Linear Statistical Models: Regression

Fixed-effects, Random-effects
& Mixed-effects Models

Updataed for Stata 11


Fixed-effects Model

  • The levels of the variables are fixed by the researcher.
  • The results can be generalized only to the levels of the variables that appear in the design.
  • The MSWcell is the error term for the main effect F-ratios.
  • The hypotheses are concerned with treatment effects (differences between means).

    Fixed-effects Hypotheses

    A Main Effect -- H0: αj = 0 for all j;   H1: αj ≠ 0 for some j

    B Main Effect -- H0: βk = 0 for all k;   H1: βk ≠ 0 for some k

    A*B Interaction -- H0: αβjk = 0 for all jk;   H1: αβjk ≠ 0 for some jk

    Fixed-Effects Expected Mean Squares

    Fixed-effects ANOVA Summary Table

    SourceSSdfMSFError Term
    A Main effect 190.000 2 95.000 1.52MS WCell
    B Main effect1543.333 2771.66612.35MS WCell
    A*B Interaction1236.667 4309.167 4.95MS WCell
    Within Cells2250.00036 62.500
    Total5220.00044

    Strength of Association

    Omega squared

    From the example above:

    Random-effects Model

  • The levels of the variables are selected by a random process.
  • The results can be generalized to all levels of the variables that were sampled.
  • The MSA*B is the error term for the main effect F-ratios.
  • The hypotheses are concerned with variances.

    Random-effects Hypotheses

    A Main Effect -- H0: σ2α = 0;   H1: σ2α ≠ 0

    B Main Effect -- H0: σ2β = 0;   H1: σ2β ≠ 0

    A*B Interaction -- H0: σ2αβ = 0;   H1: σ2αβ ≠ 0

    Random-Effects Expected Mean Squares

    Random-effects ANOVA Summary Table

    SourceSSdfMSFError Term
    A Main effect 190.000 2 95.000   .31MS A*B
    B Main effect1543.333 2771.666 2.50MS A*B
    A*B Interaction1236.667 4309.167 4.95MS WCell
    Within Cells2250.00036 62.500
    Total5220.00044

    Strength of Association

    Partial Intraclass Correlation

    Computing variance components:

    Variance components from the example above:

    Partial intraclass correlations from the example above:

    Mixed-effects Model

  • One or more variables are fixed and one or more variables are random
  • In a design with two independent variables there are two different mixed-effects models possible:

    A fixed & B random Hypotheses

    A Main Effect -- H0: αj = 0 for all j;   H1: αj ≠ 0 for some j

    B Main Effect -- H0: σ2β = 0;   H1: σ2β ≠ 0

    A*B Interaction -- H0: σ2αβ = 0;   H1: σ2αβ ≠ 0

    A fixed & B random Expected Mean Squares

    A fixed & B random ANOVA Summary Table

    SourceSSdfMSFError Term
    A Main effect 190.000 2 95.000   .31MS A*B
    B Main effect1543.333 2771.66612.35MS WCell
    A*B Interaction1236.667 4309.167 4.95MS WCell
    Within Cells2250.00036 62.500
    Total5220.00044

    Why does the fixed variable use interaction as the error term?

    According to Hays' (2nd Ed) in a two-way design in which A (the columns) is fixed and B (the rows) is random, the levels of B represent a random sample from the population of possible levels. Hays points out that the deviation of a column mean from the grand mean includes not only column effects and mean error deviation but also includes a deviation due to interaction effect of the cell mean from the grand cell mean (cj - Mc). The reason that this is so is that there is no requirement that the interaction effects sum to zero across the rows when the row treatment effects and thus the the interaction effects are only a random sample of such effects. Thus, column means formed by summing across rows contain not only fixed column effects but also some random interaction effects.

