September 30, 2005

Documentation for ANOVA test

Minn M. Soe, MD, MPH, MCTM : msoe@sph.emory.edu

Kevin M. Sullivan, PhD, MPH, MHA: cdckms@sph.emory.edu

This analysis is conducted to observe whether there is a significant difference in means between two and more independent samples, given respective standard deviation or standard error. The data input screen is as follows:

The input values requested are:

· Enter individual sample size and sample means.

· Enter standard deviation (or) standard error of respective sample mean.

The result of the calculation is shown next:

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, 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 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, ANOVA test can be used to compare the MAXFWT scores in three sample groups, assuming the data are normally distributed.

In addition, the ANOVA module displays the confidence intervals of individual sample mean in two formats, based on standard error of 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 ( Bartlett's test)

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.

February 27 2008 - fixed an error in the code where the t-value used in confidence intervals used ni rather than ni-1 for the degrees of freedom.