] 5.56
 [825,] 5.48
 [826,] 5.40
 [827,] 5.50
 [828,] 5.48
 [829,] 5.50
 [830,] 5.32
 [831,] 5.54
 [832,] 5.58
 [833,] 5.66
 [834,] 5.54
 [835,] 5.50
 [836,] 5.82
 [837,] 5.72
 [838,] 5.36
 [839,] 5.54
 [840,] 5.62
 [841,] 5.28
 [842,] 5.48
 [843,] 5.50
 [844,] 5.50
 [845,] 5.60
 [846,] 5.38
 [847,] 5.36
 [848,] 5.34
 [849,] 5.36
 [850,] 5.38
 [851,] 5.66
 [852,] 5.50
 [853,] 5.34
 [854,] 5.38
 [855,] 5.60
 [856,] 5.40
 [857,] 5.24
 [858,] 5.66
 [859,] 5.74
 [860,] 5.66
 [861,] 5.48
 [862,] 5.50
 [863,] 5.46
 [864,] 5.70
 [865,] 5.66
 [866,] 5.46
 [867,] 5.42
 [868,] 5.30
 [869,] 5.50
 [870,] 5.48
 [871,] 5.36
 [872,] 5.38
 [873,] 5.48
 [874,] 5.58
 [875,] 5.42
 [876,] 5.50
 [877,] 5.54
 [878,] 5.50
 [879,] 5.54
 [880,] 5.22
 [881,] 5.28
 [882,] 5.26
 [883,] 5.32
 [884,] 5.50
 [885,] 5.48
 [886,] 5.64
 [887,] 5.40
 [888,] 5.56
 [889,] 5.36
 [890,] 5.64
 [891,] 5.64
 [892,] 5.38
 [893,] 5.32
 [894,] 5.40
 [895,] 5.36
 [896,] 5.48
 [897,] 5.34
 [898,] 5.68
 [899,] 5.50
 [900,] 5.82
 [901,] 5.74
 [902,] 5.56
 [903,] 5.52
 [904,] 5.46
 [905,] 5.54
 [906,] 5.94
 [907,] 5.78
 [908,] 5.38
 [909,] 5.78
 [910,] 5.38
 [911,] 5.30
 [912,] 5.36
 [913,] 5.62
 [914,] 5.50
 [915,] 5.54
 [916,] 5.32
 [917,] 5.26
 [918,] 5.38
 [919,] 5.36
 [920,] 5.84
 [921,] 5.42
 [922,] 5.52
 [923,] 5.50
 [924,] 5.68
 [925,] 5.30
 [926,] 5.46
 [927,] 5.48
 [928,] 5.64
 [929,] 5.46
 [930,] 5.82
 [931,] 5.30
 [932,] 5.72
 [933,] 5.82
 [934,] 5.46
 [935,] 5.32
 [936,] 5.64
 [937,] 5.44
 [938,] 5.64
 [939,] 5.40
 [940,] 5.48
 [941,] 5.48
 [942,] 5.54
 [943,] 5.54
 [944,] 5.30
 [945,] 5.66
 [946,] 5.50
 [947,] 5.48
 [948,] 5.44
 [949,] 5.46
 [950,] 5.46
 [951,] 5.68
 [952,] 5.92
 [953,] 5.46
 [954,] 5.74
 [955,] 5.40
 [956,] 5.64
 [957,] 5.42
 [958,] 5.72
 [959,] 5.56
 [960,] 5.50
 [961,] 5.34
 [962,] 5.36
 [963,] 5.70
 [964,] 5.44
 [965,] 5.52
 [966,] 5.64
 [967,] 5.42
 [968,] 5.48
 [969,] 5.72
 [970,] 5.58
 [971,] 5.30
 [972,] 5.44
 [973,] 5.36
 [974,] 5.54
 [975,] 5.66
 [976,] 5.66
 [977,] 5.32
 [978,] 5.56
 [979,] 5.62
 [980,] 5.26
 [981,] 5.48
 [982,] 5.54
 [983,] 5.38
 [984,] 5.60
 [985,] 5.68
 [986,] 5.64
 [987,] 5.48
 [988,] 5.70
 [989,] 5.92
 [990,] 5.42
 [991,] 5.34
 [992,] 5.48
 [993,] 5.72
 [994,] 5.54
 [995,] 5.34
 [996,] 5.46
 [997,] 5.84
 [998,] 5.24
 [999,] 5.46
[1000,] 5.80

