---
title: "Residual Diagnostics"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Residual Diagnostics}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, echo=FALSE, message=FALSE}
library(olsrr)
library(ggplot2)
library(gridExtra)
library(nortest)
library(goftest)
```

# Introduction

olsrr offers tools for detecting violation of standard regression assumptions. Here we take a look 
at residual diagnostics. The standard regression assumptions include the following about residuals/errors:

- The error has a normal distribution (normality assumption).
- The errors have mean zero.
- The errors have same but unknown variance (homoscedasticity assumption).
- The error are independent of each other (independent errors assumption). 

## Residual QQ Plot

Graph for detecting violation of normality assumption.

```{r qqresid, fig.width=5, fig.height=5, fig.align='center'}
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_plot_resid_qq(model)
```

## Residual Normality Test

Test for detecting violation of normality assumption.

```{r normtest}
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_normality(model)
```

Correlation between observed residuals and expected residuals under normality.

```{r corrtest}
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_test_correlation(model)
```

## Residual vs Fitted Values Plot

It is a scatter plot of residuals on the y axis and fitted values on the x axis to detect non-linearity, unequal error variances, and outliers. 

**Characteristics of a well behaved residual vs fitted plot:**

- The residuals spread randomly around the 0 line indicating that the relationship is linear.
- The residuals form an approximate horizontal band around the 0 line indicating homogeneity of error variance.
- No one residual is visibly away from the random pattern of the residuals indicating that there are no outliers.

```{r rvsfplot, fig.width=5, fig.height=5, fig.align='center'}
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_plot_resid_fit(model)
```


## Residual Histogram

Histogram of residuals for detecting violation of normality assumption.

```{r residhist, fig.width=5, fig.height=5, fig.align='center'}
model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
ols_plot_resid_hist(model)
```