21-lta-estimation.Rmd

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, 
                      warning = FALSE,
                      message = FALSE) #Here, I have made it so that when you knit your .rmd, warnings and messages will not show up in the html markdown. 
```

# Measurement Invariance 

--------------------------------------------------------------------------------------

**Data Source**: The data used to illustrate these analyses include elementary school student *Science Attitude* survey items collected during 7th and 10th grades from the **Longitudinal Study of American Youth** (LSAY; Miller, 2015).

--------------------------------------------------------------------------------------

To install package {`rhdf5`} 

```{r}
if (!requireNamespace("BiocManager", quietly = TRUE)) 
  install.packages("BiocManager")

#BiocManager::install("rhdf5")
```

*Load packages*

```{r, eval=TRUE}
library(MplusAutomation)
library(rhdf5)
library(tidyverse)       
library(here)            
library(glue)            
library(janitor)            
library(gt) 
library(reshape2)
library(cowplot)
library(ggrepel)
library(haven)
library(modelsummary)
library(corrplot)
library(DiagrammeR)
library(filesstrings)
library(PNWColors)
```

--------------------------------------------------------------------------------------

Read in LSAY data file, `lsay_new.csv`.

```{r, eval=TRUE}

lsay_data <- read_csv(here("data","lsay_lta.csv"), na = c("9999")) %>% 
    mutate(across(everything(), as.numeric))
```

--------------------------------------------------------------------------------------

## Estimate Latent Transition Analysis (LTA) Model

--------------------------------------------------------------------------------------

### Estimate Invariant LTA Model

```{r, eval = FALSE}

lta_inv <- mplusObject(
  
  TITLE = 
    "Invariant LTA", 
  
  VARIABLE = 
     "usev = ab39m ab39t ab39u ab39w ab39x  ! 7th grade indicators
             ga33a ga33h ga33i ga33k ga33l; ! 10th grade indicators
      
      categorical = ab39m-ab39x ga33a-ga33l;

      classes = c1(4) c2(4);",
    
  ANALYSIS = 
     "estimator = mlr;
      type = mixture;
      starts = 500 100;
      processors=10;",

  MODEL = 
     "%overall%
      c2 on c1;

      MODEL c1: 
      %c1#1%
      [AB39M$1-AB39X$1] (1-5);  !!! labels that are repeated will constrain parameters to equality !!!
      %c1#2%
      [AB39M$1-AB39X$1] (6-10);
      %c1#3%
      [AB39M$1-AB39X$1] (11-15);
      %c1#4%
      [AB39M$1-AB39X$1] (16-20);

      MODEL c2:
      %c2#1%
      [GA33A$1-GA33L$1] (1-5);
      %c2#2%
      [GA33A$1-GA33L$1] (6-10);
      %c2#3%
      [GA33A$1-GA33L$1] (11-15);
      %c2#4%
      [GA33A$1-GA33L$1] (16-20);",
   
  SAVEDATA = 
   "file = LTA_Inv_CPROBS.dat;
    save = cprob;
    missflag = 9999;",

  OUTPUT = "tech1 tech15 svalues;",
  
  usevariables = colnames(lsay_data),
  rdata = lsay_data)

lta_inv_fit <- mplusModeler(lta_inv,
                 dataout=here("lta","lta_model","lta.dat"),
                 modelout=here("lta","lta_model","4-class-invariant.inp"),
                 check=TRUE, run = TRUE, hashfilename = FALSE)
```

--------------------------------------------------------------------------------------

### Estimate Non-Invariant Estimated LTA Model

```{r, eval = FALSE}

lta_non_inv <- mplusObject(
  
  TITLE = 
    "Non-Invariant LTA", 
  
  VARIABLE = 
     "usev = ab39m ab39t ab39u ab39w ab39x  ! 7th grade indicators
             ga33a ga33h ga33i ga33k ga33l; ! 10th grade indicators
      
      categorical = ab39m-ab39x ga33a-ga33l;

      classes = c1(4) c2(4);",
    
  ANALYSIS = 
     "estimator = mlr;
      type = mixture;
      starts = 500 100;
      processors=10;",

  MODEL = 
     "%overall%
      c2 on c1; !!! estimate all multinomial logistic regressions !!!
      