Original sample mean

mean(height)

[1] 5.5

12 / 23

Make a histogram of the means of the 1000 bootstrap samples

df <- data.frame(mean=result$t)
library(ggplot2)
ggplot(df, aes(x=mean)) + geom_histogram(col="white")

13 / 23

Understanding the output

result


ORDINARY NONPARAMETRIC BOOTSTRAP
Call:
boot(data = height, statistic = samplemean, R = 1000)
Bootstrap Statistics :
    original  bias    std. error
t1*      5.5 0.00528   0.1402647

mean(height)

[1] 5.5

mean(result$t) - mean(height)

[1] 0.00528

sd(result$t)

[1] 0.1402647

14 / 23

Confidence intervals via bootstrap (cont.)

Method 1:

The 95% bootstrap confidence interval is

c(sort(result$t)[25],sort(result$t)[975])

[1] 5.26 5.80

Method 2:

boot.ci(result, type="all")

Warning in boot.ci(result, type = "all"): bootstrap variances needed for
studentized intervals

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1000 bootstrap replicates
CALL : 
boot.ci(boot.out = result, type = "all")
Intervals : 
Level      Normal              Basic         
95%   ( 5.220,  5.770 )   ( 5.200,  5.739 )  
Level     Percentile            BCa          
95%   ( 5.261,  5.800 )   ( 5.280,  5.820 )  
Calculations and Intervals on Original Scale

15 / 23

Confidence intervals via bootstrap (cont.)

Normal
Basic
Percentile
BCa (“Bias Corrected and Accelerated)

Reading here: https://www.r-bloggers.com/2019/09/understanding-bootstrap-confidence-interval-output-from-the-r-boot-package/

16 / 23

Hypothesis testing

$H_{0} : μ = 5$

$H_{1} : μ \neq 5$

boot.ci(result, type="all")

Warning in boot.ci(result, type = "all"): bootstrap variances needed for
studentized intervals

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1000 bootstrap replicates
CALL : 
boot.ci(boot.out = result, type = "all")
Intervals : 
Level      Normal              Basic         
95%   ( 5.220,  5.770 )   ( 5.200,  5.739 )  
Level     Percentile            BCa          
95%   ( 5.261,  5.800 )   ( 5.280,  5.820 )  
Calculations and Intervals on Original Scale

5 is outside the interval. Hence, $H_{0}$ would be rejected under 0.05 level of significance. We can conclude that population mean is significantly different from 5.

17 / 23

Your turn

Compute bootstrap confidence interval for median.

data: heights

18 / 23

Answer

samplemedian <- function(data, indices) {
  return(median(data[indices]))
}
resultmedian <-  boot(data = height, statistic = samplemedian, R = 1000)
resultmedian


ORDINARY NONPARAMETRIC BOOTSTRAP
Call:
boot(data = height, statistic = samplemedian, R = 1000)
Bootstrap Statistics :
    original  bias    std. error
t1*      5.4  0.0348   0.1904343

19 / 23

Answer (cont.)

boot.ci(resultmedian, type="all")

Warning in boot.ci(resultmedian, type = "all"): bootstrap variances needed for
studentized intervals

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 1000 bootstrap replicates
CALL : 
boot.ci(boot.out = resultmedian, type = "all")
Intervals : 
Level      Normal              Basic         
95%   ( 4.992,  5.738 )   ( 4.700,  5.600 )  
Level     Percentile            BCa          
95%   ( 5.2,  6.1 )   ( 5.2,  5.5 )  
Calculations and Intervals on Original Scale
Some BCa intervals may be unstable

20 / 23

Jackknife Approach

Unlike bootstrap, jackknife is an iterative process. A parameter is calculated on the whole dataset and it is repeatedly recalculated by removing an element one after another.
The main application of jackknife is to reduce bias and evaluate variance for an estimator.

21 / 23

Exercise

Construct

CIs for median Sepal.Length,
CIs for median Sepal.Width and
CIs for Spearman's rank correlation coefficient between Sepal.Length and Sepal.Width

using bootstrap sampling.

Data: iris

22 / 23

Slides available at: https://thiyanga.netlify.app/courses/rmsc2020/contentr/

23 / 23

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