STAT 512
Analysis
1
ANALYSIS: CRDs & ORTHOGONALLY BLOCKED DESIGNS
DIAGNOSTICS
• Results depend on the assumption y ∼ M V N (XA β A , σ 2 I)
• Checks are based on r = y − ŷ = (I − HA )y
• Under HypA , E[r] = 0 V ar[r] = σ 2 (I − HA )
• e.g. for CRD, HA = diag( n11 Jn1 ×n1 ,
1
J
,
n2 n2 ×n2
...
1
J
)
nt nt ×nt
– individual entries are “small” if ni are “large”
– V ar(r) ≈ σ 2 I
– approximation is generally (but not always) better as rank(XA )/N
is smaller
• Primary concern about model form is lack-of fit, check with plots
– elements of r versus time (trend)
– interaction plots for CBD’s
STAT 512
Analysis
• Primary concern about variances is equality, check with plots of
the elements of r versus:
– treatment group (compare spread within groups)
– time (“wedge”)
– corresponding elements of ŷ (“wedge”)
• Bartlett Test (exact)
– not robust to non-normal errors
• Modified Levene Test (approx.)
– for CRD and other designs that include true replication
– for data group i, and observation j within the group,
– compute zi,j = |yi,j − mediani |
– apply one-way ANOVA F to z-values
– rejection implies that means of z’s vary, suggests that the
spread of y’s varies with group.
2
STAT 512
Analysis
3
3
0
2
1
2
z
4
y
6
4
8
5
6
10
Example:
1
2
group
3
1
2
group
3
STAT 512
Analysis
POWER TRANSFORMS (for variance-related-to-mean)
• Generally used with non-negative data.
• Suppose V ar(y) isn’t constant, but varies as E(y)q .
• Then, via the delta method:
V ar(y p ) ≈ V ar(y) × {py p−1 |y=E(y) }2 ∝ E(y)q+2p−2
• This suggests transformation, e.g.:
– q = 1 (e.g. Poisson) → p =
1
2
– q = 2 (e.g. exponential) → p = 0 (e.g. log)
limp→0 (y p − 1)/p = ln(y)
4
STAT 512
Analysis
• Actually, better (but related) transformations exist for both
Poisson and exponential distributions, and when we know these
are appropriate we would likely use a generalized linear model
anyay. But for empirical modeling, the Box-Cox transform:
p
y
−1
∗
y =
p
is popular and easy to fit:
QN
1. compute the geometric mean of all data, ỹ = [ i=1 yi ]1/N
2. fit y ∗∗ = y ∗ /ỹ p to your intended model form for several value
of p
3. the value of p that minimizes the residual sum of squares is the
MLE under a model that says y ∗ has mean structure as you
claim, plus i.i.d. normal errors.
5
STAT 512
Analysis
TEST OF EQUAL TREATMENT MEANS
• The bottom line here is that for all orthogonally blocked designs
Pt
– the numerator sum of squares is i=1 ni (ȳi. − ȳ)2
Pt
– the noncentrality parameter is i=1 ni (τi − τ̄ )2 /σ 2
– the denominator sum of squares comes from the fit of the full
model
– power analysis differs from what we did for CRDs only in
counting denominator df
6
STAT 512
Analysis
CONFIDENCE INTERVALS FOR C0 τ
• General interest is in invidual intervals for individual contrasts:
sX
√
α
0 τ ± t(1 −
d
c
, df) M SE
c2i /ni
2
i
• In large experiments (large t and many contrasts of interest),
multiple inference risk can be a problem, e.g.
• Suppose you have a design with t = 10 treatment groups, and
want to estimate all (45) pair-wise differences of treatment effects:


CI for τ1 − τ2 (5% risk)




 CI for τ − τ (5% risk)
1






3
...
CI for τ9 − τ10 (5% risk)
so the risk of making at least one error is much greater than 5%
7
STAT 512
Analysis
8
• Simultaneous CI’s are constructed so that, with 95% confidence
(or whatever you pick), ALL intervals are correct.
• This works by replacing the t quantile in the usual CI by one from
another distribution, e.g.:
intervals of interest
procedure
distribution/quantile
all τi − τj
Tukey intevals
studentized max|xi − xj |
all τi − τ1
Dunnett intervals
studentized max|xi − x1 |
all contrasts
Scheffe’ intervals
specified contrasts
Edwards & Berry
p
(t − 1)F (1 − α, t − 1, df)
simulation
• Intervals are typically larger than usual t-based intervals
• Intervals are typically larger for procedures made for larger
collections of intervals (e.g. Scheffe’) than for those made for
smaller collections (e.g. Dunnett)