07-example_finch_read.Rmd

# MLM, Centering/Scaling: Reading Achievement (2-levels only)



```{r, include=FALSE}
knitr::opts_chunk$set(comment     = "",
                      echo        = TRUE, 
                      warning     = FALSE, 
                      message     = FALSE,
                      fig.align   = "center", # center all figures
                      fig.width   = 6,        # set default figure width to 4 inches
                      fig.height  = 4)        # set default figure height to 3 inches
```

```{r, message=FALSE, error=FALSE}
library(tidyverse)    # all things tidy
library(haven)        # read in SPSS dataset
library(furniture)    # nice table1() descriptives
library(stargazer)    # display nice tables: summary & regression
library(texreg)       # Convert Regression Output to LaTeX or HTML Tables
library(psych)        # contains some useful functions, like headTail
library(performance)  # ICC and R2 calculations
library(effects)      # Effects for regression models
library(optimx)       # Different optimizers to solve mlm's
library(lme4)         # non-linear mixed-effects models
library(haven)        # read in SPSS dataset
```



<!-- ========================================================= -->
## Background
<!-- ========================================================= -->

```{block type='rmdlink', echo=TRUE}
The following example was included in the text "Multilevel Modeling in R" by @finch2016.

The datasets for this textbook may be downloaded from the website: http://www.mlminr.com/data-sets/.  
```


I was unable to find any documentation on this dataset in the book or online, so I contacted the authors.  There were unable to provide much either, but based on visual inspection designated the class of *factor* to thoes vairables that seem to represent categorical quantities. The labels for gender and class size are relative to the frequencies in the journal article the authors did point me to *(although the samples sizes do not match up)*.

```{block type='rmdimportant', echo=TRUE}
FOR THIS CHAPTER WE WILL IGNORE ALL LEVELS EXCEPT FOR STUDNETS BEING NESTED WITHIN SCHOOLS.
```

Read the SPSS data in with the `haven` package .

```{r}
data_raw <- haven::read_sav("http://www.mlminr.com/data-sets/Achieve.sav?attredirects=0")
```

Declare all categorical variables to be factors and apply labels where meaningful.

> Student-specific   
> * `gender` = Male or Female    
> * `age` = Age, in months    
> * `gevocab` = Vocabulary Score    
> * `geread` = Reading Score    

> Class-specific      
> * `classsize` = category of class's size    

> School-specific     
> * `senroll` = school enrollment    
> * `ses` = school's SES level    

```{r}
data_achieve <- data_raw %>% 
  dplyr::mutate_at(vars(id, region, corp, school, class), factor) %>% 
  dplyr::mutate(gender = gender %>% 
                  factor(labels = c("Female", "Male"))) %>% 
  dplyr::mutate(classize = classize %>% 
                  factor(labels = c("12-17", "18-21", 
                                    "22-26", ">26"))) %>% 
  dplyr::select(id, region, corp, school, class,           # Identifiers
                gender, age, geread, gevocab,              # Pupil-level vars
                classize,                                  # Class-Level vars
                senroll, ses)                              # School-level vars
```



### Sample Structure    


It is obvious that the sample is hiarchical in nature.  The nesting starts with `students` (level 1) nested within `class` (level 2), which are further nested within `school` (level 3), `corp` (level 4), and finally `region` (level 5).  

For this chapter we will only focus on TWO levels: **students** (level 1) are the units on which the outcome is measured and **schools** (level 2) are the units in which they are nested.

The number of ***regions*** = 9:

```{r num_reg}
num_regions <- data_achieve %>% 
  dplyr::group_by(region) %>% 
  dplyr::tally() %>% 
  nrow()

num_regions
```

The number of **corps** = 60:

```{r num_corp}
num_corps <- data_achieve %>% 
  dplyr::group_by(region, corp) %>% 
  dplyr::tally() %>% 
  nrow()

num_corps 
```

The number of **schools** = 160 

```{r num_school}
num_schools <- data_achieve %>% 
  dplyr::group_by(region, corp, school) %>% 
  dplyr::tally() %>% 
  nrow()

num_schools
```

The number of **classes** = 568
```{r num_class}
num_classes <- data_achieve %>% 
  dplyr::group_by(region, corp, school, class) %>% 
  dplyr::tally() %>% 
  nrow()

num_classes
```

The number of **students** = 10320

```{r num_kid}
num_subjects <- data_achieve %>% nrow

num_subjects
```

<!-- ========================================================= -->
## Exploratory Data Analysis
<!-- ========================================================= -->

