# estimation alternatives using the LMS approach

    library(modsem)

# Accelerated EM and Adaptive Quadrature

By default (as of `v1.0.9`) the LMS approach uses an accelerated EM procedure (`"EMA"`) that uses Quasi-Newton and Fisher Scoring optimization steps when needed. If desireable, this can be switched to the standard Expectation-Maximization (EM) algorithm, by setting `algorithm = "EM"`.

By default the LMS approach also uses a fixed Gauss-Hermite quadrature, to estimate a numerical approximation of the log-likelihood. Instead of a fixed quadrature, it is possible to use a quasi-adaptive quadrature instead. Due to performance reasons, the adaptive quadrature does not fit an individual quadrature to each participant, but instead one for the entire sample (at each EM iteration), based on the whole sample densities of the likelihood function. It essentially works by removing irrelevant nodes which don’t contribute to the integral, and increasing the number of nodes which actually contribute to the integral. This usually means that more nodes are placed towards the center of the distribution, compared to a standard fixed Gauss-Hermite quadrature. Using the EMA and adaptive quadrature might yield estimates that are closer to results from `Mplus`.

If the model struggles to converge, you can try changing the EM procedure by setting `algorithm = "EMA"`, or `algorithm = "EM"`, and `adaptive.quad = TRUE` in the `modsem()` function. Additionally it is possible to tweak these parameters:

- `max.iter`: Maximum number of iterations for the EM algorithm (default is 500).
- `max.step`: Maximum number of steps used in the Maximization step of the EM algorithm (default is 1).
- `convergence.rel`: Relative convergence criterion for the EM algorithm.
- `convergence.abs`: Absolute convergence criterion for the EM algorithm.
- `nodes`: Number of nodes for numerical integration (default is 24). Increasing this number can improve the accuracy of the estimates, especially for complex models.
- `quad.range`: Integration range for quadrature. Smaller ranges means that the integral is more focused. Applies to only when using a quasi-adaptive quadrature.
- `adaptive.frequency`: How often should the quasi-adaptive quadrature be calculated? Defaults to every third EM iteration.
- `adaptive.quad.tol`: Relative error tolerance when calculating the quasi-adaptive quadrature.

Here we can see an example using the `TPB_UK` dataaset, which is more troublesome to estimate than the simulated `TPB` dataset.

    tpb_uk <- "
    # Outer Model (Based on Hagger et al., 2007)
      ATT =~ att3 + att2 + att1 + att4
      SN =~ sn4 + sn2 + sn3 + sn1
      PBC =~ pbc2 + pbc1 + pbc3 + pbc4
      INT =~ int2 + int1 + int3 + int4
      BEH =~ beh3 + beh2 + beh1 + beh4

    # Inner Model (Based on Steinmetz et al., 2011)
      INT ~ ATT + SN + PBC
      BEH ~ INT + PBC
      BEH ~ INT:PBC
    "

    fit <- modsem(tpb_uk,
                  data = TPB_UK,
                  method = "lms",
                  nodes = 32, # Number of nodes for numerical integration
                  adaptive.quad = TRUE, # Use quasi-adaptive quadrature
                  adaptive.frequency = 3, # Update the quasi-adaptive quadrature every third EM-iteration
                  adaptive.quad.tol = 1e-12, # Relative error tolerance for quasi-adaptive quad
                  algorithm ="EMA", # Use accelerated EM algorithm (Default)
                  convergence.abs = 1e-4, # Relative convergence criterion
                  convergence.rel = 1e-10, # Relative convergence criterion
                  max.iter = 500, # Maximum number of iterations
                  max.step = 1) # Maximum number of steps in the maximization step
    summary(fit)