text:
\section{Introduction}\label{SIntro}

Statistical investigations begin by determining which  sources(s) of
information will be used to answer the motivating questions and
generate hypotheses for future exploration. Conventional approaches to
statistical inference preclude structures that facilitate incorporation
of partially informative data, imposing polarity on the data selection
process. Putatively relevant supplemental data acquired from broadly
similar therapeutic interventions, patient cohorts, previous
investigations or biological processes are often either excluded from
statistical analysis or treated as exchangeable with the primary data.
In oncology, prospective studies designed to evaluate the performance
of an experimental therapy usually ignore historical information about
the control therapy and limit enrollment to patients presenting lesions
with a particular histological subtype, grade or performance status, or
who are na\"{i}ve to prior therapy. In contrast, studies with more
liberal information inclusion rules, such as intention-to-treat
designs, often pool information from potentially heterogeneous sources
or adjust for this potential heterogeneity using simple linear regressors.

Ignoring relevant, supplemental sources of information reduces the
reproducibility and scope of the primary study. On the other hand,
using the supplemental information while neglecting to account for
heterogeneity between the sources of information obscures understanding
of the complex underlying mechanisms that produced the primary data and
may lead to severely biased inference. Relaxing this dichotomy would
improve the efficiency of the experimental process and enable
investigators to implement statistical models that use all available
information while accommodating important between-source features.
Several models have been proposed for incorporating partially
informative supplemental data. \citet{poco76} used generalized linear
models with static, data-independent borrowing using a prespecified
amount of between-source variability. \citet{ibra00} proposed
data-independent or dynamic nonhierarchical methods for partially
weighting likelihoods. Bayesian [\cite{smit95}] and frequentist
[\cite{doi11}] methods using hierarchical modeling have been developed for
estimating between-source variability for univariate observables or
repeated measures with generalized linear relations among covariates.

Bayesian hierarchical models facilitate dynamic partial pooling of
between-source information and flexibly estimate the extent of
borrowing. Here the goal is limiting bias for estimating primary
effects when between-source heterogeneity occurs, while improving
efficiency when approximate coherence, or \textit{commensurability},
occurs; see \citet{hobb12} for generalized linear mixed models, and
\citet{hobb13} and \citet{murr14} for piecewise-exponential
time-to-event models. In Section~\ref{SSupp} we discuss these methods,
which can be used to borrow strength from supplemental data in
parametric models for regression coefficients and other univariate
parameters. We then develop a Bayesian hierarchical model structure to
leverage supplemental information more generally, including for both
nonparametric regression and semiparametric hazard models with
penalized splines. The literature appears devoid of general methods for
flexibly borrowing strength from supplemental data in semi- and
nonparametric models for a group of related parameters that
characterize a complex object (e.g., a curve or surface).

Two oncological applications motivate our methodological developments.
The first, from diagnostic radiology, involves estimating prognostic
functional imaging biomarkers acquired using perfusion computed
tomography (CT) in cancerous and cancer-free liver tissue. Perfusion CT
is an emerging technology that enables observation and quantification
of characteristics pertaining to the passage of fluid through blood
vessels. Researchers have developed physiological models to quantify a
variety of perfusion characteristics derived from analysis of the
distribution of contrast enhancement in tissue acquired using repeated
CT scans during intravenous administration of contrast medium.
Investigators have used the technology in a number of organs and
tumors, including prostate, colorectal, head and neck, lung and liver.
\citet{mile03} review the clinical relevance of perfusion CT.

Our application considers three characteristics: permeability-surface
area product (PS), blood volume (BV), and blood flow (BF). Each
characteristic is measured at 7 to 13 acquisition times between 11 and
95 seconds following contrast injection. Data from 16 individuals
comprise a total of 25 regions containing pathology-verified metastases
to the liver from neuroendocrine tumors (i.e., cancerous liver tissue),
and a total of 27 regions consisting of noncancerous liver tissue. PS
and BF are rates measured as milliliters per minute per 100 grams of
liver tissue (ml/min per 100~g), whereas BV is a volume measured as
milliliters per 100 grams of liver tissue (ml per 100~g). Figure~\ref{PRawCTpPlot} displays the observed perfusion CT (CTp) curves along
with Loess estimates of the average CTp curve for each characteristic
in each tissue region. Characterization of the perfusion
characteristics in cancerous tissue has implications for constructing
biomarkers to assist in treatment monitoring, prognostication and
pathophysiological understanding of metastatic tumor vasculature. It is
important that the acquisition duration cover the range of CTp curve
instability, but once the curve has stabilized, the scan can be
terminated. To limit radiation exposure and cost, the acquisition
duration should be minimized [\cite{ng13}]. Critically, because the
tissue type is unknown prior to diagnosis, any proposed acquisition
period must ensure stable quantification of CTp characteristics for
both types of tissue before CTp can be used for detection of metastatic
sites. Though nonexchangeable, the perfusion CT data obtained in
cancer-free liver tissue may inform the shape and stabilization time of
the corresponding CTp curve in cancerous tissue. Figure~\ref{PRawCTpPlot} shows substantial heterogeneity in the shapes of the
average CTp curves by tissue region for PS, whereas for BV and BF, the
average CTp curves are more similar.

Our second application, from colorectal cancer, involves estimating
prog\-ression-free survival (PFS) with data from two consecutive
randomized phase III colorectal cancer trials, reported by \citet
{salt00} and \citet{gold04}. Both trials used PFS to assess the
efficacy of various treatment regimens for patients with previously
untreated metastatic colorectal cancer; disease progression was defined
as a 25\% increase in measurable tumor size, presence of a new lesion
or death. The initial trial [\cite{salt00}] compared three treatment
regimens: 5-Fluorouracil and Leucovorin, Irinotecan alone, and
Irinotecan and bolus Fluorouracil plus Leucovorin (IFL). The results
indicated that the IFL regimen was significantly more efficacious than
the other two regimens, and IFL became the ``standard of care'' leading
into the subsequent trial [\cite{gold04}], which then compared an
identical IFL regimen with two novel regimens: Oxaliplatin with infused
Fluorouracil plus Leucovorin (FOLFOX), and Irinotecan with Oxaliplatin (IROX).

\begin{figure}[t]

\includegraphics{840f01.eps}

\caption{Individual-by-region perfusion CT (CTp) curves are displayed
by tissue region (``Individual CTp curves'') for permeability-surface
area product (PS), blood volume (BV) and blood flow (BF), over
acquisition durations between 11 and 95 seconds after contrast
injection. The dots mark actual observations. The thick ``Average CTp
curve'' is a Loess estimate that ignores potential clustering.}\label
{PRawCTpPlot}
\end{figure}

Figure~\ref{PRawKMPlot} shows that the PFS curves for the IFL regime
are commensurate in the Saltz and Goldberg trials, with the PFS curve
in the Goldberg trial tracking just above that of the Saltz trial,
though within the 95\% CIs for nearly all of follow-up. Moreover,
Figure~\ref{PRawKMPlot} suggests that in the Goldberg trial FOLFOX is
superior to both IROX (log-rank test $p$-value of 0.006) and IFL
(log-rank test $p$-value${}<0.001$), and that IFL and IROX perform
similarly (log-rank test $p$-value of 0.404). The efficiency for
estimating the PFS curve of the IFL regimen in the Goldberg trial may
be improved by using nonexchangeable, yet relevant data on the IFL
regimen in the Saltz trial.

The commonality between our two motivating examples is the availability
of supplemental information about unknown curves that may aid our
inference in the primary investigation (i.e., CTp curves among tissue
types, and PFS curves of IFL among clinical trials). In Section~\ref{SNPReg} we investigate the borrowing properties via simulation of the
proposed hierarchical model structure for nonparametric regression
using penalized splines, and then we analyze the perfusion CT data. In
Section~\ref{SHazReg} we analyze data from the colorectal cancer
clinical trials using the proposed hierarchical structure for a
semiparametric hazard model. In Section~\ref{SDisc} we close with a
discussion and propose directions for future work.
\begin{figure}

\includegraphics{840f02.eps}

\caption{Kaplan--Meier estimates of PFS for the IFL regimen in the Saltz
trial (along with 95\% CIs), and the IFL, FOLFOX and IROX regimes in
the Goldberg trial.}\label{PRawKMPlot}
\end{figure}

\section{Leveraging supplemental information}\label{SSupp}
We restrict our attention to two Bayesian methods for leveraging
supplemental information, power priors [\cite{ibra00}] and commensurate
priors [\cite{hobb11}]. In general, Bayesian models consist of a
likelihood for the data and prior specifications for the parameters in
the likelihood [see, e.g., \cite{carl09}]. For both these modeling
approaches, the \textit{primary} and \textit{supplemental} likelihoods are
assumed to have the same structure (e.g., both Gaussian), and we denote
them by $L(\bolds{\theta}|\mathbf{D})$ and $L(\bolds{\theta}_0|\mathbf{D}_0)$,
respectively. As a result, the supplemental parameter $\bolds{\theta}_0$
is analogous to the primary parameter $\bolds{\theta}$, and information
about $\bolds{\theta}_0$ can be leveraged to aid posterior inference about
$\bolds{\theta}$.

\subsection{Existing methods}\label{SSCommPrior}
Power priors assume the sources of information are exchangeable (i.e.,
$\bolds{\theta} \equiv\bolds{\theta}_0$) and downweight the supplemental
likelihood by raising it to a prespecified power $a_0 \in[0,1]$. This
strategy works because raising the supplemental likelihood to the power
$a_0$ diffuses it. In the extreme case where $a_0 = 0$, the
downweighted supplemental likelihood is a noninformative constant.
Formally, the posterior arises as
\begin{equation}
\label{EPowPr}
p(\bolds{\theta}|\mathbf{D},\mathbf{D}_0) \propto L(
\bolds{\theta}|\mathbf{D}) L(\bolds{\theta}|\mathbf{D}_0)^{a_0}
\pi(\bolds{\theta}).
\end{equation}
Alternatively, $a_0$ can be modeled as another unknown parameter in the
model [\cite{ibra00}]. Power priors are extremely general, so nothing
explicitly prevents their use in any modeling context. However, they
employ a single parameter ($a_0$) to control between-source borrowing,
thereby making differential borrowing among components of $\bolds{\theta}$
infeasible unless $L(\bolds{\theta}|\mathbf{D})$ has a convenient
factorization, as it generally does not. This is undesirable because
the supplemental data may provide relevant information for only a
subset of $\bolds\theta$. Furthermore, when $a_0$ is assumed unknown,
power priors tend to excessively downweight the supplemental
likelihood, even when $\mathbf{D}_0$ and $\mathbf{D}$ are identical [\cite{neel10}].