### Summarize Descriptive Statistics

#### The `stargazer` package

Most posters, journal articles, and reports start with a table of descriptive statistics.  Since it tends to come first, this type of table is often refered to as Table 1.  The `stargazer()` function can be used to create such a table, but only for the entire dataset.  I haven't been able to find a way to get it to summarize subsamples and compare them in the standard format.  Also, it only summarises continuous, not categorical variables.

```{r eval = FALSE}
# Knit to Website: type = "html" 
# Knit to PDF:     type = "latex"
# View on Screen:  type = "text"

data_achieve %>% 
  dplyr::select(classize, gender, geread, gevocab, age, ses, senroll) %>% 
  data.frame() %>% 
  stargazer::stargazer(header = FALSE,
                      title = "Summary of the numeric variables with `stargazer`",
                       type = "text")
```

```{r sumtable0, echo=FALSE,results='asis'}
data_achieve %>% 
  dplyr::select(classize, gender, geread, gevocab, age, ses, senroll) %>% 
  data.frame() %>% 
  stargazer::stargazer(header = FALSE,
                      title = "Summary of the numeric variables with `stargazer`",
                       type = "html")
```


#### The `furniture` package

Tyson Barrett's **furniture** package includes the extremely useful function `table1()` which simplifies the common task of creating a stratified, comparative table of descriptive statistics.  Full documentation can be accessed by executing `?furniture::table1`.

```{r, eval=FALSE}
# Knit to Website: output = "html" 
# Knit to PDF:     output = "latex2"
# View on Screen:  output = ""text", or "markdown", "html"

data_achieve %>% 
  furniture:: table1("Reading Score"       = geread, 
                     "Vocabulary Score"    = gevocab, 
                     "Age (in months)"     = age, 
                     "School SES"          = ses, 
                     "School's Enrollment" = senroll,
                     splitby   = ~ gender,                                        # var to divide sample by
                     test      = TRUE,                                            # test groups different?
                     caption   = "Summary of the numeric variables with `table1`", 
                     output    = "html")
```

```{r sumtable1, echo=FALSE, output="asis"}
data_achieve %>% 
  furniture:: table1("Reading Score"       = geread, 
                     "Vocabulary Score"    = gevocab, 
                     "Age (in months)"     = age, 
                     "School SES"          = ses, 
                     "School's Enrollment" = senroll,
                     splitby   = ~ gender,                                        # var to divide sample by
                     test      = TRUE,                                            # test groups different?
                     caption   = "Summary of the numeric variables with `table1`", 
                     output    = "html")
```


### Visualization of Raw Data


#### Level One Plots: Disaggregate or ignore higher levels   


For a first look, its useful to plot all the data points on a single scatterplot as displayed in the previous plot.  Due to the large sample size, many points end up being plotted on top of or very near each other (*overplotted*).  When this is the case, it can be useful to use  `geom_binhex()` rather than `geom_point()` so the color saturation of the hexigons convey the number of points at that location, as seen in Figure \ref{fig:hexbin}.  

*Note: I had to manually install the package `hexbin` for the `geom_hex()` to run.*

```{r densityHex, fig.cap="Raw Data: Density, Vocab vs. Reading"}
data_achieve %>% 
  ggplot() +
  aes(x = gevocab, 
      y = geread) +
  stat_binhex(colour = "grey85", na.rm  = TRUE) +     # outlines
  scale_fill_gradientn(colors   = c("grey80","navyblue"), # fill color extremes
                       name     = "Frequency",        # legend title
                       na.value = NA) +               # color for count = 0
  theme_bw()
```


#### Multilevel plots: illustrate two nested levels  

Up to this point, all investigation of this dataset has been only at the pupil level and any nesting or clustering within schools has been ignored.  Plotting is a good was to start to get an idea of the school-to-school variability.  This figure displays four handpicked school to illustrate the degreen of school-to-school variability in the **association between vocab and reading** scores.

```{r schoolRegEx, fig.width=6, fig.height=6, fig.cap="Raw Data: Independent Single-Level Regression within each school, a few illustrative cases"}
data_achieve %>% 
  dplyr::filter(school %in% c(1321, 6181, 
                              6197, 6823)) %>%  # choose school numbers
  ggplot(aes(x = gevocab,
             y = geread))+
  geom_count() +             # creates points, size by overplotted number
  geom_smooth(method = "lm") +     # linear model (OLS) 
  facet_wrap(~ school) +           # panels by school
  theme_bw()
```

Another way to explore the **school-to-school variability** is to plot the linear model fit *independently* to each of the schools.  This next figure displays only the smooth lines without the standard error bands or the raw data in the form of points or hexagons.

```{r schoolRegAll, fig.cap="Raw Data: Independent Single-Level Regression within each school, all schools shown together"}
data_achieve %>% 
  ggplot(aes(x = gevocab,
             y = geread)) +
  geom_smooth(aes(group = school),
              method = "lm",
              se     = FALSE,      # do NOT want the SE bands
              size   = 0.3) +   
  geom_smooth(method = "lm",
              se     = FALSE,
              color = "red",       # do NOT want the SE bands
              size   = 2) +        # make the lines thinner
  theme_bw() 
```


