Balanced Multi-Way Mixed-Effect Model

Unrestricted Model

Setup

Do StatANOVA Balanced ANOVA:

(turbine01.png here)
(turbine02.png here)

Session Output

ANOVA: drop versus temperature, pressure, speed 

Factor       Type    Levels  Values
temperature  fixed        3  60, 75, 90
pressure     random       3  50, 75, 85
speed        random       4  150, 200, 225, 300


Analysis of Variance for drop

Source                      DF       SS      MS      F      P
temperature                  2  1023.36  511.68   2.30  0.171 x
pressure                     2     7.19    3.60   0.06  0.938 x
speed                        3   423.82  141.27   0.63  0.616 x
temperature*pressure         4   137.89   34.47   2.49  0.099
temperature*speed            6  1211.97  202.00  14.59  0.000
pressure*speed               6   209.47   34.91   2.52  0.081
temperature*pressure*speed  12   166.11   13.84   0.65  0.788
Error                       36   770.50   21.40
Total                       71  3950.32

x Not an exact F-test.


S = 4.62631   R-Sq = 80.50%   R-Sq(adj) = 61.53%

                                               Expected Mean Square for Each
                              Variance  Error  Term (using unrestricted
  Source                     component   term  model)
1 temperature                               *  (8) + 2 (7) + 6 (5) + 8 (4) +
                                               Q[1]
2 pressure                      -2.164      *  (8) + 2 (7) + 6 (6) + 8 (4) +
                                               24 (2)
3 speed                         -4.544      *  (8) + 2 (7) + 6 (6) + 6 (5) +
                                               18 (3)
4 temperature*pressure           2.579      7  (8) + 2 (7) + 8 (4)
5 temperature*speed             31.359      7  (8) + 2 (7) + 6 (5)
6 pressure*speed                 3.512      7  (8) + 2 (7) + 6 (6)
7 temperature*pressure*speed    -3.780      8  (8) + 2 (7)
8 Error                         21.403         (8)


* Synthesized Test.


Error Terms for Synthesized Tests

                                   Synthesis of
Source         Error DF  Error MS  Error MS
1 temperature      6.97    222.63  (4) + (5) - (7)
2 pressure         5.98     55.54  (4) + (6) - (7)
3 speed            7.09    223.06  (5) + (6) - (7)


Means

temperature   N     drop
60           24  -2.6250
75           24   5.6667
90           24  -2.0000

The following R session shows an example of approx F test using Satterthwaite approximation. The same can be accomplished in MINITAB using CalcCalculator and CalcProbability Distributions. Note that all calculations are based on the above MINITAB outputs (with MS's already rounded off). The example shows the approx F test for the fixed effect temp. The result is quite comparable to the reported F and P value in MINITAB output.

R Session Output

> MS <- c(A=511.68, AB=34.47, AC=202.00, ABC=13.84)
> DF <- c(A=2, AB=2*2, AC=2*3, ABC=2*2*3)
> dof <- function(ms, d) sum(ms)^2 / sum(ms^2/d)
> dof(MS[c("A","ABC")], DF[c("A","ABC")]) -> r1
> r1   # numerator d.o.f.
[1] 2.109399
> r2 <- dof(MS[c("AB","AC")], DF[c("AB","AC")])
> r2   # denominator d.o.f.
[1] 7.878322
> FA <- sum(MS[c("A","ABC")]) / sum(MS[c("AB","AC")])
> FA   # approx F statistic
[1] 2.222354
> pf(FA, r1, r2, lower=FALSE)   # P-val using approx F
[1] 0.1708550

Restricted Model

Setup

Do as in unrestricted model except for Options. Click Options button and select Restricted Model:

(turbine03.png here)

Session Output

ANOVA: drop versus temperature, pressure, speed 

Factor       Type    Levels  Values
temperature  fixed        3  60, 75, 90
pressure     random       3  50, 75, 85
speed        random       4  150, 200, 225, 300


Analysis of Variance for drop

Source                      DF       SS      MS      F      P
temperature                  2  1023.36  511.68   2.30  0.171 x
pressure                     2     7.19    3.60   0.10  0.904
speed                        3   423.82  141.27   4.05  0.069
temperature*pressure         4   137.89   34.47   2.49  0.099
temperature*speed            6  1211.97  202.00  14.59  0.000
pressure*speed               6   209.47   34.91   1.63  0.167
temperature*pressure*speed  12   166.11   13.84   0.65  0.788
Error                       36   770.50   21.40
Total                       71  3950.32

x Not an exact F-test.


S = 4.62631   R-Sq = 80.50%   R-Sq(adj) = 61.53%


                                Variance  Error  Expected Mean Square for Each
   Source                      component   term  Term (using restricted model)
1  temperature                                *  (8) + 2 (7) + 6 (5) + 8 (4) +
                                                 24 Q[1]
2  pressure                       -1.305      6  (8) + 6 (6) + 24 (2)
3  speed                           5.909      6  (8) + 6 (6) + 18 (3)
4  temperature*pressure            2.579      7  (8) + 2 (7) + 8 (4)
5  temperature*speed              31.359      7  (8) + 2 (7) + 6 (5)
6  pressure*speed                  2.252      8  (8) + 6 (6)
7  temperature*pressure*speed     -3.780      8  (8) + 2 (7)
8  Error                          21.403         (8)


* Synthesized Test.

Error Terms for Synthesized Tests
                                   Synthesis of
Source         Error DF  Error MS  Error MS
1 temperature      6.97    222.63  (4) + (5) - (7)

Means

temperature   N     drop
60           24  -2.6250
75           24   5.6667
90           24  -2.0000