Commensurate priors do not assume the sources of information are
exchangeable (i.e., $\bolds{\theta} \not\equiv\bolds{\theta}_0$), and
instead specify a hierarchical model where the posterior arises as
\begin{equation}
\label{ECommMod}
p(\bolds{\theta},\bolds{\theta}_0,\bolds{\eta}|
\mathbf{D},\mathbf{D}_0) \propto L(\bolds{\theta}|\mathbf{D}) L(
\bolds{\theta}_0|\mathbf{D}_0) \pi(\bolds{\theta}|
\bolds{\theta}_0,\bolds{\eta}) \pi(\bolds{\eta}) \pi(\bolds{\theta}_0).
\end{equation}
The \textit{commensurate prior}, $\pi(\bolds{\theta}|\bolds{\theta}_0,\bolds{\eta
})$, ``centers'' $\bolds{\theta}$ about $\bolds{\theta}_0$, and $\bolds{\eta}$
controls between-source borrowing to reflect the commensurability of
the two parameters. In practice, $\pi(\bolds{\theta}|\bolds{\theta}_0,\bolds{\eta
})$ is defined so that $E[\bolds{\theta}|\bolds{\theta}_0,\bolds{\eta}] = \bolds{\theta}_0$ and $\operatorname{Var}[\bolds{\theta}|\bolds{\theta}_0,
\bolds{\eta}]$ is
decreasing in $\bolds{\eta}$.

For a unidimensional real-valued parameter $\theta$, a useful approach
takes $\theta|\theta_0,\eta\sim\mathcal{N}(\theta_0,\eta^{-1})$, a
Gaussian distribution with mean $\theta_0$ and precision $\eta$.
Estimation of $\eta$ is inherently difficult, but feasible by inducing
sparsity over the precision domain using a ``spike-and-slab'' prior for
$\eta$ [\cite{hobb12}]. The spike-and-slab prior has a mixture density
\begin{equation}
\label{ESpSl1}
\pi(\eta) \equiv(1-p_0) \mathcal{U}(
\eta|s_l,s_u) + p_0 \delta _{\mathcal{R}}(
\eta),
\end{equation}
where $0 \le s_l < s_u << \mathcal{R}$ and $p_0 \in[0,1]$ are
prespecified, and $\delta_{\mathcal{R}}(\eta)$ is one at $\mathcal{R}$
and zero otherwise. The distribution in (\ref{ESpSl1}) is locally
uniform on the ``slab,'' $(s_l,s_u)$, with probability $(1-p_0)$, and
places probability mass $p_0$ at the ``spike,'' $\mathcal{R}$, otherwise.

We prefer to modify the commensurate prior construction suggested by
\citet{hobb12} as
\begin{eqnarray}
&&\theta|\theta_{0},\tau,\iota\sim \bigl[\mathcal{N}
\bigl(\theta _{0},\tau^{-1}\bigr) \bigr]^{(1-\iota)}
\bigl[\mathcal{N}\bigl(\theta _{0},\mathcal{R}^{-1}\bigr)
\bigr]^{\iota},
\nonumber
\\[-8pt]
\label{ESpSl2}
\\[-8pt]
\eqntext{\displaystyle\iota\sim \operatorname{Bern}(p_0) \mbox{ and } \tau\sim
\mathcal{U}(s_l,s_u),}
\end{eqnarray}
where $\operatorname{Bern}(p_0)$ denotes a Bernoulli distribution with $\operatorname{Pr}(\iota= 1)
= p_0$. Therefore, the commensurate prior in (\ref{ESpSl1}) induces a
two-part mixture prior distribution on~$\theta$ that consists of a
highly concentrated component [i.e.,\break $\mathcal{N}(\theta_{0},\mathcal
{R}^{-1})$] and a relatively diffuse component [i.e., $\mathcal
{N}(\theta_{0},\tau^{-1})$], both centered at $\theta_0$. The model in~(\ref{ESpSl2}) has parallels to Bayesian variable selection methods,
in which the two-part mixture prior for a regression coefficient has
one component heavily concentrated about zero and the other vague [see,
e.g., \cite{geor97}]. Hence, the commensurate prior facilitates
selective borrowing by inheriting the selective shrinkage property of
the spike-and-slab distributions used in Bayesian variable selection.
In particular, the commensurate prior strongly shrinks $\theta$ to
$\theta_0$ when evidence indicates that this difference is small,
thereby improving efficiency, and minimally affects $\theta$ otherwise,
thereby limiting bias.

We specify the spike so that if $\theta|\theta_0 \sim\mathcal{N}(\theta
_{0},\mathcal{R}^{-1})$, $\theta$ deviates negligibly from~$\theta_0$.
On the other hand, we specify the slab to contain small values that
correspond to modest shrinkage of $\theta$ toward $\theta_{0}$. Hence,
a $\mathcal{N}(\theta_{0},s_u^{-1})$ prior for $\theta$ will be weakly
informative. The tails of the prior distribution in (\ref{ESpSl2}) are
jointly controlled by $s_l$ and $s_u$: when $s_l \approx0$, smaller
values of $s_u$ provide a prior with heavier tails. In our experience,
posterior inference on $\theta$ is insensitive to modest shifts in
these hyperparameters, so we suggest specifying $s_l$ near zero, with
$s_u$ small and $\mathcal{R}$ large relative to the magnitude of a
meaningful difference between $\theta$ and $\theta_0$. Posterior
inference \textit{can} be sensitive to the fourth hyperparameter, $p_0$,
the prior probability of effective equality between $\theta$ and $\theta
_0$. We recommend choosing the value for $p_0$ to deliver satisfactory
operating characteristics at the anticipated sample size, such as
mean-squared-error properties over a range of likely true differences
between $\theta$ and $\theta_0$; see \citet{murr14}.

Specifying a value for $s_u$ that is too large can result in a model
that is not robust to between-source heterogeneity, whereas specifying
a value for $\mathcal{R}$ that is too small will result in a model that
gains little posterior precision for $\theta$ even when $\mathbf{D}_0$ and
$\mathbf{D}$ are equivalent. Specifying too small a value for $p_0$ may
also result in a model that borrows little when $\mathbf{D}_0 \equiv\mathbf{D}$, whereas specifying too large a value for $p_0$ may result in a
model that borrows too much (resulting in unacceptable bias) in the
presence of substantial between-source heterogeneity. Of course,
$p_0$'s influence on posterior inference diminishes as the primary
sample size increases.

\subsection{Generalized mixture commensurate priors}\label{SSGMCPrior}
The general model structure described in (\ref{ECommMod}) can
accommodate a latent precision parameter (component of $\bolds{\eta}$) for
each component of $\bolds\theta$. Therefore, the extent to which the
supplemental data influence estimation of the primary effects can be
allowed to differ among the components of $\bolds\theta$, yielding
flexibility and reducing bias. However, assuming mutual independence
between the components of $\bolds{\eta}$ allows strength to be borrowed
from the supplemental source(s) differentially among the components of
$\bolds\theta$, reducing efficiency. Though this may be sensible,
differential borrowing among the components of $\bolds\theta$ may not
always be appropriate.

Suppose $\bolds{\beta} = (\beta_1,\ldots,\beta_K)$ is a subset of $\bolds
{\theta}$ that characterizes a feature in the primary data (e.g., the
shape of a CTp curve), with an analogous definition for $\bolds{\beta}_0$.
We may wish to borrow from the supplemental data for $\bolds{\beta}_0$ to
a degree that reflects its coherence with $\bolds{\beta}$ in its entirety,
rather than case-by-case for each $\beta_k$. Moreover, $\bolds{\beta}$
(and $\bolds{\beta}_0$) may conventionally be assigned a prior that
induces smoothness. Ideally, (\ref{ECommMod}) will inherit the
smoothness properties of the conventional prior and simultaneously
borrow similar amounts for related subgroups of $\bolds{\theta}$ (i.e.,
$\bolds{\beta}$). We propose to model
\begin{eqnarray}
\beta_k|
\iota_k,\beta_{k,0}  &\sim&  \bigl[\pi^*(\beta_k|
\beta _{k,0}) \bigr]^{1-\iota_k} \bigl[\pi(\beta_k|\beta
_{k,0}) \bigr]^{\iota_k},\nonumber\\[-8pt]
\label{EGenMixCommPr}
\\[-8pt]
 \eqntext{\displaystyle\iota_k|\nu\sim \operatorname{Bern}(\nu),
\nu\sim\mathcal{B}(a_1,a_2), \beta_{0,k} \sim
\pi^*(\beta_{0,k}),}
\end{eqnarray}
where\vspace*{1pt} $\mathcal{B}(a_1,a_2)$ denotes a beta distribution with mean
$\frac{a_1}{a_1+a_2}$ and the specifications for $\pi^*$ mirror a
conventional analysis of these data. To facilitate borrowing of
strength, we take $\pi(\beta_k|\beta_{0,k})$ in (\ref{EGenMixCommPr})
to be heavily concentrated about $\beta_{0,k}$; we provide exact
specifications for each application later. To borrow similar amounts of
strength for each component of $\bolds{\beta}$, we assume the $\iota_k$
are identically distributed with $\operatorname{Pr}(\iota_k=1) = \nu$. Thus, the
resulting prior is composed of $K$ two-piece mixtures and generalizes
the commensurate prior defined in (\ref{ESpSl2}). Hereafter, we refer
to the general prior structure defined in (\ref{EGenMixCommPr}) as a
\textit{generalized mixture commensurate} (GMC) prior.

\section{GMC priors in nonparametric regression analysis}\label{SNPReg}
Suppose $\mathbf{y} = (y_1,\break \ldots,y_N)$ is a real-valued variable (e.g.,
log blood flow, as in Figure~\ref{PRawCTpPlot}) that depends on a
continuous covariate $\mathbf{t} = (t_1,\ldots,t_N)$ (e.g., time, as in
Figure~\ref{PRawCTpPlot}), and that we model $y_i \sim \mathcal
{N}(\phi(t_i),\sigma^2)$. Often primary interest lies in the shape,
derivatives or some other function of $\phi$, and $\phi$ requires a
smooth but flexible nonparametric specification, as opposed to a
parametric linear specification $\phi(t) = \beta_0 + \beta_1 t$.
Penalized splines are a practical choice for modeling $\phi$; see \citet{rupp03}, Chapters~3 and 14.
There are many ways to formulate a
penalized spline. Low-rank thin-plate (LRTP) splines are appealing
since they are defined by tractable radial basis functions and tend to
exhibit fast Markov chain Monte Carlo (MCMC) convergence properties in
a Bayesian context relative to truncated basis splines [\cite{crai05}].
B-splines are another reasonable Bayesian option because they too tend
to exhibit fast MCMC convergence, but they rely on a recursive
algorithm to define the basis functions, making them less tractable
than LRTP splines [\cite{eile96}]. Our method will work with either
LRTP or B-spline formulations for $\phi$, and we do not anticipate
substantial differences in the behavior of our method under either
formulation. Hereafter, we focus on a LRTP cubic spline specification
for $\phi$.

