I want to perform a screening experiment to determine which of the factors $A_1, A_2, \ldots, A_6$ do not significantly influence the response and may be omitted from future experimentation. I decided to use Definitive Screening Design (DSD) (Jones and Nachstsheim, 2011). There, each factor has three levels, as opposed to full and fractional factorial designs for two-level variables.

However, now I need to determine the sample size ($n$) and the number of measurements per subject ($r$) in the sample, so that the design is powerful enough. By "enough", I mean that power should be at least $0.8$. Type I error ($\alpha$) is chosen to be $0.05$.

Now, my question is how many replications (i.e. how many subjects in the sample, $n$) and how many repetitions (i.e. how many measurements per subject, $r$) should I have so that the above criteria are met? I guess that the estimate of population variance should be included in calculation, but I'm not sure how to do it. Each treatment (combination of different levels of $A_1, \ldots, A_6$) in the design will be applied to each of $n$ subjects for $r$ times and the response $Y$ will be measured. If $m$ denotes the number of factors, there will be $2m+1$ treatments in DSD (in this case $m=6$).

If it helps, this is how I plan to analyse data:

1) For each treatment in design I will calculate the mean value of $Y$. For example, for the $i$th treatment, the mean is: $$Y_i = \frac 1 {n\times r}\sum_{j=1}^{n}\sum_{k=1}^{r}Y_{i,j,k}$$ where $Y_{i,j,k}$ is the response recorded in the $k$th measurment of applying the $i$th treatment on the $j$th subject in the sample.

2) Then, I will construct the model for $Y$:

$$Y_i = \beta_0 + \sum_{j=1}^m\beta_j x_{i,j}+\sum_{j=1}^{m-1}\sum_{k=j+1}^{m}\beta_{j,k}(x_{i,j}x_{i,k}) + \sum_{j=1}^{m}\beta_{j,j}x_{i,j}^2 + \varepsilon_i,\quad i=1,2,\ldots,2m+1$$

Here $[Y_1, Y_2, \ldots, Y_{2m+1}]$ is the response vector, $x_{i,j}$ represents main effects, $(x_{i,j}x_{i,k})$ stands for two-factor interactions, and $x_{i,j}^2$ denotes pure quadratic effects. Let us assume that regression assumptions are not violated by the derived model.

3) After computing the values of $\beta$s, for each coefficient $\beta$ I will test the following hypotheses with $t$ test:

$$H_0 : \beta = 0$$ $$H_a : \beta \neq 0$$


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