Hypothesis Testing

F-values

If we return to the table, we now have the variances (mean squares, MS) that the table and the analysis are named after. So, how do we use them to perform ANOVA? The variance for the experimental treatment (pollination type) is our signal, and the unexplained residual variance is the noise. The signal-to-noise ratio is calculated when we divide the treatment variance by the residual error variance to produce the F-value. The F-value was originally called the variance ratio but was later renamed to be after Fisher, the inventor of ANOVA. This value is given in the fifth column and is represented by the following formula:

$$F=\frac{MS_{treatment}}{MS_{residual}}$$

In this case, the value of 5.9 means that the estimated signal is nearly six times larger than the estimated noise. In a perfect world, if there’s no signal, then the treatment variance is nothing more than another estimate of the noise—the error variance. If we could estimate both the signal and the noise perfectly, and if there was absolutely no effect of the treatment, we would be dividing noise by noise.