Without loss of generality, we take $t \in[0,1]$ and model
\begin{equation}
\label{EmLRTP}
\phi(t;\bolds{\beta}) = \beta_0 +
\beta_1 t + \sum_{k=2}^{K}
\beta _k \bigl(|t-\tilde{t}_{k-1}|^3 - |
\tilde{t}_{k-1}|^3 \bigr),
\end{equation}
where $\tilde{\mathbf{t}} = (0=\tilde{t}_0<\tilde{t}_1<\cdots<\tilde
{t}_{K-1}<\tilde{t}_K = 1)$ is a generic partition with $K$ intervals.
We discuss this choice later.
Typically, a LRTP spline model is formulated as
\begin{equation}
\label{ELRTP} \phi\bigl(t;\bolds{\beta}^*\bigr) = \beta^*_0 +
\beta^*_1 t + \sum_{k=2}^{K}
\beta^*_{k}|t-\tilde{t}_{k-1}|^3;
\end{equation}
however, in (\ref{ELRTP}), $\beta^*_0$ is not an intercept because
$\phi(0;\bolds{\beta}^*) = \beta^*_0 + \sum_{k=2}^{K} \beta
^*_{k}|\tilde{t}_{k-1}|^3$. We prefer the modified LRTP (mLRTP) model
defined in (\ref{EmLRTP}), because it simplifies interpretation and
prior elicitation when the regression includes an intercept, which will
be the case for the CTp data, and it retains an equivalent
interpretation of $(\beta_1,\ldots,\beta_K)$ and $(\beta^*_1,\ldots
,\beta^*_K)$.
In the sequel, we denote $\bolds{\beta}_{(-0)} = (\beta_1,\ldots,\beta
_K)$, that is, the vector $\bolds{\beta}$ with $\beta_0$ omitted.

A \textit{penalized} spline requires prespecification of a fine partition
$\tilde{\mathbf{t}}$, and smooths via the prior, whereas a \textit{free-knot}
spline uses complex sampling algorithms (e.g., reversible-jump MCMC) to
sample over all possible partitions. We prefer penalized splines, since
they are computationally much simpler and have proven to be competitive
with free-knot splines when the temporal variation of $\phi$ is smooth
[\cite{rupp03}, Section~3.16]. \citet{wand00} reviews methods for
free-knot spline estimation; we do not consider this approach further.
To select $\tilde{\mathbf{t}}$ for a penalized spline, we consider a set of
values for $K$ with either equally-spaced or quantile-spaced $\tilde
{t}_k$'s, and select the partition that has low posterior mean deviance
and deviance information criterion (DIC) relative to the other
partitions considered [\cite{spie02}].

Following \citet{crai05}, without supplemental data, we complete the
Bayesian specification of (\ref{EmLRTP}) by placing vague priors on
$\beta_0$ and $\beta_1$, and a $\mathcal{N}(\mathbf{0},\sigma^2_{\bolds{\beta}}\bolds{\Omega}^{-})$ prior on $(\beta_2,\ldots,\beta_K)'$, where the
$(j,k)$th entry of $\bolds{\Omega}$ is defined as $|\tilde{t}_{j-1} -
\tilde{t}_{k-1}|^3$, for $j,k=2,\ldots,K$. In practice, we apply the
transformations $b_0 = \beta_0$, $b_1 = \beta_1$, $(b_2,\ldots,b_K)' =
\bolds{\Omega}^{1/2}(\beta_2,\ldots,\beta_K)'$, and $\sigma^{2}_{\mathbf{b}} =
\sigma^2_{\bolds{\beta}}$, and then complete the model specification by assuming
\begin{eqnarray}
\pi^*(b_k) & \equiv&
\mathcal{N}\bigl(b_k|0,10^4\bigr)\qquad \mbox{for } k=0,1,
\nonumber\\
\label{ELRTPPrior}
\pi^*(b_k|\sigma_{\mathbf{b}})& \equiv & \mathcal{N}
\bigl(b_k|0,\sigma _{\mathbf{b}}^{2}\bigr)\qquad \mbox{for }
k=2,\ldots,K \quad\mbox{and}
\\
\nonumber
\pi^*(\sigma_{\mathbf{b}}) & \equiv & \mathcal{U}(\sigma
_{\mathbf{b}}|0.01,100).
\end{eqnarray}
The prior in (\ref{ELRTPPrior}) smooths $\phi$ by preferring values
for $(b_2,\ldots,b_K)$ near zero {a priori}.

\subsection{GMC prior specification}\label{SSGMCP}
With supplemental data [$\mathbf{y}_0 = (y_{0,1},\ldots,\break y_{0,n_0})$, $\mathbf{t}_0  = (t_{0,1},\ldots,t_{0,n_0})$], we also assume
$y_{0,i_0}|t_{0,i_0} \sim \mathcal{N}(\phi_0(t_{0,i_0}),\sigma
^2_0)$, for $i_0 = 1,\ldots,n_0$. To ensure that $\phi$ and $\phi_0$
are analogous, we use the same partition $\tilde{\mathbf{t}}$ for $\phi_0$,
and model $\phi_0(t;\bolds{\beta}_0)$ using (\ref{EmLRTP}) by replacing
$\bolds{\beta}$ with $\bolds{\beta}_0$. We apply the transformation\vspace*{1.5pt} $b_{0,k}
= \beta_{0,k}$, for $k=0,1$, and $(b_{0,2},\ldots,b_{0,K})' = \bolds{\Omega}^{1/2}(\beta_{0,2},\ldots,\beta_{0,K})'$, and use the prior
specifications $\pi^*(b_{0,k}) \equiv\mathcal{N}(b_{0,k}|0,10^4)$, for
$k=0,1$, $\pi^*(b_{0,k}|\sigma_{\mathbf{b}_0}) \equiv\mathcal
{N}(b_{0,k}|0,\sigma_{\mathbf{b}_0}^{2})$ for $k=2,\ldots,K$, and $\pi
^*(\sigma_{\mathbf{b}_0}) \equiv\mathcal{U}(\sigma_{\mathbf{b}_0}|0.01,100)$.

To borrow flexibly for the shape of $\phi$ from the supplemental data
for $\phi_0$, we apply the GMC prior specification defined in (\ref
{EGenMixCommPr}) to $\mathbf{b}_{(-0)}=(b_1,\ldots,b_K)$ given $\mathbf{b}_{0,(-0)}=(b_{0,1},\ldots,b_{0,K})$. We assume
\begin{eqnarray*}
&&\pi(\mathbf{b}_{(-0)}|\mathbf{b}_{0,(-0)},\bolds{\iota}_{(-0)},\sigma_{\mathbf{b}}) \\
&&\qquad= \bigl[\pi^*(b_1)
\bigr]^{(1-\iota_1)} \bigl[\pi (b_1|b_{0,1})
\bigr]^{\iota_1} \prod_{k=2}^{K}
\bigl[\pi^*(b_k|\sigma_{\mathbf{b}}) \bigr]^{(1-\iota_k)} \bigl[
\pi(b_k|b_{0,k}) \bigr]^{\iota_k},
\end{eqnarray*}
where $\bolds{\iota}_{(-0)}$ = $(\iota_1,\ldots,\iota_K)$, and $\pi
^*(b_1)$, $\pi^*(b_k|\sigma_{\mathbf{b}})$ and $\pi^*(\sigma_{\mathbf{b}})$ are
defined in (\ref{ELRTPPrior}). We set $\pi(b_k|b_{0,k})$ $\equiv$
$\mathcal{N}(b_k |b_{0,k},\mathcal{R}_{\mathbf{b}}^{-1})$, for $k=1,\ldots
,K$, where $\mathcal{R}_{\mathbf{b}}$ is a prespecified \textit{large} value.
We then specify $\iota_k|\nu \stackrel{\mathrm{iid}}{\sim} \operatorname{Bern}(\nu)$ and
$\nu$ $\sim$ $\mathcal{B}(a_1,a_2)$ with fixed, known hyperparameters
$a_1$ and $a_2$ that reflect our prior opinion about the relevance of
the supplemental data for the shape of $\phi$.

To allow differential borrowing for the intercept versus the shape of
$\phi$, we specify an independent commensurate prior distribution for
$b_0$ given $b_{0,0}$ following (\ref{ESpSl2}). We define\vspace*{1pt}
$\pi(b_0|b_{0,0},\iota_0,\tau)\equiv[\pi
^*(b_0|b_{0,0},\tau)  ]^{(1-\iota_0)}[\pi
(b_0|b_{0,0})  ]^{\iota_0}$,
where $\pi^*(b_0|b_{0,0},\tau)\equiv\mathcal{N}(b_0|b_{0,0},\tau
^{-1})$ and $\pi(b_0|b_{0,0})\equiv\mathcal
{N}(b_0|b_{0,0},\mathcal{R}^{-1})$. We then specify $\tau \sim \mathcal{U}(s_l,s_u)$ and $\iota_0\sim\operatorname{Bern}(p_0)$,
where $s_l$,
$s_u$, $\mathcal{R}$ and $p_0$ are prespecified following the guidance
in Section~\ref{SSupp}. Taken together, the full posterior arises as
\begin{eqnarray}
&&p(\mathbf{b}, \mathbf{b}_0,
 \sigma_{\mathbf{b}}, \sigma_{\mathbf{b}_0}, \bolds{\iota}, \nu, \tau|
\mathbf{D}, \mathbf{D}_0) \nonumber\\
 &&\qquad\propto  L(\mathbf{b} | \mathbf{D}) L(
\mathbf{b}_0 | \mathbf{D}_0) \pi(b_0|b_{0,0},
\iota_0,\tau) \pi(\iota_0) \pi(\tau) \pi
^*(b_{0,0})
\nonumber
\\[-8pt]
\label{EGMCPost}
\\[-8pt]
\nonumber
&&\qquad\quad{}\times
 \pi(\mathbf{b}_{(-0)}|\mathbf{b}_{0,(-0)},\bolds{\iota
}_{(-0)},\sigma_{\mathbf{b}}) \pi(\bolds{\iota}_{(-0)}|\nu)
\pi(\nu) \pi ^*(\sigma_{\mathbf{b}})\nonumber \\
&&\qquad\quad{}\times\pi^*(\mathbf{b}_{0,(-0)} |
\sigma_{\mathbf{b}_0}) \pi ^*(\sigma_{\mathbf{b}_0}).\nonumber
\end{eqnarray}

\subsection{Simulation assessment}\label{SSSim}
We now investigate via simulation the borrowing properties of the GMC
prior model in (\ref{EGMCPost}) for nonparametric regression. To do
so, we sample $y_i|t_i \sim\mathcal{N} \lbrace\mu(t_i), \sigma
^{2}  \rbrace$, $i=1,\ldots,N$, where $\mu(t) = 5t\sin\lbrace5t
\rbrace$, and sample $y_{0,i_0}|t_{0,i_0},d \sim  \mathcal{N}\lbrace
\mu_0(t_{0,i_0}|d), \sigma_{0}^{2} \rbrace$, $i_0=1,\ldots,N_0$, where
$\mu_0(t|d) = (5+d)t\sin((5+d)t)$ and $d \in[0,5]$. Hence, the primary
data always have the same true mean structure, and the supplemental
data have a mean structure that deviates from that of the primary
according to the value of $d$, the discordance parameter. When $d = 0$,
the two curves are the same, and as $d$ increases, the supplemental
curve has increasingly greater curvature than the primary curve. Figure~\ref{PSimFunc} shows curves for selected values of $d$.

\begin{figure}[b]

\includegraphics{840f03.eps}

\caption{Mean curves for the supplemental data for a subset of
discordance parameter ($d$) values. The mean curve for the primary data
is denoted by the solid line (i.e., $d=0$).}
\label{PSimFunc}
\end{figure}

In each run, we sample $d$ uniformly from the set $\lbrace$0, 0.05,
0.10, 0.20, 0.35, 0.50, 0.75, 1, 1.50, 2, 3, 4, 5$\rbrace$. Given $d$,
we then generate primary and supplemental data sets with $N = N_0 =
50$, error variances $\sigma^{2} = \sigma_{0}^{2} = 1$, and equally
spaced $t$ and $t_0$ values over $[0,1]$. For each data set pair, $(\mathbf{D} = (\mathbf{y},\mathbf{t}),\mathbf{D}_0 =
(\mathbf{y}_0,\mathbf{t}_0))$, we fit the GMC
prior model developed in Section~\ref{SSGMCP}. We use the mLRTP model
defined in (\ref{EmLRTP}) with $K=10$ and $\tilde{\mathbf{t}}$ spaced
equally over $[0,1]$ for both $\phi_0(t_0;\bolds{\beta}_0)$ and $\phi(t;\bolds{\beta})$. After preliminary investigation, we set $s_l = 0$, $s_u =
2$, $\mathcal{R} = 2000$, $p_0 = 0.50$, and $\pi(\nu) \equiv\mathcal
{B}(\nu|0.50,0.50)$. The latter two choices represent an indifferent
prior opinion about the relevance of the supplemental data for both the
shape and the intercept of $\phi$. We choose to place an extremely
vague yet indifferent prior on $\nu$ to allow the primary and
supplemental data to have substantial influence on whether to borrow
for curve shape.

