README.Rmd

---
output: github_document
---

<!-- README.md is generated from README.Rmd. Please edit that file -->

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%",
  message = F
)
```

# 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

```{r}
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)
```

#### Number of successes in a vector

```{r}
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)
```

#### Vector of successes

```{r}
library(tidyverse)
library(statBasics)
x <- rbinom(50, size = 1, prob = 0.75)
ci_1pop_bern(x, conf_level = 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

```{r}
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)
```

#### Unknown standard deviation

```{r}
library(tidyverse)
library(statBasics)
media_pop <- 10
sd_pop <- 2
x <- rnorm(100, mean = media_pop)
ci_1pop_norm(x, conf_level = 0.91)
```


### Confidence interval for a population standard deviation (normal distribution)

```{r}
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)
```

### Confidence interval for a population mean (exponential distribution)

```{r}
library(tidyverse)
library(statBasics)
media_pop <- 800
taxa_pop <- 1 / media_pop
x <- rexp(100, rate = taxa_pop)
ci_1pop_exp(x)
```

### 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.

```{r}
library(tidyverse)
library(statBasics)
media_pop <- 50
x <- rpois(100, lambda  = media_pop)
ci_1pop_general(x)
```

## 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`:

1. `alternative == "two.sided"`: `H0: mu == mean_null` and `H1: mu != mean_null`. Default value.
2. `alternative == "less"`: `H0: mu >= mean_null` and `H1: mu < mean_null`
3. `alternative == "greater"`: `H0: mu =< mean_null` and `H1: mu > mean_null`

#### Normal distribution with known standard deviation

```{r}
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")
```

#### Normal distribution with unknown standard deviation

```{r}
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")
```

### Hypothesis testing for a population standard deviation

In the examples below, `sigma_null` is the standard deviation in the null hypothesis `H0`:

1. `alternative == "two.sided"`: `H0: sigma == sigma_null` and `H1: sigma != sigma_null`. Default value.
2. `alternative == "less"`: `H0: sigma >= sigma_null` and `H1: sigma < sigma_null`
3. `alternative == "greater"`: `H0: sigma =< sigma_null` and `H1: sigma > sigma_null`

```{r}
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")

```

### Hypothesis testing for a population proportion

In the examples below, `proportion_null` is the proportion in the null hypothesis `H0`:

1. `alternative == "two.sided"`: `H0: proportion == proportion_null` and `H1: proportion != proportion_null`. Default value.
2. `alternative == "less"`: `H0: proportion >= proportion_null` and `H1: proportion < proportion_null`
3. `alternative == "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.

```{r}
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)
```

#### 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.

```{r}
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)
```

#### 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.

```{r}
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)
```

## 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.

```{r}
x <- 3
n_x <- 100
y <- 50
n_y <- 333
ci_2pop_bern(x, y, n_x, n_y)
```

#### Confidence Interval -- Vectors of 0 and 1

In this case, we have a vector of 0 and 1 (`x` and `y`) for both populations.

```{r}
x <- rbinom(100, 1, 0.75)
y <- rbinom(500, 1, 0.75)
ci_2pop_bern(x, y)
```

### 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.

```{r}
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)
```

#### 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.

```{r}
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)
```

#### 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.

```{r}
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")
```

## 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 (`x` and `y`) and the numbers of trials (`n_x` and `n_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.

```{r}
x <- rbinom(100, 1, 0.75)
y <- rbinom(500, 1, 0.75)
ht_2pop_prop(x, y)
```

#### Number of successes

In this case, we have the number of success and the number of trials for both populations.

```{r}
x <- 3
n_x <- 100
y <- 50
n_y <- 333
ht_2pop_prop(x, y, n_x, n_y)
```

## Hypothesis testing for comparing the means of two independent populations

There are three cases to be considere when comparing two means:

i. `t-test` unknown but equal variances;
i. `t-test` unknown and unequal variances;
i. `z-test` known variances.

### Comparing two means -- unknown, equal variances (`t-test`)

```{r}
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)
```

### Comparing two means -- unknown, unequal variances (`t-test`)

```{r}
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)
```

### Comparing two means -- known variances (`z-test`)

```{r}
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")
```

## Hypothesis testing for comparing variances of two independet populations

```{r}
x <- rnorm(100, sd = 2)
y <- rnorm(1000, sd = 10)
ht_2pop_var(x, y)
```