Correlation

1
Correlation

• Overview and interpretation

2
Making a Scatterplot
Line up the data in columns (eliminate missing
data)
Plot the students score on each variable
Bill
Adapted from Wiersma, W., Jurs, S. G. (1990).
Educational measurement and testing (2nd ed.).
Needham Heights, MA Allyn and Bacon.
3
Inspect the scatterplot

• The correlation coefficient (Pearson r) can only
  be interpreted for linear relationships

These are all examples of linear relationships
The strength of the correlations vary
Shavelson, R. J. (1996). Statistical reasoning
for the behavioral sciences (Third ed.). Needham
Heights, MA Allyn Bacon.
4
Inspect the scatterplot (2)

• If you see these types of distributions, you are
  dealing with a curvilinear relationship

5
Inspect the scatterplot (3)

• Students who seem to be out on their own in the
  scatter plot are called outliers
• Including outliers in the calculation can change
  the relationship

6
Pearson r correlation coefficient

• Range from -1.0 (perfect inverse correlation) to
  1.0 (perfect correlation)
• The sign (, -) shows the direction of the
  relationship
• The number shows the strength of the relationship
  (regardless of sign)
• No relationship is 0.0

7
The formula
Note that there are other equivalent formulas
also possible.
8
Assumptions of Pearson correlation

• Each pair of scores is independent
• Each set of scores is normally distributed
• The relationship between scores is linear

9
Interpreting correlation

• Correlation merely shows a relationship between
  two variables, not the meaning of the
  relationship
• Correlation is not causation
• Statistical significance does not imply
  importance
• Statistical significance merely indicates that
  the correlation strength is greater than one
  would expect by chance

10
Statistical significance of r
(Cody Smith, 1997)
Imagine a population with a zero correlation
Now, sample 10 points from this population
The resulting sample would probably have a
non-zero correlation
11
Statistical significance

• If a correlation is much larger than what one
  would expect by chance, it is considered to be
  significant
• Significant does not mean important or strong
• Significant merely means that the size of the
  correlation coefficient is larger than would be
  expected by a chance sampling from a zero
  correlation population

12
Determining significance

• Most statistical software packages will
  automatically flag significant correlations
• If checking by hand, compare the r value with the
  appropriate table
• 2-tailed decision at alpha .05 is common
• If the value of r is equal to or larger than the
  value in the table, the correlation is
  significant

13
Decide the level of certainty that you want
This is the table from the back of a statistics
book
Find the number corresponding to your N 2
Check to see if your correlation coefficient is
as large or larger than the one in the table
14
Coefficient of determination

• The coefficient of determination (r2) is a
  measure of the shared variance between the two
  variables

(Shavelson, 1996)
15
Potential problems in correlation analysis

• restriction of range
• correlation of TOEFL, GRE, etc. with grade point
  average
• skewedness
• test too easy or too difficult
• attribution of causality
• variable must be correlation to claim that they
  are causally related, but correlation alone is
  not sufficient to prove causality

16
Point-biserial correlation

• Used to correlate a dichotomous variable with a
  continuous variable
• In testing, used to correlate a persons
  performance on an item (correct, incorrect) with
  their total test score
• Used as an index of item discrimination

17
Point-biserial formula
IF for item
1 IF for item
Mean on the test for people who got item correct
Mean on the test for people who got item incorrect
Standard deviation for test
18
TAP output
Number Item Disc.
Correct Correct Point Adj. Item Key
Correct Diff. Index in High Grp in Low Grp
Biser. Pt Bis ------- ----- ------- ----- -----
----------- ----------- ------- ------- Item 01
(2 ) 22 0.44 0.72 14 (0.93) 3 (0.21)
0.64 0.60 Item 02 (4 ) 29 0.58 0.58
13 (0.87) 4 (0.29) 0.51 0.47 Item 03 (4
) 35 0.70 0.71 15 (1.00) 4 (0.29)
0.52 0.48 Item 04 (3 ) 26 0.52 0.72
14 (0.93) 3 (0.21) 0.63 0.59 Item 05 (2 )
37 0.74 0.50 15 (1.00) 7 (0.50)
0.38 0.34 Item 06 (1 ) 19 0.38 0.72
13 (0.87) 2 (0.14) 0.59 0.55 Item 07 (3 )
36 0.72 0.43 14 (0.93) 7 (0.50)
0.34 0.28 Item 08 (4 ) 23 0.46 0.79
15 (1.00) 3 (0.21) 0.63 0.59 Item 09 (4 )
23 0.46 0.79 14 (0.93) 2 (0.14)
0.61 0.56 Item 10 (4 ) 37 0.74 0.22
13 (0.87) 9 (0.64) 0.18 0.12