Ed230B/C

Linear Statistical Models

Randomized Block Factorial Designs

Updated for Stata 11


RBF-pq

Schematic with Example Data

Levelsa1a2a3
b1s1
s2
s3
s4
s5
37
42
33
29
24
39
30
34
26
21
31
21
20
18
10
b2s1
s2
s3
s4
s5
43
44
36
27
25
35
40
31
22
27
41
50
39
36
34
b3s1
s2
s3
s4
s5
48
47
29
38
28
46
36
45
27
26
64
52
53
42
49

Or in the abbreviated form,
Levela1a2a3
b1  S1
n = 5
  S1
n = 5
  S1
n = 5
b2  S1
n = 5
  S1
n = 5
  S1
n = 5
b3  S1
n = 5
  S1
n = 5
  S1
n = 5

Linear Model

Yijk = μ + αj + βk + αβjk + πi + εijk

μ = overall poulation mean
αj = the effect of treatment level j
βk = the effect of treatment level k
αβjk = the joint effects of treatment level j and k
πi = the effect of block i
εijk = experimental error

Hypotheses

  • A Main Effect

  • B Main Effect

  • A*B Interaction

  • Block (Subject) Effect

    Assumptions

    1.Independence
    2.Normality
    3.Homogeneity of variance
    4.Non-additivity
    5.Variance-covariance matrix
        a. compound symmetry
        b. circularity
        c. sphericity

    Expected Mean Squares

    E(MS a)     = σ2ε + nσ2α
    E(MS b)     = σ2ε + nσ2β
    E(MS a*b)   = σ2ε + nσs2αβ
    E(MS blks)  = σ2ε + pσ2π
    E(MS res)   = σ2ε

    ANOVA Summary Table

    SourceSSdfMSFp-valueError
    1A190.000295.000 4.79   .0152[5]
    2B1543.3332771.66638.99   .0000[5]
    3A*B1236.6674309.16715.58   .0000[5]
    4Blocks (Subjects)1615.1114403.77820.35   .0000[5]
    5Residual634.8893219.840
    Total5220.00044

    Table of the F-distribution

    Omega-Squared

    ω2A*B = (1236.6667 - 4*19.84)/(634.889 + 19.84) = 0.2209
    ω2B = (1543.333 - 2*19.84)/(634.889 + 19.84) = 0.2870
    ω2A = (1190.0 - 2*19.84)/(634.889 + 19.84) = 0.0287

    Using Stata

    input s a b y x1 x2 x3 x4 s1 s2 s3 s4
    1 1 1 37  1  1  1  1  1  1  1  1
    2 1 1 42  1  1  1  1 -1  1  1  1
    3 1 1 33  1  1  1  1  0 -2  1  1
    4 1 1 29  1  1  1  1  0  0 -3  1
    5 1 1 24  1  1  1  1  0  0  0 -4
    1 1 2 43  1  1 -1  1  1  1  1  1
    2 1 2 44  1  1 -1  1 -1  1  1  1
    3 1 2 36  1  1 -1  1  0 -2  1  1
    4 1 2 27  1  1 -1  1  0  0 -3  1
    5 1 2 25  1  1 -1  1  0  0  0 -4
    1 1 3 48  1  1  0 -2  1  1  1  1
    2 1 3 47  1  1  0 -2 -1  1  1  1
    3 1 3 29  1  1  0 -2  0 -2  1  1
    4 1 3 38  1  1  0 -2  0  0 -3  1
    5 1 3 28  1  1  0 -2  0  0  0 -4
    1 2 1 39 -1  1  1  1  1  1  1  1
    2 2 1 30 -1  1  1  1 -1  1  1  1
    3 2 1 34 -1  1  1  1  0 -2  1  1
    4 2 1 26 -1  1  1  1  0  0 -3  1
    5 2 1 21 -1  1  1  1  0  0  0 -4
    1 2 2 35 -1  1 -1  1  1  1  1  1
    2 2 2 40 -1  1 -1  1 -1  1  1  1
    3 2 2 31 -1  1 -1  1  0 -2  1  1
    4 2 2 22 -1  1 -1  1  0  0 -3  1
    5 2 2 27 -1  1 -1  1  0  0  0 -4
    1 2 3 46 -1  1  0 -2  1  1  1  1
    2 2 3 36 -1  1  0 -2 -1  1  1  1
    3 2 3 45 -1  1  0 -2  0 -2  1  1
    4 2 3 27 -1  1  0 -2  0  0 -3  1
    5 2 3 26 -1  1  0 -2  0  0  0 -4
    1 3 1 31  0 -2  1  1  1  1  1  1
    2 3 1 21  0 -2  1  1 -1  1  1  1
    3 3 1 20  0 -2  1  1  0 -2  1  1
    4 3 1 18  0 -2  1  1  0  0 -3  1
    5 3 1 10  0 -2  1  1  0  0  0 -4
    1 3 2 41  0 -2 -1  1  1  1  1  1
    2 3 2 50  0 -2 -1  1 -1  1  1  1
    3 3 2 39  0 -2 -1  1  0 -2  1  1
    4 3 2 36  0 -2 -1  1  0  0 -3  1
    5 3 2 34  0 -2 -1  1  0  0  0 -4
    1 3 3 64  0 -2  0 -2  1  1  1  1
    2 3 3 52  0 -2  0 -2 -1  1  1  1
    3 3 3 53  0 -2  0 -2  0 -2  1  1
    4 3 3 42  0 -2  0 -2  0  0 -3  1
    5 3 3 49  0 -2  0 -2  0  0  0 -4
    end
    
