Author: Po-Hsien Huang

# Semi-Confirmatory Factor Analysis

#### Po-Hsien Huang

In this example, we will show how to use `lslx` to conduct semi-confirmatory factor analysis. The example uses data `HolzingerSwineford1939` in the package `lavaan`. Hence, `lavaan` must be installed.

## Model Specification

In the following specification, `x1` - `x9` is assumed to be measurements of 3 latent factors: `visual`, `textual`, and `speed`.

    model_fa <- "visual  :=> x1 + x2 + x3
                 textual :=> x4 + x5 + x6
                 speed   :=> x7 + x8 + x9
                 visual  :~> x4 + x5 + x6 + x7 + x8 + x9
                 textual :~> x1 + x2 + x3 + x7 + x8 + x9
                 speed   :~> x1 + x2 + x3 + x4 + x5 + x6
                 visual  <=> fix(1)* visual
                 textual <=> fix(1)* textual
                 speed   <=> fix(1)* speed"

The operator `:=>` means that the LHS latent factors is defined by the RHS observed variables. In particular, the loadings are freely estimated. The operator `:~>` also means that the LHS latent factors is defined by the RHS observed variables, but these loadings are set as penalized coefficients. In this model, `visual` is mainly measured by `x1` - `x3`, `textual` is mainly measured by `x4` - `x6`, and `speed` is mainly measured by `x7` - `x9`. However, the inclusion of the penalized loadings indicates that each measurement may not be only influenced by one latent factor. The operator `<=>` means that the LHS and RHS variables/factors are covaried. If the LHS and RHS variable/factor are the same, `<=>` specifies the variance of that variable/factor. For scale setting, `visual <=> fix(1) * visual` makes the variance of `visual` to be zero. Details of model syntax can be found in the section of Model Syntax via `?lslx`.

## Object Initialization

`lslx` is written as an `R6` class. Every time we conduct analysis with `lslx`, an `lslx` object must be initialized. The following code initializes an `lslx` object named `lslx_fa`.

    library(lslx)
    lslx_fa <- lslx$new(model = model_fa, data = lavaan::HolzingerSwineford1939)

    An 'lslx' R6 class is initialized via 'data' argument.
      Response Variables: x1 x2 x3 x4 x5 x6 x7 x8 x9
      Latent Factors: visual textual speed

Here, `lslx` is the object generator for `lslx` object and `$new()` is the build-in method of `lslx` to generate a new `lslx` object. The initialization of `lslx` requires users to specify a model for model specification (argument `model`) and a data to be fitted (argument `sample_data`). The data set must contain all the observed variables specified in the given model. In is also possible to initialize an `lslx` object via sample moments (see `vignette("structural-equation-modeling")`).

To see the model specification, we may use the `$extract_specification()` method.

    lslx_fa$extract_specification()

