Chapter 13: Introduction to Analysis of Variance

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Chapter 13 Introduction to Analysis of Variance

• The Single Factor Independent Measures Design

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Concepts you will need

• Variability
• Sum of Squares (SS)
• Sample variance
• df
• The logic of hypothesis testing
• The basic logic of the t test
• t ___________________________

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Chapter Overview

• Preview Rogers, Kuiper Kirker, (1977)
  self-reference study.
• 13.1 Intro - Advantages of ANOVA over t-test,
  flexibility of ANOVA for different types of
  designs, definitions of factors and factorial
  designs
• 13.2 Logic of ANOVA
• 13.3 ANOVA notation formulas
• 13.4 Distribution of F ratios
• 13.5 Examples of hypothesis testing with ANOVA
• Calculation of Effect Size
• 13.6 Post Hoc Tests
• 13.7 Relationship between ANOVA t-tests

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Chap Preview

• Rogers, Kuiper, Kirker (1977) memory study has
  four experimental conditions (4 treatment
  conditions).
• Results in Figure 13.1 on p. 388.
• With 4 independent groups, we would need to do 6
  t-tests to analyse these data.
• Problem is

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13.1 Introduction

• Anova and t- tests do the same job.
• Both test for ..
• Problem with multiple t-tests
• Probability of making a Type 1 error ..
• ANOVA avoids increased Type 1 error by doing 1
  single test to compare

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Other advantages of ANOVA overt-tests

• ANOVA used to test for mean differences in a wide
  variety of research situations.
• See Fig. 13.2 for situation with ...
• ANOVA permits analysis of ..
• See Fig. 13.3 for this situation

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ANOVA Terminology

• Factor one independent..
• Levels individual ........
• Self-reference study had .
• Expt. with one independent variable (IV) is
  called a single factor design (like
  self-reference study and Figure 13.2).
• Expt with gt 1 factor (e.g., 2 IVs) is known as
  ..................... design (like Fig. 13.3).
• ANOVA can be used to analyse results from all
  designs
• Independent...
• Repeated ..
• Mixed design mixture of ..

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Single Factor Independent Measures Design

• Typical set-up for independent measures
  experiment shown in study investigating effect of
  room temperature on learning (see Table 13.1)
• Note separate sample for each of the (3)
  treatment conditions, i.e. ...

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Table 13-1 (p. 393)Hypothetical data from an
experiment examining learning performance under
three temperature conditions.
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Statistical Hypotheses for ANOVA

• Goal of ANOVA is to decide between the null and
  alternative hypotheses
• Ho (null) There are no differences between the
  populations (treatments).
• The observed differences..
• That is, in the population, room temperature..
• Ho u1

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Statistical Hypotheses for ANOVA

• H1(alternative) The differences between the
  sample means represent ..
• That is, the populations (treatments) really are
  ..
• The mean differences btw. samples are genuine
  not ..
• E.g. Room temperature does ..
• H1 u1 ? is one
  possibility
• General form of H1 At least one population mean
  is ..

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Test statistic for Anova The Numerator

• ANOVA similar logic and structure to t-test
• t obtained difference between the 2 sample
  means
• difference expected by chance (due to sampling
  error)
• In ANOVA we calculate a F ratio rather than a
  t-ratio.
• F ..
  between sample means
• .. expected by
  chance (due to sampling error)

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F Ratio The Numerator

• F .. between sample
  means
• .. expected by chance (due to
  sampling error)
• The greater the differences btw. sample means,
  the greater the..
• F ratio based on variance of sample means rather
  than differences btw. sample means
• But still testing for significance of differences
  btw. means.
• Why variance of means rather than difference btw.
  means?
• Because ..

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Test statistic for Anova The Denominator

• Both t and F statistics measure differences
  expected by chance
• For t -- diff. btw. means expected by chance
• For F -- ..
  expected by chance
• So smaller the value expected by chance, the
  smaller ...
• As with t, the larger the value of F, the greater
  the chance that the Ho ..
• and that we will conclude that the difference
  btw. means is due to the different ..
• Like t, we must compare obtained F to required F
  (criterion value) at chosen alpha level
• As with t, we have F tables to allow us to do
  this.

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SUMMARY

• Anova works on variation between (among) means
  rather than ..
• However, Anova uses variation among means to
  decide if the means are ..
• Both t-test and ANOVA testing for significance of
  ..
• Same purpose, different method.
• Also Anova can be used with more than just 2
  means which is a limitation of the t-test.
• ANOVA can be used with independent measures or
  repeated measures designs (Ch. 14)
• ANOVA can also be used with more than 1 factor
  (gt1 IV) (Ch. 15)

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13.2 The Logic of ANOVA

• Use room temperature and learning example (see
  Table 13.1 on next slide for data)

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Table 13-1 (p. 401)Hypothetical data from an
experiment examining learning performance under
three temperature conditions.
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ANOVA calculations

• Numbers in Table 13.1 are not all the same
• There is .
• Goal of ANOVA is to measure the amount of
  variability and to ..
• Determine the ..
• First, we calculate total variability using all
  the data
• Calculate SS ..
• Then we analyse the total variability
• Break it down into ..
• Entire analysis shown on next slide

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Figure 13-4 (p. 403)The independent-measures
analysis of variance partitions, or analyzes, the
total variability into two components variance
between treatments and variance within treatments.
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ANOVA Calculations

• Total variability broken down into 2 components
  or sources of variability
• 1. Between Treatments Variance
• Some of the variability in scores is due to ..
• 2. Within Treatments Variance
• Some of the variability in scores is due to
  differences in scores of ..
• Within treatments variance provides a measure of
  variability that is .

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Between treatments variance

• Measures how much difference exists between
  treatment conditions
• ANOVA decides between 2 explanations of between
  treatments variance
• 1. Differences between scores in the various
  treatment conditions ..
• Differences reflects naturally occurring
  differences that exist between one sample and
  another.
• Unplanned ..
• 2. Differences between scores in the various
  treatment conditions is due to ..
• Differences are too large to be due to ..

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Within treatments variance

• Differences due to chance are measured by ..
• Individual within same treatment condition are
  treated identically
• Any difference in their scores is assumed to be
  ..
• One primary sources of chance differences are .
• Different individuals in different ..
• Second source of chance differences is ..
• ..
  errors can contribute to EE