text:
\section{Introduction and motivation}

This work was conceived and carried out in the context of a task to reduce
respondent burden in a mental health study. We were interested in a range of outcome
variables, and our analytical goal was fitting regression models to explain the mental health outcomes.
The instrument collects demographic explanatory variables,
as well as scores from the Big Five Inventory \citep{john:sriv:1999:big5},
a commonly used set of five psychological traits that are often found to be correlated
with behaviors and outcomes. We expected that multiple matrix sampling would allow
us to reduce the instrument length from over an hour to about 20--25 minutes.
A key component of sampling design, sample size determination,
will be based on a linear regression power analysis.
However, complexities of regression analysis with missing data required custom
derivations of power analyses, which is what this technical paper addresses.

\section{Regression setting}

Consider a regression analysis problem where an outcome $y$ is predicted by a set of
explanatory variables $x_1, \ldots, x_p$:

\begin{equation}
    y = \beta_0 + \beta_1 x_1 + \ldots + \beta_p x_p + \varepsilon
    \label{eq:regress}
\end{equation}

In the simplest possible case of no missing data and homoskedastic normal errors $\mathbb{V}[\epsilon_i]=\sigma^2 \, \forall i$,
the maximum likelihood estimates are the OLS estimates

\begin{gather}
    \hat\beta_{\rm OLS} = (X'X)^{-1} X'Y;
    \quad \mathbb{V}[ \hat\beta_{\rm OLS} ] = \sigma^2 (X'X)^{-1};
    \notag \\
    v[ \hat\beta_{\rm OLS} ] = s^2 (X'X)^{-1};
    \quad s^2 = \frac1n \sum_{i=1}^n (y_i - x_i'\hat\beta)^2
    \label{eq:ols}
\end{gather}

Inference on regression coefficients is based on normality of coefficient estimates,
$\hat\beta \sim N(\beta,\sigma^2(X'X)^{-1})$.

An unbiased estimate of $s^2$ can be obtained by changing the denominator (degrees of freedom)
from $n$ to $n-p$. The OLS estimates hold desirable properties in more general settings,
e.g., by dropping the normality requirement. When regressors $X$ are stochastic, the OLS estimates
and their variance estimates given above only have an asymptotic justification, and require
independence of regressors $X$ and errors $\varepsilon$. When the basic assumptions are violated,
sandwich-type or resampling variance estimates need to be used.

\subsection{Power analysis and sample size determination in regression setting}
\label{subsec:reg:power}

Power analysis and sample size determination are statistical tasks of addressing,
quantifying and controling type I error. In a typical power analysis problem,
a null hypothesis, $H_0: \theta \in \Theta_0$, and an alternative, $H_1: \theta \in \Theta_1$,
are formulated; a test statistic $t(X)$ is selected, for which a critical region of level $\alpha$ is specified.
E.g., assuming that high values of the test statistic indicate disagreement between the data
and the null, as is typical with $\chi^2$ or $F$-statistic tests common in regression models,
the rejection region would have the form $T_\alpha = [c,+\infty)$ so that
$\rm{Prob} [t(X) > c] \le \alpha$ when the true value of the parameter $\theta \in \Theta_0$.
Finally, power analysis addresses the issue of Type II error, i.e., $\rm{Prob} [t(X)\le c]$ under
the alternative $\theta \in \Theta_1$. While the null hypothesis typically represents
a simple hypothesis $\theta=\theta_0$ or a subset of reasonably small dimension, the alternative
is necessarily complex. Hence researchers often formulate a measure of \textit{effect size}
$\delta$ and consider power analysis for parameter values under the alternative that are
at least $\delta$ away from the specific value $\theta_0$ or the subset $\Theta_0$ in an appropriate metric.

As the power to reject the null typically grows with the sample size $n$, the task
of sample size determination is to find the value $n_\beta$ that guarantees a given level
of power $1-\beta$. While the size of the test is often taken to be $\alpha=5\%$,
the traditional type II error rate is $\beta=20\%$ leading to $1-\beta=80\%$ power.

While testing the location parameter of two populations, for example, the natural hypotheses
might be $H_0: \mu_1 = \mu_2$ vs. the (two-sided) alternative $H_1: |\mu_1 - \mu_2| \ge \delta$,
with relatively straightforward testing based on the Student $t$ distribution, linear regression
models feature a variety of statistics that may be subject to testing and power analysis:
\begin{enumerate}
    \item Test of overall fit: $H_0: \mathbf{\beta}=0$.
    \begin{enumerate}
        \item A version of this hypothesis can be formulated as $H_0: R^2=0$. Depending
            on how easy or difficult it is to conduct inference on parameter estimates
            or regression sums of squares, one or the other may be preferred in applications.
    \end{enumerate}
    \item An increase of overall fit: $H_0: R^2 \le R_0^2$ vs. $H_1: R^2 \ge R_0^2 + \delta$.
    \item Specific regression coefficients: the coefficient of the $j$-th explanatory variable
        is zero, $H_0: \beta_j = 0$ vs. $H_1: | \beta_j | \ge \delta$.
    \item Linear hypothesis $H_0: R\beta = r_0$, which covers cases like:
    \begin{enumerate}
        \item Equality of two regression coefficient for the $j$-th and the $k$-th explanatory variable,
        $H_0: \beta_j - \beta_k = 0$, so $R = (0, \ldots, 1, \ldots, -1), r=0$
        \item No impact of a set of variables $j_1, j_2, \ldots$: $H_0: \beta_{j_1}=0, \beta_{j_2}=0, \ldots$,
        so that $R$ is a subset of rows of a unit matrix, and $r$ is a zero vector of conforming dimension.
    \end{enumerate}
    \item Tests on error variance $\sigma^2$, e.g. $H_0: \sigma_2 \le \sigma^2_0$ vs. $H_1: \sigma^2 \ge \sigma_0^2 + \delta$.
\end{enumerate}

\section{Multiple matrix sampling}

\textit{Multiple matrix sampling} of a survey questionnaire consists
of administering only a specific subset of items to a given respondent, out of all items
this respondent is potentially eligible to be asked. The name stems from representation of the data
with respondents as rows, and items as columns, so that matrix sampling concerns selecting
specific entries in the matrix to be administered, rather than the full row as is typically done.
The focus of the technique is on selecting items out of all the relevant ones that the respondent
could be asked, with the potential skip  patterns already taken into account.
Similar or equivalent techniques are also known as \textit{partitioned designs}
and \textit{questionnaire splitting}. The method originated in educational testing
\citep{shoemaker:1972}, where it was first used to select items from a large pool
of available ones. The educational testing companies have identified the need to implement
multiple matrix sampling methods to protect the integrity of their data products, so that
the students taking a standardized test are not able to get
trained on a small subset of items known to be administered on standardized tests, thus
biasing the estimates of achievement.

Given the relatively esoteric nature of the method, the existing publications
have addressed some specific niche problems in matrix sampling (and the current paper
is no exception).
\citet{gonzalez:eltinge:2007} provided a review of matrix sampling
and applications in Consumer Expenditure Quarterly Survey.
\citet{chipperfield:steel:2009} put the problem of matrix sampling into a cost
optimization framework, where proper subsets of $K$ items can be administered on
up to $2^K-1$ forms at specific cost per item rates. They demonstrated that
with two items (or groups of items),
the split questionnaire best linear unbiased estimator (BLUE; also related to the GLS
estimator) provides modest efficiency gains over a design in which all items
are administered at once, and over a two-phase design in which all items are administered
to a fraction of respondents, and one subset of items is administered to the remainder of respondents.
\citet{merkouris:2010} extended their work to provide simplified composite estimation
using the estimates based on the form-specific subsamples, where compositing is based
on the second-order probabilities of selection and the way they are utilized
in estimating the variance of the Horvitz-Thompson estimator.
\citet{eltinge:2013} discussed connections to and relations with multiple frame
and multiple systems estimation methods (e.g., integration of survey and administrative
data, where administrative data may fill some of the survey items when available).
We add to this literature by providing the asymptotic variance-covariance matrix
of the coefficient estimates under matrix sampling of regressors, assuming
that the outcome is always collected. We also discuss implications for power analysis
and sample size determination.

\subsection{A simple example}

Consider the following matrix sampling design, in which
the outcome $y$ is collected on every form, while the explanatory variables
differ between forms.

\begin{table}[!h]

\centering

\caption{Three questionnaire forms for data collection: Design 1. \label{tab:forms3:design1}}

\begin{tabular}{l|ccc|c}
    Form & $X_1$ & $X_2$ & $X_3$ & $n$ \\
    \hline
    1   & + & & & $n_1$ \\
    2   & & + & & $n_2$ \\
    3   & & & + & $n_3$ \\
\end{tabular}

\end{table}

With this design, summaries (means, totals) of all the variables
($x_1,x_2,x_3,y$) can be obtained, and the bivariate relations
between each of the regressors and the outcome $y$ can be analyzed.
However, estimation of a multiple regression model requires
estimability of all of the entries of the $(X'X)$ matrix,
which this specific matrix sampling design does not provide.

To conduct regression analysis, we need to observe the cross-entries
of the $X'X$ matrix, which necessitates the following matrix sampling design.

\begin{table}[!h]

\centering

\caption{Three questionnaire forms for data collection: Design 2. \label{tab:forms3:design2}}

\begin{tabular}{l|ccc|c}
    Form & $X_1$ & $X_2$ & $X_3$ & $n$ \\
    \hline
    1   & & + & + & $n_1$ \\
    2   & + & & + & $n_2$ \\
    3   & + & + & & $n_3$ \\
\end{tabular}

\end{table}

\subsection{Parameter estimation under matrix sampling}

Since the components of $X'X$ and/or $X'y$ necessary to obtain the OLS regression
estimates may not be jointly available, more complex estimation strategies may need to
be employed. We study two such strategies.

One possibility is to utilize structural equation modeling (SEM) with missing data,
in which the marginal regression model of interest is formulated by using the regressors
as exogenous variables, the dependent variable is introduced as the only endogenous variable
explained by the model
\citep{bollen:selv:1989},
and the existing SEM estimation methods are applied
\citep{yuan:bentler:2000,savalei:2010}.

Alternatively, since the data are missing by design, and can be treated as MCAR, multiple imputation
\citep[MI]{rubin:1996,vanbuuren:2012}
can be used to fill in the missing values, with Rubin's variance formulae used to combine
MI estimates and provide inference. Of the several existing flavors of multiple imputation,
one of the simplest strategies is
imputation under multivariate normality (which we expect to behave in ways similar to
the estimation methods for SEM with missing data under multivariate normality).
A less model-dependent method is predictive mean matching \citep{little:1988}
in which a regression model is fit for each imputed variable, a linear prediction
is obtained for each case with missing variable, and an imputation is made by choosing
the value of the dependent variable from one of the nearest neighbors in terms of
the linear prediction score.

