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```
# UI: Module 5 Step 1
span(
# Text: title
h4("Step 1: Population level MDPP"),
# Sub-goal
p(HTML("<b>Sub-goal:</b> To understand average (population-level) effects of multiple environmental factors on the phenotype.")),
conditionalPanel(
condition = "0",
uiOutput("Mod5Step1_hidden")
),
# Introduction
p(HTML(paste0("<b>Introduction:</b> Here we model multiple sources of environmental variance ($V_",NOT$envEffect,"$) in a single trait,
expressed multiple times within individuals but measured within a single population.
This step illustrates some differences between simple and multiple regression, but may
also allow simulation of more complex data structures, such as correlations between environments."))),
# Exercise
p(HTML("<b>Exercise:</b> to explore multiple sources of $V_",NOT$envEffect,"$.")),
# Input: Number of individuals
getSliderInput("Mod5Step1_NI", Modules_VAR$NI),
# Input: Among-individual variance (Vi)
getSliderInput("Mod5Step1_Vi", Modules_VAR$Vi),
# Input: NMeasurement error variance
getSliderInput("Mod5Step1_Ve", Modules_VAR$Vm),
# Input: Number of trait expression measured
getSliderInput("Mod5Step1_NR", Modules_VAR$NR),
info_msg("For now, we will assume all individuals are sampled equally often and at the same time."),
p(strong("The environment")),
p(HTML(paste0("Let's simulate phenotypes that are influenced by two factors, both of which are shared by the
whole population (e.g., spring temperature) and the values are random from one measurement period
to the next. The environment thus has an intercept effect on phenotype of 0, and a slope that
you can input (we recommend at first that the slope be >> 0). Each environment contributes
the value $[slope]^2Var(",NOT$env,")$ to the total phenotypic variance (see Table 1 in Allegue <i>et al.</i> 2016),
so by specifying the slope (positive or negative), you will affect the total phenotypic variance.
Note that in SQuID each environmental effect is standardized (i.e. mean = 0 and variance = 1)."))),
p("Enter the slope for each environmental factor. These can be either positive or negative. "),
# Input: Environment 1
getSliderInput("Mod5Step1_B1", Modules_VAR$B1.1),
# Input: Environment 2
getSliderInput("Mod5Step1_B2", Modules_VAR$B2.1),
# Simulation run button
actionButton("Mod5Step1_Run", label = Modules_VAR$Run$label, icon = Modules_VAR$Run$icon, class = "runButton"),
runningIndicator(),
sim_msg(),
p(strong("Results")),
p("Suppose we assume that there is only one environmental effect. That is, we analyse
the population we simulated using the following model:"),
# Equation
p(paste0("$$",
NOT$trait.1,"_{",NOT$time,NOT$ind,"} =
",EQ1$mean0," +
",EQ1$dev0," +
",EQ1$mean1,NOT$env,"_{1",NOT$time,NOT$ind,"} +
",NOT$error,"_{",NOT$time,NOT$ind,"}$$")),
p("A mixed statistical model estimates the parameters:"),
displayRCode("# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 1 + X1 + (1|Individual), data = sampled_data)"),
p("Statistical output:"),
# Output: Table 1
uiOutput("Mod5Step1_summary_table1"),
p(HTML(paste0("This makes the simple point, also made in Module <i>",Module_titles$mod3,"</i>,
that leaving out an important factor inflates other variance components. In this case
it was mostly the residual variance because the environment was set as random from one
measurement to the next and all individuals experienced it."))),
p("A reanalysis with the following model pulls the missing environmental variance out of the residual term:"),
# Equation
p(paste0("$$",
NOT$trait.1,"_{",NOT$time,NOT$ind,"} =
",EQ1$mean0," +
",EQ1$dev0," +
",EQ1$mean1,NOT$env,"_{1",NOT$time,NOT$ind,"} +
",EQ1$mean2,NOT$env,"_{2",NOT$time,NOT$ind,"} +
",NOT$error,"_{",NOT$time,NOT$ind,"}$$")),
displayRCode("# install.packages("lme4")<br>
LMM <- lme4::lmer(Phenotype ~ 1 + X1 + X2 + (1|Individual), data = sampled_data)"),
p("Statistical output:"),
# Output: Table 2
uiOutput("Mod5Step1_summary_table2"),
p("This is a multiple regression within a mixed model. A 3-dimensional graph helps visualize
the way in which the two x variables affect a phenotype in the 2 dimensions defined by the environment."),
# Output: Figure 1
# p(uiOutput("Mod5Step1_3D_1")),
info_msg("The grey plane in the figures represents the population mean phenotypic plane."),
p(plotlyOutput("Mod5Step1_3D_1")),
p(paste0("Individuals in this simulation vary in their intercept by the amount you entered previously in $V_{",NOT$devI,"}$.
Below we pick three individuals across the range of the intercept variance to illustrate
how each individual's plane sits in the space defined by the two environmental variables.
You can see that the three planes are parallel or very close to parallel, and differ only
in their elevation. If you play around with the number of measures within an individual,
you will see that the resolution of these planes requires fairly large sample sizes
(this is covered in more detail later).")),
p(paste0("A new 3-D graph with three individuals picked from the low end of $",NOT$devI,"$,
the middle, and from the high end of $",NOT$devI,"$.")),
# Output: Figure 2
# p(uiOutput("Mod5Step1_3D_2")),
p(plotlyOutput("Mod5Step1_3D_2")),
p(paste0("Run through this simulation several times using different values for $",EQ1$mean1,"$ and $",EQ1$mean2,"$,
including having some slopes negative. In particular, try making the two have opposite signs.
Inspect the table above and look at the two graphs so you gain a feel for how the two slopes
produce a flat plane that may be tilted in various ways.")),
p(HTML("<b>Conclusion:</b> is exercise should reinforce your understanding of where measured
and unmeasured sources of variance end up in a statistical analysis and how systematic
effects of multiple environments can be appropriately captured. In the next step,
we illustrate one important complexity.")),
p(strong("References:")),
p(HTML("Allegue, H., Araya-Ajoy, Y.G., Dingemanse, N.J., Dochtermann N.A., Garamszegi, L.Z., Nakagawa, S., Reale, D., Schielzeth, H.
and Westneat, D.F. (2016). SQuID - Statistical Quantification of Individual Differences: an educational and
statistical tool for understanding multi-level phenotypic data in linear mixed models.
<i>Methods in Ecology and Evolution</i>, 8, 257-267.
<a href='https://doi.org/10.1111/2041-210X.12659' target='_blank'>doi: 10.1111/2041-210X.12659</a>")),
div(class = "line"),
# Go to next step
actionLink("Mod5Step1GotoStep2",
label = "Next Step (2) >>",
class = "linkToModuleSteps")
)
```

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