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TitleApplied Asymptotics
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Page 123

7.3 Variance components 113

Table 7.4 Analysis of variance for regulator voltages.

Sum of squares df Mean square F value

Setting stations (SS) 4.4191 9 0.4910 9.08
Testing stations (TS) 0.7845 3 0.2615 4.84
SS×TS 0.9041 27 0.0335 0.62
Regulators within SS 9.4808 54 0.1756 3.25

Residual 8.7614 162 0.0541

setting stations implies that these should be regarded as selected at random from suitable
populations. As mentioned above, interest focuses on the variances, the �t being essential
to successful modelling but of secondary interest.
Under model (7.3) and owing to the balanced structure of the data, the quantities

ysrt−ysr· −ys·t+ys··� ysr· −ys··� ys·t−ys··� ys··� r = 1� � � � � ns� t = 1� � � � �4�

are independent normal variables for each setting station s; as usual, the overbar and
replacement of a subscript by a dot indicates averaging over that subscript. Thus the
residual sum of squares, the sum of squares between regulators within setting stations,
and the sum of squares between testing stations,

SSs =

r�t

�ysrt−ysr· −ys·t+ys··�2�

SSRs =

r�t

�ysr· −ys··�2�

SSTs =

r�t

�ys·t−ys··�2�

are independent. Their distributions are respectively

�2�23�ns−1�� ��
2+4�2$��2ns−1� �2�23����

the last being non-central chi-squared with non-centrality parameter � =∑t��t −�·�2.
The average voltage ys·· for setting station s has a normal distribution with mean �· and
variance �2�+�2$/ns+�2/�4ns�. The distribution of the sum of squares for testing stations
SSTs depends only on the �t and �

2, and so contributes only indirectly to inference on
the other variances. As there are 162 degrees of freedom available to estimate �2 from
the residual sum of squares, the gain from inclusion of SSTs will be slight, and we base a
marginal log likelihood on the other summary statistics.

Page 124

114 Further topics

Apart from constants, the contribution to the overall log marginal likelihood from the
sth setting station is therefore

�s���=− 12
[
SSs
�2

+ 3�ns−1� log�2+
SSRs

�2+4�2$
+ �ns−1� log��2+4�2$�

+ �ys·· −�·�
2

�2�+�2$/ns+�2/�4ns�
+ log{�2�+�2$/ns+�2/�4ns�}

]



The overall log marginal likelihood


s �s depends on the residual sum of squares SS =∑
s SSs, the sum of squares for regulators within setting stations SS

R =∑s SSRs , and the
average voltages ys·· for each of the setting stations, and these are mutually independent.
The argument of Section 2.4 can therefore be applied to the twelve log marginal likelihood
contributions.
Use of expression (2.11) for inference depends on computation of the local parametriza-

tion ���� defined at (2.12), and this requires the quantities Vi. In this continuous model
we can use expression (8.19) with pivots

zs =
ys·· −�·{

�2�+�2$/ns+�2/�4ns�
}1/2 � s = 1� � � � �10�

based on the averages, and z11 = SS/�2 and z12 = SSR/��2+4�2$� based on the sums of
squares. With �T = ��2���2$��2��·� this yields the following vectors:

Vs =
1
2cs

�̂zs� ẑs/ns� ẑs/�4ns�� cs�� s = 1� � � � �10�

V11 = �0�0� SS/�̂2�0��

V12 =
SSR

�̂2+4�̂2$
�0�4�1�0��

where cs = ��̂2�+ �̂2$/ns + �̂2/�4ns��1/2 and ẑs = �ys·· − �̂·�/cs. Further algebraic simpli-
fication of these expressions is possible. The corresponding likelihood derivatives with
respect to the data are



ys··
= − ys·· −�·

�2�+�2$/ns+�2/�4ns�
� s = 1� � � � �10�



SS
= − 1

2�2




SSR
= − 1

2��2+4�2$�


which are weighted by the V s to produce ����. All that is needed to obtain the local
parametrizations for the other parameters, and for functions of them, is to rewrite the