    table a, cont(freq mean y sd y) by(b)
    
    ----------+-----------------------------------
      b and a |      Freq.     mean(y)       sd(y)
    ----------+-----------------------------------
    1         |
            1 |          5          33    6.964194
            2 |          5          30    6.964194
            3 |          5          20    7.516648
    ----------+-----------------------------------
    2         |
            1 |          5          35    8.803409
            2 |          5          31    6.964194
            3 |          5          40    6.204837
    ----------+-----------------------------------
    3         |
            1 |          5          38    9.513149
            2 |          5          36    9.513149
            3 |          5          52    7.968688
    ----------+-----------------------------------
    
    histogram y, by(a b) normal
    
    
    
    anova y a b a#b s, repeated(a b)
    
    
                               Number of obs =      45     R-squared     =  0.8784
                               Root MSE      = 4.45424     Adj R-squared =  0.8328
    
                      Source |  Partial SS    df       MS           F     Prob > F
                  -----------+----------------------------------------------------
                       Model |  4585.11111    12  382.092593      19.26     0.0000
                             |
                           a |         190     2          95       4.79     0.0152
                           b |  1543.33333     2  771.666667      38.89     0.0000
                         a#b |  1236.66667     4  309.166667      15.58     0.0000
                           s |  1615.11111     4  403.777778      20.35     0.0000
                             |
                    Residual |  634.888889    32  19.8402778   
                  -----------+----------------------------------------------------
                       Total |        5220    44  118.636364   
    
    
    Between-subjects error term:  s
                         Levels:  5         (4 df)
         Lowest b.s.e. variable:  s
    
    Repeated variable: a
                                              Huynh-Feldt epsilon        =  0.7892
                                              Greenhouse-Geisser epsilon =  0.6319
                                              Box's conservative epsilon =  0.5000
    
                                                ------------ Prob > F ------------
                      Source |     df      F    Regular    H-F      G-G      Box
                  -----------+----------------------------------------------------
                           a |      2     4.79   0.0152   0.0237   0.0331   0.0438
                    Residual |     32
                  ----------------------------------------------------------------
    
    Repeated variable: b
                                              Huynh-Feldt epsilon        =  0.9493
                                              Greenhouse-Geisser epsilon =  0.6954
                                              Box's conservative epsilon =  0.5000
    
                                                ------------ Prob > F ------------
                      Source |     df      F    Regular    H-F      G-G      Box
                  -----------+----------------------------------------------------
                           b |      2    38.89   0.0000   0.0000   0.0000   0.0000
                    Residual |     32
                  ----------------------------------------------------------------
    
    Repeated variables: a#b
                                              Huynh-Feldt epsilon        =  1.4947
                                              *Huynh-Feldt epsilon reset to 1.0000
                                              Greenhouse-Geisser epsilon =  0.5901
                                              Box's conservative epsilon =  0.2500
    
                                                ------------ Prob > F ------------
                      Source |     df      F    Regular    H-F      G-G      Box
                  -----------+----------------------------------------------------
                         a#b |      4    15.58   0.0000   0.0000   0.0001   0.0043
                    Residual |     32
                  ----------------------------------------------------------------
    
            
    effectsize a#b
    
    anova effect size for a#b with dep var = y
    
    total variance accounted for
    omega2         = .22086657
    eta2           = .23690932
    Cohen's f      = .53242578
    
    partial variance accounted for
    partial omega2 = .56450679
    partial eta2   = .66076941
    
    effectsize b
    
    anova effect size for b with dep var = y
    
    total variance accounted for
    omega2         = .28696538
    eta2           = .29565773
    Cohen's f      = .63439456
    
    partial variance accounted for
    partial omega2 = .62744609
    partial eta2   = .70852887
    
    effectsize a
    
    anova effect size for a with dep var = y
    
    total variance accounted for
    omega2         = .02868779
    eta2           = .03639847
    Cohen's f      = .17185775
    
    partial variance accounted for
    partial omega2 = .14410396
    partial eta2   = .23033405
     
    quietly anova y a b a#b  /* run without s to get a plot of cell means */
     
    anovaplot b a, scatter(msym(none))
     