We used the {\tt R2jags} package to call {\tt JAGS} from {\tt R}, and
ran two chains for 20{,}000 iterations of burn-in, followed by 200{,}000
iterations for posterior estimation. We also fit the mLRTP model
defined in (\ref{EmLRTP}) with prior specifications described in (\ref
{ELRTPPrior}) to the primary data alone, as well as to the data set
obtained by simply pooling the primary and supplemental data. These
models feature lower MCMC autocorrelation, and thus required only two
chains with 2000 iterations of burn-in, followed by 20{,}000 iterations
for posterior estimation. These choices reflect preliminary
investigations to ensure acceptable MCMC convergence and relatively
small MC standard errors for the intercept and functional effect
coefficients. Estimation of the GMC model took about 45 seconds,
whereas estimation of each comparison model took about 5 seconds.

To evaluate the three models, we calculated four criteria at each run:
mean error (ME), root-mean-square error (RMSE), mean pointwise credible
interval width (CrIW), and mean pointwise coverage probability (CP).
Alternatively, we could calculate simultaneous confidence bands
following \citet{kriv10}, Section~3. We define the four criteria as
\begin{eqnarray*}
\operatorname{ME}\bigl(d^{(m)}\bigr) &=&  N^{-1}\sum
_{i=1}^{N} \bigl[\hat{\phi} \bigl(t_i|d^{(m)}
\bigr) - \mu \bigl(t_i|d^{(m)} \bigr) \bigr],
\\
\operatorname{RMSE}\bigl(d^{(m)}\bigr) &=& \Biggl\lbrace N^{-1}\sum
_{i=1}^{N} \bigl[ \hat{\phi} \bigl(t_i|d^{(m)}
\bigr) - \mu \bigl(t_i|d^{(m)} \bigr) \bigr]^2
\Biggr\rbrace^{{1}/{2}},
\\
\operatorname{CrIW}\bigl(d^{(m)}\bigr) &=& N^{-1}\sum
_{i=1}^{N} \bigl[\hat{\phi}_{0.975}
\bigl(t_i|d^{(m)} \bigr) - \hat{\phi}_{0.025}
\bigl(t_i|d^{(m)} \bigr) \bigr],
\\
\operatorname{CP}\bigl(d^{(m)}\bigr) &=& N^{-1}\sum
_{i=1}^{N} I \bigl\lbrace\mu \bigl(t_i|d^{(m)}
\bigr) \in \bigl[\hat{\phi}_{0.025} \bigl(t_i|d^{(m)}
\bigr), \hat{\phi}_{0.975} \bigl(t_i|d^{(m)}
\bigr) \bigr] \bigr\rbrace,
\end{eqnarray*}
for $m=1,\ldots,M$, where $\hat{\phi} (t|d^{(m)} )$ denotes
the posterior mean estimate for $\phi (t|d^{(m)} )$, and
$\hat{\phi}_{q} (t|d^{(m)} )$ denotes the $q$th quantile
posterior estimate of $\phi (t|d^{(m)} )$ in the $m$th run.
We then compare the sampling average of each criterion over the $M$
simulated data set pairs as a function of $d$, the discordance parameter.
Specifically, we visually compare Loess estimates of each evaluation
criteria as a function of the discordance parameter.
One simulation iteration takes about 60 seconds, which entails
generating a pair of data sets, fitting the three models, and then
calculating and saving the evaluation criteria, along with the sampled
value of $d$ in that run. We reduced overall computation time using the
{\tt snowfall} package for {\tt R} to conduct the simulation runs in
parallel. A {\tt R} program to implement this simulation is available
through the third author's software page \surl{http://www.biostat.umn.edu/\textasciitilde brad/software.\\html}.

\begin{figure}

\includegraphics{840f04.eps}

\caption{Simulation assessment of the GMC prior spline model (solid)
versus a conventional spline model fit to the primary data alone
(dashed) and to the data set obtained by pooling the supplemental and
primary data (dot-dashed). These methods are compared on sampling
averages of mean error (top left), root-mean-square error (top right),
mean credible interval width (bottom right) and mean pointwise coverage
probability (bottom left) as a function of $d$, the discordance
parameter. All results are based on $M$ = 2000 runs.}\label{PSimRes}
\end{figure}

The results of our simulation investigation are illustrated in Figure~\ref{PSimRes}. Each panel shows the Loess estimates of the sampling
average of the corresponding evaluation criteria as a function of $d$
for three models: the GMC prior model (solid), the conventional model
fit to the primary data alone (dashed), and the conventional model fit
to the pooled data (dot-dashed). As expected, inferential properties
for the GMC prior model reflect those for the pooled approach under
true concordance (i.e., $d=0$) and approach primary data alone for
increasing degrees of discordance. The primary-alone approach has the
largest average CrIW and RMSE under true concordance, but its
properties do not deteriorate for increasing degrees of discordance
(because it does not acknowledge the supplemental data). By contrast,
the pooling approach has the smallest CrIW and RMSE under true
concordance, but its performance on all four evaluation criteria
deteriorates substantially as discordance increases.

The ME plot (top left panel) in Figure~\ref{PSimRes} demonstrates that
the GMC prior model has bias properties that interpolate the two
conventional approaches for near concordance, but near $d=1$ it learns
to effectively ignore the supplemental information. Furthermore, the
RMSE (top right panel) shows that the GMC prior model learns to borrow
strongly from the supplemental information when the sources are
commensurate (i.e., slight discordance), achieving RMSE similar to that
of the pooling approach. Similarly, the CrIW plot (bottom right panel)
shows for slight discordance that the credible intervals from the GMC
prior model are nearly as tight as those for the pooling approach.
Conversely, when the supplemental curve has greater oscillation than
the primary curve, the GMC prior approach has credible interval width
similar to those of the primary-alone approach. Last, the CP plot
(bottom left panel) shows that the mean pointwise coverage
probabilities for the GMC prior model are similar to those of the
conventional spline model fit to the primary data alone.

\subsection{Application: Liver imaging study}\label{SSRadOnc}
We now apply our GMC prior model structure to estimate the temporal
features of each perfusion characteristic in cancerous liver tissue.
Recall that similar information derived from noncancerous tissue is
potentially valuable supplemental information because the shape of the
average CTp curve in noncancerous regions may provide relevant
information about the shape of the corresponding average CTp curve in
cancerous regions. The data consist of 7 to 13 readings acquired at
times between 11 and 100 seconds after contrast injection in 0 to 2
cancerous and noncancerous regions of interest (ROIs) among 16
individuals. For each perfusion characteristic, there are 687 total
readings from fifteen individuals who each contribute $N_i =
11\mbox{--}13$
readings in $M_{T,i} = 1\mbox{--}2$ cancerous and $M_{N,i} = 1\mbox{--}2$ noncancerous
ROIs, and one individual who contributes 7 readings in 1 noncancerous ROI.

Individual acquisition times $t_{i,\ell}$, $\ell=1,\ldots,N_i$, are
necessarily identical for all ROIs and all perfusion characteristics.
Thus, we let $y_{r,i,j,\ell}$, for $j$ = $1,\ldots,M_{r,i}$, denote a
reading at time $t_{i,\ell}$. We denote the average CTp curve for a
given perfusion characteristic in cancerous tissue and noncancerous
tissue by $\phi_{T}(t)$ and $\phi_{N}(t)$, respectively. We assume each
individual's CTp curve deviates from the average CTp curve smoothly
over time, and we denote these deviation curves by $\psi_{r,i}(t)$,
$r= T$, $N$ and $i = 1,\ldots,16$. We then model
\begin{equation}
\label{ECTpModel}
y_{r,i,j,\ell} \sim\mathcal{N} \bigl\lbrace\phi_{r}(t_{i,\ell})
+ \psi _{r,i}(t_{i,\ell}), \sigma^2_{e,r}
\bigr\rbrace.
\end{equation}
We use the mLRTP model defined in (\ref{EmLRTP}) for $\phi_{r}$ and
$\psi_{r,i}$, parametrized by $\bolds{\beta}_{r}$ and $\bolds{\alpha}_{r,i}$,
respectively. Therefore, the first derivative of the average CTp curve is
\begin{equation}
\label{ELRTPDeriv}
\phi'_{r}(t;\bolds{\beta}_r)
= \beta_{r,1} + \sum_{k=2}^{K}
\operatorname{sign}(t-\tilde{t}_{k-1}) 3\beta_{r,k}(t-\tilde{t}_{k-1})^2,\qquad
r=N,T.
\end{equation}
To shrink the individual deviation curves $\psi_{r,i}(t;\bolds{\alpha}_{r,i})$ toward the average CTp curve $\phi_r(t;\bolds{\beta}_r)$, we
assume $\alpha_{r,i,k} \sim\mathcal{N}(0,\sigma^2_{\mathbf{a},r,i})$, for
$k=0,\ldots,K$, $r=N,T$ and $i=1,\ldots,16$.

For our\vspace*{1pt} conventional model, we use the prior specifications on $\bolds{\beta}_{N}$ and $\bolds{\beta}_{T}$ described in (\ref{ELRTPPrior}),
and specify vague $\mathcal{N}(0,10^4)$ priors for $b_{N,0}$ and
$b_{T,0}$. For the GMC prior model, we apply the prior developed in
Section~\ref{SSGMCP} for $\bolds{\beta}_{T}$ given $\bolds{\beta}_{N}$. That
is, we specify a GMC prior for $\bolds{\beta}_{T,(-0)} = (\bolds{\beta
}_{T,1},\ldots,\bolds{\beta}_{T,K})$ given $\bolds{\beta}_{N,(-0)} = (\bolds
{\beta}_{N,1},\ldots,\bolds{\beta}_{N,K})$ and an independent commensurate
prior for $b_{T,0}$ given $b_{N,0}$. This choice enables differential
borrowing from the noncancerous tissue data for the intercept versus
the curve shape parameters, and simultaneously enables borrowing
similar amounts among curve shape parameters. Thus, if the average CTp
curve in noncancerous regions differs from the average CTp curve in
cancerous regions only for the intercept, the model still permits
borrowing for the CTp curve shape. Last, we place vague $\mathcal
{U}(0.01,100)$ priors on each of the standard deviation parameters
($\sigma_{e,r}$, $\sigma_{\mathbf{b},r}$, $\sigma_{\mathbf{a},r,i}$).

