Two-factor ANOVA: What should be done when each cell has only one observation?

Question

I have a data set, described as the measurement of insecticide found through four different extraction methods, A, B, C, and D. A measurement for each extraction method is made under the conditions when the flower is: Fresh, 1-year untreated, and 1-year treated. The data set looks like:

               A  B  C  D
Fresh:         1  2  3  2
1yr untreated: 2  3  1  3
1yr treated:   5  3  2  4

I am asked to propose a model for the data and estimate its parameters. By the description of the data, it sounds like it just has to be a two-factor ANOVA. Both the extraction method and the "freshness" seem like independent and relevant variables controlling the outcome. Moreover, I would think it's fair to suspect interaction effects. So this sounds like my proposed model ...

Except, each cell has only one observation. My textbook says that we cannot conduct a hypothesis test without at least two observations per cell. So now it sounds like perhaps I can't use this model.

On the other hand, I'm not asked to actually conduct a hypothesis test, only estimate parameters. So for instance, can I still estimate the value of $\mu \approx \hat{\mu} = X...$, which is to say average over the $i,j,k$ coordinates (here, $k=1$ always)? And likewise for the formulae to estimate the "row-" and "column-" and "interaction-effects"?

Answer 1

Problem with two-way anova without replication is that it is not obvious how to get an error sum of squares. You can use a model without interaction, but then you use for error the same sum of squares that otherwise represents interaction. So you need to assume there is no interaction, see Why no interaction in two-way ANOVA without replication More details are given at Block-treatment interaction for Randomized Block Design (RCBD) and Generalized Random Block Design (GRBD)

Some limited test of interaction is possible, use Tukey's one degree of freedom for interaction. See

• Size of Tukey's 1 Degree of Freedom Test