Introduction to Regression Analysis

Two Purposes

• Explanation– Explain (or account for) the variance in a

variable (e.g., explain why children’s test scores vary).

– We’ll cover this later.• Prediction

– Construction an equation to predict scores on some variable.

– Construct an equation that can be used in selecting individuals.

Prediction• Use a set of scores collected from a sample

to make predictions about individuals in the population (not in the sample).

• Use the scores to construct a mathematical (typically linear) equation that allows us to predict performance.

• Two types of scores are collected:– Usually a measure on one criterion (outcome,

dependent) variable.– Scores on one or more predictor (independent)

variables.

The equations

The equation for one individual’s criterion score:

The prediction equation for that individual’s score

The difference between the two equations (called a residual)

Y f X X e1 11 1 2 1 ( . . . )

( ... )Y f X X1 11 1 2

e Y Y1 1 1

The function

The linear function has the form:

Where the βs are weights (regression weights) selected such that sum of squared errors are minimized (least squares criterion)

1 1 11 2 12( ) ...f X X X

M in e 2

M in Y Y2

Multiple Correlation

Minimizing the sum of squared errors causes the correlation between the actual criterion scores and the predicted scores to be maximized (as large as possible). This correlation is called a multiple correlation. It is the correlation between the criterion variable and a linear composite of the predictor variables.

ˆyyR Maximum

Coefficient of DeterminatinThe square of the multiple correlation,

is called the coefficient of determination. It gives the proportion of shared variance (i.e., covariance) between the criterion variable and the weighted linear composite.

Hence the larger the, R2, the better the prediction equation.

R 2

Basic regression equation

The parametric regression model is given by

Yi = α + βXi + εi The model f or a sample, by

Yi = a + bXi + ei

Computing the constants in the regression equation

where

A closer look at the regression equation

bxY

XXbY

bXXbY

bXaY

)(

)(

ˆ

Partitioning the Sum ofSquares (SSy)

SSy is given by

Now, Consider the following identity

Subtracting from each side gives,

Squaring and summing gives,

2( )Y Y

)ˆ()ˆ( YYYYYY

ˆ ˆ( ) ( )Y Y Y Y Y Y

2 2

2 2

ˆ ˆ( ) [( ) ( )]

ˆ ˆ ˆ ˆ( ) ( ) 2 ( )( )

Y Y Y Y Y Y

Y Y Y Y Y Y Y Y

Simplifying the previous equation

Where SSreg = Sum of squares due to regression, and

SSres = Residual sum of squares.

Dividing through by the total sum of squares, gives:

, or

2 2 2ˆ ˆ( ) ( )

reg res

y Y Y Y Y

SS SS

Example

Y X

31045

101

-12

Calculation of squares and cross-products

Deviation squares and cross-products

Sums of squares and cross-products

Y 3 1 0 4 5

y .4

-1.6 -2.6 1.4 2.4

y2 .16

2.56 6.76 1.96 5.76

X 1 0 1

-1 2

x .4

-.6 -1.6

.4 1.4

x2 .16 .36

2.56 .16

1.96

xy .16 .96

4.16 .56

3.36

Calculation of the coefficients

The slope,

the intercept,

and the regression line.

769.12.5

2.92

x

xyb

5224.1

0776.16.2

)6.769.16.2(

XbYa

XY 769.1522.1ˆ

Calculation of SSreg

From an earlier equation…

.28.162.5

64.84

)(

)(

)(

)(

])[(

)(

2

2

22

22

2

2

2

2

x

xy

xx

xy

xb

bx

YbxY

YbxY

YYSSreg

Some additional equations for SSreg

Hence…

SSreg computed from a correlation

The formula for the Pearson correlation is…

therefore,

A Closer Look at the Equations in Regression

Analysis

The Variance

sx

nx2

2

1

The standard deviation

S sx x 2

The covariance

sxy

nxy 1

The Pearson product moment correlation

rs

s sxyxy

x y

The normal equations (for the regressions of y on x)

bxyx

s

s

xy

xy

x

2

2

a Y b Xyx -

The structural model (for an observation on individual i)

Y a b X ei yx i i

The regression equation

ˆ

( )

( )

yx

yx yx

yx

yx

Y a b X

Y b X b X

Y b X X

Y b X

Partitioning a deviation score, y

y Y Y

Y Y Y Y Y Y

Y Y Y Y

( ) ( )

( ) ( )

The score, Y, is partitioned

Hence, Y is partitioned into a deviation of a predicted score from the mean or the scores PLUS a deviation of the actual score from the predicted score.

