Mplus version 8 was used for these examples. All the files for this portion of this seminar can be downloaded here.

There are at least two reasons why a researcher might want to fix or free parameters. The first is that, while the default settings in Mplus do a fairly good job of deciding which parameters should be freed and which should be fixed, researchers often want to fit models that differ from these defaults in some way. Second, in addition fitting an overall model, researchers often wish to test hypotheses about specific parameters. Freeing and/or fixing parameters is often necessary in order to perform these tests. These commands are general in the sense that the same basic commands are used with all of the models estimated by Mplus. The dataset to be used in the following examples is worland.dat.

## 1.0 Constraining a parameter to a given value

You may want to fix paths to a given value for a variety of
reasons. Examples include needing to identify a model (although Mplus will often do this
by default) or testing a nested model. In the example below, we fix the path
from adjust to achieve to zero. We do this using the **@** symbol followed
by the number to which we wish to fix the parameter. In this example, we will set the parameter to 0 with **adjust@0**.

title: Fixing a parameter to zero data: file is worland.dat; variable: names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; model: family by ppsych ses; cog by verbal vissp mem; achieve by read arith spell; adjust by motiv extra harm stabi; achieve on cog adjust@0; adjust on family;

The MODEL RESULTS section of the output is shown below.

MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value FAMILY BY PPSYCH 1.000 0.000 999.000 999.000 SES -1.095 0.125 -8.790 0.000 COG BY VERBAL 1.000 0.000 999.000 999.000 VISSP 0.833 0.045 18.358 0.000 MEM 0.980 0.044 22.335 0.000 ACHIEVE BY READ 1.000 0.000 999.000 999.000 ARITH 0.826 0.034 24.548 0.000 SPELL 0.942 0.026 35.902 0.000 ADJUST BY MOTIV 1.000 0.000 999.000 999.000 EXTRA 0.231 0.049 4.676 0.000 HARM 0.880 0.043 20.294 0.000 STABI 0.677 0.046 14.824 0.000 ACHIEVE ON COG 1.008 0.041 24.605 0.000ADJUST 0.000 0.000 999.000 999.000ADJUST ON FAMILY -1.267 0.141 -9.015 0.000 COG WITH FAMILY -0.403 0.045 -8.897 0.000 Intercepts PPSYCH 0.000 0.045 0.000 1.000 SES 0.000 0.045 0.000 1.000 VERBAL 0.000 0.045 0.000 1.000 VISSP 0.000 0.045 0.000 1.000 MEM 0.000 0.045 0.000 1.000 READ 0.000 0.045 0.000 1.000 ARITH 0.000 0.045 0.000 1.000 SPELL 0.000 0.045 0.000 1.000 MOTIV 0.000 0.045 0.000 1.000 EXTRA 0.000 0.045 0.000 1.000 HARM 0.000 0.045 0.000 1.000 STABI 0.000 0.045 0.000 1.000 Variances FAMILY 0.235 0.047 5.017 0.000 COG 0.736 0.063 11.616 0.000 Residual Variances PPSYCH 0.763 0.054 14.247 0.000 SES 0.716 0.051 13.947 0.000 VERBAL 0.262 0.024 10.909 0.000 VISSP 0.487 0.035 13.966 0.000 MEM 0.292 0.025 11.538 0.000 READ 0.086 0.014 6.347 0.000 ARITH 0.376 0.027 13.764 0.000 SPELL 0.188 0.016 11.782 0.000 MOTIV 0.119 0.032 3.728 0.000 EXTRA 0.951 0.061 15.699 0.000 HARM 0.318 0.033 9.771 0.000 STABI 0.596 0.042 14.251 0.000 ACHIEVE 0.164 0.022 7.379 0.000 ADJUST 0.501 0.050 10.097 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.233E-02 (ratio of smallest to largest eigenvalue)

Note that the coefficient for **achieve** regressed on **adjust** is equal to zero,
as specified in the input file. Also note that the standard errors of parameters that
were fixed rather than estimated are listed as 0. When setting constraints, it
is worth checking both the parameter and its standard error, as well as the
model degrees of freedom to be sure the constraints were implemented as
desired.

