Basic Functions to Statistical Methods Course.
statBasics
R package of the course Métodos Estatísticos at Federal University of Bahia.
Confidence interval for a single population parameter
Confidence interval are computed using the methods presented by Montgomery and Runger (2010). All implemented methods are slighted modification of methods already implemented in stats. The user can compute bilateral and unilateral confidence intervals.
Confidence interval for a population proportion
In this package, there are three approaches for computing a confidence interval for a population proportion.
Number of successes in n (scalar value) trials
library(tidyverse)
library(statBasics)
size <- 1000
sample <- rbinom(size, 1, prob = 0.5)
n_success <- sum(sample)
ci_1pop_bern(n_success, size, conf_level = 0.99)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 0.475 0.557 0.99
Number of successes in a vector
library(tidyverse)
library(statBasics)
n <- c(30, 20, 10)
x <- n |> map_int(~ sum(rbinom(1, size = .x, prob = 0.75)))
ci_1pop_bern(x, n, conf_level = 0.99)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 0.500 0.833 0.99
Vector of successes
library(tidyverse)
library(statBasics)
x <- rbinom(50, size = 1, prob = 0.75)
ci_1pop_bern(x, conf_level = 0.99)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 0.658 1 0.99
Confidence interval for a populationa mean (normal distribution)
We illustrate how to compute a confidence interval for the mean of a normal distribution in two cases: 1) the standard deviation is known; 2) the standard deviation is unknown.
Known standard deviation
library(tidyverse)
library(statBasics)
media_pop <- 10
sd_pop <- 2
x <- rnorm(100, mean = media_pop, sd = sd_pop)
ci_1pop_norm(x, sd_pop = sd_pop, conf_level = 0.91)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 9.62 10.3 0.91
Unknown standard deviation
library(tidyverse)
library(statBasics)
media_pop <- 10
sd_pop <- 2
x <- rnorm(100, mean = media_pop)
ci_1pop_norm(x, conf_level = 0.91)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 9.73 10.1 0.91
Confidence interval for a population standard deviation (normal distribution)
library(tidyverse)
library(statBasics)
media_pop <- 10
sd_pop <- 2
x <- rnorm(100, mean = media_pop, sd = sd_pop)
ci_1pop_norm(x, parameter = 'variance', conf_level = 0.91)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 3.92 6.37 0.91
Confidence interval for a population mean (exponential distribution)
library(tidyverse)
library(statBasics)
media_pop <- 800
taxa_pop <- 1 / media_pop
x <- rexp(100, rate = taxa_pop)
ci_1pop_exp(x)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 739. 1095. 0.95
Confidence interval for a population mean (general case)
In the general case, a confidence interval using the t-Student distribution is still suitable, even if the distribution is not normal, as illustrated in the example bellow.
library(tidyverse)
library(statBasics)
media_pop <- 50
x <- rpois(100, lambda = media_pop)
ci_1pop_general(x)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 47.8 50.7 0.95
Hypothesis testing for a single population parameter
Next, we will illustrate how to use this package to test a statistical hypothesis about a single population parameter. All methods are already implemented in R. This package provides slight modifications for teaching purposes.
Hypothesis testing for a population mean
In the examples below, mean_null is the mean in the null hypothesis H0:
alternative == "two.sided":H0: mu == mean_nullandH1: mu != mean_null. Default value.alternative == "less":H0: mu >= mean_nullandH1: mu < mean_nullalternative == "greater": `H0: mu =< mean_null` and `H1: mu > mean_null`
Normal distribution with known standard deviation
library(tidyverse)
library(statBasics)
mean_null <- 5
sd_pop <- 2
x <- rnorm(100, mean = 10, sd = sd_pop)
ht_1pop_mean(x, mu = mean_null, conf_level = 0.95, sd_pop = sd_pop, alternative = "two.sided")
#> # A tibble: 1 × 10
#> statistic p_value critical_value critical_region alternative mu sig_level
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl> <dbl>
#> 1 24.4 0 1.96 (-Inf,-1.960)U(1… two.sided 5 0.05
#> # … with 3 more variables: lower_ci <dbl>, upper_ci <dbl>, conf_level <dbl>
Normal distribution with unknown standard deviation
library(tidyverse)
library(statBasics)
mean_null <- 5
sd_pop <- 2
x <- rnorm(100, mean = 10, sd = sd_pop)
ht_1pop_mean(x, mu = mean_null, conf_level = 0.95, alternative = "two.sided")
#> # A tibble: 1 × 10
#> statistic p_value critical_value critical_region alternative mu sig_level
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl> <dbl>
#> 1 24.5 0 1.98 (-Inf,-1.984)U(1… two.sided 5 0.05
#> # … with 3 more variables: lower_ci <dbl>, upper_ci <dbl>, conf_level <dbl>
Hypothesis testing for a population standard deviation
In the examples below, sigma_null is the standard deviation in the null hypothesis H0:
alternative == "two.sided":H0: sigma == sigma_nullandH1: sigma != sigma_null. Default value.alternative == "less":H0: sigma >= sigma_nullandH1: sigma < sigma_nullalternative == "greater": `H0: sigma =< sigma_null` and `H1: sigma > sigma_null`
library(tidyverse)
library(statBasics)
sigma_null <- 4
sd_pop <- 2
x <- rnorm(100, mean = 10, sd = sd_pop)
ht_1pop_var(x, sigma = sigma_null, conf_level = 0.95, alternative = "two.sided")
#> # A tibble: 2 × 10
#> statistic p_value critical_value critical_region alternative sigma sig_level
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl> <dbl>
#> 1 27.1 8.75e-14 73.4 (0,73.361)U(128… two.sided 4 0.05
#> 2 27.1 8.75e-14 128. (0,73.361)U(128… two.sided 4 0.05
#> # … with 3 more variables: lower_ci <dbl>, upper_ci <dbl>, conf_level <dbl>
Hypothesis testing for a population proportion
In the examples below, proportion_null is the proportion in the null hypothesis H0:
alternative == "two.sided":H0: proportion == proportion_nullandH1: proportion != proportion_null. Default value.alternative == "less":H0: proportion >= proportion_nullandH1: proportion < proportion_nullalternative == "greater": `H0: proportion =< proportion_null` and `H1: proportion > proportion_null`
Number of successes
The following example illustrates how to perform a hypothesis test when the number of successes (in a number of trials) is a scalar.
