Analysis of Variance ANOVA

1
Analysis of Variance (ANOVA)

• The F distribution
• Good for two or more groups

2
The F distribution

• F is a ratio of two independent estimates of the
  variance of the population
• Consequently, it depends on the analysis
  (separating into parts) of the variance in a set
  of scores.
• We have already analyzed the variance in a set of
  scores when we did t tests

3
The analysis of variance

• The numerator of the t test is the difference
  between two means. Since the variance is the
  average of the differences from a mean, the
  difference between two means drawn from the same
  set of scores is one estimate of variance.
• The denominator of the t test, although expressed
  as a standard deviation, is also an estimate of
  variance.

4
The F ratio

• A ratio of two estimates of variance drawn from
  the same set of scores is called an F ratio.
• F larger estimate of variance
• smaller estimate of variance
• F variance estimate 1 of s2
• variance estimate 2 of s2

5
Why use ANOVA?

• We sometimes have independent variables with more
  than two groups
• Grape Kool-Aid, Lemon Kool-Aid, and water
• One marijuana cigarette, two, three, four
• TQM in place for one month, two, three
• Four styles of government
• With more than two groups, multiple t tests would
  be necessary.
• Multiple t tests inflate the Type I error rate.

6
Logic of ANOVA

• 1. Find the total variance in a set of data.
• 2. Analyze the variance into
• a. The part due to the treatment (plus people)
• b. The part due to people without treatment
  (individual differences and error)
• 3. Form an F ratio of the two parts

7
The ANOVA summary table

• To keep track of the ANOVA process, start with a
  summary table
• Source SS df MS ( s2) F
  p Decision Between
• Within
• Total

8
Notation summary

• N is the number of scores altogether.
• n is the number of scores in a single group.
• k is the number of groups.
• g is a particular group.
• The variance, s2 , is also symbolized MS, which
  stands for Mean Square.

9
SS total

• To get SStotal, simply group all of the scores
  for all of the groups together and find the SS as
  if there were only one group
• SStotal SX2 - (SX)2 / N
• Or SStotal SX2 (G2 / N)

10
SS within groups

• SS within groups is due to individual differences
  and error.
• To compute SSwithin,calculate SS normally for
  each group separately, and add them up.
• SSwithin S(SX2 - (SX)2/ n)
• SS1SS2SSk
• SX2 - (SX1)2 (SX2)2 (SXk)2

11
SS between groups

• To get SSbetween, treat the sum of each group as
  a score, and apply the be-bop version of the sum
  of squares song
• SSB
• (SX1)2 (SX2)2 (SXk)2 -
  (SX)2 N

12
Completing the one-way ANOVA

• Source SS df MS ( s2) F
  p Between k-1 SSB/dfB
  MSB/MSW
• Within N-k SSW/dfW
• Total N-1
• To obtain p, either use the table to find the
  critical value of F, or SPSS to find the
  probability of the obtained value of F.

13
Making your decision
To decide whether the obtained F ratio is
significant, compare it to the table value (the
critical value). If the obtained F is equal to
or larger than the critical F, reject H0.
Computed value Length of walk
Fishy zone Reject H0
Table value Length of pier
.05 level
14
Decision rules for ANOVA

• Find the critical values for F from the table.
  Use dfB for the (between) column, and dfW for the
  (within) row in the table.
• The first table has the critical value at the .05
  level, and the second table for the .01 level.
• In the p column of the ANOVA summary table, enter
  lt.01 if the obtained F ratio is greater than the
  value in the second table lt.05 if it is greater
  than the value in the first table but less than
  the value in the second table and n.s. (for not
  significant) if it is less than the value in the
  first table.

15
More ANOVA items

• For two group studies, t2 F
• Two-way factorial ANOVA
• Main effects Combines two (or more) studies
• Fully crossed factors A x B
• Interaction effect The effect of one independent
  variable depends on the level of the other
  independent variable

16
Try this one

• Here are the number of errors on an analysis of
  variance problem for each of nine students after
  drinking grape Kool-Aid, Frankenberry Punch, or
  water.
• GKA FBP H2O
• 5 8 10
• 3 7 11
• 4 7 12

17
Effect size measures

• Significant t and F ratios show that there is a
  real effect of the treatment, a real difference
  between the groups that cannot be explained by
  chance.
• Effect size measures show how big the effect of
  the treatment is.
• C.O.D. is one effect size measure C.O.D. r2
  SSB /SSTotal h2

18
Effect sizes in samples vs. populations

• h2 is a sample estimate of the proportion of the
  variance in the dependent variable that is
  accounted for by the independent variable.
• For population estimates of effect size, use a
  different statistic, w2

19
Computing w2

• For the t test for independent samples,
• w2 t2 1
• t2 N 1
• For the analysis of variance,
• w2 SSB (k 1)MSW
• SSTotal MSW

20
Concluding details

• ANOVA assumptions are the same as the assumptions
  for the t-test for independent samples.
• Try to have each group/sample be the same size,
  that is, have equal ns. However, the ANOVA
  computations will work with unequal sample sizes.

21
Reporting ANOVA in APA format

• Report the mean and standard deviation for each
  sample/group in a table, and refer to it in a
  sentence reporting the F ratio
• Table 1 contains the means and standard
  deviations for the IQs, which were significantly
  different, F(3, 36) 9.47, p lt .05.
• Table 1. Tested IQ.