    A random & B fixed Hypotheses

    A Main Effect -- H0: σ2α = 0;   H1: σ2α ≠ 0

    B Main Effect -- H0: βk = 0 for all k;   H1: βk ≠ 0 for some k

    A*B Interaction -- H0: σ2αβ = 0;   H1: σ2αβ ≠ 0

    A random & B fixed Expected Mean Squares

    A random & B fixed ANOVA Summary Table

    SourceSSdfMSFError Term
    A Main effect 190.000 2 95.000  1.52MS WCell
    B Main effect1543.333 2771.666 2.50MS A*B
    A*B Interaction1236.667 4309.167 4.95MS WCell
    Within Cells2250.00036 62.500
    Total5220.00044

    Using Stata

    Example problem taken from Completely Randomized Factorial Design

    use http://www.philender.com/courses/data/crf33, clear
    
    anova y a b a#b  /* fixed effects model */
    
                               Number of obs =      45     R-squared     =  0.5690
                               Root MSE      = 7.90569     Adj R-squared =  0.4732
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                       Model |        2970     8      371.25       5.94     0.0001
                             |
                           a |         190     2          95       1.52     0.2324
                           b |  1543.33333     2  771.666667      12.35     0.0001
                         a#b |  1236.66667     4  309.166667       4.95     0.0028
                             |
                    Residual |        2250    36        62.5   
                  -----------+----------------------------------------------------
                       Total |        5220    44  118.636364 
    
    anova y a b / a#b /  /* random effects model */
    
                               Number of obs =      45     R-squared     =  0.5690
                               Root MSE      = 7.90569     Adj R-squared =  0.4732
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                       Model |        2970     8      371.25       5.94     0.0001
                             |
                           a |         190     2          95       0.31     0.7514
                           b |  1543.33333     2  771.666667       2.50     0.1979
                         a#b |  1236.66667     4  309.166667   
                  -----------+----------------------------------------------------
                         a#b |  1236.66667     4  309.166667       4.95     0.0028
                             |
                    Residual |        2250    36        62.5   
                  -----------+----------------------------------------------------
                       Total |        5220    44  118.636364 
     
    anova y  a b a#b  /* mixed effects model: a fixed; b random */
    
                               Number of obs =      45     R-squared     =  0.5690
                               Root MSE      = 7.90569     Adj R-squared =  0.4732
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                       Model |        2970     8      371.25       5.94     0.0001
                             |
                           a |         190     2          95       1.52     0.2324
                           b |  1543.33333     2  771.666667      12.35     0.0001
                         a#b |  1236.66667     4  309.166667       4.95     0.0028
                             |
                    Residual |        2250    36        62.5   
                  -----------+----------------------------------------------------
                       Total |        5220    44  118.636364    
    
    test a / a#b
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                           a |         190     2          95       0.31     0.7514
                         a#b |  1236.66667     4  309.166667  
    
    anova y  a  b a#b  /* mixed effects model: b fixed; a random */
    
                               Number of obs =      45     R-squared     =  0.5690
                               Root MSE      = 7.90569     Adj R-squared =  0.4732
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                       Model |        2970     8      371.25       5.94     0.0001
                             |
                           a |         190     2          95       1.52     0.2324
                           b |  1543.33333     2  771.666667      12.35     0.0001
                         a#b |  1236.66667     4  309.166667       4.95     0.0028
                             |
                    Residual |        2250    36        62.5   
                  -----------+----------------------------------------------------
                       Total |        5220    44  118.636364
    
    test b / a#b
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                           b |  1543.33333     2  771.666667       2.50     0.1979
                         a#b |  1236.66667     4  309.166667 
    Using Orthogonal Coding

    Example problem taken from Completely Randomized Factorial Design

    input a b y x1 x2 x3 x4
    1 1 24  1  1  1  1
    1 1 33  1  1  1  1
    1 1 37  1  1  1  1
    1 1 29  1  1  1  1
    1 1 42  1  1  1  1
    1 2 44  1  1 -1  1
    1 2 36  1  1 -1  1
    1 2 25  1  1 -1  1
    1 2 27  1  1 -1  1
    1 2 43  1  1 -1  1
    1 3 38  1  1  0 -2
    1 3 29  1  1  0 -2
    1 3 28  1  1  0 -2
    1 3 47  1  1  0 -2
    1 3 48  1  1  0 -2
    2 1 30 -1  1  1  1
    2 1 21 -1  1  1  1
    2 1 39 -1  1  1  1
    2 1 26 -1  1  1  1
    2 1 34 -1  1  1  1
    2 2 35 -1  1 -1  1
    2 2 40 -1  1 -1  1
    2 2 27 -1  1 -1  1
    2 2 31 -1  1 -1  1
    2 2 22 -1  1 -1  1
    2 3 26 -1  1  0 -2
    2 3 27 -1  1  0 -2
    2 3 36 -1  1  0 -2
    2 3 46 -1  1  0 -2
    2 3 45 -1  1  0 -2
    3 1 21  0 -2  1  1
    3 1 18  0 -2  1  1
    3 1 10  0 -2  1  1
    3 1 31  0 -2  1  1
    3 1 20  0 -2  1  1
    3 2 41  0 -2 -1  1
    3 2 39  0 -2 -1  1
    3 2 50  0 -2 -1  1
    3 2 36  0 -2 -1  1
    3 2 34  0 -2 -1  1
    3 3 42  0 -2  0 -2
    3 3 52  0 -2  0 -2
    3 3 53  0 -2  0 -2
    3 3 49  0 -2  0 -2
    3 3 64  0 -2  0 -2
    end
     