\section{Set up and notation}

All of the derivations in this paper concern the joint matrix of the first and second order moments of the data:
\begin{equation}
    \Omega =
    \mathbb{E}
    \left[
    \begin{pmatrix}
      1 \\
      \mathbf{x} \\
      y
    \end{pmatrix}
    \begin{pmatrix}
      1 &
      \mathbf{x}' &
      y
    \end{pmatrix}
    \right]
    =
    \begin{pmatrix}
      1 & \mu_\mathbf{x}' & \mu_y \\
      \mu_\mathbf{x} & \mathbb{E}[ \mathbf{x} \mathbf{x}'] & \mathbb{E}[\mathbf{x} y] \\
      \mu_y & \mathbb{E}[\mathbf{x}' y] & \mathbb{E}[y^2]
    \end{pmatrix}
    \equiv
    \begin{pmatrix}
      \omega_{00} & \Omega_{0x} & \omega_{0y} \\
      \Omega_{0x}' & \Omega_{xx} & \Omega_{xy} \\
      \omega_{0y} & \Omega_{xy}' & \omega_{yy}
    \end{pmatrix}
    \label{eq:Omega}
\end{equation}

The maximum likelihood estimates of the coefficients in the regression of $y$ on $x$
(obtained, for instance, through SEM modeling using maximum likelihood estimates with multivariate normal missing data method;
or approximated through multiple imputation) are obtained as
\begin{equation}
    \hat\beta_{\rm FIML} =
    \begin{pmatrix}
        \omega_{00} & \hat\Omega_{0x} \\
        \hat\Omega_{0x}' & \hat\Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        \hat\omega_{0y} \\
        \hat\Omega_{xy}
    \end{pmatrix}
    \label{eq:ml:regression}
\end{equation}
where $\hat\Omega$ is the maximum likelihood estimator of the joint parameter matrix:
\begin{equation}
    \hat\Omega =
    \begin{pmatrix}
        \omega_{00} & \hat\Omega_{0x} & \hat\omega_{0y} \\
        \hat\Omega_{0x}' & \hat\Omega_{xx} & \hat\Omega_{xy} \\
        \hat\omega_{0y} & \hat\Omega_{xy}' & \hat\omega_{yy}
    \end{pmatrix}
    =
    \begin{pmatrix}
            \omega_{00} & \hat\omega_{01} & \ldots & \hat\omega_{0p} & \hat\omega_{0y}  \\
        \hat\omega_{01} & \hat\omega_{11} & \ldots & \hat\omega_{1p} & \hat\omega_{y1}  \\
        \vdots & \ldots & \vdots & \vdots & \vdots  \\
        \hat\omega_{0p} & \hat\omega_{1p} & \ldots & \hat\omega_{pp} & \hat\omega_{yp}  \\
        \hat\omega_{0y} & \hat\omega_{y1} & \ldots & \hat\omega_{yp} & \hat\omega_{yy}
    \end{pmatrix}
    \label{eq:hat:Sigma}
\end{equation}
where $x_0=1$ is the regression intercept by convention, so that $\omega_{00}\equiv 1$, $\hat\omega_{0j}=\hat\mu_j$
are the (estimated) means of the $j$-th explanatory variable, and $\hat\omega_{0y}=\hat\mu_y$ is the estimated mean of $y$.

To derive the likelihood, we need the form-specific submatrices obtained by multiplying the
overall matrix by selector matrices. For instance, in Design 2 above, for the first form, the relevant covariance matrix is
\begin{equation}
    {\rm Cov}(x_2, x_3, y)' = F_1 \Omega F_1',
    \quad
    F_1 =
    \begin{pmatrix}
        0 & 0 & 1 & 0 & 0 \\
        0 & 0 & 0 & 1 & 0 \\
        0 & 0 & 0 & 0 & 1
    \end{pmatrix}
    \label{eq:selector:1}
\end{equation}
Matrices necessary to form $F_2\Omega F_2'$ and
$F_3 \Omega F_3'$ are defined in a similar way.

Define the unit selector vector
that picks up the estimates of the means $e_0=(1,0,\ldots,0)$, which is the unit vector with 1
in the ``zeroth'' position corresponding to the intercepts in the parameter matrix $\Omega$.
In addition to $e_0$ selecting the first order moments, define the unit selection vectors
$e_y=(0,0,0,0,1)'$ as the unit vector selecting the last row/column of $\Omega$ corresponding to the $y$-parameters,
and $e_j=(0,\ldots,0,1,0,\ldots)$ is a unit vector with 1 in the $j$-th position corresponding to the $j$-th variable
(with the convention of indexing starting at zero). Then we observe that
\begin{align}
    F_1' & = (e_2, e_3, e_y) \notag \\
    F_2' & = (e_1, e_3, e_y) \notag \\
    F_3' & = (e_1, e_2, e_y)
\end{align}

\section{Likelihood and derivatives}

\subsection{Likelihood}

Indexing the forms by $k$, and observations within forms by $i$, the likelihood can be written as
\begin{align}
    \ln L(\omega;X) =
        \sum_{k=1}^3 \sum_{i=1}^{n_k} &
            \Bigl\{
                - \frac12 \mathop{\rm tr} (F_k F_k') \ln(2\pi)
                - \frac12 \ln \det ( F_k \Omega F_k' )
                \notag \\ &
                - \frac12 \bigl[ (x_i',y) - e_0' \Omega F_k' \bigr] (F_k \Omega F_k')^{-1}
                        \bigl[ (x_i', y)' - F_k \Omega e_0 \bigr]
            \Bigr\}
    \label{eq:log:lkhd}
\end{align}
where $n_k$ is the number of observations on which the $k$-th form is collected,
and $F_k$ is the selector matrix corresponding to the $k$-th form.

Derivations of the asymptotic properties of the MLE estimate $\widehat\Omega$
are based on the matrix differential \citep{magn:neud:1999}
\begin{align}
    {\sf d} \Omega = &
        {\sf d} \omega_{00} \, e_0 e_0'
        + \sum_{j=1}^p {\sf d} \omega_{0j} \, (e_0 e_j' + e_j e_0')
        + {\sf d} \omega_{0y} \, (e_0 e_y' + e_y e_0')
        + \sum_{j=1}^p {\sf d} \omega_{jj} \, e_j e_j'
    \notag \\ &
        + \sum_{j=1}^p \sum_{i\neq j} {\sf d} \omega_{ij} \, (e_i e_j' + e_j e_i')
        + \sum_{j=1}^p {\sf d} \omega_{yj} \, (e_y e_j' + e_j e_y'),
    \label{eq:omega:as:sum:of:e:crossed}
\end{align}

After some tedious algebra,
the following information matrix $\mathbb{E} \nabla^2 \ln L(\omega;X)$ results.

\begin{align}
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{0s}\partial \omega_{0t}} \Bigr]
    & =
    - \sum_{k=1}^3 n_k \tau^{(k)}_{st},
    \label{eq:d2lnl:domega0:domega0}
    \\
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{0s}\partial \omega_{uu}} \Bigr]
    & = 0,
    \label{eq:d2lnl:domega0:domega:uu}
    \\
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{0s}\partial \omega_{uv}} \Bigr]
    & = 0
    \label{eq:d2lnl:domega0:domega:uv}
\end{align}

The zero expected cross-derivatives indicate that the estimates of the multivariate
normal means and the variance-covariance parameters are independent. (This may not be the case in general
if the missing data mechanism coded by the matrices $F_k$ is not MCAR, and instead related to the data values.)

\begin{align}
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{ss}\partial \omega_{uu}} \Bigr]
        & = - \frac12 \sum_{k=1}^3 n_k \bigr[ \tau^{(k)}_{su} \bigl]^2
    \label{eq:d2lnl:domega:ss:domega:uu}
    \\
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{ss}\partial \omega_{uv}} \Bigr]
    & = - \sum_{k=1}^3 n_k \tau^{(k)}_{su} \tau^{(k)}_{sv}
    \label{eq:d2lnl:domega:ss:domega:uv}
    \\
    \mathbb{E} \Bigl[ \frac {\partial^2 \ln L(\omega;X)}{\partial \omega_{st}\partial \omega_{uv}} \Bigr]
    & = - \sum_{k=1}^3 n_k \bigl[ \tau^{(k)}_{su} \tau^{(k)}_{tv} + \tau^{(k)}_{sv} \tau^{(k)}_{tu} \bigr]
    \label{eq:d2lnl:domega:st:domega:uv}
    \\
    \tau^{(k)}_{st} & = e_s' F_k' (F_k \Omega F_k')^{-1} F_k e_t
    \label{eq:tau:kst}
\end{align}
where
$\tau^{(k)}_{st}$ is the $(s,t)$-th entry of the inverse of the form-specific covariance matrix;
and indices $s,t,u,v$ can enumerate the explanatory variables $x_j$ and the response $y$.
As $x_i$ and $y$ are considered jointly multivariate normal at this point, there is no separation into
dependent and explanatory variables.

Putting these entries together into a matrix, and using
the standard maximum likelihood estimation theory results, the asymptotic variance
of the maximum likelihood estimates of $\mathop{\rm vech}\Omega$ is given by
\begin{equation}
    {\rm As.} \mathbb{V}[ \hat\omega ] = - \mathbb{E} \bigr[ \nabla^2 \ln L(\omega;X) \bigr]^{-1}
    \label{eq:asvar:omega-hat}
\end{equation}

\subsection{The delta method derivation of the asymptotic variance of $\hat\beta$}

