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In Design and analysis of experiment , Random effect is defined as :

An experimenter is frequently interested in a factor that has a large number of possible levels. If the experimenters randomly selects a of these levels from the population of factor levels , then we say that the factor is random .

In random effect model , we are randomly choosing level .

But in response surface methodology class , we have learned that choosing factor level randomly can seriously distort the result of experiment .

  • My first question is , aren't these two concept of random effect model and response surface methodology contradictory ?

  • It seems response surface methodology is based on fixed effect model , since experimenter fixed the level . Is it ?

  • If my study is the first study in that field then how can i fix appropriate levels at the very beginning ? I may have the possibility to choose such levels which do not have optimum result at all . In such case , isn't trial and error the only way to execute the study ?

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The difference is that the levels of the randomly chosen factor are assumed to be exchangeable: that is, you could swap the labels of the factor (e.g. subjects A, B, C are relabeled as B, C, A) without changing the predicted values or meaning of the experiment. This is not true if the design aspects you're choosing are the values of different continuous covariates (I can't swap "temperature=100°C" and "temperature=200°C" without changing the meaning of the data). Partly this for reason, we sometimes make a distinction between random effects grouping variables (e.g. subject) and random effects terms (the quantities that vary among levels of the grouping variables, e.g. the baseline response [intercept] or the effect of some covariate that is measured more than once per subject).

Response surface methodologies are typically based on fixed effects, although you could certainly imagine a situation where measurements were taken for a number of randomly chosen subjects (batches, chambers, etc.) at the same values of one or more continuous control variables; then subjects could be treated as a random-effects grouping variable, with the responses to to the controls varying among subjects.

As for your last question, it's true that it's hard to choose experimental levels if you are completely ignorant of the experimental system. Experimenters are rarely completely ignorant when they start.

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