Due to the high number of schools, the figure with all the school's independent linear regression lines resembles a hairball and is hard to deduce much about individual schools.  By using the `facet_grid()` layer, we can separate the schools out so better see school-to-school variability.  It also allows investigation of  higher level predictors, such as the school's SES (median split with `ntile(var, 2)`) and class size.


```{r classRegSep, fig.width=6, fig.height=8, fig.cap="Raw Data: Independent Single-Level Regression within each school, sepearated by school size and school SES"}
data_achieve %>% 
  dplyr::mutate(ses2 = ntile(ses, 2) %>%                  # median split
                  factor(labels = c("SES: Lower Half", 
                                    "SES: Upper Half"))) %>% 
  dplyr::mutate(senroll = ntile(senroll, 3) %>% 
                  factor(labels = c("Enroll: Smallest Third",
                                    "Enroll: Middle Third",
                                    "Enroll: Largest Third"))) %>% 
  ggplot(aes(x       = gevocab,
             y       = geread,
             group   = school)) +     # separates students into schools
  geom_smooth(method = "lm",
              se     = FALSE,         
              size   = 0.3,
              color = "black",
              alpha = .2) +         
  theme_bw() +
  facet_grid(senroll ~ ses2)     # makes separate panels (rows ~ columns)
```


<!-- ========================================================= -->
## Single-Level Regression
<!-- ========================================================= -->

### Fit Nested Models

Ignoring the fact that students are nested or clustered within schools, is called dissagregating.  This treats all students as independent units.

```{r}
# linear model - ignores school (for reference only)
fit_read_lm_0 <- lm(formula = geread ~ 1,               # intercept only
                    data    = data_achieve)

fit_read_lm_1 <- lm(formula = geread ~ gevocab ,        # one predictor
                    data    = data_achieve)

fit_read_lm_2 <- lm(formula = geread ~ gevocab + age,   # two predictors
                    data    = data_achieve)

fit_read_lm_3 <- lm(formula = geread ~ gevocab*age,    # interation+main effects
                    data    = data_achieve)
```


Now compare the models:


```{r regLM, results='asis'}
texreg::knitreg(list(fit_read_lm_0, 
                     fit_read_lm_1, 
                     fit_read_lm_2, 
                     fit_read_lm_3),
                custom.model.names = c("Null",
                                       "1 IV",
                                       "2 IV",
                                       "Interaction"),
                caption            = "OLS: Investigate Fixed, Pupil-level Predictors",
                caption.above      = TRUE,
                single.row         = TRUE)
```


Assess the significance of terms in the last 'best' model

```{r}
summary(fit_read_lm_3) 
```


```{r}
performance::r2(fit_read_lm_3)
```


```{r}
anova(fit_read_lm_3)
```

### Visualize the Interaction

```{r}
effects::Effect(focal.predictors = c("gevocab", "age"),     # chooses default values for
                mod = fit_read_lm_3)                        # continuous vars
```



```{r}
effects::Effect(focal.predictors = c("gevocab", "age"),     # chooses default values for
                mod = fit_read_lm_3) %>%                    # continuous vars
  data.frame() %>%  
  mutate(age = factor(age)) %>%           # must make a factor to separate lines
  ggplot(aes(x = gevocab,
             y = fit,
             color = age)) +
  geom_point() +
  geom_line() 
```


Here is a better version of the plot.

Age is in months, so we want multiples of 12 for good visualization

```{r}
summary(data_achieve$age)/12   # divide by 12 to change months to years
```

A good set set of illustrative ages could be: 7, 9, and 11:

```{r}
c(7, 9, 11) * 12   # times by 12 to change years to months
```


```{r mlInteraction}
effects::Effect(focal.predictors = c("gevocab", "age"),
                mod = fit_read_lm_3,
                xlevels = list(age = c(84, 108, 132))) %>%  # age is in months
  data.frame() %>% 
  mutate(age_yr = factor(age/12)) %>%    # it would be nice to plot age in years
  ggplot(aes(x        = gevocab,
             y        = fit,
             color    = age_yr,
             linetype = age_yr)) +
  geom_line(size = 1.25) +
  theme_bw() +
  labs(title = "Best Linear Model - Disaggregated Data (OLS)",
       x = "Vocabulary Score",
       y = "Reading Score",
       linetype = "Age (yrs)",
       color    = "Age (yrs)") +
  theme(legend.position = c(0.85, 0.2),
        legend.key.width = unit(2, "cm"),
        legend.background = element_rect(color = "black")) +
  scale_linetype_manual(values = c("solid", "longdash", "dotted")) +
  scale_x_continuous(breaks = seq(from = 0, to = 11, by = 2)) +
  scale_y_continuous(breaks = seq(from = 0, to = 11, by = 1))
```



<!-- ========================================================= -->
## MLM - Step 1: Null Model, only fixed and random intercepts
<!-- ========================================================= -->


A so called *Empty Model* only includes random intercepts.  No independent variables are involved, other the grouping or clustering variable that designates how *level 1* units are nested within *level 2* units.  For a cross-sectional study design this would be the grouping variables, where as for longitudinal or repeated measures designs this would be the subject identifier.  This **nested structure** variable should be set to have class `factor`.