                                 relation    left   right group reference matrix block  type start
    x1<-1/g                         x1<-1      x1       1     g     FALSE  alpha  y<-1  free    NA
    x2<-1/g                         x2<-1      x2       1     g     FALSE  alpha  y<-1  free    NA
    x3<-1/g                         x3<-1      x3       1     g     FALSE  alpha  y<-1  free    NA
    x4<-1/g                         x4<-1      x4       1     g     FALSE  alpha  y<-1  free    NA
    x5<-1/g                         x5<-1      x5       1     g     FALSE  alpha  y<-1  free    NA
    x6<-1/g                         x6<-1      x6       1     g     FALSE  alpha  y<-1  free    NA
    x7<-1/g                         x7<-1      x7       1     g     FALSE  alpha  y<-1  free    NA
    x8<-1/g                         x8<-1      x8       1     g     FALSE  alpha  y<-1  free    NA
    x9<-1/g                         x9<-1      x9       1     g     FALSE  alpha  y<-1  free    NA
    x1<-visual/g               x1<-visual      x1  visual     g     FALSE   beta  y<-f  free    NA
    x2<-visual/g               x2<-visual      x2  visual     g     FALSE   beta  y<-f  free    NA
    x3<-visual/g               x3<-visual      x3  visual     g     FALSE   beta  y<-f  free    NA
    x4<-visual/g               x4<-visual      x4  visual     g     FALSE   beta  y<-f   pen    NA
    x5<-visual/g               x5<-visual      x5  visual     g     FALSE   beta  y<-f   pen    NA
    x6<-visual/g               x6<-visual      x6  visual     g     FALSE   beta  y<-f   pen    NA
    x7<-visual/g               x7<-visual      x7  visual     g     FALSE   beta  y<-f   pen    NA
    x8<-visual/g               x8<-visual      x8  visual     g     FALSE   beta  y<-f   pen    NA
    x9<-visual/g               x9<-visual      x9  visual     g     FALSE   beta  y<-f   pen    NA
    x1<-textual/g             x1<-textual      x1 textual     g     FALSE   beta  y<-f   pen    NA
    x2<-textual/g             x2<-textual      x2 textual     g     FALSE   beta  y<-f   pen    NA
    x3<-textual/g             x3<-textual      x3 textual     g     FALSE   beta  y<-f   pen    NA
    x4<-textual/g             x4<-textual      x4 textual     g     FALSE   beta  y<-f  free    NA
    x5<-textual/g             x5<-textual      x5 textual     g     FALSE   beta  y<-f  free    NA
    x6<-textual/g             x6<-textual      x6 textual     g     FALSE   beta  y<-f  free    NA
    x7<-textual/g             x7<-textual      x7 textual     g     FALSE   beta  y<-f   pen    NA
    x8<-textual/g             x8<-textual      x8 textual     g     FALSE   beta  y<-f   pen    NA
    x9<-textual/g             x9<-textual      x9 textual     g     FALSE   beta  y<-f   pen    NA
    x1<-speed/g                 x1<-speed      x1   speed     g     FALSE   beta  y<-f   pen    NA
    x2<-speed/g                 x2<-speed      x2   speed     g     FALSE   beta  y<-f   pen    NA
    x3<-speed/g                 x3<-speed      x3   speed     g     FALSE   beta  y<-f   pen    NA
    x4<-speed/g                 x4<-speed      x4   speed     g     FALSE   beta  y<-f   pen    NA
    x5<-speed/g                 x5<-speed      x5   speed     g     FALSE   beta  y<-f   pen    NA
    x6<-speed/g                 x6<-speed      x6   speed     g     FALSE   beta  y<-f   pen    NA
    x7<-speed/g                 x7<-speed      x7   speed     g     FALSE   beta  y<-f  free    NA
    x8<-speed/g                 x8<-speed      x8   speed     g     FALSE   beta  y<-f  free    NA
    x9<-speed/g                 x9<-speed      x9   speed     g     FALSE   beta  y<-f  free    NA
    visual<->visual/g     visual<->visual  visual  visual     g     FALSE    phi f<->f fixed     1
    textual<->visual/g   textual<->visual textual  visual     g     FALSE    phi f<->f  free    NA
