Correlations of fixed effects in linear mixed models

Ever wondered what those correlations are in a linear mixed model? For example:

The estimated correlation between $\hat{\beta}_1$ and $\hat{\beta}_2$ is $0.988$.  Note that

$\hat{\beta}_1 = (Y_{1,1} + Y_{2,1} + \dots + Y_{10,1})/10=10.360$

and 

$\hat{\beta}_2 = (Y_{1,2} + Y_{2,2} + \dots + Y_{10,2})/10 = 11.040$

From this we can recover the correlation $0.988$ as follows:

By comparison, in the linear model version of the above:

because $Var(\hat{\beta}) = \hat{\sigma}^2 (X^T X)^{-1}$.

Linear models summary sheet

As part of my long slog towards statistical understanding, I started making notes on the very specific topic of linear models. The details are tricky and hard to keep in mind, and it is difficult to go back and forth between books and notes to try to review them. So I tried to summarize the basic ideas into a few pages (the summary sheet is not yet complete).

It's not quite a cheat sheet, so I call it a summary sheet.

Here is the current version:

https://github.com/vasishth/StatisticsNotes

Needless to say (although I feel compelled to so it), the document is highly derivative of lecture notes I've been reading. Corrections and comments and/or suggestions for improvement are most welcome.

Cauchy and determinants: when life was simple

" In Cauchy's day, when life was simple and matrices were small, determinants played a major role in analytic geometry and other parts of mathematics."

Lay, p. 202 [Linear Algebra and its Applications, 3rd Edition (Update)]

Alpha values

Recently seen quote on Gelman's blog:


“No scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.”
(Ronald A. Fisher, 1956, p. 42)


Also read about mindless statistics here: http://people.umass.edu/~bioep740/yr2009/topics/Gigerenzer-jSoc-Econ-1994.pdf

How to append R code in appendix using Sweave

In the statistics course I'm doing, the solutions to assignments have R code, but the instructor doesn't want to see them in the main text but rather in an appendix.  Here's apparently how to do that without copying and pasting the code into an appendix (I got this nifty example from here):

\begin{appendix}
\section{Appendix A}
\subsection{R session information}
<<SessionInforamtaion,echo=F,eval=T,results=tex>>=

toLatex(sessionInfo())
@

\subsection{The simulation's source code}

<>=

Stangle(file.path("Projectpath","RnwFile.Rnw"))

SourceCode <- readLines(file.path("Projectpath","Codefile.R"))

writeLines(SourceCode)

@
\end{appendix}