├── .gitignore
├── GettingStarted.Rmd
├── GettingStarted.md
├── Tutorial1-ExploringLMERobjects.Rmd
├── Tutorial1-ExploringLMERobjects.md
├── Tutorial2-ProperSpecification.Rmd
└── docs
├── GettingStarted.html
├── Tutorial1-ExploringLMERobjects.html
└── Tutorial1-ExploringLMERobjects_files
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└── tabsets.js
/.gitignore:
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1 | .Rproj.user
2 | .Rhistory
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/GettingStarted.Rmd:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Getting Started with Multilevel Modeling in R"
3 | author: "Jared E. Knowles"
4 | date: "November, 25, 2013"
5 | output:
6 | html_document:
7 | keep_md: yes
8 | ---
9 |
10 | ## Jared E. Knowles
11 |
12 | # Introduction
13 |
14 | Analysts dealing with grouped data and complex hierarchical structures in their data ranging from
15 | measurements nested within participants, to counties nested within states or students nested within
16 | classrooms often find themselves in need of modeling tools to reflect this structure of their data.
17 | In R there are two predominant ways to fit multilevel models that account for such structure in the
18 | data. These tutorials will show the user how to use both the `lme4` package in R to fit linear and
19 | nonlinear mixed effect models, and to use `rstan` to fit fully Bayesian multilevel models. The focus
20 | here will be on how to fit the models in R and not the theory behind the models. For background on
21 | multilevel modeling, see the references. [1]
22 |
23 | This tutorial will cover getting set up and running a few basic models using `lme4` in R. Future
24 | tutorials will cover:
25 |
26 | * constructing varying intercept, varying slope, and varying slope and intercept models in R
27 | * generating predictions and interpreting parameters from mixed-effect models
28 | * generalized and non-linear multilevel models
29 | * fully Bayesian multilevel models fit with `rstan` or other MCMC methods
30 |
31 | # Setting up your enviRonment
32 |
33 | Getting started with multilevel modeling in R is simple. `lme4` is the canonical package for
34 | implementing multilevel models in R, though there are a number of packages that depend on and
35 | enhance its feature set, including Bayesian extensions. `lme4` has been recently rewritten to
36 | improve speed and to incorporate a C++ codebase, and as such the features of the package are
37 | somewhat in flux. Be sure to update the package frequently.
38 |
39 | To install `lme4`, we just run:
40 |
41 | ```{r eval=FALSE, results='hide', echo=TRUE}
42 | # Main version
43 | install.packages("lme4")
44 |
45 | # Or to install the dev version
46 | library(devtools)
47 | install_github("lme4",user="lme4")
48 | ```
49 |
50 | ```{r setup, echo=FALSE, error=FALSE, message=FALSE, eval=TRUE, results='hide'}
51 | library(knitr)
52 | knitr::opts_chunk$set(dev='png', fig.width=8, fig.height=6, echo=TRUE,
53 | message=FALSE, error=FALSE, warning=FALSE)
54 | options(width = 100)
55 | ```
56 |
57 | # Read in the data
58 |
59 | Multilevel models are appropriate for a particular kind of data structure where units are nested
60 | within groups (generally 5+ groups) and where we want to model the group structure of the data. For
61 | our introductory example we will start with a simple example from the `lme4` documentation and
62 | explain what the model is doing. We will use data from Jon Starkweather at the [University of North
63 | Texas](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/). Visit the excellent tutorial
64 | [available here for
65 | more.](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/Benchmarks/LinearMixedModels_JDS_Dec2010.pdf)
66 |
67 | ```{r loadandviewdata}
68 | library(lme4) # load library
69 | library(arm) # convenience functions for regression in R
70 | lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
71 | header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
72 | #summary(lmm.data)
73 | head(lmm.data)
74 | ```
75 |
76 | Here we have data on the extroversion of subjects nested within classes and within schools.
77 |
78 | # Fit the Non-Multilevel Models
79 |
80 | Let's start by fitting a simple OLS regression of measures of openness, agreeableness, and
81 | socialability on extroversion. Here we use the `display` function in the excellent `arm` package for
82 | abbreviated output. Other options include `stargazer` for LaTeX typeset tables, `xtable`, or the
83 | `ascii` package for more flexible plain text output options.
84 |
85 | ```{r nonlmermodels}
86 | OLSexamp <- lm(extro ~ open + agree + social, data = lmm.data)
87 | display(OLSexamp)
88 | ```
89 |
90 | So far this model does not fit very well at all. The R model interface is quite a simple one with
91 | the dependent variable being specified first, followed by the `~` symbol. The righ hand side,
92 | predictor variables, are each named. Addition signs indicate that these are modeled as additive
93 | effects. Finally, we specify that datframe on which to calculate the model. Here we use the `lm`
94 | function to perform OLS regression, but there are many other options in R.
95 |
96 | If we want to extract measures such as the AIC, we may prefer to fit a generalized linear model with
97 | `glm` which produces a model fit through maximum likelihood estimation. Note that the model formula
98 | specification is the same.
99 |
100 | ```{r nonlmerglm}
101 | MLexamp <- glm(extro ~ open + agree + social, data=lmm.data)
102 | display(MLexamp)
103 | AIC(MLexamp)
104 | ```
105 |
106 | This results in a poor model fit. Let's look at a simple varying intercept model now.
107 |
108 | # Fit a varying intercept model
109 |
110 | Depending on disciplinary norms, our next step might be to fit a varying intercept model using a
111 | grouping variable such as school or classes. Using the `glm` function and the familiar formula
112 | interface, such a fit is easy:
113 |
114 | ```{r nonlmerfixedeffect}
115 | MLexamp.2 <- glm(extro ~ open + agree + social + class, data=lmm.data )
116 | display(MLexamp.2)
117 | AIC(MLexamp.2)
118 | anova(MLexamp, MLexamp.2, test="F")
119 | ```
120 |
121 | This is called a fixed-effects specification often. This is simply the case of fitting a separate
122 | dummy variable as a predictor for each class. We can see this does not provide much additional model
123 | fit. Let's see if school performs any better.
124 |
125 | ```{r nonlmerfixedeffect2}
126 | MLexamp.3 <- glm(extro ~ open + agree + social + school, data=lmm.data )
127 | display(MLexamp.3)
128 | AIC(MLexamp.3)
129 | anova(MLexamp, MLexamp.3, test="F")
130 | ```
131 |
132 | The school effect greatly improves our model fit. However, how do we interpret these effects?
133 |
134 | ```{r effectbreakdown}
135 | table(lmm.data$school, lmm.data$class)
136 |
137 | ```
138 |
139 | Here we can see we have a perfectly balanced design with fifty observations in each combination of
140 | class and school (if only data were always so nice!).
141 |
142 | Let's try to model each of these unique cells. To do this, we fit a model and use the `:` operator
143 | to specify the interaction between `school` and `class`.
144 |
145 | ```{r interaction}
146 | MLexamp.4 <- glm(extro ~ open + agree + social + school:class, data=lmm.data )
147 | display(MLexamp.4)
148 | AIC(MLexamp.4)
149 | ```
150 |
151 | This is very useful, but what if we want to understand both the effect of the school and the effect
152 | of the class, as well as the effect of the schools and classes? Unfortunately, this is not easily
153 | done with the standard `glm`.
154 |
155 | ```{r interaction2}
156 | MLexamp.5 <- glm(extro ~ open + agree + social + school*class - 1, data=lmm.data )
157 | display(MLexamp.5)
158 | AIC(MLexamp.5)
159 | ```
160 |
161 | # Exploring Random Slopes
162 |
163 | Another alternative is to fit a separate model for each of the school and class combinations. If we
164 | believe the relationsihp between our variables may be highly dependent on the school and class
165 | combination, we can simply fit a series of models and explore the parameter variation among them:
166 |
167 | ```{r}
168 | require(plyr)
169 |
170 | modellist <- dlply(lmm.data, .(school, class), function(x)
171 | glm(extro~ open + agree + social, data=x))
172 | display(modellist[[1]])
173 | display(modellist[[2]])
174 |
175 | ```
176 |
177 | We will discuss this strategy in more depth in future tutorials including how to performan inference
178 | on the list of models produced in this command.
179 |
180 | # Fit a varying intercept model with lmer
181 |
182 | Enter `lme4`. While all of the above techniques are valid approaches to this problem, they are not
183 | necessarily the best approach when we are interested explicitly in variation among and by groups.
184 | This is where a mixed-effect modeling framework is useful. Now we use the `lmer` function with the
185 | familiar formula interface, but now group level variables are specified using a special syntax:
186 | `(1|school)` tells `lmer` to fit a linear model with a varying-intercept group effect using the
187 | variable `school`.
188 |
189 | ```{r lmer1}
190 | MLexamp.6 <- lmer(extro ~ open + agree + social + (1|school), data=lmm.data)
191 | display(MLexamp.6)
192 | ```
193 |
194 | We can fit multiple group effects with multiple group effect terms.
195 |
196 |
197 | ```{r lmer2}
198 | MLexamp.7 <- lmer(extro ~ open + agree + social + (1|school) + (1|class), data=lmm.data)
199 | display(MLexamp.7)
200 | ```
201 |
202 | And finally, we can fit nested group effect terms through the following syntax:
203 |
204 | ```{r lmer3}
205 | MLexamp.8 <- lmer(extro ~ open + agree + social + (1|school/class), data=lmm.data)
206 | display(MLexamp.8)
207 |
208 |
209 | ```
210 |
211 | Here the `(1|school/class)` says that we want to fit a mixed effect term for varying intercepts `1|`
212 | by schools, and for classes that are nested within schools.
213 |
214 | # Fit a varying slope model with lmer
215 |
216 | But, what if we want to explore the effect of different student level indicators as they vary across
217 | classrooms. Instead of fitting unique models by school (or school/class) we can fit a varying slope
218 | model. Here we modify our random effect term to include variables before the grouping terms:
219 | `(1 +open|school/class)` tells R to fit a varying slope and varying intercept model for schools and
220 | classes nested within schools, and to allow the slope of the `open` variable to vary by school.
221 |
222 | ```{r lmer4}
223 | MLexamp.9 <- lmer(extro ~ open + agree + social + (1+open|school/class), data=lmm.data)
224 | display(MLexamp.9)
225 | ```
226 |
227 | # Conclusion
228 |
229 | Fitting mixed effect models and exploring group level variation is very easy within the R language
230 | and ecosystem. In future tutorials we will explore comparing across models, doing inference with
231 | mixed-effect models, and creating graphical representations of mixed effect models to understand
232 | their effects.
233 |
234 | # Appendix
235 |
236 | ```{r sessioninfo}
237 | print(sessionInfo(),locale=FALSE)
238 |
239 | ```
240 |
241 | [1] Examples include [Gelman and Hill](http://stat.columbia.edu/~gelman/arm),
242 | [Gelman et al. 2013](http://stat.columbia.edu/~gelman/book/), etc.
243 |
--------------------------------------------------------------------------------
/GettingStarted.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Getting Started with Multilevel Modeling in R"
3 | author: "Jared E. Knowles"
4 | date: "November, 25, 2013"
5 | output:
6 | html_document:
7 | keep_md: yes
8 | ---
9 |
10 | ## Jared E. Knowles
11 |
12 | # Introduction
13 |
14 | Analysts dealing with grouped data and complex hierarchical structures in their data ranging from
15 | measurements nested within participants, to counties nested within states or students nested within
16 | classrooms often find themselves in need of modeling tools to reflect this structure of their data.
17 | In R there are two predominant ways to fit multilevel models that account for such structure in the
18 | data. These tutorials will show the user how to use both the `lme4` package in R to fit linear and
19 | nonlinear mixed effect models, and to use `rstan` to fit fully Bayesian multilevel models. The focus
20 | here will be on how to fit the models in R and not the theory behind the models. For background on
21 | multilevel modeling, see the references. [1]
22 |
23 | This tutorial will cover getting set up and running a few basic models using `lme4` in R. Future
24 | tutorials will cover:
25 |
26 | * constructing varying intercept, varying slope, and varying slope and intercept models in R
27 | * generating predictions and interpreting parameters from mixed-effect models
28 | * generalized and non-linear multilevel models
29 | * fully Bayesian multilevel models fit with `rstan` or other MCMC methods
30 |
31 | # Setting up your enviRonment
32 |
33 | Getting started with multilevel modeling in R is simple. `lme4` is the canonical package for
34 | implementing multilevel models in R, though there are a number of packages that depend on and
35 | enhance its feature set, including Bayesian extensions. `lme4` has been recently rewritten to
36 | improve speed and to incorporate a C++ codebase, and as such the features of the package are
37 | somewhat in flux. Be sure to update the package frequently.
38 |
39 | To install `lme4`, we just run:
40 |
41 |
42 | ```r
43 | # Main version
44 | install.packages("lme4")
45 |
46 | # Or to install the dev version
47 | library(devtools)
48 | install_github("lme4",user="lme4")
49 | ```
50 |
51 |
52 |
53 | # Read in the data
54 |
55 | Multilevel models are appropriate for a particular kind of data structure where units are nested
56 | within groups (generally 5+ groups) and where we want to model the group structure of the data. For
57 | our introductory example we will start with a simple example from the `lme4` documentation and
58 | explain what the model is doing. We will use data from Jon Starkweather at the [University of North
59 | Texas](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/). Visit the excellent tutorial
60 | [available here for
61 | more.](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/Benchmarks/LinearMixedModels_JDS_Dec2010.pdf)
62 |
63 |
64 | ```r
65 | library(lme4) # load library
66 | library(arm) # convenience functions for regression in R
67 | lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
68 | header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
69 | #summary(lmm.data)
70 | head(lmm.data)
71 | ```
72 |
73 | ```
74 | ## id extro open agree social class school
75 | ## 1 1 63.69356 43.43306 38.02668 75.05811 d IV
76 | ## 2 2 69.48244 46.86979 31.48957 98.12560 a VI
77 | ## 3 3 79.74006 32.27013 40.20866 116.33897 d VI
78 | ## 4 4 62.96674 44.40790 30.50866 90.46888 c IV
79 | ## 5 5 64.24582 36.86337 37.43949 98.51873 d IV
80 | ## 6 6 50.97107 46.25627 38.83196 75.21992 d I
81 | ```
82 |
83 | Here we have data on the extroversion of subjects nested within classes and within schools.
84 |
85 | # Fit the Non-Multilevel Models
86 |
87 | Let's start by fitting a simple OLS regression of measures of openness, agreeableness, and
88 | socialability on extroversion. Here we use the `display` function in the excellent `arm` package for
89 | abbreviated output. Other options include `stargazer` for LaTeX typeset tables, `xtable`, or the
90 | `ascii` package for more flexible plain text output options.