    speed<->visual/g       speed<->visual   speed  visual     g     FALSE    phi f<->f  free    NA
    textual<->textual/g textual<->textual textual textual     g     FALSE    phi f<->f fixed     1
    speed<->textual/g     speed<->textual   speed textual     g     FALSE    phi f<->f  free    NA
    speed<->speed/g         speed<->speed   speed   speed     g     FALSE    phi f<->f fixed     1
    x1<->x1/g                     x1<->x1      x1      x1     g     FALSE    phi y<->y  free    NA
    x2<->x2/g                     x2<->x2      x2      x2     g     FALSE    phi y<->y  free    NA
    x3<->x3/g                     x3<->x3      x3      x3     g     FALSE    phi y<->y  free    NA
    x4<->x4/g                     x4<->x4      x4      x4     g     FALSE    phi y<->y  free    NA
    x5<->x5/g                     x5<->x5      x5      x5     g     FALSE    phi y<->y  free    NA
    x6<->x6/g                     x6<->x6      x6      x6     g     FALSE    phi y<->y  free    NA
    x7<->x7/g                     x7<->x7      x7      x7     g     FALSE    phi y<->y  free    NA
    x8<->x8/g                     x8<->x8      x8      x8     g     FALSE    phi y<->y  free    NA
    x9<->x9/g                     x9<->x9      x9      x9     g     FALSE    phi y<->y  free    NA
                        label penalty set weight
    x1<-1/g              <NA>    none   0      0
    x2<-1/g              <NA>    none   0      0
    x3<-1/g              <NA>    none   0      0
    x4<-1/g              <NA>    none   0      0
    x5<-1/g              <NA>    none   0      0
    x6<-1/g              <NA>    none   0      0
    x7<-1/g              <NA>    none   0      0
    x8<-1/g              <NA>    none   0      0
    x9<-1/g              <NA>    none   0      0
    x1<-visual/g         <NA>    none   0      0
    x2<-visual/g         <NA>    none   0      0
    x3<-visual/g         <NA>    none   0      0
    x4<-visual/g         <NA> default   1      1
    x5<-visual/g         <NA> default   1      1
    x6<-visual/g         <NA> default   1      1
    x7<-visual/g         <NA> default   1      1
    x8<-visual/g         <NA> default   1      1
    x9<-visual/g         <NA> default   1      1
    x1<-textual/g        <NA> default   1      1
    x2<-textual/g        <NA> default   1      1
    x3<-textual/g        <NA> default   1      1
    x4<-textual/g        <NA>    none   0      0
    x5<-textual/g        <NA>    none   0      0
    x6<-textual/g        <NA>    none   0      0
    x7<-textual/g        <NA> default   1      1
    x8<-textual/g        <NA> default   1      1
    x9<-textual/g        <NA> default   1      1
    x1<-speed/g          <NA> default   1      1
    x2<-speed/g          <NA> default   1      1
    x3<-speed/g          <NA> default   1      1
    x4<-speed/g          <NA> default   1      1
    x5<-speed/g          <NA> default   1      1
    x6<-speed/g          <NA> default   1      1
    x7<-speed/g          <NA>    none   0      0
    x8<-speed/g          <NA>    none   0      0
    x9<-speed/g          <NA>    none   0      0
    visual<->visual/g    <NA>    none   0      0
    textual<->visual/g   <NA>    none   0      0
    speed<->visual/g     <NA>    none   0      0
    textual<->textual/g  <NA>    none   0      0
    speed<->textual/g    <NA>    none   0      0
    speed<->speed/g      <NA>    none   0      0
    x1<->x1/g            <NA>    none   0      0
    x2<->x2/g            <NA>    none   0      0
    x3<->x3/g            <NA>    none   0      0
    x4<->x4/g            <NA>    none   0      0
    x5<->x5/g            <NA>    none   0      0
    x6<->x6/g            <NA>    none   0      0
    x7<->x7/g            <NA>    none   0      0
    x8<->x8/g            <NA>    none   0      0
    x9<->x9/g            <NA>    none   0      0

The row names show the coefficient names. The most two relevant columns are `type` which shows the type of the coefficient and `start` which gives the starting value. In `lslx`, many `extract` related methods are defined. `extract` related methods can be used to extract quantities stored in `lslx` object. For details, please see the section of Extract-Related Methods via `?lslx`.

## Model Fitting

After an `lslx` object is initialized, method `$fit()` can be used to fit the specified model to the given data.

    lslx_fa$fit(penalty_method = "mcp",
                lambda_grid = seq(.01, .60, .01),
                delta_grid = c(1.5, 3.0, Inf))

    CONGRATS: Algorithm converges under EVERY specified penalty level.
      Specified Tolerance for Convergence: 0.001
      Specified Maximal Number of Iterations: 100

The fitting process requires users to specify the penalty method (argument `penalty_method`) and the considered penalty levels (argument `lambda_grid` and `delta_grid`). In this example, the `mcp` penalty is implemented on the lambda grid `seq(.01, .60, .01)` and delta grid `c(1.5, 3, Inf)`. Note that in this example `lambda = 0` is not considered because it may result in unidentified model. All the fitting result will be stored in the `fitting` field of `lslx_fa`.

## Model Summarizing

Unlike traditional SEM analysis, `lslx` fits the model into data under all the penalty levels considered. To summarize the fitting result, a selector to determine an optimal penalty level must be specified. Available selectors can be found in the section of Penalty Level Selection via `?lslx`. The following code summarize the fitting result under the penalty level selected by Bayesian information criterion (BIC).

    lslx_fa$summarize(selector = "bic")

    General Information
       number of observations                                301
       number of complete observations                       301
       number of missing patterns                           none
       number of groups                                        1
       number of responses                                     9
       number of factors                                       3
       number of free coefficients                            30
       number of penalized coefficients                       18

    Numerical Conditions
       selected lambda                                     0.140
       selected delta                                      1.500
       selected step                                        none
       objective value                                     0.158
       objective gradient absolute maximum                 0.001
       objective Hessian convexity                         0.593
       number of iterations                                8.000
       loss value                                          0.103
       number of non-zero coefficients                    34.000
       degrees of freedom                                 20.000
       robust degrees of freedom                          20.641
       scaling factor                                      1.032

    Fit Indices
       root mean square error of approximation (rmsea)     0.043
       comparative fit index (cfi)                         0.988
       non-normed fit index (nnfi)                         0.978
       standardized root mean of residual (srmr)           0.030

    Likelihood Ratio Test
                        statistic         df    p-value
       unadjusted          30.902     20.000      0.056
       mean-adjusted       29.943     20.000      0.071

    Root Mean Square Error of Approximation Test
                         estimate      lower      upper
       unadjusted           0.043      0.000      0.075
       mean-adjusted        0.041      0.000      0.075

    Coefficient Test (Std.Error = "sandwich")
      Factor Loading
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
           x1<-visual     free     0.709      0.095    7.477    0.000  0.523  0.895
           x2<-visual     free     0.555      0.082    6.798    0.000  0.395  0.715
           x3<-visual     free     0.744      0.070   10.575    0.000  0.606  0.882
           x4<-visual      pen     0.000        -        -        -      -      -
           x5<-visual      pen    -0.102      0.073   -1.412    0.158 -0.245  0.040
           x6<-visual      pen     0.000        -        -        -      -      -
           x7<-visual      pen    -0.254      0.118   -2.147    0.032 -0.486 -0.022
           x8<-visual      pen     0.000        -        -        -      -      -
           x9<-visual      pen     0.323      0.075    4.286    0.000  0.175  0.471
          x1<-textual      pen     0.255      0.081    3.136    0.002  0.096  0.415
          x2<-textual      pen     0.000        -        -        -      -      -
          x3<-textual      pen     0.000        -        -        -      -      -
          x4<-textual     free     0.987      0.061   16.133    0.000  0.867  1.106
          x5<-textual     free     1.143      0.062   18.583    0.000  1.023  1.264
          x6<-textual     free     0.913      0.058   15.726    0.000  0.799  1.027
          x7<-textual      pen     0.000        -        -        -      -      -
          x8<-textual      pen     0.000        -        -        -      -      -
          x9<-textual      pen     0.000        -        -        -      -      -
            x1<-speed      pen     0.000        -        -        -      -      -
            x2<-speed      pen     0.000        -        -        -      -      -
            x3<-speed      pen     0.000        -        -        -      -      -
            x4<-speed      pen     0.000        -        -        -      -      -
            x5<-speed      pen     0.000        -        -        -      -      -
            x6<-speed      pen     0.000        -        -        -      -      -
            x7<-speed     free     0.824      0.105    7.815    0.000  0.617  1.030
            x8<-speed     free     0.731      0.069   10.604    0.000  0.596  0.866
            x9<-speed     free     0.499      0.063    7.961    0.000  0.376  0.622

      Covariance
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
     textual<->visual     free     0.325      0.086    3.766    0.000  0.156  0.494
       speed<->visual     free     0.374      0.100    3.733    0.000  0.178  0.570
      speed<->textual     free     0.278      0.078    3.566    0.000  0.125  0.431

      Variance
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
      visual<->visual    fixed     1.000        -        -        -      -      -
    textual<->textual    fixed     1.000        -        -        -      -      -
        speed<->speed    fixed     1.000        -        -        -      -      -
              x1<->x1     free     0.671      0.113    5.950    0.000  0.450  0.892
              x2<->x2     free     1.073      0.106   10.165    0.000  0.866  1.280
              x3<->x3     free     0.719      0.091    7.943    0.000  0.542  0.896
              x4<->x4     free     0.377      0.050    7.507    0.000  0.279  0.476
              x5<->x5     free     0.415      0.060    6.869    0.000  0.297  0.534
              x6<->x6     free     0.362      0.046    7.903    0.000  0.273  0.452
              x7<->x7     free     0.596      0.109    5.464    0.000  0.382  0.810
              x8<->x8     free     0.487      0.084    5.771    0.000  0.322  0.653
              x9<->x9     free     0.540      0.064    8.431    0.000  0.414  0.666

