the Correlation
Describing Relationship between Two Variables The descriptive statistics we have studied so far have been univariate. That is, we use them to describe the shape, center and spread of scores we have for a single variable. It is rare to study just one variable. Often we study numerous variables, and we want to understand how a variable is related to another variable. Example: What is the relationship between optimism and depression? Perhaps we propose that those who are more optimistic tend to be less depressed. We would need to do some research to see if this is true. Let's say we have a sample of 11 people and each one completes a questionnaire measuring depression and a separate questionnaire measuring optimistic thinking. Subject 1 2 3 4 5 6 7 8 9 10 11 Optimism Score 2 6 3 7 10 12 14 17 20 24 21 M = 12.36 s = 7.50 Depression Score 13 14 17 14 18 12 5 12 3 8 9 M = 11.36 s = 4.70
Techniques for Describing a Bivariate Relationship: Graphing and Correlation Coefficient Graphing Scatterplot: Graph used to show the relationship between 2 variables
Scatterplot for optimism and depression
20 15 Depression 10 5 0 0 5 10 15 Optimism 20 25 30
A. Constructing a scatterplot
1. Horizontal axis is used for independent variable (X) 2. Vertical axis is used for the dependent variable (Y)
3. Draw axes approximately same length
4. Plot a point for each pair of scores
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B. What to look for in a scatterplot
1. Form of relationship: Linear or curvilinear
Curvilinear
80 70 60 50 DV 40 30 20 10 0 0 20 40 IV 60 80
DV 80 70 60 50 40 30 20 10 0 0 20 40 IV 60 80
Linear
80 70 60 50 DV 40 30 20 10 0 0 20
Linear
40 IV
60
80
2. Strength or degree of relationship: The closer the points are to forming a single line, the stronger the relationship
Perfect
80 70 60 50 DV 30 20 10 0 0 20 40 IV 60 80 DV 40 80 70 60 50 40 30 20 10 0 0 20 40 IV 60 80
Strong
Moderate
80 70 60 50 DV DV 40 30 20 10 0 0 20 40 IV 60 80 70 60 50 40 30 20 10 0 0 20
Weak
40 IV
60
80
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3. Direction of relationship: Positive or negative
a. Positive: scores on one variable are associated with scores at a similar level on the other variable High scores on X are associated with high scores on Y Low scores on X are associated with low scores on Y b. Negative: scores on one variable are associated with scores at the opposite level on the other variable High scores on X are associated with low scores on Y Low scores on X are associated with high scores on Y
Positive Relationship
90 80 70 60 50 40 30 20 10 0 0 20 40 IV 60 80 90 80 70 60 50 40 30 20 10 0 0 20 40 IV 60 80
Negative Relationship
Note: With weak relationships, it may be hard to discern the direction of the relationship from the scatterplot.
Correlation Coefficient: Statistical measure of the relationship between two variables
1. Most common correlation coefficient is Pearson correlation coefficient (r). It is used when both variables are continuous (either interval or ratio); it measures the degree and direction of linear relationship 2. Other correlation coefficients Spearman correlation: ordinal data and/or nonlinear relationship Point biserial correlation: one variable is continuous, the other is dichotomous Phi coefficient: two dichotomous variables 3. Equation for r (Note: You are not required to memorize the computational equation, but you are expected to know the formula expressed in z scores.) Conceptually, r = degree to which X and Y vary together degree to which X and Y vary separately
KNOW THIS: Formula expressed in terms of z scores:
r=
z
x
zy
n -1
FYI: Computational equation:
r=
XY -
( X) X 2 - N
X Y
N
2
( Y )2 Y 2 - N
DV
DV
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4. Getting r ID 1 2 3 4 5 6 7 8 9 10 11 Optimism Score (X) 2 6 3 7 10 12 14 17 20 24 21 Total: 136 Hours Slept 5 7 8 6 6 10 42 zx -1.38 -0.85 -1.25 -0.71 -0.31 -0.05 0.22 0.62 1.02 1.55 1.15 0.01 Happy Mood 2 4 7 2 3 6 24 Depression (Y) 13 14 17 14 18 12 5 12 3 8 9 125 Hours Slept z -1.12 0.56 -0.56 1.68 0 zy 0.35 0.56 1.20 0.56 1.41 0.14 -1.35 0.14 -1.78 -0.71 -0.50 0.01 Happy Mood z -0.95 1.43 -0.95 -0.48 0.95 0 zx*zy -0.48 -0.48 -1.50 -0.40 -0.44 -0.01 -0.30 0.08 -1.81 -1.11 -0.58 -7.02 Cross Product 1.07 0.80 0.27 1.60 4.26
ID 1 2 3 4 5 6 TOTAL
Interpretation:
5. Understanding the value of r: Correlation coefficient values range from -1.00 to +1.00 a. Numerical portion: Regardless of sign, the value of r indicates the strength or degree of linear relationship between the two variables The following number line indicates the labels typically assigned to various values of r within psychology and other social science research:
-1.00
perfect strong
-.5
-.3
-.1
weak
0
none
+.1
weak
+.3
+.5
+1.00
perfect
moderate
moderate strong
b.
