% NOTE -- ONLY EDIT THE .Rnw FILE!!! The .tex file is % likely to be overwritten. \setkeys{Gin}{width=\textwidth} \begin{frame} \frametitle{Item Response Models} \begin{itemize} \item Models for binary (or ordered categorical) data that are cross-classified according to subject and item are sometimes called \Emph{Item Response} or \Emph{IRT} (Item Response Theory) models. \item There is a long history of models for such data with many contributors. Only recently have statisticians become aware of this literature and considered how such models could be framed in the context of GLMMs. \item Even when approaching IRT models as GLMMs they were not expressed as GLMMs with crossed random effects, because of software limitations. \item Because \code{glmer} can fit GLMMs with crossed random effects, we can approach such models as GLMMs with random effects for subject and item. \end{itemize} \end{frame} \begin{frame} \frametitle{Data from a study of verbal aggression} \begin{itemize} \item Results on a study of verbal aggression, used as an example through the book \Emph{Expanatory Item Response Models}, edited by De Boeck and Wilson (Springer, 2004) are available as the data set \code{VerbAgg}, in the ``long'' format. \item The items correspond to scenarios for which the subject was asked if they would curse, scold or shout. \item The scenarios are classified according to the behavior mode (want versus do) and according to the situation (self-to-blame versus other-to-blame). \item The subjects are classified by sex. Each subject's score on a separately administered anger index (STAXI) is given. \item The response was recorded on a three-level ordinal scale (``no'', ``perhaps'' and ``yes''). We will consider a dichotomous version, \code{r2}. \end{itemize} \end{frame} \begin{frame}[fragile] \frametitle{Structure of VerbAgg data} \begin{itemize} \item We also check that the item-level covariates and the person-level covariates are consistently defined. \end{itemize} \begin{Schunk} \begin{Sinput} > str(VerbAgg) \end{Sinput} \begin{Soutput} 'data.frame': 7584 obs. of 9 variables: $ Anger : int 20 11 17 21 17 21 39 21 24 16 ... $ Gender: Factor w/ 2 levels "M","F": 2 2 1 1 1 1 1 1 1 1 ... $ item : Factor w/ 24 levels "S1wantcurse",..: 1 1 1 1 1 1 1 1 1 1 ... $ resp : Ord.factor w/ 3 levels "no"<"perhaps"<..: 1 1 2 2 2 3 3 1 1 3 ... $ id : Factor w/ 316 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ... $ btype : Factor w/ 3 levels "curse","scold",..: 1 1 1 1 1 1 1 1 1 1 ... $ situ : Factor w/ 2 levels "other","self": 1 1 1 1 1 1 1 1 1 1 ... $ mode : Factor w/ 2 levels "want","do": 1 1 1 1 1 1 1 1 1 1 ... $ r2 : Factor w/ 2 levels "N","Y": 1 1 2 2 2 2 2 1 1 2 ... \end{Soutput} \begin{Sinput} > stopifnot(nrow(unique(subset(VerbAgg, select = c(item, + btype, situ, mode)))) == 24, nrow(unique(subset(VerbAgg, + select = c(id, Anger, Gender)))) == 316) \end{Sinput} \end{Schunk} \end{frame} \begin{frame}[fragile] \frametitle{Influence of item-level covariates} \begin{itemize} \item We can check the proportions of responses for combinations of item-level covariates \end{itemize} \begin{Schunk} \begin{Sinput} > round(100 * ftable(prop.table(xtabs(~mode + situ + + resp, VerbAgg), 1:2)), 1) \end{Sinput} \begin{Soutput} resp no perhaps yes mode situ want other 37.7 30.0 32.3 self 55.9 29.1 15.0 do other 49.8 27.2 23.0 self 66.2 23.5 10.3 \end{Soutput} \end{Schunk} \end{frame} \begin{frame}[fragile] \frametitle{Influence of person-level covariates} \includegraphics{figs/item-VerbAggxy} \end{frame} \begin{frame}[fragile] \frametitle{Initial model fit} \begin{Schunk} \begin{Soutput} Generalized linear mixed model fit by the Laplace approximation Formula: r2 ~ Anger * Gender + situ + btype + mode + (1 | id) + (1 | item) Data: VerbAgg AIC BIC logLik deviance 8156 8225 -4068 8136 Random effects: Groups Name Variance Std.Dev. id (Intercept) 1.79337 1.33917 item (Intercept) 0.11715 0.34226 Number of obs: 7584, groups: id, 316; item, 24 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.531954 0.433782 1.226 0.22008 Anger 0.058490 0.019474 3.003 0.00267 GenderF 0.402523 0.782316 0.515 0.60688 situself -1.054279 0.151192 -6.973 3.10e-12 btypescold -1.059810 0.184156 -5.755 8.67e-09 btypeshout -2.103818 0.186515 -11.280 < 2e-16 modedo -0.707055 0.151005 -4.682 2.84e-06 Anger:GenderF -0.004111 0.038170 -0.108 0.91424 \end{Soutput} \end{Schunk} \end{frame} \begin{frame}[fragile] \frametitle{Removing non-significant gender effects} \begin{Schunk} \begin{Soutput} Generalized linear mixed model fit by the Laplace approximation Formula: r2 ~ Anger + situ + btype + mode + (1 | id) + (1 | item) Data: VerbAgg AIC BIC logLik deviance 8155 8210 -4069 8139 Random effects: Groups Name Variance Std.Dev. id (Intercept) 1.81157 1.34595 item (Intercept) 0.11720 0.34235 Number of obs: 7584, groups: id, 316; item, 24 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.63927 0.38334 1.668 0.095391 Anger 0.05685 0.01682 3.380 0.000726 situself -1.05437 0.15122 -6.972 3.12e-12 btypescold -1.05973 0.18419 -5.753 8.75e-09 btypeshout -2.10392 0.18655 -11.278 < 2e-16 modedo -0.70726 0.15104 -4.683 2.83e-06 \end{Soutput} \end{Schunk} \end{frame} \begin{frame}[fragile] \frametitle{Allowing situational/behavior random effects by person} \begin{Schunk} \begin{Soutput} Generalized linear mixed model fit by the Laplace approximation Formula: r2 ~ Anger + situ + btype + mode + (1 | id:btype) + (1 | id:situ) + (1 | id:mode) + (1 | id) + (1 | item) Data: VerbAgg AIC BIC logLik deviance 7751 7827 -3865 7729 Random effects: Groups Name Variance Std.Dev. id:btype (Intercept) 1.41069 1.18772 id:mode (Intercept) 0.80916 0.89953 id:situ (Intercept) 0.61539 0.78447 id (Intercept) 1.70950 1.30748 item (Intercept) 0.17213 0.41489 Number of obs: 7584, groups: id:btype, 948; id:mode, 632; id:situ, 632; id, 316; item, 24 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.79167 0.48183 1.643 0.100375 Anger 0.07483 0.02107 3.551 0.000384 situself -1.35742 0.19199 -7.070 1.55e-12 btypescold -1.36067 0.24118 -5.642 1.68e-08 btypeshout -2.69333 0.24372 -11.051 < 2e-16 modedo -0.94095 0.19503 -4.825 1.40e-06 \end{Soutput} \end{Schunk} \end{frame} \begin{frame}[fragile] \frametitle{Item-specific random effects} \includegraphics{figs/item-itemREplot3} \end{frame} \begin{frame}[fragile] \frametitle{Person-specific random effects - Intercept} \includegraphics{figs/item-idREplot3} \end{frame} \begin{frame}[fragile] \frametitle{Correlated random effects by person} \begin{Schunk} \begin{Soutput} Generalized linear mixed model fit by the Laplace approximation Formula: r2 ~ Anger + situ + btype + mode + (1 + situ + btype + mode | id) + (1 | item) Data: VerbAgg AIC BIC logLik deviance 7727 7880 -3842 7683 Random effects: Groups Name Variance Std.Dev. Corr id (Intercept) 4.53799 2.13025 situself 1.31455 1.14654 -0.521 btypescold 1.62188 1.27353 -0.085 -0.247 btypeshout 4.03049 2.00761 -0.374 0.010 0.423 modedo 1.68341 1.29746 -0.295 0.112 0.102 item (Intercept) 0.18222 0.42687 0.104 Number of obs: 7584, groups: id, 316; item, 24 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.98354 0.47248 2.082 0.03738 Anger 0.06564 0.02041 3.217 0.00130 situself -1.37646 0.19723 -6.979 2.97e-12 btypescold -1.30375 0.23811 -5.475 4.37e-08 btypeshout -2.73191 0.25666 -10.644 < 2e-16 modedo -0.97882 0.20000 -4.894 9.88e-07 \end{Soutput} \end{Schunk} \end{frame}