    
     
    quietly anova y a b a#b s     /* rerun the original anova to get correct mse */
    
    sme a b
     
    Test of a at b(1): F(2/32)  = 11.676584
    Test of a at b(2): F(2/32)  = 5.1242562
    Test of a at b(3): F(2/32)  = 19.152958
    
    
    Critical value of F for alpha = .05 using ...
    --------------------------------------------------
    Dunn's procedure              = 4.1487813
    Marascuilo & Levin            = 4.6658516
    per family error rate         = 4.6658516
    simultaneous test procedure   = 9.4358325
    
    anovalator a b, simple fratio
    
    anovalator test of simple main effects for a at(b=1) 
    chi2(2) = 23.353168   p-value = 8.490e-06
    scaled as F-ratio = 11.676584
    
    anovalator test of simple main effects for a at(b=2) 
    chi2(2) = 10.248512   p-value = .00595064
    scaled as F-ratio = 5.1242562
    
    anovalator test of simple main effects for a at(b=3) 
    chi2(2) = 38.305915   p-value = 4.808e-09
    scaled as F-ratio = 19.152958
    
    smecriticalvalue, num(3) df1(2) df2(32) dfm(12)
    
      number of tests: 3
         numerator df: 2
       denominator df: 32
    original model df: 12
    
    Critical value of F for alpha = .05 using ...
    ------------------------------------------------
    Dunn's procedure              = 5.0408416
    Marascuilo & Levin            = 5.5808631
    per family error rate         = 4.6659053
    simultaneous test procedure   = 9.435821
    
    tkcomp a if b==1
    
    Tukey-Kramer pairwise comparisons for variable a
    studentized range critical value(.05, 3, 32) = 3.4754008
    
                                          mean 
    grp vs grp       group means          dif     TK-test
    -------------------------------------------------------
      1 vs   2    33.0000    30.0000      3.0000   1.5060 
      1 vs   3    33.0000    20.0000     13.0000   6.5261*
      2 vs   3    30.0000    20.0000     10.0000   5.0201*
    
    tkcomp a if b==3
    
    Tukey-Kramer pairwise comparisons for variable a
    studentized range critical value(.05, 3, 32) = 3.4754008
    
                                          mean 
    grp vs grp       group means          dif     TK-test
    -------------------------------------------------------
      1 vs   2    38.0000    36.0000      2.0000   1.0040 
      1 vs   3    38.0000    52.0000     14.0000   7.0281*
      2 vs   3    36.0000    52.0000     16.0000   8.0321*
      
    anovalator a, pair at(b=1) quietly
    
    anovalator pairwise comparisons for a at(b=1) 
    
    Comparison          Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
    1 vs 2                  3    2.81711     1.06   0.287    -2.521536    8.521536
    1 vs 3                 13    2.81711     4.61   0.000     7.478464    18.52154
    2 vs 3                 10    2.81711     3.55   0.000     4.478464    15.52154
    
    anovalator a, pair at(b=3) quietly
    
    anovalator pairwise comparisons for a at(b=3) 
    
    Comparison          Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
    1 vs 2                  2    2.81711      .71   0.478    -3.521536    7.521536
    1 vs 3                -14    2.81711    -4.97   0.000    -19.52154   -8.478464
    2 vs 3                -16    2.81711    -5.68   0.000    -21.52154   -10.47846
           
    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 s1 s2 s3 s4
    
    
          Source |       SS       df       MS              Number of obs =      45
    -------------+------------------------------           F( 12,    32) =   19.26
           Model |  4585.11111    12  382.092593           Prob > F      =  0.0000
        Residual |  634.888889    32  19.8402778           R-squared     =  0.8784
    -------------+------------------------------           Adj R-squared =  0.8328
           Total |        5220    44  118.636364           Root MSE      =  4.4542
    