Let us now return to the task of estimating the coefficients of the regression equation
$$y=\beta'x + \epsilon$$
via (\ref{eq:ml:regression}). The asymptotic variance-covariance matrix of $\hat\beta_{\rm FIML}$
can be obtained from the asymptotic covariance matrix of $\hat\Omega$ using the delta-method,
i.e., linearization of the relation (\ref{eq:ml:regression}):
\begin{align}
    {\sf d} \beta = &
    -
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        0 & {\sf d} \Omega_{0x} \\
        {\sf d} \Omega_{0x}' & {\sf d} \Omega_{xx}
    \end{pmatrix}
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        \omega_{0y} \\
        \Omega_{xy}
    \end{pmatrix}
    +
    \notag \\
    & \hspace{5cm} +
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        {\sf d} \omega_{0y} \\
        {\sf d} \Omega_{xy}
    \end{pmatrix}
    \label{eq:diff:beta}
\end{align}
where the individual components of ${\sf d}\Omega$ can be obtained from
(\ref{eq:omega:as:sum:of:e:crossed}).
Thus
\begin{align}
    \frac{\partial\beta}{\partial\omega_{0j}} = &
        -
        \begin{pmatrix}
            \omega_{00} & \Omega_{0x} \\
            \Omega_{0x}' & \Omega_{xx}
        \end{pmatrix}^{-1}
        \begin{pmatrix}
            0 & e_j' \\
            e_j & \underline{0}
        \end{pmatrix}
        \begin{pmatrix}
            \omega_{00} & \Omega_{0x} \\
            \Omega_{0x}' & \Omega_{xx}
        \end{pmatrix}^{-1}
        \begin{pmatrix}
            \omega_{0y} \\
            \Omega_{xy}
        \end{pmatrix}
    \notag \\
    \frac{\partial\beta}{\partial\omega_{0y}} = &
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        1 \\
        \vec{0}
    \end{pmatrix}
    \notag \\
    \frac{\partial\beta}{\partial\omega_{jj}} = &
    -
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        0 & \vec{0}' \\
        \vec{0} & e_j e_j'
    \end{pmatrix}
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        \omega_{0y} \\
        \Omega_{xy}
    \end{pmatrix}
    \notag \\
    \frac{\partial\beta}{\partial\omega_{ij}} = &
    -
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        0 & \vec{0}' \\
        \vec{0} & e_i e_j' + e_j e_i'
    \end{pmatrix}
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        \omega_{0y} \\
        \Omega_{xy}
    \end{pmatrix}
    \notag \\
    \frac{\partial\beta}{\partial\omega_{yj}} = &
    \begin{pmatrix}
        \omega_{00} & \Omega_{0x} \\
        \Omega_{0x}' & \Omega_{xx}
    \end{pmatrix}^{-1}
    \begin{pmatrix}
        0 \\
        e_j
    \end{pmatrix}
    \notag \\
    \nabla_\omega \beta = & \Bigl(
            \frac{\partial\beta}{\partial\omega_{01}}, \frac{\partial\beta}{\partial\omega_{02}}, \ldots,
            \frac{\partial\beta}{\partial\omega_{y0}},
            \frac{\partial\beta}{\partial\omega_{11}}, \frac{\partial\beta}{\partial\omega_{12}}, \ldots, \frac{\partial\beta}{\partial\omega_{y1}}, \frac{\partial\beta}{\partial\omega_{22}}, \ldots,
            \frac{\partial\beta}{\partial\omega_{yp}}, 0 \Bigr)
    \label{eq:dbeta:domega}
\end{align}
where the derivatives are with respect to the components of the vectorization $\mathop{\rm vech}\Omega$,
of which the last term is $\frac{\partial\beta}{\partial\omega_{yy}}=0$. By the standard multivariate
delta-method results \citep{newey:mcfadden:1994,vandervaart:1998},
\begin{equation}
    {\rm As.} \mathbb{V}[ \hat\beta ] = \nabla_\omega \beta \, \mathbb{V}[ \hat\omega ] \, \nabla_\omega' \beta
    \label{eq:asvar:beta-hat}
\end{equation}

\section{Example: Big Five Inventory}

In our application, we wanted to analyze the relation between mental health outcomes
and the Big Five personal traits:

\begin{itemize}
    \item Openness to experience (inventive/curious vs. consistent/cautious)
    \item Conscientiousness (efficient/organized vs. easy-going/careless)
    \item Extraversion (outgoing/energetic vs. solitary/reserved)
    \item Agreeableness (friendly/compassionate vs. challenging/detached)
    \item Neuroticism (sensitive/nervous vs. secure/confident)
\end{itemize}

These personal traits have been found in numerous studies to be related to academic performance, disorders, general health,
and many other behaviors and outcomes. The standard Big Five scale consists of 44 items,
some of which are reverse worded and reverse scored to minimize the risk of straightlining, and with
items from different subscales mixed throughout the scales. Each item is a 5 point Likert scale with a clear midpoint.

In the population of interest, the Big Five traits are expected to have the following correlations, based
on preceding research:

\begin{equation}\label{eq:bigfive:corr}
  {\rm Cov}[x] =
  \begin{pmatrix}
    1    & 0.26 & 0.47 & 0.20 & -0.16 \\
    0.26 & 1    & 0.28 & 0.46 & -0.28 \\
    0.47 & 0.28 & 1    & 0.20 & -0.35 \\
    0.20 & 0.46 & 0.20 & 1    & -0.37 \\
    -0.16 & -0.28 & -0.35 & -0.37 & 1
  \end{pmatrix}
  \equiv
  \Sigma_{\rm Big 5}
\end{equation}

We thus consider a regression model
$$
    y_i = \beta_0 + \beta_1 x_{i1} + \ldots \beta_5 x_{i5} + \varepsilon_i
$$
where $x_{i1},\ldots,x_{i5}$ are subscale scores of the Big Five traits.
Measurement error in these scores is ignored, although more accurate methods are
available to account for it \citep{skrondal:laake:2001}.

A balanced multiple matrix sampling design would consist of ten forms,
each administering the outcome $y$ and two of the Big Five subscales:

\begin{table}[!h]

\centering

\caption{Multiple matrix sampling design with five explanatory variables. \label{tab:bigfive}}

\medskip

\begin{tabular}{l|cccccccccc}
    Form & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
    \hline
       O $\vphantom{\Bigl|}$ & + & + & + & + &   &   &   &   &   &    \\
       C $\vphantom{\Bigl|}$ & + &   &   &   & + & + & + &   &   &    \\
       E $\vphantom{\Bigl|}$ &   & + &   &   & + &   &   & + & + &    \\
       A $\vphantom{\Bigl|}$ &   &   & + &   &   & + &   & + &   & +  \\
       N $\vphantom{\Bigl|}$ &   &   &   & + &   &   & + &   & + & +  \\
        $y\vphantom{\Bigl|}$ & + & + & + & + & + & + & + & + & + & +  \\
\end{tabular}

\end{table}

\section{Simulation 1: Parameter space exploration}
\label{sec:explore}

In this simulation exercise, we explore the parameter space of
regression coefficients to gauge the degree of variability of sample size
determination results. Asymptotic variance resulting from
(\ref{eq:asvar:beta-hat}) is used to obtain the sample sizes
for the tasks outlined in Section \ref{subsec:reg:power}.
Simulation 1 consists of the following steps.

\begin{enumerate}
    \item Population regression parameters are simulated from $\beta \sim N(\mathbf{0},I_5)$.
    \item To provide the scale of the residual variance, the fraction of explained variance is set to
        $R^2=0.15$, a moderate effect for behavioral and social science data, and the associated
        residual variance $\sigma_\varepsilon^2$ is calculated based on this value of $R^2$.
    \item The complete data variances stemming from (\ref{eq:ols}) are recorded.
        \label{enum:explor:full:var}
    \item The multiple-matrix-sampled data variances stemming from (\ref{eq:asvar:beta-hat}) are recorded.
        \label{enum:explor:mcar:var}
    \item Sample size to reject the test of overall significance $H_0: \beta_1 = \ldots = \beta_5 = 0$
        at 5\% level with 80\% power is recorded.
        \label{enum:explor:n:overall}
    \item Sample size to detect an increase in $R^2$ by 0.01 (i.e., from 0.15 to 0.16), through a uniform
        multiplicative increase in the values of the regression parameters, keeping the residual variance
        $\sigma_\varepsilon^2$ constant, at 5\% level with 80\% power, is recorded.
        \label{enum:explor:n:infl}
    \item Sample size to detect an increase in $R^2$ by 0.01 (i.e., from 0.15 to 0.16), through an increase
        in the value of the coefficient $\beta_j, j=1, \ldots, 5$, keeping the residual variance
        $\sigma_\varepsilon^2$ constant and other regression parameters constant, at 5\% level with 80\% power, is recorded.
        \label{enum:explor:n:poke}
    \item Fraction of missing information (FMI) is computed as one minus the ratio
        of the variance of regression parameter estimate with complete data (obtained in step \ref{enum:explor:full:var})
        to the variance of regression parameter estimate with missing data (obtained in step \ref{enum:explor:mcar:var})
        \label{enum:explor:fmi}
\end{enumerate}

1,000 Monte Carlo draws of the $\beta$ vector, and subsequent analytical computation of asymptotic variances and power, were done.
Results are presented graphically. Figure \ref{fig:explore:n} presents the sample sizes obtained in steps
\ref{enum:explor:n:overall}--\ref{enum:explor:n:poke} of the parameter exploration. A striking feature of the plot
is wide variability of the sample sizes as a function of the specific configuration of parameters. While the lower limit
of the sample size necessary to detect an overall increase in $R^2$ by $0.01$ is about $n=82K$, the median value
is $n=110K$, the 95th percentile is $n=220K$, and the maximum (worst case scenario) identified in this simulation is
$n=400K$. The patterns of the coefficients of the worst case scenarios typically indicate large coefficients of opposite signs
of the positively correlated variables ($x_1$ through $x_4$), or large coefficients of similar size of one of the positively
correlated factors ($x_1$ through $x_4$) and a high value of factor $x_5$ that is negatively correlated with all other subscales.
This wide range of variability makes it difficult to provide a definite recommendation concerning the sample
size for the study to the stakeholders. A conservative value based on a high percentile (80\% or 90\%) can be recommended,
to protect against bad population values of regression parameters at the expense of a potentially unnecessary increase in costs.

Figure \ref{fig:explore:fmi} presents the exploration distribution of the fraction of missing information
due to the missing data. FMI for the intercept is generally low, below 0.2. FMI for regression slopes are generally high,
in the range of about 70\% to 80\%. Given the structure of the missing data shown by the
multiple matrix sampling design in Table \ref{tab:bigfive},
each of the predictor variables is observed in 40\% of the data (informing the diagonal entries of the $X'X$ matrix),
and each pairwise combination of the regressors is observed in 10\% of the data (informing the off-diagonal entries).
This yields an expected information loss for the predictor variables somewhere between 60\% and 90\%.

\begin{figure}[!bh]

    \centering

    \begin{tabular}{cc}
        \includegraphics[scale=0.09]{explor_n_overall.png}
        &
        \includegraphics[scale=0.09]{explor_n_infl01.png}
        \\
        (a) & (b) \\
        \includegraphics[scale=0.09]{explor_n_poke1.png}
        &
        \includegraphics[scale=0.09]{explor_n_poke5.png}
        \\
        (c) & (d)
    \end{tabular}

    \caption{
        \label{fig:explore:n}
        Sample size to ensure the necessary detectable effect.
        (a) Overall test $H_0: R^2=0$;
        (b) $R^2$ increase due to overall explanatory power increase from $R^2=0.15$ by $0.01$;
        (c) $R^2$ increase due to an increase in explanatory power from $R^2=0.15$ by $0.01$ due to $x_1$;
        (d) $R^2$ increase due to an increase in explanatory power from $R^2=0.15$ by $0.01$ due to $x_5$.
    }

\end{figure}

\begin{figure}[!bh]

    \centering

    \includegraphics[scale=0.09]{explor_fmi0.png}

    (a)

    \includegraphics[scale=0.09]{explor_fmi1.png}

    (b)

    \includegraphics[scale=0.09]{explor_fmi5.png}

    (c)

    \caption{
        \label{fig:explore:fmi}
        Fraction of missing information:
        (a) intercept; (b) slope of $x_1$; (c) slope of $x_5$.
    }

