Home > Statistics > Calculating power using Monte Carlo simulations, part 5: Structural equation models

## Calculating power using Monte Carlo simulations, part 5: Structural equation models

In our last four posts in this series, we showed you how to calculate power for a t test using Monte Carlo simulations, how to integrate your simulations into Stata’s power command, and how to do this for linear and logistic regression models and multilevel models. In today’s post, I’m going to show you how to estimate power for structural equation models (SEM) using simulations.

Our goal is to write a program that will calculate power for a given SEM at different sample sizes. We’ll follow the same general procedure as the previous two posts, but the way we’ll go about simulating data is a bit different. Rather than individually simulating each variable for our specified model, we’ll be simulating all our variables simultaneously from a given covariance matrix. Means for each of the variables can also be used to simulate the data if your SEM has a mean structure, such as in group analysis or growth curve analysis.

There are three ways you can obtain a covariance matrix to simulate SEM data:

1. Using a covariance matrix published in a paper or other source.
2. Using the reticular action model (RAM) to derive the model-implied covariance matrix using expected parameter estimates.
3. Extracting the model-implied covariance matrix after performing an sem analysis in Stata either on your own pilot study data or on another data source.

The RAM method will be demonstrated at the end of this post. Extracting model-implied covariances after using the sem command will be demonstrated below. Whichever method you choose to obtain a covariance matrix to simulate from, the remainder of our procedure will be the same as in the previous two posts. We will work through the following steps:

1. Obtain or derive a covariance matrix (and means, if applicable) that corresponds with your hypothesized model under the alternative hypothesis.
2. Simulate a single dataset and fit the model.
3. Write a program to create the datasets, fit the models, and use simulate to test the program.
4. Write a program called power_cmd_simsem that allows you to run your simulations with power.
5. Optional: Write a program called power_cmd_simsem_init so that you can see convergence rates for your model at different sample sizes.

We are planning a new study to evaluate the interaction effect between age and sex on health. We can use the NHANES dataset to obtain a reasonable covariance matrix from which we can simulate new data. We will define health as a latent variable measured by systolic blood pressure (bpsystol), diastolic blood pressure (bpdiast), serum cholesterol (tcresult), and serum triglyercides (tgresult). Specifically, we would like to fit the following model:

Figure 1: Path diagram of the hypothesized model Step 1: Obtain or derive a covariance matrix (and means, if applicable) that corresponds with your hypothesized model

We are going to use the model-implied covariance matrix from fitting the above model to the NHANES data. First, we need to load the dataset, create the interaction variable, and then fit our model.

. webuse nhanes2

. gen ageXfemale = age*female

. sem (Health -> bpsystol bpdiast tcresult tgresult)
>    (age female ageXfemale -> Health), nolog
(5301 observations with missing values excluded)

Endogenous variables
Measurement: bpsystol bpdiast tcresult tgresult
Latent:      Health

Exogenous variables
Observed: age female ageXfemale

Structural equation model                                Number of obs = 5,050
Estimation method: ml

Log likelihood = -140667.36

( 1)  [bpsystol]Health = 1
------------------------------------------------------------------------------
|                 OIM
| Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Structural   |
Health     |
age |   .4371085   .0237304    18.42   0.000     .3905977    .4836193
female |  -20.89331    1.67081   -12.50   0.000    -24.16804   -17.61859
ageXfemale |   .3395419   .0328812    10.33   0.000     .2750961    .4039878
-------------+----------------------------------------------------------------
Measurement  |
bpsystol   |
Health |          1  (constrained)
_cons |   111.2516   1.207464    92.14   0.000      108.885    113.6182
-----------+----------------------------------------------------------------
bpdiast    |
Health |   .4404182   .0096514    45.63   0.000     .4215018    .4593346
_cons |   72.88132   .5554644   131.21   0.000     71.79263    73.97001
-----------+----------------------------------------------------------------
tcresult   |
Health |   .6630559   .0373398    17.76   0.000     .5898712    .7362406
_cons |   203.6104   1.217816   167.19   0.000     201.2235    205.9973
-----------+----------------------------------------------------------------
tgresult   |
Health |   1.122149   .0726778    15.44   0.000     .9797029    1.264594
_cons |     123.03   2.275853    54.06   0.000     118.5694    127.4906
-------------+----------------------------------------------------------------
var(e.bpsy~l)|   67.62701   8.855679                      52.31863    87.41462
var(e.bpdi~t)|    80.1877   2.164596                      76.05544    84.54446
var(e.tcre~t)|   2083.505   42.75414                      2001.372     2169.01
var(e.tgre~t)|   8724.877   176.5499                      8385.617    9077.862
var(e.Health)|   340.7899   11.08864                      319.7351    363.2312
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(11) = 1542.14        Prob > chi2 = 0.0000


