© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

HLTH 300 Biostatistics for Public Health Practice,

Raul Cruz-Cano, Ph.D.5/5/2014, Spring 2014

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e

Chapter 10: Correlation

1

2

Final Exam

• Monday 5/19/2014

• Time and Place of the class

• Chapters 9, 10 and 11

• Same format as past two exams

• No re-submission of homework

• Summer SAS Course

Differentiate between the strength and direction of a correlation

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

10.1

4

10.1

Until now, we’ve examined the presence or absence of a relationship between two or more variables

What about the strength and direction of this relationship?

• We refer to this as the correlation between variables

Strength of Correlation • This can be visualized using a scatter plot

– Strength increases as the points more closely form an imaginary diagonal line across the center

Direction of Correlation• Correlations can be described as either positive or negative

– Positive – both variables move in the same direction– Negative – the variables move in opposite directions

Correlation

10.1

Figure 10.1

10.1

Figure 10.2

Identify a curvilinear correlation

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

10.2

8

10.2

A relationship between X and Y that begins as positive and becomes negative, or begins as negative and becomes positive

Curvilinear Correlation

Figure 10.3A non-linear transformation, e.g. square root, might take care of this

Discuss the characteristics of correlation coefficients

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

10.3

11

The Correlation Coefficient10.3

Direction Strength

• The sign (either – or +) indicates the direction of the relationship

• Values close to zero indicate little or no correlation

• Values closer to -1 or +1, indicate stronger correlations

Numerically expresses both the direction and strength of a relationship between two variables

• Ranges between -1.0 and + 1.0

Calculate and test the significance of Pearson’s correlation coefficient (r)

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

10.4

13

10.4

Focuses on the product of the X and Y deviations from their respective means

– Deviations Formula:

– Computational Formula:

Pearson’s Correlation Coefficient (r)

2 2

SP

SS SSX Y

X X Y Yr

X X Y Y

2 2 2 2

XY NXYr

X NX Y NY

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10.4

The null hypothesis states that no correlation exists in the population (ρ = 0)

• To test the significance of r, a t ratio with degrees of freedom N – 2 must be calculated

A simplified method for testing the significance of r

• Compare the calculated r to a critical value found in Table H in Appendix C

Testing the Significance of Pearson’s r

2

2

1

r Nt

r

15

Exercises

Problem 6, 19, 21

Requirements for the Use of Pearson’s r Correlation Coefficient

10.4

A Straight-Line Relationship

Interval Data

Random Sampling

Normally Distributed Characteristics

Calculate the partial correlation coefficient

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

10.5

18

10.5

The correlation between two variables, X and Y, after removing the common effects of a third variable, Z

When testing the significance of a partial correlation, a slightly different t formula is used

Partial Correlation

. 2 21 1XY XZ YZ

XY Z

XZ YZ

r r rr

r r

.

2.

3

1XY Z

XY Z

r Nt

r

19

Exercise

Problem 30

20

Homework

Problems 18, 22 and 31Add interpretation

© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

Correlation allows researchers to determine the strength and direction of the relationship between two

or more variables

In a curvilinear correlation, the relationship between two variables starts out positive and turns negative,

or vice versa

The correlation coefficient numerically expresses the direction and strength of a linear relationship between

two variables

Pearson’s correlation coefficient can be calculated for two interval-level variables

The partial correlation coefficient can be used to examine the relationship between two variables, after

removing the common effect of a third variable

CHAPTER SUMMARY

10.1

10.2

10.3

10.4

10.5