91 |
92 |
93 | ```r
94 | OLSexamp <- lm(extro ~ open + agree + social, data = lmm.data)
95 | display(OLSexamp)
96 | ```
97 |
98 | ```
99 | ## lm(formula = extro ~ open + agree + social, data = lmm.data)
100 | ## coef.est coef.se
101 | ## (Intercept) 57.84 3.15
102 | ## open 0.02 0.05
103 | ## agree 0.03 0.05
104 | ## social 0.01 0.02
105 | ## ---
106 | ## n = 1200, k = 4
107 | ## residual sd = 9.34, R-Squared = 0.00
108 | ```
109 |
110 | So far this model does not fit very well at all. The R model interface is quite a simple one with
111 | the dependent variable being specified first, followed by the `~` symbol. The righ hand side,
112 | predictor variables, are each named. Addition signs indicate that these are modeled as additive
113 | effects. Finally, we specify that datframe on which to calculate the model. Here we use the `lm`
114 | function to perform OLS regression, but there are many other options in R.
115 |
116 | If we want to extract measures such as the AIC, we may prefer to fit a generalized linear model with
117 | `glm` which produces a model fit through maximum likelihood estimation. Note that the model formula
118 | specification is the same.
119 |
120 |
121 | ```r
122 | MLexamp <- glm(extro ~ open + agree + social, data=lmm.data)
123 | display(MLexamp)
124 | ```
125 |
126 | ```
127 | ## glm(formula = extro ~ open + agree + social, data = lmm.data)
128 | ## coef.est coef.se
129 | ## (Intercept) 57.84 3.15
130 | ## open 0.02 0.05
131 | ## agree 0.03 0.05
132 | ## social 0.01 0.02
133 | ## ---
134 | ## n = 1200, k = 4
135 | ## residual deviance = 104378.2, null deviance = 104432.7 (difference = 54.5)
136 | ## overdispersion parameter = 87.3
137 | ## residual sd is sqrt(overdispersion) = 9.34
138 | ```
139 |
140 | ```r
141 | AIC(MLexamp)
142 | ```
143 |
144 | ```
145 | ## [1] 8774.291
146 | ```
147 |
148 | This results in a poor model fit. Let's look at a simple varying intercept model now.
149 |
150 | # Fit a varying intercept model
151 |
152 | Depending on disciplinary norms, our next step might be to fit a varying intercept model using a
153 | grouping variable such as school or classes. Using the `glm` function and the familiar formula
154 | interface, such a fit is easy:
155 |
156 |
157 | ```r
158 | MLexamp.2 <- glm(extro ~ open + agree + social + class, data=lmm.data )
159 | display(MLexamp.2)
160 | ```
161 |
162 | ```
163 | ## glm(formula = extro ~ open + agree + social + class, data = lmm.data)
164 | ## coef.est coef.se
165 | ## (Intercept) 56.05 3.09
166 | ## open 0.03 0.05
167 | ## agree -0.01 0.05
168 | ## social 0.01 0.02
169 | ## classb 2.06 0.75
170 | ## classc 3.70 0.75
171 | ## classd 5.67 0.75
172 | ## ---
173 | ## n = 1200, k = 7
174 | ## residual deviance = 99187.7, null deviance = 104432.7 (difference = 5245.0)
175 | ## overdispersion parameter = 83.1
176 | ## residual sd is sqrt(overdispersion) = 9.12
177 | ```
178 |
179 | ```r
180 | AIC(MLexamp.2)
181 | ```
182 |
183 | ```
184 | ## [1] 8719.083
185 | ```
186 |
187 | ```r
188 | anova(MLexamp, MLexamp.2, test="F")
189 | ```
190 |
191 | ```
192 | ## Analysis of Deviance Table
193 | ##
194 | ## Model 1: extro ~ open + agree + social
195 | ## Model 2: extro ~ open + agree + social + class
196 | ## Resid. Df Resid. Dev Df Deviance F Pr(>F)
197 | ## 1 1196 104378
198 | ## 2 1193 99188 3 5190.5 20.81 3.82e-13 ***
199 | ## ---
200 | ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
201 | ```
202 |
203 | This is called a fixed-effects specification often. This is simply the case of fitting a separate
204 | dummy variable as a predictor for each class. We can see this does not provide much additional model
205 | fit. Let's see if school performs any better.
206 |
207 |
208 | ```r
209 | MLexamp.3 <- glm(extro ~ open + agree + social + school, data=lmm.data )
210 | display(MLexamp.3)
211 | ```
212 |
213 | ```
214 | ## glm(formula = extro ~ open + agree + social + school, data = lmm.data)
215 | ## coef.est coef.se
216 | ## (Intercept) 45.02 0.92
217 | ## open 0.01 0.01
218 | ## agree 0.03 0.02
219 | ## social 0.00 0.00
220 | ## schoolII 7.91 0.27
221 | ## schoolIII 12.12 0.27
222 | ## schoolIV 16.06 0.27
223 | ## schoolV 20.43 0.27
224 | ## schoolVI 28.05 0.27
225 | ## ---
226 | ## n = 1200, k = 9
227 | ## residual deviance = 8496.2, null deviance = 104432.7 (difference = 95936.5)
228 | ## overdispersion parameter = 7.1
229 | ## residual sd is sqrt(overdispersion) = 2.67
230 | ```
231 |
232 | ```r
233 | AIC(MLexamp.3)
234 | ```
235 |
236 | ```
237 | ## [1] 5774.203
238 | ```
239 |
240 | ```r
241 | anova(MLexamp, MLexamp.3, test="F")
242 | ```
243 |
244 | ```
245 | ## Analysis of Deviance Table
246 | ##
247 | ## Model 1: extro ~ open + agree + social
248 | ## Model 2: extro ~ open + agree + social + school
249 | ## Resid. Df Resid. Dev Df Deviance F Pr(>F)
250 | ## 1 1196 104378
251 | ## 2 1191 8496 5 95882 2688.2 < 2.2e-16 ***
252 | ## ---
253 | ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
254 | ```
255 |
256 | The school effect greatly improves our model fit. However, how do we interpret these effects?
257 |
258 |
259 | ```r
260 | table(lmm.data$school, lmm.data$class)
261 | ```
262 |
263 | ```
264 | ##
265 | ## a b c d
266 | ## I 50 50 50 50
267 | ## II 50 50 50 50
268 | ## III 50 50 50 50
269 | ## IV 50 50 50 50
270 | ## V 50 50 50 50
271 | ## VI 50 50 50 50
272 | ```
273 |
274 | Here we can see we have a perfectly balanced design with fifty observations in each combination of
275 | class and school (if only data were always so nice!).
276 |
277 | Let's try to model each of these unique cells. To do this, we fit a model and use the `:` operator
278 | to specify the interaction between `school` and `class`.
279 |
280 |
281 | ```r
282 | MLexamp.4 <- glm(extro ~ open + agree + social + school:class, data=lmm.data )
283 | display(MLexamp.4)
284 | ```
285 |
286 | ```
287 | ## glm(formula = extro ~ open + agree + social + school:class, data = lmm.data)
288 | ## coef.est coef.se
289 | ## (Intercept) 80.36 0.37
290 | ## open 0.01 0.00
291 | ## agree -0.01 0.01
292 | ## social 0.00 0.00
293 | ## schoolI:classa -40.39 0.20
294 | ## schoolII:classa -28.15 0.20
295 | ## schoolIII:classa -23.58 0.20
296 | ## schoolIV:classa -19.76 0.20
297 | ## schoolV:classa -15.50 0.20
298 | ## schoolVI:classa -10.46 0.20
299 | ## schoolI:classb -34.60 0.20
300 | ## schoolII:classb -26.76 0.20
301 | ## schoolIII:classb -22.59 0.20
302 | ## schoolIV:classb -18.71 0.20
303 | ## schoolV:classb -14.31 0.20
304 | ## schoolVI:classb -8.54 0.20
305 | ## schoolI:classc -31.86 0.20
306 | ## schoolII:classc -25.64 0.20
307 | ## schoolIII:classc -21.58 0.20
308 | ## schoolIV:classc -17.58 0.20
309 | ## schoolV:classc -13.38 0.20
310 | ## schoolVI:classc -5.58 0.20
311 | ## schoolI:classd -30.00 0.20
312 | ## schoolII:classd -24.57 0.20
313 | ## schoolIII:classd -20.64 0.20
314 | ## schoolIV:classd -16.60 0.20
315 | ## schoolV:classd -12.04 0.20
316 | ## ---
317 | ## n = 1200, k = 27
318 | ## residual deviance = 1135.9, null deviance = 104432.7 (difference = 103296.8)
319 | ## overdispersion parameter = 1.0
320 | ## residual sd is sqrt(overdispersion) = 0.98
321 | ```
322 |
323 | ```r
324 | AIC(MLexamp.4)
325 | ```
326 |
327 | ```
328 | ## [1] 3395.573
329 | ```
330 |
331 | This is very useful, but what if we want to understand both the effect of the school and the effect
332 | of the class, as well as the effect of the schools and classes? Unfortunately, this is not easily
333 | done with the standard `glm`.
334 |
335 |
336 | ```r
337 | MLexamp.5 <- glm(extro ~ open + agree + social + school*class - 1, data=lmm.data )
338 | display(MLexamp.5)
339 | ```
340 |
341 | ```
342 | ## glm(formula = extro ~ open + agree + social + school * class -
343 | ## 1, data = lmm.data)
344 | ## coef.est coef.se
345 | ## open 0.01 0.00
346 | ## agree -0.01 0.01
347 | ## social 0.00 0.00
348 | ## schoolI 39.96 0.36
349 | ## schoolII 52.21 0.36
350 | ## schoolIII 56.78 0.36
351 | ## schoolIV 60.60 0.36
352 | ## schoolV 64.86 0.36
353 | ## schoolVI 69.90 0.36
354 | ## classb 5.79 0.20
355 | ## classc 8.53 0.20
356 | ## classd 10.39 0.20
357 | ## schoolII:classb -4.40 0.28
358 | ## schoolIII:classb -4.80 0.28
359 | ## schoolIV:classb -4.74 0.28
360 | ## schoolV:classb -4.60 0.28
361 | ## schoolVI:classb -3.87 0.28
362 | ## schoolII:classc -6.02 0.28
363 | ## schoolIII:classc -6.54 0.28
364 | ## schoolIV:classc -6.36 0.28
365 | ## schoolV:classc -6.41 0.28
366 | ## schoolVI:classc -3.65 0.28
367 | ## schoolII:classd -6.81 0.28
368 | ## schoolIII:classd -7.45 0.28
369 | ## schoolIV:classd -7.24 0.28
370 | ## schoolV:classd -6.93 0.28
371 | ## schoolVI:classd 0.06 0.28
372 | ## ---
373 | ## n = 1200, k = 27
374 | ## residual deviance = 1135.9, null deviance = 4463029.9 (difference = 4461894.0)
375 | ## overdispersion parameter = 1.0
376 | ## residual sd is sqrt(overdispersion) = 0.98
377 | ```
378 |
379 | ```r
380 | AIC(MLexamp.5)
381 | ```
382 |
383 | ```
384 | ## [1] 3395.573
385 | ```
386 |
387 | # Exploring Random Slopes
388 |
389 | Another alternative is to fit a separate model for each of the school and class combinations. If we
390 | believe the relationsihp between our variables may be highly dependent on the school and class
391 | combination, we can simply fit a series of models and explore the parameter variation among them:
392 |
393 |
394 | ```r
395 | require(plyr)
396 |
397 | modellist <- dlply(lmm.data, .(school, class), function(x)
398 | glm(extro~ open + agree + social, data=x))
399 | display(modellist[[1]])
400 | ```
401 |
402 | ```
403 | ## glm(formula = extro ~ open + agree + social, data = x)
404 | ## coef.est coef.se
405 | ## (Intercept) 35.87 5.90
406 | ## open 0.05 0.09
407 | ## agree 0.02 0.10
408 | ## social 0.01 0.03
409 | ## ---
410 | ## n = 50, k = 4
411 | ## residual deviance = 500.2, null deviance = 506.2 (difference = 5.9)
412 | ## overdispersion parameter = 10.9
413 | ## residual sd is sqrt(overdispersion) = 3.30
414 | ```
415 |
416 | ```r
417 | display(modellist[[2]])
418 | ```
419 |
420 | ```
421 | ## glm(formula = extro ~ open + agree + social, data = x)
422 | ## coef.est coef.se
423 | ## (Intercept) 47.96 2.16
424 | ## open -0.01 0.03
425 | ## agree -0.03 0.03
426 | ## social -0.01 0.01
427 | ## ---
428 | ## n = 50, k = 4
429 | ## residual deviance = 47.9, null deviance = 49.1 (difference = 1.2)
430 | ## overdispersion parameter = 1.0
431 | ## residual sd is sqrt(overdispersion) = 1.02
432 | ```
433 |
434 | We will discuss this strategy in more depth in future tutorials including how to performan inference
435 | on the list of models produced in this command.
436 |
437 | # Fit a varying intercept model with lmer
438 |
439 | Enter `lme4`. While all of the above techniques are valid approaches to this problem, they are not
440 | necessarily the best approach when we are interested explicitly in variation among and by groups.
441 | This is where a mixed-effect modeling framework is useful. Now we use the `lmer` function with the
442 | familiar formula interface, but now group level variables are specified using a special syntax:
443 | `(1|school)` tells `lmer` to fit a linear model with a varying-intercept group effect using the
444 | variable `school`.
445 |
446 |
447 | ```r
448 | MLexamp.6 <- lmer(extro ~ open + agree + social + (1|school), data=lmm.data)
449 | display(MLexamp.6)
450 | ```
451 |
452 | ```
453 | ## lmer(formula = extro ~ open + agree + social + (1 | school),
454 | ## data = lmm.data)
455 | ## coef.est coef.se
456 | ## (Intercept) 59.12 4.10
457 | ## open 0.01 0.01
458 | ## agree 0.03 0.02
459 | ## social 0.00 0.00
460 | ##
461 | ## Error terms:
462 | ## Groups Name Std.Dev.
463 | ## school (Intercept) 9.79
464 | ## Residual 2.67
465 | ## ---
466 | ## number of obs: 1200, groups: school, 6
467 | ## AIC = 5836.1, DIC = 5788.9
468 | ## deviance = 5806.5
469 | ```
470 |
471 | We can fit multiple group effects with multiple group effect terms.
472 |
473 |
474 |
475 | ```r
476 | MLexamp.7 <- lmer(extro ~ open + agree + social + (1|school) + (1|class), data=lmm.data)
477 | display(MLexamp.7)
478 | ```
479 |
480 | ```
481 | ## lmer(formula = extro ~ open + agree + social + (1 | school) +
482 | ## (1 | class), data = lmm.data)
483 | ## coef.est coef.se
484 | ## (Intercept) 60.20 4.21
485 | ## open 0.01 0.01
486 | ## agree -0.01 0.01
487 | ## social 0.00 0.00
488 | ##
489 | ## Error terms:
490 | ## Groups Name Std.Dev.
491 | ## school (Intercept) 9.79
492 | ## class (Intercept) 2.41
493 | ## Residual 1.67
494 | ## ---
495 | ## number of obs: 1200, groups: school, 6; class, 4
496 | ## AIC = 4737.9, DIC = 4683.3
497 | ## deviance = 4703.6
498 | ```
499 |
500 | And finally, we can fit nested group effect terms through the following syntax:
501 |
502 |
503 | ```r
504 | MLexamp.8 <- lmer(extro ~ open + agree + social + (1|school/class), data=lmm.data)
505 | display(MLexamp.8)
506 | ```
507 |
508 | ```
509 | ## lmer(formula = extro ~ open + agree + social + (1 | school/class),
510 | ## data = lmm.data)
511 | ## coef.est coef.se
512 | ## (Intercept) 60.24 4.01
513 | ## open 0.01 0.00
514 | ## agree -0.01 0.01
515 | ## social 0.00 0.00
516 | ##
517 | ## Error terms:
518 | ## Groups Name Std.Dev.
