Mplus version 7.4 was used for these examples.
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.
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_data.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 0.000 ADJUST 0.000 0.000 999.000 999.000 ADJUST 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: worland_data.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 Means ADJUST 0.000 0.000 999.000 999.000 Intercepts 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 ADJUST 1.000 0.000 999.000 999.000 Residual 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_data.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.
5.0 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_data.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 0.813 Wald Test of Parameter Constraints Value 21.395 Degrees of Freedom 2 P-Value 0.0000 Loglikelihood 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;
4.0 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_data.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 0.000 999.000 999.000 VISSP 0.910 0.037 24.352 0.000 MEM 0.910 0.037 24.352 0.000 ACHIEVE 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.
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.