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Stat203 of 28 Fall2011 – Week 9, Lecture 1

Topics for Today

Scatterplots Relationship between 2 Continuous Variables Pearson’s Correlation Facts and Myths Correlation as a Statistic


Two Continuous Variables

Using the 2-sample Chi-square test we were able to investigate the relationship between two discrete variables. Eg: - Radio format and age

- weather and city Now we will examine the relationship between two __________ variables. The first tool we will discuss is called ___________.


but even before that … Scatter Plots Shows the relationship between 2 continuous variables measured on the same ___________. Values of the one variable (X) are plotted on the horizontal axis and values of the other variable (Y) are plotted on the vertical axis. Each individual appears as a single point. Let’s look at this in SPSS …


Let’s look at a dataset called Detroit that has information from the city for years 1961 to 1973. It contains 6 variables:

- year - homicide rate (per 100,000 population) - # of police (per 100,000 population) - unemployment rate (%) - # registered handguns (per 10,000 population) - average weekly income ($)


Let’s create a scatterplot of two of these variables.


A scatterplot of the # of registered handguns and the # of police officers:


let’s look at the first row of the data table, and then identify that point (circle it) in the scatterplot on the previous page:

Each row in the data table corresponds to exactly one point in the scatter plot. What sort of relationship between the # of registered handguns and the # of police officers does this scatterplot show?


Correlation The term ___________ is often used in common language and has a general interpretation as implying a ____________ between two events … including two discrete events: “Autism is correlated with vaccination” … or things that can’t really be measured “there’s a correlation between my mood and my partner’s behavior” However in statistics the term correlation means something specific.


Statistical Correlation

___________ measures the _________ and ________ of a ______ relationship between two continuous variables (X and Y). Pearson’s correlation is the most commonly used:

!

r =(xi " x

i=1

n# )(yi " y )

(xi " x i=1

n# )2$ % &

' ( ) (yi " y

i=1

n# )2$ % &

' ( )

Note:

- this is ONLY a linear relationship - there are many types of relationships that are not

linear


I only give you the formula for completeness; we will not be calculating it by hand (it is extremely tedious).

In this class as in every time you analyze data in the future,

we will make the software calculate the correlation.

However, it is important that you understand that it’s just another statistic calculated from the data, just like the

mean, the standard deviation, or the odds-ratio.


Some Facts about Correlation

1. Correlation can only be used when both variables are interval or ratio level

2. Correlation does not change when we change the units

of measurement of X and Y Height in cm or in will give same correlation to weight in kg or lbs

3. Positive correlation indicates positive association

between the variables and negative correlation indicates negative association

4. Correlation is always between __ and _. Values near 0

indicate a very ____ relationship -1 or 1 will occur only if points fall on a straight line


Examples The following are scatter plots of two variables with the correlation between the two listed above the plot.


Pearson Correlation of 1

As in the definition, correlation is the strength of the linear relationship. All of these figures have the ____ correlation!

Important note! The strength of the correlation doesn’t depend on the slope of the line, just how _______ clustered the points are to a _____________ … any straight line!


Examples of a relationship with Pearson Correlation of 0


Facts in a video

http://www.youtube.com/watch?v=Ypgo4qUBt5o


Let’s do some examples – Correlation guessing

Q15, pg 370 – correlation between poverty and rates of teen pregnancy in 8 US states.

a) [-0.95, -0.5) b) [-0.5, 0) c) (0, 0.5) d) [0.5, 0.95)


Q16, pg 370 (edited) – Hours studied and exam grade

a) [-0.95, -0.5) b) [-0.5, 0) c) (0, 0.5) d) [0.5, 0.95)


Q19, pg 371 – Hours watching TV vs # books read

a) [-0.95, -0.5) b) [-0.5, 0) c) (0, 0.5) d) [0.5, 0.95)


An Example

In which of these two scatter plots is the correlation higher?

-3 -2 -1 0 1 2

-4-2

02

x

y

-5 0 5

-50

5

x

y


The correlation of the x and y in the two figures is _________, only the _____ of the axes is different! Don’t trust your eye, always calculate the correlation. … but don’t trust the correlation … always check by eye.


Myths about Correlation

1. Correlation implies causation

There could be a third, unknown variable which influences both X and Y

2. A correlation coefficient of zero implies no relationship between two variables

WRONG! it only implies no LINEAR relationship!

Remember the funky shaped figures!


Myths explained in video http://www.youtube.com/watch?v=MTbZoKEOkUg http://www.youtube.com/watch?v=VW1IEqKuf6s

(Only to 2:48)


Correlation as a statistic

As with the mean, the Odds Ratio and the other statistics we have looked at, a correlation is a characteristic of a population that we estimate with our ______:

Population (Parameter)

Sample (Statistic)

Mean µ

!

X Proportion

!

p

!

ˆ p Odds Ratio OR Correlation _ _


The r tells part of the story Remember, the correlation (r) we calculate from a sample is only one of the _____________ correlations we could have obtained one of many possible _______. It’s possible that the true population correlation, ρ, has another value … say 0, or ρ0. So … there is some variability of our estimate r, it’s standard error.

!

s ˆ e (r) =1" r2

n " 2


Hypotheses for Associations between Continuous Variables

H0: there is no linear relationship between X and Y

Ha: there is a linear relationship between X and Y

Is the same as:

H0: H0: ρ = 0

Ha: H0: ρ ≠ 0 And as in our other hypotheses tests, we will use a _________ (r ) to approximate a _________ (ρ).


Testing for Correlation = 0 Recall our hypothesis tests for the µ= 0, we used a t-test.

!

t =x " 0se(x )

=x

s / n

If both X and Y are normally distributed, the test for H0: ρ = 0 is very similar:

!

t =r " 0se(r)

=r

1" r2n " 2

and we look up our t value in the appropriate table to find the p-value!


New Topics Covered Today

Pearson’s Correlation • Most commonly calculated correlation statistic • No definition of response or predictor • Always between -1 and 1

Hypothesis testing for Correlation • Does a correlation exist? Reject null = a non-zero correlation

Reading: Chapter 10 up to page 360

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