College of Education

School of Continuing and Distance Education 2014/2015 – 2016/2017

PSYC 331

STATISTICS FOR PSYCHOLOGISTS

Session 4– A PARAMETRIC STATISTICAL

TEST FOR MORE THAN TWO

POPULATIONS

Lecturer: Dr. Paul Narh Doku, Dept of Psychology, UG Contact Information: pndoku@ug.edu.gh

godsonug.wordpress.com/blog

Session Overview

This session builds upon previous sessions and

provides further insight into some parametric

statistical concepts that will help in the

testing of hypotheses.

The goal of this session is to equip students

with the ability to

explain the terminology of analysis of variance

(ANOVA) ;

compute Fobs; Why Fobs should equal 1 if H0 is

true, and why it is greater than 1 if H0 is false;

Dr. P. N. Doku, Slide 2

Session Outline

The key topics to be covered in the session are as follows:

• The analysis of variance (ANOVA) procedure

• The general logic of ANOVA

• Computational procedures

• Post-hoc analysis: Multiple comparisons following the ANOVA test

• Worked example and exercises based on the One-Way ANOVA test

• Introduction to Two-Way analysis of variance (Two-Way ANOVA)

Slide 3

Reading List

• Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 85 - 109

Slide 4

Analysis of Variance

• The analysis of variance is the parametric

procedure for determining whether significant differences occur in an experiment with three or more sample means

• However, in a research study of experiment involving only two conditions of the independent variable (two samples means), you may use

either a t-test or the ANOVA and the outcome of the analysis will be the same.

Slide 5

Experiment-Wise Error

Slide 6

• The probability of making a Type I error over a series of individual statistical tests or comparisons in an experiment is called the

experiment-wise error rate

• When we use a t-test to compare only two means in an experiment, the experiment-

wise error rate equals

Experiment-Wise Error

Slide 7

• When there are more than two means in an

experiment, the multiple t-tests result in an

experiment-wise error rate much larger than

the we have selected

• Using the ANOVA allows us to make all our decisions and keep the experiment-wise error

rate equal to

An Overview of One Way ANOVA

• ANalysis Of VAriance is abbreviated as ANOVA

• ANOVA is also called the F ratio

• There is a single independent variable, hence

called One-Way

• An independent variable is also called a factor

• Each condition of the independent variable is called a level or treatment

• Differences produced by the independent variable are treatment effect

Slide 8

Requirements for using the F ratio

1) Must be a comparison between three or more

means.

2) Must be working with interval data.

3) Our sample must have been collected randomly from

the research population.

4) We can/must assume that the sample characteristics

are normally distributed.

5) We must assume that the variance between samples

are all equal. Slide 9

10

Between-Subjects

• A one-way ANOVA is performed when one

independent variable is tested in the experiment

• When an independent variable is studied using

independent samples in all conditions, it is called a

between-subjects factor

• A between-subjects factor involves using the formulas

for a between-subjects ANOVA

11

Within-Subjects Factor

• When a factor is studied using related (dependent) samples in all levels, it is called a within-subjects

factor

• This involves a set of formulas called a within-subjects

ANOVA

12

Diagram of a Study Having

Three Levels of One

Factor

13

Null and Alternate Hypotheses

• Null hypothesis

H0 : 1 2 ... k

• Alternate hypothesis: states that at least the means of two of the populations differ.

H a : not all k are equal

The ANOVA (F) Test

• The statistic for the ANOVA is F

• When Fobs is significant, it indicates only that somewhere among the means at least two of them differ significantly

• It does NOT indicate which specific means

differ significantly

• When the F-test is significant, we perform post hoc comparisons to determine which specific means differ significantly

Computation of the ANOVA (F) Test

The Analysis of Variance is a multi-step process.

1. Sum of Squares 2. Mean Square 3. F Ratio

Slide 15

Sum of Squares

• The computations for the ANOVA require the

use of several sums of squared deviations

• The sum of squares is simply the sum of the

squared deviations of a set of scores around

the mean of those scores

• Adding them up.

• It is symbolized as SS

Sum of Squares

Comparing Groups: When groups are compared, there are more than one type of sum of squares.

Total Sum of Squares (SS total)

Between Groups Sum of Squares (SS between)

Within Groups Sum of Squares (SS within)

Each type represents the sum of squared

deviations from a mean.

Computational Formulae for SS

Slide 18

X X 2 N SS T

2

X12 X 2 2 Xk 12 Xk 2 X 2

N nk nk 1 n2 n1

SSB ...

X12 X 2 2 Xk 12 Xk 2

k

k 1 Xk

... Xk 1

SSW X1

X 2

n n n n

2 2

2

2

1

2

The Computational Formulas for Sum of Squares: worked example

The Computational Formulas for Sum of Squares: worked example

The Computational Formulas for Sum of Squares: Summary

Mean Squares

NOTE: The value of the sum of squares becomes larger as variation increases. The sum of squares also increases with sample size.

Because of this, the SS cannot be viewed as a true measure of variation. Another measure of variation that we can use is the Mean Square.

• The mean square within groups describes the variability in

scores within the conditions of an experiment. It is symbolized

by MSW.

• The mean square between groups describes the differences

between the means of the conditions in a factor. It is

symbolized by MSB.

