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I used a Definitive Screening Design plan to examine which of the parameters $A_1, A_2, \ldots, A_6$ have significant influence on the response $Y$. The design plan consists of $13$ treatments. A treatment is a combination of low, medium and high levels of the factors. I have repeated all the treatments on the same set of subjects. The number of subjects is $n=30$. Let $Y_{i,j}$ denote the response measured in the $j$th subject when exposed to the $i$th treatment.

I am not certain how to proceed. Can I compute the average response for each treatment: $$\overline{Y_i} = \frac{1}{n}\sum_{j=1}^{n}Y_{i,j}$$ and then try to fit a linear model to these $13$ data points? For example: $$\beta_0 + \beta_1A_1 + \beta_2A_2 + \beta_{11}A_1^2 + \beta_{34}A_3A_4$$ I would then test the hypotheses $H_0: \beta = 0$ vs $H_a: \beta \neq 0$ for each $\beta$. A DSD plan that I used allows us to estimate main effects and also pure quadratic and two-way interaction terms.

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If I understood you correctly and you have 13 sets of A1,A2....and Y for your 13 treatments, I think one way is to dummy code each of your treatments ( for low, medium and high) and then include each 13 treatments in your linear model function (one by one) to see if they have an effect... other methods of interest here could be 'Moderation'- as you want to know if the treatment moderates the relationships of all your A1, A2... on your Y response variable.

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  • $\begingroup$ Treatments are combinations of low, medium and high values of the factors, chosen in a specific way to order to allow estimation of main effects, two-way interactions and pure quadratic effects of the factors. So, I need to find out how each of the factors $A_1, A_2, \ldots, A_6$ influences $Y$. This is why I don't think that dummy coding treatments would help, or at least, I don't get why it would. :( Can you, please, elaborate more on that and 'moderation'? $\endgroup$ – Milos Oct 21 '16 at 14:16
  • $\begingroup$ Are you saying that your treatments and factors are the same parameters and by treatments you mean different level of each of your factors? (A1; low, medium and high)? if that's the case, can't you create a factor variable for each treatments, call it the level of that treatment (i.e A1 level: A1 none, A1 Low, A1 Medium and A1 high) and then include this as the interaction term in your linear equation?). Moderation is basically the interaction (product term)? this is similar to dummy coding approach again and you will control how your A1 level leading to different Y as opposed to an average Y. $\endgroup$ – RomRom Oct 21 '16 at 23:54
  • $\begingroup$ That is: Y= β0+β1A1*A1level +β2A2*A2level..... $\endgroup$ – RomRom Oct 21 '16 at 23:56
  • $\begingroup$ No, factors are $A_1, A_2, \ldots, A_6$, and a treatment is a combination of their values, e.g.: [Low $A_1$, Low $A_2$, High $A_3$, Medium $A_4$, Low $A_5$, High $A_6$]. :) $\endgroup$ – Milos Oct 22 '16 at 1:13
  • $\begingroup$ Then i think you would have to dummy code each of A1, A2... into dummy codes low, medium, high and insert them all into your linear function. i.e If a treatment is low, the dummy will be zero for high and medium and 1 for the low. and if its combination of Low and medium you will have 1 and 1 and zero on high... sorry i can't think of any other way.. $\endgroup$ – RomRom Oct 23 '16 at 9:00

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