© 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.4/14/2014, Spring 2014

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

Chapter 8: Analysis of Variance

1

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

Understand the logic of analysis of variance

Understand and calculate the sum of squares

Understand and calculate the mean square

Understand and calculate the F ratio

Conduct a multiple comparison of means to determine where a significant difference lies

CHAPTER OBJECTIVES

8.1

8.2

8.3

8.4

8.5

8.6 Understand the requirements for using the F ratio

Understand the logic of analysis of variance

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

8.1

Comparing More than Two Groups8.1

Multiple t Ratios

Analysis of Variance

vs.

Combing t ratios increases the chance of making a Type I error

A single overall decision as to whether a significant difference exists

The Logic of Analysis of Variance 8.1

Total Variation

VariationbetweenGroups

Variationwithin Groups

8.1

Figure 8.1

8.1

Figure 8.2

The Logic of Analysis of Variance 8.1

Understand and calculate the sum of squares

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

8.2

8.2

10

The sum of squared deviations from the mean• Total sum of squares• Between-group sum of squares• Within-group sum of squares

The Sum of Squares

2

total total

2

within group

2

between group group total

SS

SS

SS

X X

X X

N X X

Definitional Formulas Computational Formulas

2 2total total total total

2 2within total group group

2 2between group group total total

SS

SS

SS

X N X

X N X

N X N X

Understand and calculate the mean square

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

8.3

8.3

12

The value of the sum of squares grows as variation and sample size increase

• The mean square (or variance) controls for these influences

The Mean Square

betweenbetween

between

withinwithin

within

SSMS

df

SSMS

df

number of groupsk

between

within total

df 1

df

k

N k

Understand and calculate the F ratio

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

8.4

8.4

14

A comparison of the variation between groups and variation within groups

• It must be evaluated for significance– The larger the F ratio, the more likely it is to be statistically significant – Interpret the F ratio with Table D in Appendix C (pages 554-555)

The F ratio

between

within

MS

MSF

8.4

Table 8.4

Conduct a multiple comparison of means to determine where a significant difference lies

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

8.5

8.5

17

A significant F ratio does not tell us where the difference lies

We can use Tukey’s Honesty Significant Difference (HSD) to determine this

• Allows for a comparison of any two means against HSD• Takes into account Type I error

A Multiple Comparison of Means

found in Table Eq

within

group

MSHSD q

N

A set of several t-test are also acceptable but take more time

Now just compare the values of the difference between each pair of means and HSD. A difference is significant if it is greater than the HSD

(Page 556)

Understand the requirements for using the F ratio

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

8.6

Requirements for Using the F Ratio8.6

A Comparison between Three or More Independent Means

Interval Data

Random Sampling

A Normal Distribution

Equal Variances

Example

20

Problem 17-18Problem 37

Homework

21

Chapter 8 Problems: 28 and 34-35Also Chapter 7 Problem 42

Exam #2

22

Confidence IntervalsHypothesis Testing (two-sided only)ANOVA

I’ll give you some statistics

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

Analysis of variance examines the variance between and within groups to determine if a significant

difference exists

The sum of squares examines the total deviation, the between-group deviation and the within-group

deviation

The mean square controls for the increase in the sum of squares that naturally occurs as variation and

sample size increase

The F ratio compares variation between groups and variation within groups

CHAPTER SUMMARY

8.1

8.2

8.3

8.4

Tukey’s HSD is used to determine where a significant difference lies once a significant F ratio has been found

There are several requirements that must be met in order to conduct an analysis of variance test

8.5

8.6