Lecture4Slides.Rmd

---
title: "Stat 201 - Lecture 4"
author: "Prof. Heggeseth"
date: "September 22, 2015"
output: slidy_presentation
---

```{r global_options, include=FALSE}
knitr::opts_chunk$set(dev='png',fig.path='Figs/',echo=TRUE,message=FALSE, warning=FALSE,message=FALSE,tidy=TRUE,fig.width = 6,fig.height=6)
```

# Recap

## Numerical and Graphical Summaries
- One Categorical (frequency tables, bar plots)
- One Quantitative (histogram, boxplot, mean, median, sd, IQR, etc.)
- Two Categorical (contingency table, mosaic plot)
- Categorical and Quantitative (side-by-side boxplot, mean/sd/median/IQR within groups)
- Two Quantitative (TODAY!)

## Intro to Sampling Variability
- Simulating from Population (Babies census)
- Bootstrapping from Sample (Trump poll)

--------

# Normal Model
## Normal Curve
Unimodal and symmetric histograms often look like the Normal curve

$$f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}$$

where $\mu$ and $\sigma$ are parameters of this function that control the center and spread of the curve (corresponding to mean and sd). 

```{r, echo=FALSE}
x = seq(-5,5,by=.01)
plot(x,dnorm(x),type='l',ylab='f(x)',ylim=c(0,.8),lwd=2)
lines(x,dnorm(x,mean=1,sd=1),col='red',lwd=2)
lines(x,dnorm(x,mean=1,sd=2),col='blue',lwd=2)
lines(x,dnorm(x,mean=0,sd=0.5),col='purple',lwd=2)
legend('topright',legend=c(expression(mu==0 ~','~ sigma==1),expression(mu==0~","~sigma==0.5),expression(mu==1~","~ sigma==1),expression(mu==1~","~sigma==2)),col=c('black','purple','red','blue'),lty=1,lwd=2)
```

--------

#Normal Q-Q plots
To check how close a histogram is to a Normal curve, we could use a Normal Quantile-Quantile plot (in addition to eye-balling the histogram). 
```{r}
x = rnorm(100,mean=50,sd=10) #randomly generating data 
qqnorm(x)
qqline(x)
```

Closer to the straight line, the closer to Normal curve.

## Construction (ignore axis labels)
- Y-value: variable value 
- X-value: corresponding "z-score" if histogram was exactly Normal 

**68-95-99.7 rule in action**

- "z-score" = -2: 2.5th percentile (95% in the middle)
- "z-score" = -1: 16th percentile (68% in the middle)
- "z-score" = 0: 50th percentile
- "z-score" = 1: 84th percentile (68% in the middle)
- "z-score" = 2: 97.5th percentile (95% in the middle)

## How to Interpret
- If points are ABOVE the line, they are greater (higher) than expected if it was Normal.

- If points are BELOW the line, they are less than expected if it was Normal.


**Match the Histogram with the Q-Q plot.**

```{r,echo=FALSE}
par(mfrow=c(3,2))
x = rnorm(100)
y = rbeta(100,1,5)
z = rbeta(100,5,1)

hist(x)
qqnorm(y)
qqline(y)

hist(y)
qqnorm(x)
qqline(x)

hist(z)
qqnorm(z)
qqline(z)
```

--------

#Summarizing and Displaying Two Quantitative Variables


```{r}
par(mfrow=c(1,1))
bodyfat <- read.delim("http://sites.williams.edu/rdeveaux/files/2014/09/bodyfat.txt")
bodyfat$Shirt = bodyfat$Neck/2.54 + 0.5
with(bodyfat,plot(Shirt~Weight))
#plot(bodyfat$Shirt~bodyfat$Weight)
#plot(x=bodyfat$Weight,y=bodyfat$Shirt)
```

What do we look for in a relationship between two quantitative variables?

* Direction (postive? negative? flat?)
* Form (linear? curved? nothing? other?)
* Strength 

The relationship between *Weight* and *Shirt Size* is positive, linear, and quite strong. 

I want to quantify the strength of the relationship...
--------

#Changing Units

Suppose instead of *Weight* in pounds and *Shirt Size* in inches, 

I switched to metric and plotted *Weight* in kilograms and *Shirt Size* in cm. 

