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I would like advice on how to obtain the most (and no more)
genetic information from a designed experiment. A description of the
experiment and proposed formulation are attached.
TIA,
Gerald (Jerry) Walton
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Dear list readers:
The following diagram represents a mating program that was designed and
executed by an entomologist. I have been asked to provide an analysis that can
be used to estimate genetic contributions and heritability parameters. I am a
statistician, and am comfortable with random and fixed effects models, but
pretty much at sea regarding the use and interpretation of genetics parameters
based on variance component estimates. The experimental design is not my
doing, so please, no flames about what can't be changed. I am hoping someone
will have the time and patience to help me with advice or references so I can
make the most effective use of this data, and will be terribly grateful for
their assistance.
In the following, A(i), i=1,2, is a fixed effects factor (diet) applied to male
parents, and B(j), j=1,2 is a fixed effects factor (diet) applied to female
parents. The diagram below shows matings (diagonal lines) and experimental
factors for both parents (blocks in the diagram) for a single "group", G(k),
k=1,14. S(k,l), l=1,2 for any group represents the family (egg mass) from
which the male came, where the first subscript, k, denotes nesting within
group. The notation and diagram may be cumbersome, but at least it is done in
simple ASCII.
Layout for group k, k=1,14
Diet for males A=1 A=2 A=1 A=2
Male Family (E mass) S=1 S=1 S=2 S=2
------------------------------------------
Within female fam. (EM) Q=1 Q=2 Q=1 Q=2
--- --- --- ---
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
--- --- --- --- --- --- --- ---
Female diet B=1 B=2 B=1 B=2 B=1 B=2 B=1 B=2
----------------------------------------------
Female family R=2 R=2 R=2 R=2 R=1 R=1 R=1 R=1
Betw fem. same sire T=1 T=2 T=1 T=2 T=1 T=2 T=1 T=2
Betw fem. diff.sire U=1 U=1 U=2 U=2 U=1 U=1 U=2 U=2
The factors mentioned so far are sufficient to describe the experiment in a
linear model. Measurements were made on progeny of the matings, but no
corresponding measurements are available for the parents. Using C for the
population mean, the model for the measurement taken of on one of M offspring
from each mating can be written
y(i,j,k,l,m)=C+A(i)+B(j)+G(k)+S(k,l)+e(i,j,k,l,m)+
+AB(ij)+AG(ik)+ABG(ijk)+AS(ikl)+BS(jkl)+ABS(ijkl)
where the first line contains the main effects and error term, the second all
possible interactions, and only A and B are fixed effects. The ANOVA for this
model is (and I certainly hope this is correct):
Source EMS df
A 56V(A)+4V(AG)+2V(AS)+V(e) 1 AG 4(AG)+2V(AS)+V(e) 13
AS +2V(AS)+V(e) 14
B 56V(B)+4V(BG)+2V(BS)+V(e) 1
BG 4V(BG)+2V(BS)+V(e) 13
BS 2V(BS)+V(e) 14
AB 28V(AB)+2V(ABG)+V(ABS)+V(e) 1
ABG 2V(ABG)+V(ABS)+V(e) 13
ABS V(ABS)+V(e) 14
G 8V(G)+4V(S)+V(e) 13
S 4V(S)+V(e) 14
ERR V(e) 112(M-1)
The model as shown is not at all satisfactory for an analysis of genetic
properties, and it is in this area that I seek help. I am about to jump into
the deep end of the pool by proposing what seems reasonable to me for
incorporating the genetic information contained in the experiment into the
model. I don't think all is lost. First, the "groups" were formed simply to
select mates, and do not correspond to a blocking factor as though, say, a
group of insects was reared together. If groups were the only consideration,
any V containing a (G) could be considered zero. However, there are other
considerations. One is that the 2 families from which the males were selected
were the same 2 families from which the females were selected, within each
group. (Of course, matings were not incestuous.) I don't believe this needs
to be accounted for in the analysis, at least I hope not, since I see no way to
incorporate this constraint into the model.
Now, to expose my ignorance, provoke your pity and hope that you will have time
to help me make the best of this, but not necessarily time for tact :). I
refer to the diagram as justification for a number of guesses. First, R, the
"between female family component when mating with males in the same family"
component, is entirely confounded with S, and I believe that the term V(S)
should be replaced by a linear combination of variance components with
subscripts involving S and R. However, it is not clear to me that either R or
S would "interact" with the fixed effects, so I would guess that V(AS) and
V(ABS) could be considered zero.
The diet, A, is also confounded with the "within male family" term, Q, but if I
assume no group effect, then V(AG) would, it seems to me, be the only member of
the interaction term, so that one could simply replace V(AG) with V(Q). By a
similar argument, V(BG) is confounded with U, the "between females of the same
family when mated different males (of the same family)" component, and I feel
V(BG) may be replace by V(U). Finally, V(ABG) I think can be replaced with
V(T), a "within female family within (the same) male" variance component term.
Invoking these various suppositions leads to the reduced ANOVA table
Source df EMS
A 1 56V(A)+4V(Q)+V(e)
Q 13 4V(Q)+V(e)
B 1 56V(B)+4V(U)+V(e)
U 13 4V(U)+V(e)
AB 1 28V(AB)+2V(T)+V(e)
T 13 2V(T)+V(e)
? 14 4V(?)+V(e) error 112M-57 V(e)
Throwing all self-respect to the wind, I humbly ask you, what kind of
"heritability" terms might be of interest to a geneticist?
Since I will be asked to estimate heritabilty, and since the error terms are
not likely to be "normally" distributed, there is the question of standard
errors or confidence intervals for the variance and heritability estimates. I
have a couple of references which I have not pursued (Foulley, J.L, 1994 and
Burdick and Graybill, 1992), but would appreciate more leads that might be
particularly apt for this problem (however it finally comes out). The error
estimates (residuals) can be calculated for Q, U, T, ? and, of course, the
"usual" residual error. (I have not decided on ML, RML or OLS, but assume OLS
and method of moments, for simplicity, for the moment.) These are independent
between rows in the ANOVA table, if the experiment is balanced and normality
holds (actually, under less stringent conditions) under ordinary least-squares
estimation. I would be willing to make the independence assumption (I have a
trick in mind), but don't want to rely on normality. This means that any
estimates that are functions of ms could be based on a resampling scheme that
involves separate sampling (with replacement) of the residuals on each line in
the ANOVA table, computing the estimate, and using the properties of the
re-sampling to come up with standard errors or confidence intervals for the
parameters. Some might call this bootstrapping, but it is more of resampling
permutation test. This seems to me a reasonable procedure, but if the
geneticists have better methods, or something similar, I would appreciate
references to the literature.
There are a number of other statistical issues, but I feel that I can deal with
these. My primary concern is with the genetics. Once I am clear on is how the
genetic parameters relate to the experimental design model, and what is and is
not possible and/or interesting to a geneticist, I believe I can satisfactorly
resolve the remaining issues.
And thank you all who have taken the time to read this, and I do hope some of
you will have something to contribute to my edification. Replies to this list
or to me by e-mail at "/S=G.WALTON/OU1=S24L07A at MHS-FSWA.ATTMAIL.COM" will be
welcome. I will summarize to the list if there is interest and may initiate a
dialogue in case of seemingly conflicting advice, unless asked not to do so in
a reply.
Gerald S. (Jerry) Walton
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