Correlation tests:

Correlation Coefficient:

A succinct measure of the strength of the relationship between two variables (e.g. height and weight, age and reaction time, IQ and exam score).

There are various types of correlation coefficient, for different purposes:

1. Pearson's "r":Used when both X and Y variables are(a) continuous;(b) (ideally) measurements on interval or ratio scales;(c) normally distributed - e.g. height, weight, IQ.

2. Spearman's rho:In same circumstances as (1), except that data need only be on an ordinal scale - e.g. attitudes, personality scores.

r is a parametric test: the data have to have certain characteristics (parameters) before it can be used.

rho is a non-parametric test - less fussy about the nature of the data on which it is performed.

Correlations vary between:+1 (perfect positive correlation: as X increases, so does Y):

Y

X

... and -1 (perfect negative correlation: as X increases, Y decreases, or vice versa).

Y

X

r = 0 means no correlation between X and Y: changes in X are not associated with systematic changes in Y, or vice versa.

Calculating Pearson's r: a worked example:

Is there a relationship between the number of parties a person gives each month, and the amount of flour they purchase from Vinny Millar?

Month: Flour production (X):

No. of parties (Y):

X2 Y2 XY

A 37 75 1369 5625 2775

B 41 78 1681 6084 3198

C 48 88 2304 7744 4224

D 32 80 1024 6400 2560

E 36 78 1296 6084 2808

F 30 71 900 5041 2130

G 40 75 1600 5625 3000

H 45 83 2025 6889 3735

I 39 74 1521 5476 2886

J 34 74 1156 5476 2516

N=10 ΣX = 382 ΣY =776 ΣX2 = 14876 ΣY2 = 60444 ΣXY = 29832

1077660444

1038214876

10 776382

29832r 22

N

YY

N

XX

N

YXXY

r2

2

2

2

Using our values (from the bottom row of the table:)

N=10 ΣX = 382 ΣY =776 ΣX2 = 14876 ΣY2 = 60444 ΣXY = 29832

7455.391.25380.188

40.22660.283

80.188r

60.602176044440.1459214876

20.2964329832r

r is .75 . This is a positive correlation: people who buy a lot of flour from Vinny Millar also hold a lot of parties (and vice versa).

How to interpret the size of a correlation:

r2 is the "coefficient of determination". It tells us what proportion of the variation in the Y scores is associated with changes in X.

 

e.g., if r is .2, r2 is 4% (.2 * .2 = .04 = 4%).

Only 4% of the variation in Y scores is attributable to Y's relationship with X. Thus, knowing a person's Y score tells you essentially nothing about what their X score might be.

Our correlation of .75 gives an r2 of 56%.

 

An r of .9, gives an r2 of (.9 * .9 = .81) = 81%.

 

Note that correlations become much stronger the closer they are to 1 (or -1).

Correlations of .6 or -.6 (r2 = 36%) are much better than correlations of .3 or -.3 (r2 = 9%), not merely twice as strong!

Spearman's rho:

Measures the degree of monotonicity rather than linearity in the relationship between two variables - i.e., the extent to which there is some kind of change in X associated with changes in Y:

Hence, copes better than Pearson's r when the relationship is monotonic but non-linear - e.g.:

But not:

Spearman's rho - worked example:

Is there a correlation between the number of vitamin treatments a person has, and their score on a memory test?

Subj: No.vitamin teatments (X):

Memory test score (Y):

Vitamin treatment ranks (X):

Memory ranks (Y):

D

(= X-Y) D2

A 2 22 2 1 +1 1

B 1 34 1 2 -1 1

C 3 36 3.5 3 +0.5 0.25

D 4 49 5 5 0 0

E 3 42 3.5 4 -0.5 0.25

F 6 57 7 6 +1 1

G 5 82 6 7.5 -1.5 2.25

H 8 82 8 7.5 +0.5 0.25

N = 8

ΣD2 = 6.0

NN

D61rho

3

2

1NN

D61rho

2

2

OR

Step 1: assign ranks to the raw data, for each variable separately.

Rules for ranking:(a) Give the lowest score a rank of 1; next lowest a rank of 2; etc.

(b) If two or more scores are identical, this is a "tie": give them the average of the ranks they would have obtained had they been different.

The next score that is different, gets the rank it would have had if the tied scores had not occurred.

e.g.: raw score 12 15 15 16 17"original"rank 1 2 3 4 5actual rank: 1 2.5 2.5 4 5

Rank for the tied scores is (2+3)/2 = 2.5

raw score 3 18 18 18 100"original"rank 1 2 3 4 5actual rank: 1 3 3 3 5

Rank for the tied scores is (2+3+4)/3 = 3

Step 2: Subtract one set of ranks from the other, to get a set of differences, D.

Step 3: Square each of these differences, to get D2.

Step 4:Add up the values of D2 , to get ΣD2. Here, ΣD2 = 6.0N = 8.

929.050436

1

85120.66

1NN

D61rho

3

2

Step 5:

rho = .93.

There is a strong positive correlation between the number of vitamin treatments a person has, and their memory test score.

Pearson's r on the same data = .86.

Using SPSS/PASW to obtain scatterplots: (a) simple scatterplot:Graphs > Legacy Dialogs > Scatter/Dot...

Using SPSS/PASW to obtain scatterplots: (a) simple scatterplot:Graphs > Chartbuilder

1. Pick ScatterDot 2. Drag "Simple scatter" icon into chart preview window.

3. Drag X and Y variables into x-axis and y-axis boxes in chart preview window

Using SPSS/PASW to obtain scatterplots: (b) scatterplot with regression line:Analyze > Regression > Curve Estimation...

Model Summary and Parameter Estimates

Dependent Variable: memory score

.736 16.741 1 6 .006 17.167 8.333EquationLinear

R Square F df1 df2 Sig.

Model Summary

Constant b1

Parameter Estimates

The independent variable is number of vitamin treatments.

"Constant" is the intercept, "b1" is the slope

Using SPSS/PASW to obtain correlations:

Analyze > Correlate > Bivariate...

Correlations

1 .858**

.006

8 8

.858** 1

.006

8 8

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

number of vitamintreatments

memory score

number ofvitamin

treatmentsmemory

score

Correlation is significant at the 0.01 level (2-tailed).**.

Correlations

1.000 .928**

. .001

8 8

.928** 1.000

.001 .

8 8

Correlation Coefficient

Sig. (2-tailed)

N

Correlation Coefficient

Sig. (2-tailed)

N

number of vitamintreatments

memory score

Spearman's rho

number ofvitamin

treatmentsmemory

score

Correlation is significant at the 0.01 level (2-tailed).**.