diff --git a/slides/lead_longitudinal.pdf b/slides/lead_longitudinal.pdf
index 337b067..bf98cb6 100644
Binary files a/slides/lead_longitudinal.pdf and b/slides/lead_longitudinal.pdf differ
diff --git a/slides/lead_longitudinal.tex b/slides/lead_longitudinal.tex
index ce186aa..25ec3ae 100644
--- a/slides/lead_longitudinal.tex
+++ b/slides/lead_longitudinal.tex
@@ -173,7 +173,7 @@ xyplot(Reaction ~ Days | Subject,
   \begin{itemize}
 \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 $\upsilon_{0i} \overset{iid}{\sim} N(0, \sigma^2_{\upsilon})$,
       $\varepsilon_{ij} \overset{iid}{\sim} N(0, \sigma^2_\varepsilon)$
@@ -455,7 +455,7 @@ with
     \end{column}
   \begin{column}{.4\textwidth}
   \[
-    y_{ij} = \beta_0 + \beta_1 time + \upsilon_{0i} + \upsilon_{1i} time +
+    y_{ij} = \beta_0 + \beta_1 t_{ij} + \upsilon_{0i} + \upsilon_{1i} t_{ij} +
     \varepsilon_{ij}
   \]
 with
@@ -466,7 +466,8 @@ with
         \sigma^2_{\upsilon_0} & \sigma_{\upsilon_0 \upsilon_1} \\
         \sigma_{\upsilon_0 \upsilon_1} & \sigma^2_{\upsilon_1} \\
       \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})
 \end{align*}
   \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_2} & \sigma_{\upsilon_1 \upsilon_2} & \sigma^2_{\upsilon_2}\\
       \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{itemize}
 \vspace{-.5cm}
 \end{frame}
 
-\begin{frame}[fragile]{Model with quadratic trend}
-  \begin{lstlisting}
-library("lme4")
-
-# random intercept model
-lme1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE)
-
-# random slope model
-lme2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE)
-
-# model with quadratic time trend
-lme3 <- lmer(hamd ~ week + I(week^2) + (week + I(week^2) | id),
-             data = dat, REML = FALSE)
-  \end{lstlisting}
-\end{frame}
+% \begin{frame}[fragile]{Model with quadratic trend}
+%   \begin{lstlisting}
+% library("lme4")
+%
+% # random intercept model
+% lme1 <- lmer(hamd ~ week + (1 | id), data = dat, REML = FALSE)
+%
+% # random slope model
+% lme2 <- lmer(hamd ~ week + (week | id), data = dat, REML = FALSE)
+%
+% # model with quadratic time trend
+% lme3 <- lmer(hamd ~ week + I(week^2) + (week + I(week^2) | id),
+%              data = dat, REML = FALSE)
+%   \end{lstlisting}
+% \end{frame}
 
 \begin{frame}{Model predictions}
 \begin{columns}
@@ -575,7 +577,6 @@ xyplot(
      ~ week | id,
   data = dat,
   type = c("p", "l", "g"),
-  pch = 16,
   distribute.type = TRUE,
   ylab = "HDRS score",
   xlab = "Time (Week)")
@@ -663,29 +664,21 @@ xyplot(
   \end{block}
 \end{frame}
 
-\begin{frame}[fragile]{Analysis with centered time variable}
-\begin{lstlisting}
-dat$week_c <- dat$week - mean(dat$week)
-cor(dat$week, dat$week^2)       # 0.96
-cor(dat$week_c, dat$week_c^2)   # 0.01
-
-# random slope model
-lme2c <- lmer(hamd ~ week_c + (week_c | id), data = dat, REML = FALSE)
-
-# model with quadratic time trend
-lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id),
-              data = dat, REML = FALSE)
-\end{lstlisting}
-%  \begin{itemize}
-%    \item When comparing the estimated parameters, it becomes obvious that not
-%      only the intercept changes but the estimates for the (co)variances do as
-%      well
-%    \item Why?\pause
-%      ~Be sure to make an informed choice when centering your
-%      variables!
-%  \end{itemize}
-  \nocite{Alday2025, Hedeker2006}
-\end{frame}
+% \begin{frame}[fragile]{Analysis with centered time variable}
+% \begin{lstlisting}
+% dat$week_c <- dat$week - mean(dat$week)
+% cor(dat$week, dat$week^2)       # 0.96
+% cor(dat$week_c, dat$week_c^2)   # 0.01
+%
+% # random slope model
+% lme2c <- lmer(hamd ~ week_c + (week_c | id), data = dat, REML = FALSE)
+%
+% # model with quadratic time trend
+% lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id),
+%               data = dat, REML = FALSE)
+% \end{lstlisting}
+%   \nocite{Alday2025, Hedeker2006}
+% \end{frame}
 
 \begin{frame}[fragile]{Investigating random effects structure}
   \begin{itemize}
@@ -697,8 +690,12 @@ lme3c <- lmer(hamd ~ week_c + I(week_c^2) + (week_c + I(week_c^2)|id),
   \end{itemize}
   \vfill
 \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")
-dotplot(ranef(lme3), scales = list( x = list(relation = "free")))$id
+dotplot(ranef(m), scales = list( x = list(relation = "free")))$id
 \end{lstlisting}
 \end{frame}