# Dynamic structural equation models

#### James T. Thorson

    library(tinyVAST)
    set.seed(101)
    options("tinyVAST.verbose" = FALSE)

`tinyVAST` includes features to fit a dynamic structural equation model. We here show this using a bivariate vector autoregressive model for wolf and moose abundance on Isle Royale.

    data(isle_royale, package="dsem")

    # Convert to long-form
    data = expand.grid( "time"=isle_royale[,1], "var"=colnames(isle_royale[,2:3]) )
    data$logn = unlist(log(isle_royale[2:3]))

    # Define cross-lagged DSEM
    dsem = "
      # Link, lag, param_name
      wolves -> wolves, 1, arW
      moose -> wolves, 1, MtoW
      wolves -> moose, 1, WtoM
      moose -> moose, 1, arM
      #wolves -> moose, 0, corr
      wolves <-> moose, 0, corr
    "

    # fit model
    mytiny = tinyVAST( spacetime_term = dsem,
                     data = data,
                     times = isle_royale[,1],
                     variables = colnames(isle_royale[,2:3]),
                     formula = logn ~ 0 + var )
    mytiny
    #> Call:
    #> tinyVAST(formula = logn ~ 0 + var, data = data, spacetime_term = dsem,
    #>     times = isle_royale[, 1], variables = colnames(isle_royale[,
    #>         2:3]))
    #>
    #> Run time:
    #> Time difference of 0.3489952 secs
    #>
    #> Family:
    #> $obs
    #>
    #> Family: gaussian
    #> Link function: identity
    #>
    #>
    #>
    #>
    #> sdreport(.) result
    #>               Estimate   Std. Error
    #> alpha_j     3.32526197 2.483494e-01
    #> alpha_j     6.44165343 2.116035e-01
    #> beta_z      0.89304266 8.420632e-02
    #> beta_z      0.01420908 1.279150e-01
    #> beta_z     -0.13239996 3.454997e-02
    #> beta_z      0.86147627 7.107386e-02
    #> beta_z     -0.01285890 5.063467e-02
    #> beta_z      0.37727131 3.504314e-02
    #> beta_z      0.17062266 1.584742e-02
    #> log_sigma -12.62352160 2.012258e+04
    #> Maximum gradient component: 3.60018e-05
    #>
    #> Proportion conditional deviance explained:
    #> [1] 1
    #>
    #> spacetime_term:
    #>   heads     to   from parameter start lag    Estimate  Std_Error    z_value
    #> 1     1 wolves wolves         1  <NA>   1  0.89304266 0.08420632 10.6054118
    #> 2     1 wolves  moose         2  <NA>   1  0.01420908 0.12791496  0.1110823
    #> 3     1  moose wolves         3  <NA>   1 -0.13239996 0.03454997 -3.8321293
    #> 4     1  moose  moose         4  <NA>   1  0.86147627 0.07107386 12.1208601
    #> 5     2  moose wolves         5  <NA>   0 -0.01285890 0.05063467 -0.2539545
    #> 6     2 wolves wolves         6  <NA>   0  0.37727131 0.03504314 10.7659099
    #> 7     2  moose  moose         7  <NA>   0  0.17062266 0.01584742 10.7665881
    #>        p_value
    #> 1 2.812223e-26
    #> 2 9.115511e-01
    #> 3 1.270389e-04
    #> 4 8.189516e-34
    #> 5 7.995307e-01
    #> 6 4.986648e-27
    #> 7 4.950067e-27
    #>
    #> Fixed terms:
    #>           Estimate Std_Error  z_value       p_value
    #> varwolves 3.325262 0.2483494 13.38945  6.969636e-41
    #> varmoose  6.441653 0.2116035 30.44210 1.524034e-203

    # Deviance explained relative to both intercepts
    # Note that a process-error-only estimate with have devexpl -> 1
    deviance_explained( mytiny,
                        null_formula = logn ~ 0 + var )
    #> [1] 1

    # See summary
    knitr::kable( summary(mytiny,"spacetime_term"), digits=3 )

heads | to | from | parameter | start | lag | Estimate | Std_Error | z_value | p_value
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
1 | wolves | wolves | 1 | NA | 1 | 0.893 | 0.084 | 10.605 | 0.000
1 | wolves | moose | 2 | NA | 1 | 0.014 | 0.128 | 0.111 | 0.912
1 | moose | wolves | 3 | NA | 1 | -0.132 | 0.035 | -3.832 | 0.000
1 | moose | moose | 4 | NA | 1 | 0.861 | 0.071 | 12.121 | 0.000
2 | moose | wolves | 5 | NA | 0 | -0.013 | 0.051 | -0.254 | 0.800
2 | wolves | wolves | 6 | NA | 0 | 0.377 | 0.035 | 10.766 | 0.000
2 | moose | moose | 7 | NA | 0 | 0.171 | 0.016 | 10.767 | 0.000

And we can specifically inspect the estimated interaction matrix:

| wolves | moose
--- | --- | ---
wolves | 0.893 | -0.132
moose | 0.014 | 0.861

We can then compare this with package `dsem`

    library(dsem)

    # Keep in wide-form
    dsem_data = ts( log(isle_royale[,2:3]), start=1959)
    family = c("normal","normal")

    # fit without delta0
    # SEs aren't available because measurement errors goes to zero
    mydsem = dsem::dsem( sem = dsem,
                 tsdata = dsem_data,
                 control = dsem_control(getsd=FALSE),
                 family = family )
    #>   Coefficient_name Number_of_coefficients   Type
    #> 1           beta_z                      7  Fixed
    #> 2        lnsigma_j                      2  Fixed
    #> 3             mu_j                      2 Random
    #> 4             x_tj                    122 Random
    mydsem
    #> $par
    #>        beta_z        beta_z        beta_z        beta_z        beta_z
    #>   0.895834720   0.007358847  -0.124879928   0.874884847  -0.014394603
    #>        beta_z        beta_z     lnsigma_j     lnsigma_j
    #>   0.378522244   0.172997994 -17.248978101 -12.636038113
    #>
    #> $objective
    #> [1] 7.739638
    #> attr(,"logarithm")
    #> [1] TRUE
    #>
    #> $convergence
    #> [1] 0
    #>
    #> $iterations
    #> [1] 74
    #>
    #> $evaluations
    #> function gradient
    #>       96       75
    #>
    #> $message
    #> [1] "relative convergence (4)"

    # See summary
    knitr::kable( summary(mydsem), digits=3 )

path | lag | name | start | parameter | first | second | direction | Estimate
--- | --- | --- | --- | --- | --- | --- | --- | ---
wolves -> wolves | 1 | arW | NA | 1 | wolves | wolves | 1 | 0.896
moose -> wolves | 1 | MtoW | NA | 2 | moose | wolves | 1 | 0.007
wolves -> moose | 1 | WtoM | NA | 3 | wolves | moose | 1 | -0.125
moose -> moose | 1 | arM | NA | 4 | moose | moose | 1 | 0.875
wolves <-> moose | 0 | corr | NA | 5 | wolves | moose | 2 | -0.014
wolves <-> wolves | 0 | V[wolves] | NA | 6 | wolves | wolves | 2 | 0.379
moose <-> moose | 0 | V[moose] | NA | 7 | moose | moose | 2 | 0.173

where we again inspect the estimated interaction matrix:

| wolves | moose
--- | --- | ---
wolves | 0.896 | -0.125
moose | 0.007 | 0.875