### Fit the Model

```{r}
fit_read_0ml <- lme4::lmer(geread ~ 1 + (1|school), 
                           data = data_achieve,
                           REML = FALSE)                  # fit via ML (not the default)

fit_read_0re <- lme4::lmer(geread ~ 1 + (1|school) , 
                           data = data_achieve,
                           REML = TRUE)                   # fit = REML (the default)
```


Compare the two models to OLS:


```{r mlmNull, results='asis'}
texreg::knitreg(list(fit_read_lm_0, 
                     fit_read_0ml, 
                     fit_read_0re),
                custom.model.names = c("OLS", 
                                       "MLM-ML", 
                                       "MLM-REML"),
                caption            = "MLM: NULL Model,two estimation methods",
                caption.above      = TRUE,
                single.row         = TRUE)
```

Notice that the estimate for the intercept is nearly the same in the linear regression and intercept only models, but the standard errors are quite different.  When there is clustering in sample, the result of ignoring it is under estimation of the standard errors and over stating the significance of associations.  This table was made with the `screenreg()` function in the self named package.  I tend to prefer this display over `stargazer()`.


### Estimate the ICC

First, ask for the variance compenents:

```{r}
lme4::VarCorr(fit_read_0re) %>% 
  print(comp = c("Variance", "Std.Dev"),
        digits = 4)
```




```{r}
insight::get_variance(fit_read_0re)
```



$$
\begin{align*}
\text{schools}                      \rightarrow \; & \sigma^2_{u0} = 0.6257^2 = 0.392 \\
\text{students within schools}      \rightarrow \; &   \sigma^2_{e}  = 2.2461^2 = 5.045 \\
\end{align*}
$$


```{block type='genericEq', echo=TRUE}
**Intraclass Correlation (ICC) Formula**
$$
\overbrace{\rho}^{\text{ICC}} = 
\frac{\overbrace{\sigma^2_{u0}}^{\text{Random Intercept}\atop\text{Variance}}}
     {\underbrace{\sigma^2_{u0}+\sigma^2_{e}}_{\text{Total}\atop\text{Variance}}}
\tag{Hox 2.9}
$$
```


Then you can manually calculate the ICC.

```{r}
0.392 / (0.392 + 5.045)
```

Or you can use the `icc()` function in the `performance` package.

```{r}
performance::icc(fit_read_0re)
```


*Note: On page 45  [@finch2016], the authors substituted standard deviations into the formula, rather than variances.  The mistake is listed on their webpage errata (http://www.mlminr.com/errata) and is repeated through the text.*


<!-- ========================================================= -->
## MLM - Step 2: Add Lower-level explanatory variables, fixed, ML
<!-- ========================================================= -->

**Variance Component** models (steps 2 and 3) - decompose the INTERCEPT variance into different variance compondents for each level.  The regression intercepts are assumed to varry ACROSS the groups, while the slopes are assumed fixed (no random effects).


Fixed effects selection should come **prior** to random effects.  You should use *Maximum Likelihood (ML)* estimation when fitting these models.


```{block type='rmdlightbulb', echo=TRUE}
* IF: only level 1 predictors and random intercepts are incorporated  
* Then: MLM \textit{(ML)} $\approx$ ANCOVA \textit{OLS}.
```



### Add pupil's vocab score as a fixed effects predictor

```{r}
fit_read_1ml <- lme4::lmer(geread ~ gevocab + (1|school), 
                           data = data_achieve,
                           REML = FALSE)            # to compare fixed var sig

fit_read_1re <- lme4::lmer(geread ~ gevocab + (1|school), 
                           data = data_achieve,
                           REML = TRUE)             # for R-sq calcs
```



```{r, results='asis'}
texreg::knitreg(list(fit_read_0ml, 
                     fit_read_1ml),
                custom.model.names = c("Null", 
                                       "w Pred"),
                caption = "MLM: Investigate a Fixed Pupil-level Predictor",
                caption.above = TRUE,
                digits = 4)
```


#### Assess Significance of Effects

Likelihood Ratio Test (LRT)

Since models 0 and 1 are nested models, only differing by the the inclusion or exclusion of the fixed effects predictor `gevocab`, AND both models were fit via Maximum Likelihood, we can compare the model fit may be compared via the *Likilihood-Ratio Test (LRT)*.  The *Likelihood Ratio* value *(L. Ratio)* is found by subtracting the two model's  `-2 * logLik` or `deviance` values.  Significance is judged by the Chi Squared distribution, using the difference in the number of parameters fit as the degrees of freedom. 


```{r}
anova(fit_read_0ml, fit_read_1ml)
```

What does the model look like?

```{r}
effects::Effect(focal.predictors = c("gevocab"), 
                mod = fit_read_1ml) %>% 
  data.frame() %>% 
  ggplot(aes(x = gevocab,
             y = fit)) +
  geom_ribbon(aes(ymin = lower,
                  ymax = upper),
                alpha = .3) +
  geom_line() +
  theme_bw()
```

#### Proportion of Variance Explained 

Extract the variance-covariance estimates:

