OneWay Analysis of Variance

1
One-Way Analysis of Variance

• Chapter 16

2
ANOVA

• A statistical technique for testing for
  differences in the means of several groups
• Useful for a couple of reasons
• Unlike a t-test, it can test more than 2 sets of
  means
• Allows us to discuss 2 or more IVs
  simultaneously
• One-Way ANOVA
• An analysis of variance in which the groups are
  defined on only 1 independent variable

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Hypotheses ANOVA

• Null Hypothesis
• All conditions will be equal
• H0 ?1 ?2 ?3 ?4 ?5
• Alternative Hypothesis
• At least 1 of these means is different from the
  others
• It does not specify which means are different
• H1 ?1 ? ?2 ? ?3 ? ?4 ? ?5 or
• H1 ?1 ?2 ? ?3 ? ?4 ? ?5 or
• H1 ?1 ?2 ? ?3 ? ?4 ?5 etc.

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Assumptions

• Assumption of Normality
• Scores in each population are normally
  distributed around the population mean (?j)
• Assumption of Homogeneity of Variance
• Each population of scores has the same variance
• ?21 ?22 ?23 ?24 ?25 ?26 ?2e
• ?2e is the common variance held by all of the
  values
• the e stands for error

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Assumptions

• A useful aspect of the ANOVA is that small
  violations have little impact on the results of
  the ANOVA
• This means the ANOVA is robust to violations of
  the normality and homogeneity of variance
  assumptions
• Assumption of Independence of Observations
• Each score is independent of the others
• It is a serious problems if not independent
• If we use random assignment, this is usually not
  a problem

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Logic of ANOVA

• The basic idea with an ANOVA is that we are
  computing a ratio between the variability among
  the group means and the variability among
  subjects in the same group
• Both sources of variability are an estimate of
  the population variance
• If there is no difference between the group means
  (H0 is true), then we expect the two estimates to
  be roughly equal and the ratio to roughly equal 1
• If there is a difference (H0 is false) then we
  expect the ratio to be larger than 1

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Logic of ANOVA

• We know that s2 is an estimate of ?2
• Because of the HOV assumption, we can say ?21
  ?22 ?23 ?24 ?25
• Which we call ?2e
• Another way of saying this is that the average of
  the sample variances is an estimate of ?2e
• ?2e s21 s22 s23 s24 s25
• n
• ?2e can also be called the mean squared error
  (MSerror)
• This is the variability among subjects in the
  same group

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Logic of ANOVA

• If H0 is true then each of the sample means comes
  from the same population
• In this case the variance of each sample mean
  (?2X ) is ?2e / n
• We can rearrange this equation and substitute s2X
  for ?2X
• ?2e is estimated by n s2X
• This is our estimate of variability between group
  means (MSgroup)

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Terms

• Sum of Squares
• The sum of the squared deviations around some
  point (usually a mean)
• SSTOTAL
• The sum of squares of all scores, regardless of
  group membership
• SSTOTAL can be partitioned into SS due to
  variation between groups and the SS that is due
  to variation within groups
• SSGroup
• The sum of squared deviations of group means from
  the grand mean multiplied by the sample size
• SSERROR
• The sum of the squared residuals or the sum of
  the squared deviations within each group

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ANOVA Calculations I

• SSTOTAL ?X2total - (?Xtotal)2
• Ntotal
• SSGroup (?X1)2 (?X2)2 ... (?Xk)2 -
  (?Xtotal)2
• n1 n2
  nk Ntotal
• SSERROR SSTOTAL - SSGroup
• ?X21 - (?X1)2 ?X22 - (?X2)2 ?X2k -
  (?Xk)2
• n1
  n2 nk

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ANOVA Calculations II

• dftotal N-1
• dfgroup k - 1
• k is the number of groups
• dfERROR N - k
• MSgroup SSGroup / dfgroup
• MSerror SSerror / dferror

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F Statistic

• The F statistic is the ratio between the
  variability due to groups and the variability due
  to subjects in the same group
• F MSgroup / MSerror

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Example

• We are interested in whether setting goals
  affects individuals performance on statistics
  tests. We randomly assigned 20 people to four
  groups Group 1 was given a hard goal, group 2 an
  easy goal, group three was told to do their best,
  and group four was told nothing.
• We recorded the final exam score for every
  individual
• Using a .05 level of significance, conduct an
  ANOVA

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Summary Table

• Source df SS MS F
• Group 3 538.55 179.52 29.92
• Error 16 96.00 6.00
• Total 19 634.55
• What does a F(3,16) 29.92 tell us to decide
  about the null hypothesis?

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Appendix E.3 E.4

• This table is the F distribution
• This table will give us an F critical value to
  compare to our obtained F
• We have a tables for ? .05 ? .01
• To use the table
• Across the top is our dfgroup
• Down the left is our dferror
• Report critical value as
• F?(dfgroup , dferror) Critical Value

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Comparisons Between Means I

• When we find a significant F value, we only know
  that at LEAST one mean was different
• We dont know which ones or how many are
  different
• What we want to do is use a t-test to test the
  differences between the means
• However, we cant just start comparing all of the
  means at the .05 level because conducting the
  multiple t-tests will increase the probability of
  a Type I error
• We are actually increasing that probability with
  each of the comparisons we undertake

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Comparisons Between Means II

• In order to maintain the an acceptable
  probability of a Type I error, we use special
  sets of techniques to make the comparisons
  between the means
• These are often called post-hoc tests
• We only conduct these tests if we have found a
  significant F value

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Tukeys HSD Test

• HSD Honestly Significant Difference
• HSD q? sqrt(MSERROR / n)
• q? is the q-statistic which we will look up in a
  table using alpha, the dferror , and the number
  of observations in each group (n)
• q? is found by using the studentized range table
• The HSD value is a critical value to compare to
  the difference between 2 means

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Tukeys HSD Test

• Step 1
• Make a table with the groups along the top and on
  the left
• Step 2
• Take the differences between the means
• Step 3
• Compute the HSD value compare the difference
  between the means with the HSD value
• Step 4
• Any difference that is larger than the HSD is a
  significant difference

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Example

• Using our example of goal setting and test
  performance use Tukeys HSD test to compare the
  means of the groups

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Magnitude of effect

• A measure of the degree to which variability
  among observations can be attributed to
  conditions
• ?2 (Eta squared) -
• A BIASED estimate (tends to overestimate)
• ?2 SSGROUP / SSTOTAL
• this is telling us the amount of variation due to
  condition/group effects
• ?2 (Omega squared)
• A less biased measure of the magnitude of effect
• ?2 SSGROUP - (k-1) MSERROR
• SSTOTAL MSERROR

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Example

• Using our example of goal setting and test
  performance compute ?2 and ?2

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Final Example

• We are interested in whether listening to music
  affects the cognitive development of infants. We
  randomly assign 30 infants to one of 6
  conditions Group 1 listens to Mozart, Group 2
  listens to Brittany Spears, Group 3 listens to
  Metallica, Group 4 listens to Eminem, Group 5
  listens to white noise, Group 6 listens to
  nothing
• We give each infant an IQ test after 3 months
• Using a .05 level of significance, conduct an
  ANOVA, compute both measures of effect size, and
  compute Tukeys HSD