(c) 2007 IUPUI SPEA K300 (4392)

Outline

Correlation and CovarianceBivariate Correlation CoefficientTypes of CorrelationCorrelation Coefficient FormulaCorrelation Coefficient ComputationShort-cut FormulaLinear Function (Intercept and Slope)

(c) 2007 IUPUI SPEA K300 (4392)

Correlation and Covariance

It asks how two variables are related When x changes, how does y change?Underlying information is covarianceCov(x,y)=E[(x-xbar)(y-ybar)]Cov(x,y)=Cov(y,x)Cov(x,x)=Var(x), variance is a special

type of covariance (covariance of a variable and itself)

(c) 2007 IUPUI SPEA K300 (4392)

Bivariate Correlation Coefficient

(Karl Pearson product moment) correlation coefficient

Bivariate correlation coefficient (BCC) for two interval/ratio variables

Differentiated from Spearman’s rank correlation coefficient (nonparametric)

Differentiated from partial correlation coefficient that controls the impact of other variables

No causal relationship imposed. XY or YX BCC is used for prediction

(c) 2007 IUPUI SPEA K300 (4392)

Bivariate Correlation Coefficient

BCC ranges from -1 to 1 (So does Gamma γ) Covariance component can be negative + means positive relationship; when x

increases 1 unit, y increases r unit 0 means no relationship. - means negative relationship; when x

increases 1 unit, y decreases r unit. http://noppa5.pc.helsinki.fi/koe/corr/cor7.html

(c) 2007 IUPUI SPEA K300 (4392)

Positive relationship

01

23

45

y

0 1 2 3 4 5x

r=1.0 (positive relationship)

(c) 2007 IUPUI SPEA K300 (4392)

Negative relationship

01

23

45

y

0 1 2 3 4 5x

r=-1.0 (negative relationship)

(c) 2007 IUPUI SPEA K300 (4392)

No relationship

1.5

3.5

y

0 1 2 3 4 5x

r=.0 (No relationship)

(c) 2007 IUPUI SPEA K300 (4392)

Correlation Coefficient

Ratio of the covariance component of x and y to the square root of variance components of x and y

n

xxxxxxxxSS iii

n

iiixx

2

22

1

)())((

n

yyyyyyyySS iii

n

iiiyy

2

22

1

)())((

22)()(

))((

yyxx

yyxx

SSSS

SPr

ii

ii

yyxx

xy

n

yxyxyyxxSP iiii

n

iiixy

)())((1

(c) 2007 IUPUI SPEA K300 (4392)

Correlation Coefficient (short-cut)

n

yyn

n

yySS iiiiyy

222

2

n

xxn

n

xxSS iiiixx

222

2

2222

)(

iiii

iiii

yyxx

xy

yynxxn

yxyxn

SSSS

SPr

Textbook suggests a short-cut formula below but it is not recommended.

n

yxyxn

n

yxyxSP iiiiiiiixy

)(

)(

(c) 2007 IUPUI SPEA K300 (4392)

Illustration: example 10-2, p.526

No x y (x-xbar) (y-ybar) (x-xbar)^2 (y-ybar)^2 (x-xbar)(y-ybar)

1 43 128 -14.5 -8.5 210.25 72.25 123.25

2 48 120 -9.5 -16.5 90.25 272.25 156.75

3 56 135 -1.5 -1.5 2.25 2.25 2.25

4 61 143 3.5 6.5 12.25 42.25 22.75

5 67 141 9.5 4.5 90.25 20.25 42.75

6 70 152 12.5 15.5 156.25 240.25 193.75

Sum 345 819     561.5 649.5 541.5

Mean 57.5 137          

SSxx SSyy SPxy

Correlation coefficient 0.8967      

(c) 2007 IUPUI SPEA K300 (4392)

Hypothesis Test

How reliable is a correlation coefficient? r is a random variable drawn from the

sample; ρ is its corresponding parameter H0: ρ =0, Ha: ρ ≠ 0TS follows the t distribution with df=n-2If H0 is not rejected, r is not reliable

regardless of its magnitude (ρ =0)

)2(~1

22

ntr

nrtr

(c) 2007 IUPUI SPEA K300 (4392)

Illustration: Example 10-3, p.529

Step 1. H0: ρ =0, Ha: ρ ≠ 0Step 2. α=.05, df=4 (=6-2), CV=2.776 Step 3. TS=4.059, r=.897Step 4. TS>CV, reject H0 at the .05 levelStep 5. ρ ≠ 0

)2(~059.4897.1

26897.

1

222

ntr

nrtr

(c) 2007 IUPUI SPEA K300 (4392)

Linear function

A function transforms input into output in its own way

Ex: y=square_root(x). Whey you put x (input) into the funciton square_root(), you will get y (output).

Linear function consists of a intercept and linear combinations of variables and their slops. Y= a + bX + cX2…

Slopes are constant

(c) 2007 IUPUI SPEA K300 (4392)

Intercept and Slope of a function

A linear model: Y = a + b XDependent variable Y to be explainedIndependent variable X that explains YY-Intercept a: the coordinate of the point

at which the line intersects Y axis. Slope b: the change of dependent

variable Y per unit change in independent variable X

(c) 2007 IUPUI SPEA K300 (4392)

Illustration

1

.5

12

45

3y

-1 0 1 2 3 4 5x

Y = 2 +.5X