It rather sounds as though data are already in hand, rather than yet to
be collected. That being the case, as I shall assume, your 2nd model has
half the data that your 1st model has, and it is not clear whether this
reflects the discarding of half the available data, or the averaging
together of pairs of observations from the first model to fit the second.
My preference would therefore be to start with the 1st model,
since everything you want to know from the 2nd can be deduced from the
1st, unless I have badly misconstrued your descriptions.
You write,
> In each of the above months I have 3 sampling dates (which are actually
> 3 different years) and in each date I have 10 estimates of A density.
This appears to entail an assumption that the measured density is
constant across years. Is this assumption realistic?
(If not, it would presumably be desirable to seek evidence on the point,
which would call for a more complex model or for certain specific sets
of contrasts not so far described.)
> The hypothesis states that there should be a decrease in
> density in winter, an increase in spring (growing season)
> and a possible decrease in summer (summer disturbance is
> not supposed to be as effective as the winter one).
The corresponding null hypothesis in ANOVA is that there are no changes
in density at all. But you want also to be able to detect patterns of
differences, should they exist, other than the patterns your research
hypothesis lead you to expect. Hence, in the first instance at least,
do not omit sources of variation unaccounted for in your hypothesis.
The model you describe leads to the summary table below, where the 1-df
subdivisions of "Periods" are orthogonal contrasts. You claim to be
interested only in the first 3 of these, which is fair from a theoretical
perspective; but from an empirical perspective it is possible your
hypothesis is, if not wrong, incomplete, and you surely would wish to
detect this shokuld it be the case. Treat the last two d.f. as a single
source orthogonal to the other 3 d.f., and see whether the associated SS
is large enough to make this source empirically interesting. If it is,
you can use standard post hoc comparisons (Scheffe' contrasts, e.g.) to
pursue whatever patterns there may be. No need to decide beforehand, in
the absence of evidence.
> Source of Variation DF Test against
> --------------------------------------------------
> Periods 5 P*L
> Between winter months 1 P*L
> Between spring months 1 P*L
> Between summer months 1 P*L
> Winter vs Others 1 P*L > Combine these, with
> Summer vs Spring 1 P*L > 2 d.f.
> Locations 3 D(P*L)
> Periods*Locations 15 D(P*L)
> Dates(Periods*Locations) 48 Residual
> Residual 648
> ---------------------------------------------------
> TOTAL 719
You do not mention the possibilities of interesting interactions, which
would lead to the pursuit of other details. Presumably these would be
pursued via post hoc contrasts, depending on the apparent pattern(s),
among the 24 P*L means.
On Wed, 17 May 2000, A. Murias Santos wrote, inter alia:
> I've got two possible models of ANOVA to test for differences
> in space occupancy between periods.
>
> Lets assume population A as a dominant space occupier. It
> outcompete other species, growing over them, but never
> monopolizes space (Density < 100%, and is usually below 60%).
> The hypothesis is that external disturbance events prevent
> it from reaching 100% of space occupancy.
>
> Disturbance events are seasonal. Lets assume W as a winter
> disturbance, beginning in November--December and ending
> in March--April; S is a summer disturbance, beginning in
> June--July and ending in November--December; P stands for
> spring season, begins in April--March and ends in June--July,
> and no disturbance occurs between these periods.
>
> In each of the above months I have 3 sampling dates (which
> are actually 3 different years) and in each date I have
> 10 estimates of A density.
>
> The hypothesis states that there should be a decrease in
> density in winter, an increase in spring (growing season)
> and a possible decrease in summer (summer disturbance is
> not supposed to be as effective as the winter one).
>
> First model:
>
> Factors are locations (4, random), periods (6, fixed)
> and sampling dates (3, nested, random) with 10 replicates
> per combination. Periods are:
>
> March (end of winter),
> April (beginning of spring),
> June(end of spring),
> July (beginning of summer),
> November (end of summer),
> December (beginning of winter)
>
> The model goes like this
>
> Source of Variation DF Test against
> --------------------------------------------------
> Periods 5 P*L
> Between winter months 1 P*L
> Between spring months 1 P*L
> Between summer months 1 P*L
> Winter vs Others 1 P*L
> Summer vs Spring 1 P*L
> Locations 3 D(P*L)
> Periods*Locations 15 D(P*L)
> Dates(Periods*Locations) 48 Residual
> Residual 648
> ---------------------------------------------------
> TOTAL 719
> There is no point in using "unplanned comparisons" because the
> interesting contrasts are known before the experiment.
Well, the _theoretically_ interesting contrasts, anyway. There might be
some other _empirically_ interesting contrasts to pursue if your theory
is rather badly out to lunch.
> As you see, there is a number of planned comparisons, but I'm really
> interested only in the first 3... the others are there just because they
> sum up to the SS of "Periods". They could have been Spring vs Others and
> Summer vs Winter or Summer vs Others and Spring vs Winter... whatever...
< snip, the rest >
-- DFB.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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