Age, sex, and their interaction are all significant predictors of Health when performed on 5,050 observations, but perhaps you don’t have the resources to collect a sample that large. How large does your sample need to be to have 80% power to detect an interaction between age and sex? To simulate data for our power analysis, we can use the model-implied covariance reported with the fitted option of the estat framework command. This will provide a lot of output, so we’ll run it quietly and just return what we’re interested in: the fitted covariance and the fitted means. Our hypothesized SEM doesn’t include mean structure, so including the means for our power analysis is unnecessary. However, I will include them in all the steps below so that our code can be generalized to other types of SEMs.

. quietly estat framework, fitted

. matrix list r(mu)

r(mu)[1,8]
Observed:   Observed:   Observed:   Observed:     Latent:   Observed:
bpsystol     bpdiast    tcresult    tgresult      Health         age
mu   129.84614   81.070693    215.9396   143.89584   18.594545   47.941386

Observed:   Observed:
female  ageXfemale
mu   .51663366   24.836832

. matrix list r(Sigma)

symmetric r(Sigma)[8,8]
Observed:   Observed:   Observed:   Observed:
bpsystol     bpdiast    tcresult    tgresult
Observed:bpsystol   532.05333
Observed:bpdiast   204.54181   170.27163
Observed:tcresult   307.94061   135.62265   2287.6872
Observed:tgresult   521.15538   229.52632   345.55515   9309.6905
Latent:Health   464.42632   204.54181   307.94061   521.15538
Observed:age   179.49835    79.05434   119.01744   201.42383
Observed:female  -1.1112203  -.48940167  -.73680119  -1.2469544
Observed:ageXfemale   64.672663   28.483019   42.881591   72.572343

Latent:   Observed:   Observed:   Observed:
Health         age      female  ageXfemale
Latent:Health   464.42632
Observed:age   179.49835   293.25241
Observed:female  -1.1112203   .06869774   .24972332
Observed:ageXfemale   64.672663   155.35796   12.005288   729.20189


Notice that both our covariance and our means include observed as well as latent variables. We just want the information that corresponds with the observed variables, so we’ll need to extract just those portions of these matrices and save them to a new matrix. We can specify a range of rows or columns using .. between the starting and ending row or column. When concatenating matrices together, commas are used to signify a new column and backslashes (\) are used to signify a new row.

. matrix mu = r(mu)[1,1..4],r(mu)[1,6..8]

. matrix Sigma = r(Sigma)[1..4,1..4],r(Sigma)[1..4,6..8]\r(Sigma)[6..8,1..4],
>    r(Sigma)[6..8,6..8]

. matrix list mu

mu[1,7]
Observed:   Observed:   Observed:   Observed:   Observed:   Observed:
bpsystol     bpdiast    tcresult    tgresult         age      female
mu   129.84614   81.070693    215.9396   143.89584   47.941386   .51663366

Observed:
ageXfemale
mu   24.836832

. matrix list Sigma

symmetric Sigma[7,7]
Observed:   Observed:   Observed:   Observed:
bpsystol     bpdiast    tcresult    tgresult
Observed:bpsystol   532.05333
Observed:bpdiast   204.54181   170.27163
Observed:tcresult   307.94061   135.62265   2287.6872
Observed:tgresult   521.15538   229.52632   345.55515   9309.6905
Observed:age   179.49835    79.05434   119.01744   201.42383
Observed:female  -1.1112203  -.48940167  -.73680119  -1.2469544
Observed:ageXfemale   64.672663   28.483019   42.881591   72.572343

Observed:   Observed:   Observed:
age      female  ageXfemale
Observed:age   293.25241
Observed:female   .06869774   .24972332
Observed:ageXfemale   155.35796   12.005288   729.20189


Now we have the covariance matrix and means we need to simulate our data!

Step 2: Simulate a single dataset assuming the alternative hypothesis, and fit the model

Next we create a simulated dataset from our covariance matrix (and means) using the drawnorm command. drawnorm simulates a variable or set of variables based on sample size, means, and covariance. Here we’ll use a sample size of 200.

