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### Chapter 13

### Two-FactorAnalysis of Variance

### The GLMRepeated Measures

INTRODUCTION TO ANALYSIS OF VARIANCE

INTRODUCTION

- Analysis of variance (ANOVA) is a hypothesis testing procedure that is used to evaluate mean differences between two or more treatment
- ANOVA has a tremendous advantage over t-test
- The major advantage is that it can be used to compare two or more treatments

TERMINOLOGY

- When a researcher manipulates a variable to create treatment conditions, the variable is called an independent variable
- When a researcher uses non-manipulated variable to designate groups, the variable is called a quasi independent variable
- An independent variable or a quasi independent variable is called a factor
- The individual groups or treatment condition that are used to make up a factor are called the levels of the factor

Like the t test , ANOVA can be used with either an independent measures or a repeated measures design

- An independent-measures design means that there is a separate sample for each of treatments
- A repeated-measures design means that the same sample is tested in all of the different treatment condition
- ANOVA can be used to evaluate the results from a research study that involves more than one factor

STATISTICAL HYPOTHESES FOR ANOVA

- Suppose that a psychologist examined learning performance under three temperature conditions: 150 C, 250 C, and 350 C
- Three samples of subjects are selected, one sample for each treatment condition
- The purpose of the study is to determine whether room temperature affects learning performance

The HYPOTHESES

- H0 : µ1 = µ2 = µ3

In words, the null hypothesis states the temperature has no effect on performance

- H1 : at least one condition mean is different from another

In general, H1 states that the treatment conditions are not all the same; that is, there is a real treatment effect

The TEST STATISTIC FOR ANOVA

Variance (differences) between samples means

F =

Variance (differences) expected by chance (error)

Note that the F-ratio is based on variances instead of sample mean difference

One-Way ANOVA

- The One-Way ANOVA procedure produces a one-way analysis of variance for a quantitative dependent variable by a single factor (independent) variable.
- Analysis of variance is used to test the hypothesis that several means are equal. This technique is an extension of the two-sample t-test.

One-Way ANOVA

- Adakah pengaruh kelembapan terhadap kecepatan mengetik?
- Bandingkan dengan t-test!
- Adakah perbedaan kecepatan mengetik berdasarkan temperatur udara? Pada temperatur berapakah kecepatan mengetik yang paling cepat?

One-Way ANOVA

- Bersifat between subjects
- Contoh: Pengaruh temperatur udara terhadap kecepatan mengetik

Analyze >> Compare Means >> One-Way Anova

Independent Measures

preview

Imagine that you are seated at your desk, ready to take the final exam in statistics. Just before the exam are handed out, a television crew appears and set up a camera and lights aimed directly at you. They explain they are filming students during exams for a television special. You are told to ignore the camera and go ahead with your exam.

Would the presence of a TV camera affect your performance on your exam?

example

- Shrauger (1972) tested participants on a concept formation task. Half the participants work alone (no audience), and half with an audience of people who claimed to be interested in observing the experiment.
- Shrauger also divided the participants into two groups on the basis of personality: those high in self-esteem and those low in self-esteem
- The dependent variable for this experiment was the numbers of errors on the concept formation task

result

With Audience

10

8

6

4

2

No Audience

No Audience

With Audience

Mean number of errors

HIGH

Self-Esteem

LOW

result

- Notice that the audience had no effect on the high-self-esteem participants
- However, the low-self-esteem participants made nearly twice as many errors with an audience as when working alone

Shrauger’s study have two independent variables, which are:

- Audience (present or absent)
- Self-esteem (high or low)
- The result of this study indicate that the effect of one variable depends on another variable
- To determine whether two variables are interdependent, it is necessary to examine both variables together in single study

Most of us find it difficult to think clearly or to work efficiently on hot days

- If you listen to people discussing this problem, you will occasionally hear comments like, “It’s not the heat; it’s the humidity”
- To evaluate this claim scientifically, you will need to design a study in which both heat and humidity are manipulated within the same experiment and then observe behavior under a variety of different heat and humidity combinations

