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# UI: Module 4 Step 4
span(
# Text: title
h4("Step 4: The Bivariate mixed-effects model."),
# Sub-goal
p(HTML("<b>Sub-goal:</b> Understanding the difference between a univariate and a multivariate mixed-effects model,
and how level-specific correlations are modelled.")),
# conditionalPanel(
# condition = "0",
# uiOutput("Mod4Step4_hidden")
# ),
# Introduction
p(HTML(paste0("<b>Introduction:</b> In the previous modules (",Module_titles$mod1,", ",Module_titles$mod3,", and ",
Module_titles$mod6,") we have considered univariate mixed-effects models. As you have seen,
the univariate mixed-effects model enabled us to estimate the variance attributable to variation within-
and among-individuals in a single trait (",NOT$trait.1,") with the following equation:"))),
p(paste0("$$",NOT$trait.1,"_{",NOT$time,NOT$ind,"} =
(",EQ3$mean0," + ",NOT$devI,"_",NOT$ind,") +
",NOT$error,"_{",NOT$time,NOT$ind,"}$$")),
p(HTML(paste0("The variation in intercepts ($V_",NOT$devI,"$) among individuals was assumed to be normally distributed (N)
with a mean of zero and a variance ($\\Omega_{",NOT$devI,"}$) and is called the <i>among-individual variance</i>
(estimated as $V_",NOT$devI,"$: the variance across random intercepts of individuals):"))),
p(paste0("$$[",NOT$devI,"_",NOT$ind,"] \\sim N(0, \\Omega_{",NOT$devI,"}): \\Omega_{",NOT$devI,"} = [V_",NOT$devI,"]$$")),
p(HTML(paste0("A residual error ($",NOT$error,"_{",NOT$time,NOT$ind,"}$) was also assumed to be normally distributed, with zero mean
and a variance ($\\Omega_{",NOT$error,"}$) representing the <i>within-individual variance</i>:"))),
p(paste0("$$[",NOT$error,"_{",NOT$time,NOT$ind,"}] \\sim N(0, \\Omega_{",NOT$error,"}): \\Omega_{",NOT$error,"} = [V_",NOT$error,"]$$")),
p(HTML(paste0("In the bivariate mixed-effects models, we are estimating these parameters simultaneously for two traits.
That is, the model can be formulated as a set of two phenotypic equations (one for $",NOT$trait.1,"$ and one for $",NOT$trait.2,"$):"))),
p(paste0("$$",
NOT$trait.1,"_{",NOT$time,NOT$ind,"} =
(",EQ$mean0.1," + ",EQ$dev0.1,") +
",NOT$error,"_{",NOT$trait.1,NOT$time,NOT$ind,"}$$
$$",
NOT$trait.2,"_{",NOT$time,NOT$ind,"} =
(",EQ$mean0.2," + ",EQ$dev0.2,") +
",NOT$error,"_{",NOT$trait.2,NOT$time,NOT$ind,"}
$$")),
p(paste0("As was the case for univariate models, the random intercepts ($",NOT$devI,"_",NOT$ind,"$)
and the within-individual contributions ($",NOT$error,"_{",NOT$time,NOT$ind,"}$) to ",NOT$trait.1," and ",NOT$trait.2," are modelled as
having means of zero. However, in this bivariate case, neither the random
intercepts nor the residual errors are independent. Instead, the random intercepts
are now distributed assuming a multivariate normal distribution with a variance-covariance structure
($\\Omega_{",NOT$devI,"}$) specifying the among-individual variances ($V_{",NOT$devI,"_",NOT$trait.1,"}$ and $V_{",NOT$devI,"_",NOT$trait.2,"}$)
and the among-individual covariance between the two attributes ($Cov_{",NOT$devI,"_",NOT$trait.1,",",NOT$devI,"_",NOT$trait.2,"}$): ")),
p(paste0(
"$$ \\Omega_{",NOT$devI,"}=
\\begin{pmatrix}
V_{",NOT$devI,"_",NOT$trait.1,"} & Cov_{",NOT$devI,"_",NOT$trait.1,",",NOT$devI,"_",NOT$trait.2,"} \\\\
Cov_{",NOT$devI,"_",NOT$trait.1,",",NOT$devI,"_",NOT$trait.2,"} & V_{", NOT$devI,"_",NOT$trait.2,"}\\\\
\\end{pmatrix}
$$")),
p("The residual errors ($",NOT$error,"_{",NOT$time,NOT$ind,"}$) are likewise assumed to be drawn from a multivariate normal distribution,
with means of zero, within-individual variances ($V_{",NOT$error,"_",NOT$trait.1,"}$ and $V_{",NOT$error,"_",NOT$trait.2,"}$), and within-individual
covariances ($Cov_{",NOT$error,"_{",NOT$trait.1,"},",NOT$error,"_{",NOT$trait.2,"}}$):"),
p(paste0(
"$$ \\Omega_{",NOT$error,"}=
\\begin{pmatrix}
V_{",NOT$error,"_",NOT$trait.1,"} & Cov_{",NOT$error,"_",NOT$trait.1,",",NOT$error,"_",NOT$trait.2,"} \\\\
Cov_{",NOT$error,"_",NOT$trait.1,",",NOT$error,"_",NOT$trait.2,"} & V_{", NOT$error,"_",NOT$trait.2,"} \\\\
\\end{pmatrix}
$$")),
p("From these estimated matrices, we can calculate the phenotypic variances for
each trait by adding up the variances estimated at each level:"),
p(paste0("$$V_{",NOT$total,"_",NOT$trait.1,"} = V_{",NOT$devI,"_",NOT$trait.1,"} + V_{",NOT$error,"_",NOT$trait.1,"}$$
$$V_{",NOT$total,"_",NOT$trait.2,"} = V_{",NOT$devI,"_",NOT$trait.2,"} + V_{",NOT$error,"_",NOT$trait.2,"}$$")),
p("In the same fashion, we can calculate the phenotypic covariance between the
two traits by adding up the covariances estimated at each level:"),
p(paste0("$$Cov_{",NOT$total,"_",NOT$trait.1,", ",NOT$total,"_",NOT$trait.2,"} =
Cov_{",NOT$devI,"_",NOT$trait.1,",",NOT$devI,"_",NOT$trait.2,"} +
Cov_{",NOT$error,"_",NOT$trait.1,",",NOT$error,"_",NOT$trait.2,"}$$")),
p("With this information in hand, we can now calculate the overall phenotypic correlation in the data."),
p(paste0("$$r_{",NOT$total,"_",NOT$trait.1,", ",NOT$total,"_",NOT$trait.2,"} =
\\frac{Cov_{",NOT$total,"_",NOT$trait.1,",",NOT$total,"_",NOT$trait.2,"}}
{\\sqrt{V_{",NOT$total,"_",NOT$trait.1,"}V_{",NOT$total,"_",NOT$trait.2,"}}}$$")),
strong("Conclusion: "),
p("Bivariate mixed-effects models differ distinctly from univariate mixed-effects
models as the former assumes multivariate normality while the latter assumes univariate normality.
Bivariate mixed-effects models estimate variances and covariances within and among each specified
level from which overall phenotypic variances and covariances, as well as correlation,
can be subsequently derived."),
div(class = "line"),
actionLink("Mod4Step4GotoStep3", label = "<< Previous Step (3)", class = "linkToModuleSteps"), # Go to previous step
span(Modules_VAR$StepLink$sep, class = "step-Link"),
actionLink("Mod4Step4GotoStep5", label = "Next Step (5) >>", class = "linkToModuleSteps") # Go to next step
)
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