We let $t$ $\in$ $(0,1]$ by taking $t = \frac{t^*-t^*_{\min}}{t^*_{\max}
- t^*_{\min}}$, where $t^*$ is the original timescale. Preliminary
analysis of the noncancerous regions was used to specify the
hyperparameters $\mathcal{R} = 500$, $s_l = 0.01$, $s_u = 0.50$, $p_0 =
0.10$, $a_1 = 0.10$ and $a_2 = 0.90$. To select a partition, we fit our
conventional model (i.e., independent spline models for the average CTp
curve in each tissue type) for each perfusion parameter using $K$ = 5,
10, 15 and 25 with $\tilde{\mathbf{t}}$ equally spaced over $[0,1]$, and
using $K$ = 5, 10 and 15 with $\tilde{\mathbf{t}}$ placed at equally spaced
quantiles of the observed acquisition times. The model using $K=10$
with quantile-spaced knots resulted in low deviance and low DIC
relative to models with the other partitions for all three perfusion
characteristics, so we chose to conduct our analysis using this
partition. We estimated the conventional model using 40{,}000 posterior
samples after 2000 burn-in samples from two MCMC chains. We estimated
the GMC model posterior using 200{,}000 posterior samples after 20{,}000
burn-in samples from two MCMC chains; there was greater autocorrelation
for the basis coefficient parameters than the conventional model.

\begin{table}[b]
\tabcolsep=0pt
\caption{Posterior borrowing parameter estimates (i.e., $\iota_k$'s).
$\iota_0$ corresponds to the CTp curve intercept, and the remaining
$\iota_k$'s correspond to CTp curve shape. Values near one indicate
strong borrowing}\label{TCTpResults}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccccccccccc@{}}
\hline
\textbf{Perfusion characteristic} & $\bolds{\iota_0}$ & $\bolds{\iota_1}$ &
$\bolds{\iota_2}$ & $\bolds{\iota
_3}$ & $\bolds{\iota_4}$ & $\bolds{\iota_5}$ & $\bolds{\iota_6}$ &
$\bolds{\iota_7}$ & $\bolds{\iota_8}$ &
$\bolds{\iota_9}$ & $\bolds{\iota_{10}}$ \\
\hline
PS & 0.00 & 0.23 & 0.00 & 0.00 & 0.24 & 0.43 & 0.24 & 0.05 & 0.15 &
0.03 & 0.34 \\
BV & 0.20 & 0.92 & 0.95 & 0.92 & 0.82 & 0.86 & 0.90 & 0.82 & 0.88 &
0.96 & 0.96 \\
BF & 0.02 & 0.96 & 0.98 & 0.92 & 0.92 & 0.93 & 0.94 & 0.92 & 0.96 &
0.98 & 0.99 \\
\hline
\end{tabular*}
\end{table}

Table~\ref{TCTpResults} reports the posterior mean borrowing
parameters (i.e., $\iota_k$'s) from the GMC model for the three
perfusion characteristics. Values near 1 indicate strong borrowing for
the corresponding basis parameter. The first column reports the
posterior mean for $\iota_0$, which corresponds to the intercept. PS
and BF show virtually zero borrowing for the intercept, whereas BV
shows little borrowing. The remaining columns illustrate borrowing for
the basis parameters that control the shape (and thus, first
derivative) of the CTp curve, which we have restricted to have similar
magnitude by assuming $\iota_k|\nu  \sim  \operatorname{Bern}(\nu)$, for $k =
1,\ldots,10$, {a priori}. For PS, these parameters all have
posterior means less than 0.43, and many less than 0.25, which
indicates little borrowing for CTp curve shape. In contrast, these
parameters all exceed 0.82 for BV, and 0.92 for BF. Thus, for BF and BV
the GMC model borrows substantially from the supplemental information
in noncancerous tissue for CTp curve shape in cancerous tissue.

\begin{figure}

\includegraphics{840f05.eps}

\caption{The first column displays posterior mean CTp curves from the
conventional model analysis for cancerous (solid black) and
noncancerous regions (dotted grey), along with 95\% pointwise CrIs.
The second column displays the same results for the GMC model analysis.
The third column compares the posterior mean first derivative of the
CTp curve in cancerous tissue from the conventional model (solid black)
and the GMC prior model (dotted grey), along with 95\% pointwise
CrIs.}\label{PCTpResults}
\end{figure}

The first row in Figure~\ref{PCTpResults} indicates that both the
shape and intercept of the PS curve differ substantially by tissue type
and that the results of the conventional model analysis (first column)
and the GMC prior model analysis (second column) are virtually
indistinguishable. Consequently, the first derivative of the CTp curve
in cancerous tissue (third column) is estimated with similar precision
in either the conventional or GMC model analysis. In fact, the GMC
prior model results in 95\% pointwise credible intervals (CrIs) that
are 4\% wider on average than those of the conventional model analysis.
Regardless, the PS CTp curve in cancerous tissue has not stabilized
after 95 seconds, so longer acquisition durations for PS appear
necessary, which is consistent with the findings by \citet{ng13}.

The second row in Figure~\ref{PCTpResults} corresponds to BV. Here the
GMC prior model, relative to the conventional model, results in a
noncancerous CTp curve that is more similar for both the intercept and
the shape to that of the cancerous tissue CTp curve. Moreover, the GMC
model gains precision over the conventional model for the first
derivative estimate of the cancerous tissue CTp curve via borrowing
from the noncancerous tissue CTp curve shape. The average 95\%
pointwise CrIs for the GMC model are 18\% tighter than those of the
conventional model. The third row corresponds to BF and illustrates
that the posterior CTp curve estimates are similar for the two modeling
approaches. However, the first derivative of the cancerous tissue CTp
curve is estimated more precisely for the GMC model than the
conventional model, resulting in 27\% tighter 95\% pointwise CrIs on
average. For both BF and BV, we can infer a shorter acquisition
duration using the GMC prior model than using the conventional model,
although we did not impose precise stability criteria here.

\section{GMC priors in semiparametric survival analysis}\label{SHazReg}
We now describe a flexible semiparametric survival model and then apply
our GMC prior technology to the colorectal cancer clinical trials data
described in Section~\ref{SIntro}. For simplicity, assume each
observation consists of a possibly right-censored time $t \in(0,1]$, a~binary event indicator $c$ and a binary treatment indicator~$z$.

\subsection{Piecewise-exponential proportional hazards model}\label{SSPwExp}
A flexible survival model commonly favored by Bayesians is the
piecewise-exponential proportional hazards model, which is constructed
by partitioning the time axis into $K$ intervals $\tilde{\mathbf{t}} =
(0=\tilde{t}_0<\tilde{t}_1<\cdots<\tilde{t}_{K-1}<\tilde{t}_K=1)$ and
assuming the baseline hazard is constant in each interval [cf. \cite
{ibra01}, Section~3.1]. Under this model, the likelihood for an
observation $(t,c,z)$ is given by $h(t|z;\bolds{\gamma},\rho)^{c}\exp
\lbrace- \int_{0}^{t} h(s|z;\bolds{\gamma},\rho)\,ds  \rbrace$, where
\begin{equation}
\label{EPWE}
h(t|z;\bolds{\gamma},\rho) = \exp(\gamma_k + \rho z)
\qquad\mbox{for } t \in I_k = (\tilde{t}_{k-1},
\tilde{t}_k ], k=1,\ldots,K.
\end{equation}
Thus, $h(t|z;\bolds{\gamma},\rho)$ is assumed to be piecewise-constant,
where $\gamma_k$ denotes the log-hazard in the $k$th interval of the
time-axis partition $\tilde{\mathbf{t}}$ for treatment assignment $z=0$,
and $\rho$ denotes the log-hazard ratio for treatment assignment $z=1$
relative to $z=0$. Following \citet{ibra01}, we select $K$ using a DIC
comparison over a small set of partitions. To resist overfitting, we
specify a correlated prior process for $\pi^*(\bolds{\gamma})$. We focus
on the random-walk prior process for $\pi^*(\bolds\gamma)$ introduced by
\citet{fahr01}. Formally, we specify
\begin{eqnarray}
\pi^*(\gamma_1) &\equiv &
\mathcal{N}\bigl(\gamma_1|0,10^4\bigr),
\nonumber\\
\label{ERWPP}
\pi^*(\gamma
_k|\gamma_{k-1},\sigma_{\bolds{\gamma}}) & \equiv & \mathcal{N}
\bigl(\gamma_k|\gamma _{k-1},\sigma_{\bolds{\gamma}}^{2}
\bigr)\qquad \mbox{for } k=2,\ldots,K \quad\mbox{and}
\\
\nonumber
 \pi^*(\sigma_{\bolds{\gamma}})  &\equiv & \mathcal {U}(
\sigma_{\bolds{\gamma}}|0.01,100).
\end{eqnarray}
The prior process in (\ref{ERWPP}) smooths adjacent $\gamma_k$'s
toward each other by assuming their first differences are exchangeable.
We also assume $\rho$ is independent of $\bolds{\gamma}$ {a~priori},
and specify a vague $\mathcal{N}(0,10^4)$ prior. The model defined in
(\ref{EPWE}) is useful in the absence of supplemental data, and the
parameter space can be decomposed into separable subsets, $\bolds{\gamma}$
characterizes the baseline hazard and $\rho$ characterizes the
treatment effect.

\subsection{Application: Colorectal cancer trials}\label{SSColCan}
Extending \citet{hobb13}, we apply our GMC prior model structure to
supplement the inference on the progression-free survival (PFS) curve
among all three regimens in the trial reported by \citet{gold04} using
the historical information on the IFL regimen from the trial reported
by \citet{salt00}. We use the piecewise-exponential
proportional-hazards model to estimate the PFS curves for the two
trials from these data. During the first two years of follow-up, the
primary (Goldberg) data set contains 197 progression events among 211
persons treated with IFL, 190 events among 216 persons treated with
FOLFOX, and 189 events among 206 persons treated with IROX. For the
primary data, we have two binary treatment indicators for assignment to
the FOLFOX ($z_{F}$) and IROX ($z_{I}$) regimens, respectively, so
following (\ref{EPWE}) we model
\[
h(t|\mathbf{z};\bolds{\gamma},\bolds{\rho}) = \exp(\gamma_k +
\rho_{F} z_{F} + \rho_{I} z_{I})
\qquad\mbox{for } t \in(\tilde{t}_{k-1},\tilde{t}_{k}],
\]
where $k=1,\ldots,K$. The supplemental (Saltz) data contain 172
progression events during the first two years of follow-up among 224
persons treated with IFL. The model for the supplemental data is
defined analogously, though $z_{F} = z_{I} = 0$ for all the
supplemental observations. Therefore, the supplemental hazard model is
completely parametrized by $\bolds{\gamma}_0$, which, as required, is
analogous to $\bolds{\gamma}$.

