Model inspection

After you fitted a model,

model_fit = sem_fit(model)

you end up with an object of type SemFit.

You can get some more information about it by using the sem_summary function:

sem_summary(model_fit)


Fitted Structural Equation Model

--------------------------------- Properties ---------------------------------

Optimization algorithm:      L-BFGS
Convergence:                 true
No. iterations/evaluations:  189

Number of parameters:        31
Number of observations:      75.0

----------------------------------- Model -----------------------------------

Structural Equation Model
- Loss Functions
   SemML
- Fields
   observed:    SemObservedData
   imply:       RAM
   optimizer:   SemOptimizerOptim

To compute fit measures, we use

fit_measures(model_fit)

Dict{Symbol, Union{Missing, Float64}} with 8 entries:
  :minus2ll => 3106.66
  :AIC      => 3168.66
  :BIC      => 3240.5
  :df       => 35.0
  :χ²       => 37.6169
  :p_value  => 0.350263
  :RMSEA    => 0.0315738
  :n_par    => 31.0

or compute them individually:

AIC(model_fit)

3168.655879277305

A list of available Fit measures is at the end of this page.

To inspect the parameter estimates, we can update a ParameterTable object and call sem_summary on it:

update_estimate!(partable, model_fit)

sem_summary(partable)


--------------------------------- Variables ---------------------------------

Latent variables:    ind60 dem60 dem65
Observed variables:  x1 x2 x3 y1 y2 y3 y4 y5 y6 y7 y8

---------------------------- Parameter Estimates -----------------------------

Loadings:

ind60

  to   estimate   identifier   value_fixed   start   free   from    type

  x1   0.0        const        1.0           0.0     0.0    ind60   →
  x2   2.18       θ_1          0.0           0.0     1.0    ind60   →
  x3   1.82       θ_2          0.0           0.0     1.0    ind60   →

dem60

  to   estimate   identifier   value_fixed   start   free   from    type

  y1   0.0        const        1.0           0.0     0.0    dem60   →
  y2   1.26       θ_3          0.0           0.0     1.0    dem60   →
  y3   1.06       θ_4          0.0           0.0     1.0    dem60   →
  y4   1.26       θ_5          0.0           0.0     1.0    dem60   →

dem65

  to   estimate   identifier   value_fixed   start   free   from    type

  y5   0.0        const        1.0           0.0     0.0    dem65   →
  y6   1.19       θ_6          0.0           0.0     1.0    dem65   →
  y7   1.28       θ_7          0.0           0.0     1.0    dem65   →
  y8   1.27       θ_8          0.0           0.0     1.0    dem65   →

Directed Effects:

  from        to      estimate   identifier   value_fixed   start   free

  ind60   →   dem60   1.48       θ_9          0.0           0.0     1.0
  dem60   →   dem65   0.84       θ_10         0.0           0.0     1.0
  ind60   →   dem65   0.57       θ_11         0.0           0.0     1.0

Variances:

  from        to      estimate   identifier   value_fixed   start   free

  x1      ↔   x1      0.08       θ_12         0.0           0.0     1.0
  x2      ↔   x2      0.12       θ_13         0.0           0.0     1.0
  x3      ↔   x3      0.47       θ_14         0.0           0.0     1.0
  y1      ↔   y1      1.92       θ_15         0.0           0.0     1.0
  y2      ↔   y2      7.47       θ_16         0.0           0.0     1.0
  y3      ↔   y3      5.14       θ_17         0.0           0.0     1.0
  y4      ↔   y4      3.19       θ_18         0.0           0.0     1.0
  y5      ↔   y5      2.38       θ_19         0.0           0.0     1.0
  y6      ↔   y6      5.02       θ_20         0.0           0.0     1.0
  y7      ↔   y7      3.48       θ_21         0.0           0.0     1.0
  y8      ↔   y8      3.3        θ_22         0.0           0.0     1.0
  ind60   ↔   ind60   0.45       θ_23         0.0           0.0     1.0
  dem60   ↔   dem60   4.01       θ_24         0.0           0.0     1.0
  dem65   ↔   dem65   0.17       θ_25         0.0           0.0     1.0

