Universityof DurhamD

Dr Robert CoeUniversity of Durham School of

Education

Tel: (+44 / 0) 191 334 4184Fax: (+44 / 0) 191 334 4180

E-mail: r.j.coe@dur.ac.ukhttp://www.dur.ac.uk/r.j.coe

Correlation and Regression

Doctor of Education (EdD)Analysing, Interpreting and Using Educational Research (Research

Methodology)

© 2005 Robert Coe, University of Durham 2

r = 0.0

Examples of correlation coefficients

© 2005 Robert Coe, University of Durham 3

r = 0.3

© 2005 Robert Coe, University of Durham 4

r = 0.5

© 2005 Robert Coe, University of Durham 5

r = 0.7

© 2005 Robert Coe, University of Durham 6

r = 0.9

© 2005 Robert Coe, University of Durham 7

Grammar School selection

40

60

80

100

120

140

160

40 60 80 100 120 140 160

11+ score

Ac

hie

ve

me

nt

at

16

+

A test selects the top 25% at age 11:

11% passed who should not have

18%

rightly passed

60% rightly failed

11% failed who should not have

Based on a correlation of 0.7

© 2005 Robert Coe, University of Durham 8

Variance accounted for

Cognitive measure of prior attainment / aptitude

Aca

dem

ic a

chie

vem

ent

r = 0.7(r2 = 0.49)

© 2005 Robert Coe, University of Durham 9

Measure of socioeconomic status

r = 0.3(r2 = 0.09)

Aca

dem

ic a

chie

vem

ent

High SESLow SES

© 2005 Robert Coe, University of Durham 10

Aggregated or Individual?“Ecological Fallacy”

Socioeconomic statusSchool 1 +School 2 +School 3 +School 4 +

Aca

dem

ic a

chie

vem

ent

Correlations for:

Individual students = 0.3

School means = 0.9

© 2005 Robert Coe, University of Durham 11

r = 0.03

(n = 30)

Beware small samples:

But add one extreme point ...

r = 0.33

© 2005 Robert Coe, University of Durham 12

Restricted range:

r = 0.7

© 2005 Robert Coe, University of Durham 13

r = 0.5

Universityof DurhamD

Dr Robert CoeUniversity of Durham School of

Education

Tel: (+44 / 0) 191 334 4184Fax: (+44 / 0) 191 334 4180

E-mail: r.j.coe@dur.ac.ukhttp://www.dur.ac.uk/r.j.coe

Regression

© 2005 Robert Coe, University of Durham 15

A*

A

B

C

D

E

F

G

U

0

2

17

25

14

18

11

8

3

One school’s maths GCSE grades:

45%

How good are they?

% FSM?

Socioeconomic status?

Students’ability?

Prior attainment?

=15%

School % 5A*-C?

=56%

Subject difficulty?

© 2005 Robert Coe, University of Durham 16

A*

A

B

C

D

E

F

G

U 0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80

YELLIS test score

Average performance for people with that test score

Average residual = 0.26

Value Added as we know it:

RESIDUAL

© 2005 Robert Coe, University of Durham 17

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90

Aptitude (YELLIS test score)

Mat

hs

GC

SE

gra

de

Cognitive

© 2005 Robert Coe, University of Durham 18

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6

SES (parents' educational level and occupation)

Mat

hs

GC

SE

gra

de

Social

© 2005 Robert Coe, University of Durham 19

0.10

0.54

0.55

0.55

0.56

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SES

YELLIS test

YELLIS test and SES

YELLIS test, SES andHOME

YELLIS test, SES,HOME and ETHNIC

Proportion of variance accounted for

Explained

Unexplained

What variables explain the variance in GCSE Maths performance?

© 2005 Robert Coe, University of Durham 20

Output from SPSS:

Model Summary

.761a .579 .579 1.09Model1

R R SquareAdjustedR Square

Std. Errorof the

Estimate

Predictors: (Constant), SES, Yellis test totala.

Coefficientsa

-.231 .030 -7.795 .000

8.653E-02 .001 .728 165.888 .000

.152 .008 .089 20.254 .000

(Constant)

Yellis test total

SES

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: Maths GCSE (passes only)a.

© 2005 Robert Coe, University of Durham 21

Issues in regression

Check residuals are Normally distributed Check for collinearity in explanatory

variables Use adjusted R2

Which explanatory variables to include?

Universityof DurhamD

Dr Robert CoeUniversity of Durham School of

Education

Tel: (+44 / 0) 191 334 4184Fax: (+44 / 0) 191 334 4180

E-mail: r.j.coe@dur.ac.ukhttp://www.dur.ac.uk/r.j.coe

Regression to the mean

Doctor of Education (EdD)Analysing, Interpreting and Using Educational Research (Research

Methodology)

© 2005 Robert Coe, University of Durham 23

Measures with less than perfect reliability

A test with test-retest correlation r=0.7 is repeated after an interval. What would you expect for the TEST 2 scores of A person who achieved a very high score on

TEST 1 A person who achieved a very low score on

TEST 1

How will the overall distribution of scores on the two tests compare?

© 2005 Robert Coe, University of Durham 24

TEST 1

TE

ST

2

© 2005 Robert Coe, University of Durham 25

Two subgroups with different means

Students with high SES tend to get higher test scores.

Two students have the same TEST 1 scores, but one is high SES, the other low SES. What would you expect their TEST 2 scores to be?

© 2005 Robert Coe, University of Durham 26

TEST 1

TE

ST

2

Low SES

High SES

© 2005 Robert Coe, University of Durham 27

Is social class more important than early ability?

Feinstein (2003)

© 2005 Robert Coe, University of Durham 28

Or is it just regression to the mean?

Comparison of the apparent effect of social class with the effect of unreliable initial ability measures

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

22 months 120 months

Feinstein's data: High SES

Feinstein's data: Low SES

Predicted data: High SES

Predicted data: Low SES

Reliability = 0.15