519 | ## class:school (Intercept) 2.86
520 | ## school (Intercept) 9.69
521 | ## Residual 0.98
522 | ## ---
523 | ## number of obs: 1200, groups: class:school, 24; school, 6
524 | ## AIC = 3568.6, DIC = 3507.6
525 | ## deviance = 3531.1
526 | ```
527 |
528 | Here the `(1|school/class)` says that we want to fit a mixed effect term for varying intercepts `1|`
529 | by schools, and for classes that are nested within schools.
530 |
531 | # Fit a varying slope model with lmer
532 |
533 | But, what if we want to explore the effect of different student level indicators as they vary across
534 | classrooms. Instead of fitting unique models by school (or school/class) we can fit a varying slope
535 | model. Here we modify our random effect term to include variables before the grouping terms:
536 | `(1 +open|school/class)` tells R to fit a varying slope and varying intercept model for schools and
537 | classes nested within schools, and to allow the slope of the `open` variable to vary by school.
538 |
539 |
540 | ```r
541 | MLexamp.9 <- lmer(extro ~ open + agree + social + (1+open|school/class), data=lmm.data)
542 | display(MLexamp.9)
543 | ```
544 |
545 | ```
546 | ## lmer(formula = extro ~ open + agree + social + (1 + open | school/class),
547 | ## data = lmm.data)
548 | ## coef.est coef.se
549 | ## (Intercept) 60.26 3.46
550 | ## open 0.01 0.01
551 | ## agree -0.01 0.01
552 | ## social 0.00 0.00
553 | ##
554 | ## Error terms:
555 | ## Groups Name Std.Dev. Corr
556 | ## class:school (Intercept) 2.61
557 | ## open 0.01 1.00
558 | ## school (Intercept) 8.33
559 | ## open 0.00 1.00
560 | ## Residual 0.98
561 | ## ---
562 | ## number of obs: 1200, groups: class:school, 24; school, 6
563 | ## AIC = 3574.9, DIC = 3505.6
564 | ## deviance = 3529.3
565 | ```
566 |
567 | # Conclusion
568 |
569 | Fitting mixed effect models and exploring group level variation is very easy within the R language
570 | and ecosystem. In future tutorials we will explore comparing across models, doing inference with
571 | mixed-effect models, and creating graphical representations of mixed effect models to understand
572 | their effects.
573 |
574 | # Appendix
575 |
576 |
577 | ```r
578 | print(sessionInfo(),locale=FALSE)
579 | ```
580 |
581 | ```
582 | ## R version 3.5.3 (2019-03-11)
583 | ## Platform: x86_64-w64-mingw32/x64 (64-bit)
584 | ## Running under: Windows 10 x64 (build 17134)
585 | ##
586 | ## Matrix products: default
587 | ##
588 | ## attached base packages:
589 | ## [1] stats graphics grDevices utils datasets methods base
590 | ##
591 | ## other attached packages:
592 | ## [1] plyr_1.8.4 arm_1.10-1 MASS_7.3-51.4 lme4_1.1-21 Matrix_1.2-17 knitr_1.22
593 | ##
594 | ## loaded via a namespace (and not attached):
595 | ## [1] Rcpp_1.0.1 lattice_0.20-38 digest_0.6.18 grid_3.5.3 nlme_3.1-139 magrittr_1.5
596 | ## [7] coda_0.19-2 evaluate_0.13 stringi_1.4.3 minqa_1.2.4 nloptr_1.2.1 boot_1.3-22
597 | ## [13] rmarkdown_1.12 splines_3.5.3 tools_3.5.3 stringr_1.4.0 abind_1.4-5 xfun_0.6
598 | ## [19] yaml_2.2.0 compiler_3.5.3 htmltools_0.3.6
599 | ```
600 |
601 | [1] Examples include [Gelman and Hill](http://stat.columbia.edu/~gelman/arm),
602 | [Gelman et al. 2013](http://stat.columbia.edu/~gelman/book/), etc.
603 |
--------------------------------------------------------------------------------
/Tutorial1-ExploringLMERobjects.Rmd:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Fun with merMod Objects"
3 | author: "Jared E. Knowles"
4 | date: "Saturday, May 17, 2014"
5 | output:
6 | html_document:
7 | keep_md: yes
8 | self_contained: no
9 | ---
10 |
11 |
12 |
13 | ```{r setup, echo=FALSE, error=FALSE, message=FALSE, eval=TRUE, results='hide'}
14 | library(knitr)
15 | opts_chunk$set(dev='svg', fig.width=8, fig.height=6, echo=TRUE,
16 | message=FALSE, error=FALSE, warning=FALSE)
17 | options(width = 100)
18 | ```
19 |
20 |
21 | # Introduction
22 |
23 | First of all, be warned, the terminology surrounding multilevel models is vastly inconsistent. For
24 | example, multilevel models themselves may be referred to as hierarchical linear models, random
25 | effects models, multilevel models, random intercept models, random slope models, or pooling models.
26 | Depending on the discipline, software used, and the academic literature many of these terms may be
27 | referring to the same general modeling strategy. In this tutorial I will attempt to provide a user
28 | guide to multilevel modeling by demonstrating how to fit multilevel models in R and by attempting to
29 | connect the model fitting procedure to commonly used terminology used regarding these models.
30 |
31 | We will cover the following topics:
32 |
33 | - The structure and methods of `merMod` objects
34 | - Extracting random effects of `merMod` objects
35 | - Plotting and interpreting `merMod` objects
36 |
37 | If you haven't already, make sure you head over to the [Getting Started With Multilevel Models
38 | tutorial](http://jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r)
39 | in order to ensure you have set up your environment correctly and installed all the necessary
40 | packages. The tl;dr is that you will need:
41 |
42 | - A current version of R (2.15 or greater)
43 | - The `lme4` package (`install.packages("lme4")`)
44 |
45 | # Read in the data
46 |
47 | Multilevel models are appropriate for a particular kind of data structure where units are nested
48 | within groups (generally 5+ groups) and where we want to model the group structure of the data. For
49 | our introductory example we will start with a simple example from the `lme4` documentation and
50 | explain what the model is doing. We will use data from Jon Starkweather at the [University of North
51 | Texas](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/). Visit the excellent tutorial
52 | [available here for
53 | more.](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/Benchmarks/LinearMixedModels_JDS_Dec2010.pdf)
54 |
55 | ```{r loadandviewdata}
56 | library(lme4) # load library
57 | library(arm) # convenience functions for regression in R
58 | lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
59 | header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
60 | #summary(lmm.data)
61 | head(lmm.data)
62 | ```
63 |
64 | Here we have data on the extroversion of subjects nested within classes and within schools.
65 |
66 | Let's understand the structure of the data a bit before we begin:
67 |
68 | ```{r explore}
69 | str(lmm.data)
70 | ```
71 |
72 | Here we see we have two possible grouping variables -- `class` and `school`. Let's
73 | explore them a bit further:
74 |
75 | ```{r exploregroups}
76 | table(lmm.data$class)
77 | table(lmm.data$school)
78 | table(lmm.data$class, lmm.data$school)
79 | ```
80 |
81 | This is a perfectly balanced dataset. In all likelihood you aren't working with a perfectly balanced
82 | dataset, but we'll explore the implications for that in the future. For now, let's plot the data a
83 | bit. Using the excellent `xyplot` function in the `lattice` package, we can explore the relationship
84 | between schools and classes across our variables.
85 |
86 | ```{r xyplot1}
87 | require(lattice)
88 | xyplot(extro ~ open + social + agree | class, data = lmm.data,
89 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
90 | layout = c(4,1), main = "Extroversion by Class")
91 | ```
92 |
93 | Here we see that within classes there are clear stratifications and we also see that the `social`
94 | variable is strongly distinct from the `open` and `agree` variables. We also see that class `a` and
95 | class `d` have significantly more spread in their lowest and highest bands respectively. Let's next
96 | plot the data by `school`.
97 |
98 |
99 | ```{r xyplot2}
100 | xyplot(extro ~ open + social + agree | school, data = lmm.data,
101 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
102 | layout = c(3, 2), main = "Extroversion by School")
103 | ```
104 |
105 | By school we see that students are tightly grouped, but that school `I` and school `VI` show
106 | substantially more dispersion than the other schools. The same pattern among our predictors holds
107 | between schools as it did between classes. Let's put it all together:
108 |
109 |
110 | ```{r xyplot3}
111 | xyplot(extro ~ open + social + agree | school + class, data = lmm.data,
112 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
113 | main = "Extroversion by School and Class")
114 | ```
115 |
116 | Here we can see that school and class seem to closely differentiate the relationship between our
117 | predictors and extroversion.
118 |
119 |
120 | ## Exploring the Internals of a merMod Object
121 |
122 | In the last tutorial we fit a series of random intercept models to our nested data. We will examine
123 | the `lmerMod` object produced when we fit this model in much more depth in order to understand how
124 | to work with mixed effect models in R. We start by fitting a the basic example below grouped by
125 | class:
126 |
127 | ```{r lmer1}
128 | MLexamp1 <- lmer(extro ~ open + agree + social + (1|school), data=lmm.data)
129 | class(MLexamp1)
130 | ```
131 |
132 | First, we see that `MLexamp1` is now an R object of the class `lmerMod`. This `lmerMod` object is an
133 | **S4** class, and to explore its structure, we use `slotNames`:
134 |
135 | ```{r lmermod}
136 | slotNames(MLexamp1)
137 | #showMethods(classes="lmerMod")
138 | ```
139 |
140 | Within the `lmerMod` object we see a number of objects that we may wish to explore. To look at any
141 | of these, we can simply type `MLexamp1@` and then the slot name itself. For example:
142 |
143 | ```{r slotnames1}
144 | MLexamp1@call # returns the model call
145 | MLexamp1@beta # returns the betas
146 | class(MLexamp1@frame) # returns the class for the frame slot
147 | head(MLexamp1@frame) # returns the model frame
148 | ```
149 |
150 | The `merMod` object has a number of methods available -- too many to enumerate here. But, we will go
151 | over some of the more common in the list below:
152 |
153 |
154 | ```{r modMethods}
155 | methods(class = "merMod")
156 |
157 | ```
158 |
159 | A common need is to extract the fixed effects from a `merMod` object. `fixef` extracts a named
160 | numeric vector of the fixed effects, which is handy.
161 |
162 | ```{r fixef}
163 | fixef(MLexamp1)
164 |
165 | ```
166 |
167 | If you want to get a sense of the p-values or statistical significance of these parameters, first
168 | consult the `lme4` help by running `?mcmcsamp` for a rundown of various ways of doing this. One
169 | convenient way built into the package is:
170 |
171 | ```{r confintr}
172 | confint(MLexamp1, level = 0.99)
173 |
174 | ```
175 |
176 | From here we can see first that our fixed effect parameters overlap 0 indicating no evidence of an
177 | effect. We can also see that `.sig01`, which is our estimate of the variability in the random
178 | effect, is very large and very widely defined. This indicates we may have a lack of precision
179 | between our groups - either because the group effect is small between groups, we have too few groups
180 | to get a more precise estimate, we have too few units within each group, or a combination of all of
181 | the above.
182 |
183 | Another common need is to extract the residual standard error, which is necessary for calculating
184 | effect sizes. To get a named vector of the residual standard error:
185 |
186 | ```{r sigmaef}
187 | sigma(MLexamp1)
188 | ```
189 |
190 | For example, it is common practice in education research to standardize fixed effects into "effect
191 | sizes" by dividing the fixed effect paramters by the residual standard error, which can be
192 | accomplished in `lme4` easily:
193 |
194 | ```{r effsize}
195 | fixef(MLexamp1) / sigma(MLexamp1)
196 |
197 | ```
198 |
199 | From this, we can see that our predictors of openness, agreeableness and social are virtually
200 | useless in predicting extroversion -- as our plots showed. Let's turn our attention to the random
201 | effects next.
202 |
203 | ## Explore Group Variation and Random Effects
204 |
205 | In all likelihood you fit a mixed-effect model because you are directly interested in the
206 | group-level variation in your model. It is not immediately clear how to explore this group level
207 | variation from the results of `summary.merMod`. What we get from this output is the variance and the
208 | standard deviation of the group effect, but we do not get effects for individual groups. This is
209 | where the `ranef` function comes in handy.
210 |
211 | ```{r ranef1}
212 | ranef(MLexamp1)
213 |
214 | ```
215 |
216 | Running the `ranef` function gives us the intercepts for each school, but not much additional
217 | information -- for example the precision of these estimates. To do that, we need some additional
218 | commands:
219 |
220 | ```{r ranef2}
221 | re1 <- ranef(MLexamp1, condVar=TRUE) # save the ranef.mer object
222 | class(re1)
223 | attr(re1[[1]], which = "postVar")
224 |
225 | ```
226 |
227 | The `ranef.mer` object is a list which contains a data.frame for each group level. The dataframe
228 | contains the random effects for each group (here we only have an intercept for each school). When we
229 | ask `lme4` for the conditional variance of the random effects it is stored in an `attribute` of
230 | those dataframes as a list of variance-covariance matrices.
231 |
232 | This structure is indeed *complicated*, but it is powerful as it allows for nested, grouped, and
233 | cross-level random effects. Also, the creators of `lme4` have provided users with some simple
234 | shortcuts to get what they are really interested in out of a `ranef.mer` object.
235 |
236 | ```{r ranef3}
237 | re1 <- ranef(MLexamp1, condVar=TRUE, whichel = "school")
238 | print(re1)
239 | dotplot(re1)
240 | ```
241 |
242 | This graphic shows a `dotplot` of the random effect terms, also known as a caterpillar plot. Here
243 | you can clearly see the effects of each school on `extroversion` as well as their standard errors to
244 | help identify how distinct the random effects are from one another. Interpreting random effects is
245 | notably tricky, but for assistance I would recommend looking at a few of these resources:
246 |
247 | - Gelman and Hill 2006 - [Data Analysis Using Regression and Multilevel/Hierarchical Techniques](http://www.stat.columbia.edu/~gelman/arm/)
248 | - John Fox - An R Compantion to Applied Regression [web appendix](http://socserv.mcmaster.ca/jfox/Books/Companion-1E/appendix-mixed-models.pdf)
249 |
250 | ## Using Simulation and Plots to Explore Random Effects
251 |
252 | A common econometric approach is to create what are known as **empirical Bayes** estimates of the
253 | group-level terms. Unfortunately there is not much agreement about what constitutes a proper
254 | standard error for the random effect terms or even how to consistently define **empirical Bayes**
255 | estimates.[^1] However, in R there are a few additional ways to get estimates of the random effects
256 | that provide the user with information about the relative sizes of the effects for each unit and the
257 | precision in that estimate. To do this, we use the `sim` function in the `arm` package.[^2]
258 |
259 | ```{r arm1}
260 | # A function to extract simulated estimates of random effect paramaters from
261 | # lme4 objects using the sim function in arm
262 | # whichel = the character for the name of the grouping effect to extract estimates for
263 | # nsims = the number of simulations to pass to sim
264 | # x = model object
265 | REsim <- function(x, whichel=NULL, nsims){
266 | require(plyr)
267 | mysim <- sim(x, n.sims = nsims)
268 | if(missing(whichel)){
269 | dat <- plyr::adply(mysim@ranef[[1]], c(2, 3), plyr::each(c(mean, median, sd)))
270 | warning("Only returning 1st random effect because whichel not specified")
271 | } else{
272 | dat <- plyr::adply(mysim@ranef[[whichel]], c(2, 3), plyr::each(c(mean, median, sd)))
273 | }
274 | return(dat)
275 | }
276 |
277 | REsim(MLexamp1, whichel = "school", nsims = 1000)
278 |
279 | ```
280 |
281 | The `REsim` function returns for each school the level name `X1`, the estimate name, `X2`, the mean
282 | of the estimated values, the median, and the standard deviation of the estimates.
