# Three-way MANOVA Interpretation

I'm interpreting a three-way MANOVA as part of my research, and have come across an interpretation challenge, with three independent variables (Factor A, Factor B, and Factor C) and two dependent variables (Outcome 1, Outcome 2). My multivariate tests indicate a significant interaction (Factor A x Factor B) and a significant main effect (Factor B). Tests of between-subjects reveal that the interaction effect is significant for only one of the dependent variables (Outcome 1) but not the other (Outcome 2). The main effect of Factor B, however, is significant for Outcome 2 only.

I take this to mean that Outcome 1 is impacted by the interaction of Factor A and B, while Outcome 2 is only impacted by Factor B. However, I know that in an ANOVA, if there is a significant interaction, we should ignore any main effects. Is this also true for a MANOVA? Should I be ignoring the main effect of Factor B on Outcome 2?

While you may receive this reply very late, but I saw it now. About your question, I did it on a part of minitab datasets i.e. Exh_mvar dataset and created a trial with 3 factors as A, B, C in 2 replications (R).

The dataset and the outputs are brought about here. Shortly in my opinion, generally a similar trend is expected in both anova and manova.
So you could decide as anova for manova. i.e. when A*B effect is significant, we may ignore the significance of main factors (A or B) and so on.
And when interactions (either in anova or in manova) are significant, we can ignore the main effets, and vice versa, i.e. when a main effects is significant and the interaction is nonsign., then we focus on main effect. I wish I have told you the correct things. Because I have not seen any reports regarding multi-level manova.

data and outputs:

Data Display

Row  A  B  C  R  HeatFlux  Insolation
1  1  1  1  1     271.8      783.35
2  1  1  1  2     264.0      748.45
3  1  1  2  1     238.8      684.45
4  1  1  2  2     230.7      827.80
5  1  2  1  1     251.6      860.45
6  1  2  1  2     257.9      875.15
7  1  2  2  1     263.9      909.45
8  1  2  2  2     266.5      905.55
9  2  1  1  1     229.1      756.00
10  2  1  1  2     239.3      769.35
11  2  1  2  1     258.0      793.50
12  2  1  2  2     257.6      801.65
13  2  2  1  1     267.3      819.65
14  2  2  1  2     267.0      808.55
15  2  2  2  1     259.6      774.95
16  2  2  2  2     240.4      711.85


Analysis of Variance for HeatFlux, using Adjusted SS for Tests

Source  DF   Seq SS   Adj SS   Adj MS      F      P
R        1    17.43    17.43    17.43   0.40  0.547
A        1    45.23    45.23    45.23   1.04  0.343
B        1   450.50   450.50   450.50  10.32  0.015
C        1    66.02    66.02    66.02   1.51  0.258
A*B      1    15.41    15.41    15.41   0.35  0.571
A*C      1   212.43   212.43   212.43   4.87  0.063
B*C      1     2.03     2.03     2.03   0.05  0.835
A*B*C    1  1778.73  1778.73  1778.73  40.76  0.000
Error    7   305.48   305.48    43.64
Total   15  2893.25

S = 6.60611   R-Sq = 89.44%   R-Sq(adj) = 77.37%

Analysis of Variance for Insolation, using Adjusted SS for Tests

R        1     277     277     277  0.15  0.710
A        1    8062    8062    8062  4.38  0.075
B        1   15691   15691   15691  8.52  0.022
C        1       9       9       9  0.00  0.947
A*B      1   16387   16387   16387  8.89  0.020
A*C      1    1080    1080    1080  0.59  0.469
B*C      1     788     788     788  0.43  0.534
A*B*C    1    6012    6012    6012  3.26  0.114
Error    7   12897   12897    1842
Total   15   61202

S = 42.9238   R-Sq = 78.93%   R-Sq(adj) = 54.84%


Unusual Observations for Insolation

Obs  Insolation      Fit  SE Fit  Residual  St Resid
3     684.450  751.966  32.193   -67.516     -2.38 R
4     827.800  760.284  32.193    67.516      2.38 R


R denotes an observation with a large standardized residual.