Coputation of Mean Squares

• Between • Within

MSwithin

between

SSbetween

between df

MS

MSbetween = between group mean

square

SSbetween = between group sum of

squares

dfbetween = between group degrees of

freedom

within

SSwithin df

MSwithin = within group mean square

SSwithin = within group sum of

squares

dfwithin = within group degrees of

freedom

Degrees of Freedom

Use the following equations to obtain the correct degrees of freedom:

k df within N total

k 1 df be twe e n

k = number of groups

Critical Value of F (F critical)

The Critical value of F (Fcrit) depends on:

• The degrees of freedom (both the dfbn = k – 1 and the dfwn

= N – k)

• The selected

To obtain the Fcrit from the F statistical table:

Use the dfB (the numerator) across the top of the table.

Use the dfW (the denominator) along the side of the

table.

Worked example of Mean

Square Computation Calculating the Mean Square using Table 8.2 data in the previous example

Computing Fobs

The analysis of variance yields an F ratio.

The F ratio is the variance between groups and variation within groups compared.

The larger our calculated F ratio, the increased likelihood that we

will have a statistically significant result.

M S within ( wn )

M S be twe e n (bn )

F obs

• Illustration of another way

of computing the Sum of Squares and Mean Squares

using the mean method

Dr. Richard Boateng, UGBS Slide 28

Example: does family size vary by religious affiliation?

Step 1: Find the mean for each sample

Step 2:Cal. (1) Sum of scores, (2) sum of sq. scores, (3) number of subjs., (4) and mean

computations

computations

computation

DECISION

•To reject the null hypothesis at the .05 significance level with 2 and 12 degrees of freedom, our calculated F ratio

must exceed 3.88.

•From the computation, our obtained F ratio of 8.24, is

clearly greater than the F critical, hence we must reject the null hypothesis.

•Interpretation: At 0.05 significant level, it is indeed true

that Family size does vary by religion.

Post Hoc Comparisons

• When the F-test is significant, we perform post hoc comparisons

• Post hoc comparisons are like t-tests

• We compare all possible pairs of level means

from a factor, one pair at a time to

determine which means differ significantly

from each other

Examples: The protected t test method and

Fisher Least Significant (LSD) method

The Protected t Test method

The null hypothesis for comparing any pair of means tested with the formula:

and is

Mserror = MSw

where MSw is simply taken from the ANOVA results and n1

and n2 are the sizes of the two samples whose means we are

comparing. The computed value of t is referred to the t tables at α = 0.05 for a two-tailed test with

the degrees of freedom (df) associated with the MSw (= N - k) and a decision is taken as to whether or not Ho should be

rejected

X1 X 2 X1 X 2

1 2 n1 n2

1

1 MSerror MSerror

MSerror

t

n n

Fisher LSD (Least Significant Difference)

method

• Used when all the groups have equal sample sizes, i.e. n1=n2=n3

• Then the denominator of the protected t test becomes a constant for all pairwise comparisons. In such a situation, it becomes possible to determine

what least significant difference (LSD) between means is needed to reject Ho at any given level of significance.

=

Note that t here refers to the critical value of t with N-k df in a two-tailed test

X1 X 2 X1 X 2

1 2 n1 n2

1

1 MSerror MSerror MSerror

t

n n

Two- way ANOVA- overview

• We have learned how to test for the effects of independent variables considered one at a time.

• However, much of human behavior is determined

by the influence of several variables operating at the same time.

• Sometimes these variables combine to influence performance.

Two- way ANOVA

• We need to test for the independent and combined effects

of multiple variables on performance. We do this with a

Two- way ANOVA that asks:

(i)how different from each other are the means for levels

of Variable A?

(ii)how different from each other are the means for levels

of Variable B?

(iii)how different from each other are the means for the

treatment combinations produced by A and B

together?

Two way ANOVA

• The first two of those questions are questions about main

effects of the respective independent variables.

• The third question is about the interaction effect,

the

effect of the two variables considered simultaneously.

MAIN vs INTERACTION EFFECTS

• Main effect •

–

A main effect is the effect on

performance of one treatment variable considered in isolation

(ignoring other variables in the

study)

Slide 42

Interaction Effect

– an

interaction effect occurs

when the effect of one

variable is different across

levels of one or more other variables

Illustration

• In order to detect interaction effects, we must use ͞factorial͟ designs.

– In a factorial design each variable is tested at every level of

all of the other variables.

– Below represent two variables A and B both with two

levels A1,A2 and B1,B2 respectively. A1 A2

B1 i ii

B2 iii iv

Illustration

I.vs III

– Effect of B at level A1 of variable A

II.vs IV

– Effect of B at A2

• If these are different, then we say that A and B interact

ALTERNATIVELY

I vs II

– Effect of A at B1

III vs IV

– Effect of A at B2

• If these are different, then we say that A and B interact

Illustration

B1

B1

B2

B2

A1 A2 A1 A2

•In the graphs above, the effect of A varies at levels of B, and the effect of B varies at levels of A. How you say it is a

matter of preference (and your theory).

•In each case, the interaction is the whole pattern. No part of the graph shows the interaction. It can only be seen in

the entire pattern (here, all 4 data points).

Computation of F ratios in Two- Way ANOVA

• In a Two-Way ANOVA, three F ratios are computed:

• One F ratio is computed for the factor represented along the

rows;

• a second F ratio is computed for the factor represented

along the columns; and

• a third F ratio is computed for the interaction between the

factors represented along the rows and columns.

• The various F ratios are each referred to the F tables with

the appropriate degrees of freedom associated with each F

ratio under a specified decision rule and a decision is taken

as to whether or not Ho should be rejected in each case.

Slide 46