      !!! The above syntax can also be written as: !!!
               ! c2#1 on c1#1 c1#2 c1#3; !  
               ! c2#2 on c1#1 c1#2 c1#3; !
               ! c2#3 on c1#1 c1#2 c1#3; !

      MODEL c1: !!! the following syntax will allow item thresholds to be estimated for each class (e.g. noninvariance) !!!
      
      %c1#1%
      [AB39M$1-AB39X$1]; 
      %c1#2%
      [AB39M$1-AB39X$1];
      %c1#3%
      [AB39M$1-AB39X$1];
      %c1#4%
      [AB39M$1-AB39X$1];

      MODEL c2:
      %c2#1%
      [GA33A$1-GA33L$1];
      %c2#2%
      [GA33A$1-GA33L$1];
      %c2#3%
      [GA33A$1-GA33L$1];
      %c2#4%
      [GA33A$1-GA33L$1];",

  OUTPUT = "tech1 tech15 svalues;",
  
  usevariables = colnames(lsay_data),
  rdata = lsay_data)

lta_non_inv_fit <- mplusModeler(lta_non_inv,
                     dataout=here("lta","lta_model","lta.dat"),
                     modelout=here("lta","lta_model","4-class-non-invariant.inp"),
                     check=TRUE, run = TRUE, hashfilename = FALSE)
```

------------------------------------------------------------------------


### Nested Model Testing: Sattorra-Bentler adjusted Log Likelihood Ratio Difference Testing

-   non-invariant (comparison): This model has **more** parameters.

-   invariant (nested): This model has **less** parameters.

```{r}

# *0 = null or nested model & *1 = comparison  or parent model

lta_models <- readModels(here("lta","lta_model"), quiet = TRUE)

# Log Likelihood Values
L0 <- lta_models[["X4.class.invariant.out"]][["summaries"]][["LL"]]
L1 <- lta_models[["X4.class.non.invariant.out"]][["summaries"]][["LL"]] 

# LRT equation
lr <- -2*(L0-L1) 

# Parameters
p0 <- lta_models[["X4.class.invariant.out"]][["summaries"]][["Parameters"]] 
p1 <- lta_models[["X4.class.non.invariant.out"]][["summaries"]][["Parameters"]]

# Scaling Correction Factors
c0 <- lta_models[["X4.class.invariant.out"]][["summaries"]][["LLCorrectionFactor"]]
c1 <- lta_models[["X4.class.non.invariant.out"]][["summaries"]][["LLCorrectionFactor"]]

# Difference Test Scaling correction
cd <- ((p0*c0)-(p1*c1))/(p0-p1)

# Chi-square difference test(TRd)
TRd <- (lr)/(cd)

# Degrees of freedom
df <- abs(p0 - p1)


# Significance test
(p_diff <- pchisq(TRd, df, lower.tail=FALSE))

```

**RESULT**: The Log Likelihood $\chi^2$ difference test comparing the invariant and non-invariant LTA models was, $\chi^2 (20) = 21.542, p = .624$.

[Reference](https://stats.idre.ucla.edu/mplus/faq/how-can-i-compute-a-chi-square-test-for-nested-models-with-the-mlr-or-mlm-estimators/)

#### Alternative: Nested Model Testing Using `compareModels` in MplusAutomation

```{r}
compareModels(lta_models$X4.class.invariant.out, lta_models$X4.class.non.invariant.out, show="all", diffTest=TRUE)
```

--------------------------------------------------------------------------------------

Read invariance model and extract parameters (intercepts and multinomial regression coefficients)

```{r, eval=TRUE}

lta_inv1 <- readModels(here("lta","lta_model","4-Class-Invariant.out" ), quiet = TRUE)

par <- as_tibble(lta_inv1[["parameters"]][["unstandardized"]]) %>% 
  select(1:3) %>% 
  filter(grepl('ON|Means', paramHeader)) %>% 
  mutate(est = as.numeric(est))
```

Manual method to calculate transition probabilities:

Although possible to extract transition probabilities directly from the output the following code illustrates how the parameters are used to calculate each transition. This is useful for conducting advanced LTA model specifications such as making specific constraints within or between transition matrices, or testing the equivalence of specific transition probabilities.