BL = BAseline: The Null Model (fit via REML)
```{r}
lme4::VarCorr(fit_read_0re) %>% 
  print(comp = c("Variance", "Std.Dev"),
        digits = 4)
```


$$
\sigma^2_{u0-BL} = 0.392 \\
\sigma^2_{e-BL}  = 5.045 
$$



MC = Model to Compare: Model with Predictor (fit via REML)

```{r}
lme4::VarCorr(fit_read_1re) %>% 
  print(comp = c("Variance", "Std.Dev"),
        digits = 4)
```

$$
\sigma^2_{u0-MC} = 0.100 \\
\sigma^2_{e-MC}  = 3.766 
$$


**Level 1 $R^2$ - Snijders and Bosker**

Found on page 47 [@finch2016], the proportion of variance in the outcome explained by predictor on level one is given by:   

```{block type='genericEq', echo=TRUE}
**Snijders and Bosker Formula - Level 1 ** 
$$
R^2_1 = 1 - \frac{\sigma^2_{e-MC} + \sigma^2_{u0-MC}}
                 {\sigma^2_{e-BL} + \sigma^2_{u0-BL}}
$$
```




*Note: This formula also apprears in the Finch errata.  The subscripts in the denominator of the fraction should be for model 0, not model 1. The formula is given correctly here. They did substitute in the correct values.*


Calculate the value by hand:
```{r}
1 - (0.100 + 3.766)/(0.392 + 5.045)
```


Or use the `performance` package to help out:  (Note it is using a difference method...)

```{r}
performance::r2(fit_read_1re)
```


This means nearly 30% of the variance in reading scores, above and beyond that accounted for by school membership *(i.e. school makeup or school-to-school variation)*, is attributable to vocabulary scores. 



**Level 1 $R^2$ - Raudenbush and Bryk**

@hox2017 presents this formula on page 58 of chapter 2


```{block type='genericEq', echo=TRUE}
**Raudenbush and Bryk Approximate Formula - Level 1 ** 
$$
approx\; R^2_1 = \frac{\sigma^2_{e-BL} - \sigma^2_{e-MC}}
              {\sigma^2_{e-BL} }
\tag{Hox 4.8}
$$
```



Calculate the value by hand:
```{r}
(5.045 - 3.766) / 5.045
```

Although slightly different in value and meaning, this value also conveys that vocabulary scores are highly associated with reading scores.





**Level 2 $R^2$ - Snijders and Bosker Formula Extended ** 



```{block type='genericEq', echo=TRUE}
**Snijders and Bosker Formula Extended - Level 2 ** 
$$
R^2_2 = 1 - \frac{\frac{\sigma^2_{e-MC}}{B} + \sigma^2_{u0-MC}}
                 {\frac{\sigma^2_{e-BL}}{B} + \sigma^2_{u0-BL}}
$$

$B$ is the average size of the Level 2 units (schools).  Technically, you should use the *harmonic mean*, but unless the clusters differ greatly in size, it doesn't make a huge difference.
```




* Average sample cluster size
```{r}
num_subjects / num_schools
```


* Calculate by hand:
```{r}
1 - ((3.766 / 64.5) + 0.100) / ((5.045 / 64.5) + 0.391)
```

This means that over two-thirds in school mean reading levels may be explained by their student's vocabulary scores.


**Level 2 $R^2$ - Raudenbush and Bryk**

```{block type='genericEq', echo=TRUE}
**Raudenbush and Bryk Approximate Formula - Level 2 ** 
$$
R^2_1 = \frac{\sigma^2_{u0-BL} - \sigma^2_{u0-MC}}
                 {\sigma^2_{u0-BL} }
\tag{Hox 4.9}
$$
```


```{r}
(0.392 - 0.100)/(0.392)
```


Remeber that these 'variance accounted for' estimations are not as straight forwards as we would like.  



### Investigate More Level 1 Predictors

Part of investigating lower level explanatory variables, is checking for interactions between these variables.  The interaction between fixed effects is also considered to be a fixed effect, so we need to employ *Maximum Likelihood* estimation to compare nested models.

```{r}
fit_read_2ml <- lmer(geread ~ gevocab + age + (1 | school), # add main effect of age
                     data = data_achieve,
                     REML = FALSE)

fit_read_3ml <- lmer(geread ~ gevocab*age + (1 | school), # add interaction between vocab and age
                     data = data_achieve,
                     REML = FALSE)
```




```{r, results='asis'}
texreg::knitreg(list(fit_read_1ml, 
                     fit_read_2ml, 
                     fit_read_3ml),
                custom.model.names = c("Only Vocab", 
                                       "Both Main Effects", 
                                       "Interaction"),
                caption = "MLM: Investigate Other Fixed Pupil-level Predictors",
                caption.above = TRUE,
                digits = 4)
```

#### Assess Significance of Effects

Likelihood Ratio Test (LRT)

```{r}
anova(fit_read_1ml, fit_read_2ml, fit_read_3ml)
```


Not only is student's age predictive of their reading level (I could have guessed that), but that age moderated the relationship between vocabulary and reading.


