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Distribution_Sampling.md

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Distribution Sampling and Hypothesis Testing

Jesse Cambon 02 February, 2021

References: * http://appliedpredictivemodeling.com/data * http://faculty.marshall.usc.edu/gareth-james/ISL/data.html

library(tidyverse)
library(bayestestR)
library(BayesFactor)
library(jcolors)
library(infer)
library(broom)
library(knitr)

set.seed(42) # for reproducibility

Perform sampling

bernouli_sample <- rbernoulli(10,p=0.9) # T/F
uniform = runif(10,-4,4)

num_rows <- 1000

dist <- 
  tibble(
    cauchy=rcauchy(num_rows,0,0.5),
    norm_sample = rnorm(num_rows,0,0.5),
    beta_sample = rbeta(num_rows,0,1)
) %>%
  pivot_longer(everything(),values_to='value',names_to='distribution')


# Distributions used for count data
count_dist <- tibble(poisson= rpois(num_rows,2),
                      `negative binomial`=rnbinom(num_rows,1,mu=2),
                      binom_sample = rbinom(num_rows,9,.25),
                      weibull=rweibull(num_rows,1.4)
                      ) %>%
  pivot_longer(everything(),values_to='value',names_to='distribution')

Compare some distributions

ggplot(data=dist,aes(x=value,color=distribution)) + 
#  facet_wrap(~distribution,ncol=1) +
  scale_x_continuous(limits =c(-3,3)) +
  theme_minimal() +
  theme(legend.position='top') +
  geom_density(alpha=0.8) +
  scale_color_jcolors('default') + 
  xlab('') + ylab('')
## Warning: Removed 116 rows containing non-finite values (stat_density).

Poisson v Neg Binomial v Weibull

ggplot(data=count_dist,aes(x=value,color=distribution)) + 
#  facet_wrap(~distribution,ncol=1) +
  scale_x_continuous(limits =c(0,8)) +
  theme_minimal() +
  theme(legend.position='top') +
  geom_density(alpha=0.8) +
  scale_color_jcolors('default') + 
  xlab('') + ylab('')
## Warning: Removed 25 rows containing non-finite values (stat_density).

Significance Testing

T-test (Frequentist version)

t.test(trees$Height)
## 
##  One Sample t-test
## 
## data:  trees$Height
## t = 66.41, df = 30, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  73.6628 78.3372
## sample estimates:
## mean of x 
##        76

Simulate some data and run more T-tests

compare_norms <- rnorm(100,25,10) %>%
  as_tibble() %>% rename(sample1=value) %>%
  mutate(sample2 = rnorm(100,28,10))

results <- t.test(compare_norms$sample1,compare_norms$sample2)
results
## 
##  Welch Two Sample t-test
## 
## data:  compare_norms$sample1 and compare_norms$sample2
## t = 1.4176, df = 197.32, p-value = 0.1579
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.7904857  4.8324013
## sample estimates:
## mean of x mean of y 
##  28.30323  26.28227

Tidy T-test (infer package)

https://infer.netlify.app/

compare_norms_long <- 
  compare_norms %>%
  pivot_longer(everything(),names_to='sample',values_to='value')

compare_norms_long %>%
  t_test(value ~ sample,order=c('sample1','sample2')) %>%
  kable()
statistic t_df p_value alternative lower_ci upper_ci
1.417581 197.3181 0.1578903 two.sided -0.7904857 4.832401

Bayesian T-test

https://easystats.github.io/bayestestR/articles/example2.html

bayes_result <- BayesFactor::ttestBF(compare_norms$sample1,compare_norms$sample2)
bayes_result
## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 0.3932028 ±0%
## 
## Against denominator:
##   Null, mu1-mu2 = 0 
## ---
## Bayes factor type: BFindepSample, JZS

describe_posterior(bayes_result) %>% kable()
Parameter Median CI CI_low CI_high pd ROPE_CI ROPE_low ROPE_high ROPE_Percentage BF Prior_Distribution Prior_Location Prior_Scale
Difference -1.885353 89 -4.149145 0.2915764 0.91525 89 -1.010632 1.010632 0.232519 0.3932028 cauchy 0 0.7071068