Simple Linear Regression and Measures of Correlation

8/4/2019 Simple Linear Regression and Measures of Correlation

1/33

SIMPLE LINEAR

REGRESSION ANDMEASURES OF

CORRELATION

Prepared by:

Carmela S. Obayan


2/33

TOPIC OUTLINE

1. Simple Linear RegressionA. The Scatter Diagram

B. The Least Squares Linear Regression Equation

C. The Standard Error of Estimate

2. Measures of Correlation

A. Description of Correlation

B. Correlation Between Interval Data1. Pearson r from Raw Scores

2. Pearson r Computed from Standard Scores

3. Pearson r Computed by the Method Difference


3/33

TOPIC OUTLINE

C. Correlation Between Ordinal Data1. Spearman Rho

2. Gamma

D. Correlation Between an Interval and NominalData

1. The Correlation Ratio

2. The point-Biseral Ordinal Data

3. The Z-test

4. The t- test


4/33

SCATTERDIAGRAM

A graphical approach in

solving problems that

concern estimation and

forecasting


5/33

SCATTERDIAGRAM

A graphical approach in

solving problems that concern

estimation and forecasting.

Consist of joining points

corresponding to the pairedscores of dependent and

independent variables.


6/33

EXAMPLE: Working Experience and Income

of 8 Employees

Employees Years of WorkingExperience (X) Income in Thousandof Pesos(Y)

A 2 8

B 8 10C 4 11

D 11 13

E 5 9F 13 17

G 4 8

H 15 14


7/33

SCATTERDIAGRAM

Working Experience and Income of 8 Employees


8/33

THE LEAST SQUARE LINEARREGRESSION

If a straight line appears to

describe the relationship, the

regression formula can be used.

Y = a + bx


9/33

THE LEAST SQUARE LINEAR

REGRESSION

b = (X X)(Y Y)

(X X)2

a = Y - bX


10/33

STANDARD ERRORESTIMATE

The distance of the Y values

from Y1.

Se = Y2a(Y)b(XY)

n - 2


11/33

MEASURES OF CORRELATION

When the degree of

relationship is measured,correlation is basically

the test of measurement.


12/33


The two variables tend to

vary together; the

presence of one indicatesthe presence of the

other; one can be

predicted from thepresence of the other.


13/33


The degree of relationship

between variables is expressed

into:1. Perfect Correlation

2. Some degree of Correlation

3. No Correlation


14/33


REMINDERS:

The relationship of two

variables does notnecessarily mean that

one is the cause or the

effect of the other

variable.


15/33


REMINDERS: When the computed r is high,

it does not necessarily mean

that one factor is stronglydependent on the other.

On the other hand, when the

computed r is small it doesnot necessarily mean thatone factor has no dependenceon the other factor.


16/33

EXAMPLE: Given set of scores X and Y,

find Pearson r

X YA 18 10

B 16 14

C 14 8

D 13 12

E 12 10

F 10 8

G 10 7

H 8 6

I 6 5

J 3 0


17/33

THE PEARSON R FROMTHE

RAWSCORES

r

r = NXY XY[NX2-(X)2][NY2-(Y)2]


18/33


STANDARD SCORES

r

Step 1 = Solve X and Y

X = XN

Y = YN


19/33


STANDARD SCORES

r

Step 3 = Find the standard

deviation of X and Y

Sx = X2

N

Sy = Y2

N


20/33


STANDARD SCORES

r

Step 4 = Find the values standard

scores of X and Y

Zx = X-X

Zy = Y-Y

S

S


21/33


STANDARD SCORES

r

Step 5 = Multiply ZX and ZY

r = ZX-ZY

N


22/33

THE PEARSON R FROMBY THE

METHOD OF DIFFERENCES

r

r = Sx2

+ Sy2

Sd2

2SxSy


23/33



r

X2= X2(X)2

N

Y2= Y2(Y)2

N


24/33


STANDARD SCORES

r

Sx = X2

N

Sy = Y2

N

T P R F T


25/33


STANDARD SCORES

r

D2 = D2 (D)2N

T P R F B T


26/33



r

r = Sx2

+ Sy2

Sd2

2SxSy


27/33

Spearman Rho

o This is the Spearman

rank-order correlation

coefficient (Rho).

CORRELATION BETWEEN ORDINAL

DATA


28/33


DATA

o For cases of 30 or

less, Spearmanpisthe most widely

used of the rank

correlationmethods.


29/33

p= 1 - 6D2N(N2 -

1)


DATA


30/33

Gamma

oAn alternative to therank-order

correlation

coefficient is the

Goodmans and

Kruskals gamma (G).


DATA


31/33

G = Ns N1Ns + N1


DATA


32/33

Step 1

Arrange the ordering for one of

the two characteristics from the

highest to the highest to the

lowest or vice versa from the top

to the bottom through the rows

and for the other characteristicfrom the highest to the lowest or

vice versa from left to right

through the column.


DATA


33/33

Step 2

Compute Ns by multiplying the

frequency in every cell by the

series of the frequencies in all


DATA

Simple Linear Regression and Measures of Correlation

Documents

Transcript of Simple Linear Regression and Measures of Correlation