KIN 1G03 Lecture 6
Dr R Balasubramaniam SimpleANOV A:Comparing the
Means ofThree or More Groups ANOVAT erminology
• The purpose anANOVA (analysis of variance) is to
compare the effects of a variable with multiple
factors (training intensity:low,med,high) on a single
variable (VO 2.
• The independent variable Intensity ofraining is
called a FACTOR.
• The FACTOR in this example has 3 LEVELS (low,
med,high)
• The dependent variable in this experiment isVO 2
• ANOVA allows for multiple comparisons while still
keeping alpha at 0.05. Familywise Error Rate:Why doANOVA?
Now let us consider comparing the effects of NUMBER OF
DAYSTRAINING PERWEEK (1,2,3,4,5,6) on STRENGTH.
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;2
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 aType I
ERROR.
ANOVA fixes this problem by doing only one test. Example data set Significance of results plot Assumptions ofANOVA Example data set Sources ofVariance
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 Groups Deviations Sum of SquaredWithin Deviations
SS w 7.40 + 16.86 + 4.00 + 13.40 + 5.71 = 47.37 Between Groups Deviations
SS B 7.26 X 7 = 50.82 Mean Square and F Ratio Degrees of freedom
• Dfw= N – k (35 -5) =30 (total number of data
points – total number of groups)
• Df = k -1 = 4 (total number of groups – 1)
b
• Mean Sum of squares within
MS w SS /wf =w47.37/30 = 1.58
Mean Sum of squares between
MS = SS /df = 50.82/4 = 12.71
b b b
F = MS /MS (referred to as Fisher’s F value).
b w F Statistic is a Ratio ofVariances
50.74 =12.69 Each Sum of Squares is divided by its df
4
47.43 to produce a Mean Square.
=1.58
30 F ratio is the ratio of variances
12.69 =8.03
1.58 F = MS /bMS e CriticalValues of F Statistic R (also called eta ) and ω " 2
R or eta are rough
estimates the size of
the effect.
2
▯ is a more exact
test of the Effect. Chapters from the text
• Correlation and • Chapter 7
Bivariate Regression
• Multiple Regression • Chapter 7
• Chapter 8
• t test/means
comparisons (only one-
tailed test) • Chapter 9
• simpleANOVA
• Nonparametrics • Chapter 13 Notes re:final exam
• Several short multiple choice and fill-in-the-blank
questions
• Problem sets on each of the major exercises that
we have done Pearson correlation,bivariate
regression,ANOVA,t-test,chi-squared,Spearman
correlation and some questions from pre-mid
term including frequency analyses. Multiple Regression (MR)
• Predicting one DV from a set
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