283 |
284 | Another convenience function can help us plot these results to see how they compare to the results
285 | of `dotplot`:
286 |
287 | ```{r ebplot}
288 | # Dat = results of REsim
289 | # scale = factor to multiply sd by
290 | # var = character of "mean" or "median"
291 | # sd = character of "sd"
292 | plotREsim <- function(dat, scale, var, sd){
293 | require(eeptools)
294 | dat[, sd] <- dat[, sd] * scale
295 | dat[, "ymax"] <- dat[, var] + dat[, sd]
296 | dat[, "ymin"] <- dat[, var] - dat[, sd]
297 | dat[order(dat[, var]), "id"] <- c(1:nrow(dat))
298 | ggplot(dat, aes_string(x = "id", y = var, ymax = "ymax",
299 | ymin = "ymin")) +
300 | geom_pointrange() + theme_dpi() +
301 | labs(x = "Group", y = "Effect Range", title = "Effect Ranges") +
302 | theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
303 | axis.text.x = element_blank(), axis.ticks.x = element_blank()) +
304 | geom_hline(yintercept = 0, color = I("red"), size = I(1.1))
305 | }
306 |
307 | plotREsim(REsim(MLexamp1, whichel = "school", nsims = 1000), scale = 1.2,
308 | var = "mean", sd = "sd")
309 | ```
310 |
311 |
312 | This presents a more conservative view of the variation between random effect components. Depending
313 | on how your data was collected and your research question, alternative ways of estimating these
314 | effect sizes are possible. However, proceed with caution.[^3]
315 |
316 | Another approach recommended by the authors of `lme4` involves the `RLRsim` package. Using this
317 | package we can test whether or not inclusion of the random effects improves model fit and we can
318 | evaluate the p-value of additional random effect terms using a likelihood ratio test based on
319 | simulation.[^4]
320 |
321 | ```{r rlrsim}
322 | library(RLRsim)
323 | m0 <- lm(extro ~ agree + open + social, data =lmm.data) # fit the null model
324 | exactLRT(m = MLexamp1, m0 = m0)
325 |
326 | ```
327 |
328 | Here `exactLRT` issues a warning because we originally fit the model with REML instead of full
329 | maximum likelihood. Fortunately, the `refitML` function in `lme4` allows us to easily refit our
330 | model using full maximum likelihood to conduct an exact test easily.
331 |
332 | ```{r rlrsim2}
333 | mA <- refitML(MLexamp1)
334 | exactLRT(m= mA, m0 = m0)
335 | ```
336 |
337 | Here we can see that the inclusion of our grouping variable is significant, even though the effect
338 | of each individual group may be substantively small and/or imprecisely measured. This is important
339 | in understanding the correct specification of the model. Our next tutorial will cover specification
340 | tests like this in more detail.
341 |
342 | ## What do Random Effects Matter?
343 |
344 | How do interpret the *substantive* impact of our random effects? This is often critical in
345 | observation work trying to use a multilevel structure to understand the impact that the grouping can
346 | have on the individual observation. To do this we select 12 random cases and then we simulate their
347 | predicted value of `extro` if they were placed in each of the 6 schools. Note, that this is a very
348 | simple simulation just using the mean of the fixed effect and the conditional mode of the random
349 | effect and not replicating or sampling to get a sense of the variability. This will be left as an
350 | exercise to the reader and/or a future tutorial!
351 |
352 | ```{r ranefplotsetup}
353 | # Simulate
354 | # Let's create 12 cases of students
355 | #
356 | #sample some rows
357 | simX <- sample(lmm.data$id, 12)
358 | simX <- lmm.data[lmm.data$id %in% simX, c(3:5)] # get their data
359 | # add an intercept
360 | simX[, "Intercept"] <- 1
361 | simX <- simX[, c(4, 1:3)] # reorder
362 | simRE <- REsim(MLexamp1, whichel = "school", nsims = 1000) # simulate randome effects
363 | simX$case <- row.names(simX) # create a case ID
364 | # expand a grid of case IDs by schools
365 | simDat <- expand.grid(case = row.names(simX), school = levels(lmm.data$school))
366 | simDat <- merge(simX, simDat) # merge in the data
367 | # Create the fixed effect predictor
368 | simDat[, "fepred"] <- (simDat[, 2] * fixef(MLexamp1)[1]) + (simDat[, 3] * fixef(MLexamp1)[2]) +
369 | (simDat[, 4] * fixef(MLexamp1)[3]) + (simDat[, 5] * fixef(MLexamp1)[4])
370 | # Add the school effects
371 | simDat <- merge(simDat, simRE[, c(1, 3)], by.x = "school", by.y="X1")
372 | simDat$yhat <- simDat$fepred + simDat$mean # add the school specific intercept
373 | ```
374 |
375 | Now that we have set up a simulated dataframe, let's plot it, first by case:
376 |
377 | ```{r bycaseplot}
378 | qplot(school, yhat, data = simDat) + facet_wrap(~case) + theme_dpi()
379 |
380 | ```
381 |
382 | This plot shows us that within each plot, representing a case, there is tremendous variation by
383 | school. So, moving each student into a different school has large effects on the extroversion score.
384 | But, does each case vary at each school?
385 |
386 | ```{r byschool}
387 | qplot(case, yhat, data = simDat) + facet_wrap(~school) + theme_dpi() +
388 | theme(axis.text.x = element_blank())
389 |
390 | ```
391 |
392 | Here we can clearly see that within each school the cases are relatively the same indicating that
393 | the group effect is larger than the individual effects.
394 |
395 | These plots are useful in demonstrating the relative importance of group and individual effects in a
396 | substantive fashion. Even more can be done to make the the graphs more informative, such as placing
397 | references to the total variability of the outcome and also looking at the distance moving groups
398 | moves each observation from its true value.
399 |
400 | # Conclusion
401 |
402 | `lme4` provides a very powerful object-oriented toolset for dealing with mixed effect models in R.
403 | Understanding model fit and confidence intervals of `lme4` objects requires some diligent research
404 | and the use of a variety of functions and extensions of `lme4` itself. In our next tutorial we will
405 | explore how to identify a proper specification of a random-effect model and Bayesian extensions of
406 | the `lme4` framework for difficult to specify models. We will also explore the generalized linear
407 | model framework and the `glmer` function for generalized linear modeling with multi-levels.
408 |
409 | # Appendix
410 |
411 | ```{r sessioninfo}
412 | print(sessionInfo(),locale=FALSE)
413 |
414 | ```
415 |
416 | [^1]: [See message from `lme4` co-author Doug Bates on this subject]. (https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q4/002984.html)
417 | [^2]: Andrew Gelman and Yu-Sung Su (2014). arm: Data Analysis Using Regression and Multilevel/Hierarchical Models. R package version
418 | 1.7-03. http://CRAN.R-project.org/package=arm
419 | [^3]: [WikiDot FAQ from the R Mixed Models Mailing List](http://glmm.wikidot.com/faq)
420 | [^4]: There are also an extensive series of references available in the `References`
421 | section of the help by running `?exactLRT` and `?exactRLRT`.
422 |
--------------------------------------------------------------------------------
/Tutorial1-ExploringLMERobjects.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Fun with merMod Objects"
3 | author: "Jared E. Knowles"
4 | date: "Saturday, May 17, 2014"
5 | output:
6 | html_document:
7 | keep_md: yes
8 | self_contained: no
9 | ---
10 |
11 |
12 |
13 |
14 |
15 |
16 | # Introduction
17 |
18 | First of all, be warned, the terminology surrounding multilevel models is vastly inconsistent. For
19 | example, multilevel models themselves may be referred to as hierarchical linear models, random
20 | effects models, multilevel models, random intercept models, random slope models, or pooling models.
21 | Depending on the discipline, software used, and the academic literature many of these terms may be
22 | referring to the same general modeling strategy. In this tutorial I will attempt to provide a user
23 | guide to multilevel modeling by demonstrating how to fit multilevel models in R and by attempting to
24 | connect the model fitting procedure to commonly used terminology used regarding these models.
25 |
26 | We will cover the following topics:
27 |
28 | - The structure and methods of `merMod` objects
29 | - Extracting random effects of `merMod` objects
30 | - Plotting and interpreting `merMod` objects
31 |
32 | If you haven't already, make sure you head over to the [Getting Started With Multilevel Models
33 | tutorial](http://jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r)
34 | in order to ensure you have set up your environment correctly and installed all the necessary
35 | packages. The tl;dr is that you will need:
36 |
37 | - A current version of R (2.15 or greater)
38 | - The `lme4` package (`install.packages("lme4")`)
39 |
40 | # Read in the data
41 |
42 | Multilevel models are appropriate for a particular kind of data structure where units are nested
43 | within groups (generally 5+ groups) and where we want to model the group structure of the data. For
44 | our introductory example we will start with a simple example from the `lme4` documentation and
45 | explain what the model is doing. We will use data from Jon Starkweather at the [University of North
46 | Texas](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/). Visit the excellent tutorial
47 | [available here for
48 | more.](http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/Benchmarks/LinearMixedModels_JDS_Dec2010.pdf)
49 |
50 |
51 | ```r
52 | library(lme4) # load library
53 | library(arm) # convenience functions for regression in R
54 | lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
55 | header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
56 | #summary(lmm.data)
57 | head(lmm.data)
58 | ```
59 |
60 | ```
61 | ## id extro open agree social class school
62 | ## 1 1 63.69356 43.43306 38.02668 75.05811 d IV
63 | ## 2 2 69.48244 46.86979 31.48957 98.12560 a VI
64 | ## 3 3 79.74006 32.27013 40.20866 116.33897 d VI
65 | ## 4 4 62.96674 44.40790 30.50866 90.46888 c IV
66 | ## 5 5 64.24582 36.86337 37.43949 98.51873 d IV
67 | ## 6 6 50.97107 46.25627 38.83196 75.21992 d I
68 | ```
69 |
70 | Here we have data on the extroversion of subjects nested within classes and within schools.
71 |
72 | Let's understand the structure of the data a bit before we begin:
73 |
74 |
75 | ```r
76 | str(lmm.data)
77 | ```
78 |
79 | ```
80 | ## 'data.frame': 1200 obs. of 7 variables:
81 | ## $ id : int 1 2 3 4 5 6 7 8 9 10 ...
82 | ## $ extro : num 63.7 69.5 79.7 63 64.2 ...
83 | ## $ open : num 43.4 46.9 32.3 44.4 36.9 ...
84 | ## $ agree : num 38 31.5 40.2 30.5 37.4 ...
85 | ## $ social: num 75.1 98.1 116.3 90.5 98.5 ...
86 | ## $ class : Factor w/ 4 levels "a","b","c","d": 4 1 4 3 4 4 4 4 1 2 ...
87 | ## $ school: Factor w/ 6 levels "I","II","III",..: 4 6 6 4 4 1 3 4 3 1 ...
88 | ```
89 |
90 | Here we see we have two possible grouping variables -- `class` and `school`. Let's
91 | explore them a bit further:
92 |
93 |
94 | ```r
95 | table(lmm.data$class)
96 | ```
97 |
98 | ```
99 | ##
100 | ## a b c d
101 | ## 300 300 300 300
102 | ```
103 |
104 | ```r
105 | table(lmm.data$school)
106 | ```
107 |
108 | ```
109 | ##
110 | ## I II III IV V VI
111 | ## 200 200 200 200 200 200
112 | ```
113 |
114 | ```r
115 | table(lmm.data$class, lmm.data$school)
116 | ```
117 |
118 | ```
119 | ##
120 | ## I II III IV V VI
121 | ## a 50 50 50 50 50 50
122 | ## b 50 50 50 50 50 50
123 | ## c 50 50 50 50 50 50
124 | ## d 50 50 50 50 50 50
125 | ```
126 |
127 | This is a perfectly balanced dataset. In all likelihood you aren't working with a perfectly balanced
128 | dataset, but we'll explore the implications for that in the future. For now, let's plot the data a
129 | bit. Using the excellent `xyplot` function in the `lattice` package, we can explore the relationship
130 | between schools and classes across our variables.
131 |
132 |
133 | ```r
134 | require(lattice)
135 | xyplot(extro ~ open + social + agree | class, data = lmm.data,
136 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
137 | layout = c(4,1), main = "Extroversion by Class")
138 | ```
139 |
140 | 
141 |
142 | Here we see that within classes there are clear stratifications and we also see that the `social`
143 | variable is strongly distinct from the `open` and `agree` variables. We also see that class `a` and
144 | class `d` have significantly more spread in their lowest and highest bands respectively. Let's next
145 | plot the data by `school`.
146 |
147 |
148 |
149 | ```r
150 | xyplot(extro ~ open + social + agree | school, data = lmm.data,
151 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
152 | layout = c(3, 2), main = "Extroversion by School")
153 | ```
154 |
155 | 
156 |
157 | By school we see that students are tightly grouped, but that school `I` and school `VI` show
158 | substantially more dispersion than the other schools. The same pattern among our predictors holds
159 | between schools as it did between classes. Let's put it all together:
160 |
161 |
162 |
163 | ```r
164 | xyplot(extro ~ open + social + agree | school + class, data = lmm.data,
165 | auto.key = list(x = .85, y = .035, corner = c(0, 0)),
166 | main = "Extroversion by School and Class")
167 | ```
168 |
169 | 
170 |
171 | Here we can see that school and class seem to closely differentiate the relationship between our
172 | predictors and extroversion.