```{r, eval=TRUE}
# Name each parameter individually to make the subsequent calculations more readable
a1 <- unlist(par[13,3]); a2 <- unlist(par[14,3]); a3 <- unlist(par[15,3]); b11 <- unlist(par[1,3]);
b21 <- unlist(par[4,3]); b31 <- unlist(par[7,3]); b12 <- unlist(par[2,3]); b22 <- unlist(par[5,3]);
b32 <- unlist(par[8,3]); b13 <- unlist(par[3,3]); b23 <- unlist(par[6,3]); b33 <- unlist(par[9,3])

# Calculate transition probabilities from the logit parameters
t11 <- exp(a1+b11)/(exp(a1+b11)+exp(a2+b21)+exp(a3+b31)+exp(0))
t12 <- exp(a2+b21)/(exp(a1+b11)+exp(a2+b21)+exp(a3+b31)+exp(0))
t13 <- exp(a3+b31)/(exp(a1+b11)+exp(a2+b21)+exp(a3+b31)+exp(0))
t14 <- 1 - (t11 + t12 + t13)

t21 <- exp(a1+b12)/(exp(a1+b12)+exp(a2+b22)+exp(a3+b32)+exp(0))
t22 <- exp(a2+b22)/(exp(a1+b12)+exp(a2+b22)+exp(a3+b32)+exp(0))
t23 <- exp(a3+b32)/(exp(a1+b12)+exp(a2+b22)+exp(a3+b32)+exp(0))
t24 <- 1 - (t21 + t22 + t23)

t31 <- exp(a1+b13)/(exp(a1+b13)+exp(a2+b23)+exp(a3+b33)+exp(0))
t32 <- exp(a2+b23)/(exp(a1+b13)+exp(a2+b23)+exp(a3+b33)+exp(0))
t33 <- exp(a3+b33)/(exp(a1+b13)+exp(a2+b23)+exp(a3+b33)+exp(0))
t34 <- 1 - (t31 + t32 + t33)

t41 <- exp(a1)/(exp(a1)+exp(a2)+exp(a3)+exp(0))
t42 <- exp(a2)/(exp(a1)+exp(a2)+exp(a3)+exp(0))
t43 <- exp(a3)/(exp(a1)+exp(a2)+exp(a3)+exp(0))
t44 <- 1 - (t41 + t42 + t43)

```

### Create Transition Table

```{r, eval=TRUE}

t_matrix <- tibble(
  "Time1" = c("C1=Anti-Science","C1=Amb. w/ Elevated","C1=Amb. w/ Minimal","C1=Pro-Science"),
  "C2=Anti-Science" = c(t11,t21,t31,t41),
  "C2=Amb. w/ Elevated" = c(t12,t22,t32,t42),
  "C2=Amb. w/ Minimal" = c(t13,t23,t33,t43),
  "C2=Pro-Science" = c(t14,t24,t34,t44))

t_matrix %>% 
  gt(rowname_col = "Time1") %>%
  tab_stubhead(label = "7th grade") %>% 
  tab_header(
    title = md("**Student transitions from 7th grade (rows) to 10th grade (columns)**"),
    subtitle = md("&nbsp;")) %>% 
  fmt_number(2:5,decimals = 2) %>% 
  tab_spanner(label = "10th grade",columns = 2:5) %>% 
  tab_footnote(
    footnote = md(
    "*Note.* Transition matrix values are the identical to Table 5, however Table 5 
    has the values rearranged by class for interpretation purposes. Classes may be arranged
    directly through Mplus syntax using start values."), 
    locations = cells_title())

```


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