## 2.0 Freeing a parameter

Although parameters can be fixed manually, to simplify model
input, Mplus often fixes parameters to a given value by default. By
convention, these values are typically zero or one.
For example, by default, Mplus fixes the path loading for the first observed
variable to 1 in order to identify the latent variable. To free a parameter
that is otherwise fixed by default, an asterisk (*) is placed immediately
after the parameter. In the example below, an alternative method of
identifying a latent variable in a confirmatory factor analysis is used.
Instead of using the default method of identifying the
latent variable by fixing the first path coefficient (**motiv**) to 1, we
free this parameter using **motiv*** in the line **adjust by motiv*
extra harm stabi;**. Nothing else in that line of syntax is changed. In
order to identify this model, the mean of the latent variable (**adjust**) is fixed to 0
and its variance to 1. Note that listing the name of a variable in brackets
refers to its mean, intercept, or threshold, while listing the variable name
without brackets refers to its variance or residual variance.

title: Freeing a parameter data: file is worland.dat ; variable: names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; usevariables are motiv extra harm stabi; model: adjust by motiv* extra harm stabi; [adjust@0]; adjust@1;

Below the MODEL RESULTS section of the output is shown.

<output omitted> MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value ADJUST BY MOTIV 0.901 0.041 22.031 0.000 EXTRA 0.190 0.048 3.916 0.000 HARM 0.860 0.041 20.823 0.000 STABI 0.650 0.042 15.407 0.000 MeansADJUST 0.000 0.000 999.000 999.000Intercepts MOTIV 0.000 0.045 0.000 1.000 EXTRA 0.000 0.045 0.000 1.000 HARM 0.000 0.045 0.000 1.000 STABI 0.000 0.045 0.000 1.000 VariancesADJUST 1.000 0.000 999.000 999.000Residual Variances MOTIV 0.187 0.041 4.506 0.000 EXTRA 0.962 0.061 15.693 0.000 HARM 0.259 0.040 6.499 0.000 STABI 0.575 0.041 14.055 0.000 QUALITY OF NUMERICAL RESULTS Condition Number for the Information Matrix 0.729E-01 (ratio of smallest to largest eigenvalue)

Looking at the above output, we see that the factor loading for **motiv** is
equal to .901 rather than 1, and that it has a non-zero standard error, indicating that this parameter
has been estimated rather than fixed. Looking further down, the mean of the latent variable
**adjust** is exactly zero, and a variance of exactly 1, the values specified in the input file.
Note that both parameters have a standard error of 0, which is consistant with a parameter
that has been fixed rather than freed.

## 3.0 Assigning names to parameters

Assigning names to parameters allows you to
refer them in either the **model test** or **model constraint** commands,
allowing the user to perform custom hypothesis tests or impose complex
constraints. Names are assigned to parameters by placing a name in parentheses after the
parameter is listed (similar to the way in which one assigns
constraints). Parameter names must begin with a letter; can contain
only letters, numbers, and the underscore symbol; and must be 8 characters or
less. Similar to the assignment of constraints discussed above,
the label must be the last item on a line and will be applied to all
parameters listed on that line. Putting a single label at the end of a line
with multiple parameters listed applies that label to all of the parameters
and, in the process constrains, the parameters to equality. Below we assign
the label **a1** to the estimate of the
covariance between **family** and **cog**, the label **a2** to the
covariance between **family** and **achieve**, and **a3** to the covariance between
**family** and **adjust**.

title: Assigning parameter labels data: file is worland.dat; variable: names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; model: family by ppsych ses; cog by verbal vissp mem; achieve by read arith spell; adjust by motiv extra harm stabi; family with cog (a1) achieve (a2) adjust (a3); cog with achieve adjust; achieve with adjust;

Assigning parameter labels without specifying any tests or constraints results in output that is identical to the output for this model without specifying the parameter labels. Next we will use these labels to test hypotheses about these parameters.