#### Visulaize the Interation

Visulaizations are extremely helpful to interpred interactions.

```{r}
summary(data_achieve$age)
```


```{r}
effects::Effect(focal.predictors = c("gevocab", "age"),   # variables involved in the interaction
                mod = fit_read_3ml,
                xlevels = list(age = c(84, 108, 132))) %>%  # age is in months
  data.frame() %>% 
  mutate(age_yr = factor(age/12)) %>%    # it would be nice to plot age in years 
  ggplot(aes(x = gevocab,
             y = fit,
             color = age_yr)) +
  geom_line() +
  theme_bw()
```

There is a positive association between vocabulary and reading, but it is strongest for older childred.  Among younger children, reading scores are more stable across vocabulary differences. 


<!-- ========================================================= -->
## MLM - Step 3: Higher-level explanatory variables, fixed, ML
<!-- ========================================================= -->

School enrollment (`senroll`) applies to each school as a whole.  When a variable is measured at a higher level, all units in the same group have the same value.  In this case, all student in the same school have the same value for `senroll`.  

```{r}
fit_read_4ml <- lme4::lmer(geread ~ gevocab*age + senroll + (1 | school), 
                           data = data_achieve,
                           REML = FALSE)
```




```{r, results='asis'}
texreg::knitreg(list(fit_read_0ml, 
                     fit_read_3ml, 
                     fit_read_4ml),
                custom.model.names = c("Null", 
                                       "Level 1 only", 
                                       "Level 2 Pred"),
                caption            = "MLM: Investigate a Fixed School-Level Predictor",
                caption.above      = TRUE,
                single.row         = TRUE)
```


### Assess Significance of Effects

Likelihood Ratio Test (LRT)
```{r}
anova(fit_read_0ml, fit_read_3ml, fit_read_4ml)
```

School enrollment (or size) does not seem be related to reading scores.


<!-- ========================================================= -->
## MLM - Step 4: Explanatory variables predict Slopes, random, REML
<!-- ========================================================= -->

**Random Coefficient** models - decompose the SLOPE variance BETWEEN groups. 

The fixed effect of the predictor captures the overall association it has with the outcome (intercept), while the random effect of the predictor captures the group-to-group variation in the association (slope).  *Note: A variable can be fit as BOTH a fixed and random effect.*  

```{r}
fit_read_3re <- lme4::lmer(geread ~ gevocab*age + (1 | school), # refit the previous 'best' model via REML
                     data = data_achieve,
                     REML = TRUE)

#fit_read_5re <- lmer(geread ~ gevocab + (gevocab | school), 
#                     data = achieve,
#                     REML = TRUE)         # failed to converge :(

fit_read_5re <- lme4::lmer(geread ~ gevocab*age + (gevocab | school), 
                           data = data_achieve,
                           REML = TRUE,
                           control = lmerControl(optimizer = "optimx",    # get it to converge
                                                 calc.derivs = FALSE,
                                                 optCtrl = list(method = "nlminb",
                                                                starttests = FALSE,
                                                                kkt = FALSE))) 
```



```{r, results='asis'}
texreg::knitreg(list(fit_read_3re, 
                       fit_read_5re),
                  custom.model.names = c("Rand Int", 
                                         "Rand Int and Slopes"),
                  caption            = "MLM: Investigate Random Effects",
                  caption.above      = TRUE,
                  single.row         = TRUE)
```





#### Assess Significance of Effect


```{block type='rmdlightbulb', echo=TRUE}
Likelihood Ratio Test (LRT) for Random Effects

You can use the Chi-squared LRT test based on deviances even though we fit our modesl with REML, since the models only differ in terms of including/exclusing of a random effects; they have same fixed effects.  Just make sure to include the `refit = FALSE` option.
```


```{r}
anova(fit_read_3re, fit_read_5re, refit = FALSE)
```

There is evidence the effect child vocabulary has on reading varies across schools.


#### Visualize the Model

What does the model look like?

```{r}
effects::Effect(focal.predictors = c("gevocab", "age"), 
                mod = fit_read_5re,                              # just different model
                xlevels = list(age = c(84, 108, 132))) %>% 
  data.frame() %>% 
  dplyr::mutate(age_yr = factor(age/12)) %>% 
  ggplot(aes(x = gevocab,
             y = fit,
             color = age_yr)) +
  geom_line() +
  theme_bw()
```

We are seeming much the same trends, but perhaps more separation between the lines.

<!-- ========================================================= -->
## MLM - Step 5: Cross-Level interactions between explanatory variables - fixed, ML
<!-- ========================================================= -->

Cross-level interacitons involve variables at different levels.  Here we will investigate the school-level enrollment moderating vocabulary's effect since we say that vocab's effect differs across schools (step 4).