. set seed 12345

. clear

. drawnorm bpsystol bpdiast tcresult tgresult age female ageXfemale,
>    n(200) means(mu) cov(Sigma)
(obs 200)


We can then use sem to fit the hypothesized model to our simulated data.

. sem (Health -> bpsystol bpdiast tcresult tgresult)
>    (age female ageXfemale -> Health), nolog

Endogenous variables
Measurement: bpsystol bpdiast tcresult tgresult
Latent:      Health

Exogenous variables
Observed: age female ageXfemale

Structural equation model                                  Number of obs = 200
Estimation method: ml

Log likelihood = -5599.7049

( 1)  [bpsystol]Health = 1
------------------------------------------------------------------------------
|                 OIM
| Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Structural   |
Health     |
age |   .5016981   .1243038     4.04   0.000     .2580671    .7453291
female |  -18.12394   8.529735    -2.12   0.034    -34.84191   -1.405964
ageXfemale |   .2329537   .1611524     1.45   0.148    -.0828992    .5488066
-------------+----------------------------------------------------------------
Measurement  |
bpsystol   |
Health |          1  (constrained)
_cons |   111.2897   6.191827    17.97   0.000     99.15397    123.4255
-----------+----------------------------------------------------------------
bpdiast    |
Health |   .4507904   .0627156     7.19   0.000     .3278701    .5737107
_cons |   73.00124   3.087596    23.64   0.000     66.94967    79.05282
-----------+----------------------------------------------------------------
tcresult   |
Health |   .5630107    .183409     3.07   0.002     .2035357    .9224857
_cons |    207.841   6.011418    34.57   0.000     196.0588    219.6231
-----------+----------------------------------------------------------------
tgresult   |
Health |   .8698218   .3926678     2.22   0.027     .1002072    1.639436
_cons |   126.6153    11.5315    10.98   0.000     104.0139    149.2166
-------------+----------------------------------------------------------------
var(e.bpsy~l)|   137.2281   53.00586                      64.36608    292.5695
var(e.bpdi~t)|   75.17043   12.98479                      53.58117    105.4586
var(e.tcre~t)|   2235.489   226.4431                      1832.949    2726.432
var(e.tgre~t)|   8917.533   902.1103                      7313.681     10873.1
var(e.Health)|   303.1287   58.65825                      207.4491    442.9376
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(11) = 13.10          Prob > chi2 = 0.2866


To test the null hypothesis that the interaction term equals zero, it will be easier to conduct a Wald test using the test command with sem than using the lrtest command used in previous posts. In our case, we could have extracted the p-value from the table above, but this step can be adapted to test any hypothesis of interest.

. test ageXfemale

( 1)  [Health]ageXfemale = 0

chi2(  1) =    2.09
Prob > chi2 =    0.1483


The p-value for our test is 0.148, so we would not be able to reject the null hypothesis that the interaction term equals zero.

As discussed in the previous post, the model won’t always converge when fit to simulated datasets. Like mixed, sem will store a value of 1 to e(converged) if the model converges and 0 otherwise. We will store the value of e(converged) from the model to a local macro named conv to keep track of the convergence of our models. We’ll also create a local variable, reject, that will keep track of whether the estimated interaction term is significant. The value that is stored in reject is determined by the conditional function cond(). If the model converged, then r(p)<0.05 will be evaluated. A value of 1 will be stored to reject if the Wald test p-value, r(p), is less than the alpha level, here specified as 0.05, and 0 otherwise. If the model failed to converge, then a missing value will be stored in reject.

. sem (Health -> bpsystol bpdiast tcresult tgresult)
>    (age female ageXfemale -> Health)

. local conv = e(converged)

. test ageXfemale

. local reject = cond(conv'==1, r(p)<.05, .)


Step 3: Write a program to create the datasets, fit the models, and use simulate to test the program

Next let’s write a program that creates datasets under the alternative hypothesis, fits sem models, tests the null hypothesis of interest, and uses simulate to run many iterations of the program. The primary difference between this program and the programs we have written in previous posts is that this program needs to accept matrices as input arguments. This can be a little tricky. What we need to do is specify the type of input as a string rather than a matrix. Then within the program we will define our matrices by their name, passed to the program as a string, that is, mat C = cov’. Finally, we can use these matrices to simulate our data with drawnorm, fit our model, and test the null hypothesis. We’ve also added capture in front of the sem command to capture errors in case the model doesn’t converge. The code block below contains the syntax for this program, called simsem.

