# [R] Bootstrap P-Value

Greg Snow 538280 @end|ng |rom gm@||@com
Fri Nov 6 18:34:48 CET 2020

```A p-value is for testing a specific null hypothesis, but you do not

It is the null value that needs to be subtracted from the bootstrap
differences, not the observed difference.  By subtracting the observed
difference you are setting a situation where the p-value will always
be about 0.5 or about 1 (depending on 1 tailed or 2 tailed).  If
instead you subtract a null value (such as 0), then the p-values will
be closer to what you are expecting.

On Fri, Nov 6, 2020 at 9:44 AM AbouEl-Makarim Aboueissa
<abouelmakarim1962 using gmail.com> wrote:
>
> *Dear All:*
>
> *I am trying to compute the p-value of the bootstrap test; please see
> below.*
>
> *In example 1 the p-value agrees with the confidence interval.*
> *BUT, in example 2  the p-value DOES NOT agree with the confidence
> interval. In Example 2, the p-value should be zero or close to zero.*
>
> *I am not sure what went wrong, or not sure if I missed something.*
>
> *any help would be appreciated.*
>
>
> *with many thanks*
> *abou*
>
>
>
> #####  Two - Sample Bootstrap
>
> #####  Source:
> http://www.ievbras.ru/ecostat/Kiril/R/Biblio_N/R_Eng/Chernick2011.pdf
>
> #####  Example 1:
> #####  ----------
>
>
>
> set.seed(1)
>
> n1 <- 29
> n1
> x1 <- rnorm(n1, 1.143, 0.164) #some random normal variates: mean1 = 1.143
> x1
>
> n2 <- 33
> n2
> x2 <- rnorm(n2, 1.175, 0.169) #2nd random sample: mean2 = 1.175
> x2
>
> obs.diff.theta <- mean(x1) - mean(x2)
> obs.diff.theta
>
> theta <- as.vector(NULL) #### vector to hold difference estimates
>
> iterations <- 1000
>
> for (i in 1:1000) {                        #bootstrap resamples
>  xx1 <- sample(x1, n1, replace = TRUE)
>  xx2 <- sample(x2, n2, replace = TRUE)
>  theta[i] <- mean(xx1) - mean(xx2)
>  }
>
>
>
> ##### Confidence Interval:
> ##### --------------------
>
>
> quantile(theta, probs = c(.025,0.975)) #Efron percentile CI on difference
> in means
>
> ##### 2.5% 97.5%
> ##### - 0.1248539 0.0137601
>
>
> ##### P-Value
> ##### -------
>
> p.value <- (sum (abs(theta) >= obs.diff.theta) + 1)/ (iterations+1)
>
> #####  p.value <- (sum (theta >= obs.diff.theta) + 1)/ (iterations+1)
>
> p.value
>
>
>
> #### R OUTPUT
>
> #### > quantile(theta, probs = c(.025,0.975))
> ####        2.5%       97.5%
> #### -0.12647744  0.02099391
>
> #### > p.value <- (sum (abs(theta) >= obs.diff.theta) + 1)/ (iterations+1)
> #### > p.value
> #### [1] 1
>
> #####  Example 2:
> #####  ----------
>
>
> set.seed(5)
>
> n1 <- 29
> ### n1
> x1 <- rnorm(n1, 10.5, 0.15) ######   sample 1 with mean1 = 10.5
> ### x1
>
> n2 <- 33
> ### n2
> x2 <- rnorm(n2, 1.5, 0.155) #####  Sample 2 with mean2 = 1.5
> ### x2
>
> obs.diff.theta <- mean(x1) - mean(x2)
> obs.diff.theta
>
> theta <- as.vector(NULL) #### vector to hold difference estimates
>
> iterations <- 1000
>
> #####   bootstrap resamples
>
> for (i in 1:1000) {
>  xx1 <- sample(x1, n1, replace = TRUE)
>  xx2 <- sample(x2, n2, replace = TRUE)
>  theta[i] <- mean(xx1) - mean(xx2)
>  }
>
>
>
> ##### Confidence Interval:
> ##### --------------------
>
>
> ######  CI on difference in means
>
> quantile(theta, probs = c(.025,0.975))
>
>
>
> ##### P-Value
> ##### -------
>
> p.value <- (sum (abs(theta) >= obs.diff.theta) + 1)/ (iterations+1)
>
> ##### p.value <- (sum (theta >= obs.diff.theta) + 1)/ (iterations+1)
>
> p.value
>
> ##### R OUTPUT
>
> ####   > ######  CI on difference in means
> ####   >
> ####   > quantile(theta, probs = c(.025,0.975))
> ####       2.5%    97.5%
> ####   8.908398 9.060601
>
> ####   > ##### P-Value
> ####   > p.value <- (sum (abs(theta) >= obs.diff.theta) + 1)/ (iterations+1)
>
> ####   > p.value
> ####   [1] 0.4835165
>
> ______________________
>
>
> *AbouEl-Makarim Aboueissa, PhD*
>
> *Professor, Statistics and Data Science*
>
> *Department of Mathematics and Statistics*
> *University of Southern Maine*
>
>         [[alternative HTML version deleted]]
>
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