developmentalScaling.qmd

---
title: "Developmental Scaling"
---

# Preamble

## Install Libraries

```{r}
#install.packages("remotes")
#remotes::install_github("DevPsyLab/petersenlab")
```

## Load Libraries

```{r}
library("lavaan")
library("semTools")
library("semPlot")
library("lavaanPlot")
library("lavaangui")
library("sirt")
library("kableExtra")
library("tidyverse")
```

## Import data

```{r}
mydata_wide <- read.csv("./data/Bliese-Ployhart-2002-indicators-4.csv")
```

## Data Processing

```{r}
mydata_wide$sad1 <- mydata_wide$JOBSAT11
mydata_wide$sad2 <- mydata_wide$JOBSAT21
mydata_wide$sad3 <- mydata_wide$JOBSAT31

mydata_wide$down1 <- mydata_wide$JOBSAT12
mydata_wide$down2 <- mydata_wide$JOBSAT22
mydata_wide$down3 <- mydata_wide$JOBSAT32

mydata_wide$depressed1 <- mydata_wide$JOBSAT13
mydata_wide$depressed2 <- mydata_wide$JOBSAT23
mydata_wide$depressed3 <- mydata_wide$JOBSAT33
```

# Approaches

1. [Partial strong longitudinal factorial invariance](#sec-partialStrongInvariance) (paired with second-order growth functions—i.e., a second-order growth curve model)
1. [Alignment optimization](#sec-alignment) (paired with second-order growth functions)
1. [Moderated nonlinear factor analysis](#sec-mnlfa) (MNLFA) (paired with second-order growth functions)
1. [Item response theory (IRT) with anchor items](#sec-irt) (potentially with regularization/Bayesian penalty methods on the priors)
    - [separate calibration](#sec-irtSeparateCalibration)
    - [concurrent calibration](#sec-irtConcurrentCalibration)

# Partial Strong Longitudinal Factorial Invariance {#sec-partialStrongInvariance}

The example is adapted from the section on [longitudinal measurement invariance](#sec-longitudinalMI).

Two primary options for identifying a latent factor include:

1. standardizing the latent factor(s) at all timepoints (fixing their intercepts to zero and their variances to one)
1. standardizing the latent factor(s) at T1 (fixing their intercepts to zero and their variances to one) and constraining the factor loadings and intercepts of one item to be equal across time

(Option #2 may be preferable in alignment optimization.)

## Standardizing the Latent Factors at All Timepoints

## Standardizing the Latent Factors at T1 while Constraining Loadings and Intercepts of One Item Across Time

### Configural Invariance With Correlated Residuals Within-Indicator Across Time

1. Standardize the latent factor(s) at T1 (i.e., fix the mean to zero and the variance to one)
1. For each latent construct, constrain the first indicator's factor loading to be the same across time
1. For each latent construct, constrain the first indicator's intercept to be the same across time
1. Allow within-indicator residuals to covary across time

```{r}
configuralInvarianceStandardizeT1Latent_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load1*sad1 + load12*down1 + load13*depressed1
  latent_t2 =~ NA*load1*sad2 + load22*down2 + load23*depressed2
  latent_t3 =~ NA*load1*sad3 + load32*down3 + load33*depressed3
  
  # Factor Identification: Standardize Factors at T1
  
  ## Fix Factor Means at T1 to Zero
  latent_t1 ~ 0*1
  
  ## Fix Factor Variances at T1 to One
  latent_t1 ~~ 1*latent_t1
  
  # Freely Estimate Factor Means at T2 and T3 (relative to T1)
  latent_t2 ~ 1
  latent_t3 ~ 1
  
  # Freely Estimate Factor Variances at T2 and T3 (relative to T1)
  latent_t2 ~~ latent_t2
  latent_t3 ~~ latent_t3
  
  # Fix Intercepts of Indicator 1 Across Time
  sad1 ~ inta*1
  sad2 ~ inta*1
  sad3 ~ inta*1
  
  # Free Intercepts of Remaining Manifest Variables
  down1 ~ intb1*1
  down2 ~ intb2*1
  down3 ~ intb3*1
  
  depressed1 ~ intc1*1
  depressed2 ~ intc2*1
  depressed3 ~ intc3*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
'

configuralInvarianceStandardizeT1Latent_fit <- cfa(
  configuralInvarianceStandardizeT1Latent_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)

summary(
  configuralInvarianceStandardizeT1Latent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)
```