## 4.0 Testing hypotheses about parameter estimates using Wald tests

Mplus will allow you to test hypotheses about individual parameters (e.g., b=2),
hypotheses about groups of parameters (e.g., b1=0 and b2=0), and tests about the equality of parameters
(e.g., b1=b2). The **model test** command tells Mplus that we would like to
test a series of parameter constraints. The lines following the **model
test** command give the specific constraints to be tested. In the input
file below, three parameters are assigned names (a1, a2 and a3), then the **
model test** command is used to test the null hypothesis that **a1 = a2;**
and **a1 = a3;**. This corresponds to the null hypothesis a1 = a2 = a3;
that is, that the covariances of **family** with **cog**, **family** with **
achieve**, and **family** with **adjust** are all equal. It is
important to note that all of the constraints listed in the **model test**
command are tested simultaneously, so if the null hypothesis was a1 = a2
(without worrying about a3), this hypothesis would need to be tested separately.

title: Using Wald tests data: file is worland.dat; variable: names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; model: family by ppsych ses; cog by verbal vissp mem; achieve by read arith spell; adjust by motiv extra harm stabi; family with cog (a1) achieve (a2) adjust (a3); cog with achieve adjust; achieve with adjust; model test: a1 = a2; a1 = a3;

The abridged output associated with this model is shown below.

<output omitted> THE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 600.106 Degrees of Freedom 48 P-Value 0.0000 Chi-Square Test of Model Fit for the Baseline Model Value 4124.707 Degrees of Freedom 66 P-Value 0.0000 CFI/TLI CFI 0.864 TLI 0.813Wald Test of Parameter Constraints Value 21.395 Degrees of Freedom 2 P-Value 0.0000Loglikelihood H0 Value -6745.325 H1 Value -6445.272 Information Criteria Number of Free Parameters 42 Akaike (AIC) 13574.649 Bayesian (BIC) 13751.663 Sample-Size Adjusted BIC 13618.352 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.152 90 Percent C.I. 0.141 0.163 Probability RMSEA <= .05 0.000 SRMR (Standardized Root Mean Square Residual) Value 0.063 MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value FAMILY BY PPSYCH 1.000 0.000 999.000 999.000 SES -1.107 0.115 -9.657 0.000 COG BY VERBAL 1.000 0.000 999.000 999.000 VISSP 0.833 0.045 18.393 0.000 MEM 0.972 0.044 22.326 0.000 ACHIEVE BY READ 1.000 0.000 999.000 999.000 ARITH 0.842 0.034 24.840 0.000 SPELL 0.954 0.027 35.622 0.000 ADJUST BY MOTIV 1.000 0.000 999.000 999.000 EXTRA 0.233 0.048 4.813 0.000 HARM 0.857 0.042 20.295 0.000 STABI 0.662 0.045 14.615 0.000 FAMILY WITH COG -0.411 0.046 -8.852 0.000 ACHIEVE -0.363 0.044 -8.151 0.000 ADJUST -0.245 0.040 -6.099 0.000 <output omitted>

The output associated with the **model test** command appears
under TESTS OF MODEL FIT with the heading Wald Test of Parameter Constraints.
The test statistic (labeled Value), degrees of freedom, and p-value for the
test are given. Note that under MODEL RESULTS the estimates for the
parameters we have tested are identical to those from an input file that did
not include the **model test** command. This is because the Wald test is
performed after the model has been estimated and therefore does not influence
the parameter estimates.

