# Two Numerical Variables (Part 2)

Rebekah Robinson, Georgetown College

Always remember to make sure the necessary packages are loaded:

require(mosaic)
require(tigerstats)


The datasets that will be discussed in this section are:

data(m111survey)
View(m111survey)
help(m111survey)

data(ucdavis1)
View(ucdavis1)
help(ucdavis1)

data(pennstate1)
View(pennstate1)
help(pennstate1)


## Correlation

### Strength of Association

It is important to consider the strength of association as well as the direction.

In the m111survey data, the variables height and fastest have a positive association.

However, height and ideal_ht have a stronger positive association. The points are less scattered.

### Correlation Coefficient

The correlation coefficient is the numerical measure of the direction and strength of the linear association between two numerical variables.

The correlation coefficient, $$r$$, is

$r=\sum (z_x)(z_y)/(n-1)$

where

• $$z_x=(x-\bar{x})/s_x$$

• $$z_y=(y-\bar{y})/s_y$$.

### r in R

The correlation coefficient for height and fastest is

cor(fastest~height,
data=m111survey,
use="na.or.complete")

[1] 0.1709


### r in R

The correlation coefficient for height and ideal_ht is

cor(ideal_ht~height,
data=m111survey,
use="na.or.complete")

[1] 0.832


### Properties of r

• $$r$$ always falls between -1 and 1

• The sign of $$r$$ indicates the direction of the relationship.

• $$r>0$$ indicates a positive linear association.
• $$r<0$$ indicates a negative linear association.
• The magnitude of $$r$$ indicates the strength of the relationship.

• $$r=1$$ indicates a perfect positive linear relationship.
• $$r=-1$$ indicates a perfect negative linear relationship.
• $$r=0$$ indicates no linear relationship.

### Further Investigation

You can further investigate the values of $$r$$ with the following app:

require(manipulate)
VaryCorrelation()


### Idea for Investigation

Research Question: At UC Davis, how is a student's mother's height related to their father's height?

• Response variable: momheight (numerical)

xyplot(momheight~dadheight,data=ucdavis1,
col="blue",pch=19)


### Graphical Investigation

It appears that students with tall dads have tall moms. Students with short dads have short moms.

Since the points do not form a tight cluster, the positive assocation does not appear to be very strong.

### Numerical Investigation

cor(momheight~dadheight,data=ucdavis1,
use="na.or.complete")

[1] 0.2572


## Regression Equation

### Regression Analysis

A linear relationship can be explained using the equation of a line.

Regression equation - the equation of a line used to predict the value of the response variable from a known value of the explanatory variable.

$\hat{y}=a+bx$

• $$a$$ is the $$y$$-intercept.
• $$b$$ is the slope.
• $$x$$ is the known value of the explanatory variable.
• $$\hat{y}$$ is the predicted value of the response variable.

### Regression Line

RQ: At Penn State, how is a student's right handspan related to his/her height?

Each point on the scatterplot, $$(x,y)$$, is a known observation.

Each point on the line, $$(x,\hat{y})$$, is a predicted response for a value of the explanatory variable.

### Residuals

A residual is the vertical distance between a point and the regression line.

$\mbox{residual}=y-\hat{y}$

### Regression Line and Residuals

The regression line is the line that minimizes the sum of the squared residuals.

$\mbox{Sum of Squares} = \sum (\mbox{residuals})^2= \sum(y_i-\hat{y})^2$

Investigate this further with the following app:

require(manipulate)
FindRegLine()


### Linear Model Function

The built-in function, lmGC, outputs the

• correlation coefficient,

• equation of the regression line,

• gives the option to display the graph of the regression line,

• and more.

### Finding the Regression Equation

lmGC(Height~RtSpan,data=pennstate1)


Simple Linear Regression

Correlation coefficient r =  0.6314

Equation of Regression Line:

Height = 41.96 + 1.239 * RtSpan

Residual Standard Error:    s   = 3.149


### Graphing the Regression Line

Including the argument graph=TRUE will output the regression line and provide a graph.

lmGC(Height~RtSpan,data=pennstate1,
graph=TRUE)


### The Result


Simple Linear Regression

Correlation coefficient r =  0.6314

Equation of Regression Line:

Height = 41.96 + 1.239 * RtSpan

Residual Standard Error:    s   = 3.149


### Predictions

What is the predicted height of a Penn State student with a right handspan of 22 cm?

We can use the regression equation

$\hat{y}=41.96+1.239x$

to predict this height by plugging in $$x=22$$.

$\hat{y}=41.96+1.239(22)=69.2$

A Penn State student with a right handspan of 22 cm is predicted to be 69.2 inches tall.

### Predictions using the Predict Function

The predict function will also compute this. The predict function requires two inputs:

• a linear model

• an $$x$$-value.

handheightmod<-lmGC(Height~RtSpan,
data=pennstate1)

predict(handheightmod,x=22)

[1] 69.23


### Interpretation of Regression Line

The regression line is $$\hat{y}=41.9593+1.2394x$$.

The intercept is 41.9593.

The slope is 1.2394.

What do these numbers mean?

### Interpretation of Slope and Intercept

Intercept: 41.9593 is the predicted height of a Penn State student whose right handspan is 0 cm.

The interpretation of the intercept does not always make logical sense!

Slope: The predicted height of a Penn State student changes by 1.2394 inches as right handspan increases by 1 centimeter.

For every one centimeter increase in right handspan, the predicted height increases by 1.2394 inches.

### How well does our line fit?

When there is variation in the data, a line is not a perfect explanation of the relationship.

The variation is measured two ways:

• Residual Standard Error (RSE)

• Squared Correlation ($$r^2$$)

### Residual Standard Error

Let's return to the Penn State data of right handspans and heights.

lmGC(Height~RtSpan,data=pennstate1)


RSE measures the spread of the residuals.

$\mbox{RSE}=3.149$

Warning: RSE is directly affected by a change in scale.

### RSE for Different Units

RSE=3.149


RSE=0.262


### Squared Correlation

The squared correlation, $$r^2$$, is

• another measurement of the explained variation in the scatterplot.

• the proportion of variation in the response variable that is explained by the explanatory variable.

• unaffected by a change in scale.

### Properties of r-squared

• $$r^2$$ is always between 0 and 1.

• $$r^2=1$$ implies perfect correlation between explanatory and response variables.

• $$r^2=0$$ implies no correlation.

### Next Topic

Part 3 will discuss topics of caution when examining relationships between two numerical variables.