Correlation

1. Important Concepts
   1. Correlation
   2. Correlation Coefficient
   3. Scatterplot
2. Range of a Correlation Coefficient
3. Pearson's r
   1. Rationale for Computation
   2. Computational Formula & Example - [Minitab]
4. Spearman's rho
   1. Example 1 - [Minitab]
   2. Example 2 - [Minitab]
5. Important Issues With Correlation
   1. Factors Influencing It
   1. Linearity
   2. Limited (Restricted & Truncated) Ranges
   3. Extreme Groups
   4. An Extreme Score
   2. Relation to Causality
   3. Some Specific Uses of it

Homework

I. Important Concepts

1. Correlation
   Earlier in the semester we noted that scientists are interested in relationships between variables. When two variables vary together (a change in one is accompanied by a change in the other), we say they are correlated.

2. Correlation Coefficient
   Expresses quantitatively the extent to which two variables are related. There are several. We will learn about two.

3. Scatterplot
   A graph of a collection of pairs of scores. Example:
   
   
   

   Note that in scatterplots, the X and Y axes are equal in length and thus this type of graph does not obey the 3/4 high rule.

II. Range of a Correlation Coefficient

Is best illustrated with examples:

1. Perfect positive (all points fall on a straight line)
   
   
   

   More realistic example
   
   

   

2. Perfect negative
   
   
   

   More realistic example
   
   

   

3. No correlation
   
   
   

   More realistic example
   
   

   

Summary



r =	±	(0 « 1)
	Sign	Magnitude
	Gives direction	Gives strength

III. Pearson's r

• Termed Pearson's Product Moment Correlation Coefficient.
• Good with metric data.
• Probably the most popular correlation coefficient.
• It is required that both variables involved be normally distributed.
• Represents quantitatively the extent to which scores on two variables occupy the same relative position.

1. Rationale for Computation
   We have seen that z scores provide information about the relative position of a score compared to other scores in the distribution. Pearson’s r uses this:
   
   Thus, r is the mean of the sum of the products of the z scores for the two variables. What follows is a demonstration of why this works in the case of perfect positive relationship (variables X & Y) and in the case of a perfect negative relationship (variables X & W).
   
   First, the perfect positive relationship between X & Y.

   

   X	ZX	ZX2		Y	ZY	ZXZY
   3	-1.42	2.02		1	-1.42	2.02
   5	-.71	.50		2	-.71	.50
   7	0	0		3	0	0
   9	.71	.50		4	.71	.50
   11	1.42	2.02		5	1.42	2.02
   s=2.82	åZX2=5=N			s=1.41	åZXZY=åZX2
   m=7				m=3
   N=5

   If the relative position of the scores on the two variables is the same (as in the present case), then the z scores of each of the variables will be the same and å(Z_XZ_Y) would be equal to åZ_X². As we saw above, åZ_X² is equal to N and thus r would equal N/N or 1.
   
   Now for the perfect negative relationship between X & W.
   
   

   X	ZX	ZX2		W	ZW	ZXZW
   3	-1.42	2.02		5	1.42	-2.02
   5	-.71	.50		4	.71	-.5
   7	0	0		3	0	0
   9	.71	.50		2	-.71	-.5
   11	1.42	2.02		1	-1.42	-2.02
   s=2.82	åZX2=5=N			åZXZW=-5
   m=7
   N=5

   The scores again have the same relative position, but this time the relationship is indirect. In this case, å(Z_XZ_W) would be equal to -N and r would be equal to -N/N or -1.

1. Computational Formula & Example
   Since the standard score formula is cumbersome, a computational formula was developed which doesn’t require the calculation of z scores for all of the scores.

   

   Example: Scores on 20 point math and science quizzes. [Minitab]

   Person	Math (X)	Science (Y)
   A	11	11
   B	13	10
   C	18	17
   D	12	13
   E	16	14
   N=5

   First step would be to create a scatterplot:
   
   
   
   Since the scatterplot looks promising (suggests a strong positive relationship), create the necessary grid for the computations.

   Person	Math (X)	Science (Y)	XY	X2	Y2
   A	11	11	121	121	121
   B	13	10	130	169	100
   C	18	17	306	324	289
   D	12	13	156	144	169
   E	16	14	224	256	196
   N=5	åX=70	åY=65	åXY=937	åX2=1014	åY2=875

   
   Then perform the computations:

As was suggested by the scatterplot, there is indeed a strong positive correlation between the math and science scores.

