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Nora Wickelmaier
2026-04-27 12:32:08 +02:00
parent 7b2649be60
commit 3cd066ccde
2 changed files with 41 additions and 44 deletions
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slides/lead_longitudinal.tex
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@@ -173,7 +173,7 @@ xyplot(Reaction ~ Days | Subject,
\begin{itemize} \begin{itemize}
\item The random intercept model adds a random intercept for each subject \item The random intercept model adds a random intercept for each subject
\[ \[
y_{ij} = \beta_0 + \beta_1\,Days_{ij} + \upsilon_{0i} + \varepsilon_i y_{ij} = \beta_0 + \beta_1\,Days_{ij} + \upsilon_{0i} + \varepsilon_{ij}
\] \]
with $\upsilon_{0i} \overset{iid}{\sim} N(0, \sigma^2_{\upsilon})$, with $\upsilon_{0i} \overset{iid}{\sim} N(0, \sigma^2_{\upsilon})$,
$\varepsilon_{ij} \overset{iid}{\sim} N(0, \sigma^2_\varepsilon)$ $\varepsilon_{ij} \overset{iid}{\sim} N(0, \sigma^2_\varepsilon)$
@@ -455,7 +455,7 @@ with
\end{column} \end{column}
\begin{column}{.4\textwidth} \begin{column}{.4\textwidth}
\[ \[
y_{ij} = \beta_0 + \beta_1 time + \upsilon_{0i} + \upsilon_{1i} time + y_{ij} = \beta_0 + \beta_1 t_{ij} + \upsilon_{0i} + \upsilon_{1i} t_{ij} +
\varepsilon_{ij} \varepsilon_{ij}
\] \]
with with
@@ -466,7 +466,8 @@ with
\sigma^2_{\upsilon_0} & \sigma_{\upsilon_0 \upsilon_1} \\ \sigma^2_{\upsilon_0} & \sigma_{\upsilon_0 \upsilon_1} \\
\sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} \\ \sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} \\
\end{pmatrix} \right)\\ \end{pmatrix} \right)\\
\boldsymbol{\varepsilon}_i &\overset{iid}\sim N(\mathbf{0}, \, \sigma^2 %\boldsymbol{\varepsilon}_i &\overset{iid}\sim N(\mathbf{0}, \, \sigma^2
\varepsilon_{ij} &\overset{iid}{\sim} N(0, \sigma^2_\varepsilon)
\mathbf{I}_{n_i}) \mathbf{I}_{n_i})
\end{align*} \end{align*}
\end{column} \end{column}
@@ -518,27 +519,28 @@ with
\sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} & \sigma_{\upsilon_1 \upsilon_2}\\ \sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} & \sigma_{\upsilon_1 \upsilon_2}\\
\sigma_{\upsilon_0 \upsilon_2} & \sigma_{\upsilon_1 \upsilon_2} & \sigma^2_{\upsilon_2}\\ \sigma_{\upsilon_0 \upsilon_2} & \sigma_{\upsilon_1 \upsilon_2} & \sigma^2_{\upsilon_2}\\
\end{pmatrix} \right) \\ \end{pmatrix} \right) \\
\boldsymbol{\varepsilon}_i &\overset{iid}{\sim} N(\mathbf{0}, \, \sigma^2 \mathbf{I}_{n_i}) %\boldsymbol{\varepsilon}_{ij} &\overset{iid}{\sim} N(\mathbf{0}, \, \sigma^2 \mathbf{I}_{n_i})
\varepsilon_{ij} &\overset{iid}{\sim} N(0, \sigma^2_\varepsilon)
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\vspace{-.5cm} \vspace{-.5cm}
\end{frame} \end{frame}
\begin{frame}[fragile]{Model with quadratic trend} % \begin{frame}[fragile]{Model with quadratic trend}
\begin{lstlisting} % \begin{lstlisting}
library("lme4") % library("lme4")
%
# random intercept model % # random intercept model
lme1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE) % lme1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE)
%
# random slope model % # random slope model
lme2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE) % lme2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE)
%
# model with quadratic time trend % # model with quadratic time trend
lme3 <- lmer(hamd ~ week + I(week^2) + (week + I(week^2) | id), % lme3 <- lmer(hamd ~ week + I(week^2) + (week + I(week^2) | id),
data = dat, REML = FALSE) % data = dat, REML = FALSE)
\end{lstlisting} % \end{lstlisting}
\end{frame} % \end{frame}
\begin{frame}{Model predictions} \begin{frame}{Model predictions}
\begin{columns} \begin{columns}
@@ -575,7 +577,6 @@ xyplot(
~ week | id, ~ week | id,
data = dat, data = dat,
type = c("p", "l", "g"), type = c("p", "l", "g"),
pch = 16,
distribute.type = TRUE, distribute.type = TRUE,
ylab = "HDRS score", ylab = "HDRS score",
xlab = "Time (Week)") xlab = "Time (Week)")
@@ -663,29 +664,21 @@ xyplot(
\end{block} \end{block}
\end{frame} \end{frame}
\begin{frame}[fragile]{Analysis with centered time variable} % \begin{frame}[fragile]{Analysis with centered time variable}
\begin{lstlisting} % \begin{lstlisting}
dat$week_c <- dat$week - mean(dat$week) % dat$week_c <- dat$week - mean(dat$week)
cor(dat$week, dat$week^2) # 0.96 % cor(dat$week, dat$week^2) # 0.96
cor(dat$week_c, dat$week_c^2) # 0.01 % cor(dat$week_c, dat$week_c^2) # 0.01
%
# random slope model % # random slope model
lme2c <- lmer(hamd ~ week_c + (week_c | id), data = dat, REML = FALSE) % lme2c <- lmer(hamd ~ week_c + (week_c | id), data = dat, REML = FALSE)
%
# model with quadratic time trend % # model with quadratic time trend
lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id), % lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id),
data = dat, REML = FALSE) % data = dat, REML = FALSE)
\end{lstlisting} % \end{lstlisting}
% \begin{itemize} % \nocite{Alday2025, Hedeker2006}
% \item When comparing the estimated parameters, it becomes obvious that not % \end{frame}
% only the intercept changes but the estimates for the (co)variances do as
% well
% \item Why?\pause
% ~Be sure to make an informed choice when centering your
% variables!
% \end{itemize}
\nocite{Alday2025, Hedeker2006}
\end{frame}
\begin{frame}[fragile]{Investigating random effects structure} \begin{frame}[fragile]{Investigating random effects structure}
\begin{itemize} \begin{itemize}
@@ -697,8 +690,12 @@ lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id),
\end{itemize} \end{itemize}
\vfill \vfill
\begin{lstlisting} \begin{lstlisting}
# model with quadratic time trend
m <- lmer(hamd ~ week + I(week^2) + (week + I(week^2) | id),
data = dat, REML = FALSE)
library("lattice") library("lattice")
dotplot(ranef(lme3), scales = list( x = list(relation = "free")))$id dotplot(ranef(m), scales = list( x = list(relation = "free")))$id
\end{lstlisting} \end{lstlisting}
\end{frame} \end{frame}