### VARIANCE Statement

Subsections:

`VARIANCE` assignment <, assignment …> ;

where assignment represents var-list < =parameter-spec>

The VARIANCE statement specifies the factor variance parameters in connection with the FACTOR statement. Notice that the VARIANCE statement is different from the VAR statement, which specifies variables for analysis. You can list factors only in the var-list of the VARIANCE statement.

In each assignment of the VARIANCE statement, you include the var-list whose variances you want to specify. Optionally, you can provide a list of parameter specifications (parameter-spec) after an equal sign for each var-list.

You can specify the following five types of the parameters for the variances of the latent factor in the VARIANCE statement:

• an unnamed free parameter

• an initial value

• a fixed value

• a free parameter with a name provided

• a free parameter with a name and initial value provided

Consider a confirmatory model that has the latent factors `F1`, `F2`, `F3`, `F4`, and `F5`.

The following VARIANCE statement illustrates the five types of parameter specifications in five assignments:

```variance
F1 ,
F2 = (.5),
F3 = 1.0,
F4 = fvar,
F5 = fvar(0.7);
```

In this statement, the variance of `F1` is specified as an unnamed free parameter. The variance of `F2` is an unnamed free parameter that has an initial value of 0.5. The variance of `F3` is a fixed value of 1.0. This value stays the same during the estimation. The variance of `F4` is a free parameter named `fvar1`. The variance of `F5` is a free parameter named `fvar2` that has an initial value of 0.7.

#### Mixed Parameter Lists

You can specify different types of parameters for the list of variances. For example, the following statement uses a list of parameters that have mixed types:

```variance
F1-F6 = vp1 vp2(2.0) vp3  4. (.3) vp6(.4);
```

This is equivalent to the following statement:

```variance
F1 = vp1
F2 = vp2(2.0),
F3 = vp3,
F4 = 4. ,
F5 = (.3),
F6 = vp6(.4);
```

As you can see, an initial value that follows a parameter name is associated with the free parameter. For example, in the original mixed list specification, the specification `(2.0)` after `vp2` is interpreted as the initial value of the parameter `vp2`, but not as the initial estimate of the variance of `F3`.

However, if you indeed want to specify that `vp2` is a free parameter without an initial value and 2.0 is an initial estimate of the variance of `F3` (while keeping all other things the same), you can use a null initial value specification for the parameter `vp2`, as shown in the following statement:

```variance
F1-F6 = vp1 vp2() (2.0)  4. (.3) vp6(.4);
```

This way, 2.0 becomes the initial estimate of the variance of `F3`. Because a, parameter list that contains mixed parameter types might be confusing, you can break down the specifications into separate assignments to remove ambiguities. For example, you can use the following equivalent statement:

```variance
F1 = vp1
F2 = vp2,
F3 = (2.),
F4 = 4. ,
F5 = (.3),
F6 = vp6(.4);
```

#### Shorter and Longer Parameter Lists

If you provide fewer parameters than the number of variances in the var-list, all the remaining parameters are treated as unnamed free parameters. For example, the following statement assigns a fixed value of 1.0 to the variance of F1 while treating the other three variances as unnamed free parameters:

```variance
F1-F4 = 1.0;
```

This statement is equivalent to the following statement:

```variance
F1 = 1.0, F2-F4;
```

If you intend to fill up all values with the last parameter specification in the list, you can use the continuation syntax `[...]`, `[..]`, or `[.]`, as shown in the following example:

```variance
F1-F100 = 1.0 psi [...];
```

This means that the variance of `F1` is fixed at 1.0 and that the variances of `F1``F100` are all free parameters named `psi`. All variances except that for `F1` are thus constrained to be equal by having the same parameter name.

However, you must be careful not to provide too many parameters. For example, the following statement results in an error:

```variance
F1-F6 = 1.0 psi2-psi6 extra;
```

The parameters after `psi6` are excessive.

#### Default Variance Parameters

In the IRT model, by default, the factor variances are fixed at ones. You can also use the VARIANCE statement to override these default variance parameters in situations where you want to specify parameter constraints, provide initial or fixed values, or make parameter references.