173 |
174 |
175 | ## Exploring the Internals of a merMod Object
176 |
177 | In the last tutorial we fit a series of random intercept models to our nested data. We will examine
178 | the `lmerMod` object produced when we fit this model in much more depth in order to understand how
179 | to work with mixed effect models in R. We start by fitting a the basic example below grouped by
180 | class:
181 |
182 |
183 | ```r
184 | MLexamp1 <- lmer(extro ~ open + agree + social + (1|school), data=lmm.data)
185 | class(MLexamp1)
186 | ```
187 |
188 | ```
189 | ## [1] "lmerMod"
190 | ## attr(,"package")
191 | ## [1] "lme4"
192 | ```
193 |
194 | First, we see that `MLexamp1` is now an R object of the class `lmerMod`. This `lmerMod` object is an
195 | **S4** class, and to explore its structure, we use `slotNames`:
196 |
197 |
198 | ```r
199 | slotNames(MLexamp1)
200 | ```
201 |
202 | ```
203 | ## [1] "resp" "Gp" "call" "frame" "flist" "cnms" "lower" "theta" "beta"
204 | ## [10] "u" "devcomp" "pp" "optinfo"
205 | ```
206 |
207 | ```r
208 | #showMethods(classes="lmerMod")
209 | ```
210 |
211 | Within the `lmerMod` object we see a number of objects that we may wish to explore. To look at any
212 | of these, we can simply type `MLexamp1@` and then the slot name itself. For example:
213 |
214 |
215 | ```r
216 | MLexamp1@call # returns the model call
217 | ```
218 |
219 | ```
220 | ## lmer(formula = extro ~ open + agree + social + (1 | school),
221 | ## data = lmm.data)
222 | ```
223 |
224 | ```r
225 | MLexamp1@beta # returns the betas
226 | ```
227 |
228 | ```
229 | ## [1] 59.116514199 0.009750941 0.027788360 -0.002151446
230 | ```
231 |
232 | ```r
233 | class(MLexamp1@frame) # returns the class for the frame slot
234 | ```
235 |
236 | ```
237 | ## [1] "data.frame"
238 | ```
239 |
240 | ```r
241 | head(MLexamp1@frame) # returns the model frame
242 | ```
243 |
244 | ```
245 | ## extro open agree social school
246 | ## 1 63.69356 43.43306 38.02668 75.05811 IV
247 | ## 2 69.48244 46.86979 31.48957 98.12560 VI
248 | ## 3 79.74006 32.27013 40.20866 116.33897 VI
249 | ## 4 62.96674 44.40790 30.50866 90.46888 IV
250 | ## 5 64.24582 36.86337 37.43949 98.51873 IV
251 | ## 6 50.97107 46.25627 38.83196 75.21992 I
252 | ```
253 |
254 | The `merMod` object has a number of methods available -- too many to enumerate here. But, we will go
255 | over some of the more common in the list below:
256 |
257 |
258 |
259 | ```r
260 | methods(class = "merMod")
261 | ```
262 |
263 | ```
264 | ## [1] anova as.function coef confint cooks.distance deviance
265 | ## [7] df.residual display drop1 extractAIC extractDIC family
266 | ## [13] fitted fixef formula getL getME hatvalues
267 | ## [19] influence isGLMM isLMM isNLMM isREML logLik
268 | ## [25] mcsamp model.frame model.matrix ngrps nobs plot
269 | ## [31] predict print profile qqmath ranef refit
270 | ## [37] refitML rePCA residuals rstudent se.coef show
271 | ## [43] sigma.hat sigma sim simulate standardize summary
272 | ## [49] terms update VarCorr vcov weights
273 | ## see '?methods' for accessing help and source code
274 | ```
275 |
276 | A common need is to extract the fixed effects from a `merMod` object. `fixef` extracts a named
277 | numeric vector of the fixed effects, which is handy.
278 |
279 |
280 | ```r
281 | fixef(MLexamp1)
282 | ```
283 |
284 | ```
285 | ## (Intercept) open agree social
286 | ## 59.116514199 0.009750941 0.027788360 -0.002151446
287 | ```
288 |
289 | If you want to get a sense of the p-values or statistical significance of these parameters, first
290 | consult the `lme4` help by running `?mcmcsamp` for a rundown of various ways of doing this. One
291 | convenient way built into the package is:
292 |
293 |
294 | ```r
295 | confint(MLexamp1, level = 0.99)
296 | ```
297 |
298 | ```
299 | ## 0.5 % 99.5 %
300 | ## .sig01 4.91840325 23.88757695
301 | ## .sigma 2.53286648 2.81455985
302 | ## (Intercept) 46.27750884 71.95609747
303 | ## open -0.02464506 0.04414924
304 | ## agree -0.01163700 0.06721354
305 | ## social -0.01492690 0.01062510
306 | ```
307 |
308 | From here we can see first that our fixed effect parameters overlap 0 indicating no evidence of an
309 | effect. We can also see that `.sig01`, which is our estimate of the variability in the random
310 | effect, is very large and very widely defined. This indicates we may have a lack of precision
311 | between our groups - either because the group effect is small between groups, we have too few groups
312 | to get a more precise estimate, we have too few units within each group, or a combination of all of
313 | the above.
314 |
315 | Another common need is to extract the residual standard error, which is necessary for calculating
316 | effect sizes. To get a named vector of the residual standard error:
317 |
318 |
319 | ```r
320 | sigma(MLexamp1)
321 | ```
322 |
323 | ```
324 | ## [1] 2.670886
325 | ```
326 |
327 | For example, it is common practice in education research to standardize fixed effects into "effect
328 | sizes" by dividing the fixed effect paramters by the residual standard error, which can be
329 | accomplished in `lme4` easily:
330 |
331 |
332 | ```r
333 | fixef(MLexamp1) / sigma(MLexamp1)
334 | ```
335 |
336 | ```
337 | ## (Intercept) open agree social
338 | ## 22.1336707437 0.0036508262 0.0104041726 -0.0008055176
339 | ```
340 |
341 | From this, we can see that our predictors of openness, agreeableness and social are virtually
342 | useless in predicting extroversion -- as our plots showed. Let's turn our attention to the random
343 | effects next.
344 |
345 | ## Explore Group Variation and Random Effects
346 |
347 | In all likelihood you fit a mixed-effect model because you are directly interested in the
348 | group-level variation in your model. It is not immediately clear how to explore this group level
349 | variation from the results of `summary.merMod`. What we get from this output is the variance and the
350 | standard deviation of the group effect, but we do not get effects for individual groups. This is
351 | where the `ranef` function comes in handy.
352 |
353 |
354 | ```r
355 | ranef(MLexamp1)
356 | ```
357 |
358 | ```
359 | ## $school
360 | ## (Intercept)
361 | ## I -14.090991
362 | ## II -6.183368
363 | ## III -1.970700
364 | ## IV 1.965938
365 | ## V 6.330710
366 | ## VI 13.948412
367 | ##
368 | ## with conditional variances for "school"
369 | ```
370 |
371 | Running the `ranef` function gives us the intercepts for each school, but not much additional
372 | information -- for example the precision of these estimates. To do that, we need some additional
373 | commands:
374 |
375 |
376 | ```r
377 | re1 <- ranef(MLexamp1, condVar=TRUE) # save the ranef.mer object
378 | class(re1)
379 | ```
380 |
381 | ```
382 | ## [1] "ranef.mer"
383 | ```
384 |
385 | ```r
386 | attr(re1[[1]], which = "postVar")
387 | ```
388 |
389 | ```
390 | ## , , 1
391 | ##
392 | ## [,1]
393 | ## [1,] 0.0356549
394 | ##
395 | ## , , 2
396 | ##
397 | ## [,1]
398 | ## [1,] 0.0356549
399 | ##
400 | ## , , 3
401 | ##
402 | ## [,1]
403 | ## [1,] 0.0356549
404 | ##
405 | ## , , 4
406 | ##
407 | ## [,1]
408 | ## [1,] 0.0356549
409 | ##
410 | ## , , 5
411 | ##
412 | ## [,1]
413 | ## [1,] 0.0356549
414 | ##
415 | ## , , 6
416 | ##
417 | ## [,1]
418 | ## [1,] 0.0356549
419 | ```
420 |
421 | The `ranef.mer` object is a list which contains a data.frame for each group level. The dataframe
422 | contains the random effects for each group (here we only have an intercept for each school). When we
423 | ask `lme4` for the conditional variance of the random effects it is stored in an `attribute` of
424 | those dataframes as a list of variance-covariance matrices.
425 |
426 | This structure is indeed *complicated*, but it is powerful as it allows for nested, grouped, and
427 | cross-level random effects. Also, the creators of `lme4` have provided users with some simple
428 | shortcuts to get what they are really interested in out of a `ranef.mer` object.
429 |
430 |
431 | ```r
432 | re1 <- ranef(MLexamp1, condVar=TRUE, whichel = "school")
433 | print(re1)
434 | ```
435 |
436 | ```
437 | ## $school
438 | ## (Intercept)
439 | ## I -14.090991
440 | ## II -6.183368
441 | ## III -1.970700
442 | ## IV 1.965938
443 | ## V 6.330710
444 | ## VI 13.948412
445 | ##
446 | ## with conditional variances for "school"
447 | ```
448 |
449 | ```r
450 | dotplot(re1)
451 | ```
452 |
453 | ```
454 | ## $school
455 | ```
456 |
457 | 
458 |
459 | This graphic shows a `dotplot` of the random effect terms, also known as a caterpillar plot. Here
460 | you can clearly see the effects of each school on `extroversion` as well as their standard errors to
461 | help identify how distinct the random effects are from one another. Interpreting random effects is
462 | notably tricky, but for assistance I would recommend looking at a few of these resources:
463 |
464 | - Gelman and Hill 2006 - [Data Analysis Using Regression and Multilevel/Hierarchical Techniques](http://www.stat.columbia.edu/~gelman/arm/)
465 | - John Fox - An R Compantion to Applied Regression [web appendix](http://socserv.mcmaster.ca/jfox/Books/Companion-1E/appendix-mixed-models.pdf)
466 |
467 | ## Using Simulation and Plots to Explore Random Effects
468 |
469 | A common econometric approach is to create what are known as **empirical Bayes** estimates of the
470 | group-level terms. Unfortunately there is not much agreement about what constitutes a proper
471 | standard error for the random effect terms or even how to consistently define **empirical Bayes**
472 | estimates.[^1] However, in R there are a few additional ways to get estimates of the random effects
473 | that provide the user with information about the relative sizes of the effects for each unit and the
474 | precision in that estimate. To do this, we use the `sim` function in the `arm` package.[^2]
475 |
476 |
477 | ```r
478 | # A function to extract simulated estimates of random effect paramaters from
479 | # lme4 objects using the sim function in arm
480 | # whichel = the character for the name of the grouping effect to extract estimates for
481 | # nsims = the number of simulations to pass to sim
482 | # x = model object
483 | REsim <- function(x, whichel=NULL, nsims){
484 | require(plyr)
485 | mysim <- sim(x, n.sims = nsims)
486 | if(missing(whichel)){
487 | dat <- plyr::adply(mysim@ranef[[1]], c(2, 3), plyr::each(c(mean, median, sd)))
488 | warning("Only returning 1st random effect because whichel not specified")
489 | } else{
490 | dat <- plyr::adply(mysim@ranef[[whichel]], c(2, 3), plyr::each(c(mean, median, sd)))
491 | }
492 | return(dat)
493 | }
494 |
495 | REsim(MLexamp1, whichel = "school", nsims = 1000)
496 | ```
497 |
498 | ```
499 | ## X1 X2 mean median sd
500 | ## 1 I (Intercept) -14.088205 -14.287477 3.923270
501 | ## 2 II (Intercept) -6.183570 -6.369473 3.937168
502 | ## 3 III (Intercept) -1.956252 -2.078317 3.925270
503 | ## 4 IV (Intercept) 1.968866 1.843241 3.930132
504 | ## 5 V (Intercept) 6.344076 6.142423 3.925949
505 | ## 6 VI (Intercept) 13.954781 13.770879 3.919504
506 | ```
507 |
508 | The `REsim` function returns for each school the level name `X1`, the estimate name, `X2`, the mean
509 | of the estimated values, the median, and the standard deviation of the estimates.
510 |
511 | Another convenience function can help us plot these results to see how they compare to the results
512 | of `dotplot`:
513 |
514 |
515 | ```r
516 | # Dat = results of REsim
517 | # scale = factor to multiply sd by
518 | # var = character of "mean" or "median"
519 | # sd = character of "sd"
520 | plotREsim <- function(dat, scale, var, sd){
521 | require(eeptools)
522 | dat[, sd] <- dat[, sd] * scale
523 | dat[, "ymax"] <- dat[, var] + dat[, sd]
524 | dat[, "ymin"] <- dat[, var] - dat[, sd]
525 | dat[order(dat[, var]), "id"] <- c(1:nrow(dat))
526 | ggplot(dat, aes_string(x = "id", y = var, ymax = "ymax",
527 | ymin = "ymin")) +
528 | geom_pointrange() + theme_dpi() +
529 | labs(x = "Group", y = "Effect Range", title = "Effect Ranges") +
530 | theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
531 | axis.text.x = element_blank(), axis.ticks.x = element_blank()) +
532 | geom_hline(yintercept = 0, color = I("red"), size = I(1.1))
533 | }
534 |
535 | plotREsim(REsim(MLexamp1, whichel = "school", nsims = 1000), scale = 1.2,
536 | var = "mean", sd = "sd")
537 | ```
538 |
539 | 
540 |
541 |
542 | This presents a more conservative view of the variation between random effect components. Depending
543 | on how your data was collected and your research question, alternative ways of estimating these
544 | effect sizes are possible. However, proceed with caution.[^3]
545 |
546 | Another approach recommended by the authors of `lme4` involves the `RLRsim` package. Using this
547 | package we can test whether or not inclusion of the random effects improves model fit and we can
548 | evaluate the p-value of additional random effect terms using a likelihood ratio test based on
549 | simulation.[^4]
550 |
551 |
552 | ```r
553 | library(RLRsim)
554 | m0 <- lm(extro ~ agree + open + social, data =lmm.data) # fit the null model
555 | exactLRT(m = MLexamp1, m0 = m0)
556 | ```
557 |
558 | ```
559 | ##
560 | ## simulated finite sample distribution of LRT. (p-value based on 10000 simulated values)
561 | ##
562 | ## data:
563 | ## LRT = 2957.7, p-value < 2.2e-16
564 | ```
565 |
566 | Here `exactLRT` issues a warning because we originally fit the model with REML instead of full
567 | maximum likelihood. Fortunately, the `refitML` function in `lme4` allows us to easily refit our
568 | model using full maximum likelihood to conduct an exact test easily.
569 |
570 |
571 | ```r
572 | mA <- refitML(MLexamp1)
573 | exactLRT(m= mA, m0 = m0)
574 | ```
575 |
576 | ```
577 | ##
578 | ## simulated finite sample distribution of LRT. (p-value based on 10000 simulated values)
579 | ##
580 | ## data:
581 | ## LRT = 2957.8, p-value < 2.2e-16
582 | ```
583 |
584 | Here we can see that the inclusion of our grouping variable is significant, even though the effect
585 | of each individual group may be substantively small and/or imprecisely measured. This is important
586 | in understanding the correct specification of the model. Our next tutorial will cover specification
587 | tests like this in more detail.
588 |
589 | ## What do Random Effects Matter?
590 |
591 | How do interpret the *substantive* impact of our random effects? This is often critical in
592 | observation work trying to use a multilevel structure to understand the impact that the grouping can
593 | have on the individual observation. To do this we select 12 random cases and then we simulate their
594 | predicted value of `extro` if they were placed in each of the 6 schools. Note, that this is a very
595 | simple simulation just using the mean of the fixed effect and the conditional mode of the random
596 | effect and not replicating or sampling to get a sense of the variability. This will be left as an
597 | exercise to the reader and/or a future tutorial!
