Hi, Margaret. I've given some thought to your problem; here's a
restatement of it, and a few thoughts.
Recapitulating, in case I've misunderstood a small point or three,
you have "a 3-factor experiment", by which I assume you mean a
complete, balanced, crossed design: R(AxBxC) for observations (R)
within factors A, B, C, which in turn are crossed with each other.
R is a random effect; A,B,C are fixed effects.
You have measurements (Y, response variable) taken at 97 equal
(2-hour) intervals over time (T, whose values range from 0 to 192).
It is not clear whether T is one of the three factors.
Y is known or assumed to be a monotonic increasing function of T.
There exists a value Ymax such that when Y >= Ymax for some value T,
say T = Tmax, no observations are made for T > Tmax.
I had a similar situation once, years ago.
Investigating the error between a reported covariance and the known
true value of the covariance, in an experiment where two factors were
manipulated (A: the number of digits carried in arithmetic computations;
B: the number of digits required to represent the data), I made the
logical error of using data for which the true covariance was exactly
expressible as an integer. Consequently I couldn't tell, for some cells
of the design (where A was large and B was small), whether the
disagreement between computed and theoretical values was actually zero,
or merely smaller than was detectable because the theoretical value was
an integer. In the event, I was fortunate that discarding the rows or
columns in which the observed error was zero left me with the bulk of
the design intact.
In your case, I gather (between the lines) that discarding cells that
have missing data does rather more damage to the design, and is
therefore not an option.
First question: If Y = f(T), f a monotone increasing function,
as you assert, is the form of f known?
If not, may we assume it's of the same form for all cells of the design?
(If f is linear, the problem reduces to an analysis of covariance
of some sort, possibly with some missing cells, and can be addressed
in a straightforward way by multiple regression analysis.
If for some transformation of Y and/or T the corresponding f
is linear, the same is true for the transformed variable(s).)
(In my problem, when the precision of the reported covariance was
expressed as the negative logarithm of the absolute relative error,
the relationship between the transformed Y, A, and B turned out to be
markedly linear (even with unit coefficients, when all three variables
were expressed to the same base). Took a while to discover that,
though.)
In each cell of your design, you have a number of values (<=97) of (Y,T).
You can therefore estimate the parameters of the function f in each
cell for which n is large enough; this may not even entail discarding
any cells for insufficient data, although if you have cells for which
Tmax << 97 you may not be able to get a _good_ estimate of some of the
parameters of f in those cells.
(Although since f is monotone, it may be possible to find a
transformation for which f is linear, which would vastly simplify the
estimating process.)
At this point I don't know whether the cells of your design are for a
2-way or a 3-way design (if T is one of the three factors, it's 2-way);
but you can then carry out the obvious ANOVA (possibly unbalanced) on
the parameters of f to see how that varies across vales of your
factors. You can also discover how much it costs you to assume that one
or more parameters remain constant across the design (like the slope of a
regression line in analysis of covariance, under homogeneity of
regression).
Hope this has been helpful.
-- Don.
Donald F. Burrill [EMAIL PROTECTED]
184 Nashua Road, Bedford, NH 03110 603-471-7128
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