## 7.7 Hypotheses regarding spatial structure

We will evaluate the data support for the following hypotheses about the population structure:

• H1: stock 3 subpopulations defined by management units
• H2: coast+PS 2 subpopulations defined by coastal versus WA inland
• H3: N+S 2 subpopulations defined by north and south split in the middle of Oregon
• H4:NC+strait+PS+SC 4 subpopulations defined by N coastal, S coastal, SJF+Georgia Strait, and Puget Sound
• H5: panmictic All regions are part of the same panmictic population
• H6: site Each of the 11 regions is a subpopulation

These hypotheses translate to these $$\mathbf{Z}$$ matrices (H6 not shown; it is an identity matrix): $\begin{equation*} \begin{array}{rcccc} &H1&H2&H4&H5\\ &\text{pnw ps ca}&\text{coast pc}&\text{nc is ps sc}&\text{pan}\\ \hline \begin{array}{r}\text{Coastal Estuaries}\\ \text{Olympic Peninsula} \\ \text{Str. Juan de Fuca} \\ \text{San Juan Islands} \\ \text{Eastern Bays} \\ \text{Puget Sound} \\ \text{CA Mainland} \\ \text{CA Channel Islands} \\ \text{OR North Coast} \\ \text{OR South Coast} \\ \text{Georgia Strait} \end{array}& \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}& \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}& \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 \end{bmatrix}& \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \end{array} \end{equation*}$

To tell MARSS() the form of $$\mathbf{Z}$$, we construct the same matrix in R. For example, for hypotheses 1, we can write:

Z.model <- matrix(0, 11, 3)
Z.model[c(1, 2, 9, 10), 1] <- 1 # which elements in col 1 are 1
Z.model[c(3:6, 11), 2] <- 1 # which elements in col 2 are 1
Z.model[7:8, 3] <- 1 # which elements in col 3 are 1

Or we can use a short-cut by specifying $$\mathbf{Z}$$ as a factor that has the name of the subpopulation associated with each row in $$\mathbf{y}$$. For hypothesis 1, this is

Z1 <- factor(c("pnw", "pnw", rep("ps", 4), "ca", "ca", "pnw",
"pnw", "ps"))

Notice it is 11 elements in length; one element for each row of data.