Correlation Regression

Causation

Causation

Causation is any cause that produces an effect.
This means that when

Correlation

Correlation measures the relationship between two things.
Positive correlations happen when one

Correlation

Correlations happen when:
A causes B
B causes A
A and B are consequences

Causation and Correlation

Causation and correlation can happen at the same time.
But

Correlation or Causation?

As people’s happiness level increases, so does their helpfulness.

This

Correlation or Causation?

Dogs pant to cool themselves down.

This would be a

Correlation or Causation?

Among babies, those who are held more tend to

Let's think of our own

Correlation:

Causation:

Quick Review

Causation is any cause that produces an effect.

Correlation measure the

Correlation

The Question

Are two variables related?
Does one increase as the other increases?
e.

Scatterplots

Graphically depicts the relationship between two variables in two dimensional space.

Direct Relationship

Inverse Relationship

An Example

Does smoking cigarettes increase systolic blood pressure?
Plotting number of cigarettes

Trend?

Smoking and BP

Note relationship is moderate, but real.
Why do we care

Heart Disease and Cigarettes

Data on heart disease and cigarette smoking in

The Data

Surprisingly, the U.S. is the first country on the list--the

Scatterplot of Heart Disease

CHD Mortality goes on Y axis
Why?
Cigarette consumption on

{X = 6, Y = 11}

What Does the Scatterplot Show?

As smoking increases, so does coronary heart

Correlation

Co-relation
The relationship between two variables
Measured with a correlation coefficient
Most popularly seen

Types of Correlation

Positive correlation
High values of X tend to be associated

Correlation Coefficient

A measure of degree of relationship.
Between 1 and -1
Sign refers

Covariance

The formula for co-variance is:
How this works, and why?
When would covXY

Example

Example

What the heck is a covariance?
I thought we were talking

Correlation Coefficient

Pearson’s Product Moment Correlation
Symbolized by r
Covariance ÷ (product of the

Calculation for Example

CovXY = 11.12
sX = 2.33
sY = 6.69

Example

Correlation = .713
Sign is positive
Why?
If sign were negative
What would it mean?
Would

Factors Affecting r

Range restrictions
Looking at only a small portion of the

Factors Affecting r

Outliers
Overestimate Correlation
Underestimate Correlation

Countries With Low Consumptions

Outliers

Testing Correlations

So you have a correlation. Now what?
In terms of magnitude,

Regression

„Regression” refers to the process of fitting a simple line to

What is regression?

How do we predict one variable from another?
How does

Linear Regression

A technique we use to predict the most likely score

Linear Regression: Parts

Y - the variables you are predicting
i.e. dependent variable
X

Why Do We Care?

We may want to make a prediction.
More likely,

An Example

Cigarettes and CHD Mortality again
Data repeated on next slide
We want

The Data

Based on the data we have what would we predict

For a country that smokes 6 C/A/D…

We predict a CHD rate

Regression Line

Formula
= the predicted value of Y (e.g. CHD mortality)
X

Regression Coefficients

“Coefficients” are a and b
b = slope
Change in predicted

Calculation

Slope
Intercept

For Our Data

CovXY = 11.12
s2X = 2.332 = 5.447
b = 11.12/5.447

Note:

The values we obtained are shown on printout.
The intercept is the

Making a Prediction

Second, once we know the relationship we can predict
We

Accuracy of Prediction

Finnish smokers smoke 6 C/A/D
We predict:
They actually have 23

Cigarette Consumption per Adult per Day

12

10

8

6

4

2

CHD Mortality per 10,000

30

20

10

0

Residual

Prediction

Residuals

When we predict Ŷ for a given X, we will sometimes

Minimizing Residuals

Again, the problem lies with this definition of the mean:
So,