**Would the strength change?**

```{r}
par(mfrow=c(1,2))
with(bodyfat,plot(Shirt~Weight,xlab="Wt (lbs)",ylab="Shirt (in)"))
with(bodyfat,plot(I(Shirt*2.54)~I(Weight/2.2),xlab="Wt (kg)",ylab="Shirt (cm)")) #I() allows us to do math in the y~x framework

with(bodyfat,plot(Shirt~Weight,xlab="Wt (lbs)",ylab="Shirt (in)",xlim=c(0,250),ylim=c(0,50)))
with(bodyfat,plot(I(Shirt*2.54)~I(Weight/2.2),xlab="Wt (kg)",ylab="Shirt (cm)",xlim=c(0,250),ylim=c(0,50)))
```

--------

#Standardizing
     
Since shifting and scaling make no difference, let's standardize both variables into z-scores and replot (and add some color):

```{r, echo=FALSE}
par(mfrow=c(1,1))
zShirt=with(bodyfat,scale(Shirt))
zWeight=with(bodyfat, scale(Weight))

plot(zShirt~zWeight,type="n")
abline(h=0)
abline(v=0)
points(zShirt[zWeight>0 & zShirt>0]~zWeight[zWeight>0 & zShirt>0],col="blue",pch=19)
points(zShirt[zWeight<0 & zShirt<0]~zWeight[zWeight<0 & zShirt<0],col="blue",pch=19)
points(zShirt[zWeight>0 & zShirt<0]~zWeight[zWeight>0 & zShirt<0],col="red",pch=19)
points(zShirt[zWeight<0 & zShirt>0]~zWeight[zWeight<0 & zShirt>0],col="red",pch=19)
```

**If we were to have a weaker positive relationship, how would this plot change?**

**If we were to have a negative relationship, how would this plot change?**

--------

#Make this Formal
We want one number to represent **strength** and **direction**.

- Points in the $1^{st}$ and $3^{rd}$ quadrants (blue) have the **same sign** (where both $z's$ are positive or both are negative).

- Points in the $2^{nd}$ and $4^{th}$ (red) have opposite signs.

**What if we took the product of the $z$-scores? What would the values look like?**

--------

#Correlation Coefficient

The almost *average* of products of the $z$-scores is the **correlation coefficient,** 

$$ r = \frac{\sum z_x z_y}{n-1} $$

**Which points contribute the most to this average?**

##Calculation in R
```{r}
sum(zShirt*zWeight)/(length(zShirt)-1) #By formula
cor(zShirt,zWeight) #correlation function 
```

--------

#Other expressions for r:

$$ r = \frac{\sum z_x z_y}{n-1}  $$
$$ = \frac{\sum{\frac{(x_i-\bar{x})}{s_x}\frac{(y_i-\bar{y})}{s_y}}}{n-1}$$
$$= \frac{\sum{(x_i-\bar{x})(y_i-\bar{y})}}{(n-1) s_x s_y}$$
$$= \frac{\sum{(x_i-\bar{x})(y_i-\bar{y})}}{{(n-1)\sqrt{\sum{\frac{(x_i-\bar{x})^2}{n-1}}}}{\sqrt{\sum{\frac{(y_i-\bar{y})^2}{n-1}}}}}$$
$$=\frac{\sum{(x_i-\bar{x})(y_i-\bar{y})}}{{\sqrt{\sum{(x_i-\bar{x})^2}}}{\sqrt{\sum{(y_i-\bar{y})^2}}}}$$
$$=\frac{\sum{(x_i-\bar{x})(y_i-\bar{y})}}{{\sqrt{\sum{(x_i-\bar{x})^2\sum{(y_i-\bar{y})^2}}}}}$$

--------

#Properties of Correlation

* $-1 \leq r \leq 1$

* Sign of r goes with the direction of the relationship.

* r(x,y) = r(y,x), it doesn't matter which is x and which is y.

* r(ax+b, cy+d) = r(x,y)   Change of scale (linear) don't affect r. Why? (look at original formula)

* r measures strength of *linear* relationship -- not curved

* one outlier can completely change r

--------

#Guess the Correlation 
http://www.rossmanchance.com/applets/GuessCorrelation.html

--------

#Is correlation always the right way to judge strength?

Here is the plot of brownie quality by temperature.
The correlation is near 0, but it doesn't mean that there's no relationship -- there's no **linear** relationship.


![Brownies](http://sites.williams.edu/rdeveaux/files/2014/09/Brownies.png)


----------

#Outliers !!

And be careful of outliers. One point can turn a near zero correlation large or a very high correlation can change sign with the addition of one outlying point:


![Bozo](http://sites.williams.edu/rdeveaux/files/2014/09/Bozo.png)

----------

#Correlation and Causality

http://xkcd.com/552/

http://www.tylervigen.com/

http://io9.com/on-correlation-causation-and-the-real-cause-of-auti-1494972271