data_exploration.Rmd

---
title: "Data exploration: the importance of plotting"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Simple example

Consider that you have two variables `x` and `y`. 

```{r create data}
x <- 1:100
y <-  20 * x - 0.2 * x^2 + rnorm(100, 0, 30)
```


You are interested to understand if `x` explains variation in `y`.
How will you approach this? 

## Should you build a linear model first or plot the variables first and visually explore?

Let's examine the outcome and inference if **we do not visually examine our data**
and only rely on regression modeling. 

```{r model}
mod_lin <- lm(y ~ x)
summary(mod_lin)
```

The above model summary table would lead us to believe that there is no relationship
between `x` and `y` which we know is false because we created the variables above.

## Why is it our regression table leading us to an incorrect inference? 

Because our model is systematically mis-representing the functional form of the
relationship between `x` and `y` which we defined to be a quadratic relationship. 
This would have been obvious if we first graphed `y` and `x`

```{r}
plot(y ~ x)
```

This simple step would indicate to us that `y` is not just a linear function of 
`x` but it is a quadratic function, such that a more appropriate model is: 

```{r}
mod_quad <- lm(y ~ x + I(x^2))
summary(mod_quad)
```

To further articulate why the regression only approach failed we should 
overlay the models and the data. Unfortunately, this critical step is often
overlooked.

```{r}
plot(y ~ x)
lines(x, predict(mod_lin), col='red')
lines(x, predict(mod_quad), col='blue')
legend('bottom', c('linear', 'quadratic'), col=c('red', 'blue'), lty=1, bty='n')
```

Here we just looked at two variables so it may be obvious that graphing them is wise, 
but even in multivariate scenarios where a graphical exploration may be more 
tedious it can still be very helpful and illuminating prior to model fitting.