\end{figure}

\section{Simulation 2: Performance in finite samples}

To study the performance of estimation methods based on SEM estimation with missing data, and on multiple imputation procedures,
a simulation with microdata was also performed. For each simulation draw, the following steps were taken.

\begin{enumerate}
    \item Sample size is set to $n=1,000$ (i.e., $100$ observations per form).
    \item Multivariate non-normal factor scores are simulated:
    \begin{enumerate}
        \item The non-normal principal components of $x_1, \ldots, x_5$ are simulated as
            \begin{align}
                f_1 & = -\ln u_1-1, \quad u_1 \sim U[0,1] \\
                f_2 & = (2 b-1) (-\ln u_2-1), \quad b \sim {\rm Bernoulli}(0.5), u_2 \sim U[0,1] \\
                f_3, f_4, f_4 & \sim N(0,1)
            \end{align}
            so that each principal component has a mean of 0 and variance of 1,
            with all the underlying random variables being drawn independently of each other.
            The first component $f_1$ has a marginal exponential distribution with a heavy right tail,
            ensuring the overall skewness of each factor. The second component has a bimodal distribution
            with two exponential components and heavy tails. The remaining three components are normal.
        \item The factor values are reconstructed as
            \begin{equation}
                \begin{pmatrix}
                    x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5
                \end{pmatrix}
                =
                \sum_{j=1}^5 \mathbf{u}_j \sqrt{\lambda_j} f_j
                \label{eq:Sigma:big5:eigenproblem}
            \end{equation}
            where $\Sigma_{\rm Big 5} = U' \Lambda U$ is the eigenvalue decomposition of the target
            covariance matrix (\ref{eq:bigfive:corr}) of the Big Five factors.
    \end{enumerate}
        \item The outcome is obtained as $y=0.3x_1 + 0.3x_4 + \varepsilon, \varepsilon \sim N(0,1.248602)$
            where the specific value of the residual variance was chosen to ensure that $R^2=0.15$ in the population.
        \item The regression model with the complete data is fit to obtain the benchmark for FMI calculation.
        \item The values of regressors were deleted in accordance with
            the multiple matrix sampling design in Table \ref{tab:bigfive}.
        \item The normal theory based SEM model for missing data was fit; regression parameter estimates
            and their asymptotic standard errors based on the inverse Hessian were recorded.
        \item $M=50$ complete data sets were imputed using multivariate normal imputation model.
        \item The regression model was estimated using the first $M=5$ data sets, in accordance with the traditional
            recommendation regarding the number of imputed data sets. Regression parameter estimates
            and their asymptotic standard errors based on the Rubin's rules were recorded.
        \item The regression model was estimated using all of the $M=50$ data sets. Regression parameter estimates
            and their asymptotic standard errors based on the Rubin's rules were recorded.
        \item $M=50$ complete data sets were imputed using predictive mean matching imputation model for each of the missing variables.
        \item The regression model was estimated using the first $M=5$ data sets, in accordance with the traditional
            recommendation regarding the number of imputed data sets. Regression parameter estimates
            and their asymptotic standard errors based on the Rubin's rules were recorded.
        \item The regression model was estimated using all of the $M=50$ data sets. Regression parameter estimates
            and their asymptotic standard errors based on the Rubin's rules were recorded.
\end{enumerate}

There were 1,200 Monte Carlo samples drawn.

\begin{figure}[!bh]
  \centering
  \includegraphics[scale=0.2]{simul_hist_b1_cens.png}
  \caption{Sampling distributions of the parameter estimates $\hat\beta_1$ across different methods.}\label{fig:simul:b1}
\end{figure}

Figure \ref{fig:simul:b1} reports the simulated distributions of the estimates of parameter $\beta_1$.
The population value of 0.3 is shown as a vertical line on the plot. As expected, the complete data
regression model demonstrates higher efficiency. Estimates based on the multivarate normal methods
are biased up, while those based on MI with predictive mean matching are biased down.
Distributions of the estimates based on the multivariate normal methods are more spread out
than the asymptotic variance based on (\ref{eq:asvar:beta-hat}), while those based
on PMM MI are less spread out, with apparent efficiency gains extracted from higher moments of the data.
The plots in Figure \ref{fig:simul:b1} are truncated, with about 3\% of the Monte Carlo simulations
outside the right range of the plot (the value of $\beta_1=0.6$), and about 1\% of the Monte Carlo simulations
outside the left range of the plot (the value of $\beta_1=0$) for each of the methods based on multivariate normality.
Details for $\hat\beta_1$ and other regression coefficient estimates are provided in Table \ref{tab:simul:b1}.

\begin{table}[!th]
  \centering
  \caption{Monte Carlo means, [95\% confidence intervals] for the means and $\langle$standard deviations$\rangle$ for regression parameter estimates.}
  \label{tab:simul:b1}

  \begin{tabular}{l|ccccc}
     Method & $\hat\beta_1$ & $\hat\beta_2$ & $\hat\beta_3$ & $\hat\beta_4$ & $\hat\beta_5$ \\
     \hline
     Complete & 0.3002	& 0.0016	& 0.0006	& 0.3002	& 0.0015  \\
     data & [0.298,0.303]	& [-0.001,0.004]	& [-0.002,0.003]	& [0.298,0.303]	& [-0.001,0.004] \\
     regression	& $\langle$ 0.0418	$\rangle$ & $\langle$ 0.0408 $\rangle$ & $\langle$ 0.0440 $\rangle$ & $\langle$ 0.0414 $\rangle$ & $\langle$ 0.0413 $\rangle$
     \medskip
     \\
     SEM with & 0.3277	& -0.0203	& -0.0130	& 0.3324	& 0.0096  \\
     MVN & [0.320,0.336]	& [-0.028,-0.013]	& [-0.022,-0.004]	& [0.324,0.340]	& [0.003,0.017]	\\
     missing data & $\langle$ 0.1414	$\rangle$ & $\langle$ 0.1356 $\rangle$ & $\langle$ 0.1588 $\rangle$ & $\langle$ 0.1429 $\rangle$ & $\langle$ 0.1249 $\rangle$
     \medskip
     \\
     MI using & 0.3369	& -0.0253	& -0.0173	& 0.3393	& 0.0091  \\
     MVN model, & [0.329,0.345]	& [-0.033,-0.017]	& [-0.027,-0.008]	& [0.331,0.347]	& [0.002,0.016] \\
     $M=5$ & $\langle$ 0.1390	$\rangle$ & $\langle$ 0.1415 $\rangle$ & $\langle$ 0.1645 $\rangle$ & $\langle$ 0.1435 $\rangle$ & $\langle$ 0.1259 $\rangle$
     \medskip
     \\
     MI using & 0.3430	& -0.0314	& -0.0208	& 0.3466	& 0.0109  \\
     MVN model, & [0.334,0.352]	& [-0.040,-0.023]	& [-0.031,-0.011]	& [0.338,0.355]	& [0.003,0.018] \\
     $M=50$ & $\langle$ 0.1556	$\rangle$ & $\langle$ 0.1507 $\rangle$ & $\langle$ 0.1760 $\rangle$ & $\langle$ 0.1531 $\rangle$ & $\langle$ 0.1336 $\rangle$
     \medskip
     \\
     MI using & 0.2661	& 0.0356	& 0.0261	& 0.2666	& -0.0056 \\
     PMM model, & [0.262,0.270]	& [0.032,0.039]	& [0.022,0.030]	& [0.263,0.271]	& [-0.009,-0.002] \\
     $M=5$ & $\langle$ 0.0707	$\rangle$ & $\langle$ 0.0679 $\rangle$ & $\langle$ 0.0758 $\rangle$ & $\langle$ 0.0707 $\rangle$ & $\langle$ 0.0631 $\rangle$
     \medskip
     \\
     MI using & 0.2678	& 0.0361	& 0.0251	& 0.2671	& -0.0043 \\
     PMM model, & [0.264,0.272]	& [0.032,0.040]	& [0.021,0.029]	& [0.263,0.271]	& [-0.008,-0.001] \\
     $M=50$ & $\langle$ 0.0676	$\rangle$ & $\langle$ 0.0656 $\rangle$ & $\langle$ 0.0719 $\rangle$ & $\langle$ 0.0665 $\rangle$ & $\langle$ 0.0591 $\rangle$ \\
     \hline
     Population & 0.3 & 0 & 0 & 0.3 & 0 \\
     & $\langle$ 0.0791	$\rangle$ & $\langle$ 0.0856 $\rangle$ & $\langle$ 0.0926 $\rangle$ & $\langle$ 0.0824 $\rangle$ & $\langle$ 0.0832 $\rangle$ \\
  \end{tabular}
\end{table}

Figure \ref{fig:simul:se1} provides the Monte Carlo distributions of the standard errors reported for the missing data methods.
The dotted vertical line is the asymptotic standard error based on (\ref{eq:asvar:beta-hat}), 0.0791. The dashed lines are empirical
means of the standard errors. All distributions are skewed with heavy right tails. The distributions of the standard errors
based on multivariate data contain outliers outside the range of the plot (3\% of the SEM with missing data results; 6\% of
the results for MI using the multivariate normal model with $M=5$; 8\% of the results for MI using the multivariate normal model with $M=50$;
the range of the plots is from 0 to $3\times$ the asymptotic standard error, 0.0791). Distributions of the standard errors
for the multivariate normal methods are significantly higher that this asymptotic standard error, which reflects, to some extent,
the greater variability of the estimates observed above in Figure \ref{fig:simul:b1} and Table \ref{tab:simul:b1}.
Distributions of the standard errors
for the PMM MI method are significantly lower that the asymptotic standard error, which reflects, to some extent,
the lower variability of the estimates based on this method.
A higher number of multiple imputations $M=50$ vs. $M=5$ helps to stabilize the variance estimates, particularly in the case of PMM.

\begin{figure}[!bh]
  \centering
  \includegraphics[scale=0.2]{simul_se1_sep.png}
  \caption{Sampling distributions of the standard errors of $\hat\beta_1$ across different methods.}\label{fig:simul:se1}
\end{figure}

Coverage of the nominal 95\% confidence intervals is analyzed in Table \ref{tab:simul:cover95}. Despite the shortcomings
of both the point estimates and the standard errors noted above, things seem to balance out and provide confidence interval coverage
fairly close to the target.

\begin{table}[!th]
  \centering
  \caption{Coverage of the nominal 95\% coverage intervals.}\label{tab:simul:cover95}

  \begin{tabular}{l|ccccc}
     Method & $\hat\beta_1$ & $\hat\beta_2$ & $\hat\beta_3$ & $\hat\beta_4$ & $\hat\beta_5$ \\
     \hline \\
     Complete data regression	 & 95.5\% & 95.4\% & 95.1\% & 95.8\% & 97.3\% \\
     SEM with MVN missing data	& 97.8\% & 98.6\% & 97.3\% & 98.7\% & 98.4\% \\
     MI using MVN model, $M=5$	 &93.3\% & 93.8\% & 93.0\% & 93.3\% & 96.8\% \\
     MI using MVN model, $M=50$	&92.8\% & 93.4\% & 93.2\% & 93.1\% & 97.9\% \\
     MI using PMM model, $M=5$	 &94.4\% & 95.6\% & 94.5\% & 95.0\% & 94.2\% \\
     MI using PMM model, $M=50$	&96.3\% & 96.6\% & 95.8\% & 96.8\% & 97.2\% \\
  \end{tabular}
\end{table}