We specify a GMC prior to flexibly borrow strength from the
supplemental information on the IFL regimen. Namely, we specify the
random-walk prior process defined by (\ref{ERWPP}) for $\pi^*(\bolds{\gamma}_0)$, and apply the GMC prior structure on $\bolds{\gamma}$. In
this\vspace*{1pt} setting, the prior specification follows as
$\gamma_1|\gamma_{0,1},\iota_1 \sim[\mathcal{N}(\gamma
_1|0,10^4) ]^{1-\iota_1}  [\mathcal{N}(\gamma
_1|\gamma_{0,1},\mathcal{R}_{\bolds{\gamma}}^{-1}) ]^{\iota
_1}$, $\gamma_k|\gamma_{k-1}, \gamma_{0,k},\iota_k$, $\sigma_{\bolds{\gamma}}
\sim[\mathcal{N}(\gamma_k| \gamma_{k-1},\break\sigma^{2}_{\bolds{\gamma}}) ]^{1-\iota_k} [\mathcal{N}(\gamma
_k| \gamma_{0,k},\mathcal{R}_{\bolds{\gamma}}^{-1}) ]^{\iota
_k}$, for $k=2,\ldots,K$, and $\iota_k|\nu_{\bolds{\gamma}} \sim
\operatorname{Bern}(\nu_{\bolds{\gamma}})$, for $k=1,\ldots,K$,
where $\mathcal{R}_{\bolds{\gamma}}$ is prespecified. Next, we place a
$\mathcal{B}(a_1,a_2)$ prior on $\nu_{\bolds{\gamma}}$, where $a_1$ and
$a_2$ are also prespecified. Last, we place vague $\mathcal
{U}(0.01,100)$ priors on each of the standard deviation parameters
(i.e., $\sigma_{\bolds{\gamma}}$ and $\sigma_{\bolds{\gamma}_0}$) and vague
$\mathcal{N}(0,10^4)$ priors on the treatment effect parameters (i.e.,
$\rho_{F}$ and $\rho_{I}$).

We transformed the timescale so that $t$ $\in$ $(0,1]$, then selected
$\mathcal{R}_{\bolds{\gamma}} = 10{,}000$ and specified a vague $\mathcal
{B}(0.10,0.90)$ prior for $\nu_{\bolds{\gamma}}$, which represents a
vague, yet skeptical prior opinion about the relevance of the
supplemental data. For estimation, we used 200{,}000 posterior samples
for estimation after 50{,}000 iterations of burn-in from two MCMC chains.
For comparison, we also fit conventional piecewise-exponential
proportional hazards models with the same time-axis partition and
random-walk prior process to the primary (Goldberg) data alone,
supplemental (Saltz) data alone, and the data set obtained by naively
pooling the two sources of information. To estimate these conventional
models, we used 20{,}000 posterior estimation draws after 2000 burn-in
draws from two MCMC chains, again the shorter chain length justified by
faster convergence and lower post-convergence autocorrelations.

\begin{figure}[b]

\includegraphics{840f06.eps}

\caption{The left panel displays the posterior mean PFS curve for the
IFL regimen along with 95\% pointwise CrIs from the conventional
piecewise-constant-hazard analysis of the primary (Goldberg) data alone
and the supplemental (Saltz) data alone. The middle panel displays the
same results from the GMC prior model analysis and the conventional
model analysis of the dataset obtained by pooling the two sources of
information. The third panel displays a posterior histogram for $\nu
_{\bolds{\gamma}}$, which controls borrowing for the baseline hazard among
trials. For reference, the prior assigned to $\nu_{\bolds{\gamma}}$ is
also displayed (grey line).}\label{PColCanPlots}
\end{figure}

We begin with a comparison of the estimated PFS curves for the IFL
regimen from the four analyses. The left panel in Figure~\ref{PColCanPlots} indicates that the estimated PFS curve from the
Goldberg data (solid, with shaded grey 95\% pointwise CrIs) overlaps
substantially with the PFS curve estimated from the Saltz data (dashed,
with dotted 95\% pointwise CrIs), suggesting the Saltz data provide
relevant information. The middle panel in Figure~\ref{PColCanPlots}
illustrates that the GMC analysis (solid, with shaded grey 95\%
pointwise CrIs) results in a PFS estimate nearly indistinguishable from
that of the conventional analysis of the pooled data set (dashed, with
dotted 95\% pointwise CrIs), though shifted very slightly toward the
primary information. The GMC prior model achieved a PFS curve estimate
with noticeably greater precision than that of the conventional
analysis that ignores the supplemental data without requiring bold {a priori} assumptions regarding the relevance of the Saltz data.
Finally, the rightmost panel in Figure~\ref{PColCanPlots} displays a
posterior histogram of $\nu_{\bolds{\gamma}}$ that has much of its mass
shifted toward one. In fact, the $\iota_k$'s had posterior means of at
least 0.87, with an average of 0.94, indicating that the GMC prior
model has learned that the supplemental FOLFOX data from the Saltz
trial are indeed relevant.

\begin{table}
\caption{Hazard-ratio estimates for the FOLFOX and IROX regimens
relative to the IFL regimen, and estimates of median days to disease
progression for each treatment regimen from the four analyses of the
Goldberg data and Saltz data}
\label{THRandMSTab}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccc@{}}
\hline
\multicolumn{1}{@{}l}{\textbf{Drug regimen}} & \textbf{IFL} & \textbf{FOLFOX} & \textbf{IROX} \\
\hline
\multicolumn{1}{@{}l}{\textit{Analysis}} & &\multicolumn{2}{c@{}}{\textit{Hazard
ratios (\textup{95}\% CrIs)}} \\
Goldberg alone & -- & 0.70 (0.57, 0.86) & 0.92 (0.75, 1.12) \\
Pooled analysis & -- & 0.66 (0.56, 0.79) & 0.87 (0.73, 1.04) \\
GMC prior analysis & -- & 0.66 (0.56, 0.79) & 0.87 (0.73, 1.04) \\[6pt]
& \multicolumn{3}{c@{}}{\textit{Median days to disease progression (\textup{95}\% CrIs)}} \\
Saltz alone & 200 (168, 236) & -- & -- \\
Goldberg alone & 205 (182, 227) & 262 (235, 293)& 217 (194, 242)\\
Pooled analysis & 200 (182, 217) & 263 (245, 286) & 221 (201, 240)\\
GMC prior analysis & 200 (184, 216) & 265 (245, 286) & 221 (203, 240) \\
\hline
\end{tabular*}
\end{table}

Turning to the comparative effectiveness of three regimens, Table~\ref{THRandMSTab} shows that the hazard ratios (95\% CrIs) from the GMC
prior model analysis for FOLFOX and IROX versus IFL are 0.66
(0.56,0.79) and 0.87 (0.73,1.04), respectively. The
hazard ratios from the conventional analysis of the
Goldberg data alone are slightly larger with notably wider CrI widths,
0.70 (0.57,0.86) for FOLFOX and 0.92 (0.75,1.12) for IROX. By contrast,
the hazard-ratio estimates and CrI widths from the conventional
analysis of the pooled data set are indistinguishable from the GMC
prior model analysis. Table~\ref{THRandMSTab} also contains the
posterior estimates from each analysis of median PFS for the IFL,
FOLFOX and IROX regimens. The conventional analysis of the Saltz data
alone estimates median PFS in the IFL regimen to be 200 (168, 236)
days, broadly similar to the estimate of 205 (182, 227) days provided
by the conventional analysis of the Goldberg data alone. The estimates
of median days to disease progression for each regimen from the pooled
analysis of these data are also nearly identical to the estimates from
the GMC prior analysis, namely, 200, 265 and 221 days for the IFL,
FOLFOX and IROX regimens, respectively. The estimates from the GMC
prior model are consistent with the results reported by \citet{gold04},
who found FOLFOX to be the superior regimen, and significantly better
than IFL. In addition, the GMC prior analysis yields stronger evidence
that IROX may be better than IFL for PFS, though the difference remains
statistically insignificant.

\section{Discussion and future work}\label{SDisc}
Our proposed methods for prior specification in functional and survival
data models with penalized splines facilitate data-dependent borrowing
that is robust to biased estimation of primary effects when conflict
among information sources occurs. The simulation study illustrates the
beneficial flexible borrowing properties the proposed methods offer.
The application in perfusion CT illustrates potential gains in CTp
curve estimation from using supplemental data collected concurrently.
By contrast, the colorectal cancer application illustrates the use of
these methods for semiparametric survival modeling to flexibly borrow
from supplemental data on a control therapy collected in a previous
trial. A hierarchical model with the proposed structure allows the
degree of borrowing to be estimated differentially for each feature
(e.g., CTp curve intercept versus shape). The amount of strength being
borrowed between sources reflects the evidence of commensurability for
that feature. When substantial evidence indicates that sources differ
for a feature, the proposed method will learn to effectively ignore the
supplemental data for that feature, yet possibly still borrow strength
for another feature. For minor discordance or concordance, the proposed
method also facilitates partial (rather than full) pooling of
information from the supplemental source, hence a modest yet
justifiable gain in efficiency. Our general modeling strategy enables
data-driven estimation of heterogeneity among information sources for
complex functional relationships, and thus provides a sensible and
justifiable synthesis of clinical information.

In future work, we plan to extend the commensurate prior approach to
settings that have \textit{multiple} supplemental sources of information.
This setting substantially complicates the construction of a
hierarchical model that facilitates flexible borrowing for each
supplemental source. Ideally, the method will facilitate data-dependent
differential borrowing from various sources, learning from which to
borrow and which to ignore as primary data accumulate. We also hope to
develop concise, interpretable summaries that quantify the amount of
strength being borrowed from each information source, thereby allowing
the use of these models in an adaptive trial [see \cite{hobb13}]. We
will also consider using posterior summaries of $\nu$ in (\ref{EGenMixCommPr}) to determine whether the curves are commensurate
among information sources, for example, by assessing $\operatorname{Pr}(\nu> 0.80|\mathbf
{D},\mathbf{D}_0) > 0.90$. Furthermore, we are currently studying the
properties of the piecewise-exponential model used in the colorectal
cancer application, which relies on a simple, yet flexible
piecewise-constant assumption for the baseline hazard. We are
developing extensions that use a piecewise-linear model for the
baseline hazard function, relax the proportional hazards assumption,
and allow functional covariate effects with shape constraints (e.g.,
monotonicity).

\section*{Acknowledgments}
We thank the Associate Editor and referees for their well thought-out
and detailed comments during revision that substantially improved the
clarity of presentation in this manuscript and the model we use in the
CTp analysis.

redpajama_set_name: RedPajamaArXiv


text:
El falangero de cola anillada (Pseudocheirus peregrinus, griego para "mano falsa" y latín para "peregrino") es una especie de marsupial diprotodonto de la familia Pseudocheiridae propia de Australia. Es herbívoro y de hábitos nocturnos. Pesa entre 550 g y 1.100 g. Tiene pelo gris con partes blancas detrás de los ojos y generalmente un estómago de color crema. Tiene una larga cola prensil con una distintiva punta blanca de aproximadamente un 25% del tamaño total de la cola. Sus pies traseros son sindáctilos, lo que le facilita subir los árboles.