Remember that an interaction beween fixed effects is also fixed.  

```{r}
fit_read_5ml <- lme4::lmer(geread ~ gevocab*age + (gevocab | school), 
                           data = data_achieve,
                           REML = FALSE,
                           control = lmerControl(optimizer = "optimx", 
                                                 calc.derivs = FALSE,
                                                 optCtrl = list(method = "nlminb",
                                                                starttests = FALSE,
                                                                kkt = FALSE)))

fit_read_6ml <- lme4::lmer(geread ~ gevocab*age + senroll + (gevocab | school), 
                           data = data_achieve,
                           REML = FALSE,
                           control = lmerControl(optimizer ="Nelder_Mead"))

fit_read_7ml <- lme4::lmer(geread ~ gevocab*age + gevocab*senroll + (gevocab | school), 
                           data = data_achieve,
                           REML = FALSE)

fit_read_8ml <- lme4::lmer(geread ~ gevocab*age*senroll + (gevocab | school), 
                           data = data_achieve,
                           REML = FALSE)
```

```{block type='rmdlightbulb', echo=TRUE}
If you get the`lmer()` message: **Some predictor variables are on very different scales: consider rescaling**, you can trust your results, but you really should try re-scaling your variables.
```

We are getting this message since `gevoab` is on mostly a single digit scale,0 to 11.2, and age (in months) ranges in the low thripe-digits, 82 through 135, while school enrollment is in the mid-hundreds, 112-916.  When we compute the interactions we get much, much larger values.  Having variables on such widely different ranges of values can cause estimation problems.

```{r}
data_achieve %>% 
  dplyr::select(gevocab, age, senroll) %>% 
  summary()
```

For now, let us look at the results.

```{r, results='asis'}
texreg::knitreg(list(fit_read_5ml, 
                     fit_read_6ml, 
                     fit_read_7ml,
                     fit_read_8ml),
                custom.model.names = c("Level 1 only", 
                                       "Both Levels", 
                                       "Cross-Level",
                                       "3-way"),
                caption            = "MLM: Investigate a Fixed Cross-level Interaction",
                caption.above      = TRUE,
                digits = 3,
                bold = TRUE)
```


There is no evidence school enrollment moderates either of age or vocabulary's effects.


#### Assess Significance of Effects

Likelihood Ratio Test (LRT)

When you have a list of sequentially nested models, you can test them in order with one call to the `anova()` funtion.

```{r}
anova(fit_read_5ml, fit_read_6ml)
```
```{r}
anova(fit_read_5ml, fit_read_7ml)
```

```{r}
anova(fit_read_5ml, fit_read_8ml)
```

<!-- ========================================================= -->
## Centering Predictors: Change Center
<!-- ========================================================= -->

Centering variables measured on the lowest level only involves subtacting the mean from every value.  The spread or standard deviation is not changed.

```{block type='rmdlightbulb', echo=TRUE}
Although there are functions to automatically center and standardize variables, it is beneficial to manually create these variables, as it is more transparent and facilitates un-centering them later.
```


```{r}
data_achieve %>% 
  dplyr::select(gevocab, age, senroll) %>% 
  summary()
```


### Use Centered Variables


NOTE:  The models with CENTERED variables are able to be fit with the default optimizer settings and do not return the error: `"unable to evaluate scaled gradientModel failed to converge: degenerate  Hessian with 1 negative eigenvalues"`

```{r}
fit_read_5ml_c <- lme4::lmer(geread ~ I(gevocab - 4.494) * I(age - 107.5) + 
                               (I(gevocab - 4.494) | school), 
                             data = data_achieve,
                             REML = FALSE)
```




#### Compare the Models

```{r, results='asis'}
texreg::knitreg(list(fit_read_5ml, 
                     fit_read_5ml_c),
                custom.model.names = c("Raw Units", 
                                       "Grand Mean Centered"),
                groups = list("Raw Scale" = 2:4,
                              "Mean Centered" = 5:7),
                caption            = "MLM: Investigate Centering Variables Involved in an Interaction",
                caption.above      = TRUE,
                single.row         = TRUE,
                digits             = 3)
```


Notice that the interactions yield the exact same parameter estimates and significances, but the main effects (including the interactions) are different.  Model fit statistics include $-2LL$ are exactly the same, too.



```{r}
performance::compare_performance(fit_read_5ml, fit_read_5ml_c)
```


#### Visualize the Model

What does the model look like?

First plot the model fit to the centered variables with all defaut settings.

```{r}
interactions::interact_plot(model = fit_read_5ml_c,
                            pred  = gevocab,
                            modx  = age)
```

```{r}
interactions::interact_plot(model = fit_read_5ml_c,
                            pred  = gevocab,
                            modx  = age,
                            modx.values = c(80, 110, 150),
                            interval = TRUE)
```



<!-- ========================================================= -->
## Rescaling Predictors: Change Units or Standardize
<!-- ========================================================= -->


Where centering variables involved subtracting a set value, scalling a varaibles involves dividing by a set amount.  When we both center to the mean and divide by the standard deviation, the new resulting varaible is said to be standardized (not to be confusing with normalizing, which is does not do).  

To retain meaningful units, you can multiply or divide all the measured values of a variable by a set amount, like a multiple of 10.  This retains the meaning behind the units while still bringing them into line with other variables in the model and can avoid some convergence issues.