Estimated fractions of missing information reported by the software are shown on Figure \ref{fig:simul:fmi}.
The dotted line is the value based on asymptotic variance, 73.6\%. Dashed lines are the empirical FMI,
based on the ratios of the Monte Carlo variance of $\hat\beta_1$ based on a given missing data method
to the variance of $\hat\beta_1$ based on the complete data. The latter empirical FMI is greater than
the theoretical one for the MI methods based on the multivariate normality assumption, and lower than
the theoretical one for the PMM MI methods. The methods based on multivariate normality appear to underestimate
FMI, as the distributions of the reported empirical FMI appear to the left of the true value (dashed line).
The FMI that come out of PMM MI appear to be more accurate. An increase in the number of completed data sets
from $M=5$ to $M=50$ helps to improve stability of the FMI estimates, making the distributions of the empirical FMI
more concentrated.

\begin{figure}[!bh]
  \centering
  \includegraphics[scale=0.2]{simul_fmi1_sep.png}
  \caption{Reported fraction of missing information.}\label{fig:simul:fmi}
\end{figure}

\section{Concluding remarks}

This paper provides an analytical framework for analysis of regression models
(and, more generally, other statistical methods that are based on the covariance matrices of observed
items or scales)
that allows for quick power analysis avoiding computationally intensive simulations.

Revisiting the initial motivation of burden reduction, the results are underwhelming.
Is burden really reduced by multiple matrix sampling in the example considered?
Out of five explanatory variables (based on approximately 8 survey items each) and one outcome,
only three variables are collected on each of the matrix sampled instrument forms.
This translates to about 50\% burden reduction per respondent. However,
given that the loss of information quantified by the fraction of missing information
(FMI) is about 75–-80\%, the data collection sample sizes would
need to be about 4--5 times larger compared to the traditional data collection
of all items at once.
Unless the response rate drops sharply by a factor of more than two due to
the increase in questionnaire length, the total public burden is increased.

The sample sizes necessary to detect the required effect sizes in increased $R^2$
demonstrate long tails in the exploration of parameter spaces. These long tails
make it difficult to plan for the worst-case scenarios associated with ``unfortunate''
regression parameter configurations. Should a specific decision need to be made
based on the parameter explorations akin to those undertaken in Section \ref{sec:explore},
the trade-off between the survey costs due to large sample sizes and risks of having
an underpowered study should the coefficient estimates be found to have an ``unfortunate''
configuration should be carefully discussed with the survey stakeholders to find
the most appropriate course of action.

We conducted a finite sample simulation with non-normal data and several missing data methods,
and determined that the methods that assume multivariate normality generally perform poorly,
and generate a non-negligible proportion of really bad outliers. In comparison,
semiparametric multiple imputation by predictive mean matching with sufficiently large
number of imputed data sets seem to work best.

Our work can be extended in a number of additional dimensions. The derivations of asymptotic
variances are based on the working assumption of multivariate normality and using
the inverse information matrix to estimate variances. With non-normal data, the problem
can be formulated in terms of estimating equations, and sandwich variance estimators
should be formed. As our simulation demonstrated, asymptotic standard errors based on
inverse information matrix are inadequate for the analysis methods that we used,
leading to underestimates with misspecified normality-based methods, and overestimates with
a more accurate semiparametric method.

The current paper assumed independence of respondents. In practice, complex survey features
such as strata, clusters, unequal probabilities of selection, and weight calibration would
affect asymptotic properties of the estimates. In particular, the sandwich variance estimation
will be required. Many practical survey statistics issues may also interact with multiple
matrix sampling in unusual ways. How would differential nonresponse by form affect the results?
What should we do when a stratum has fewer than two cases of a given form? These and other
questions related to design-based inference would need to be answered when multiple matrix sampling
is applied in practice.

Finally, in terms of ensuring adequate measurement properties, we note
that psychometric properties are usually established and validated for scales,
but not necessarily subscales that respondents are exposed to in multiple matrix sampling instruments.
In particular, if the order of the items, or the degree of mixing of items from the different
subscales of the Big Five Inventory is important for the validity of the scale and its subscales,
these properties may be violated when shorter subscales are administered that require
the respondent to answer similar questions more frequently.

\clearpage

redpajama_set_name: RedPajamaArXiv


text:
Student Organizes 'Sleepout' to Raise Money for Homelessness
Jesse Bernstein
Nitan Shanas, center, organized a speaking program and sleepout at Rutgers-Camden. From left: Nyeema Watson, Shawn Sheekey, Shanas, Tricia Bradly and Tania Martinez (Photo by Daoud Moon)
Nitan Shanas had to do a lot of convincing to get his recent project off the ground.
He had to lobby campus authorities at Rutgers University-Camden to let him host it on campus, and he had to plead with the Office of Civic Engagement to allow it to happen in less-than-ideal weather. He had to convince other student groups to pitch in, too.
After clearing those hurdles, all Shanas had left was to do the most difficult job of persuasion: convincing his peers to join him for a night sleeping outside on the Rutgers-Camden campus to raise money and awareness for homeless people. A so-called "sleepout."
"I said, 'Nitan, if it's 30, and will remain 30 throughout the night, I'll let this happen,'" Nyeema Watson, associate chancellor for civic engagement at Rutgers-Camden, said with a laugh. "'If it dips, if I see that it's going to dip, you can't do that.'"
Though Shanas believes it may have dipped for a moment overnight from Nov. 14 to Nov. 15, it didn't stop him and 20 other students from braving the cold and raising awareness and thousands of dollars for Joseph's House of Camden, one of the city's largest overnight homeless shelters.
Shanas, who is part of a program known as Civic Scholars, was born in Jerusalem, and spent his first few years in Gan Yavne, just east of Ashdod. In 2009, his family moved to Cherry Hill, New Jersey, when he was 10. He was an Eagle Scout growing up, and his time in the scouts gave him his first exposure to homelessness and food insecurity.
"What I really loved about Boy Scouts, specifically volunteering-wise, was that every year, we'd put bags on people's doors to collect food for the local food pantry, and then, we'd collect that next week," he said. "It made me realize how much I loved volunteering."
He also recalls trips to New York with his mother and an evening in Baltimore with friends, each of which ended with Shanas giving his leftover meals to homeless people. These early experiences awoke him to the realities of homelessness, and set him on his path.
Shanas followed his older brother to Rutgers-Camden. He applied to the Rutgers-Camden Civic Scholars program, a branch of a national program based at Princeton University. Applicants submit an idea for a long-term service project in and around Camden, explaining how it dovetails with their studies. Successful applicants typically have some experience in their chosen area, which Shanas did. Those who are accepted commit to 300 service hours per school year, and receive a $2,000 scholarship.
Though many have been accepted to the program over the years, said Watson, who oversees the program, Shanas' participation has been a little different, as best highlighted by the sleepout program.
"This is probably the most expansive project I've ever seen a student a part of the Civic Scholars program do, or even just a student here generally at Rutgers-Camden," she said.
During his freshman year, Shanas read "Evicted" by Matthew Desmond, which follows eight Milwaukee families as they struggle to keep their homes during the 2007-2008 financial crisis. As he learned more about the economic, social and emotional toll of eviction, he stepped up his involvement with Camden's homeless shelters, where he routinely volunteers his time.
Volunteering at the since-closed New Visions Homeless Day Shelter and Joseph's House of Camden, Shanas began to think about what an awareness and fundraising event could do for the people he served. So, he began researching how like-minded groups conducted sleepouts before.
With guidance from Watson, the plan came together over many months. On Nov. 14, 80 college students came and listened to hours of lectures and testimony from formerly homeless people and employees from Joseph's House. The students took part in workshops designed to help them think more consciously about the way the homeless are treated in their city. Finally, it was time to head outside.
For one cold night, with Rutgers University Police looking on, Shanas and 20 students bundled up on cardboard and slept outside. It was illuminating, Shanas believes, but not for the reason you might expect; harshness of the situation aside, each of the students knew that a quick warm-up or a bathroom in a campus building was steps away. What would it be like not to have those options? In the morning, students reconvened for reflection.
Shanas was happy with the event, though he continues on his fundraising campaign with a goal of $10,000. For Watson, the event was another example of Shanas' commitment to service.
"This event was better than some of what I've seen our own faculty or staff put together," she said.
[email protected]; 215-832-0740
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Jesse Bernstein is from Dresher, Pennsylvania, and graduated from the University of Rochester. His writing has appeared in The Philadelphia Inquirer, Tablet, The Forward, and the Washington Jewish Week. He has worked for the Jewish Exponent since 2018.
Homeless IDF Vet Finds a Home, at Least for Now
Homeless Photographer, IDF Veteran Seeks the American Dream
Ask Miriam | Passing a Passed-Out Person Posed a Problem

redpajama_set_name: RedPajamaCommonCrawl


text:
Citizen Activism Helps Fend off Privatization of St. Louis Water
November 19, 2014; St. Louis American
In the midst of the Detroit water crisis, there were active and credible suspicions that the city was thinking about privatizing its water system by selling to a corporate entity. Organizers placed the potential privatization of Detroit Water in the public spotlight, which may have ultimately compelled the department to abandon those ideas in favor of converting to a regional but still public water system, at least for the moment.
A new report from Corporate Accountability International, "Troubled Waters: Misleading Industry PR and the Case for Public Water," addresses the privatization juggernaut to describe how some cities have dealt with corporate pressure, especially since some of it is predicated on the needs of U.S. water systems for as much as $4.8 trillion in investment in the next 20 years, as private companies, such as the French multinational Veolia Water North America, hinting that privatization would help create the needed capitalization.
The report describes elements of Veolia's strategy in St. Louis. One example is their offer of consulting services (through Veolia's Peer Performance Solutions) that would cut public water system costs, but in reality would be a foot in the door toward privatization. After years of pitching, Veolia got the city, including Mayor Francis Slay, to approve a Peer Performance Solutions contract with Veolia, but community activists and nonprofits challenged the idea. Activists formed the St. Louis Dump Veolia coalition to oppose the contract. The Great Rivers Environmental Law Center did its own analysis of the proposed contract, finding that the "contract will have the effect of privatizing the city's Water Division, and will make city residents captive to Veolia." According to the Corporate Accountability International report author, Emanuele Lobina, the terms of the Veolia contract would make Veolia "the private owner of all ideas for improving the St. Louis Water Division."
Slay did everything he could to get the Board of Alderman to approve the contract with Veolia, but the aldermen didn't go along, and ultimately, Veolia withdrew. Maybe the St. Louis alderman were cognizant of the number of U.S. cities—currently 33—that have "re-municipalized" their water systems after being disappointed with the performance and follow-through of the private water companies.
Just outside of St. Louis is the suburb of Ferguson, where privatization has been in full swing for some time. Mark Ames of PandoDaily reported in September that Ferguson has been implementing privatization schemes designed and promoted by the Reason Foundation's Bob Poole, a longtime associate of the Koch brothers, who provided the support for the Foundation's creation. A number of members of the Ferguson Commission do not appear to be diehard opponents of privatization, and may actually support it in a reflexive way, as some people almost always think that private sector activity is somehow more efficient and effective than nonprofit or public. In light of the St. Louis Water story and Ames's detailed description of the influence of Poole's ideas in and around Ferguson, activists had better be prepared to watch out for and confront unnecessary or inappropriate municipal privatization as Ferguson fixes.—Rick Cohen
More about: Activism
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Part 1: Illustrate the essentials of partnering with communities to gain an understanding of the health status and priorities of the population in order to inform options for health promotion action.
Skill Competencies: Demonstrate effective partnering with communities to gain an understanding of the health status and priorities of the population in order to inform options for health promotion action.
Knowledge Competencies: Discuss the essentials of partnering with communities to gain an understanding of the health status and priorities of the population in order to inform options for health promotion action.


