September 20, 2010
Documentation for ANOVA test
Kevin M. Sullivan, PhD, MPH, MHA: cdckms@sph.emory.edu
This analysis is conducted to observe whether there is a significant difference in means among two or more independent samples, given values for standard deviation or standard error.
The input values requested are, for each group, sample size, sample means, and either standard deviation or the standard error of the sample mean.
The results of the calculation are shown below:
The interpretation is that there is an overall significant difference among the mean MAXFWT scores (maximum finger-wrist tapping score) in the three groups. It is noteworthy that before interpreting as above, the Chi square test for the equality of variances from these independent, normally distributed samples should be first checked. If the sample variances are significantly different, i.e. p-value from test for equality of variance is <0.05, the use of the ANOVA test may not be justified and the alternative non-parametric test should be done (e.g. Kruskal-Wallis test). In the example above, the sample variances are not significantly different, i.e. p-value from test of equality of variance is 0.138, and therefore, the ANOVA test can be used to compare the MAXFWT scores in the three sample groups, assuming the data are normally distributed.
The ANOVA module also displays the confidence intervals of individual sample means in two formats, based on the standard error of the observed samples and on Mean Square Within (Mean Square Error).
The formulae and definitions for one-way ANOVA test are from Bernard Rosner. Fundamentals of Biostatistics
(5th edition).
Equation 12.5
Short Computational Form for the
between SS and Within SS
Y.. = sum of the observations across all groups-ie, the grand total of all observations over all
groups; n= total number of
observations over all groups.
Definition 12.5: Between
Mean Square= Between MS= Between SS/(k-1). (k=
number of comparison groups)
Definition 12.6: Within Mean Square= Within MS= Within SS/(n-k).
Display of one-way ANOVA results
Source of variation |
Sum of Square |
df |
Mean Square |
F statistic |
p-value |
Between |
|
|
|
|
|
Within |
|
|
|
|
|
Total |
Between SS + Within SS |
|
|
|
|
Statistical formula for equality
of variance (
In the above, Si2 is the
variance of the ith group, ni is the sample size of the ith group, k is the number of groups, and S2
is the pooled variance. The pooled variance is a weighted average of the group
variances and is defined as:
Calculation of
95% confidence intervals are
based on t distribution.
·
X
± t(df,alpha=0.5)
* standard error of each sample
·
X
± t(df,alpha=0.5) *
(Mean square within/ni )½ , assuming
equal variance.
Reference:
Bernard Rosner. Fundamentals of Biostatistics (5th edition).
Acknowledgement:
Default values were obtained from table 12.5 (pg. 533) described in 'Fundamentals of Biostatistics' (5th edition) by Bernard Rosner.