MANOVA for A
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.59203  2.067    2      6  0.208
Lawley-Hotelling    0.68911  2.067    2      6  0.208
Pillai's            0.40797  2.067    2      6  0.208
Roy's               0.68911

HeatFlux  Insolation
HeatFlux       45.23       603.8
Insolation    603.82      8061.8

HeatFlux  Insolation
HeatFlux       305.5       340.1
Insolation     340.1     12897.2

Partial Correlations for the Error SSCP Matrix

HeatFlux  Insolation
HeatFlux     1.00000     0.17135
Insolation   0.17135     1.00000

EIGEN Analysis for A

Eigenvalue  0.6891  0.00000
Proportion  1.0000  0.00000
Cumulative  1.0000  1.00000

Eigenvector         1          2
HeatFlux     0.017702   0.055310
Insolation   0.007920  -0.004143

MANOVA for B
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.30305  6.899    2      6  0.028
Lawley-Hotelling    2.29980  6.899    2      6  0.028
Pillai's            0.69695  6.899    2      6  0.028
Roy's               2.29980

HeatFlux  Insolation
HeatFlux       450.5        2659
Insolation    2658.7       15691

EIGEN Analysis for B

Eigenvalue  2.300  0.00000
Proportion  1.000  0.00000
Cumulative  1.000  1.00000

Eigenvector         1         2
HeatFlux     0.039855  -0.04224
Insolation   0.005353   0.00716

MANOVA for C
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.82029  0.657    2      6  0.552
Lawley-Hotelling    0.21908  0.657    2      6  0.552
Pillai's            0.17971  0.657    2      6  0.552
Roy's               0.21908

HeatFlux  Insolation
HeatFlux       66.02      23.867
Insolation     23.87       8.629

EIGEN Analysis for C

Eigenvalue  0.2191  0.00000
Proportion  1.0000  0.00000
Cumulative  1.0000  1.00000

Eigenvector          1          2
HeatFlux      0.057985  -0.003209
Insolation   -0.001043   0.008877

MANOVA for A*B
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.40810  4.351    2      6  0.068
Lawley-Hotelling    1.45037  4.351    2      6  0.068
Pillai's            0.59190  4.351    2      6  0.068
Roy's               1.45037

HeatFlux  Insolation
HeatFlux        15.4      -502.4
Insolation    -502.4     16387.2

EIGEN Analysis for A*B

Eigenvalue  1.450  0.00000
Proportion  1.000  0.00000
Cumulative  1.000  1.00000

Eigenvector         1        2
HeatFlux     -0.02045  0.05436
Insolation    0.00878  0.00167

MANOVA for A*C
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.52969  2.664    2      6  0.149
Lawley-Hotelling    0.88789  2.664    2      6  0.149
Pillai's            0.47031  2.664    2      6  0.149
Roy's               0.88789

HeatFlux  Insolation
HeatFlux       212.4      -479.0
Insolation    -479.0      1079.9

EIGEN Analysis for A*C

Eigenvalue  0.8879  0.00000
Proportion  1.0000  0.00000
Cumulative  1.0000  1.00000

Eigenvector          1         2
HeatFlux      0.055267  0.017834
Insolation   -0.004162  0.007910

MANOVA for B*C
s = 1    m = 0.0    n = 2.0

Test             DF
Criterion         Statistic      F  Num  Denom      P
Wilks'              0.92862  0.231    2      6  0.801
Lawley-Hotelling    0.07687  0.231    2      6  0.801
Pillai's            0.07138  0.231    2      6  0.801
Roy's               0.07687

HeatFlux  Insolation
HeatFlux        2.03      -39.99
Insolation    -39.99      787.50

EIGEN Analysis for B*C

Eigenvalue  0.07687  0.00000
Proportion  1.00000  0.00000
Cumulative  1.00000  1.00000

Eigenvector         1        2
HeatFlux     -0.02634  0.05176
Insolation    0.00854  0.00263

MANOVA for A*B*C
s = 1    m = 0.0    n = 2.0

Test              DF
Criterion         Statistic       F  Num  Denom      P
Wilks'              0.14498  17.692    2      6  0.003
Lawley-Hotelling    5.89736  17.692    2      6  0.003
Pillai's            0.85502  17.692    2      6  0.003
Roy's               5.89736

HeatFlux  Insolation
HeatFlux        1779        3270
Insolation      3270        6012

EIGEN Analysis for A*B*C

Eigenvalue  5.897  0.00000
Proportion  1.000  0.00000
Cumulative  1.000  1.00000

Eigenvector         1         2
HeatFlux     0.055731  -0.01633
Insolation   0.001006   0.00888