A collection of potentially useful models
We've already seen two very common mixed models:
for subsampling for designed experiments with multiple experimental units
Here are three more general classes
Stat 511
Homework 10 Solution - revised
Spring 2013
All questions are 1 point, with 1 free point.
1. Barley-Fungus studies
(a) e.u.=chamber, o.u.=trays
Source
fungus
barley
fungus*barley
chamber
tray
2)
a) 1 pt for F statistic, 1 for theoretical distribution, and 1 for p-value and conclusion.
F = 2.86, F distribution with 4 and 65 degrees of freedom, p = 0.0301. evidence that the mean
rating for a
Stat 511: HW 2 answers
1 point on each part, except point each on 1a, 1b, 1d, 1e, 3f, 3g, 3i and 3k
1) a)2 b)plane
c)1/3
d)2 e)Plane
f) the same as c)
g) The estimate for 1 or +1 is the average of the
Stat 511, HW 3 Answers
Parts are worth one point, except where marked otherwise.
1. Orthogonal polynomials
(a) testing the hypothesis that all treatments have the same mean. There are 4 treatment,
so
Stat 511 HW 5 answers
1.
I used the shifted t formula derived from =(t(1-/2).df+ t(1-).df)s.e.,
Sample sizes using non-central distributions are at the end of the question. They are exactly the
same.
Stat 511
Homework 8 Answers
Spring 2013
1. Split plot experiment
(a)
Yijkl = + i + j + ij + ik +
ijkl
where i, j, k, l = 1, 2. i indexes temperatures, j indexes catalysts, k indexes growth
chamber, an
Stat 511
Homework 9 Solution
Spring 2011
1. New Zealand pasture fertilization study.
(a) Degrees of freedom: Source
df
variety
3
block
3
v*b
9
fertilizer 1
v*f
3
residual 12
c. total
31
(b) E MS for v
Stat 511
Final - answers
6 May 2013
1. Growth rates of pigs
(a) 10 pts. Heres my skeleton ANOVA / model. Others were accepted if they 1) included
terms for the consistency of treatments across locatio
Stat 511
Final
6 May 2013
Remember, I want your name only on the back of the last page of your answers. If you have
answers written on that page, please put your name on a clean sheet of paper.
R and
Stat 511
Homework 9 Solution
Spring 2011
1. New Zealand pasture fertilization study.
(a) Degrees of freedom: Source
df
variety
3
block
3
v*b
9
fertilizer 1
v*f
3
residual 12
c. total
31
(b) E MS for v
Stat 511
Homework 8 Answers
Spring 2013
1. Split plot experiment
(a)
Yijkl = + i + j + ij + ik +
ijkl
where i, j, k, l = 1, 2. i indexes temperatures, j indexes catalysts, k indexes growth
chamber, an
Stat 511 HW 5 answers
1. Sample size calculations
I used the shifted t formula derived from =(t(1-/2).df+ t(1-).df)s.e.,
Sample sizes using non-central distributions are at the end of the question. Th
Stat 511, HW 3 Answers
Parts are worth one point, except where marked otherwise.
1. Orthogonal polynomials
(a) testing the hypothesis that all treatments have the same mean. There are 4 treatment,
so
Stat 511: HW 2 answers
1 point on each part, except point each on 1a, 1b, 1d, 1e, 3f, 3g, 3i and 3k
1) a)2 b)plane
c)1/3
d)2 e)Plane
f) the same as c)
g) The estimate for 1 or +1 is the average of the
2)
a) 1 pt for F statistic, 1 for theoretical distribution, and 1 for p-value and conclusion.
F = 2.86, F distribution with 4 and 65 degrees of freedom, p = 0.0301. evidence that the mean
rating for a
Stat 511 Spring 2013 HW 6 answers corrected (1d)
Most parts are 1 point each. Exceptions are noted.
1. Temperature trends
(a) 2 pts.
Estimate Std. Error t value Pr(>|t|)
(Intercept) -8.7867143 0.67957
Stat 511
Homework 10 Solution - revised
Spring 2013
All questions are 1 point, with 1 free point.
1. Barley-Fungus studies
(a) e.u.=chamber, o.u.=trays
Source
fungus
barley
fungus*barley
chamber
tray
Stat 511
Midterm 1 - Answers
25 Feb 2013
1. Small experiment, hand calculations
(a) F = 7.69, F distribution with 4,5 df.
Using the SS provided:
Source
Trt
Error
C.total
2
i ni (y i y . )
2
ij (yij y
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS
A model for expt. data with subsampling yijk = + i + uij + eijk , (i = 1, ., t; j = 1, ., n; k = 1, ., m) = (, i , ., t ) , u = (u11 ,
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELS
This is the commonly-used model for a CRD with t treatments, n experimental units per treatment, and m observations per experimental u
Experimental Designs and LME's
LME models provide one way to model correlations among observations Very useful for experimental designs where there is more than one size of experimental unit Or design
Experimental Designs and LME's
One example:
LME models provide one way to model correlations among observations
Very useful for experimental designs where there is more than one size of experimental u
LINEAR MIXED-EFFECT MODELS
Studies / data / models seen previously in 511 assumed a single source of "error" variation y = X + . are fixed constants (in the frequentist approach to inference) is the o
LINEAR MIXED-EFFECT MODELS
Seedling weight in 2 genotype study from Aitken model section. Seedling weight measured on each seedling. Two (potential) sources of variation: among flats and among seedlin
Randomization/permutation tests
Bootstrapping preserves the fixed effects
Randomization / Permutation tests
Resampling from a single pool of observations tests HoR : F1 (x) = F2 (x)
Notice a subtle po