### Metric ("Weak Factorial") Invariance

1. Standardize the latent factor(s) at T1 (i.e., fix the mean to zero and the variance to one)
1. For each latent construct, constrain the first indicator's intercept to be the same across time
1. Allow within-indicator residuals to covary across time
1. **For each indicator, constrain its factor loading to be the same across time**

```{r}
metricInvarianceStandardizeT1Latent_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load1*sad1 + load2*down1 + load3*depressed1
  latent_t2 =~ NA*load1*sad2 + load2*down2 + load3*depressed2
  latent_t3 =~ NA*load1*sad3 + load2*down3 + load3*depressed3
  
  # Factor Identification: Standardize Factors at T1
  
  ## Fix Factor Means at T1 to Zero
  latent_t1 ~ 0*1
  
  ## Fix Factor Variances at T1 to One
  latent_t1 ~~ 1*latent_t1
  
  # Freely Estimate Factor Means at T2 and T3 (relative to T1)
  latent_t2 ~ 1
  latent_t3 ~ 1
  
  # Freely Estimate Factor Variances at T2 and T3 (relative to T1)
  latent_t2 ~~ latent_t2
  latent_t3 ~~ latent_t3
  
  # Fix Intercepts of Indicator 1 Across Time
  sad1 ~ inta*1
  sad2 ~ inta*1
  sad3 ~ inta*1
  
  # Free Intercepts of Remaining Manifest Variables
  down1 ~ intb1*1
  down2 ~ intb2*1
  down3 ~ intb3*1
  
  depressed1 ~ intc1*1
  depressed2 ~ intc2*1
  depressed3 ~ intc3*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
'

metricInvarianceStandardizeT1Latent_fit <- cfa(
  metricInvarianceStandardizeT1Latent_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)

summary(
  metricInvarianceStandardizeT1Latent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)

anova(
  configuralInvarianceStandardizeT1Latent_fit,
  metricInvarianceStandardizeT1Latent_fit
)
```

### Scalar ("Strong Factorial") Invariance

1. Standardize the latent factor(s) at T1 (i.e., fix the mean to zero and the variance to one)
1. Allow within-indicator residuals to covary across time
1. For each indicator, constrain its factor loading to be the same across time
1. **For each indicator, constrain its intercept to be the same across time**

```{r}
scalarInvarianceStandardizeT1Latent_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load1*sad1 + load2*down1 + load3*depressed1
  latent_t2 =~ NA*load1*sad2 + load2*down2 + load3*depressed2
  latent_t3 =~ NA*load1*sad3 + load2*down3 + load3*depressed3
  
  # Factor Identification: Standardize Factors at T1
  
  ## Fix Factor Means at T1 to Zero
  latent_t1 ~ 0*1
  
  ## Fix Factor Variances at T1 to One
  latent_t1 ~~ 1*latent_t1
  
  # Freely Estimate Factor Means at T2 and T3 (relative to T1)
  latent_t2 ~ 1
  latent_t3 ~ 1
  
  # Freely Estimate Factor Variances at T2 and T3 (relative to T1)
  latent_t2 ~~ latent_t2
  latent_t3 ~~ latent_t3
  
  # Fix Intercepts Across Time
  sad1 ~ inta*1
  sad2 ~ inta*1
  sad3 ~ inta*1
  
  down1 ~ intb*1
  down2 ~ intb*1
  down3 ~ intb*1
  
  depressed1 ~ intc*1
  depressed2 ~ intc*1
  depressed3 ~ intc*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
'

scalarInvarianceStandardizeT1Latent_fit <- cfa(
  scalarInvarianceStandardizeT1Latent_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)

summary(
  scalarInvarianceStandardizeT1Latent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)

anova(
  scalarInvarianceStandardizeT1Latent_fit,
  metricInvarianceStandardizeT1Latent_fit
)
```

