From 3cd066ccde782d805e8c68d59bebdcbb15a86fc0 Mon Sep 17 00:00:00 2001 From: nwickel Date: Mon, 27 Apr 2026 12:32:08 +0200 Subject: [PATCH] Update slides --- slides/lead_longitudinal.pdf | Bin 760123 -> 744376 bytes slides/lead_longitudinal.tex | 85 +++++++++++++++++------------------ 2 files changed, 41 insertions(+), 44 deletions(-) diff --git a/slides/lead_longitudinal.pdf b/slides/lead_longitudinal.pdf index 337b06798b6d0dbea46dfd337d7e82298cdd2647..bf98cb6b6e12c97e820621f722b4fa4aa264b583 100644 GIT binary patch delta 87536 zcmafbbwE`66EEz-?$X_zDo89FbO{C`sfZ%oh@^ms;1Vj0NFC`iXhC8LMUW7XR8Wx+ z6chpJ5)j_mUA))(yZ62KkM23&bIy0_J2Rh|(T1newRoxETjaMORnS8j!Ig+yH%OcImqX+D2KZ76lRpGtg4XHrn4Iv@9=;#0TE-B&c`oHle&>E?zu zS3gawbR~TQCC_;74r$J(iM^Zjveu_{%aZMnxyGGJ6Y(4W8JN?bjmg67SW08(=#h%y;3bKTGS0@!`ChJu z4_QWCXG^WK7*A0R_pLll*DbAV{xSdRaA47ujETEtn-M=#@wh~JR@)gf``FvHm76mc zT<=8TbRD$qUA63ghEmk^^ClG(cz(>LeQX!?T=pXJ$jQp)nMbsFr3Kc&xetw7b_(Bu zWP)?xsh-W!;{BfAhpnH@l9ACLyJWS`8#C;!ZRy&#$a}UYTSORtY{BDpn4pn1l|A2K 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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}