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Could somebody please explain why this model is "just identified"

enter image description here

As I see it, there are 5 * 4 / 2 = 10 variances/covariances, 4 observed means, giving 14 available degrees of freedom

5 DF are used on the 5 paths

2 intercepts are estimated

2 error variances are estimated

2 endogenous variances are estimated

2 exogenous variances are estimated

Giving 13 used DF. What am I missing ?

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Instead of "2 endogenous variances" you should have "2 exogenous means". Also, the missing DF is the covariance between exogenous variables.

Let me elaborate:

There are k(k+3)/2 = 14 available DF as you said.

These are used as follows

  • 5 paths
  • 1 covariance between the two exogenous variables hs and col
  • 2 variances of the exogenous variables
  • 2 means for the exogenous variables
  • 2 residual variances for the endogenous variables
  • intercepts for the endogenous variables

= 14 DF used, so the model is just identified.

The point of confusion here is that, in the path diagram, there is no curved 2-headed arrow between the exogenous variables. Mplus assumes that exogenous variables are correlated. Furthermore, the estimated means and variances of these are also not shown in the output, so the output shown in the link is somewhat unhelpful for calculating the model degrees of freedom.

MODEL RESULTS

                                                Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 GRE      ON
    HS                 0.309      0.065      4.756      0.000
    COL                0.400      0.071      5.625      0.000

 GRAD     ON
    HS                 0.372      0.075      4.937      0.000
    COL                0.123      0.084      1.465      0.143
    GRE                0.369      0.078      4.754      0.000

 Intercepts
    GRE               15.534      2.995      5.186      0.000
    GRAD               6.971      3.506      1.989      0.047

 Residual Variances
    GRE               49.694      4.969     10.000      0.000
    GRAD              59.998      6.000     10.000      0.000

If we explicity add the covariance term to the model:

Model:
    gre on hs col;
    grad on hs col gre;
    hs with col;

...then the output becomes

                                                    Two-Tailed
                    Estimate       S.E.  Est./S.E.    P-Value

 GRE      ON
    HS                 0.309      0.065      4.756      0.000
    COL                0.400      0.071      5.626      0.000

 GRAD     ON
    HS                 0.372      0.075      4.937      0.000
    COL                0.123      0.084      1.465      0.143
    GRE                0.369      0.078      4.755      0.000

 HS       WITH
    COL               63.297      8.106      7.809      0.000

 Means
    HS                52.230      0.723     72.223      0.000
    COL               52.645      0.661     79.670      0.000

 Intercepts
    GRE               15.534      2.995      5.186      0.000
    GRAD               6.971      3.506      1.989      0.047

 Variances
    HS               104.597     10.460     10.000      0.000
    COL               87.329      8.733     10.000      0.000

 Residual Variances
    GRE               49.694      4.969     10.000      0.000
    GRAD              59.998      6.000     10.000      0.000

and from this it is clear that 14 parameters have been estimated, and so the model is just identified.

Note that the estimates of all the other parameters are unchanged

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