Please note: The following example is for illustrative purposes only. The data presented is not meant to recommend or encourage the estimation of random effects on categorical variables with very few unique levels.

Consider the following nested experiment: A study was conducted measuring the thickness of the oxide layer on silicon wafers. The wafers were produced on two different machines (source). Four lots of wafers were selected at random from each machine. From each lot three wafers were selected at random to be measured. Finally, on each wafer three positions were selected. So, we have position nested in wafer, wafer nested in lot which is nested in source. The primary concern of this experiment is to
determine whether the two machines (source) differ in the thickness of their oxide layers.

Let’s load the data and look at our sample.

use https://stats.idre.ucla.edu/stat/data/thickness, clearlist in 1/10

     +--------------------------------------------+
     | source   lot   wafer   position   thickn~s |
     |--------------------------------------------|
  1. |      1     1       1          1       2006 |
  2. |      1     1       1          2       1999 |
  3. |      1     1       1          3       2007 |
  4. |      1     1       2          1       1980 |
  5. |      1     1       2          2       1988 |
     |--------------------------------------------|
  6. |      1     1       2          3       1982 |
  7. |      1     1       3          1       2000 |
  8. |      1     1       3          2       1998 |
  9. |      1     1       3          3       2007 |
 10. |      1     2       1          1       1991 |
     +--------------------------------------------+ 

tabstat thickness, by(source) stat(n mean sd)

Summary for variables: thickness
     by categories of: source 

  source |         N      mean        sd
---------+------------------------------
       1 |        36  1995.111  7.531943
       2 |        36  2005.194  14.86668
---------+------------------------------
   Total |        72  2000.153  12.75518
----------------------------------------

Next, we will need to create a variable that indicates lot nested in source. We
will do this using the egen command with thegroup function.

egen lotinsource = group(lot source), labeltab lotinsource

   group(lot |
    source) |      Freq.     Percent        Cum.
------------+-----------------------------------
        1 1 |          9       12.50       12.50
        1 2 |          9       12.50       25.00
        2 1 |          9       12.50       37.50
        2 2 |          9       12.50       50.00
        3 1 |          9       12.50       62.50
        3 2 |          9       12.50       75.00
        4 1 |          9       12.50       87.50
        4 2 |          9       12.50      100.00
------------+-----------------------------------
      Total |         72      100.00

From the table above it looks lot is crossed withsource. This is not the case since a lot
drawn from source1 is a different from a lot that is drawn from source2. Fortunately,
mixed will be able to sort this out for us. Here is one way to parameterize this model.

mixed thickness i.source || lotinsource: || wafer:, var

Performing gradient-based optimization: 

Iteration 0:   log likelihood = -228.43197  
Iteration 1:   log likelihood = -228.43197  

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =         72

-------------------------------------------------------------
                |     No. of       Observations per Group
 Group Variable |     Groups    Minimum    Average    Maximum
----------------+--------------------------------------------
    lotinsource |          8          9        9.0          9
          wafer |         24          3        3.0          3
-------------------------------------------------------------

                                                Wald chi2(1)      =       2.03
Log likelihood = -228.43197                     Prob > chi2       =     0.1537

------------------------------------------------------------------------------
   thickness |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    2.source |   10.08333   7.068711     1.43   0.154    -3.771085    23.93775
       _cons |   1995.111   4.998333   399.16   0.000     1985.315    2004.908
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
lotinsource: Identity        |
                  var(_cons) |   86.58149    50.1892      27.79739    269.6783
-----------------------------+------------------------------------------------
wafer: Identity              |
                  var(_cons) |   35.86577   14.18759      16.51834    77.87428
-----------------------------+------------------------------------------------
               var(Residual) |   12.56944   2.565726      8.424908    18.75282
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 100.65                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

Note that the test for differences in source is not significant. Also, note that the variable position does not appear in the model. That’s because variability due to position is accounted for by the residual variance. In the output above, lots nested
in source (lotinsource) has a variance of 86.58, wafer has a variance of 35.87 and position
(residual) has a variance of 12.57.

There is an alternative way to parameterize this model that is somewhat more efficient.

mixed thickness i.source || lotinsource: || _all: R.wafer, var

Performing EM optimization: 

Performing gradient-based optimization: 

Iteration 0:   log likelihood = -228.43197  
Iteration 1:   log likelihood = -228.43197  

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =         72

-------------------------------------------------------------
                |     No. of       Observations per Group
 Group Variable |     Groups    Minimum    Average    Maximum
----------------+--------------------------------------------
    lotinsource |          8          9        9.0          9
           _all |          8          9        9.0          9
-------------------------------------------------------------

                                                Wald chi2(1)      =       2.03
Log likelihood = -228.43197                     Prob > chi2       =     0.1537

------------------------------------------------------------------------------
   thickness |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    2.source |   10.08333   7.068711     1.43   0.154    -3.771085    23.93775
       _cons |   1995.111   4.998333   399.16   0.000     1985.315    2004.908
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
lotinsource: Identity        |
                  var(_cons) |   86.58149    50.1892      27.79739    269.6783
-----------------------------+------------------------------------------------
_all: Identity               |
                var(R.wafer) |   35.86577   14.18759      16.51834    77.87427
-----------------------------+------------------------------------------------
               var(Residual) |   12.56944   2.565726      8.424908    18.75282
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 100.65                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

All of the results as the same as in our first model, however some of the labels for the
variance components differ.

This design is completely balanced so the mixed results will be identical to those
using the anova command.

anova thickness source / lot|source wafer|lot|source

                           Number of obs =      72     R-squared     =  0.9478
                           Root MSE      = 3.54534     Adj R-squared =  0.9227

                  Source |  Partial SS    df       MS           F     Prob > F
        -----------------+----------------------------------------------------
                   Model |  10947.9861    23  475.999396      37.87     0.0000
                         |
                  source |    1830.125     1    1830.125       1.53     0.2629
              lot|source |  7195.19444     6  1199.19907   
        -----------------+----------------------------------------------------
        wafer|lot|source |  1922.66667    16  120.166667       9.56     0.0000
                         |
                Residual |  603.333333    48  12.5694444   
        -----------------+----------------------------------------------------
                   Total |  11551.3194    71   162.69464

For more information on nested models see: “Multilevel and Longitudinal Modeling Using Stata” by Sophia Rabe-Hesketh and Anders Skrondal (2012)