      Intercept
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
                x1<-1     free     4.936      0.067   73.473    0.000  4.804  5.067
                x2<-1     free     6.088      0.068   89.855    0.000  5.955  6.221
                x3<-1     free     2.250      0.065   34.579    0.000  2.123  2.378
                x4<-1     free     3.061      0.067   45.694    0.000  2.930  3.192
                x5<-1     free     4.341      0.074   58.452    0.000  4.195  4.486
                x6<-1     free     2.186      0.063   34.667    0.000  2.062  2.309
                x7<-1     free     4.186      0.063   66.766    0.000  4.063  4.309
                x8<-1     free     5.527      0.058   94.854    0.000  5.413  5.641
                x9<-1     free     5.374      0.058   92.546    0.000  5.260  5.488

In this example, we can see that most penalized coefficients are estimated as zero under the selected penalty level except for `x9<-visual`, which shows the benefit of using the semi-confirmatory approach. The `summarize` method also shows the result of significance tests for the coefficients. In `lslx`, the default standard errors are calculated based on sandwich formula whenever raw data is available. It is generally valid even when the model is misspecified and the data is not normal. However, it may not be valid after selecting an optimal penalty level.

## Visualization

`lslx` provides four methods for visualizing the fitting results. The method `$plot_numerical_condition()` shows the numerical condition under all the penalty levels. The following code plots the values of `n_iter_out` (number of iterations in outer loop), `objective_gradient_abs_max` (maximum of absolute value of gradient of objective function), and `objective_hessian_convexity` (minimum of univariate approximate hessian). The plot can be used to evaluate the quality of numerical optimization.

    lslx_fa$plot_numerical_condition()

The method `$plot_information_criterion()` shows the values of information criteria under all the penalty levels.

    lslx_fa$plot_information_criterion()

The method `$plot_fit_index()` shows the values of fit indices under all the penalty levels.

    lslx_fa$plot_fit_index()

The method `$plot_coefficient()` shows the solution path of coefficients in the given block. The following code plots the solution paths of all coefficients in the block `y<-f`, which contains all the regression coefficients from latent factors to observed variables (i.e., factor loadings).

    lslx_fa$plot_coefficient(block = "y<-f")

## Objects Extraction

In `lslx`, many quantities related to SEM can be extracted by extract-related method. For example, the loading matrix can be obtained by

    lslx_fa$extract_coefficient_matrix(selector = "bic", block = "y<-f")

    $g
       visual textual speed
    x1  0.709   0.255 0.000
    x2  0.555   0.000 0.000
    x3  0.744   0.000 0.000
    x4  0.000   0.987 0.000
    x5 -0.102   1.143 0.000
    x6  0.000   0.913 0.000
    x7 -0.254   0.000 0.824
    x8  0.000   0.000 0.731
    x9  0.323   0.000 0.499

The model-implied covariance matrix and residual matrix can be obtained by

    lslx_fa$extract_implied_cov(selector = "bic")

    $g
           x1     x2     x3    x4    x5    x6     x7    x8    x9
    x1 1.3560 0.4393 0.5892 0.479 0.474 0.444 0.0756 0.246 0.423
    x2 0.4393 1.3806 0.4129 0.178 0.149 0.165 0.0299 0.152 0.283
    x3 0.5892 0.4129 1.2728 0.239 0.200 0.221 0.0401 0.203 0.379
    x4 0.4792 0.1779 0.2386 1.350 1.095 0.901 0.1443 0.200 0.240
    x5 0.4741 0.1492 0.2002 1.095 1.656 1.013 0.1617 0.204 0.226
    x6 0.4435 0.1646 0.2208 0.901 1.013 1.196 0.1336 0.185 0.222
    x7 0.0756 0.0299 0.0401 0.144 0.162 0.134 1.1828 0.533 0.381
    x8 0.2456 0.1517 0.2034 0.200 0.204 0.185 0.5326 1.022 0.453
    x9 0.4234 0.2828 0.3792 0.240 0.226 0.222 0.3812 0.453 1.014

    lslx_fa$extract_residual_cov(selector = "bic")