Covariances:

  from       to   estimate   identifier   value_fixed   start   free

  y1     ↔   y5   0.63       θ_26         0.0           0.0     1.0
  y2     ↔   y4   1.33       θ_27         0.0           0.0     1.0
  y2     ↔   y6   2.18       θ_28         0.0           0.0     1.0
  y3     ↔   y7   0.81       θ_29         0.0           0.0     1.0
  y8     ↔   y4   0.35       θ_30         0.0           0.0     1.0
  y8     ↔   y6   1.37       θ_31         0.0           0.0     1.0

We can also update the ParameterTable object with other information via update_partable!. For example, if we want to compare hessian-based and bootstrap-based standard errors, we may write

se_bs = se_bootstrap(model_fit; n_boot = 20)
se_he = se_hessian(model_fit)

update_partable!(partable, model_fit, se_he, :se_hessian)
update_partable!(partable, model_fit, se_bs, :se_bootstrap)

sem_summary(partable)

Number of nonconverged models: 0.0

--------------------------------- Variables ---------------------------------

Latent variables:    ind60 dem60 dem65
Observed variables:  x1 x2 x3 y1 y2 y3 y4 y5 y6 y7 y8

---------------------------- Parameter Estimates -----------------------------

Loadings:

ind60

  to   estimate   identifier   value_fixed   start   free   from    type   se_bootstrap   se_hessian

  x1   0.0        const        1.0           0.0     0.0    ind60   →      0.0            0.0
  x2   2.18       θ_1          0.0           0.0     1.0    ind60   →      0.17           0.14
  x3   1.82       θ_2          0.0           0.0     1.0    ind60   →      0.14           0.15

dem60

  to   estimate   identifier   value_fixed   start   free   from    type   se_bootstrap   se_hessian

  y1   0.0        const        1.0           0.0     0.0    dem60   →      0.0            0.0
  y2   1.26       θ_3          0.0           0.0     1.0    dem60   →      0.14           0.19
  y3   1.06       θ_4          0.0           0.0     1.0    dem60   →      0.14           0.15
  y4   1.26       θ_5          0.0           0.0     1.0    dem60   →      0.16           0.15

dem65

  to   estimate   identifier   value_fixed   start   free   from    type   se_bootstrap   se_hessian

  y5   0.0        const        1.0           0.0     0.0    dem65   →      0.0            0.0
  y6   1.19       θ_6          0.0           0.0     1.0    dem65   →      0.19           0.17
  y7   1.28       θ_7          0.0           0.0     1.0    dem65   →      0.2            0.16
  y8   1.27       θ_8          0.0           0.0     1.0    dem65   →      0.21           0.16

Directed Effects:

  from        to      estimate   identifier   value_fixed   start   free   se_bootstrap   se_hessian

  ind60   →   dem60   1.48       θ_9          0.0           0.0     1.0    0.35           0.4
  dem60   →   dem65   0.84       θ_10         0.0           0.0     1.0    0.13           0.1
  ind60   →   dem65   0.57       θ_11         0.0           0.0     1.0    0.3            0.24

Variances:

  from        to      estimate   identifier   value_fixed   start   free   se_bootstrap   se_hessian

  x1      ↔   x1      0.08       θ_12         0.0           0.0     1.0    0.02           0.02
  x2      ↔   x2      0.12       θ_13         0.0           0.0     1.0    0.07           0.07
  x3      ↔   x3      0.47       θ_14         0.0           0.0     1.0    0.05           0.09
  y1      ↔   y1      1.92       θ_15         0.0           0.0     1.0    0.5            0.48
  y2      ↔   y2      7.47       θ_16         0.0           0.0     1.0    1.15           1.37
  y3      ↔   y3      5.14       θ_17         0.0           0.0     1.0    0.94           0.99
  y4      ↔   y4      3.19       θ_18         0.0           0.0     1.0    0.84           0.77
  y5      ↔   y5      2.38       θ_19         0.0           0.0     1.0    0.5            0.5
  y6      ↔   y6      5.02       θ_20         0.0           0.0     1.0    1.16           0.91
  y7      ↔   y7      3.48       θ_21         0.0           0.0     1.0    0.56           0.74
  y8      ↔   y8      3.3        θ_22         0.0           0.0     1.0    0.96           0.72
  ind60   ↔   ind60   0.45       θ_23         0.0           0.0     1.0    0.04           0.09
  dem60   ↔   dem60   4.01       θ_24         0.0           0.0     1.0    0.64           0.96
  dem65   ↔   dem65   0.17       θ_25         0.0           0.0     1.0    0.25           0.22