### Partial Strong Factorial Invariance

1. Standardize the latent factor(s) at T1 (i.e., fix the mean to zero and the variance to one)
1. Allow within-indicator residuals to covary across time
1. For some indicators (i.e., the indicators with invariant factor loadings, called "anchor items"), constrain its factor loading to be the same across time
1. For some indicators (i.e., the indicators with invariant intercepts, called "anchor items"), constrain its intercept to be the same across time

```{r}
partialStrongInvarianceStandardizeT1Latent_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load1*sad1 + load2*down1 + load3*depressed1
  latent_t2 =~ NA*load1*sad2 + load2*down2 + load3*depressed2
  latent_t3 =~ NA*load1*sad3 + load2*down3 + load33*depressed3
  
  # Factor Identification: Standardize Factors at T1
  
  ## Fix Factor Means at T1 to Zero
  latent_t1 ~ 0*1
  
  ## Fix Factor Variances at T1 to One
  latent_t1 ~~ 1*latent_t1
  
  # Freely Estimate Factor Means at T2 and T3 (relative to T1)
  latent_t2 ~ 1
  latent_t3 ~ 1
  
  # Freely Estimate Factor Variances at T2 and T3 (relative to T1)
  latent_t2 ~~ latent_t2
  latent_t3 ~~ latent_t3
  
  # Fix Some Intercepts Across Time
  sad1 ~ inta*1
  sad2 ~ inta*1
  sad3 ~ inta*1
  
  down1 ~ intb*1
  down2 ~ intb2*1
  down3 ~ intb*1
  
  depressed1 ~ intc*1
  depressed2 ~ intc2*1
  depressed3 ~ intc*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
'

partialStrongInvarianceStandardizeT1Latent_fit <- cfa(
  partialStrongInvarianceStandardizeT1Latent_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)

summary(
  partialStrongInvarianceStandardizeT1Latent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)

anova(
  configuralInvarianceStandardizeT1Latent_fit,
  partialStrongInvarianceStandardizeT1Latent_fit
)
```

### Partial Strong Factorial Invariance With Second-Order Growth Curve {#sec-secondOrderLGCM}

#### Model Syntax

```{r}
partialStrongInvarianceGrowthCurve_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load1*sad1 + load2*down1 + load3*depressed1
  latent_t2 =~ NA*load1*sad2 + load2*down2 + load3*depressed2
  latent_t3 =~ NA*load1*sad3 + load2*down3 + load33*depressed3
  
  # Factor Identification: Standardize Factors at T1
  
  ## Fix Factor Means at T1 to Zero
  latent_t1 ~ 0*1
  
  ## Fix Factor Variances at T1 to One
  latent_t1 ~~ 1*latent_t1
  
  # Fix Some Intercepts Across Time
  sad1 ~ inta*1
  sad2 ~ inta*1
  sad3 ~ inta*1
  
  down1 ~ intb*1
  down2 ~ intb2*1
  down3 ~ intb*1
  
  depressed1 ~ intc*1
  depressed2 ~ intc2*1
  depressed3 ~ intc*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
  
  # Linear Growth Model
  i =~ 1*latent_t1 + 1*latent_t2 + 1*latent_t3
  s =~ 0*latent_t1 + 1*latent_t2 + 2*latent_t3
  
  # Variance-covariances of intercepts and slopes
  i ~~ i
  s ~~ s
  i ~~ s
  
  # Means of level and slope
  i ~ 1
  s ~ 1
'
```

