Math 217—One-Factor ANOVA Example
Many
studies have suggested that there is a link between exercise and healthy bones.
Exercise stresses the bones and this causes them to get stronger. One study
(done at
Variable
Treatment N Mean
StDev
Minimum Q1
Median Q3 Maximum
Bone Density (mg/cm3) 1 - Control 10 601.10 27.36 554.00 587.00
601.50 615.75 653.00
2 - Low Jump 10 612.50 19.33 588.00 595.50
606.00 632.75 638.00
3 - High Jump 10 638.70 16.59 622.00 625.00
637.00 650.00 674.00
Per usual, the first step in any
statistical analysis is to summarize the variables graphically and numerically
(this gives you a first look descriptively that allows you, among other things,
to see if there are any issues with the data—for example, outliers or mis-recorded observations).
We’d like to test 
against 
at least two of the 
are different. First let’s assess the equal-variances condition
of the ANOVA test. From the boxplots, the variability
in bone densities looks somewhat similar for the three treatment groups (not
exactly the same, but not wildly different). Using our rule of thumb, the
largest standard deviation (27.36 mg/cm3) is less than twice the
smallest standard deviation (16.59 mg/cm3), so it seems feasible
that the equal-variance condition on the errors is met).
From Minitab we get the following
information, as well as graphs (histogram and normal probability plot) of the
residuals.
One-way ANOVA: Bone Density (mg/cm3)
versus Treatment
Source DF SS
MS F P
Treatment 2 7434 3717 7.98
0.002
Error 27 12580
466
Total 29 20013
S
= 21.58 R-Sq = 37.14% R-Sq(adj) = 32.49%
Recall
the residual plots allow us to assess the normality assumption (since checking
the normality separately for each treatment group can be difficult if there are
few observations). The residuals seem to roughly follow a normal distribution,
so this assumption seems reasonable. Note the P-value for our test is 0.002.
That is, assuming the average bone density is the same for all rats regardless
of treatment, there is only a 0.002 chance of getting our particular sample
results or more extreme results. This gives strong evidence that there is a
difference in average bone density for at least two of the treatment groups. (Although the experimental treatment only explains 37% of the
variation in bone densities.) This begs the
question: between which treatment groups is there a significant difference in
average bone density?
Multiple Comparisons
Since we found
a significant difference between at least two means, it’s appropriate to now
compare the means pairwise. Included below is the
Minitab output for Tukey’s method (which adjusts for
multiple comparisons, and has a family error rate of 5%) and for Fisher’s
method (which doesn’t adjust for multiple comparisons). Not adjusting for
multiple comparisons (Fisher’s method) is only appropriate if you’re simply
exploring the data looking for interesting effects to investigate in another
experiment.
Notice that, in
this case, regardless of whether we adjust for multiple comparisons we see a
significant difference in average bone density between the control group and
the high-jump group and between the low-jump group and high-jump group (but not
between the control group and the low-jump group). We can also use these
intervals to assess the practical
significance of the differences.
Tukey 95%
Simultaneous Confidence Intervals
All Pairwise
Comparisons among Levels of Treatment
Individual confidence level = 98.04%
Treatment = 1-Control
subtracted from:
Treatment
2-Low Jump -12.56
11.40 35.36 (-------*-------)
3-High Jump 13.64
37.60 61.56 (-------*-------)
-------+---------+---------+---------+--
-30 0 30 60
Treatment = 2-Low Jump
subtracted from:
Treatment
3-High Jump 2.24
26.20 50.16 (-------*-------)
-------+---------+---------+---------+--
-30 0 30 60
Fisher 95% Individual Confidence Intervals
All Pairwise
Comparisons among Levels of Treatment
Simultaneous confidence level = 88.07%
Treatment = 1-Control
subtracted from:
Treatment
2-Low Jump -8.41
11.40 31.21 (------*-----)
3-High Jump 17.79
37.60 57.41 (------*-----)
-----+---------+---------+---------+----
-30 0 30 60
Treatment = 2-Low Jump
subtracted from:
Treatment
3-High Jump 6.39
26.20 46.01 (------*-----)
-----+---------+---------+---------+----
-30 0 30 60
Relationship to Regression
The ANOVA table
for a one-way analysis of variance is
exactly the same as the ANOVA table in a regression analysis when using
indicator variables (i.e., “dummy” variables). If indicator variables are
created for the first two levels of the experimental treatment (control and
low-jump), and then a regression is run with bone density as the response and
control-treatment and low-jump-treatment as the predictors, the Minitab ANOVA
information is exactly the same as that from the one-way ANOVA analysis:
Analysis of Variance
Source DF SS
MS F P
Regression 2
7433.9 3716.9 7.98
0.002
Residual Error 27
12579.5 465.9
Total 29 20013.4
S
= 21.5849 R-Sq = 37.1% R-Sq(adj) = 32.5%