# Trend analysis from aggregate data

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Often, people collect data for time with replication. For example, the LTRM program collects fish, aquatic vegetation, and water quality data through time. Multiple samples are collected for each year. However, these observations are not independent and failure to consider this would be pseudoreplication. Aggregating (or taking the mean) of data within a year can be one method to prevent pseudoreplication. Aggregating comes with a trade-off of losing information about the raw data. State-space models may be a method to recover this information.

State-space models describe a true, but unknown and un-measurable “state” (e.g., the “true” population of catfish in the Upper Mississippi River) and the observation error associated with collecting the data. Kalman Fileters can be used to fit these model such as the MARSS package in R can be used to fit these models.

We were interesting in comparing state-space models from the MARSS package to other methods such as simple linear regression and auto-regressive models (publication here). Using simulated data and observed data from the LTRM, we found that the simpler models performed better than the state-space models likely because the LTRM data was not long enough for the state-space models.

# tikz in LaTeX and Structural Equation Modeling

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During grad school, I attended an ESA Workshop on Structural Equation Modeling (SEM) let by Jim Grace. The approach allows for multivariate analysis with multiple predictors, multiple response variables, and latent variables. Up until now, my research never required using the method and I never bought the software he recommended at the time because the GUI program recommended by Grace was too expensive for my limited needs.

Recently, I had a need to use SEM at work. We had two response variables: environmental DNA (eDNA) and the ash-free dry weight of an aquatic organism (AFDW). Both were predicted by multiple environmental variables and AFDW predicted eDNA. A perfect problem for SEM.

To refresh myself of SEM, I revisited Grace’s work. I discovered that he maintains an excellent tutorial about SEM. The pages provide a nice introduction, as does his (slightly outdated) book, his classic book, and a recent Ecoshephere article.

However, I did not have a nice way to plot my results. I did not want to use a WYSIWYG tool like Inkscape or Power Point. But I remembered the tikz package in LaTeX. Here’s the figure I created:

Example SEM plot.

I created the figure using this LaTeX code:

 \documentclass{article}

 \usepackage[paperheight =11.3cm, paperwidth =9.5cm, margin = 0.1cm]{geometry} \usepackage{tikz} \usetikzlibrary{arrows} \usetikzlibrary{positioning} \begin{document} \pagenumbering{gobble} \begin{tikzpicture}[ -> , >=stealth',auto,node distance=3.5cm, thick,main node/.style={rectangle,draw, font=\sffamily}] \node[main node] (1) {Lake}; \node[main node] (2) [below of=1] {Depth}; \node[main node] (3) [below of=2] {Non-habitat}; \node[main node] (4) [below of=3] {Habitat}; \node[main node] (6) [below right of=2, align = center] {AFDW\\ $$r^2 = 0.223$$}; \node[main node] (7) [right of=6, align = center] {eDNA\\ $$r^2 = 0.384$$}; \path[every node/.style={font=\sffamily\small}] (1) edge node [above = 40pt] {\textbf{0.497}} (6) (2) edge node [left = 10pt] {\textbf{-0.370}} (6) (3) edge node [above] {0.094} (6) (4) edge node [left = 10pt] {0.116} (6) (1) edge[bend left] node [above = 10 pt] {\textbf{0.385}} (7) (2) edge[bend left] node [above = 5pt ] {0.197} (7) (3) edge[bend right] node [above = 0pt] {-0.298} (7) (4) edge[bend right] node [below = 5pt] {0.204} (7) (6) edge node [ ] {-0.180} (7); \end{tikzpicture} 

\end{document}