Correlation Coefficients

• Pearson’s Product Moment Correlation

Coefficient

interval or ratio data only

• What about ordinal data?

Spearman’s Rank Correlation Coefficient

rs = 1 - di

2i=1

i=n

n3 - n

6

http://www.mnstate.edu/wasson/ed602spearcorr.htm

Spearman’s Rank Correlation Coefficient: Example

Spearman’s Rank Correlation Coefficient: Example

A Significance Test for rs

SErs =

1

n -1

ttest = rs

SErs

= rs n -1

df = n - 1

http://www.mnstate.edu/wasson/ed602spearcorr.htm

Spearman’s Rank Correlation Coefficient: Example

Pearson’s r - Assumptions

1. Interval or ratio scale data

2. Selected randomly

3. Linear

4. Joint bivariate normal distribution

S-Plus (qqnorm)

Spearman’s Rank Correlation Coefficient

• Ordinal data

• already in a ranked form

• Interval or ratio data

• convert it to rankings

Spearman’s Rank Correlation Coefficient

TVDI (x)0.2740.5420.4190.2860.3740.4890.6230.5060.7680.725

Rank (x)17423586

109

Theta (y)0.4140.3590.3960.4580.3500.3570.2550.1890.1710.119

Rank (y)978

10564321

Difference (di)

-80-4-8-2-14388

A Significance Test for rs

S-Plus

http://www.mnstate.edu/wasson/ed602spearcorr.htm

TVDI (x)0.2740.5420.4190.2860.3740.4890.6230.5060.7680.725

Theta (y)0.4140.3590.3960.4580.3500.3570.2550.1890.1710.119

• Correlation Direction & Strength

• We might wish to go a little further

• Rate of change

• Predictability

Correlation Regression

• Deterministic

perfect knowledge

• Probabilistic

estimate

not with absolute accuracy

(or certainty)

Two Sorts of Bivariate Relationships

• Travel at a constant speed

• Deterministic time spent driving vs. distance traveled

A Deterministic Relationship

s = s0 + vt

s: distance traveleds0: initial distancev: speedt: time traveledtime (t)

distance (s)

slope (v)

intercept (s0)

• Truly deterministic rare

• More often probabilistic

• e.g., ages vs. heights (2 – 20 yrs)

A Probabilistic Relationship

age (years)

height (meters)

• Good relationship

• Unpredictability or error

Sampling and Regression

• Our expectation (less than perfect)

• Collecting data measurement errors

height

• Other factors (not accounted for in the model)

plant growth vs. T

Simple vs. Multiple Regression

• Simple linear regression

y

x

• Multiple linear regression

y

x1, x2, … xn

• Model

y = a + bx + e

Simple Linear Regression

x: independent variable

y: dependent variable

b: slope

a: intercepte: error term

x (independent)

y (dependent)

b

a

error:

• Scatterplot fitting a line

Fitting a Line to a Set of Points

x (independent)

y (dependent)

• Least squares method

• Minimize the error term e

Sampling and Regression

• Sampled data model

y = a + bx + e

• Attempt to estimate a “true” regression line

y = + x +

• Multiple samples several similar regression

lines the population regression line

• Minimize the error term e

• The line of best fit

ŷ = a + b

Least Squares Method

y

ŷ = a + bxŷ

(y - ŷ)

Estimates and Residuals

• Errors

e = y – ŷ

• Residuals

Underestimate

Overestimate

• Errors (residuals)

e = (y - ŷ)

• Overall error

Simply sum these error terms 0

Square the differences and then sum them up to

create a useful estimate

Minimizing the Error Term

SSE = (y - ŷ)2

i = 1

n

Minimizing the SSE

(y - ŷ)2

i = 1

n

mina,b

n

(yi - a - bxi)2

i = 1

mina,b

=

• Least squares method

Finding Regression Coefficients

(xi - x) (yi - y)i = 1

n

b =

(xi - x)2

i = 1

n

a = y - bx

Interpreting Slope (b)

• Slope of the line (b the change in y due to a unit change in x

b > 0 b < 0

Regression Slope and Correlation

(xi - x)(yi - y)i=1

i=n

(n - 1) sXsY

r =(xi - x) (yi - y)i = 1

n

b =

(xi - x)2

i = 1

n

b = rsy

sx