To examine the effect of a quantitative factor temperature on yield,the researcher has a plan to use the following model for the analysis:
$$y_{ix}=\beta_0+\beta_1 x+\epsilon_{ix}$$
where $y_{ix}$ is the response corresponding to the $ith$ experimental unit which is assigned to the coded treatment level $x$ (where $x$ can take any value between $-1$ to $1$), $\beta_0$ and $\beta_1$ are intercept and slope of the model, and $\epsilon$ is the random error term. The researcher wants to use one of the following two designs, each of which has $12$ experimental units.
Design $A:$ Out of total $12$ experimental units, $6$ units are assigned to $x=-1$ and $x=+1$ each.
Design $B:$ Out of total $12$ experimental units, $4$ units are assigned to $x=-1$, $x=0$ and $x=+1$ each.
Which design you would suggest to the experimenter and why?
I think , Design $B$ is preferable to Design $A$. Since by adding the level $x=0$ , the researcher can know whether there is curvature in the response function.
The researcher's hypothesis of the response function is that the yield has a linear relationship to the temperature and he wants to set up an experimental design to the test this relationship. The coded treatment level of X would correspond to some temperature whose relationship has not been established by this model. Here X seems to be a dummy variable. Assuming that each treatment level could secondarily be transformed into a temperature scale, the reasons why Design B is better than Design A: by having 6 experiments at X = 1 and 6 experiments at X = -1 in Design A, effectively you have worked to improve the precision of the experiment at each of these points. Only two X's do not provide any information on the linearity. The linearity in a range of X is confirmed by as many X data points as possible. Design B is better to confirm the linearity and reject curvature by making sure the regressed line falls within a certain standard deviation of errors (obtained by 4 datapoints at each X) around each X. A nicer design would have been to split 12 experiements into 3 each ( which is pretty normal in experimental research) for 4 X's and would help establish this relationship. In Design B, having 4 datapoints for each X has reduced the possibility of outliers. Design B will result in better representation of $R^2$ to make statistical inference on the relationship.