Judging the adequacy of models

Let us return now to a question asked at the beginning of this chapter, What can be learned about the behavior of the system from two experiments, each carried out at a different level of a single factor  

For the data used in this section, the unconstrained straight line model y, = Po + Pi- i, + i, fits exactly. Although the constrained model y, = p,x, + r, does not fit exactly, it does explain the observed data better than the simple model yi, = Po + r,/ (sj values of 1/5 and 2, respectively). Thus, the response of the system would seem to increase as the level of the factor x, increases. 

Is it justifiable to conclude that the factor x, has an influence on the output y, The answer requires a knowledge of the purely experimental uncertainty of the response, the variability that arises from causes other than intentional changes in the factor levels. This variance due to purely experimental uncertainty is given the symbol a, and its estimate is denoted sL. 

If the purely experimental uncertainty is small compared with the difference in 

Unfortunately, two experiments at two different levels of a single factor cannot provide an estimate of the purely experimental uncertainty. The difference in the two observed responses might be due to experimental uncertainty, or it might be caused by a sloping response surface, or it might be caused by both. For this particular experimental design, the effects are confused (or confounded) and there is no way to separate the relative importance of these two sources of variation. 

---