MAIN EFFECT

- The main differences among the level of one-factor are referred to as the main effect of the factor
- When the design of the research study is represented as a matrix of one factor determining the rows and the second factor determining the columns, then the mean differences among the row describe the main effect of one factor, and the mean differences among the column describe the main effect for the second factor

INTERACTION

An interaction between two factors occurs whenever the mean differences between individual treatment condition, or cells, are different from what would be predicted from the overall main effects of the factors

Factorial ANOVA

- is used when we have two or more independent variables (hence it called factorial)
- Several types of factorial design:
- Unrelated factorial design
- Related factorial design
- Mixed design

Several Types Factorial ANOVA

- Unrelated factorial design

This type of experiment is where there are several IV and each has been measured using different subject

- Related factorial design

An experiment in which several IV have been measures, but the same subjects have been used in all conditions (repeated measures)

- Mixed design

A design in which several independent variables have been measured; some have been measured with different subject whereas other used the same subject

Factorial ANOVA

- Bersifat between subject
- Contoh: Pengaruh golongan darah dan jenis kelamin terhadap kemampuan meyelam

Analyze >> General Linear Model >> Univariat

ONE INDEPENDENT VARIABLE

What is…

- ‘Repeated Measures’ is a term used when the same subjects participate in all condition of an experiment
- For example, you might test the effects of alcohol on enjoyment of a party
- Some people can drink a lot of alcohol without really feelings the consequences, whereas other only have to sniff a pint of lager and they fall to the floor and pretend to be a fish

Repeated ANOVA

- Bersifat within subjects
- Contoh: Pengaruh waktu (pagi/siang/malam) terhadap kemampuan push-up

Analyze >> General Linear Model >> Repeated Measures

Advantages…

- It reduces the unsystematic variability and so provides greater power to detect effects
- More economical because fewer subjects are required

Disadvantages…

- In between-groups ANOVA, the accuracy of the F-test depends upon the assumption that scores in different conditions are independent. When repeated measures are used this assumption is violated: scores taken under different experimental condition are related because they come from the same subjects
- As such, the conventional F-test will lack accuracy

SPHERICITY

- The relationship between scores in different treatment condition means that an additional assumption has to be made and, put simplistically, we assume that the relationship between pairs of experimental condition is similar
- This assumption is called the assumption of sphericity

What is SPHERICITY?

- Most of us are taught that is crucial to have homogeneity of variance between conditions when analyzing data from different subjects, but often we are left to assume that this problem ‘goes away’ in repeated measure design
- Sphericity refers to the equality of variances of the differences between treatment level
- So, if you were to take each pair of treatment levels, and calculate the difference between each pair of scores, then it is necessary that differences have equal variance

How is sphericity measured?

variance A-B ≈ variance A-C ≈ variance B-C

Assessing the severity of departures from sphericity

- SPSS produces a test known as Mauchly’s, which tests the hypothesis that the variances of the differences between conditions are equal
- Therefore, if Mauchly’s test statistic is significant, we should conclude that there are significant differences between the variance differences, ergo the condition of sphericity is not met

Mixed ANOVA

- A design in which several independent variables have been measured; some have been measured with different subject whereas other used the same subject
- Minimal ada 2 IV
- Bersifat between subject

Analyze >> General Linear Model >> Repeated Measures

Mixed ANOVA

- Contoh: Pengaruh waktu (pagi/siang/malam) dan jenis kelamin terhadap kemampuan push-up
- Semua subjek dilihat kemampuan push-up di pagi, siang, dan malam. Tetapi ada dua kelompok yang sama sekali berbeda, yaitu kelompok laki-laki dan perempuan

THE LOGIC OF ANALYSIS OF VARIANCE

150 C 250 C 350 C

One obvious characteristic of the data is that the scores are not all the same. Our goal is to measure the amount of variability and to explain where it comes from

0

1

3

1

0

M = 1

4

3

6

3

4

M = 4

1

2

2

0

0

M = 1

* Note that there are three separate samples, with n = 5 in each sample. The dependent variable is the number of problems solved correctly

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