    ------------------------------------------------------------------------------
               y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
    -------------+----------------------------------------------------------------
              x1 |        1.5   .8132297     1.84   0.074    -.1564948    3.156495
              x2 |  -1.166667   .4695184    -2.48   0.018    -2.123044    -.210289
              x3 |  -3.833333   .8132297    -4.71   0.000    -5.489828   -2.176839
              x4 |       -3.5   .4695184    -7.45   0.000    -4.456378   -2.543622
              x5 |       -.25   .9959989    -0.25   0.803    -2.278783    1.778783
              x6 |        .25   .5750403     0.43   0.667    -.9213187    1.421319
              x7 |   3.083333   .5750403     5.36   0.000     1.912015    4.254652
              x8 |   1.916667   .3319996     5.77   0.000     1.240406    2.592928
              s1 |   1.222222   1.049875     1.16   0.253    -.9163033    3.360748
              s2 |   1.962963   .6061457     3.24   0.003     .7282847    3.197641
              s3 |   2.509259   .4286097     5.85   0.000      1.63621    3.382309
              s4 |   1.972222   .3319996     5.94   0.000     1.295961    2.648483
           _cons |         35   .6639993    52.71   0.000     33.64748    36.35252
    ------------------------------------------------------------------------------
    
    test x1 x2
    
     ( 1)  x1 = 0
     ( 2)  x2 = 0
    
           F(  2,    32) =    4.79
                Prob > F =    0.0152
    
    test x3 x4
    
     ( 1)  x3 = 0
     ( 2)  x4 = 0
    
           F(  2,    32) =   38.89
                Prob > F =    0.0000
    
    test x5 x6 x7 x8
    
     ( 1)  x5 = 0
     ( 2)  x6 = 0
     ( 3)  x7 = 0
     ( 4)  x8 = 0
    
           F(  4,    32) =   15.58
                Prob > F =    0.0000
    
    test s1 s2 s3 s4
    
     ( 1)  s1 = 0
     ( 2)  s2 = 0
     ( 3)  s3 = 0
     ( 4)  s4 = 0
    
           F(  4,    32) =   20.35
                Prob > F =    0.0000
    
    regress y i.a##i.b i.s
    
          Source |       SS       df       MS              Number of obs =      45
    -------------+------------------------------           F( 12,    32) =   19.26
           Model |  4585.11111    12  382.092593           Prob > F      =  0.0000
        Residual |  634.888889    32  19.8402778           R-squared     =  0.8784
    -------------+------------------------------           Adj R-squared =  0.8328
           Total |        5220    44  118.636364           Root MSE      =  4.4542
    
    ------------------------------------------------------------------------------
               y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
    -------------+----------------------------------------------------------------
               a |
              2  |         -3    2.81711    -1.06   0.295    -8.738266    2.738266
              3  |        -13    2.81711    -4.61   0.000    -18.73827   -7.261734
                 |
               b |
              2  |          2    2.81711     0.71   0.483    -3.738266    7.738266
              3  |          5    2.81711     1.77   0.085    -.7382661    10.73827
                 |
             a#b |
            2 2  |         -1   3.983996    -0.25   0.803    -9.115134    7.115134
            2 3  |          1   3.983996     0.25   0.803    -7.115134    9.115134
            3 2  |         18   3.983996     4.52   0.000     9.884866    26.11513
            3 3  |         27   3.983996     6.78   0.000     18.88487    35.11513
                 |
               s |
              2  |  -2.444444    2.09975    -1.16   0.253    -6.721496    1.832607
              3  |  -7.111111    2.09975    -3.39   0.002    -11.38816    -2.83406
              4  |  -13.22222    2.09975    -6.30   0.000    -17.49927   -8.945171
              5  |  -15.55556    2.09975    -7.41   0.000    -19.83261    -11.2785
                 |
           _cons |   40.66667   2.394083    16.99   0.000     35.79008    45.54326
    ------------------------------------------------------------------------------
    
    anovalator a b, main 2way fratio
    
    anovalator main-effect for a  
    chi2(2) = 9.5764788   p-value = .00832711
    scaled as F-ratio = 4.7882394
    
    anovalator main-effect for b  
    chi2(2) = 77.787889   p-value = 1.284e-17
    scaled as F-ratio = 38.893945
    
    anovalator two-way interaction for a#b  
    chi2(4) = 62.331117   p-value = 9.383e-13
    scaled as F-ratio = 15.582779
    
    anovalator s, main fratio
    
    anovalator main-effect for s  
    chi2(4) = 81.40567   p-value = 8.773e-17
    scaled as F-ratio = 20.351418


    Linear Statistical Models Course

    Phil Ender, 17sep10, 25apr06, 12Feb98