598 |
599 |
600 | ```r
601 | # Simulate
602 | # Let's create 12 cases of students
603 | #
604 | #sample some rows
605 | simX <- sample(lmm.data$id, 12)
606 | simX <- lmm.data[lmm.data$id %in% simX, c(3:5)] # get their data
607 | # add an intercept
608 | simX[, "Intercept"] <- 1
609 | simX <- simX[, c(4, 1:3)] # reorder
610 | simRE <- REsim(MLexamp1, whichel = "school", nsims = 1000) # simulate randome effects
611 | simX$case <- row.names(simX) # create a case ID
612 | # expand a grid of case IDs by schools
613 | simDat <- expand.grid(case = row.names(simX), school = levels(lmm.data$school))
614 | simDat <- merge(simX, simDat) # merge in the data
615 | # Create the fixed effect predictor
616 | simDat[, "fepred"] <- (simDat[, 2] * fixef(MLexamp1)[1]) + (simDat[, 3] * fixef(MLexamp1)[2]) +
617 | (simDat[, 4] * fixef(MLexamp1)[3]) + (simDat[, 5] * fixef(MLexamp1)[4])
618 | # Add the school effects
619 | simDat <- merge(simDat, simRE[, c(1, 3)], by.x = "school", by.y="X1")
620 | simDat$yhat <- simDat$fepred + simDat$mean # add the school specific intercept
621 | ```
622 |
623 | Now that we have set up a simulated dataframe, let's plot it, first by case:
624 |
625 |
626 | ```r
627 | qplot(school, yhat, data = simDat) + facet_wrap(~case) + theme_dpi()
628 | ```
629 |
630 | 
631 |
632 | This plot shows us that within each plot, representing a case, there is tremendous variation by
633 | school. So, moving each student into a different school has large effects on the extroversion score.
634 | But, does each case vary at each school?
635 |
636 |
637 | ```r
638 | qplot(case, yhat, data = simDat) + facet_wrap(~school) + theme_dpi() +
639 | theme(axis.text.x = element_blank())
640 | ```
641 |
642 | 
643 |
644 | Here we can clearly see that within each school the cases are relatively the same indicating that
645 | the group effect is larger than the individual effects.
646 |
647 | These plots are useful in demonstrating the relative importance of group and individual effects in a
648 | substantive fashion. Even more can be done to make the the graphs more informative, such as placing
649 | references to the total variability of the outcome and also looking at the distance moving groups
650 | moves each observation from its true value.
651 |
652 | # Conclusion
653 |
654 | `lme4` provides a very powerful object-oriented toolset for dealing with mixed effect models in R.
655 | Understanding model fit and confidence intervals of `lme4` objects requires some diligent research
656 | and the use of a variety of functions and extensions of `lme4` itself. In our next tutorial we will
657 | explore how to identify a proper specification of a random-effect model and Bayesian extensions of
658 | the `lme4` framework for difficult to specify models. We will also explore the generalized linear
659 | model framework and the `glmer` function for generalized linear modeling with multi-levels.
660 |
661 | # Appendix
662 |
663 |
664 | ```r
665 | print(sessionInfo(),locale=FALSE)
666 | ```
667 |
668 | ```
669 | ## R version 3.5.3 (2019-03-11)
670 | ## Platform: x86_64-w64-mingw32/x64 (64-bit)
671 | ## Running under: Windows 10 x64 (build 17134)
672 | ##
673 | ## Matrix products: default
674 | ##
675 | ## attached base packages:
676 | ## [1] stats graphics grDevices utils datasets methods base
677 | ##
678 | ## other attached packages:
679 | ## [1] RLRsim_3.1-3 eeptools_1.2.2 ggplot2_3.1.1 plyr_1.8.4 lattice_0.20-38 arm_1.10-1
680 | ## [7] MASS_7.3-51.4 lme4_1.1-21 Matrix_1.2-17 knitr_1.22
681 | ##
682 | ## loaded via a namespace (and not attached):
683 | ## [1] zoo_1.8-5 tidyselect_0.2.5 xfun_0.6 purrr_0.3.2 splines_3.5.3
684 | ## [6] colorspace_1.4-1 htmltools_0.3.6 yaml_2.2.0 mgcv_1.8-28 rlang_0.3.4
685 | ## [11] nloptr_1.2.1 pillar_1.3.1 foreign_0.8-71 glue_1.3.1 withr_2.1.2
686 | ## [16] sp_1.3-1 stringr_1.4.0 munsell_0.5.0 gtable_0.3.0 coda_0.19-2
687 | ## [21] evaluate_0.13 labeling_0.3 maptools_0.9-5 lmtest_0.9-37 vcd_1.4-4
688 | ## [26] Rcpp_1.0.1 scales_1.0.0 abind_1.4-5 digest_0.6.18 stringi_1.4.3
689 | ## [31] dplyr_0.8.0.1 grid_3.5.3 tools_3.5.3 magrittr_1.5 lazyeval_0.2.2
690 | ## [36] tibble_2.1.1 crayon_1.3.4 pkgconfig_2.0.2 data.table_1.12.2 assertthat_0.2.1
691 | ## [41] minqa_1.2.4 rmarkdown_1.12 R6_2.4.0 boot_1.3-22 nlme_3.1-139
692 | ## [46] compiler_3.5.3
693 | ```
694 |
695 | [^1]: [See message from `lme4` co-author Doug Bates on this subject]. (https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q4/002984.html)
696 | [^2]: Andrew Gelman and Yu-Sung Su (2014). arm: Data Analysis Using Regression and Multilevel/Hierarchical Models. R package version
697 | 1.7-03. http://CRAN.R-project.org/package=arm
698 | [^3]: [WikiDot FAQ from the R Mixed Models Mailing List](http://glmm.wikidot.com/faq)
699 | [^4]: There are also an extensive series of references available in the `References`
700 | section of the help by running `?exactLRT` and `?exactRLRT`.
701 |
--------------------------------------------------------------------------------
/Tutorial2-ProperSpecification.Rmd:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Tutorial2-ProperSpecificationofREmodels"
3 | author: "Jared Knowles"
4 | date: "Saturday, May 17, 2014"
5 | output: html_document
6 | ---
7 |
8 |
14 |
15 | ```{r setup, echo=FALSE, error=FALSE, message=FALSE, eval=TRUE, results='hide'}
16 | library(knitr)
17 | opts_chunk$set(dev='svg', fig.width=6, fig.height=6, echo=TRUE,
18 | message=FALSE, error=FALSE, warning=FALSE)
19 |
20 | ```
21 |
22 |
23 | ```{r loadandviewdata}
24 | library(lme4) # load library
25 | library(arm) # convenience functions for regression in R
26 | lmm.data <- read.table("http://bayes.acs.unt.edu:8083/BayesContent/class/Jon/R_SC/Module9/lmm.data.txt",
27 | header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
28 | #summary(lmm.data)
29 | head(lmm.data)
30 | ```
31 |
32 | ## Fully Interacted Models
33 |
34 | ```{r fullinteract}
35 |
36 | classEff <- lmList(extro ~ open + agree + social | class, data = lmm.data)
37 | schoolEff <- lmList(extro ~ open + agree + social | school, data = lmm.data)
38 |
39 |
40 |
41 | ```
42 |
43 | ## Non-nested Classes
44 |
45 | ```{r lmer2}
46 | MLexamp.7 <- lmer(extro ~ open + agree + social + (1|school) + (1|class), data=lmm.data)
47 | display(MLexamp.7)
48 | ```
49 |
50 | ## Nested group effects
51 |
52 | And finally, we can fit nested group effect terms through the following syntax:
53 |
54 | ```{r lmer3}
55 | MLexamp.8 <- lmer(extro ~ open + agree + social + (1|school/class), data=lmm.data)
56 | display(MLexamp.8)
57 |
58 |
59 | ```
60 |
61 | Here the `(1|school/class)` says that we want to fit a mixed effect term for varying
62 | intercepts `1|` by schools, and for classes that are nested within schools.
63 |
64 |
65 | ```{r}
66 | MLexamp2 <- update(MLexamp1, .~ . + (1|class)) # add a intercept for class
67 | MLexamp2 <- lmer(extro ~ open + agree + social + (1|class) + (1|school),
68 | data = lmm.data)
69 | exactRLRT(m = MLexamp1, mA=MLexamp2,
70 | m0 = )
71 | ```
72 |
73 | ## Non-correlated random effects
74 |
75 | ```{r lmer4}
76 | MLexampUnCor <- lmer(extro ~ open + agree + social + (1|school/class) + (0 + open|school/class),
77 | data = lmm.data)
78 | MLexampCor <- lmer(extro ~ open + agree + social + (1+open|school/class),
79 | data = lmm.data)
80 |
81 | VarCorr(MLexampUnCor)
82 | VarCorr(MLexampCor)
83 | ```
84 |
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/docs/Tutorial1-ExploringLMERobjects_files/bootstrap-3.3.5/css/bootstrap-theme.css:
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1 | /*!
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3 | * Copyright 2011-2015 Twitter, Inc.
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5 | */
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231 | background-image: -webkit-linear-gradient(top, #f0ad4e 0%, #eb9316 100%);
232 | background-image: -o-linear-gradient(top, #f0ad4e 0%, #eb9316 100%);
233 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f0ad4e), to(#eb9316));
234 | background-image: linear-gradient(to bottom, #f0ad4e 0%, #eb9316 100%);
235 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff0ad4e', endColorstr='#ffeb9316', GradientType=0);
236 | filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
237 | background-repeat: repeat-x;
238 | border-color: #e38d13;
239 | }
240 | .btn-warning:hover,
241 | .btn-warning:focus {
242 | background-color: #eb9316;
243 | background-position: 0 -15px;
244 | }
245 | .btn-warning:active,
246 | .btn-warning.active {
247 | background-color: #eb9316;
248 | border-color: #e38d13;
249 | }
250 | .btn-warning.disabled,
251 | .btn-warning[disabled],
252 | fieldset[disabled] .btn-warning,
253 | .btn-warning.disabled:hover,
254 | .btn-warning[disabled]:hover,
255 | fieldset[disabled] .btn-warning:hover,
256 | .btn-warning.disabled:focus,
257 | .btn-warning[disabled]:focus,
258 | fieldset[disabled] .btn-warning:focus,
259 | .btn-warning.disabled.focus,
260 | .btn-warning[disabled].focus,
261 | fieldset[disabled] .btn-warning.focus,
262 | .btn-warning.disabled:active,
263 | .btn-warning[disabled]:active,
264 | fieldset[disabled] .btn-warning:active,
265 | .btn-warning.disabled.active,
266 | .btn-warning[disabled].active,
267 | fieldset[disabled] .btn-warning.active {
268 | background-color: #eb9316;
269 | background-image: none;
270 | }
271 | .btn-danger {
272 | background-image: -webkit-linear-gradient(top, #d9534f 0%, #c12e2a 100%);
273 | background-image: -o-linear-gradient(top, #d9534f 0%, #c12e2a 100%);
274 | background-image: -webkit-gradient(linear, left top, left bottom, from(#d9534f), to(#c12e2a));
275 | background-image: linear-gradient(to bottom, #d9534f 0%, #c12e2a 100%);
276 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffd9534f', endColorstr='#ffc12e2a', GradientType=0);
277 | filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
278 | background-repeat: repeat-x;
279 | border-color: #b92c28;
280 | }
281 | .btn-danger:hover,
282 | .btn-danger:focus {
283 | background-color: #c12e2a;
284 | background-position: 0 -15px;
285 | }
286 | .btn-danger:active,
287 | .btn-danger.active {
288 | background-color: #c12e2a;
289 | border-color: #b92c28;
290 | }
291 | .btn-danger.disabled,
292 | .btn-danger[disabled],
293 | fieldset[disabled] .btn-danger,
294 | .btn-danger.disabled:hover,
295 | .btn-danger[disabled]:hover,
296 | fieldset[disabled] .btn-danger:hover,
297 | .btn-danger.disabled:focus,
298 | .btn-danger[disabled]:focus,
299 | fieldset[disabled] .btn-danger:focus,
300 | .btn-danger.disabled.focus,
301 | .btn-danger[disabled].focus,
302 | fieldset[disabled] .btn-danger.focus,
303 | .btn-danger.disabled:active,
304 | .btn-danger[disabled]:active,
305 | fieldset[disabled] .btn-danger:active,
306 | .btn-danger.disabled.active,
307 | .btn-danger[disabled].active,
308 | fieldset[disabled] .btn-danger.active {
309 | background-color: #c12e2a;
310 | background-image: none;
311 | }
312 | .thumbnail,
313 | .img-thumbnail {
314 | -webkit-box-shadow: 0 1px 2px rgba(0, 0, 0, .075);
315 | box-shadow: 0 1px 2px rgba(0, 0, 0, .075);
316 | }
317 | .dropdown-menu > li > a:hover,
318 | .dropdown-menu > li > a:focus {
319 | background-color: #e8e8e8;
320 | background-image: -webkit-linear-gradient(top, #f5f5f5 0%, #e8e8e8 100%);
321 | background-image: -o-linear-gradient(top, #f5f5f5 0%, #e8e8e8 100%);
322 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f5f5f5), to(#e8e8e8));
323 | background-image: linear-gradient(to bottom, #f5f5f5 0%, #e8e8e8 100%);
324 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff5f5f5', endColorstr='#ffe8e8e8', GradientType=0);
325 | background-repeat: repeat-x;
326 | }
327 | .dropdown-menu > .active > a,
328 | .dropdown-menu > .active > a:hover,
329 | .dropdown-menu > .active > a:focus {
330 | background-color: #2e6da4;
331 | background-image: -webkit-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
332 | background-image: -o-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
333 | background-image: -webkit-gradient(linear, left top, left bottom, from(#337ab7), to(#2e6da4));
334 | background-image: linear-gradient(to bottom, #337ab7 0%, #2e6da4 100%);
335 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff337ab7', endColorstr='#ff2e6da4', GradientType=0);
336 | background-repeat: repeat-x;
337 | }
338 | .navbar-default {
339 | background-image: -webkit-linear-gradient(top, #fff 0%, #f8f8f8 100%);
340 | background-image: -o-linear-gradient(top, #fff 0%, #f8f8f8 100%);
341 | background-image: -webkit-gradient(linear, left top, left bottom, from(#fff), to(#f8f8f8));
342 | background-image: linear-gradient(to bottom, #fff 0%, #f8f8f8 100%);
343 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffffffff', endColorstr='#fff8f8f8', GradientType=0);
344 | filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
345 | background-repeat: repeat-x;
346 | border-radius: 4px;
347 | -webkit-box-shadow: inset 0 1px 0 rgba(255, 255, 255, .15), 0 1px 5px rgba(0, 0, 0, .075);
348 | box-shadow: inset 0 1px 0 rgba(255, 255, 255, .15), 0 1px 5px rgba(0, 0, 0, .075);
349 | }
350 | .navbar-default .navbar-nav > .open > a,
351 | .navbar-default .navbar-nav > .active > a {
352 | background-image: -webkit-linear-gradient(top, #dbdbdb 0%, #e2e2e2 100%);
353 | background-image: -o-linear-gradient(top, #dbdbdb 0%, #e2e2e2 100%);
354 | background-image: -webkit-gradient(linear, left top, left bottom, from(#dbdbdb), to(#e2e2e2));
355 | background-image: linear-gradient(to bottom, #dbdbdb 0%, #e2e2e2 100%);
356 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffdbdbdb', endColorstr='#ffe2e2e2', GradientType=0);
357 | background-repeat: repeat-x;
358 | -webkit-box-shadow: inset 0 3px 9px rgba(0, 0, 0, .075);
