Consider the Design
SSCP Matrices
Tests of Significance
Wilks' Lambda
where Se represents the error SSCP matrix and Sh represents the hypothesis SSCP matrix.
For Example
In a fixed effects model, Sw is the Se for all effects.
While in the randoms effects model Sab is the Se for the main effects and Sw for the interaction.
If A is fixed and B is random th Sab is the Se for A main effect and Sw is the Se for the B main effect and the interaction.
Rao's F Approximation
Degreees of Freedom
Special Note Concerning s
If either the numerator or the deminator of s equals 0 set s = 1.
Other Test Criteria
Hotelling's Trace Criterion
Roy's Largest Latent Root
Pillai's Trace Criterion
Which of these is "best?"
Stata Manova Example
use http://www.gseis.ucla.edu/courses/data/hsb2
table prog female, cont(freq mean read mean write mean math)
------------------------------
type of | female
program | male female
----------+-------------------
general | 21 24
| 52.95238 46.95833
| 49.14286 53.25
| 50.19048 49.875
|
academic | 47 58
| 56.2766 56.06897
| 54.61702 57.58621
| 57.12766 56.41379
|
vocation | 23 27
| 45.65217 46.66667
| 41.82609 50.96296
| 46.91304 46
------------------------------
corr read write math
(obs=200)
| read write math
-------------+---------------------------
read | 1.0000
write | 0.5968 1.0000
math | 0.6623 0.6174 1.0000
manova read write math = prog female prog*female
Number of obs = 200
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
------------+--------------------------------------------------
Model | W 0.5808 5 15.0 530.4 7.69 0.0000 a
| P 0.4796 15.0 582.0 7.38 0.0000 a
| L 0.6206 15.0 572.0 7.89 0.0000 a
| R 0.3762 5.0 194.0 14.59 0.0000 u
|--------------------------------------------------
Residual | 194
------------+--------------------------------------------------
prog | W 0.7305 2 6.0 384.0 10.88 0.0000 e
| P 0.2712 6.0 386.0 10.09 0.0000 a
| L 0.3666 6.0 382.0 11.67 0.0000 a
| R 0.3602 3.0 193.0 23.17 0.0000 u
|--------------------------------------------------
female | W 0.8238 1 3.0 192.0 13.69 0.0000 e
| P 0.1762 3.0 192.0 13.69 0.0000 e
| L 0.2139 3.0 192.0 13.69 0.0000 e
| R 0.2139 3.0 192.0 13.69 0.0000 e
|--------------------------------------------------
prog*female | W 0.9321 2 6.0 384.0 2.29 0.0347 e
| P 0.0691 6.0 386.0 2.30 0.0338 a
| L 0.0716 6.0 382.0 2.28 0.0356 a
| R 0.0381 3.0 193.0 2.45 0.0646 u
|--------------------------------------------------
Residual | 194
------------+--------------------------------------------------
Total | 199
---------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
We can look at the simultaneous confidence intervals for all pairwise combinations
of the means by converting the 3x2 design into a one-way design with six levels. We
can then use the Heck Charts with s=3, m=.5, n=95 and cv=.095.
egen grp = group(prog female)
tablist grp prog female, sort(v) nolabel clean /* findit tablist */
grp prog female Freq
1 1 0 21
2 1 1 24
3 2 0 47
4 2 1 58
5 3 0 23
6 3 1 27
manova read write math = grp
Number of obs = 200
W = Wilks' lambda L = Lawley-Hotelling trace
P = Pillai's trace R = Roy's largest root
Source | Statistic df F(df1, df2) = F Prob>F
-----------+--------------------------------------------------
grp | W 0.5808 5 15.0 530.4 7.69 0.0000 a
| P 0.4796 15.0 582.0 7.38 0.0000 a
| L 0.6206 15.0 572.0 7.89 0.0000 a
| R 0.3762 5.0 194.0 14.59 0.0000 u
|--------------------------------------------------