Subespecies
Se reconocen las siguientes subespecies:
 Pseudocheirus peregrinus peregrinus
 Pseudocheirus peregrinus occidentalis
 Pseudocheirus peregrinus pulcher
 Pseudocheirus peregrinus convolutor

Referencias

Enlaces externos

Marsupiales de Australia
Pseudocheirus
Taxones descritos por Pieter Boddaert
Animales descritos en 1785

redpajama_set_name: RedPajamaWikipedia


text:
Does Tommy Togiai's bench press prowess predict NFL success for the former Ohio State defensive tackle?
Updated: Apr. 26, 2021, 2:45 p.m. | Published: Apr. 26, 2021, 5:30 a.m.
Ohio State defensive tackle Tommy Togiai augmented his Pro Day bench press performance with strong efforts in other events.AP
By Nathan Baird, cleveland.com
COLUMBUS, Ohio — Tommy Togiai leaned into his signature move at Ohio State football's Pro Day, making his bench press one of the showcase events of the pre-draft process.
That performance was never going to tell NFL teams something they did not know about the defensive tackle.
"Everyone thinks of me as just strong and big and that's it," Togiai said. "But I've also got the speed to along with it. I can move. I just kind of wanted to put that on showcase, too."
Togiai fell short of his goal of 50 bench press reps of 225 pounds to break the previous NFL Scouting Combine record. He only made it to 40, tying with Pittsburgh's Jaylen Twyman for the best of the scattered Pro Day performances in 2021.
That effort raised over $3,000 for Ohio State's LiFEsports program, providing opportunities for youth sports and rec programs. That also was the most crucial outcome of Togiai's bench press attempt. Historically, the event has not been a reliable indicator of NFL success.
• The NFL Combine bench press record Togiai hoped to beat belongs to former Oregon State defensive tackle Stephen Paea. He hoisted 49 reps in 2011, helping convince the Chicago Bears to draft him in the second round. He also played for Washington, the Browns and the Dallas Cowboys in his seven-year career, primarily as a backup.
• Ohio State defensive lineman Mike Kudla, who put up 45 reps in 2006, never played in the NFL.
• Norweigan defensive tackle Leif Larsen played 16 games over two seasons with the Buffalo Bills after his 45 reps at the combine.
• Arkansas guard Mitch Petrus played in the New York Giants' Super Bowl XLVI victory during his three-year career, mostly as a backup.
Ohio State defensive tackle Tommy Togiai achieved 40 bench presses during the Buckeyes' Pro Day on March 30.AP
Of the other players who rank in the top 10 combine bench press performances, only two had truly memorable NFL careers. Both, like Togai, played nose tackle.
Florida State's Brodrick Bunkley played 130 games over eight seasons. Memphis' Dontari Poe went a step further, making two Pro Bowls and starting six playoff games with the Kansas City Chiefs and Atlanta Falcons over his nine seasons.
That is the sort of career to which Togiai now aspires. If he achieves it, some of his other Pro Day numbers will likely lead the way.
Togiai ran the 40-yard dash in 4.97 seconds. That ranked 13th among players who weighed 290 or more, Only six of them reached even 30 bench press reps.
That 32-inch vertical jump? Eight among players in his weight range or higher — and again, none of those players topped 33 bench reps.
Togiai's 7.20 seconds in the 3-cone drill ranked third among people pushing 300 pounds. His 4.49-second shuttle time ranked sixth.
Perhaps most importantly, Togiai converted his athleticism into production. According to Pro Football Focus, he graded among the best defensive tackles in the nation in pass rush win rate, run-stop rate and overall run stop win grade.
"That's one of the things that helped me change my mindset — I saw Tommy do that last year," defensive end Zach Harrison said. "He had goals and he wanted to reach them. You could see at every workout he went hard in sprints. Every practice he would give 100 %, and that's something I tried to incorporate."
Togiai has other questions to answer as a pro. He does not have special height and size by NFL standards. He only spent one season as a starter — and a pandemic-shortened one at that.
The athleticism, though, shows up. Combined with the raw strength he has displayed since high school, Togiai may succeed where other bench press standouts fell short.
Get Ohio State Sugar Bowl champs & CFP gear: Check out shirts, hats and more merchandise commemorating Ohio State's Sugar Bowl win over Clemson, as well as gear on the Buckeyes advancing to the College Football Playoff national championship game.
Buy Buckeyes gear: Fanatics, Nike, Amazon, Lids
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redpajama_set_name: RedPajamaCommonCrawl


It's a good thing that the U.S. Olympic team flag bearer did not dip Old Glory as it passed the reviewing stand during the opening ceremonies of the Winter Olympics.
More than this directive, our banner represents the best the USA has to offer: in business, in battle, and on the sports field.
Waving the Stars and Stripes is never out of fashion. During the Olympics, I think it is always appropriate to salute and cheer this sacred symbol.
Some dreamy-eyed liberals, still in a swoon celebrating the end of the Cold War, think we should have dipped the flag and, worse, that we should even suppress our patriotic pride during international sporting events. Through the Olympics, they say, our allegiance to something called a "new world order" ought to replace old-fashioned love of country.
Want to know just how goofy that sentiment is? Ask the newly liberated Baltic athletes to repress their ethnic pride during this first Olympic Games in a half century as a free people. Or, conversely, ask the Tibetans how they appreciate being part of the Chinese Communist delegation and not having their own historic flag.
Obviously squashing nationalistic pride would put the Olympics — and any other international sporting meet — out of business.
Personally, I hope we win the gold at every event for something more than simple athletic prowess. I want my kids to chant "We're Number One" and feel as if they are an important part of the team just by waving the Stars and Stripes.


Stiff designer carrier bags, plush carpets, floor-to-ceiling mirrors and service with a smile: how do you replicate the luxury experience in an app? Nothing one's iPhone can serve up has yet come close to emulating the hallowed temples and hushed reverence of a Bond Street flagship. Yet, in spite of this, customers increasingly want to worship remotely.


text:
George Woods ist der Name folgender Personen:
* George Woods (Leichtathlet) (1943–2022), US-amerikanischer Leichtathlet
 George A. Woods (1926–1988), US-amerikanischer Literaturkritiker und Kinderbuchautor
 George Austin Woods (1828–1905), britischer Offizier, Premier des Königreichs Fiji
 George D. Woods (1901–1982), US-amerikanischer Ökonom
 George Lemuel Woods (1832–1890), US-amerikanischer Politiker

Siehe auch:
 George Wood


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text:
Óscar De La Hoya warns that Munguía will not have an easy rival
Former world champion and promoter Óscar de la Hoya assured this Saturday that Mexican Jaime Munguía, super welterweight champion of the World Boxing Organization (WBO), must be careful against his rival, Ghanaian Patrick Allotey.
«The rival is an African, so from the beginning he is strong and difficult, I faced several boxers from that part of the world and I really suffered because most of them have a strong jaw and ample resistance, so Munguía knows that he will not have it simple "De la Hoya told Efe.
Jaime Munguía (33-0-0, 26 ko's) will have his fifth defense of the WBO super welterweight championship on September 14 in Carson, California against Allotey who has accumulated 40 wins by 3 losses and has not lost since 2015.
De La Hoya said he hopes to see an improvement in Munguía's boxing quality by having former Mexican champion Erick 'Terrible' Morales support him in his corner.
You can also read: VIDEO: The best fights during "Black Friday" shopping
«The good thing is that Jaime is a disciplined fighter, he trains hard, I came to see him in Tijuana and his progress surprised me. Now with Erick Morales he wants to polish his blows and his strategy, so he has everything in place to show that he will stay as world champion for a long time.
Munguía, 22, will fight for the third time so far in 2019. In January he beat Takeshi Inoue and in April he beat Dennis Hogan. After fighting Allotey, he will consider whether to drop the 154-pound category, Oscar de la Hoya said.
"We will see how he feels after the weigh-in, on September 13, that is where you notice if he can continue to maintain his weight. Munguía is disciplined, he listens to the doctors, he listens to diets, but it all depends on the development of the body, because he is young and has yet to grow, so he may move up a category ».
You can also read: Cotto promises to "finish" Foreman
De la Hoya trusted that Munguía can fill the arena in Carson, California, on a special date for Mexicans who celebrate independence and traditionally attend boxing to see stars from this country.
«Since Julio César Chávez there has been a tradition. Last year it was Saúl 'Canelo' Álvarez, but now it's Munguía's turn. They say he is defensive, however I think the opposite, that he is obedient and respects the times of a fight, he does not get carried away, so for those who like fine boxing they will have a good display ».
Óscar de la Hoya, promoter of Golden Boy Productions, manages the career of Saúl 'Canelo' Álvarez, who will not fight on the second Saturday in September to commemorate the independence of Mexico as he does not have a suitable rival for it.
You can also read: Boxing receives donation of sports equipment
The promoter assured that a date will be given shortly for Álvarez's return to the ring.
"It was a shame not to be able to carry out a grand function in Las Vegas this year as Mexicans are used to seeing, but it was disrespectful if we put a lesser rival against Álvarez. We work to present better fights to the public and take care of quality, "he concluded. EFE
Medvedev, Paris Masters 1000 finalist for the first time
Kluber returns strong as Yankees displace Blue Jays for AL wild card


It is on the north bank of the River Dee, which here marks the parish and country boundaries (Peterculter ABD to the north, Maryculter KCD to the south). Close to St Peter's Church.
Peter's Heugh 1843, OS 6 inch first edn.