IV. Spearman’s Rho

A variant of Pearson’s r which is used with rank data is called Spearman’s Rho (r_s). This correlation coefficient is appropriate when either of the following two conditions are met:

1. One variable is an ordinal scale and the other is an ordinal scale or higher.
2. One of the distributions is markedly skewed.

In either case, both scales must be converted to ranks. And if we computed Pearson's r on the ranked data, it would give Spearman's Rho. However, for computations by hand, there is a simpler formula:

Where D= Rank of X – Rank of Y (i.e., a Difference score).

1. Example 1. Beauty & Sociability. [Minitab]
   
   

   Person	Beauty	Sociability
   A	3	3
   B	1=most	2
   C	2	1=most
   D	5	4
   E	4	5
   N=5

   First step would be to create a scatterplot.
   
   
   
   Since the scatterplot looks promising (suggests a strong positive relationship), create the necessary grid for the computations.

   Person	Beauty	Sociability	D	D2
   A	3	3	0	0
   B	1=most	2	-1	1
   C	2	1=most	1	1
   D	5	4	1	1
   E	4	5	-1	1
   N=5			åD=0	åD2=4

   Then perform the computations:

   
   

2. Example 2. Beauty & Science scores. [Minitab]
   
   Since the science score is a ratio variable, it makes sense to rank it from low to high, that is, where low ranks represent low scores. If we are going to correlate beauty with this score, it makes sense to rerank the beauty scores so that they go from low to high as well.
   
   

   Person	Beauty	Beauty (reranked)	Science	Science (ranked)
   A	3	3	11	2
   B	1=most	5=most	10	1
   C	2	4	17	5=most
   D	5	1	13	3
   E	4	2	14	4
   N=5

   
   

   Then we would create a scatterplot of the ranked scores.
   
   
   The data do not look very promising, but let's prepare the grid for the computations anyway.

   Person	Beauty (reranked)	Science (ranked)	D	D2
   A	3	2	1	1
   B	5=most	1	4	16
   C	4	5=most	-1	1
   D	1	3	-2	4
   E	2	4	-2	4
   N=5			åD=0	åD2=26

   Then perform the computations:
   
   

   
   

   So as the scatter plot indicated, there wasn't much of a correlation.

   Note tied ranks would get the average of the tie, for example:

Person	X	Y	Y (rank)
A	3	11	4.5
B	1	11	4.5
C	2	17	1
D	5	13	3
E	4	14	2
N=5

V. Important Issues With Correlation

1. Factors Influencing the Correlation

   These are the reasons why it is important to create a scatterplot.

   1. Linearity
      A linear (or monotonic) relationship is best characterized by a straight line. Both r and r_s assume this.
      Example linear relationship:
      

      

      Example of a curvilinear (or nonmonotonic) relationship:

      

      In general, curvilinearity in a relationship will result in an r that underestimates the true relationship.

   2. Limited (Restricted & Truncated) Ranges
      Refer to situations in which the sample is somehow limited. In both cases, it results in an underestimated r.

      Example of a Restricted Range - Foot size and age in 6 year olds:
      

      

      Example of a Truncated Range - ACT scores and GPA in college students:
      
      

      

   3. Extreme Groups
      Results in an overestimated r. Consider looking at the relationship of reading ability and IQ, but only in poor and excellent readers:

      
      

   4. An Extreme Score
      Also results in an overestimated r. Is more of a problem when using small sample sizes. Example:

      
      

2. Relation to Causality

Possible causal relationships between X and Y if they are correlated include:

Possibility	Symbols	Explanation
a.	X ® Y	X causes Y
b.	X ¬ Y	Y causes X
c.	X ¬ A ® Y	A causes both X & Y
d.	B ® C ® X   B ® Y	Etc.

Main point is that correlation doesn’t tell us much about causality. It should be noted that inferring causality from a correlation is an error that is all too common.

3. Some Specific Uses of Correlation

1. Determining Reliabilities
   Compare two raters (interobserver) or the same raters (intraobserver) observations of behavior to see if they agree.
2. Determining Validities
   If ACT scores are highly correlated with GPA's then we can say that ACT scores are a valid predictor of GPA's.
3. For Prediction
   A set of procedures similar to correlation called regression is used for predicting one variable from one or more other variables.