OneWay Analysis of Variance

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One-Way Analysis of Variance

• COMPS Review

2
Weve Already Learned

• How to test hypotheses about TWO population MEANS
• One-Sample differences b/w sample mean test
  value
• Paired Samples differences b/w two RELATED
  sample means
• Independent Samples differences b/w two means
  from TWO samples

3
What Were Doing Now

• NOW, we want to compare MORE THAN 2 population
  means using ANOVA
• 4 teaching methods, compare test scores for 4
  groups
• 7 treatments, compare results
• Testing the NULL hypothesis that several
  independent population means are equal (no
  difference)
• Chapter Sample Data gssft.sav (version 10)
  (The one different outcome from version 11 is in
  the Bonferroni)

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New Research Question

• We found that college grads work b/w 46 and 49
  hours a week
• Now, we want to compare the average number of
  hours worked per week for everyone, by education
  category (degree)
• Null Hypothesis Population means for all 5
  groups are the same (no difference)
• Alternative There is a difference

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Analysis of Variance

• ANOVA examines the variability in sample values
• How much the scores within each group vary, and
  how much the group means vary
• Based on these 2 estimates of variability, we can
  draw conclusions about population means
• One-Way ANOVA, cases are assigned to groups
  based on their values for ONE variable (factor
  grouping variable)

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First Step Descriptive Statistics
Analyze gt General Linear Model gt Univariate Enter
the DV and the factor Select your POST HOC and
OPTIONS
Sample Means range from 43.69 to 50.27, and the
total average is 46.29 Notice variability ranges
from 8.72 to 12.89 S.E. How much sample means
vary in repeated samples (SD/sq.rt.n)
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95 Confidence Intervals
95 CI H.S. has smallest interval and the
largest n. Jr. College has the largest CI, and
only n54. Most CIs overlap, except for the
Graduate group it doesnt overlap the Less than
HS and HS groups. None of the 95 CIs contain 40
(You can reject the 40-hr work week hypothesis)
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Necessary Assumptions

• Independent random samples
• No pairing, repeated measures
• No changing conditions
• Normal populations
• OK unless data are extremely non-normal
• Use histograms or Q-Q plots
• Population variances are all equal
• Can use the Levenes Test
• If number of cases per group is similar, then OK

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Analyzing Variability

• Partitioned Variability
• Conclusions about means are based on variability
  of sample means (we determine if the observed
  mean is outside the usual range of means)
• In ANOVA, compare observed variability to the
  variability youd expect if the null is true
• More variance than youd expect reject null
• If only 2 groups, same results using equal var. t

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Partitioned Variability

• To find out if sample means vary more than youd
  expect if the null were true
• Within-Groups Variability Avg. the variances in
  each group to get a single number
• Variances must be taken individually by groups
• Sum (variance)(n-1)
• Between-Groups Variability How much sample
  means vary b/w groups (from S.E. of the mean)
• SD of observations (S.E.)(sq.rt. N)
• Sum (n)(grp. mean total mean)2

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The F Ratio to Test the Null

• Comparing the between-groups and within-groups
  estimates of variability
• The between-groups If null is false, will be
  large
• The within-groups Always a good estimate
• If b/w-groups is sufficiently larger than the
  w/in-groups estimate of variability, youll
  reject the null (i.e., Large F value small
  Sig.)
• If null of no difference is true, the F value1

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Second Step SPSS ANOVA Results
Ratio of mean squares (BGSS/WGSS) 3.646,
Significance .006 Sample means varied more than
expected. Only 6x/1,000 would you expect this
much variance if null were true. Reject NULL
HYPOTHESIS of no difference in hours worked b/w
groups.
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F Ratio Up Close

• F BGMS/WGMS
• Within-Groups
• Sum of squares (group variance and n-1)
• Degrees of freedom (for ea. Group, n-1)
• WG Mean square (WGSS/df)
• Between-Groups
• Sum of squares (group mean, overall mean, and n)
• Degree of freedom (number of groups 1)
• BG Mean square (BGSS/df)

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Observed Significance Level

• Calculated by comparing observed F ratio to
  values of the F distribution
• Calculated by SPSS
• Sig. level depends on F ratio and df
• When reporting, must give both
• F Distribution is defined mathematically
• Indexed by 2 values for df

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Third Step Follow-up Tests

• Rejecting null means are not equal, but which
  ones are different?
• To pinpoint which means are different, need to do
  a Multiple Comparisons Procedure
• One example Bonferroni
• Adjusts sig. by number of comparisons made
  (.05/5)
• If we cannot assume equal variance, use Dunnetts
  C

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Bonferroni
Each row is a comparison of 2 groups. The
difference in hours worked is labeled Mean
Difference. Pairs of significantly different
means are marked with an asterisk. People w/
Graduate Degrees work sig. longer than people
with Less than HS and HS. Group n Important!
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From Green Book

• Run the ANOVA differently
• Analyze gt General Linear Model gt Univariate
• Under Options, for grouping variable, display
  means
• Also click Homogeneity tests, Estimates of effect
  size, and Descriptive statistics, then Post Hoc
  tests
• Effect Size Statistic eta square (?2)
• The proportion of variance in D.V. related to
  factor
• .01, small / .06, medium / .14, large
• APA Write-up

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Summary

• ANOVA when comparing more than 2 population
  means
• Assumptions Independence, Normal, Var.
• F Ratio BGMS/WGMS
• Follow-up test Bonferroni (which one depends on
  the Levenes results)
• APA Write-up should include F(df1,df2), p-value,
  and ?2 as well as multiple comparison results