text:
Q: Proof of upper semi-continuity of sheaf cohomology $\DeclareMathOperator{\Spec}{Spec}$
$\DeclareMathOperator{\im}{im}$
Hello Math.Stackexchange.com-Community.
Sorry for asking two questions at the same time, however they are part of one single step in a proof and hence closely related.
My questions arise from the following context:
Let $X\to Y$ be a proper morphism between locally noetherian Schemes and let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module on $X$ which is flat over $\mathcal{O}_Y$.
Then for all $p\geq 0$ the function $h^p(-,\mathcal{F})\colon Y\to\mathbb{N}$ which is defined by
$h^p(y,\mathcal{F}):=\dim_{\kappa(y)}H^p(X_y,\mathcal{F}|_{X_y})$
is upper semi-continuous in the sense that for all $c\in\mathbb{R}$ the set $\{y\in Y\mid h^p(y,\mathcal{F})\leq c\}$ is open.
In order to prove this, the following lemma was invoked:
Lemma:
Let $A$ be a noetherian ring.
Let $\delta\colon M\to F$ be a morphism between finitely-generated $A$-modules, of which $F$ is free.
Then, the function $g\colon \Spec(A)\to \mathbb{N}$ defined by
$g(\mathfrak{p}):=\dim_{\kappa(\mathfrak{p})}(\ker(M\otimes_A \kappa(\mathfrak{p})\overset{\delta_{\mathfrak{p}}:=\delta\otimes_A \kappa(\mathfrak{p})}{\to} F\otimes_A \kappa(\mathfrak{p}))$ is upper semi-continuous.
My question arises in the proof of this lemma, which goes as follows:
Proof
Let $\mu_1,\cdots,\mu_s\in M$ be elements, such that the images $\overline{\mu_1},\cdots,\overline{\mu_s}\in M\otimes_A\kappa(\mathfrak{p})$ form a basis of that vector space, and such that $\delta_{\mathfrak{p}}(\overline{\mu_i})$ vanish for $i=1,\cdots,r=g(\mathfrak{p})$ and form a basis of $\im(\delta_{\mathfrak{p}})$ for $i=r+1,\cdots,s$.
Here comes the part that I don't understand, and also my first question:
Since the assertion is local near any $\mathfrak{p}$ we may replace $A$ by $A_f$ for $f\not \in\mathfrak{p}$ and assume that $M$ is generated by the $\mu_1,\cdots\mu_s$ as an $A$ module.
The last assertion is clear to me. This is a Nakayama-Style argument.
It is however not clear to me, what "local near $\mathfrak{p}$" means, and why this implies, that we may replace $A$ by $A_f$. Aside from this argument, it is also not clear to me, how to prove by hand (i.e. without using the "near $\mathfrak{p}$"-Argument) why we may replace $A$ by $A_f$.
I tried and failed as follows:
Assume that we may find a covering by $D(f_j)$'s of $\Spec(A)$ such that the morphisms $g_j\colon Spec(A_{f_j})\to\mathbb{N}, g_j(\mathfrak{p}_j)=\dim_{\kappa(\mathfrak{p}_j)}(\ker(M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j})\to F_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j}))$ is EDIT: locally constant upper semi-continuous, where by $\mathfrak{p}_j$ I mean the prime ideal in $\Spec(A_{f_j})$ corresponding to $\mathfrak{p}\in D(f_j)$.
Then it would suffice if one could show that this implies, that the restrictions of $g$ to the $D(f_j)$ are upper semi-continuous.
In order to show this one would have to show, that for every $c\in\mathbb{R}$ we have $\dim_{\kappa(\mathfrak{p}_j)}(\ker(M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j})\to F_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j}))\leq c$ for all $j$ iff $\dim_{\kappa(\mathfrak{p})}(\ker(M\otimes_A \kappa(\mathfrak{p})\overset{\delta\otimes_A \kappa(\mathfrak{p})}{\to} F\otimes_A \kappa(\mathfrak{p}))\leq c$ for all prime ideals $\mathfrak{p}\in D(f_j)$.
This is the point where I don't see how to continue.
My second question is why we may construct a basis for $M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p_{f_j}})$ out of the basis for $M\otimes_A \kappa(\mathfrak{p})$ such that it still has the properties we set up in the beginning of the proof, so that we may carry on with the proof after replacing $A$ by $A_f$.
Thank You for your participation!

A: Let me answer your first question. I will use the following setup:
$A$ a noetherian ring, $M$ a finitely generated $A$-module, and $p$ a prime ideal of $A$. We show that if $M_p$ is generated by the images of $x_1,\dots, x_n \in M$, then there exists $f \in A$ such that $M_f$ is generated by the images of the same set of elements.
Let $N$ be the $A$-submodule of $M$ generated by $x_1,\dots, x_n$, and let $K$ be the cokernel of the inclusion map $N \to M$.
Our goal is to find an element $f \in A$ such that $K_f = 0$.
If $K$ is $0$, then there is nothing to do. Assume that $K \neq 0$.
Consider the support of $K$. Since $K$ is finitely generated $Supp(K)$ is determined by an $A$-ideal $I$, and since $p$ is not in it, $I$ is a proper ideal. Hence we may choose any nonzero element in $I$ for $f$.

redpajama_set_name: RedPajamaStackExchange


text:
Q: How can I throw an exception in C? I typed this into Google, but I only found how-tos in C++.
How can I do it in C?

A: On Windows with Microsoft Visual C++ (MSVC) there's __try ... __except ..., but it's really horrible and you don't want to use it if you can possibly avoid it. Better to say that there are no exceptions.

A: C doesn't have exceptions.
There are various hacky implementations that try to do it (one example at: http://adomas.org/excc/).

A: In C you could use the combination of the setjmp() and longjmp() functions, defined in setjmp.h. Example from Wikipedia
#include <stdio.h>
#include <setjmp.h>

static jmp_buf buf;

void second(void) {
    printf("second\n");         // prints
    longjmp(buf,1);             // jumps back to where setjmp
                                //   was called - making setjmp now return 1
}

void first(void) {
    second();
    printf("first\n");          // does not print
}

int main() {
    if ( ! setjmp(buf) ) {
        first();                // when executed, setjmp returns 0
    } else {                    // when longjmp jumps back, setjmp returns 1
        printf("main");         // prints
    }

    return 0;
}

Note: I would actually advise you not to use them as they work awful with C++ (destructors of local objects wouldn't get called) and it is really hard to understand what is going on. Return some kind of error instead.

A: As mentioned in numerous threads, the "standard" way of doing this is using setjmp/longjmp. I posted yet another such solution to https://github.com/psevon/exceptions-and-raii-in-c
This is to my knowledge the only solution that relies on automatic cleanup of allocated resources. It implements unique and shared smartpointers, and allows intermediate functions to let exceptions pass through without catching and still have their locally allocated resources cleaned up properly.

A: There's no built-in exception mechanism in C; you need to simulate exceptions and their semantics. This is usually achieved by relying on setjmp and longjmp.
There are quite a few libraries around, and I'm implementing yet another one. It's called exceptions4c; it's portable and free. You may take a look at it, and compare it against other alternatives to see which fits you most.

A: Plain old C doesn't actually support exceptions natively.
You can use alternative error handling strategies, such as:

*

*returning an error code

*returning FALSE and using a last_error variable or function.

See http://en.wikibooks.org/wiki/C_Programming/Error_handling.

A: C doesn't support exceptions. You can try compiling your C code as C++ with Visual Studio or G++ and see if it'll compile as-is. Most C applications will compile as C++ without major changes, and you can then use the try... catch syntax.

A: C is able to throw C++ exceptions. It is machine code anyway.
For example, in file bar.c:
#include <stdlib.h>
#include <stdint.h>

extern void *__cxa_allocate_exception(size_t thrown_size);
extern void __cxa_throw (void *thrown_exception, void* *tinfo, void (*dest) (void *) );
extern void * _ZTIl; // typeinfo of long

int bar1()
{
   int64_t * p = (int64_t*)__cxa_allocate_exception(8);
   *p = 1976;
   __cxa_throw(p, &_ZTIl, 0);
  return 10;
}

In file a.cc,
#include <stdint.h>
#include <cstdio>
extern "C" int bar1();

void foo()
{
  try
  {
    bar1();
  }
  catch(int64_t x)
  {
    printf("good %ld", x);
  }
}

int main(int argc, char *argv[])
{
  foo();
  return 0;
}

To compile it:
gcc -o bar.o -c bar.c && g++ a.cc bar.o && ./a.out

Output
good 1976

https://itanium-cxx-abi.github.io/cxx-abi/abi-eh.html has more detail info about __cxa_throw.
I am not sure whether it is portable or not, and I test it with 'gcc-4.8.2' on Linux.

A: There are no exceptions in C. In C the errors are notified by the returned value of the function, the exit value of the process, signals to the process (Program Error Signals (GNU libc)) or the CPU hardware interruption (or other notification error form the CPU if there is)(How processor handles the case of division by zero).
Exceptions are defined in C++ and other languages though. Exception handling in C++ is specified in the C++ standard "S.15 Exception handling", there is no equivalent section in the C standard.

A: This question is super old, but I just stumbled across it and thought I'd share a technique: divide by zero, or dereference a null pointer.
The question is simply "how to throw", not how to catch, or even how to throw a specific type of exception. I had a situation ages ago where we needed to trigger an exception from C to be caught in C++. Specifically, we had occasional reports of "pure virtual function call" errors, and needed to convince the C runtime's _purecall function to throw something. So we added our own _purecall function that divided by zero, and boom, we got an exception that we could catch on C++, and even use some stack fun to see where things went wrong.

