POWER: an example of use of non-central F

Example Consider the following AOV of yield data from a randomized block experiment to measure yields of 8 lucerne varieties from 3 replicates.

The statistical model is :-

$$y_{ij} = \mu + \underbrace{\beta_i}_{\mbox{block effect}} + ... ...lon_{ij}}_{\mbox{error}} \ \ \epsilon_{ij} \sim N(0,\sigma^2)$$

Table 6.1: AOV of lucerne variety yields

Source	df	SS	MS	F	E(MS)
replicate	2	41,091	20,545	2.4	$\sigma^2 + 8 \sum_{i=1}^3 \beta_i^2/2$
variety	7	75,437	10,777	1.26	$\sigma^2 + 3 \sum_{i=1}^3 \tau_i^2/7$
residuals	14	120,218	8,587		$\sigma^2$

The variety effects are:-

1	2	3	4	5	6	7	8
0	-69	-46	90	81	71	60	22

Is there sufficient evidence to say that the observed differences are due to systematic effects? If there is not sufficient evidence to say this, we are obliged to take the position that the observed differences could have arisen from random sampling from a population for which there were no variety effects.

If the null hypothesis that $\tau_j=0, \forall j$ is true,

$$\mbox{variety MS} \sim \sigma^2 \chi^2_7/7$$

and

$$\mbox{residual MS} \sim \sigma^2 \chi^2_{14}/14$$

The ratio of these 2 mean squares is a random variable whose distribution is named the $F$ distribution. We assess the null hypothesis using the $F$ statistic which in this case is 1.26 and the probability of getting this or more extreme by sampling from a population with $\tau_j=0$ is 0.34. Thus we have no strong evidence to support the alternate hypothesis that not all $\tau_j=0$.

Given the effort of conducting an experiment, this is a disappointing result. Specific contrasts amongst the varieties should be tested but if there were no contrasts specified at the design stage, this ``data-snooping'' may also be misleading. A post mortem poses the following questions.

• Was the experiment designed to account for natural variation? If the blocking is not effective, the systematic component $\tau_j$ is masked by the random component $\epsilon_{ij}$.
• Suppose that genuine differences do exist in the population. What was the probability of detecting them with such a design?

The quantile of $F$ demarking the 5% critical region is $F_{crit}=2.67$. To be 95% sure that the observed differences were not due to chance, the observed $F$ has to exceed $F_{crit}$. The probability of rejection of the null hypothesis when the differences are actually not zero is called POWER and is measured by the area under the non-central $F$ density to the right of $F_{crit}$.

The $F$ curves for this example are shown in Figure 6.1 where the vertical lines is $F_{crit}$. The calculated value of power is 0.1 which is very poor. A better design which will reduce experiment error is required.

Figure 6.1: central and non-central F densities

For post mortems of AOV's, the non-centrality parameter is estimated by

$$\hat \lambda ={{\mbox{df}_1 \times(\mbox{Trt MS} - \mbox{Error MS})} \over {\mbox{Error MS}}}$$

It remains a student exercise to compare this with previous definitions (eg. 6.9).

We revisit the topic of POWER in Statistical Inference.