capture program drop simsem
program simsem, rclass
version 17
// PARSE INPUT
syntax, n(integer)      ///
cov(string)         ///
[ means(string)     ///
alpha(real 0.05) ]
// COMPUTE POWER
quietly {
drop _all
mat C =cov'
mat m = means'
drawnorm bpsystol bpdiast tcresult tgresult age female ageXfemale, ///
n(n') means(m) cov(C)
capture sem (Health -> bpsystol bpdiast tcresult tgresult)         ///
(age female ageXfemale -> Health)
local conv = e(converged)
test ageXfemale
local reject = cond(conv'==1, r(p)<alpha', .)
}
// RETURN RESULTS
return scalar reject = reject'
return scalar conv = conv'
end

We then use simulate to run simsem 10 times using our covariance matrix and means for a sample size of 200.

. simulate reject=r(reject) converged=r(conv), reps(10) seed(12345):
>    simsem, n(200) means(mu) cov(Sigma)

Command:  simsem, n(200) means(mu) cov(Sigma)
reject:  r(reject)
converged:  r(conv)

Simulations (10)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
...x...x..


simulate saved two variables: reject and converged. The mean of converged is the convergence rate. The mean of reject is the power to test the null hypothesis that the age by sex interaction term equals zero.

. summarize reject

Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
reject |          8          .5    .5345225          0          1

. summarize converged

Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
converged |         10          .8     .421637          0          1


With a sample size of 200, the model converged in 8 of the 10 repetitions (80%) and had a power of 50% to detect the interaction effect on Health.

Step 4: Write a program called power_cmd_simsem, which allows you to run your simulations with power

We could stop with our quick simulation if we were interested only in a specific set of assumptions. But it’s easy to write an additional program named power_cmd_simsem that will allow us to use Stata’s power command to create tables and graphs for a range of sample sizes. We just need to include the input syntax as we did in the simsem command, simulate with the simsem command, and return the results.

capture program drop power_cmd_simsem
program power_cmd_simsem, rclass
version 17
// PARSE INPUT
syntax, n(integer)                       ///
cov(string)                       ///
[ alpha(real 0.05)                ///
means(string)                     ///
reps(integer 100) ]
// COMPUTE POWER
quietly simulate reject=r(reject) converged=r(conv), reps(reps'):  ///
simsem, n(n') means(means') cov(cov') alpha(alpha')
// RETURN RESULTS
return scalar N = n'
return scalar alpha = alpha'
quietly summarize reject
return scalar power = r(mean)
quietly summarize converged
return scalar conv_rate = r(mean)
end

Step 5: (Optional) Write a program called power_cmd_simsem_init so that you can see convergence rates for your model at different sample sizes.

Convergence is often an issue in SEM simulation. If your model is having trouble converging at smaller sample sizes, it may be useful to add a convergence rate column to the power output table.

capture program drop power_cmd_simsem_init
program power_cmd_simsem_init, sclass
version 17
sreturn clear
// ADD COLUMNS TO THE OUTPUT TABLE
sreturn local pss_colnames "conv_rate"
end

Using power simsem

At last, we can use power simsem to simulate power and convergence rate for a range of sample sizes. The example below simulates power for sample sizes ranging from 200 to 500, using 100 repetitions each.

. power simsem, n(200(100)500) means(mu) cov(Sigma) reps(100)
>    table(N conv_rate power) graph
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Estimated power
Two-sided test

+---------------------------+
|       N conv_rate   power |
|---------------------------|
|     200       .86      .5 |
|     300       .92   .7283 |
|     400       .93   .8602 |
|     500       .97   .8866 |
+---------------------------+


Figure 2: Estimated power for a structural equation model The table and graph above indicate that 80% power is achieved around a sample size of 350.

The procedure demonstrated here can be used to perform a power analysis on any SEM. You just need to define your model of interest and simulate data based on a covariance matrix. Including the previous posts in this series, we have now given examples of how you can use Stata to perform power analysis by simulation for a variety of models. You can use similar methods for virtually any power analysis you need to perform.

Deriving covariances based on SEM path coefficients using the RAM method

As discussed at the beginning of this post, you can use the RAM to directly specify population parameters or use results from a publication to obtain a covariance matrix. The RAM method uses a set of matrices to represent the model, which can be combined using matrix algebra to derive the corresponding covariance matrix. Three matrices are required: $$\mathbf{A}$$, $$\mathbf{S}$$, and $$\mathbf{F}$$. These contain the path coefficients, factor loadings, variances, and covariances of the model you would like to simulate data from. Additionally, a matrix of means, $$\mathbf{M}$$, is required if the SEM involves mean structure.

