Statistics 850 yandell
Homework # 6-Due Mon 3 Apr
Scientists who study social insects, such as wasps and ants, have long
noted that queens and workers can have very different sizes and
shapes. For some species, the queens are simply larger, suggesting
that they continue to grow with the same basic ``plan'' as workers.
They are just fed for longer. An entomologist on campus examined a
wasp species which does not follow this pattern. Your job is to
provide statistical evidence to support the claim that queens are not
simply larger workers - they likely differed in development at a very
early stage. The data consist of measurements on 50 workers and 50
queens for p=13 multiple responses. The measurements have names
which code where they were taken - head (H), thorax (T) or
gonadium (G) - and what kind of measurement - width (W),
height (H) and length (L). The gonadium has two length and 3
height measurements.
1. The first task is to reduce the ``dimensionality'' of the data.
(a) Overall Test: Conduct univariate analyses on all responses to assess
worker-queen differences. Conduct a multivariate ANOVA for same.
Report test results.
(b) Use a stepwise discriminant analysis procedure (e.g. stepdisc or
ANCOVA) to reduce to a subset of significant responses as
discriminators. Briefly summarize these results.
(c) Head width (HW) and gonadium width (e.g. G1Wa) appear to be
the most significant discriminators, as found in part b. Plot them
against each other to note that one is larger while the other is
smaller in queens than workers. Use analyses so far, along with
correlations and partial correlations (adjusting for group) among the
multiple responses to argue that these two responses appear to capture
the main features.
(d) Conduct a canonical discriminant analysis to examine the ``best''
linear combination of responses to discriminate between workers and
queens. How much better does it do than any one response? What is
the canoncial variate correlation (see Total Canonical Structure in
SAS output) with the two responses considered in (c)? Why is one
positive and one negative?
(e) Plot the canonical variate against the two responses in (c).
Briefly interpret in light of (d) above.
2. The scientist thought it might be important to first remove
``size'' in order to isolate ``shape'' differences between workers and
queens. He suggested using TL, as it is the poorest discriminator
- that is, the distributions of TL substantially overlap for queens
and workers.
(a) Conduct an analysis of covariance of the two responses HW and
G1Wa on size (TL) and group. Are there group differences after
adjusting for size?
(b) Remove the effect of size and conduct a stepwise DA. Compare
results with 1(b).
(c) Conduct canonical DA for the residuals with size removed. How
do the canoncial variate correlations compare with 1(d)?
(d) Plot canonical variate and the residuals for the two responses
from 1(c) against each other. Briefly compare and contrast with
earlier plots.
The data can be found in hwk6.dat, with ideas on analysis
shown in hwk6.sas and hwk6.prt. [It is much easier
with current tools to use SAS for this problem.]