    $g
             x1       x2       x3        x4       x5        x6        x7       x8        x9
    x1 -0.00235  0.03195  0.00931 -0.025619  0.03351 -0.011278 -0.009183 -0.01820 -0.034894
    x2  0.03195 -0.00117 -0.03818 -0.031048 -0.06185 -0.082916  0.126662  0.04202  0.038746
    x3  0.00931 -0.03818 -0.00209  0.030390  0.08786 -0.023315 -0.048226 -0.00890  0.005370
    x4 -0.02562 -0.03105  0.03039 -0.000309 -0.00291  0.005236 -0.075404  0.07476 -0.003009
    x5  0.03351 -0.06185  0.08786 -0.002905 -0.00331 -0.001209  0.018725  0.02353 -0.068971
    x6 -0.01128 -0.08292 -0.02331  0.005236 -0.00121 -0.000255 -0.010488  0.02000 -0.013542
    x7 -0.00918  0.12666 -0.04823 -0.075404  0.01872 -0.010488 -0.000334 -0.00262  0.007874
    x8 -0.01820  0.04202 -0.00890  0.074757  0.02353  0.019999 -0.002621 -0.00045 -0.004238
    x9 -0.03489  0.03875  0.00537 -0.003009 -0.06897 -0.013542  0.007874 -0.00424 -0.000963

## Wrapper Function `plsem()` and `S3` Methods

After version 0.6.3, the previous analysis can be equivalently conducted by `plsem()` with `lavaan` style model syntax.

    model_fa <- "visual  =~ x1 + x2 + x3
                 textual =~ x4 + x5 + x6
                 speed   =~ x7 + x8 + x9
                 pen() * visual  =~ x4 + x5 + x6 + x7 + x8 + x9
                 pen() * textual =~ x1 + x2 + x3 + x7 + x8 + x9
                 pen() * speed   =~ x1 + x2 + x3 + x4 + x5 + x6
                 visual  ~~ 1 * visual
                 textual ~~ 1 * textual
                 speed   ~~ 1 * speed"

    lslx_fa <- plsem(model = model_fa,
                     data = lavaan::HolzingerSwineford1939,
                     penalty_method = "mcp",
                     lambda_grid = seq(.01, .60, .01),
                     delta_grid = c(1.5, 3.0, Inf))

    An 'lslx' R6 class is initialized via 'data' argument.
      Response Variables: x1 x2 x3 x4 x5 x6 x7 x8 x9
      Latent Factors: visual textual speed
    CONGRATS: Algorithm converges under EVERY specified penalty level.
      Specified Tolerance for Convergence: 0.001
      Specified Maximal Number of Iterations: 100

    summary(lslx_fa, selector = "bic")

    General Information
       number of observations                                301
       number of complete observations                       301
       number of missing patterns                           none
       number of groups                                        1
       number of responses                                     9
       number of factors                                       3
       number of free coefficients                            30
       number of penalized coefficients                       18

    Numerical Conditions
       selected lambda                                     0.140
       selected delta                                      1.500
       selected step                                        none
       objective value                                     0.158
       objective gradient absolute maximum                 0.001
       objective Hessian convexity                         0.593
       number of iterations                                8.000
       loss value                                          0.103
       number of non-zero coefficients                    34.000
       degrees of freedom                                 20.000
       robust degrees of freedom                          20.641
       scaling factor                                      1.032

    Fit Indices
       root mean square error of approximation (rmsea)     0.043
       comparative fit index (cfi)                         0.988
       non-normed fit index (nnfi)                         0.978
       standardized root mean of residual (srmr)           0.030

    Likelihood Ratio Test
                        statistic         df    p-value
       unadjusted          30.904     20.000      0.056
       mean-adjusted       29.944     20.000      0.071

    Root Mean Square Error of Approximation Test
                         estimate      lower      upper
       unadjusted           0.043      0.000      0.075
       mean-adjusted        0.041      0.000      0.075