Our next step is to square the deviation,

and sum over all the scores.

y Y Y

y Y Y

Partitioning the sum of squared deviations (sum of squares,

SSy)

y Y Y

Y Y Y Y

Y Y Y Y

SS SS

2 2

2

2 2

( )

[( ) ( )]

( ) ( )

reg res

What happened to the term,

Showing that reduces

to zero requires some complicated

algebra, recalling that

and that

ˆ ˆ2 ( )( )?Y Y Y Y ˆ ˆ2 ( )( )Y Y Y Y

ˆyxY a b X

2/ .yxb xy x

Calculation of proportions of sums of squares due to regression and due to

error (or residual)

y

y

SS

y

SS

y

SS

y

SS

y

2

2

21

reg

2

res

2

reg

2

res

Alternative formulas for computing the sums of squares due to regression

SS Y Y

Y bx Y

bx

b x

xy

xx

xy

x

xy

xxy

b xy

reg

( )

( )

( )

( )

( )

( )

2

2

2

2 2

2

2 22

2

2

2

Test of the regression coefficient, byx, (i.e. test the null hypothesis that byx =

0)First compute the variance of estimate

s est

Y Y

N kSS

N k

y x y

2 2

2

1

1

( )

( )

res

Test of the regression coefficient, byx, (i.e. test the null hypothesis that byx =

0)Then obtain the standard error of estimate

Then compute the standard error of the regression coefficient, Sb

s sy x y x 2

ss

x n

s

x Nb

y x y x

2

2 21 1( ) / ( ) ( ) / ( )

The test of significance of the regression coefficient (byx)

The significance of the regression coefficient is tested using a t test with (N-k-1) degrees of freedom:

tb

s

b

S

S n

yx

b

yx

y x

x

1

Computing regression using correlations

The correlation, in the population, is given by

The population correlation coefficient, ρxy, is estimated by the

sample correlation coefficient, rxy

xy

x yN

rz z

Ns

s s

xy

x y

xyx y

xy

x y

2 2

Sums of squares, regression (SSreg)

Recalling that R2 gives the proportion of variance of Y accounted for (or explained) by X, we can obtain

or, in other words, SSreg is that portion of SSy predicted or explained by the regression of Y on X.

SS r y

SS r y

reg

res

2 2

2 21( )

Standard error of estimate

From SSres we can compute the variance of estimate and standard error of estimate as

(Note alternative formulas were given earlier.)

sr y

N ks s

y x

y x y x

22 21

1( )

Testing the Significance of r

The significance of a correlation coefficient, r, is tested using a t test:

With N-2 degrees of freedom.

21

2

r

Nrt

Testing the difference between two correlations

To test the difference between two Pearson correlation coefficients, use the “Comparing two correlation coefficients” calculator on my web site.

Testing the difference between two regression coefficients

This, also, is a t test:

Where

was given earlier.

22

21

21 bb SS

bbt

2bS

Point-biserial and Phi correlation

These are both Pearson Product-moment correlationsThe Point-biserial correlation is used when

on variable is a scale variable and the other represents a true dichotomy.For instance, the correlation between a

performance on an item—the dichotomous variable—and the total score on a test—the scaled variable.

Point-biserial and Phi correlation

The Phi correlation is used when both variables represent a true dichotomy.For instance, the correlation between two

test items.

Biserial and Tetrachoric correlation

These are non-Pearson correlations.Both are rarely used anymore.The biserial correlation is used when

one variable is truly a scaled variable and the other represents an artificial dichotomy.

The Tetrachoric correlation is used when both variables represent an artificial dichotomy.

Spearman’s Rho Coefficient and Kendall’s Tau Coefficient

Spearman’s rho is used to compute the correlation between two ordinal (or ranked) variables.It is the correlation between two sets of

ranks.