As discussed above, a variety of specific null hypotheses can be tested using the
**model test** command. Below are a few examples of different hypotheses and how to test them.
One potential hypothesis is that all three of the covariances (i.e., **family**
with **cog**, **family** with **achieve**, and **family** with **adjust**) are equal to
0. The **model test** command shown below tests the
hypothesis that all three parameters (**a1**, **a2** and **a3**) are simultaneously equal
to 0. (Note that the tests of the coefficients under MODEL RESULTS test the
hypothesis that the coefficients are equal to 0 individually.)

model test: a1=0; a2=0; a3=0;

The **model test** command can also be used to test whether a
parameter (or multiple parameters if one wants to test the
parameters simultaneously) is equal to some non-zero value. The **model test**
command shown below tests the hypothesis that a1 = 1.

model test: a1=1;

## 5.0 Constraining parameters to equality

In some cases, rather than specifying that a parameter take on a
given value, you may want to constrain two parameters to be equal. This is particularly common in multiple
group models where a parameter may be fixed to equality across groups. Below
we constrain the path coefficients for **vissp** and **mem** to be the
same. We do this by placing the same number within parentheses following the
specification of the parameter (e.g., **vissp (1)**) in the **Model**
command. There are two important things to remember. First, the number in
parentheses must be the last thing on a given line (but not necessarily the
last part of the command). This is why **mem (1);** appears on a separate line.
Note that the semicolon (;) that ends the command does not appear until after
**mem (1)**. Second, when a single number is listed at the end of a line,
it constrains all the (previously free) parameters on that line to equality.
This means that if you only want to constrain one parameter in a command, that
parameter needs to be on a line by itself. The exception to this is
parameters fixed to some value in order to identify the model (e.g., fixing
the factor loading of the first variable listed after **by** to one);
these parameters will continue to be fixed at their typical value unless
they are specifically freed using an asterisk ( * ). Additional parameters could
be fixed to the same value as **vissp** and **mem**
by adding the **(1)** in the appropriate place. If we wanted to fix another pair of
variables to the same value (but not the same value as **vissp** and **
mem**) we could do this by placing a **(2)** (or other number) in the
appropriate places. Note that while we don’t cover it here, list operators
can be used to assign constraints to lists of variables.

title: Constraining parameters to equality data: file is worland.dat; variable: names are ppsych ses verbal vissp mem read arith spell motiv extra harm stabi; model: family by ppsych ses; cog by verbal vissp (1) mem (1); achieve by read arith spell; adjust by motiv extra harm stabi; achieve on cog adjust; adjust on family;

Because the a constraint applies to all parameters on the line, the following produces the same constraint.

cog by verbal vissp mem (1);

The abridged output generated by the input file above is shown below.

<output omitted> MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value FAMILY BY PPSYCH 1.000 0.000 999.000 999.000 SES -1.093 0.119 -9.167 0.000 COG BY VERBAL 1.000 0.000 999.000 999.000VISSP 0.910 0.037 24.352 0.000 MEM 0.910 0.037 24.352 0.000ACHIEVE BY READ 1.000 0.000 999.000 999.000 ARITH 0.835 0.034 24.670 0.000 SPELL 0.950 0.027 35.676 0.000 ADJUST BY MOTIV 1.000 0.000 999.000 999.000 EXTRA 0.232 0.049 4.744 0.000 HARM 0.868 0.043 20.207 0.000 STABI 0.669 0.046 14.683 0.000 <output omitted>

Note that the path coefficients and standard errors for
**vissp** and **mem** are identical, indicating that they have been
fixed to equality.

## Data source

The data for these examples is based on a correlation matrix published in Worland et. al., 1984. Although the correlation matrix would have been sufficient to specify these models, 500 cases were randomly drawn from the distribution described by the published correlation matrix. The models above do not necessarily match those specified in Worland et. al., 1984, they are intended as examples only.

Worland, Julien, David G. Weeks, Cynthia L. Janes, and Strock, Barbara D. (1984)
Intelligence, classroom behavior, and academic achievement in children at high
and low risk for psychopathology: A structural equation analysis. *Journal of Abnormal Child Psychology*
Vol. 12, No. 3, pp. 437-454.