On Tue, 20 Jun 2000, Murtagh wrote:
> Firstly, thank you for your comments. Am I right in saying that the two
> (equivalent) options I have are:
>
> 1. ANOVA
>
> Yijk = mew + Ai + Bj + ABij + Eijk
>
> Ai: a fixed factor representing the treatments (2 levels)
> Bj: a fixed factor representing prior perfromance (2 levels)
> ABij: an interaction between Ai and Bj
> Yijk: the score of the kth child who received treatment i and is from
> group j
> Eijk: random error
>
> I suspect that this model is inapporpriate, as the Eijk term will represent
> between subjects (children) variation, which is not usually included in the
> estimate of random error.
I do not understand this comment. What source(s) of random error exist
in this design APART from variation between subjects within cells?
Between-subjects variation (as residuals from the model) defines the
standard error-variance term against which the variability in the
systematic effects is tested.
> 2. MLR
>
> Y = Bo + B1*X1 + B2*X2 + B3*X3 + E
>
> X1: prior performance (0 => weak, 1 => strong)
> X2: treatment (0 => treament A, 1 => treatment B)
> X3: treatment*prior performance
-- hence with the coding shown for X1 and X2, 1 =>
strong prior performance and treatment B, 0 => all other conditions.
And E = Eijk of the ANOVA model. B1 is a straightforward function
(depending on the coding of X1, of course) of the Ai in the ANOVA model,
B2 of the Bj (and depends on the coding of X2), and B3 of Ai, Bj, and
ABij.
> I appreciate that prior performance is probably better considered as a
> continuum, rather than a dichotomy.
_I_ would consider it so. In fact, the first thing I'd do is ask for
scatterplots of post-performance vs. pre-performance for all the cells
in the design I was considering. (In what you've described, that's two
cells.) THEN decide whether it appeared to make better sense to divide
the continuum into two (or more) pieces, or to model it AS a continuum,
possibly with non-linear functions.
> >> 1. If there are children of different sexes, you may be able to
> >> consider a three-way design, although I suspect it would be
> >> unbalanced, which (I also suspect!) may induce serious difficulties
> >> for you.
> You mean that there would not be the same numbers in each group?
Yes.
> I can't see why this should cause problems, but then that's probably
> due to my relative ignorance of linear models!
Doesn't cause problems in one-way designs. But in 2-way designs (let
alone 3-way, 4-way, ...) unequal n's induce association of some kind
between the design factors. People who do multiple regression don't have
much problem with this, it's their normal situation; but people who try
to do formal ANOVA design-of-experiments (and are therefore accustomed to
the notion that the factors are mutually independent (and therefore are
orthogonal)) are sometimes boggled by (1) the fact that the sums of
squares for the several sources of variation do not simply add to the
total sum of squares about the grand mean, or (2) the fact that the
sums of squares reported depend on the order in which the factors are
considered. And many of the standard packages for doing multi-factor
ANOVA use algorithms that require the design to be balanced.
(A GLM -- general linear model -- program does not usually have such
constraints, and may even produce output patterned after the form of a
standard balanced ANOVA, but one needs to be aware of (1) and (2) above.)
> >> 2. Your Performance information you have chosen to dichotomize,
> >> although it is presumably (quasi-)continuous to start with. You
> >> might find out something useful by treating it as a continuous
> >> predictor rather than as a dichotomy: in effect carrying out an
> >> analysis of covariance with pre-treatment reading score as the
> >> covariate, whether you used an "Analysis of Covariance" program or
> >> a "Multiple Regression" program or a "General Linear Model" (GLM)
> >> program to do the arithmetic.
>
> Presumably, this could achieved by simply using the pre-treatment score
> itself (rather than 0 or 1) for the value of X1 in the suggested MLR
> model above?
Right.
And if the pre-post relationship should turn out to be detectably
nonlinear, you can substitute some candidate nonlinear function(s) of X1
and see if that helps.
There may be nonlinearity to be EXPECTED: in the nature of a reading
test, there is a highest possible score (all items right, e.g.) and a
lowest possible score (no items right, e.g.). Students who perform well
pre-treatment cannot have change scores that would put them above the
highest possible score at post-treatment; so it would not be surprising
if (a) change correlates negatively with pre-treatment, (b) post scores
were censored at the maximum (and negatively skewed), (c) pre scores were
censored at the minimum (and positively skewed), and/or (d) the post vs.
pre scatterplot showed curvature at one end or the other (or both).
One way of dealing with nonlinearity, in the absence of strong theory
that would predict a particular form of nonlinear function, is to divide
the subjects into groups based on pre-treatment performance (and on the
observed post-vs.-pre relationship); the optimum number of groups might
not be two, however.
-- 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
===========================================================================
This list is open to everyone. Occasionally, less thoughtful
people send inappropriate messages. Please DO NOT COMPLAIN TO
THE POSTMASTER about these messages because the postmaster has no
way of controlling them, and excessive complaints will result in
termination of the list.
For information about this list, including information about the
problem of inappropriate messages and information about how to
unsubscribe, please see the web page at
http://jse.stat.ncsu.edu/
===========================================================================