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text:
Jewish Organizations to Trump Administration: Families Belong Together!
"As Jews, we understand the plight of being an immigrant fleeing violence and oppression. We believe that the United States is a nation of immigrants and how we treat the stranger reflects on the moral values and ideals of this nation."
Alarmed by the U.S. government's "zero tolerance" policy of separating children from their migrant parents when they cross the border, the Reform Movement joined with 26 national Jewish organizations to send a letter to the Administration expressing our strong opposition to the cruel practice. We are urging the Administration to immediately rescind the "zero tolerance" policy and uphold the values of family unity and justice on which our nation was built.
Reports indicate that over 1,000 children have already been separated from their parents since the policy was implemented. Mounting evidence suggests that many children have been threatened, treated poorly, and held in dangerous conditions. In response to such cruel policy, Jews across our community and throughout the country are joining together to voice our Jewish values of caring for the stranger in our midst, which requires that we immediately end family separation.
How we treat the stranger reflects on the moral values and ideals of this nation.
Read our letter:
Dear Attorney General Sessions and Secretary Nielsen,
On behalf of the 26 undersigned national Jewish organizations and institutions, we write to express our strong opposition to the recently expanded "zero-tolerance" policy that includes separating children from their migrant parents when they cross the border. This policy undermines the values of our nation and jeopardizes the safety and well-being of thousands of people.
Many of these migrant families are seeking asylum in the United States to escape violence in Central America. Taking children away from their families is unconscionable. Such practices inflict unnecessary trauma on parents and children, many of whom have already suffered traumatic experiences. This added trauma negatively impacts physical and mental health, including increasing the risk of early death.
Separating families is a cruel punishment for children and families simply seeking a better life and exacerbates existing challenges in our immigration system. It adds to the backlog of deportation cases and legal challenges in federal courts, places thousands more immigrants in detention facilities and shelters, endangers the lives of more children, and instills additional fear in people seeking safety in our country. In addition, those seeking asylum or other legal protection face numerous obstacles to making a claim, especially from detention. Separating family members at the border would force families into two or more immigration cases instead of a single case for each family, harming their ability to present a successful case.
Our Jewish faith demands of us concern for the stranger in our midst. Our own people's history as "strangers" reminds us of the many struggles faced by immigrants today and compels our commitment to an immigration system in this country that is compassionate and just. We urge you to immediately rescind the "zero tolerance" policy and uphold the values of family unity and justice on which our nation was built.
American Conference of Cantors
B'nai B'rith International
Bend the Arc Jewish Action
Hadassah, The Women's Zionist Organization of America, Inc.
HIAS
Jewish Council for Public Affairs
Jewish Labor Committee
Jewish Women International
National Association of Jewish Legislators
National Council of Jewish Women
Network of Jewish Human Service Agencies
NFTY – The Reform Jewish Youth Movement
Reconstructing Judaism
Reconstructionist Rabbinical Association
The Workmen's Circle
United Synagogue of Conservative Judaism
Uri L'Tzedek, The Orthodox Social Justice Movement
Together the rabbis of the Conference combine our voices to unify, elevate, raise up and exalt the world in which we live.
Rabbi Steve Fox
As we read our ancient story of oppression in Parashat Bo, we commemorate Dr. Martin Luther King, Jr. who fought against oppression in this very lifetime. @RavShmuly brings our sacred text alive through social justice, echoing our imperative to work toward a world MLK envisioned. pic.twitter.com/i5uG…


SEC is examining whether the rules for the fees and their disclosure are being followed and what funds are paying intermediaries for.
The 12b-1 fee on a fund can range from 0.25% to 1%.
The Securities and Exchange Commission is examining fees paid to intermediaries by mutual funds "to discern" if the agency's 12b-1 rule is "being adhered to" and "what funds are paying intermediaries for," Paula Drake, chief counsel in the agency's Office of Compliance Inspections and Examinations, said Thursday.
Distribution fees, or so-called 12b-1 fees, are paid by a mutual fund out of fund assets to cover distribution expenses and sometimes shareholder service expenses. The SEC's 12b-1 fee rule, which was promulgated in 1980, permits a fund to pay distribution fees out of fund assets only if the fund has adopted a 12b-1 plan authorizing their payment.
"The plan has to be written and adopted by the board and it has to be disclosed to the shareholders," Drake said during her Thursday remarks.
Distribution fees include fees paid for marketing and selling fund shares, such as compensating brokers and other financial professionals who sell fund shares, and paying for advertising, the printing and mailing of prospectuses to new investors, and the printing and mailing of sales literature.
The 12b-1 fee is considered an operational expense and, as such, is included in a fund's expense ratio, and is generally between 0.25% and 1% (the maximum allowed) of a fund's net assets.
Wyderko then asked Drake how precise the SEC expects fund boards to be in allocating fees to shareholder servicing or distribution.
Fund Democracy and other groups like the Consumer Federation of America told White in a March 12 letter to resurrect the plan that the agency put forth early in the Obama administration for 12b-1 fee reform.
— Check out SEC Urged to Reform 12b-1 Fees, Block Risky ETFs on ThinkAdvisor.


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Alexander SokoloskyAttorney
Alex Sokolosky is an associate in the firm's Tulsa office and a member of the Banking & Financial Institutions, Bankruptcy & Creditor's Rights, and Litigation & Trial Practice Groups. He represents businesses and individuals in a wide range of commercial disputes in both federal and state court with a focus on bankruptcy, breach of contract, collection, and foreclosure matters. Alex also serves on the board of directors of First Bank of Owasso, where he worked prior to attending law school.
Alex previously served as a law clerk to the Honorable Ronald A. White and the Honorable John F. Heil, III, of the United States District Court for the Eastern District of Oklahoma. While in law school, he served as a student clerk to the Honorable Yvonne Kauger of the Supreme Court of Oklahoma.
Alex graduated with honors from the University of Oklahoma College of Law and was selected into the Order of the Coif. He also received seven American Jurisprudence awards and was on the editorial board of the Oklahoma Law Review. In addition, Alex received his Bachelor of Arts from the University of Oklahoma.
Bankruptcy & Creditor's Rights
With Honors
Order of the Coif
Order of the Scribes
Oklahoma Law Review
Comfort Scholarship
Frank and Edna Asper Elkouri Scholarship
Nathalie Pierrepont Scholarship
B.A., Philosophy, 2014
Oklahoma, 2018
U.S. District Court Northern District of Oklahoma, 2018
U.S. District Court Eastern District of Oklahoma, 2018
U.S. District Court Western District of Oklahoma, 2018
U.S. Bankruptcy Court Northern District of Oklahoma, 2018
U.S. Bankruptcy Court Eastern District of Oklahoma, 2018
U.S. Bankruptcy Court Western District of Oklahoma, 2018
Federal Bar Association (Northern & Eastern Oklahoma Chapters), Board of Directors Member
American Bankruptcy Institute, Member
American Bar Association, Member
Oklahoma Bar Association, Member
Tulsa County Bar Association, Member
Listed in Best Lawyers in Bankruptcy and Creditor Debtor Rights / Insolvency, Reorganization Law, and Litigation – Bankruptcy (2021)
NextGen Under 30 (2020)
*Crowe & Dunlevy has no input in the rating methodologies used by Best Lawyers.
"Indian Child Welfare Act Held Unconstitutional," The Oklahoman, March 20, 2019
Alexander Sokolosky, Rebuilding Trust? The Sand Creek Massacre and the Federal-Tribal Trust Relationship in Flute v. United States, 70 Okla. L. Rev. 985 (2018)
"In Rem Jurisdiction and Tribal Sovereign Immunity - Upper Skagit Indian Tribe v. Lundgren," Sovereignty Symposium XXXI - Infrastructure (2018)
alex.sokolosky@crowedunlevy.com
222 North Detroit Avenue
Alexander Sokolosky on LinkedIn
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Insolvency worksheet form fill online printable fillable blank sample of completed 982 for insolvency. Irs form 982 insolvency worksheet davezan abitlikethis. Cancellation of debt insolvency worksheet davezan canceled stacie 39 s more tax tips. Step by instructions on how to fill out insolvency worksheet worksheet. Insolvency worksheet form fill online printable fillable blank worksheet.


Sadly, the end of a pregnancy can come suddenly and unexpected, or can be a planned scenario. Either way, I am here to support you. During loss, a mother and family may be overwhelmed with what is happening, how to deal with emotions as well make decisions that are not often thought of. Subsequent pregnancies may be more stressful, anxious and worrisome. If you are a current client and you find yourself in this situation, I am a phone call away. If you are not a current client I can offer in person support at $25/ hour (Min $100 ~Max $300) If you'd like to learn more about my experience with pregnancy loss please refer to my blog or check out our local resources below.
IN OUR HEARTS ~ local monthly gathering for women who have miscarried, terminated, had a stillbirth or neonatal death.


Modern genomics labs are looking for high-throughput, quality-data workstations to perform PCR reactions at a reduced cost. PCR reaction setup is a tedious and time consuming process as it requires mixing multiple reagents in specific ratios to achieve the proper master mix for amplification. Moreover, the exact volumes of the reagents and buffers depend upon the total number of samples to be amplified and the desired final concentration of sample DNA. Automation of such procedures eliminates both human error and contamination problems generally associated with manual process. To provide an automated solution to such a cumbersome protocol, Aurora Biomed's VERSA Mini PCR Setup Workstation was developed for the preparation of reagents and samples used in an amplification protocol in 96-well format.


In case you missed the two or three 12-second reports on the news: Shaul Mofaz, deputy prime minister of Israel, said Friday that an attack on Iran's nuclear sites is "unavoidable" if Tehran refuses to halt it's alleged nuclear weapons program.
He didn't say "alleged," however. That's my word, a proper modifier, but not something he or the Bushies would use, any more than they talked about "alleged" WMDs in Iraq.
Mofaz failed to specify how Iran can now halt a program that all U.S. and European intelligence agencies and top scientists agree was halted in 2003.
The Israel minister also said that the attack will occur if international diplomacy fails to make Iran stop that already stopped weapons program. But, of course, Israel and the Bush administration flatly refuse to engage in "international diplomacy" with Iran, although Iran has several times offered to negotiate without restrictions on topics.
In case you haven't quite got it: Israel, with timing almost certainly coordinated with the American White House, is pushing for an attack on Iran. If Israel attacks, we'll soon take over the major portion of the killing, of course.
Isn't it interesting to live in a very big and powerful country that allows a much smaller country to call the shots on key portions of its foreign policy?
OK, hard to tell who's manipulating whom sometimes, but that this country's politicians are frequently the intimidated stand-ins for Israel's right-wing government and its big-buck U.S. backers is not in question.


We are pleased to announce that the Benchmade 535 Bugout is the KnifeNews Dealers' Choice for Best New Manual Folder 2017. Votes cast by dealers indicated that other great new blades in this category included the Fox Knives TUR, Kizer Feist, Spyderco Para 3, and Zero Tolerance 0920.
The Bugout is also a tribute to the Axis Lock itself. Benchmade's signature locking mechanism doesn't need much real estate to operate. Even with its small footprint, it brings the muscle, able to withstand the rigors and abuse of outdoors cutting. It's the perfect lightweight folder lock. The Bugout platform showcases one of the knife world's favorite locks at the top of its game.
Visit KnifeNews.com tomorrow to find out the winner of the KnifeNews Dealers' Choice Award for Most Innovative New Knife 2017 (See the nominees below).
Cold Steel Golden Eye: Cold Steel aimed to impress with the HTR opening mechanism, which debuted on the Golden Eye and combines a thumb ramp and opening hole.
CRKT Daktyl: Tom Hitchock's Daktyl design is a double shot of innovation: the Hole in One and Slide Lock mechanisms.
ZT 0055: Gustavo C. Cecchini's SLT hidden flipper tab saw its first-ever production iteration with a knife inspired by the GTC Airborne.


U n s c r i p t e d fabulous. As our clients have grown up in the business, we've grown up with them. That's one of the wonder- ful things about this company and how the people here work.
Keith Hodne (KH): As soon as we heard that this was coming through the door, all of us were very excited. It was the biggest trial of the century and we all remembered it. I was a little bit younger at the time, and I thought that I knew everything about the trial. But then when you start to get into the meat and potatoes of it, there was this whole world that we really didn't know about.
We kept hearing about it growing from three hours to five hours, to a seven-and-a-half-hour masterpiece. We work on super cool stuff all day, every day, but when you know before you even touch something that it's going to be something really successful and that it could be a bit of a le popping or crackling, just a hair of it there to give you some of that nostalgic feeling that brings you back to when you saw it originally. It's kind of the equivalent of when you put a record on and you hear that vinyl, it takes you back to a different place. So what we wanted to do was keep as much of that content there as possible while still scaling it back so that it fits with all our modern elements and nothing stands out as being strange or odd or wrong.