library(tidyverse)
library(statBasics)
proportion_null <- 0.1
p0 <- 0.75
x <- rbinom(1, size = 1000, prob = p0)
ht_1pop_prop(x, 1000, proportion = p0, alternative = "two.sided", conf_level = 0.95)
#> # A tibble: 1 × 10
#> statistic p_value critical_value critical_region alternative proportion
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl>
#> 1 1.61 0.108 1.96 (-Inf,-1.960)U(1.960,… two.sided 0.75
#> # … with 4 more variables: sig_level <dbl>, lower_ci <dbl>, upper_ci <dbl>,
#> # conf_level <dbl>
Number of successes in a vector
The example below shows how to perform a hypothesis test when the number of successes (in a number of trials) is a vector. The vector of number of trials must also be provided.
library(tidyverse)
library(statBasics)
proportion_null <- 0.9
p0 <- 0.75
n <- c(10, 20, 30)
x <- n |> map_int(~ rbinom(1, .x, prob = p0))
ht_1pop_prop(x, n, proportion = p0, alternative = "less", conf_level = 0.99)
#> # A tibble: 1 × 10
#> statistic p_value critical_value critical_region alternative proportion
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl>
#> 1 -2.98 0.00143 -1.64 (-Inf,-1.645) less 0.75
#> # … with 4 more variables: sig_level <dbl>, lower_ci <dbl>, upper_ci <dbl>,
#> # conf_level <dbl>
Vector of successes (0 or 1)
The following example shows how to perform a hypothesis test when the number of successes (in a number of trials) is a vector of zeroes and ones.
library(tidyverse)
library(statBasics)
proportion_null <- 0.1
p0 <- 0.75
x <- rbinom(1000, 1, prob = p0)
ht_1pop_prop(x, proportion = p0, alternative = "greater", conf_level = 0.95)
#> # A tibble: 1 × 10
#> statistic p_value critical_value critical_region alternative proportion
#> <dbl> <dbl> <dbl> <chr> <chr> <dbl>
#> 1 -0.803 0.789 1.64 (1.645, Inf) greater 0.75
#> # … with 4 more variables: sig_level <dbl>, lower_ci <dbl>, upper_ci <dbl>,
#> # conf_level <dbl>
Confidence interval for two populations
Bernoulli distribution
In this package, there are two approaches for computing a confidence interval for the difference in proportions.
Confidence Interval – Number of successes
In this case, we have the number of trials (n_x and n_y) and the number of success (x and y) for both popuations.
x <- 3
n_x <- 100
y <- 50
n_y <- 333
ci_2pop_bern(x, y, n_x, n_y)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 -0.232 -0.00840 0.95
Confidence Interval – Vectors of 0 and 1
In this case, we have a vector of 0 and 1 (x and y) for both populations.
x <- rbinom(100, 1, 0.75)
y <- rbinom(500, 1, 0.75)
ci_2pop_bern(x, y)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 -0.119 0.0954 0.95
Normal distribution
In this case, we can build the interval for the difference in means of two populations with known or unknown standard deviations, and we can build the ratio of the variances of two populations. DÚVIDA!
