Experimental designs and Analysis of Variance Introduction

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Experimental designsandAnalysis of
VarianceIntroduction
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Introduction

• Mark Huisman
• Room 185, Heymans building
• Tel. [phone]
• Email j.m.e.huisman_at_rug.nl
• Questions Tuesdays 16 17 (or via Nestor)
• Rivka de Vries
• Lab classes

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Overview

• Univariate Analysis of Variance
• Refresher (Moore McCabe Chap. 12 13)
• Principles
• The ANOVA model
• Next lectures
• Assumptions and power
• Multiple comparisons (contrasts and post hoc
  tests)

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Principles of one-way ANOVA

• I independent populations ? comparison of I
  groups
• In each population dependent variable y N(µj,s
  )
• The SDs are equal in all populations s
• I samples of size nj
• Observations yij of person i in group j
• Test H0 µ1 µI vs. Ha not all µs are
  equal
• F-test based on partitioning of the (total)
  variance of y
• SST SSG SSE

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Principles of one-way ANOVA
partitioning of variance between groups and
within groups
F ratio between / within
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Experimental error

• Differences between means due to
• Treatment effect
• Chance (motivation, attention, measurement, etc)
• Estimate extent to which differences are due to
  experimental error ? evaluate hypothesis of equal
  group means
• Variability within treatment groups of subjects
  provides estimate
• If null hypothesis is true and subjects are
  randomly assigned to treatments, than variability
  between treatment groups also provides estimate
• If null hypothesis is false there is a treatment
  effect (systematic differences), than variability
  between treatment groups reflects treatment
  effects and experimental error

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Logic of hypothesis testing

• Experimental error reflected in
• differences (variability) among subjects given
  same treatment
• differences (variability) among groups of
  subjects given different treatment
• Inspect the ratio of variabilities
• If H0 is true
• If H0 is false

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Partitioning the variance
One-way ANOVA population model
In sample (based on observations)
Partitioning the variance
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The ANOVA table
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Example Sesame Street

• Sesame street data set evaluating impact of the
  first year of Sesame street television series
• n 240 children, aged 35
• y test on knowledge about numbers (POSTNUMB,
  054)
• factor viewing categories (VIEWCAT) coded 1
  (rarely) to 4 (often more than 5 times a week) ?
  4 groups

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Principles of one-way ANOVA
partitioning of variance between groups and
within groups
µT
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The one-way ANOVA model
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The one-way ANOVA model
One-way ANOVA population model
Testing H0 µ1 µI µT
equals H0 aj 0 for all groups j
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Expected Mean Squares

• F ratio
• from the partitioning (ANOVA table)
• Properties? Look at long-term average (expected
  value) and use ANOVA model

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Example Table 14.1

• Keppel Wickens Chapter 14
• I 3 groups, n1 3, n2 6, n3 4 (unbalanced)
• Test H0 all group means are equal
• If H0 true
• Model

Example
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Example Table 14.1

• Model
• Error
• Sum of squares
• unexplained variation (error) in H0 model
• What happens if H0 is false, or what happens if
  Ha is true?
• Then group means are not equal and best guess
  for each group is its sample mean
• Model

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Example Table 14.1

• Model
• Error
• Sum of squares
• unexplained variation (error) in Ha model
• Improvement 86.28 24.00

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Model-based comparison

• Treatment SS difference in SS of H0 and Ha
  model
• Error SS unexplained SS of Ha model (reflects
  only sampling error)
• Degrees of freedom are equal to the number of
  observations minus number of parameters

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Significant result

• Significant result what does it mean?
• at least 1 group differs from the other groups,
  based on one or more effects (main/interaction)
• Which groups differ (w.r.t. mean)?
• further inspection
• visual inspection ? no statistical proof
• (multiple) comparisons (tests or CIs)
• planned ? contrasts
• post hoc comparisons

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Visual inspection Means plot
Check overlap
? statistical proof for significant
differences tests or CIs