#### Fit Model

```{r}
partialStrongInvarianceGrowthCurve_fit <- cfa(
  partialStrongInvarianceGrowthCurve_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)
```

#### Model Summary

```{r}
summary(
  partialStrongInvarianceGrowthCurve_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)
```

Parameter estimates:

```{r}
lavaan::parameterEstimates(partialStrongInvarianceGrowthCurve_fit) %>%
  subset(
    lhs %in% c("i", "s") & substr(rhs, 1, 6) != "latent",
    -label
  )
```

# Alignment Optimization {#sec-alignment}

The example is adapted from the section on [longitudinal measurement invariance](#sec-longitudinalMI).
Two primary options for identifying a latent factor include:

1. standardizing the latent factor(s) at all timepoints (fixing their intercepts to zero and their variances to one)
1. standardizing the latent factor(s) at T1 (fixing their intercepts to zero and their variances to one) and constraining the factor loadings and intercepts of one item to be equal across time

(Option #1 may be preferable in alignment optimization.)

## Standardizing the Latent Factors at All Timepoints

### Configural Invariance With Correlated Residuals Within-Indicator Across Time

1. Standardize the latent factor(s) at all timepoints (i.e., fix the mean to zero and the variance to one)
1. Allow within-indicator residuals to covary across time

#### Model Syntax

```{r}
configuralInvarianceStandardizeAllLatent_syntax <- '
  # Factor Loadings
  latent_t1 =~ NA*load11*sad1 + load12*down1 + load13*depressed1
  latent_t2 =~ NA*load21*sad2 + load22*down2 + load23*depressed2
  latent_t3 =~ NA*load31*sad3 + load32*down3 + load33*depressed3
  
  # Factor Identification: Standardize Factors at All Timepoints (using std.lv = TRUE)
  
  # Free Intercepts of Manifest Variables
  sad1 ~ inta1*1
  sad2 ~ inta2*1
  sad3 ~ inta3*1
  
  down1 ~ intb1*1
  down2 ~ intb2*1
  down3 ~ intb3*1
  
  depressed1 ~ intc1*1
  depressed2 ~ intc2*1
  depressed3 ~ intc3*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
'
```

#### Fit Model

```{r}
configuralInvarianceStandardizeAllLatent_fit <- cfa(
  configuralInvarianceStandardizeAllLatent_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  std.lv = TRUE,
  fixed.x = FALSE)
```

#### Model Summary

```{r}
summary(
  configuralInvarianceStandardizeAllLatent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)
```

### Longitudinal Alignment Optimization

Adapted from Lai (2013; [code](https://github.com/marklhc/awc-growth-supp/blob/main/ex_MIDUS_growth.Rmd)):

```{r}
# Extract loadings and intercepts for alignment
lam_mat <- lavInspect(configuralInvarianceStandardizeAllLatent_fit, what = "est")$lambda
nu_vec <- lavInspect(configuralInvarianceStandardizeAllLatent_fit, what = "est")$nu

# Put them into T x p matrices
num_items <- 3
num_waves <- 3
lam_config <- crossprod(lam_mat, rep(1, num_waves) %x% diag(num_items))
nu_config <- matrix(nu_vec, nrow = num_waves, ncol = num_items, byrow = TRUE)

# Add indicator names
colnames(lam_config) <- colnames(nu_config) <- c("sad", "down", "depressed")

# Alignment optimization
aligned_pars <- sirt::invariance.alignment(
  lambda = lam_config,
  nu = nu_config,
  fixed = TRUE
)
```

The aligned loadings and intercepts are below:

```{r}
alignedParameters <- rbind(
  "Loadings" = rep(NA, num_items),
  `rownames<-`(aligned_pars$lambda.aligned,
               paste0("Time", 1:num_waves)),
  "Intercepts" = rep(NA, num_items),
  `rownames<-`(aligned_pars$nu.aligned,
               paste0("Time", 1:num_waves))
)

#opts <- options(knitr.kable.NA = "")
#
#knitr::kable(
#  alignedParameters,
#  format = "simple",
#  digits = 3L
#)

print(alignedParameters, na.print = "")

aligned_pars$lambda.aligned
aligned_pars$nu.aligned
```

