BHS 307 Statistics for the Behavioral Sciences

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BHS 307 Statistics for the Behavioral Sciences

• Chapter 16 One-Factor Analysis of Variance
  (ANOVA)

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Fishers F-Test (ANOVA)
Ronald Fisher
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Testing Yields in Agriculture


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ANOVA

• Analysis of variance (ANOVA) a test of more
  than two population means.
• One-way ANOVA only one factor or independent
  variable is manipulated.
• ANOVA compares two sources of variability.

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Two Sources of Variability

• Treatment effect the existence of at least one
  difference between the population means defined
  by IV.
• Between groups variability variability among
  subjects receiving different treatments
  (alternative hypothesis).
• Within groups variability variability among
  subjects who receive the same treatment (null
  hypothesis).

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F-Test

• If the null hypothesis is true, the numerator and
  denominator of the F-ratio will be the same.
• F random error / random error
• If the null hypothesis is false, the numerator
  will be greater than the denominator and F gt 1.
• F random error treatment effect random
  error

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Difference vs Error

• Difference on the top and the error on the
  bottom
• Difference is the variability between the groups,
  expressed as the sum of the squares for the
  groups.
• Error is the variability within all of the
  subjects treated as one large group.
• When the difference exceeds the variability, the
  F-ratio will be large.

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F-Ratio

• F MSbetween MSwithin
• MS SS df
• SS is the sum of the squared differences from the
  mean.

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F-Ratio

• F MSbetween MSwithin
• MSbetween treats each of the group means as a
  data set and calculates the sum of squares for
  it.
• MSwithin combines the groups into one large group
  and calculates the sum of squares for the whole
  group.

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Testing Hypotheses

• If there is a true difference between the groups,
  the numerator will be larger than the
  denominator.
• F will be greater than 1
• Writing hypotheses
• H0 m1 m2 m3
• H1 H0 is false

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Calculating F

• SSbetween S T2 G2 n N
• Where T2 is the total for each group
• G2 is the grand total
• Sswithin S X2 - S T2 N
• SStotal S X2 G2/N (error in
  textbook)

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F-Distribution
Common retain null
Rare reject null
Critical value
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ANOVA Assumptions

• Assumptions for the F-test are the same as for
  the t-test
• Underlying populations are assumed to be normal
  with equal variances.
• Results are still valid with violations of
  normality if
• All sample sizes are close to equal
• Samples are gt 10 per group
• Otherwise use a different test

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Cautions

• The ANOVA presented in the text assumes
  independent samples.
• With matched samples or repeated measures use a
  different form of ANOVA.
• The sample sizes shown in the text are small in
  order to simplify calculations.
• Small samples should not be used.