Covariances:

  from       to   estimate   identifier   value_fixed   start   free   se_bootstrap   se_hessian

  y1     ↔   y5   0.63       θ_26         0.0           0.0     1.0    0.55           0.38
  y2     ↔   y4   1.33       θ_27         0.0           0.0     1.0    0.71           0.71
  y2     ↔   y6   2.18       θ_28         0.0           0.0     1.0    0.79           0.74
  y3     ↔   y7   0.81       θ_29         0.0           0.0     1.0    0.58           0.63
  y8     ↔   y4   0.35       θ_30         0.0           0.0     1.0    0.27           0.47
  y8     ↔   y6   1.37       θ_31         0.0           0.0     1.0    0.83           0.58

Export results

You may convert a ParameterTable to a DataFrame and use the DataFrames package for further analysis (or to save it to your hard drive).

using DataFrames

parameters_df = DataFrame(partable)

34×10 DataFrame

Row	parameter_type	se_hessian	from	value_fixed	identifier	start	estimate	to	se_bootstrap	free
	Symbol	Float64	Symbol	Float64	Symbol	Float64	Float64	Symbol	Float64	Bool
1	→	0.0	ind60	1.0	const	0.0	0.0	x1	0.0	false
2	→	0.139945	ind60	0.0	θ_1	0.0	2.18035	x2	0.174679	true
3	→	0.153212	ind60	0.0	θ_2	0.0	1.81848	x3	0.144255	true
4	→	0.0	dem60	1.0	const	0.0	0.0	y1	0.0	false
5	→	0.186751	dem60	0.0	θ_3	0.0	1.25672	y2	0.137071	true
6	→	0.14933	dem60	0.0	θ_4	0.0	1.05773	y3	0.135911	true
7	→	0.151741	dem60	0.0	θ_5	0.0	1.26476	y4	0.158868	true
8	→	0.0	dem65	1.0	const	0.0	0.0	y5	0.0	false
9	→	0.172509	dem65	0.0	θ_6	0.0	1.18572	y6	0.192544	true
10	→	0.161464	dem65	0.0	θ_7	0.0	1.27954	y7	0.203132	true
11	→	0.16432	dem65	0.0	θ_8	0.0	1.26596	y8	0.213715	true
12	→	0.399936	ind60	0.0	θ_9	0.0	1.48297	dem60	0.350699	true
13	→	0.0994512	dem60	0.0	θ_10	0.0	0.837319	dem65	0.127864	true
14	→	0.235306	ind60	0.0	θ_11	0.0	0.572344	dem65	0.295112	true
15	↔	0.0201157	x1	0.0	θ_12	0.0	0.0826497	x1	0.0208137	true
16	↔	0.0713895	x2	0.0	θ_13	0.0	0.12143	x2	0.0721299	true
17	↔	0.0909945	x3	0.0	θ_14	0.0	0.473005	x3	0.0474256	true
18	↔	0.478275	y1	0.0	θ_15	0.0	1.91688	y1	0.501061	true
19	↔	1.3733	y2	0.0	θ_16	0.0	7.47324	y2	1.15016	true
20	↔	0.987867	y3	0.0	θ_17	0.0	5.13557	y3	0.944243	true
21	↔	0.771184	y4	0.0	θ_18	0.0	3.19053	y4	0.83622	true
22	↔	0.498679	y5	0.0	θ_19	0.0	2.38271	y5	0.503207	true
23	↔	0.913706	y6	0.0	θ_20	0.0	5.0211	y6	1.15585	true
24	↔	0.742606	y7	0.0	θ_21	0.0	3.47771	y7	0.555314	true
25	↔	0.721402	y8	0.0	θ_22	0.0	3.29808	y8	0.961604	true
26	↔	0.0885061	ind60	0.0	θ_23	0.0	0.4545	ind60	0.038695	true
27	↔	0.963906	dem60	0.0	θ_24	0.0	4.00973	dem60	0.644566	true
28	↔	0.224793	dem65	0.0	θ_25	0.0	0.174844	dem65	0.251257	true
29	↔	0.376487	y1	0.0	θ_26	0.0	0.632078	y5	0.553719	true
30	↔	0.713059	y2	0.0	θ_27	0.0	1.33103	y4	0.711435	true
31	↔	0.741161	y2	0.0	θ_28	0.0	2.18224	y6	0.787832	true
32	↔	0.633761	y3	0.0	θ_29	0.0	0.805662	y7	0.579235	true
33	↔	0.467172	y8	0.0	θ_30	0.0	0.352869	y4	0.270135	true
34	↔	0.583634	y8	0.0	θ_31	0.0	1.37452	y6	0.825226	true