To demonstrate how to use model results to build these matrices, we will use the results from our data analysis on the NHANES datset. We will use the noxconditional option in the sem command because we need the estimated variances or covariances of the exogenous variables in order to create our matrices.

. webuse nhanes2, clear

. gen ageXfemale = age*female

. sem (Health -> bpsystol bpdiast tcresult tgresult)
>    (age female ageXfemale -> Health), noxconditional nolog
(5301 observations with missing values excluded)

Endogenous variables
Measurement: bpsystol bpdiast tcresult tgresult
Latent:      Health

Exogenous variables
Observed: age female ageXfemale

Structural equation model                                Number of obs = 5,050
Estimation method: ml

Log likelihood = -140667.36

( 1)  [bpsystol]Health = 1
------------------------------------------------------------------------------
|                 OIM
| Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
Structural   |
Health     |
age |   .4371086   .0237305    18.42   0.000     .3905977    .4836194
female |  -20.89331    1.67081   -12.50   0.000    -24.16804   -17.61858
ageXfemale |   .3395419   .0328812    10.33   0.000      .275096    .4039878
-------------+----------------------------------------------------------------
Measurement  |
bpsystol   |
Health |          1  (constrained)
_cons |   111.2516   1.207464    92.14   0.000      108.885    113.6182
-----------+----------------------------------------------------------------
bpdiast    |
Health |   .4404182   .0096514    45.63   0.000     .4215018    .4593346
_cons |   72.88132   .5554643   131.21   0.000     71.79263    73.97001
-----------+----------------------------------------------------------------
tcresult   |
Health |   .6630558   .0373398    17.76   0.000     .5898711    .7362405
_cons |   203.6104   1.217816   167.19   0.000     201.2235    205.9973
-----------+----------------------------------------------------------------
tgresult   |
Health |   1.122149   .0726778    15.44   0.000     .9797027    1.264594
_cons |     123.03   2.275853    54.06   0.000     118.5694    127.4906
-------------+----------------------------------------------------------------
mean(age)|   47.94139   .2409767   198.95   0.000     47.46908    48.41369
mean(female)|   .5166337   .0070321    73.47   0.000      .502851    .5304163
mean(ageXf~e)|   24.83683   .3799953    65.36   0.000     24.09205    25.58161
-------------+----------------------------------------------------------------
var(e.bpsy~l)|   67.62698    8.85568                       52.3186    87.41459
var(e.bpdi~t)|    80.1877   2.164596                      76.05545    84.54447
var(e.tcre~t)|   2083.505   42.75414                      2001.372     2169.01
var(e.tgre~t)|   8724.877   176.5499                      8385.617    9077.862
var(e.Health)|     340.79   11.08864                      319.7351    363.2313
var(age)|   293.2524   5.835941                      282.0344    304.9166
var(female)|   .2497233   .0049697                      .2401704    .2596562
var(ageXfe~e)|   729.2019   14.51166                      701.3071    758.2062
-------------+----------------------------------------------------------------
cov(age,|
female)|   .0686977   .1204256     0.57   0.568     -.167332    .3047275
cov(age,|
ageXfemale)|    155.358   6.864694    22.63   0.000     141.9034    168.8125
cov(female,|
ageXfemale)|   12.00529   .2541636    47.23   0.000     11.50714    12.50344
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(11) = 1542.14        Prob > chi2 = 0.0000


The first matrix we will create is the $$\mathbf{A}$$ matrix. The $$\mathbf{A}$$ matrix is an asymmetric matrix that stores the single-headed path coefficients in the model. The rows and columns of our matrix represent each of the observed and latent variables. The cells represent paths from the column variable to the row variable. I will start by creating a matrix of 0s of the size that I need using J(r,c,z), an $$r \times c$$ matrix containing elements z. Then I will substitute blocks of cells with the paths from the model into the $$\mathbf{A}$$ matrix. We only need to specify the upper left cell of the cells we are substituting. Finally, I will add row and column names to this matrix to make it easier to interpret.