A: If you write code with the happy path design pattern (for example, for an embedded device) you may simulate exception error processing (AKA deffering or finally emulation) with operator "goto".
int process(int port)
{
    int rc;
    int fd1;
    int fd2;

    fd1 = open("/dev/...", ...);
    if (fd1 == -1) {
      rc = -1;
      goto out;
    }

    fd2 = open("/dev/...", ...);
    if (fd2 == -1) {
      rc = -1;
      goto out;
    }

    // Do some with fd1 and fd2 for example write(f2, read(fd1))

    rc = 0;

   out:

    //if (rc != 0) {
        (void)close(fd1);
        (void)close(fd2);
    //}

    return rc;
}

It is not actually an exception handler, but it takes you a way to handle error at function exit.
P.S.: You should be careful to use goto only from the same or more deep scopes and never jump a variable declaration.

A: Implementing exceptions in C by Eric Roberts.
Chapter 4 of C Interfaces and Implementations by Hanson.
A Discipline of Error Handling by Doug Moen
Implementing Exceptions in C (details the article of E. Roberts)

A: In C we can't use try case to handle the error.
but if you can use Windows.h so you can:
#include <stdio.h>
#include <Windows.h>
#include <setjmp.h>
jmp_buf Buf;

NTAPI Error_Handler(struct _EXCEPTION_POINTERS *ExceptionInfo)
{
    printf("co loi roi ban oi.!!!\r\n");
    longjmp(Buf, 1);
}

void main()
{
    AddVectoredExceptionHandler(1, Error_Handler);
    int x = 0;
    printf("start main\r\n");
    if (setjmp(Buf) == 0)
    {
        int y = 1 / x;
    }
    printf("end main\r\n");

}


text:
Q: How do I resynchronise two git branches? I have two branches, master, and test.  When I try merging master into test, git thinks everything is up to date, but if I run git diff master test, I get a difference that I don't expect.  At this particular moment in time, I expect them to be completely identical.
I'm trying to figure out how to get these to be identical again.  One though I have is to delete the test branch and simply recreate it from master.  However, test is a shared branch, and exists also on the test server which automatically pulls the test branch based on commit hooks.  I expect that recreating the test branch will break the test server's attempt to pull?
Is there a cleaner way to resynchronise the two branches, or should I just bite the bullet and manually resync the test server here?
I'm also expecting a reasonable chance of having to do the same thing to the production branch at some point, which makes it all that much more important that I understand what I'm doing before going and doing it.
Note that if I ask git for any merges in one branch not in the other, it comes back with nothing - all merges are present.  It may have been a bad merge commit or something which got us to this point.

A: Is test based branched off master?
The reason I can think of is in your current scenario your master branch is behind test branch, meaning that test branch already has master's commits and it's own.
If you're using github or gitlab you could check your network graph to see what's up with them.
Also, make sure you didn't change your gitignore at some point.
If master is behind test, the only reasonable way I can think of you sync'ing them is by merging test into master, but that'd be when you want to take that to production.

A: Not a complete answer, because it's not enough information for the complete answer, just few suggestions:

*

*run git log test --not master. Thus you get the list of commits which are in test but not in master.

*Reverse the arguments (... master --not test). Likely you'd get an empty list here

*Certainly you shouldn't voluntarily delete published branches, thus you can create various problems for you and/or your co-workers. But if you clearly understand what the state of your repository is you could use git push -f. But think twice before forcibly override the state of a remote remository.

*A good method to understand the current state of a repo is to visualize it e.g. with a local tool like gitk --all or a remote service (each popular hosting like github or gitlab has their own webtools for this).

*I would suggest to use git blame <file> (or even git gui blame <file>) to understand from what commit a particular line comes from. Alternatively you may git log -p to get commits list with their changes.

*In any case I would estimate the need of manual revamp the repo as extremely low. Likely you have a relatively simple problem, and simply don't have enough information to completely understand it.


V-KOOL X15 is the ideal solution for customers who want to control visible light reflections while reducing or minimizing the heat of the vehicle.
It is the industry's leading film technology that provides excellent visibility compared to films with typical dark visible light transmittance while achieving only 12% visible light reflectance.
Help to reduce glare while delivering the benefit of privacy protection.
Protect skin by blocking out more than 99% of UV rays.
Enjoy reduced heat transmission because of the infra-red reflection capabilities.
Experience superior quality product technology that was developed in Silicon Valley, USA.


text:
Human Rights Council: LWF and member churches in Namibia and Sierra Leone submit statements defending human rights
The ceiling of the Human Rights Council, in the Palais des Nations where sessions typically take place. Due to the COVID-19 crisis, statements to the 48th session were submitted virtually. Photo: LWF/C. Kästner
LWF with member churches in Namibia and Sierra Leone engages governments through UPR mechanism
(LWI) - In statements to the 48th regular session of the United Nations (UN) Human Rights Council, the Lutheran World Federation (LWF) and two of its member churches call upon the governments in Sierra Leone and Namibia, through the Universal Periodic Review (UPR) mechanism, to continue improving the state of human rights in their country.
This was the first time the two churches engaged in human rights advocacy through the UPR mechanism to promote civil and political as well as economic, social, and cultural rights in their countries. The process drew on the unique capacity and outreach of the church to be present at all levels of society. Most of the recommendations advanced by the churches were well received in the UPR working group outcome reports of both countries. They were subsequently adopted by the Human Rights Council for the governments to follow-up on until the next review in four years.
Implementation of the recommendations however will require continued close collaboration of governments with all stakeholders. All national and local stakeholders working closely to implement the accepted recommendations would ensure tangible results and collaboration, including between state and faith based actors and would mean an expanded community involvement.
Both member churches also called on the international community to provide the necessary resources to ensure the implementation of the accepted recommendations.
Uhuru Dempers, the director of the Social Development Desk at Evangelical Lutheran Church in the Republic of Namibia (ELCRN) stated that "It was indeed worth participating in this process as churches and civil society in Namibia. In terms of the way forward, the proposal to mobilize resources ensures that we maintain the momentum and intensify our efforts with government."
Director Dempers added, "Our thanks and appreciation to the Lutheran World Federation for inviting us and supporting us to participate in this process,. We look forward to LWF's continued support and collaboration as we continue with the next critical stage of this platform. ELCRN would not have successfully participated in this intervention without the active involvement and support of the member churches of the Council of Churches in Namibia and other civil society organizations. We are therefore grateful for their support and active participation."
The UPR is most effective when all relevant stakeholders, including rights holders at the grassroots level are involved. Faith based actors have the comparative advantage of being close to people and communities, including in rural areas where government presence is often scant. Churches are also able to leverage their moral integrity and credibility with Governments at the national level.
— Dr Emmanuel Gore, LWF Action for Justice Advocacy Officer for Human Rights
Moses Foray the focal person for the UPR engagement at the Evangelical Lutheran Church in Sierra Leone (ELCSR) thanked and appreciated the support LWF has provided to the team in Sierra Leone to engage with the UPR process. "We have gathered more experience during the period under review and we are open to more learning during the process with the hope that it will make tangible difference to our communities on the ground."
"The UPR is most effective when all relevant stakeholders, including rights holders at the grassroots level are involved. Faith based actors have the comparative advantage of being close to people and communities, including in rural areas where government presence is often scant. Churches are also able to leverage their moral integrity and credibility with Governments at the national level," commented Dr Emmanuel Gore, LWF Action for Justice Advocacy Officer for Human Rights.
Read the full statement for Namibia here
Read the full statement for Sierra Leone here
LWF/OCS
Namibia Sierra Leone
Human Rights Council
Human Rights Council: LWF welcomes appointment of special rapporteur for climate
Human Rights Council: Upholding the right to freedom of religion or belief
Protect civic space in post-pandemic recovery


The beach at the Cold Spring Harbor Beach Club has been reopened for bathing after recent samples showed the water quality to be within acceptable levels, according to the Suffolk County Department of Health Services.
Bayberry Cove Beach in Strongs Neck is still closed for bathing, the health department said.
For updates on affected beaches, call the county's beach hotline at 631-852-5822 or check the website at www.suffolkcountyny.gov/health.


text:
NS_CC_EXTRA_BEGIN

void Native::showActivityIndicator(void)
{
	NativeWin32::sharedInstance()->showActivityIndicator();
}

void Native::hideActivityIndicator(void)
{
	NativeWin32::sharedInstance()->hideActivityIndicator();
}

void Native::createAlert(const char* title,
                           const char* message,
                           const char* cancelButtonTitle)
{
	NativeWin32::sharedInstance()->createAlertView(title, message, cancelButtonTitle);
}

int Native::addAlertButton(const char* buttonTitle)
{
	return NativeWin32::sharedInstance()->addAlertButton(buttonTitle);
}

#if CC_LUA_ENGINE_ENABLED > 0
int Native::addAlertButtonLua(const char* buttonTitle)
{
    return addAlertButton(buttonTitle) + 1;
}
#endif

void Native::showAlert(AlertViewDelegate* delegate)
{
	NativeWin32::sharedInstance()->showAlertViewWithDelegate(delegate);
}

#if CC_LUA_ENGINE_ENABLED > 0
void Native::showAlertLua(cocos2d::LUA_FUNCTION listener)
{
	NativeWin32::sharedInstance()->showAlertViewWithLuaListener(listener);
}
#endif

void Native::cancelAlert(void)
{
	NativeWin32::sharedInstance()->cancelAlertView();
}

const std::string Native::getOpenUDID(void)
{
    return NativeWin32::sharedInstance()->getUDID();
}

void Native::openURL(const char* url)
{
    if (!url) return;
}

const std::string Native::getInputText(const char* title, const char* message, const char* defaultValue)
{
	return NativeWin32::sharedInstance()->getInputText(title, message, defaultValue);
}

const string Native::getDeviceName(void)
{
    return "Win32";
}

void Native::vibrate()
{
}

NS_CC_EXTRA_END

redpajama_set_name: RedPajamaGithub


You'll need these pictures to follow the tutorial.
Open up "car" resource image.
We need to separate the car from the background. I'm using Pen Tool (P) but you can use whatever suits you better. Either novice or experienced I'd suggest you use Pen Tool because it produces best results.
Once the car is selected, copy it (CTRL+C) then create a new layer and paste it onto the new layer (CTRL+V) and delete the background layer. This is my car cutout.
Open up your "road" resource image.
Zoom in on the little truck.
Using Clone Stamp Tool (S) remove it from the picture.
I used Sharpen Tool when I was satisfied with my result.
Place car inside your document.
Transform it, as you wish.
I arranged the layers a bit. Changed the names. Made a new layer between "car" and "road" layer. Named it "shadow".
Using black-to-nothing radial gradient we will create a shadow. Gradient as shown.
Using Transform Tool (CTRL+T) Transform it like this.
Now I created a new layer, "Shadow 2". Using soft black brush with 30% opacity I painted underneath the car, as shown on this picture.
Now again I created a new layer, "Shadow 3". By holding CTRL and clicking on car's layer image in layers window you make it a selection. Now on "Shadow 3" just paint on the line where the car touches the ground. Again with soft black low opacity brush. Experiment with opacity, scale, hardness to achieve different results.
Two new layers, "Tire shadow 1" and "Tire shadow 2". Just paint a bit underneath tires.
Duplicate "Road" layer by RIGHT CLICK > Duplicate or by selecting it than hitting CTRL+J.
Image > Adjustments > Gradient Map. Black to white.
Image > Adjustments > Photo Filter. Cooling Filter (LBB) on 43%.
Put that layer to "Hard Light" blending mode and around 80% opacity.
Add a mask to it by clicking on add mask button. Radial gradient it as shown. Black to white.
Transform it to fit on the light.
Put it on Lighten blending mode.
Play a bit with eraser, try to get it looking realistic.
Duplicate light layer (CTRL+J) and transform it to fit the little light.
Again, play a bit with Eraser Tool (E).
Now we duplicated the light again but this time we transformed it this way.
CTRL+Click on car's layer icon to make a selection. After that delete light parts so this last light only is visible on the road.
Erase it a bit more.
Duplicate light. Transform it this way.
Make a car selection again and delete light that's visible on the car.
Again , delete unnecessary parts of light.
Right click on selection > Feather.
Fill it a few times with any color on low opacity (30%).
Image > Adjustments > Gradient Map.
Soft Light on blending mode, 60% opacity.
And that's it for today. When designing something just try to be slow as possible so the quality of your work increases.


text:
On réserve, en général, le nom de microcalorimètre aux dispositifs permettent la mesure d'un dépôt d'énergie impulsionnel comme celui engendré par une interaction X-gamma dans un absorbeur et le nom de microbolomètre aux dispositifs permettant la mesure de flux comme c'est le cas dans l'astronomie infrarouge.

Principe
Le principe du calorimètre repose sur le fait que la capacité calorifique des matériaux tend vers 0 quand on descend à très basse température.

Types de microcalorimétrie
Il existe plusieurs types de microcalorimétrie. Le titrage calorimétrique isotherme (ITC) et la calorimétrie à balayage différentiel (DSC) en sont deux exemples.

Le titrage calorimétrique isotherme est une technique qui permet notamment d'étudier avec une grande précision la liaison d'un ligand à une macromolécule ou l'interaction entre 2 macromolécules. Cette technique mesure le changement d'enthalpie lors de la liaison de biomolécules à température constante.

La calorimétrie à balayage différentiel permet d'étudier l'interaction entre des biomolécules, la stabilité des protéines, des transitions de phase lipidique, la micellisation de surfactants, la fusion d'acide nucléique et la stabilité de certains principes actifs (sous forme liquide).

Limites de détection
La limite de détection pour l'ITC se situe à environ 50µJ

Notes et références

Calorimétrie

redpajama_set_name: RedPajamaWikipedia


We are committed to meeting the evolving health & social needs of our community.
The Dare County Department of Health & Human Services is the primary department responsible for protecting the health of all Dare County residents and protecting, strengthening and helping families and individuals achieve self-sufficiency.
The Department of Health & Human Services partners with individuals, families and the community to strengthen their efforts towards healthy living, independence, permanence and safety. The department is comprised of a public health division, social services division, and administrative division.
The Dare County Health & Human Services Board is comprised of members selected by the Dare County Board of Commissioners. This board is responsible for policymaking and rulemaking in accordance with North Carolina General Statute Laws and meets at least quarterly throughout the year. The board meets every other month on the 3rd Tuesday of the month unless otherwise noted. All meetings of the board are subject to the provisions of North Carolina's open meeting law.
The strategic planning process is meant to strengthen the organization internally, and focus on a range of agency- level organizational goals, strategies, objectives, and new initiatives.
In November of 2013, the Dare County Board of Commissioners voted unanimously to merge the Department fo Public Health and the Department of Social Services to form the Dare County Department of Health & Human Services.
In efforts to support consolidation between Social Services and Health, a team was tasked with creating a Consolidated Strategic Plan.
The plan is designed to be implemented until the end of June 2019 and the department has made a summary and the entire report available to the community.


text:
Leiodes brunnea är en skalbaggsart som först beskrevs av Sturm 1807.  Leiodes brunnea ingår i släktet Liodes, och familjen mycelbaggar. Arten har ej påträffats i Sverige.

Källor

Mycelbaggar
Liodes (Leiodidae)


When I tell someone that I made a pen, exactly what do I mean? Over 90% of creating a pen has to do with producing the body of the pen. This is where most of the time, effort, and experience is involved. The pen I'm crafting below is the Acadia. This is a medium-sized pen that takes a Parker-style refill. The pen blank for this pen is Azurite-Malachite Tru-stone (the colored block in the picture below). This is a man-made material consisting of 85% natural stone that has been pulverized to a powder then mixed with pigments and acrylic resin, The result is a material that looks and feels like stone, but can be turned on a lathe with typically available turning tools.
The core of the pen are the two brass tubes seen below the Tru-stone. These go inside the Tru-stone and are sized to accept the various mechanical and trim components. Above the Tru-stone from left to right are the nib, centerband, transmission, finial holder, clip, and top finial. The trim pieces for this pen are electro-plated with chromium, one of the hardest pen platings available. The transmission converts the twisting motion of the pen upper half to an in-out motion that extends and retracts the refill.
The first step is to cut out two pieces from the blank, each slightly longer than the brass tube that's going to be placed inside. While not apparent above, the brass tube for the lower part of the pen is slightly longer than the upper pen tube. I use my bandsaw to cut the Tru-stone. This material is pretty rough on blades, so I don't use my new saw blade for this.
I then mount each Tru-stone block in the chuck on my lathe so I can drill a hole through the middle of the block. A Jacob's chuck mounted in the tailstock holds the drill bit. The drill bit remains stationary while the block rotates.
These pictures show my Jet 1642-EVS2 lathe, which is a full size lathe that can turn large bowls and pretty much anything I want. If you just want to turn pens and smaller items, you can purchase a much smaller midi or mini lathe. They take up less room and cost a lot less. I would highly recommend you get one with variable speed. I use this feature all the time on mine and find it indispensable.
I use a drill diameter a few thousandths of an inch larger than the brass tube so that the tube can slide easily into the hole. If you drill slowly and frequently empty the cuttings from the hole, it's not a problem drilling the Tru-stone.
Here are the drilled Tru-stone blocks. The brass tubes can now be epoxied into the holes.
I first rough up the outside of the tubes with sandpaper to get better adhesion and plug the ends of the tubes with dental wax to prevent any epoxy from getting into the tubes. It's critical that you get full epoxy coating between the tube and the block. I coat both the inside of the hole as well as the outside of the brass tube with epoxy. The tubes are inserted so that a small amount of Tru-stone protrudes beyond each end of the tube. I let the epoxy harden overnight. Cut-up cereal boxes make great mixing cards for the epoxy.
I next sand the ends of the blocks on my belt sander.
Yes, that is an alligator on top of my drill press. A souvenir of our 1963 family vacation to Florida, pre-Disney. What 9 year old wouldn't want a stuffed alligator?
Here's a better view of the sanding operation. It's key that the block is held square to the sanding belt, because you want the sanded surface perpendicular to the hole.
You continue sanding, a little at a time, until you reach the brass tube. Be careful not to sand away any of the brass tube.
Here is a close-up of the sanded end. The dental wax is still plugging the brass tube.
I also sand away the corners of the blocks to make them easier to turn. Then I remove the dental wax plugs and clean out the inside of the brass tubes using an appropriate sized brass wire brush. If any epoxy does adhere to the inside of the tube ends, this can be removed using a Vargus deburring tool.
For my initial turning, I use bushings at each end of the block. These are specially designed for making pens. Part of the bushing is sized to the brass tube and slips inside. The part of the bushing outside is sized to the desired diameter that you want to turn the block down to. I mount these bushings on my lathe using a cone drive center (left) and revolving cone center in the tailstock (right).
Here is what the piece looks like after rough turning. I find that a skew works well for more rapid material removal. The Tru-stone will quickly dull the tool though. A carbide-tip scraper (shown) will do a great job in shaping and the carbide can stand up to the Tru-stone.
Once I get close to the final dimensions, I remove the bushings and place the brass tube directly between the cone centers. This is called "between-center turning" and I find it to be more accurate in keeping the turning precisely aligned around the central axis of the brass tube. Now it's a matter of doing the final shaping and dimensioning of the piece, stopping frequently to check measurements with the calipers. I measure the trim pieces used for this particular pen and custom fit the body to precisely fit the trim pieces. You'd be surprised at how much variability there can be in a supposedly uniform part.
Next I mount my home-made squaring block. This is a piece of hard maple mounted onto a faceplate. I trued-up the face of this piece and stuck on a velcro backing pad. I can then easily place hook-and-loop sandpaper onto the block. Using a Jacobs chuck in the tailstock, I place a center punch that is just small enough to fit inside the brass tube. Now I slide my Tru-stone body piece with brass tube onto the center punch. This holds the piece exactly parallel with the turning axis. When I slide the Tru-stone piece up against the sandpaper, it accurately squares the end of the piece. This makes for a precise fit with the trim components.
Here are the body pieces all ready to polish. Note the visible sanding marks from the 400 grit sandpaper.
The piece goes back on the lathe between centers for wet sanding with Micro-mesh pads. I go from 1800 to 3600 pads (equivalent to 1350 grit sandpaper), which starts bringing a gloss to the surface.
I next polish the body pieces using my buffer. I order my buffing supplies from Caswell Plating. Their 6" Canton Flannel wheels work very nicely. I use their Formax 6163 and 6165 polishing compounds on my two wheels. I'm able to get a high gloss, glass-like finish from the buffer.
Here is a comparison of the two body pieces. Guess which one has been micro-meshed and buffed?
Now both pieces have been polished. Here is an end view of the body piece that abuts the nib. Notice how the Tru-stone is about 1 mm thick at this point. This is why it's critical that the Tru-stone be turned around the precise center axis of the brass tube. Any variation at all will result in varying thickness around the brass tube and a sloppy fit.
Pen assembly is next. I use my drill press as a pen press. The various components are pressure fitted into the ends of the brass tubes. I use a little Loctite to ensure that these fitting stay tight. I always debur and wire brush the the brass tubes before assembly to smooth the assembly process.
After assembly, I'll put on a light coat of Renaissance wax and buff with a soft cloth. The wax helps prevent fingerprints.
And here is the completed pen.
While the concepts of making such a pen are fairly simple, there is a lot of equipment, technical know-how, and skill required to get a pen that is perfect. I make my pens one at a time and derive my enjoyment by making each one as perfect as I can get it.