359 | box-shadow: inset 0 3px 9px rgba(0, 0, 0, .075);
360 | }
361 | .navbar-brand,
362 | .navbar-nav > li > a {
363 | text-shadow: 0 1px 0 rgba(255, 255, 255, .25);
364 | }
365 | .navbar-inverse {
366 | background-image: -webkit-linear-gradient(top, #3c3c3c 0%, #222 100%);
367 | background-image: -o-linear-gradient(top, #3c3c3c 0%, #222 100%);
368 | background-image: -webkit-gradient(linear, left top, left bottom, from(#3c3c3c), to(#222));
369 | background-image: linear-gradient(to bottom, #3c3c3c 0%, #222 100%);
370 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff3c3c3c', endColorstr='#ff222222', GradientType=0);
371 | filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
372 | background-repeat: repeat-x;
373 | border-radius: 4px;
374 | }
375 | .navbar-inverse .navbar-nav > .open > a,
376 | .navbar-inverse .navbar-nav > .active > a {
377 | background-image: -webkit-linear-gradient(top, #080808 0%, #0f0f0f 100%);
378 | background-image: -o-linear-gradient(top, #080808 0%, #0f0f0f 100%);
379 | background-image: -webkit-gradient(linear, left top, left bottom, from(#080808), to(#0f0f0f));
380 | background-image: linear-gradient(to bottom, #080808 0%, #0f0f0f 100%);
381 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff080808', endColorstr='#ff0f0f0f', GradientType=0);
382 | background-repeat: repeat-x;
383 | -webkit-box-shadow: inset 0 3px 9px rgba(0, 0, 0, .25);
384 | box-shadow: inset 0 3px 9px rgba(0, 0, 0, .25);
385 | }
386 | .navbar-inverse .navbar-brand,
387 | .navbar-inverse .navbar-nav > li > a {
388 | text-shadow: 0 -1px 0 rgba(0, 0, 0, .25);
389 | }
390 | .navbar-static-top,
391 | .navbar-fixed-top,
392 | .navbar-fixed-bottom {
393 | border-radius: 0;
394 | }
395 | @media (max-width: 767px) {
396 | .navbar .navbar-nav .open .dropdown-menu > .active > a,
397 | .navbar .navbar-nav .open .dropdown-menu > .active > a:hover,
398 | .navbar .navbar-nav .open .dropdown-menu > .active > a:focus {
399 | color: #fff;
400 | background-image: -webkit-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
401 | background-image: -o-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
402 | background-image: -webkit-gradient(linear, left top, left bottom, from(#337ab7), to(#2e6da4));
403 | background-image: linear-gradient(to bottom, #337ab7 0%, #2e6da4 100%);
404 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff337ab7', endColorstr='#ff2e6da4', GradientType=0);
405 | background-repeat: repeat-x;
406 | }
407 | }
408 | .alert {
409 | text-shadow: 0 1px 0 rgba(255, 255, 255, .2);
410 | -webkit-box-shadow: inset 0 1px 0 rgba(255, 255, 255, .25), 0 1px 2px rgba(0, 0, 0, .05);
411 | box-shadow: inset 0 1px 0 rgba(255, 255, 255, .25), 0 1px 2px rgba(0, 0, 0, .05);
412 | }
413 | .alert-success {
414 | background-image: -webkit-linear-gradient(top, #dff0d8 0%, #c8e5bc 100%);
415 | background-image: -o-linear-gradient(top, #dff0d8 0%, #c8e5bc 100%);
416 | background-image: -webkit-gradient(linear, left top, left bottom, from(#dff0d8), to(#c8e5bc));
417 | background-image: linear-gradient(to bottom, #dff0d8 0%, #c8e5bc 100%);
418 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffdff0d8', endColorstr='#ffc8e5bc', GradientType=0);
419 | background-repeat: repeat-x;
420 | border-color: #b2dba1;
421 | }
422 | .alert-info {
423 | background-image: -webkit-linear-gradient(top, #d9edf7 0%, #b9def0 100%);
424 | background-image: -o-linear-gradient(top, #d9edf7 0%, #b9def0 100%);
425 | background-image: -webkit-gradient(linear, left top, left bottom, from(#d9edf7), to(#b9def0));
426 | background-image: linear-gradient(to bottom, #d9edf7 0%, #b9def0 100%);
427 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffd9edf7', endColorstr='#ffb9def0', GradientType=0);
428 | background-repeat: repeat-x;
429 | border-color: #9acfea;
430 | }
431 | .alert-warning {
432 | background-image: -webkit-linear-gradient(top, #fcf8e3 0%, #f8efc0 100%);
433 | background-image: -o-linear-gradient(top, #fcf8e3 0%, #f8efc0 100%);
434 | background-image: -webkit-gradient(linear, left top, left bottom, from(#fcf8e3), to(#f8efc0));
435 | background-image: linear-gradient(to bottom, #fcf8e3 0%, #f8efc0 100%);
436 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fffcf8e3', endColorstr='#fff8efc0', GradientType=0);
437 | background-repeat: repeat-x;
438 | border-color: #f5e79e;
439 | }
440 | .alert-danger {
441 | background-image: -webkit-linear-gradient(top, #f2dede 0%, #e7c3c3 100%);
442 | background-image: -o-linear-gradient(top, #f2dede 0%, #e7c3c3 100%);
443 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f2dede), to(#e7c3c3));
444 | background-image: linear-gradient(to bottom, #f2dede 0%, #e7c3c3 100%);
445 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff2dede', endColorstr='#ffe7c3c3', GradientType=0);
446 | background-repeat: repeat-x;
447 | border-color: #dca7a7;
448 | }
449 | .progress {
450 | background-image: -webkit-linear-gradient(top, #ebebeb 0%, #f5f5f5 100%);
451 | background-image: -o-linear-gradient(top, #ebebeb 0%, #f5f5f5 100%);
452 | background-image: -webkit-gradient(linear, left top, left bottom, from(#ebebeb), to(#f5f5f5));
453 | background-image: linear-gradient(to bottom, #ebebeb 0%, #f5f5f5 100%);
454 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffebebeb', endColorstr='#fff5f5f5', GradientType=0);
455 | background-repeat: repeat-x;
456 | }
457 | .progress-bar {
458 | background-image: -webkit-linear-gradient(top, #337ab7 0%, #286090 100%);
459 | background-image: -o-linear-gradient(top, #337ab7 0%, #286090 100%);
460 | background-image: -webkit-gradient(linear, left top, left bottom, from(#337ab7), to(#286090));
461 | background-image: linear-gradient(to bottom, #337ab7 0%, #286090 100%);
462 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff337ab7', endColorstr='#ff286090', GradientType=0);
463 | background-repeat: repeat-x;
464 | }
465 | .progress-bar-success {
466 | background-image: -webkit-linear-gradient(top, #5cb85c 0%, #449d44 100%);
467 | background-image: -o-linear-gradient(top, #5cb85c 0%, #449d44 100%);
468 | background-image: -webkit-gradient(linear, left top, left bottom, from(#5cb85c), to(#449d44));
469 | background-image: linear-gradient(to bottom, #5cb85c 0%, #449d44 100%);
470 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5cb85c', endColorstr='#ff449d44', GradientType=0);
471 | background-repeat: repeat-x;
472 | }
473 | .progress-bar-info {
474 | background-image: -webkit-linear-gradient(top, #5bc0de 0%, #31b0d5 100%);
475 | background-image: -o-linear-gradient(top, #5bc0de 0%, #31b0d5 100%);
476 | background-image: -webkit-gradient(linear, left top, left bottom, from(#5bc0de), to(#31b0d5));
477 | background-image: linear-gradient(to bottom, #5bc0de 0%, #31b0d5 100%);
478 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5bc0de', endColorstr='#ff31b0d5', GradientType=0);
479 | background-repeat: repeat-x;
480 | }
481 | .progress-bar-warning {
482 | background-image: -webkit-linear-gradient(top, #f0ad4e 0%, #ec971f 100%);
483 | background-image: -o-linear-gradient(top, #f0ad4e 0%, #ec971f 100%);
484 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f0ad4e), to(#ec971f));
485 | background-image: linear-gradient(to bottom, #f0ad4e 0%, #ec971f 100%);
486 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff0ad4e', endColorstr='#ffec971f', GradientType=0);
487 | background-repeat: repeat-x;
488 | }
489 | .progress-bar-danger {
490 | background-image: -webkit-linear-gradient(top, #d9534f 0%, #c9302c 100%);
491 | background-image: -o-linear-gradient(top, #d9534f 0%, #c9302c 100%);
492 | background-image: -webkit-gradient(linear, left top, left bottom, from(#d9534f), to(#c9302c));
493 | background-image: linear-gradient(to bottom, #d9534f 0%, #c9302c 100%);
494 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffd9534f', endColorstr='#ffc9302c', GradientType=0);
495 | background-repeat: repeat-x;
496 | }
497 | .progress-bar-striped {
498 | background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, .15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, .15) 50%, rgba(255, 255, 255, .15) 75%, transparent 75%, transparent);
499 | background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, .15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, .15) 50%, rgba(255, 255, 255, .15) 75%, transparent 75%, transparent);
500 | background-image: linear-gradient(45deg, rgba(255, 255, 255, .15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, .15) 50%, rgba(255, 255, 255, .15) 75%, transparent 75%, transparent);
501 | }
502 | .list-group {
503 | border-radius: 4px;
504 | -webkit-box-shadow: 0 1px 2px rgba(0, 0, 0, .075);
505 | box-shadow: 0 1px 2px rgba(0, 0, 0, .075);
506 | }
507 | .list-group-item.active,
508 | .list-group-item.active:hover,
509 | .list-group-item.active:focus {
510 | text-shadow: 0 -1px 0 #286090;
511 | background-image: -webkit-linear-gradient(top, #337ab7 0%, #2b669a 100%);
512 | background-image: -o-linear-gradient(top, #337ab7 0%, #2b669a 100%);
513 | background-image: -webkit-gradient(linear, left top, left bottom, from(#337ab7), to(#2b669a));
514 | background-image: linear-gradient(to bottom, #337ab7 0%, #2b669a 100%);
515 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff337ab7', endColorstr='#ff2b669a', GradientType=0);
516 | background-repeat: repeat-x;
517 | border-color: #2b669a;
518 | }
519 | .list-group-item.active .badge,
520 | .list-group-item.active:hover .badge,
521 | .list-group-item.active:focus .badge {
522 | text-shadow: none;
523 | }
524 | .panel {
525 | -webkit-box-shadow: 0 1px 2px rgba(0, 0, 0, .05);
526 | box-shadow: 0 1px 2px rgba(0, 0, 0, .05);
527 | }
528 | .panel-default > .panel-heading {
529 | background-image: -webkit-linear-gradient(top, #f5f5f5 0%, #e8e8e8 100%);
530 | background-image: -o-linear-gradient(top, #f5f5f5 0%, #e8e8e8 100%);
531 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f5f5f5), to(#e8e8e8));
532 | background-image: linear-gradient(to bottom, #f5f5f5 0%, #e8e8e8 100%);
533 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff5f5f5', endColorstr='#ffe8e8e8', GradientType=0);
534 | background-repeat: repeat-x;
535 | }
536 | .panel-primary > .panel-heading {
537 | background-image: -webkit-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
538 | background-image: -o-linear-gradient(top, #337ab7 0%, #2e6da4 100%);
539 | background-image: -webkit-gradient(linear, left top, left bottom, from(#337ab7), to(#2e6da4));
540 | background-image: linear-gradient(to bottom, #337ab7 0%, #2e6da4 100%);
541 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff337ab7', endColorstr='#ff2e6da4', GradientType=0);
542 | background-repeat: repeat-x;
543 | }
544 | .panel-success > .panel-heading {
545 | background-image: -webkit-linear-gradient(top, #dff0d8 0%, #d0e9c6 100%);
546 | background-image: -o-linear-gradient(top, #dff0d8 0%, #d0e9c6 100%);
547 | background-image: -webkit-gradient(linear, left top, left bottom, from(#dff0d8), to(#d0e9c6));
548 | background-image: linear-gradient(to bottom, #dff0d8 0%, #d0e9c6 100%);
549 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffdff0d8', endColorstr='#ffd0e9c6', GradientType=0);
550 | background-repeat: repeat-x;
551 | }
552 | .panel-info > .panel-heading {
553 | background-image: -webkit-linear-gradient(top, #d9edf7 0%, #c4e3f3 100%);
554 | background-image: -o-linear-gradient(top, #d9edf7 0%, #c4e3f3 100%);
555 | background-image: -webkit-gradient(linear, left top, left bottom, from(#d9edf7), to(#c4e3f3));
556 | background-image: linear-gradient(to bottom, #d9edf7 0%, #c4e3f3 100%);
557 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffd9edf7', endColorstr='#ffc4e3f3', GradientType=0);
558 | background-repeat: repeat-x;
559 | }
560 | .panel-warning > .panel-heading {
561 | background-image: -webkit-linear-gradient(top, #fcf8e3 0%, #faf2cc 100%);
562 | background-image: -o-linear-gradient(top, #fcf8e3 0%, #faf2cc 100%);
563 | background-image: -webkit-gradient(linear, left top, left bottom, from(#fcf8e3), to(#faf2cc));
564 | background-image: linear-gradient(to bottom, #fcf8e3 0%, #faf2cc 100%);
565 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fffcf8e3', endColorstr='#fffaf2cc', GradientType=0);
566 | background-repeat: repeat-x;
567 | }
568 | .panel-danger > .panel-heading {
569 | background-image: -webkit-linear-gradient(top, #f2dede 0%, #ebcccc 100%);
570 | background-image: -o-linear-gradient(top, #f2dede 0%, #ebcccc 100%);
571 | background-image: -webkit-gradient(linear, left top, left bottom, from(#f2dede), to(#ebcccc));
572 | background-image: linear-gradient(to bottom, #f2dede 0%, #ebcccc 100%);
573 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff2dede', endColorstr='#ffebcccc', GradientType=0);
574 | background-repeat: repeat-x;
575 | }
576 | .well {
577 | background-image: -webkit-linear-gradient(top, #e8e8e8 0%, #f5f5f5 100%);
578 | background-image: -o-linear-gradient(top, #e8e8e8 0%, #f5f5f5 100%);
579 | background-image: -webkit-gradient(linear, left top, left bottom, from(#e8e8e8), to(#f5f5f5));
580 | background-image: linear-gradient(to bottom, #e8e8e8 0%, #f5f5f5 100%);
581 | filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffe8e8e8', endColorstr='#fff5f5f5', GradientType=0);
582 | background-repeat: repeat-x;
583 | border-color: #dcdcdc;
584 | -webkit-box-shadow: inset 0 1px 3px rgba(0, 0, 0, .05), 0 1px 0 rgba(255, 255, 255, .1);
585 | box-shadow: inset 0 1px 3px rgba(0, 0, 0, .05), 0 1px 0 rgba(255, 255, 255, .1);
586 | }
587 | /*# sourceMappingURL=bootstrap-theme.css.map */
588 |
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/docs/Tutorial1-ExploringLMERobjects_files/bootstrap-3.3.5/css/bootstrap-theme.min.css:
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1 | /*!