. matrix A =  J(8,8,0)

. matrix A[1,8] = (1.00\0.44\0.66\1.12)

. matrix A[8,5] = (0.44,-20.89,0.34)

. matrix rownames A = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix colnames A = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix list A

A[8,8]
bpsystol     bpdiast    tcresult    tgresult         age
bpsystol           0           0           0           0           0
bpdiast           0           0           0           0           0
tcresult           0           0           0           0           0
tgresult           0           0           0           0           0
age           0           0           0           0           0
female           0           0           0           0           0
ageXfemale           0           0           0           0           0
Health           0           0           0           0         .44

female  ageXfemale      Health
bpsystol           0           0           1
bpdiast           0           0         .44
tcresult           0           0         .66
tgresult           0           0        1.12
age           0           0           0
female           0           0           0
ageXfemale           0           0           0
Health      -20.89         .34           0


The $$\mathbf{S}$$ matrix is a symmetric matrix that stores double-headed path coefficients, variances, and covariances of our model. I start by creating a diagnal matrix of all the estimated variances from the model, and then I add the covariances of the exogenous variables.

. matrix S = diag((67.63,80.19,2083.51,8724.88,0,0,0,340.79))

. matrix S[5,5] = (293.25,0.07,155.36\0.07,0.25,12.01\155.36,12.01,729.20)

. matrix rownames S = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix colnames S = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix list S

symmetric S[8,8]
bpsystol     bpdiast    tcresult    tgresult         age
bpsystol       67.63
bpdiast           0       80.19
tcresult           0           0     2083.51
tgresult           0           0           0     8724.88
age           0           0           0           0      293.25
female           0           0           0           0         .07
ageXfemale           0           0           0           0      155.36
Health           0           0           0           0           0

female  ageXfemale      Health
female         .25
ageXfemale       12.01       729.2
Health           0           0      340.79


The $$\mathbf{F}$$ matrix is a rectangular matrix to distinguish observed and latent variables. The rows represent each observed variable, and the columns represent each observed and latent variable. Each row gets a 1′ in the cell of that variable’s corresponding column, creating a diagonal submatrix. A diagonal matrix of 1s is also called the identity matrix. We can specify an identity matrix with I(n) where n is the number of rows/columns. I add a column vector of 0s with J().

. matrix F = I(7),J(7,1,0)

. matrix rownames F = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale

. matrix colnames F = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix list F

F[7,8]
bpsystol     bpdiast    tcresult    tgresult         age
bpsystol           1           0           0           0           0
bpdiast           0           1           0           0           0
tcresult           0           0           1           0           0
tgresult           0           0           0           1           0
age           0           0           0           0           1
female           0           0           0           0           0
ageXfemale           0           0           0           0           0

female  ageXfemale      Health
bpsystol           0           0           0
bpdiast           0           0           0
tcresult           0           0           0
tgresult           0           0           0
age           0           0           0
female           1           0           0
ageXfemale           0           1           0


Finally, if your model involves mean structure, an $$\mathbf{M}$$ matrix must also be created to obtain the model-implied means. This is just a column vector of estimated means or intercepts for the observed and latent variables in the model. Unless explicitly estimated, you can assume all the latent variable means are 0.

. matrix M = (111.25\72.88\203.61\123.03\47.94\.52\24.84\0)

. matrix rownames M = bpsystol bpdiast tcresult tgresult age female
>    ageXfemale Health

. matrix list M

M[8,1]
c1
bpsystol  111.25
bpdiast   72.88
tcresult  203.61
tgresult  123.03
age   47.94
female     .52
ageXfemale   24.84
Health       0


Now we can use matrix algebra to derive the model-implied means and covariance. The dimension of I() will need to be changed based on the number of observed and latent variables in the model. Here it’s 8 to include the seven observed variables and the one latent variable.

. *model-implied means
. matrix mu = F*inv(I(8)-A)*M

. matrix list mu

mu[7,1]
c1
bpsystol   129.9264
bpdiast  81.097616
tcresult  215.93642
tgresult  143.94757
age      47.94
female        .52
ageXfemale      24.84

. *model-implied covariance
. matrix Sigma = F*inv(I(8)-A)*S*inv(I(8)-A)'*F'

. matrix list Sigma

symmetric Sigma[7,7]
bpsystol     bpdiast    tcresult    tgresult         age
bpsystol   533.17918
bpdiast   204.84164   170.32032
tcresult   307.26246   135.19548   2286.3032
tgresult   521.41508   229.42264   344.13395   9308.8649
age    180.3901   79.371644   119.05747   202.03691      293.25
female     -1.1083    -.487652    -.731478   -1.241296         .07
ageXfemale     65.3975     28.7749    43.16235     73.2452      155.36

female  ageXfemale
female         .25
ageXfemale       12.01       729.2
`

We see the covariance matrix implied by the model with the supplied model parameters.

Categories: Statistics Tags: