One-way Analysis of Variance ANOVA  Analysis of Variance is used with quantitative and qualitative data to find whether each input has significant effect on the system’s response.

Estimates of variance are often called “mean squares”

Consider the simplest case of analyzing a randomized experiment in which only one factor is being investigated with two or more replicates being used for each level of treatment. There will be three or more levels of treatment. The null hypothesis will be that all treatments produce equal results (i.e. all population means for the various treatments are equal). The alternative hypothesis will be that at least two treatment means are not equal.

m different levels of treament r different observations for each level of treatment yik is the k th observation from the i th treatment is the mean observation for treatment i is the mean of all N observations, where N=m×r Then:

and Total sum of squares of the deviations from the mean of all the observations (SST): Treatment sum of squares of the deviations of the treatment means from the mean of all of the observations (SSA):

Residual sum of squares of the deviations from the means within treatments: It can be shown algebraically: or: SST=SSA+SSR

The degrees of freedom are partitioned in a similar method The degrees of freedom are partitioned in a similar method. The total number of degrees of freedom is N-1. The number of degrees of freedom between treatment means is m-1. Therefore, the number of degrees of freedom within treatments must be:

Therefore, the estimate of variance within treatments is: The estimate of variance between treatments is:

Are the two estimates of variance (sR2 and sA2) compatible with each other? If the populations means are not equal, the true population variance between treatments will be larger that the true population variance within treatments. Is sA2 significantly larger than sR2 (one tailed test at 5% level of significance). Before this test can be conducted, diagnostic plots must be checked to see that the necessary assumptions have been met.

Diagnostic Plots: Stem-and-leaf display (or equivalent) of residuals Plot of residuals against estimated values of y Plot of residuals against time sequence of measurement Plot of residuals against any variable which might affect results Residuals are the differences between the observations and the estimates of the true values according to the mathematical model. In our case the residual is: Example 39