API - model inspection

StructuralEquationModels.sem_summary — Function

(1) sem_summary(sem_fit::SemFit; show_fitmeasures = false)

(2) sem_summary(partable::AbstractParameterTable)

Print information about (1) a fitted SEM or (2) a parameter table to stdout.

Extended help

Addition keyword arguments

digits = 2: controls precision of printed estimates, standard errors, etc.
color = :light_cyan: color of some parts of the printed output. Can be adjusted for readability.
secondary_color = :light_yellow
show_variables = true

source

StructuralEquationModels.update_estimate! — Function

update_estimate!(
    partable::AbstractParameterTable, 
    sem_fit::SemFit)

Write parameter estimates from sem_fit to the :estimate column of partable

source

StructuralEquationModels.update_partable! — Function

update_partable!(partable::AbstractParameterTable, sem_fit::SemFit, vec, column)

Write vec to column of partable.

Arguments

vec::Vector: has to be in the same order as the model parameters

source

Additional functions

Additional functions that can be used to extract information from a SemFit object:

StructuralEquationModels.SemFit — Type

SemFit

Fitted structural equation model.

Interfaces

minimum(::SemFit) -> minimum objective value
solution(::SemFit) -> parameter estimates
start_val(::SemFit) -> starting values
model(::SemFit)
optimization_result(::SemFit)
optimizer(::SemFit) -> optimization algorithm
n_iterations(::SemFit) -> number of iterations
convergence(::SemFit) -> convergence properties

source

Fit measures

StructuralEquationModels.fit_measures — Function

fit_measures(sem_fit, args...)

Return a default set of fit measures or the fit measures passed as arg....

source

StructuralEquationModels.AIC — Function

AIC(sem_fit::SemFit)

Return the akaike information criterion.

source

StructuralEquationModels.BIC — Function

BIC(sem_fit::SemFit)

Return the bayesian information criterion.

source

StructuralEquationModels.χ² — Function

χ²(sem_fit::SemFit)

Return the χ² value.

source

StructuralEquationModels.df — Function

df(sem_fit::SemFit)
df(model::AbstractSem)

Return the degrees of freedom.

source

StructuralEquationModels.minus2ll — Function

minus2ll(sem_fit::SemFit)

Return the negative 2* log likelihood.

source

StructuralEquationModels.n_man — Function

n_man(sem_fit::SemFit)
n_man(model::AbstractSemSingle)

Return the number of manifest variables.

source

StructuralEquationModels.n_obs — Function

n_obs(sem_fit::SemFit)
n_obs(model::AbstractSemSingle)
n_obs(model::SemEnsemble)

Return the number of observed data points.

For ensemble models, return the sum over all submodels.

source

StructuralEquationModels.n_par — Function

n_par(sem_fit::SemFit)
n_par(model::AbstractSemSingle)
n_par(model::SemEnsemble)
n_par(identifier::Dict)

Return the number of parameters.

source

StructuralEquationModels.p_value — Function

p(sem_fit::SemFit)

Return the p value computed from the χ² test statistic.

source

StructuralEquationModels.RMSEA — Function

RMSEA(sem_fit::SemFit)

Return the RMSEA.

source