#### Effect Size of Noninvariance ($d_{MACS}$)

Lai (2023) suggests using a "50/30/20" rule of thumb for using $d_{MACS}$ to evaluate the appropriateness of the alignment optimization method:
Alignment optimization within CFA "is trustworthy when (a) no more than 50% of items have one or more $d_{MACS}$ > .20 and (b) no more than 30% of the pairwise $d_{MACS}$ > .20.
Lai (2023) suggests reporting the range of $d_{MACS}$ values, the proportion of $d_{MACS}$ > .20, and the proportion of items with at least one $d_{MACS}$ > .20.

```{r}
# Function for dMACS
dmacs <- function(loadings, intercepts, pooled_item_sd,
                  latent_mean = 0, latent_var = 1) {
  dloading <- diff(loadings)
  dintercept <- diff(intercepts)
  integral <- dintercept^2 + 2 * dintercept * dloading * latent_mean +
    dloading^2 * (latent_var + latent_mean^2)
  sqrt(integral) / pooled_item_sd
}

# Use item SDs at first time point
item_sds_wave1 <-
  apply(
    mydata_wide[c("sad1", "down1", "depressed1")],
    2, sd
  )

dmacs_pairwise <- function(loading_mat, intercept_mat, pooled_item_sd,
                           latent_mean = 0, latent_var = 1) {
  ngroups <- nrow(loading_mat)
  pairs <- combn(ngroups, 2)
  out <- matrix(NA, nrow = ncol(pairs), ncol = ncol(loading_mat))
  for (i in seq_len(ncol(pairs))) {
    out[i, ] <- dmacs(loading_mat[pairs[, i], ],
      intercepts = intercept_mat[pairs[, i], ],
      pooled_item_sd,
      latent_mean,
      latent_var
    )
  }
  rownames(out) <- apply(pairs, 2, paste, collapse = " vs ")
  colnames(out) <- colnames(loading_mat)
  out
}

# All pairwise dMACS
dmacs_pairwise(aligned_pars$lambda.aligned,
  intercept_mat = aligned_pars$nu.aligned,
  pooled_item_sd = item_sds_wave1,
  latent_mean = 0,
  latent_var = 1
)
```

### An Equivalent Configural Model with Aligned Loadings

For the reference indicator (for which the loadings and intercepts are fixed), it is preferable to select an indicator with a strong factor loading (Lai, 2023).

```{r}
equivalentConfiguralModelAligned_syntax <- '
  # Factor Loadings (Loadings of first indicator fixed to alignment solution)
  latent_t1 =~ 0.9904750*load11*sad1 + load12*down1 + load13*depressed1
  latent_t2 =~ 0.9880715*load21*sad2 + load22*down2 + load23*depressed2
  latent_t3 =~ 0.9863225*load31*sad3 + load32*down3 + load33*depressed3
  
  # Intercepts of Manifest Variables (Intercepts of first indicator fixed to alignment solution)
  sad1 ~ 3.319192*inta1*1
  sad2 ~ 3.332276*inta2*1
  sad3 ~ 3.345631*inta3*1
  
  down1 ~ intb1*1
  down2 ~ intb2*1
  down3 ~ intb3*1
  
  depressed1 ~ intc1*1
  depressed2 ~ intc2*1
  depressed3 ~ intc3*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
  
  # Free latent means
  latent_t1 + latent_t2 + latent_t3 ~ NA*1
'

equivalentConfiguralModelAligned_fit <- cfa(
  equivalentConfiguralModelAligned_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)
```

Compare model fit (they're equivalent):

```{r}
anova(
  configuralInvarianceStandardizeAllLatent_fit,
  equivalentConfiguralModelAligned_fit
)
```