Sign: Direction of relationship
Page 5 of 8 6. Describing correlations: Putting the value of r into words a. First, state the direction and strength of the relationship in a sentence like the following: There is a (strength) b. (direction) relationship between (name of IV) and . (name of DV)
Then, elaborate to give more detail about how the variables covary. i. Suggestions for negative relationships:
As (name of IV) People who have more
increases, (name of DV)
decreases.
tend to have less (name of IV) (name of DV)
.
People who are more tend to be less . (adjective form of IV) (adjective form of DV)
ii. Suggestions positive for relationships:
As (name of IV) People who have more
increases, (name of DV) tend to have more (name of DV)
also increases.
.
(name of IV) People who are more
tend to be more . (adjective form of IV) (adjective form of DV)
Examples: 1. Optimism and depression: r = -.70 There is a strong negative relationship between optimism and depression. People who are more optimistic tend to be less depressed. 2. Parental involvement and children's school achievement: r = .23 There is a weak positive relationship between parental involvement and children's school achievement. Children with more involved parents tend to have higher school achievement.
Practice Problems: Practice estimating correlations http://istics.net/stat/Correlations/ Practice
Correlation Practice 1 and 2 on Blackboard
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Table 1: Correlations of procrastination with measures of stress, health, and academic performance Early semester Grade on Term paper Score on Lay's General Procrastination Scale -.29 Exam grade -.64 Symptoms of illness -.45 Stress -.31 Late semester Symptoms of illness .65 Stress .68
The
Source: Tice, D.M, & Baumeister, R.F. (1997). Longitudinal study of procrastination, performance, stress, and health: costs and benefits of dawdling. Psychological Science, 8, 454-458.
Table 2: Correlations Among Peer Relationship Variables, Victimization, and Behavioral Vulnerability Measures
1 Peer Measures 1. Total acceptance 2. Reciprocated friends 3. Friendship quality Victimization measures 4. Overt victimization 5. Relational victimization Behavioral vulnerability measures 6. Externalizing problems 7. Internalizing problems 8. Attentional problems -0.40 0.40 -0.30 -0.51 -0.33 -0.11 -0.26 2 -0.49 -0.38 -0.43 -0.32 -0.19 -0.16 3 4 5 6 7 8
--0.53 -0.44 -0.42 -0.36 -0.24 -0.73 0.63 0.35 0.25 -0.59 0.39 0.19 -0.07 0.30 --0.12
--
Source: Jensen-Campbell, L.A., & Malcolm, K.T. (2007). The importance of conscientiousness in adolescent interpersonal relationships. Personality and Social Psychology Bulletin, 33, 368-383.
Figure 1: The mediating influence of attentional and externalizing problems on the conscientiousness-peer relationship link
Source: Jensen-Campbell, L.A., & Malcolm, K.T. (2007). The importance of conscientiousness in adolescent interpersonal relationships. Personality and Social Psychology Bulletin, 33, 368-383.
Page 7 of 8 6. Extraneous factors that can affect r a. Range of values on one or both variables Restricted can range distort r
b. Outliers: Extreme scores on one or both variables
Outliers are especially influential if sample size is small 7. Explaining r: Why are two variables correlated? a. Correlation is not causation If two variables are correlated, then there are three possible reasons for that relationship 1. X causes Y 2. Y causes X 3. A third variable may be involved as a mediating variable Mediating variable: A variable that accounts for the relationship between X and Y; it is an essential link in the causal pathway connecting X and Y b. Correlation may be squared to measure proportion of variance explained by the relationship r2 or R2 = coefficient of determination = proportion of variance in one variable that is accounted for by the relationship with the other variable; also known as proportion of explained variance 1 R2 = coefficient of nondetermination = proportion of variance not accounted for Example: Parental involvement and children's school achievement: r = .23 so r2 = .05 This would mean that 5% of the variance in children's school achievement is explained by the relationship with parental involvement. 95% of the variance is unexplained. Additional examples: i. Procrastination score and late semester stress: r = .68 There is a strong positive correlation between procrastination and late semester stress. R^2=0.46; nearly half of the variance in late semester stress could be explained by procrastination. ii. Attentional problems and friendship quality: r = -.12 There is a weak negative relationship between attentional problems and friendship quality. R^2=0.01, meaning that only 1% of friendship qualities can be explained by attentional problems.
8. Uses of correlation In addition to measuring the strength and direction of the relationship between two variables, correlation is important for other aspects of research. a. Validity of a test or measurement procedure: Does it measure what it is supposed to measure? Validity can be assessed with a correlation coefficient measuring the relationship between scores on a new measure/test and scores on an established measure/test of the same construct. b. Reliability of a test or measurement procedure: consistency of measurement Reliability is assessed by a correlation coefficient. Types of reliability include: Interrater : Correlation of two sets of ratings made by independent observers Test-retest : Correlation of the test given at one time with the same test given again Internal consistency : Correlation of one part of a test with another part c. Prediction: If we find that two variables are related, then we can use that information to create a model to predict the value of one variable given a known value of another variable.
Page 8 of 8 Because our predictive model is based on correlation, the accuracy of predictions made using that model will be directly related to the strength of the correlation.