I have the following dataset:

urecovery time temp acid
1     86.88   -1   -1   -1
2     88.96   -1    0    0
3     93.68   -1    1    1
4     88.92    0   -1    1
5     86.96    0    0   -1
6     91.26    0    1    0
7     88.20    1   -1    0
8     93.20    1    0    1
9     88.75    1    1   -1

Where urecovery is the response variable and time, temp, acid are categorial variables. The value -1 corresponds to level low, 0, medium, and 1 high.

Would this be a three way ANOVA or 2 way, not all combinations seem to be tested here, does this mean this is not a three-way case?

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    $\begingroup$ Is that the complete dataset? $\endgroup$ – gung Sep 20 '13 at 4:25
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    $\begingroup$ Is that from a certain designed experiment? The set up is not full factorial (each combination of the factor levels is not present). There are specialized methods for analyzing such setups, but this is still basically a three-way analysis, since there are three different independent variables. $\endgroup$ – JTT Sep 20 '13 at 8:03
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    $\begingroup$ I'm afraid that the number of samples will be hardly sufficient for an ANOVA. Three variables with three categories each, that makes six out of eight degrees of freedom used up already, not counting interactions. Given that the independent variables are ordinal rather than categorical, I'd rather use a regression anyway. $\endgroup$ – January Sep 20 '13 at 10:58
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    $\begingroup$ With only 9 subjects, you should not look at all three independent variables at once, because of the danger of overfitting the model. $\endgroup$ – Peter Flom Sep 20 '13 at 10:58
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    $\begingroup$ @Gavin: I agree; in this case with orthogonal predictors, examining each individually will give the same coefficient estimates, but inflate the error variance (assuming there are effects). I'm not sure how that's supposed to deal with overfitting in any case. Model reduction might be appropriate to improve predictive accuracy, if that's what you want. The next step if this were an industrial experiment might just be to bump up the temperature & splash in more acid, leaving time alone; then monitor the process to confirm it's doing what's expected. $\endgroup$ – Scortchi Sep 20 '13 at 15:23

This is clearly a designed experiment, & looks very much as if it was intended as a $3^{3-1}$ design for continuous predictors. The coding suggests that, & the names of the predictors suggest things that could be measured quantitatively, or at least treated as such with a pinch of salt. If so: the three main effects are orthogonal, each partially confounded with a second-order interaction between the other two predictors; each quadratic effect is orthogonal to the two other predictors; & there are no true replicates. The aim may have been to get independent estimates of the main effects, with 5 degrees of freedom for the residuals, while allowing an independent check for curvature for each predictor separately, in the confident assumption of no interactions. Perhaps to estimate some or all of the quadratic terms, noting that the inclusion of, say, temp squared won't affect the coefficient estimates for acid or time.

Such designs aren't too common; it's rare to be concerned with curvature in each predictor separately but not with interactions between predictors. More often you'd see a full factorial design (for the three-predictor case—a fractional factorial when there are more predictors) along with some centre points to check for curvature & provide a pure error estimate; an orthogonal block of axial points (plus some more centre points) being added when necessary to allow estimation of the quadratic effects.

If you're sure you want to treat each predictor as categorical then the best you can do is fit just the main effects (which will still be orthogonal), leaving only 2 degrees of freedom for the residuals as @January points out.


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