Lecture 6 Notes
Simple ANOVA: Comparing the Means of Three or More Groups
- The purpose of AONVA (analysis of variance) is to compare the effects of a variable with
multiple factors (training intensity: low, med, high) on a single variable (VO2)
- The independent variable Intensity of Training is called a FACTOR
- The FACTOR in this example has 3 LEVELS (low, med, high)
- The dependent variable in this experiment is VO2
- ANOVA allows for multiple comparisons while still keeping α = 0.05
Faimlywise Error Rate: Why do ANOVA?
Now let us consider comparing the effects of NUMBER OF DAYS TRAINING PER WEEK (1,2,3,4,5,6) on
The number of days training is a factor with 6 levels. We could use multiple t-tests to compare (1 v 2, 1 v
3, 1 v 4, 1 v 5, 1 v 6; 1 v 3, 2 v 4, 2 v 5, 2 v 6, 3 v 4, 3 v 5, 3 v 6, 4 v 5, 4 v 6, 5 v 6). That would require 15 t-
tests. This would cause alpha to inflate from 0.05 to 0.26 greatly increasing the probability of making a
Type 1 ERROR.
ANOVA fixes this problem by doing only one test.
FW α 1 – (1-α)
FWα = 1 – (1-0.5) = 0.26
Assumptions of ANOVA
- The populations from which the samples are drawn in normally distributed. Violation ofthis
assumption has little effect on the F value among the samples. The F test produces valid results
even when the population is not normally distributed. For this reason, it is considered to be
robust. (samples come from normal distribution)
- The variability of the samples in the experiment is equal or nearly so (homogeneity of variance).
As with the assumption of normality, violation of this assumption does not radically change the
F value. However, as a general rule, the largest group variance should not be more than two
times the smallest group variance. (one group doesn’t have a huge amt of variability compared
to another, similar SD range, variance across columns are approx. equal)
- The scores in all the groups are independent; that is, the scores in each group are not
dependent on, not correlated with, or not taken from the sample subjects as the scores in any
other group. The samples have been randomly selected rom the population and randomly assigned to conditions. If there is a known relationship among the scores of subjects in the
several groups, use repeated measures analysis of variance (independence across samples)
- The data are based on a parametric scale, either interval or ratio.
Sources of Variance
- Between Groups, variance is the deviation of the group means from the grand MEAN
- Within Groups, variance is the deviation of individual scores from their Group means.
Within Group Deviations
Note: These values are derived by subtracting the group mean from table 9.1 from the individual score
(for the first score (X) in group X1; 4.00 – 5.29 = -1.29).
Sum of Squared Within Deviations
- Add the squares of the values from above.. SSw = 7.40 + 16.86 + 4.00 + 13.40 + 5.71 = 47.37
Between Group Deviations
- Mean of each group (5.29, 7.86, 5.00, 6.29, 4.43) subtract the grand mean (5.77), squared,
multiplied by the # in each group (to give you total number squares between..?)
- SSB = 7.26 x 7 = 50.82 Mean Square and F Ratio
Degrees of Freedom
- Df = N – k (35-5) = 30 (total number of data points – total number of groups)
- Dfb= k – 1 = 4 (total number of groups – 1 )
- Mean Sum of squares within
- MS = SS / df = 47.37/30 = 1.58
w w w
- Mean Sum o squares between
- MS b SS /bdf =b50.82/4 = 12.71
- F = MS bMS (rwferred to as Fisher’s F Value) F Statistic i