Page 245

Index 235

likelihood root, 5, 11, 139, 187, 206
modified, 6, 11, 171, 173, 187

decomposition of, 12, 42, 97, 155
linear model, 2, 58, 61, 84, 108, 117, 156–158,

184, 193, 194, 196, 206, 207
heteroscedastic, 117, 132
mixed, 132, 202, 203
non-normal, 163, 171, 175
normal, 58, 59, 108, 111, 168, 208

link function, 89, 107, 199
location model, 17, 138, 141, 142, 150, 186,

205
location-scale model, 84, 138, 145, 168, 207
log-linear model, 37, 54, 177
logistic regression, 37, 39, 54, 57, 90, 93, 94,

172, 177, 191
logistic response function, 96, 107
Lugannani–Rice approximation, 7, 18, 30, 147,

156, 164, 216, 218, 219

marginal inference, 59
Markov chain Monte Carlo, 54, 57, 132
matched pairs, 47, 52, 172, 192
maximum likelihood estimator, 5, 135, 205

asymptotic distribution, 138
consistency, 138, 185
constrained, 10, 74, 136
density approximation, 140

measurement error, 100, 201
Metropolis–Hastings algorithm, 95
mid-P-value, 21, 26, 35, 154, 187
model selection, 85
multivariate analysis, 133

non-convergence, 38, 39, 89, 93
nonlinear model, 59, 66, 72, 84, 85, 143, 146,

159–161, 172, 176, 180, 197, 198, 205,
207, 208

diagnostics, 99, 100
heteroscedastic, 60
logistic, 199
mixed, 132, 203

nuisance parameter, 10, 35, 136, 151–154, 168
numerical instability, 149, 175

odds, 24, 47, 187
order statistics, 60
orthogonal parameters, 12, 74, 164
outlier, 3, 58, 100, 190, 200
overdispersion, 193

parametrization, 167
parc jurassien, 91
partial likelihood, 133
permutation test, 129

pivot, 7, 8, 144, 145, 153, 167
approximate, 7

pivot profiling, 170, 174–176
posterior distribution, 209
power-of-the-mean variance function, 73
prior distribution, 123

Jeffreys, 187
matching, 110, 163, 168, 187, 191
non-informative, 163

probability difference, 24, 28, 49, 187
probability integral transform, 144
probability ratio, 24, 26, 187
profile plot, 7, 60, 73
profile trace, 60, 73
proportional hazards model, 127, 133

random effects, 54
regression-scale model, 14, 138, 168, 177, 195,

206
regularity conditions, 135
relative error, 10
REML, see likelihood, restricted
reparametrization, 136
residual, 99, 200
residual sum of squares, 59–61
robust inference, 184

saddlepoint approximation, 2, 129, 132, 167,
168, 207, 214, 218, 219

double, 133
sequential, 133, 207

sample mean, 212–213
sample space derivative, 102, 137, 153, 160
score equation, 135, 151, 205, 206
score pivot, see score statistic
score statistic, 5, 138, 139, 193
serial dependence, 120
serial dilution assay, 199
signed likelihood ratio statistic, see likelihood

root
significance function, 6, 7
skewness, 213
Skovgaard approximation, 70, 73, 85, 100, 102,

121, 123, 160, 166, 168, 184, 196, 200,
203, 208, 209

spline, 175
split-plot experiment, 55, 91
statistical-no-effect dose (SNED), 103, 107
stress-strength reliability, 188
sufficient statistic, 137, 141
summation convention, 165, 210
symbolic computation, 170, 184
symbolic differentation, 180–182

Page 246

236 Index

tail area approximation, 147–161, 167,
168

tangent exponential model, see exponential
family, tangent

tensor calculus, 184
time series, 120
transform-both-sides, 66, 96, 107, 198, 199,

201
transformation family, 137, 141, 167
transformation-invariance, 9
truncation, 188

variance components, 111, 132

variance function, 100, 198
error-in-variables, 100, 102, 107
power-of-x, 100, 102, 107, 199

variance independence, 136
variance inhomogeneity, see heteroscedasticity
variance parameters, 76
variance stabilization, 66, 73, 199

Wald pivot, see Wald statistic
Wald statistic, 5, 11, 36, 75, 152, 187, 205

zero inflated distribution, 192

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