    generate x5 = x1*x3
    generate x6 = x1*x4
    generate x7 = x2*x3
    generate x8 = x2*x4
     
    regress y x1 x2 x3 x4 x5 x6 x7 x8
    
      Source |       SS       df       MS                  Number of obs =      45
    ---------+------------------------------               F(  8,    36) =    5.94
       Model |     2970.00     8      371.25               Prob > F      =  0.0001
    Residual |     2250.00    36       62.50               R-squared     =  0.5690
    ---------+------------------------------               Adj R-squared =  0.4732
       Total |     5220.00    44  118.636364               Root MSE      =  7.9057
    
    ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
    ---------+--------------------------------------------------------------------
          x1 |        1.5   1.443376      1.039   0.306      -1.427302    4.427302
          x2 |  -1.166667   .8333333     -1.400   0.170      -2.856745    .5234117
          x3 |  -3.833333   1.443376     -2.656   0.012      -6.760635   -.9060318
          x4 |       -3.5   .8333333     -4.200   0.000      -5.190078   -1.809922
          x5 |       -.25   1.767767     -0.141   0.888      -3.835198    3.335198
          x6 |        .25   1.020621      0.245   0.808      -1.819915    2.319915
          x7 |   3.083333   1.020621      3.021   0.005       1.013419    5.153248
          x8 |   1.916667   .5892557      3.253   0.002       .7216008    3.111733
       _cons |         35   1.178511     29.698   0.000       32.60987    37.39013
    ------------------------------------------------------------------------------
     
    test x1 x2 /*  using residual as error */
    
     ( 1)  x1 = 0.0
     ( 2)  x2 = 0.0
    
           F(  2,    36) =    1.52
                Prob > F =    0.2324
     
    test x3 x4 /*  using residual as error */
    
     ( 1)  x3 = 0.0
     ( 2)  x4 = 0.0
    
           F(  2,    36) =   12.35
                Prob > F =    0.0001
     
    test x5 x6 x7 x8 /*  using residual as error */
    
     ( 1)  x5 = 0.0
     ( 2)  x6 = 0.0
     ( 3)  x7 = 0.0
     ( 4)  x8 = 0.0
    
           F(  4,    36) =    4.95
                Prob > F =    0.0028
    
     
    regress y x1 x2 
    
      Source |       SS       df       MS                  Number of obs =      45
    ---------+------------------------------               F(  2,    42) =    0.79
       Model |      190.00     2       95.00               Prob > F      =  0.4590
    Residual |     5030.00    42  119.761905               R-squared     =  0.0364
    ---------+------------------------------               Adj R-squared = -0.0095
       Total |     5220.00    44  118.636364               Root MSE      =  10.944
    
    ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
    ---------+--------------------------------------------------------------------
          x1 |        1.5   1.998015      0.751   0.457      -2.532157    5.532157
          x2 |  -1.166667   1.153554     -1.011   0.318      -3.494634      1.1613
       _cons |         35   1.631372     21.454   0.000       31.70776    38.29224
    ------------------------------------------------------------------------------
     
    regress y x3 x4 
    
      Source |       SS       df       MS                  Number of obs =      45
    ---------+------------------------------               F(  2,    42) =    8.82
       Model |  1543.33333     2  771.666667               Prob > F      =  0.0006
    Residual |  3676.66667    42  87.5396825               R-squared     =  0.2957
    ---------+------------------------------               Adj R-squared =  0.2621
       Total |     5220.00    44  118.636364               Root MSE      =  9.3563
    
    ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
    ---------+--------------------------------------------------------------------
          x3 |  -3.833333   1.708212     -2.244   0.030      -7.280645   -.3860213
          x4 |       -3.5   .9862369     -3.549   0.001      -5.490307   -1.509693
       _cons |         35    1.39475     25.094   0.000       32.18528    37.81472
    ------------------------------------------------------------------------------
     
    regress y x5 x6 x7 x8 
    
      Source |       SS       df       MS                  Number of obs =      45
    ---------+------------------------------               F(  4,    40) =    3.10
       Model |  1236.66667     4  309.166667               Prob > F      =  0.0257
    Residual |  3983.33333    40  99.5833333               R-squared     =  0.2369
    ---------+------------------------------               Adj R-squared =  0.1606
       Total |     5220.00    44  118.636364               Root MSE      =  9.9791
    
    ------------------------------------------------------------------------------
           y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
    ---------+--------------------------------------------------------------------
          x5 |       -.25   2.231405     -0.112   0.911      -4.759837    4.259837
          x6 |        .25   1.288302      0.194   0.847      -2.353756    2.853756
          x7 |   3.083333   1.288302      2.393   0.021       .4795777    5.687089
          x8 |   1.916667   .7438015      2.577   0.014       .4133877    3.419946
       _cons |         35   1.487603     23.528   0.000       31.99344    38.00656
    ------------------------------------------------------------------------------
     
    
    Regression Results Summarized
    
    model m0:      R-square       0.5690                            
    model m1:      R-square       0.0364
    model m2:      R-square       0.2957
    model m3:      R-square       0.2369
    

    Random-Effects F-ratios Using Regression

  • F-ratio for A*B interaction

    With 4 & 36 degrees of freedom

  • F-ratio for A main effect

    With 2 & 4 degrees of freedom

  • F-ratio for B main effect

    With 2 & 4 degrees of freedom

    Random & Mixed Effects Using test2

    The test2 command (available from ATS) can be used to perform tests with different error terms after using regress. The test2 command is an alternative to computing the F-ratio manually.

    regress y x1 x2 x3 x4 x5 x6 x7 x8
    
          Source |       SS       df       MS              Number of obs =      45
    -------------+------------------------------           F(  8,    36) =    5.94
           Model |     2970.00     8      371.25           Prob > F      =  0.0001
        Residual |     2250.00    36       62.50           R-squared     =  0.5690
    -------------+------------------------------           Adj R-squared =  0.4732
           Total |     5220.00    44  118.636364           Root MSE      =  7.9057
    
    ------------------------------------------------------------------------------
               y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
    -------------+----------------------------------------------------------------
              x1 |        1.5   1.443376     1.04   0.306    -1.427302    4.427302
              x2 |  -1.166667   .8333333    -1.40   0.170    -2.856745    .5234117
              x3 |  -3.833333   1.443376    -2.66   0.012    -6.760635   -.9060318
              x4 |       -3.5   .8333333    -4.20   0.000    -5.190078   -1.809922
              x5 |       -.25   1.767767    -0.14   0.888    -3.835198    3.335198
              x6 |        .25   1.020621     0.24   0.808    -1.819915    2.319915
              x7 |   3.083333   1.020621     3.02   0.005     1.013419    5.153248
              x8 |   1.916667   .5892557     3.25   0.002     .7216008    3.111733
           _cons |         35   1.178511    29.70   0.000     32.60987    37.39013
    ------------------------------------------------------------------------------
     
    test2 x1 x2 / x5 x6 x7 x8  /* using A*B as error */
    
    Testing: x1 x2 
    Error term: x5 x6 x7 x8
    
         F(  2,   4)  =  .31
             Prob > F = 0.7514
     
    test2 x3 x4 / x5 x6 x7 x8  /* using A*B as error */
     
    Testing: x3 x4 
    Error term: x5 x6 x7 x8
    
         F(  2,   4)  =  2.50
             Prob > F =  0.1979
     
    test x5 x6 x7 x8  /* using residual as error */
    
     ( 1)  x5 = 0.0
     ( 2)  x6 = 0.0
     ( 3)  x7 = 0.0
     ( 4)  x8 = 0.0
    
           F(  4,    36) =    4.95
                Prob > F =    0.0028


    Linear Statistical Models Course

    Phil Ender, 24sep10, 29apr06, 12Feb98