    Coefficient Test (Std.Error = "sandwich")
      Factor Loading
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
           x1<-visual     free     0.709      0.095    7.477    0.000  0.523  0.895
           x2<-visual     free     0.555      0.082    6.797    0.000  0.395  0.715
           x3<-visual     free     0.744      0.070   10.575    0.000  0.606  0.882
           x4<-visual      pen     0.000        -        -        -      -      -
           x5<-visual      pen    -0.102      0.073   -1.411    0.158 -0.245  0.040
           x6<-visual      pen     0.000        -        -        -      -      -
           x7<-visual      pen    -0.254      0.118   -2.147    0.032 -0.486 -0.022
           x8<-visual      pen     0.000        -        -        -      -      -
           x9<-visual      pen     0.323      0.075    4.289    0.000  0.175  0.471
          x1<-textual      pen     0.255      0.081    3.136    0.002  0.096  0.415
          x2<-textual      pen     0.000        -        -        -      -      -
          x3<-textual      pen     0.000        -        -        -      -      -
          x4<-textual     free     0.987      0.061   16.133    0.000  0.867  1.106
          x5<-textual     free     1.143      0.062   18.583    0.000  1.023  1.264
          x6<-textual     free     0.913      0.058   15.726    0.000  0.799  1.027
          x7<-textual      pen     0.000        -        -        -      -      -
          x8<-textual      pen     0.000        -        -        -      -      -
          x9<-textual      pen     0.000        -        -        -      -      -
            x1<-speed      pen     0.000        -        -        -      -      -
            x2<-speed      pen     0.000        -        -        -      -      -
            x3<-speed      pen     0.000        -        -        -      -      -
            x4<-speed      pen     0.000        -        -        -      -      -
            x5<-speed      pen     0.000        -        -        -      -      -
            x6<-speed      pen     0.000        -        -        -      -      -
            x7<-speed     free     0.823      0.105    7.816    0.000  0.617  1.030
            x8<-speed     free     0.731      0.069   10.604    0.000  0.596  0.866
            x9<-speed     free     0.499      0.063    7.960    0.000  0.376  0.622

      Covariance
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
     textual<->visual     free     0.325      0.086    3.766    0.000  0.156  0.494
       speed<->visual     free     0.374      0.100    3.733    0.000  0.178  0.570
      speed<->textual     free     0.278      0.078    3.566    0.000  0.125  0.431

      Variance
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
      visual<->visual    fixed     1.000        -        -        -      -      -
    textual<->textual    fixed     1.000        -        -        -      -      -
        speed<->speed    fixed     1.000        -        -        -      -      -
              x1<->x1     free     0.671      0.113    5.951    0.000  0.450  0.892
              x2<->x2     free     1.073      0.106   10.164    0.000  0.866  1.280
              x3<->x3     free     0.719      0.091    7.943    0.000  0.542  0.896
              x4<->x4     free     0.377      0.050    7.507    0.000  0.279  0.476
              x5<->x5     free     0.415      0.060    6.869    0.000  0.297  0.534
              x6<->x6     free     0.362      0.046    7.903    0.000  0.273  0.452
              x7<->x7     free     0.596      0.109    5.469    0.000  0.383  0.810
              x8<->x8     free     0.487      0.084    5.769    0.000  0.322  0.653
              x9<->x9     free     0.540      0.064    8.430    0.000  0.414  0.666

      Intercept
                          type  estimate  std.error  z-value  P(>|z|)  lower  upper
                x1<-1     free     4.936      0.067   73.473    0.000  4.804  5.067
                x2<-1     free     6.088      0.068   89.855    0.000  5.955  6.221
                x3<-1     free     2.250      0.065   34.579    0.000  2.123  2.378
                x4<-1     free     3.061      0.067   45.694    0.000  2.930  3.192
                x5<-1     free     4.341      0.074   58.452    0.000  4.195  4.486
                x6<-1     free     2.186      0.063   34.667    0.000  2.062  2.309
                x7<-1     free     4.186      0.063   66.766    0.000  4.063  4.309
                x8<-1     free     5.527      0.058   94.854    0.000  5.413  5.641
                x9<-1     free     5.374      0.058   92.546    0.000  5.260  5.488