BIVARIATE CORRELATION



Correlation

Research Problem:

What is the relationship between two variables?

Relationship between hours studying (X)

and grades on a midterm (Y)?

Relationship between stressful life events (X)

and number of illness symptoms (Y)?

Correlation = Direction and strength of (linear) relationship between two variables

I. The Scatterplot

What is the relationship between hours studying (X) and scores on a quiz (Y)?

|Student |Hours |Score |

|A |1 |1 |

|B |1 |3 |

|C |3 |4 |

|D |4 |5 |

|E |6 |4 |

|F |7 |6 |

II. Pearson Correlation Coefficeint

Symbol: r

r can range from -1.0 to +1.0

Sign (+/-) indicates “direction”

Value indicates “strength”

Measures a “linear” relationship only

(a) Direction of relationship between X, Y

Positive (+r) = As X goes up, Y goes up

Negative (-r) = As X goes up, Y goes down

(b) Strength of a relationship between X, Y

Closer to ( 1.0, stronger

Closer to 0, weaker

when r = 0 ( X,Y relationship not defined

by a straight line

Pearson Correlation Coefficient

-1.0 0 +1.0

1. Closer to 0 = weaker

2. Closer to (1.0 = stronger

3. r close to (1.0 very rare in social science

4. r ( ( .30 considered important

5. r ( 0, no linear relationship between X & Y

What does r represent?

r = degree to which X & Y vary together

degree to which X & Y vary separately

r = covariance of X & Y

variance of X & Y

Definitional Formula for Pearson r:

r = [pic]

SP = “Sum of Products”

SS = Sum of Squared Deviations

SP = ((X-[pic])(Y-[pic])

SSX=((X-[pic])2

SSy=((Y-[pic])2

Variance interpretation of r :

r 2 = % of variance in Y explained by its linear

relationship with X (and vice versa)

r 2 = “Coefficient of determination”

% of shared variance between X & Y

% of variance in Y predicted by X

III. Factors that affect the size of r

6. r ( 0 could mean many things:

• No relationship at all between X & Y

Non-linear relationship between X & Y

• Restricted range on X and/or Y

• Outlier may be causing problem

• Non-linear relationships

Curvilinear relationship

• Restricted range

Low variability on X and/or Y

• Outliers

Extreme value on X and/or Y

Examples of how restricted range can distort a correlation

Example of how an outlier can distort a correlation

IV. Correlation vs. Causality:

• Correlation tells you two variables are related

• Does NOT tell you why!!

• Do not draw causal inferences from a correlation

X ( Y

Y ( X

example:

r = -.30 #friends, depression

r = +.40 hours studying, grades

• Causal inferences require an “experiment”

V. Other Correlation Coefficients

Pearson r used when X & Y are at least interval level

Many types of correlation coefficients for other data

Spearman ( ordinal (rank) data

Point-biserial ( nominal X, interval/ratio Y

Phi ( nominal X & Y

-----------------------

Y

(Score)

X (Hours Studying)

No Linear Relationship

Perfect Positive Relationship

Perfect Negative Relationship

X

Y

Z

Third variable problem

Figure 16-3 (p. 524)

Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.

Figure 16-5 (p. 525)

Examples of different values for linear correlations: (a) shows a strong positive relationship, approx +.90; (b) shows a relatively weak negative correlation, approx –.40; (c) shows a perfect negative correlation, correlation = –1.0; (d) shows no linear trend, correlation = 0.0.

r = .90

r = -.40

r = -1.0

r = .00

(a) In this example, the full range of X and Y values shows a strong, positive correlation, but the restricted range of scores produces a correlation near zero.

(b) An example in which the full range of X and Y values shows a correlation near zero, but the scores in the restricted range produce a strong, positive correlation.

A demonstration of how one extreme data point (an outlier) can influence the value of a correlation.

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