2 | * Bootstrap v3.3.5 (http://getbootstrap.com)
3 | * Copyright 2011-2015 Twitter, Inc.
4 | * Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE)
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1 | /*! Respond.js v1.4.2: min/max-width media query polyfill * Copyright 2013 Scott Jehl
2 | * Licensed under https://github.com/scottjehl/Respond/blob/master/LICENSE-MIT
3 | * */
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5 | // Only run this code in IE 8
6 | if (!!window.navigator.userAgent.match("MSIE 8")) {
7 | !function(a){"use strict";a.matchMedia=a.matchMedia||function(a){var b,c=a.documentElement,d=c.firstElementChild||c.firstChild,e=a.createElement("body"),f=a.createElement("div");return f.id="mq-test-1",f.style.cssText="position:absolute;top:-100em",e.style.background="none",e.appendChild(f),function(a){return f.innerHTML='',c.insertBefore(e,d),b=42===f.offsetWidth,c.removeChild(e),{matches:b,media:a}}}(a.document)}(this),function(a){"use strict";function b(){u(!0)}var c={};a.respond=c,c.update=function(){};var d=[],e=function(){var b=!1;try{b=new a.XMLHttpRequest}catch(c){b=new a.ActiveXObject("Microsoft.XMLHTTP")}return function(){return b}}(),f=function(a,b){var c=e();c&&(c.open("GET",a,!0),c.onreadystatechange=function(){4!==c.readyState||200!==c.status&&304!==c.status||b(c.responseText)},4!==c.readyState&&c.send(null))};if(c.ajax=f,c.queue=d,c.regex={media:/@media[^\{]+\{([^\{\}]*\{[^\}\{]*\})+/gi,keyframes:/@(?:\-(?:o|moz|webkit)\-)?keyframes[^\{]+\{(?:[^\{\}]*\{[^\}\{]*\})+[^\}]*\}/gi,urls:/(url\()['"]?([^\/\)'"][^:\)'"]+)['"]?(\))/g,findStyles:/@media *([^\{]+)\{([\S\s]+?)$/,only:/(only\s+)?([a-zA-Z]+)\s?/,minw:/\([\s]*min\-width\s*:[\s]*([\s]*[0-9\.]+)(px|em)[\s]*\)/,maxw:/\([\s]*max\-width\s*:[\s]*([\s]*[0-9\.]+)(px|em)[\s]*\)/},c.mediaQueriesSupported=a.matchMedia&&null!==a.matchMedia("only all")&&a.matchMedia("only all").matches,!c.mediaQueriesSupported){var g,h,i,j=a.document,k=j.documentElement,l=[],m=[],n=[],o={},p=30,q=j.getElementsByTagName("head")[0]||k,r=j.getElementsByTagName("base")[0],s=q.getElementsByTagName("link"),t=function(){var a,b=j.createElement("div"),c=j.body,d=k.style.fontSize,e=c&&c.style.fontSize,f=!1;return b.style.cssText="position:absolute;font-size:1em;width:1em",c||(c=f=j.createElement("body"),c.style.background="none"),k.style.fontSize="100%",c.style.fontSize="100%",c.appendChild(b),f&&k.insertBefore(c,k.firstChild),a=b.offsetWidth,f?k.removeChild(c):c.removeChild(b),k.style.fontSize=d,e&&(c.style.fontSize=e),a=i=parseFloat(a)},u=function(b){var c="clientWidth",d=k[c],e="CSS1Compat"===j.compatMode&&d||j.body[c]||d,f={},o=s[s.length-1],r=(new Date).getTime();if(b&&g&&p>r-g)return a.clearTimeout(h),h=a.setTimeout(u,p),void 0;g=r;for(var v in l)if(l.hasOwnProperty(v)){var w=l[v],x=w.minw,y=w.maxw,z=null===x,A=null===y,B="em";x&&(x=parseFloat(x)*(x.indexOf(B)>-1?i||t():1)),y&&(y=parseFloat(y)*(y.indexOf(B)>-1?i||t():1)),w.hasquery&&(z&&A||!(z||e>=x)||!(A||y>=e))||(f[w.media]||(f[w.media]=[]),f[w.media].push(m[w.rules]))}for(var C in n)n.hasOwnProperty(C)&&n[C]&&n[C].parentNode===q&&q.removeChild(n[C]);n.length=0;for(var D in f)if(f.hasOwnProperty(D)){var E=j.createElement("style"),F=f[D].join("\n");E.type="text/css",E.media=D,q.insertBefore(E,o.nextSibling),E.styleSheet?E.styleSheet.cssText=F:E.appendChild(j.createTextNode(F)),n.push(E)}},v=function(a,b,d){var e=a.replace(c.regex.keyframes,"").match(c.regex.media),f=e&&e.length||0;b=b.substring(0,b.lastIndexOf("/"));var g=function(a){return a.replace(c.regex.urls,"$1"+b+"$2$3")},h=!f&&d;b.length&&(b+="/"),h&&(f=1);for(var i=0;f>i;i++){var j,k,n,o;h?(j=d,m.push(g(a))):(j=e[i].match(c.regex.findStyles)&&RegExp.$1,m.push(RegExp.$2&&g(RegExp.$2))),n=j.split(","),o=n.length;for(var p=0;o>p;p++)k=n[p],l.push({media:k.split("(")[0].match(c.regex.only)&&RegExp.$2||"all",rules:m.length-1,hasquery:k.indexOf("(")>-1,minw:k.match(c.regex.minw)&&parseFloat(RegExp.$1)+(RegExp.$2||""),maxw:k.match(c.regex.maxw)&&parseFloat(RegExp.$1)+(RegExp.$2||"")})}u()},w=function(){if(d.length){var b=d.shift();f(b.href,function(c){v(c,b.href,b.media),o[b.href]=!0,a.setTimeout(function(){w()},0)})}},x=function(){for(var b=0;b
2 |
234 |
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/docs/Tutorial1-ExploringLMERobjects_files/highlightjs-9.12.0/default.css:
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1 | .hljs-literal {
2 | color: #990073;
3 | }
4 |
5 | .hljs-number {
6 | color: #099;
7 | }
8 |
9 | .hljs-comment {
10 | color: #998;
11 | font-style: italic;
12 | }
13 |
14 | .hljs-keyword {
15 | color: #900;
16 | font-weight: bold;
17 | }
18 |
19 | .hljs-string {
20 | color: #d14;
21 | }
22 |
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/docs/Tutorial1-ExploringLMERobjects_files/highlightjs-9.12.0/textmate.css:
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1 | .hljs-literal {
2 | color: rgb(88, 72, 246);
3 | }
4 |
5 | .hljs-number {
6 | color: rgb(0, 0, 205);
7 | }
8 |
9 | .hljs-comment {
10 | color: rgb(76, 136, 107);
11 | }
12 |
13 | .hljs-keyword {
14 | color: rgb(0, 0, 255);
15 | }
16 |
17 | .hljs-string {
18 | color: rgb(3, 106, 7);
19 | }
20 |
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/docs/Tutorial1-ExploringLMERobjects_files/navigation-1.1/codefolding.js:
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1 |
2 | window.initializeCodeFolding = function(show) {
3 |
4 | // handlers for show-all and hide all
5 | $("#rmd-show-all-code").click(function() {
6 | $('div.r-code-collapse').each(function() {
7 | $(this).collapse('show');
8 | });
9 | });
10 | $("#rmd-hide-all-code").click(function() {
11 | $('div.r-code-collapse').each(function() {
12 | $(this).collapse('hide');
13 | });
14 | });
15 |
16 | // index for unique code element ids
17 | var currentIndex = 1;
18 |
19 | // select all R code blocks
20 | var rCodeBlocks = $('pre.r, pre.python, pre.bash, pre.sql, pre.cpp, pre.stan, pre.julia');
21 | rCodeBlocks.each(function() {
22 |
23 | // create a collapsable div to wrap the code in
24 | var div = $('');
25 | if (show)
26 | div.addClass('in');
27 | var id = 'rcode-643E0F36' + currentIndex++;
28 | div.attr('id', id);
29 | $(this).before(div);
30 | $(this).detach().appendTo(div);
31 |
32 | // add a show code button right above
33 | var showCodeText = $('' + (show ? 'Hide' : 'Code') + '');
34 | var showCodeButton = $('');
35 | showCodeButton.append(showCodeText);
36 | showCodeButton
37 | .attr('data-toggle', 'collapse')
38 | .attr('data-target', '#' + id)
39 | .attr('aria-expanded', show)
40 | .attr('aria-controls', id);
41 |
42 | var buttonRow = $('');
43 | var buttonCol = $('');
44 |
45 | buttonCol.append(showCodeButton);
46 | buttonRow.append(buttonCol);
47 |
48 | div.before(buttonRow);
49 |
50 | // update state of button on show/hide
51 | div.on('hidden.bs.collapse', function () {
52 | showCodeText.text('Code');
53 | });
54 | div.on('show.bs.collapse', function () {
55 | showCodeText.text('Hide');
56 | });
57 | });
58 |
59 | }
60 |
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/docs/Tutorial1-ExploringLMERobjects_files/navigation-1.1/sourceembed.js:
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1 |
2 | window.initializeSourceEmbed = function(filename) {
3 | $("#rmd-download-source").click(function() {
4 | var src = $("#rmd-source-code").html();
5 | var a = document.createElement('a');
6 | a.href = "data:text/x-r-markdown;base64," + src;
7 | a.download = filename;
8 | document.body.appendChild(a);
9 | a.click();
10 | document.body.removeChild(a);
11 | });
12 | };
13 |
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/docs/Tutorial1-ExploringLMERobjects_files/navigation-1.1/tabsets.js:
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1 |
2 |
3 | /**
4 | * jQuery Plugin: Sticky Tabs
5 | *
6 | * @author Aidan Lister
7 | * adapted by Ruben Arslan to activate parent tabs too
8 | * http://www.aidanlister.com/2014/03/persisting-the-tab-state-in-bootstrap/
9 | */
10 | (function($) {
11 | "use strict";
12 | $.fn.rmarkdownStickyTabs = function() {
13 | var context = this;
14 | // Show the tab corresponding with the hash in the URL, or the first tab
15 | var showStuffFromHash = function() {
16 | var hash = window.location.hash;
17 | var selector = hash ? 'a[href="' + hash + '"]' : 'li.active > a';
18 | var $selector = $(selector, context);
19 | if($selector.data('toggle') === "tab") {
20 | $selector.tab('show');
21 | // walk up the ancestors of this element, show any hidden tabs
22 | $selector.parents('.section.tabset').each(function(i, elm) {
23 | var link = $('a[href="#' + $(elm).attr('id') + '"]');
24 | if(link.data('toggle') === "tab") {
25 | link.tab("show");
26 | }
27 | });
28 | }
29 | };
30 |
31 |
32 | // Set the correct tab when the page loads
33 | showStuffFromHash(context);
34 |
35 | // Set the correct tab when a user uses their back/forward button
36 | $(window).on('hashchange', function() {
37 | showStuffFromHash(context);
38 | });
39 |
40 | // Change the URL when tabs are clicked
41 | $('a', context).on('click', function(e) {
42 | history.pushState(null, null, this.href);
43 | showStuffFromHash(context);
44 | });
45 |
46 | return this;
47 | };
48 | }(jQuery));
49 |
50 | window.buildTabsets = function(tocID) {
51 |
52 | // build a tabset from a section div with the .tabset class
53 | function buildTabset(tabset) {
54 |
55 | // check for fade and pills options
56 | var fade = tabset.hasClass("tabset-fade");
57 | var pills = tabset.hasClass("tabset-pills");
58 | var navClass = pills ? "nav-pills" : "nav-tabs";
59 |
60 | // determine the heading level of the tabset and tabs
61 | var match = tabset.attr('class').match(/level(\d) /);
62 | if (match === null)
63 | return;
64 | var tabsetLevel = Number(match[1]);
65 | var tabLevel = tabsetLevel + 1;
66 |
67 | // find all subheadings immediately below
68 | var tabs = tabset.find("div.section.level" + tabLevel);
69 | if (!tabs.length)
70 | return;
71 |
72 | // create tablist and tab-content elements
73 | var tabList = $('
');
74 | $(tabs[0]).before(tabList);
75 | var tabContent = $('');
76 | $(tabs[0]).before(tabContent);
77 |
78 | // build the tabset
79 | var activeTab = 0;
80 | tabs.each(function(i) {
81 |
82 | // get the tab div
83 | var tab = $(tabs[i]);
84 |
85 | // get the id then sanitize it for use with bootstrap tabs
86 | var id = tab.attr('id');
87 |
88 | // see if this is marked as the active tab
89 | if (tab.hasClass('active'))
90 | activeTab = i;
91 |
92 | // remove any table of contents entries associated with
93 | // this ID (since we'll be removing the heading element)
94 | $("div#" + tocID + " li a[href='#" + id + "']").parent().remove();
95 |
96 | // sanitize the id for use with bootstrap tabs
97 | id = id.replace(/[.\/?&!#<>]/g, '').replace(/\s/g, '_');
98 | tab.attr('id', id);
99 |
100 | // get the heading element within it, grab it's text, then remove it
101 | var heading = tab.find('h' + tabLevel + ':first');
102 | var headingText = heading.html();
103 | heading.remove();
104 |
105 | // build and append the tab list item
106 | var a = $('' + headingText + '');
107 | a.attr('href', '#' + id);
108 | a.attr('aria-controls', id);
109 | var li = $('');
110 | li.append(a);
111 | tabList.append(li);
112 |
113 | // set it's attributes
114 | tab.attr('role', 'tabpanel');
115 | tab.addClass('tab-pane');
116 | tab.addClass('tabbed-pane');
117 | if (fade)
118 | tab.addClass('fade');
119 |
120 | // move it into the tab content div
121 | tab.detach().appendTo(tabContent);
122 | });
123 |
124 | // set active tab
125 | $(tabList.children('li')[activeTab]).addClass('active');
126 | var active = $(tabContent.children('div.section')[activeTab]);
127 | active.addClass('active');
128 | if (fade)
129 | active.addClass('in');
130 |
131 | if (tabset.hasClass("tabset-sticky"))
132 | tabset.rmarkdownStickyTabs();
133 | }
134 |
135 | // convert section divs with the .tabset class to tabsets
136 | var tabsets = $("div.section.tabset");
137 | tabsets.each(function(i) {
138 | buildTabset($(tabsets[i]));
139 | });
140 | };
141 |
142 |
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