Confidence Interval – Comparing means when standard deviations are unknown
Next, we illustrate how to compute a confidence interval for the difference in means of two populations when population standard deviations are unknown.
x <- rnorm(1000, mean = 0, sd = 2)
y <- rnorm(1000, mean = 0, sd = 1)
# unknown variance and confidence interval for difference of means
ci_2pop_norm(x, y)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 -0.178 0.0962 0.95
Confidence Interval – Comparing means when standard deviations are known
The example below illustrates how to obtain a confidence interval for the difference in means of two populations when population standard deviations are known.
x <- rnorm(1000, mean = 0, sd = 2)
y <- rnorm(1000, mean = 0, sd = 3)
# known variance and confidence interval for difference of means
ci_2pop_norm(x, y, sd_pop_1 = 2, sd_pop_2 = 3)
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 -0.309 0.138 0.95
Confidence Interval – Comparing standard deviations
In thsi case, a confidence interval is obtained by considering the ratio of standard deviations (or variances) of the two populations.
x <- rnorm(1000, mean = 0, sd = 2)
y <- rnorm(1000, mean = 0, sd = 3)
# confidence interval for the variance ratio of 2 populations
ci_2pop_norm(x, y, parameter = "variance")
#> # A tibble: 1 × 3
#> lower_ci upper_ci conf_level
#> <dbl> <dbl> <dbl>
#> 1 0.360 0.461 0.95
Hypothesis testing for two populations
Comparing proportions in two populations
There two approaches to compare the proportions in two populations:
- We have the numbers of sucecss (
xandy) and the numbers of trials (n_xandn_y) for both populations; - We have vector of 1 (success) and 0 (failure) for both populations.
Vector of 1 and 0
In this case, we have a vector of 1 (success) and 0 (failure) for both populations.
x <- rbinom(100, 1, 0.75)
y <- rbinom(500, 1, 0.75)
ht_2pop_prop(x, y)
#> [1] FALSE
#> # A tibble: 1 × 6
#> statistic p_value critical_value critical_region delta alternative
#> <dbl> <dbl> <dbl> <chr> <dbl> <chr>
#> 1 1.23 0.220 1.96 (-Inf,-1.960)U(1.960,Inf) 0 two.sided
Number of successes
In this case, we have the number of success and the number of trials for both populations.
x <- 3
n_x <- 100
y <- 50
n_y <- 333
ht_2pop_prop(x, y, n_x, n_y)
#> [1] FALSE
#> # A tibble: 1 × 6
#> statistic p_value critical_value critical_region delta alternative
#> <dbl> <dbl> <dbl> <chr> <dbl> <chr>
#> 1 -3.21 0.00131 1.96 (-Inf,-1.960)U(1.960,Inf) 0 two.sided
Hypothesis testing for comparing the means of two independent populations
There are three cases to be considere when comparing two means:
t-testunknown but equal variances;t-testunknown and unequal variances;z-testknown variances.
Comparing two means – unknown, equal variances (t-test)
x <- rnorm(1000, mean = 10, sd = 2)
y <- rnorm(500, mean = 5, sd = 2)
# H0: mu_1 - mu_2 == -1 versus H1: mu_1 - mu_2 != -1
ht_2pop_mean(x, y, delta = -1, var_equal = TRUE)
#> # A tibble: 1 × 6
#> statistic p_value critical_value critical_region delta alternative
#> <dbl> <dbl> <dbl> <chr> <dbl> <chr>
#> 1 54.3 0 1.96 (-Inf,-1.962)U(1.962, Inf) -1 two.sided
Comparing two means – unknown, unequal variances (t-test)
x <- rnorm(1000, mean = 10, sd = 2)
y <- rnorm(500, mean = 5, sd = 1)
# H0: mu_1 - mu_2 == -1 versus H1: mu_1 - mu_2 != -1
ht_2pop_mean(x, y, delta = -1)
#> # A tibble: 1 × 6
#> statistic p_value critical_value critical_region delta alternative
#> <dbl> <dbl> <dbl> <chr> <dbl> <chr>
#> 1 77.8 0 1.96 (-Inf,-1.962)U(1.962, Inf) -1 two.sided
Comparing two means – known variances (z-test)
x <- rnorm(1000, mean = 10, sd = 3)
x <- rnorm(500, mean = 5, sd = 1)
# H0: mu_1 - mu_2 >= 0 versus H1: mu_1 - mu_2 < 0
ht_2pop_mean(x, y, delta = 0, sd_pop_1 = 3, sd_pop_2 = 1, alternative = "less")
#> # A tibble: 1 × 6
#> statistic p_value critical_value critical_region delta alternative
#> <dbl> <dbl> <dbl> <chr> <dbl> <chr>
#> 1 -0.368 0.357 -1.64 (-Inf, -1.645) 0 less
Hypothesis testing for comparing variances of two independet populations
x <- rnorm(100, sd = 2)
y <- rnorm(1000, sd = 10)
ht_2pop_var(x, y)
#> # A tibble: 2 × 7
#> statistic p_value critical_vale ratio alternative lower_ci upper_ci
#> <dbl> <dbl> <dbl> <dbl> <chr> <dbl> <dbl>
#> 1 0.0532 1.86e-43 0.733 1 two.sided 0.0535 0.0535
#> 2 0.0532 1.86e-43 1.32 1 two.sided 0.0535 0.0535