### Scale Varaibles

#### Divide by a Meaningful Value

```{r}
data_achieve %>% 
  dplyr::select(gevocab, age, ses) %>% 
  summary()
```



For this situation, lets only divide SES by ten.



### Use Scaled Variables

Using the new versions of our variables, investigate is SES has an effect, either in 2-way or 3-way interactions with age and vocabulary.

```{r}
fit_read_5ml_s <- lme4::lmer(geread ~ I(gevocab - 4.494) * I(age - 107.5) + 
                                     (I(gevocab - 4.494) | school), 
                             data   = data_achieve,
                             REML   = FALSE)

fit_read_6ml_s <- lme4::lmer(geread ~ I(gevocab - 4.494) * I(age - 107.5) + 
                                      I(gevocab - 4.494) * I(ses/10) +
                                     (I(gevocab - 4.494) | school), 
                             data   = data_achieve,
                             REML   = FALSE,
                             control = lmerControl(optimizer ="Nelder_Mead"))

fit_read_7ml_s <- lme4::lmer(geread ~ I(gevocab - 4.494) * I(age - 107.5) * I(ses/10) +
                                     (I(gevocab - 4.494) | school),  
                             data   = data_achieve,
                             REML  = FALSE,
                             control = lmerControl(optimizer ="Nelder_Mead"))
```



```{r, results='asis'}
texreg::knitreg(list(fit_read_5ml_s, 
                     fit_read_6ml_s, 
                     fit_read_7ml_s),
                custom.model.names = c("no SES", 
                                       "2-way", 
                                       "3-way"),
                caption            = "MLM: Investigate More Complex Fixed Interactions (with Scalling)",
                caption.above      = TRUE,
                digits = 3)
```


```{r}
anova(fit_read_5ml_s, fit_read_6ml_s, fit_read_7ml_s)
```

There is evidence that SES moderates the main effect of vocabulary, after accounting for the interaction between age and vocabulary.  But there is NOT evidence of a three-way interaction between vobaculary, age, and SES.


## Final Model

Always refit the final model via REML.

```{r}
fit_read_6re_s <- lme4::lmer(geread ~ I(gevocab - 4.494) * I(age - 107.5) + 
                                      I(gevocab - 4.494) * I(ses/10) +
                                     (I(gevocab - 4.494) | school), 
                             data   = data_achieve,
                             REML   = TRUE)
```

### Table

```{r, results='asis'}
texreg::knitreg(fit_read_6re_s,
                caption            = "MLM: Final Model",
                caption.above      = TRUE,
                single.row         = TRUE,
                digits = 3)
```


#### Visualize the Model

Recall the scales that the revised variables are now on:

```{r}
data_achieve %>% 
  dplyr::select(gevocab, age, ses) %>% 
  summary()
```

```{r}
interactions::interact_plot(model = fit_read_6ml_s,  # model name
                            pred  = gevocab,         # x-axis, main independent variable (continuous, ordinal)
                            modx  = age,             # lines by moderator, another independent variable
                            mod2 = ses)              # panel by 2nd moderator, another indep var
```

Define values for the moderator(s):

```{r}
interactions::interact_plot(model = fit_read_6ml_s,
                            pred  = gevocab,
                            modx  = age,
                            mod2 = ses,
                            modx.values = c(90, 110, 130),
                            mod2.values = c(20, 55, 90),
                            interval = TRUE)
```

Swap the moderators:

```{r}
interactions::interact_plot(model = fit_read_6ml_s,
                            pred  = gevocab,
                            mod2  = age,
                            modx = ses,
                            mod2.values = c(90, 110, 130),
                            modx.values = c(20, 55, 90),
                            interval = TRUE)
```

### Interpretation of model

There is strong evidence that higher vocabulary scores correlate with higher reading scores.  This relationship is strongest in low SES schools and among older students.  This relationship is also weaker in younger students and those attending high SES schools.


See: @nakagawa2013 for more regarding model $R^2$

From the `performance::r2()` documentation, for mixed models:    

* **marginal r-squared** considers only the variance of the fixed effects    
* **conditional r-squared** takes both the fixed and random effects into account

```{r}
performance::r2(fit_read_6re_s)
```

Over 32% of the variance in student reading is attributable vocab, ses, age, and school-to-school differences, R^2 = .322.

---- 

Helpful links:

http://maths-people.anu.edu.au/~johnm/r-book/xtras/mlm-ohp.pdf

http://ase.tufts.edu/gsc/gradresources/guidetomixedmodelsinr/mixed%20model%20guide.html 

http://web.stanford.edu/class/psych252/section_2015/Section_week9.html 

https://www.r-bloggers.com/visualizing-generalized-linear-mixed-effects-models-with-ggplot-rstats-lme4/ 

https://www.r-bloggers.com/visualizing-generalized-linear-mixed-effects-models-part-2-rstats-lme4/ 

http://www.strengejacke.de/sjPlot/sjp.lmer/