### Alignment-Within-CFA (AwC) Approach for Growth Modeling

#### Model Syntax

For the reference indicator (for which the loadings and intercepts are fixed), it is preferable to select an indicator with a strong factor loading (Lai, 2023).

```{r}
awcGrowthModel_syntax <- '
  # Factor Loadings (Loadings of first indicator fixed to alignment solution)
  latent_t1 =~ 0.9904750*load11*sad1 + load12*down1 + load13*depressed1
  latent_t2 =~ 0.9880715*load21*sad2 + load22*down2 + load23*depressed2
  latent_t3 =~ 0.9863225*load31*sad3 + load32*down3 + load33*depressed3
  
  # Intercepts of Manifest Variables (Intercepts of first indicator fixed to alignment solution)
  sad1 ~ 3.319192*inta1*1
  sad2 ~ 3.332276*inta2*1
  sad3 ~ 3.345631*inta3*1
  
  down1 ~ intb1*1
  down2 ~ intb2*1
  down3 ~ intb3*1
  
  depressed1 ~ intc1*1
  depressed2 ~ intc2*1
  depressed3 ~ intc3*1
  
  # Residual Covariances Within Indicator Across Time
  sad1 ~~ sad2
  sad2 ~~ sad3
  sad3 ~~ sad3
  
  down1 ~~ depressed2
  down2 ~~ depressed3
  down3 ~~ depressed3
  
  depressed1 ~~ depressed2
  depressed2 ~~ depressed3
  depressed3 ~~ depressed3
  
  # Linear Growth Model
  i =~ 1*latent_t1 + 1*latent_t2 + 1*latent_t3
  s =~ 0*latent_t1 + 1*latent_t2 + 2*latent_t3
  
  # Variance-covariances of intercepts and slopes
  i ~~ i
  s ~~ s
  i ~~ s
  
  # Means of level and slope
  i ~ 1
  s ~ 1
  
  # Fixed disturbances of latent outcomes to zero
  latent_t1 ~ 0*1
  latent_t2 ~ 0*1
  latent_t3 ~ 0*1
'
```

#### Fit Model

```{r}
awcGrowthModel_fit <- cfa(
  awcGrowthModel_syntax,
  data = mydata_wide,
  missing = "ML",
  estimator = "MLR",
  meanstructure = TRUE,
  #std.lv = TRUE,
  fixed.x = FALSE)
```

#### Model Summary

```{r}
summary(
  configuralInvarianceStandardizeAllLatent_fit,
  fit.measures = TRUE,
  standardized = TRUE,
  rsquare = TRUE)
```

Parameter estimates:

```{r}
lavaan::parameterEstimates(awcGrowthModel_fit) %>%
  subset(
    lhs %in% c("i", "s") & substr(rhs, 1, 6) != "latent",
    -label
  ) %>%
  knitr::kable(format = "simple", digits = 3L)
```

## Standardizing the Latent Factors at T1 while Constraining Loadings and Intercepts of One Item Across Time

### Configural Invariance With Correlated Residuals Within-Indicator Across Time

1. Standardize the latent factor(s) at T1 (i.e., fix the mean to zero and the variance to one)
1. For each latent construct, constrain the first indicator's factor loading to be the same across time
1. For each latent construct, constrain the first indicator's intercept to be the same across time
1. **Allow within-indicator residuals to covary across time**

# Moderated Nonlinear Factor Analysis (MNLFA) {#sec-mnlfa}

# Item Response Theory (IRT) with Anchor Items {#sec-irt}

It can be potentially helpful to consider regularization/Bayesian penalty methods (e.g., using priors such as Horseshoe, Lasso, or Spike-and-slab).

For examples of IRT models, see [here](irt.qmd).

## Separate Calibration {#sec-irtSeparateCalibration}

## Concurrent Calibration {#sec-irtConcurrentCalibration}