Residual | 194
-----------+--------------------------------------------------
Total | 199
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
manovatest, showorder
Order of columns in the design matrix
1: _cons
2: (grp==1)
3: (grp==2)
4: (grp==3)
5: (grp==4)
6: (grp==5)
7: (grp==6)
simulci read write math, by(grp) cv(.095)
s=3 m=.5 n=95 cv= .095
group variable: grp
pairwise simultaneous
comparison difference confidence intervals
dv: read
grp 1 vs grp 2 5.994 -2.401 14.390
grp 1 vs grp 3 -3.324 -11.720 5.071
grp 1 vs grp 4 -3.117 -11.512 5.279
grp 1 vs grp 5 7.300 -1.095 15.696
grp 1 vs grp 6 6.286 -2.110 14.681
grp 2 vs grp 3 -9.318* -17.714 -0.923
grp 2 vs grp 4 -9.111* -17.506 -0.715
grp 2 vs grp 5 1.306 -7.089 9.702
grp 2 vs grp 6 0.292 -8.104 8.687
grp 3 vs grp 4 0.208 -8.188 8.603
grp 3 vs grp 5 10.624* 2.229 19.020
grp 3 vs grp 6 9.610* 1.214 18.005
grp 4 vs grp 5 10.417* 2.021 18.812
grp 4 vs grp 6 9.402* 1.007 17.798
grp 5 vs grp 6 -1.014 -9.410 7.381
dv: write
grp 1 vs grp 2 -4.107 -11.566 3.351
grp 1 vs grp 3 -5.474 -12.933 1.984
grp 1 vs grp 4 -8.443* -15.902 -0.985
grp 1 vs grp 5 7.317 -0.142 14.775
grp 1 vs grp 6 -1.820 -9.279 5.638
grp 2 vs grp 3 -1.367 -8.826 6.091
grp 2 vs grp 4 -4.336 -11.795 3.122
grp 2 vs grp 5 11.424* 3.965 18.882
grp 2 vs grp 6 2.287 -5.171 9.746
grp 3 vs grp 4 -2.969 -10.428 4.489
grp 3 vs grp 5 12.791* 5.332 20.249
grp 3 vs grp 6 3.654 -3.804 11.113
grp 4 vs grp 5 15.760* 8.302 23.219
grp 4 vs grp 6 6.623 -0.835 14.082
grp 5 vs grp 6 -9.137* -16.595 -1.678
dv: math
grp 1 vs grp 2 0.315 -7.196 7.827
grp 1 vs grp 3 -6.937 -14.449 0.575
grp 1 vs grp 4 -6.223 -13.735 1.289
grp 1 vs grp 5 3.277 -4.234 10.789
grp 1 vs grp 6 4.190 -3.321 11.702
grp 2 vs grp 3 -7.253 -14.765 0.259
grp 2 vs grp 4 -6.539 -14.051 0.973
grp 2 vs grp 5 2.962 -4.550 10.474
grp 2 vs grp 6 3.875 -3.637 11.387
grp 3 vs grp 4 0.714 -6.798 8.226
grp 3 vs grp 5 10.215* 2.703 17.727
grp 3 vs grp 6 11.128* 3.616 18.640
grp 4 vs grp 5 9.501* 1.989 17.013
grp 4 vs grp 6 10.414* 2.902 17.926
grp 5 vs grp 6 0.913 -6.599 8.425
Even though Stata has a general purpose manova command that can do factorial designs. We will
demonstrate the use of mvreg (multivariate regression) along with the mvtest
command by David E. Moore of the University of Cincinnati (findit mvtest).
xi3: mvreg read write math = r.female*r.prog
r.female _Ifemale_0-1 (naturally coded; _Ifemale_0 omitted)
r.prog _Iprog_1-3 (naturally coded; _Iprog_1 omitted)
r.fem~e*r.prog _IfemXpro_#_# (coded as above)
Equation Obs Parms RMSE "R-sq" F P
----------------------------------------------------------------------
read 200 6 9.301994 0.1976 9.553455 0.0000
write 200 6 8.263856 0.2590 13.56062 0.0000
math 200 6 8.32305 0.2306 11.62587 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read |
_Ifemale_1 | -1.729062 1.415205 -1.22 0.223 -4.520224 1.0621
_Iprog_2 | 6.217423 1.662715 3.74 0.000 2.938105 9.496742
_Iprog_3 | -6.904649 1.559758 -4.43 0.000 -9.980908 -3.828389
_IfemXpro_~2 | 5.786417 3.32543 1.74 0.083 -.7722193 12.34505
_IfemXpro_~3 | 4.115332 3.119515 1.32 0.189 -2.037187 10.26785
_cons | 50.76252 .7076023 71.74 0.000 49.36694 52.1581
-------------+----------------------------------------------------------------
write |
_Ifemale_1 | 5.404401 1.257262 4.30 0.000 2.924744 7.884059
_Iprog_2 | 4.905186 1.477149 3.32 0.001 1.991852 7.818519
_Iprog_3 | -7.254496 1.385683 -5.24 0.000 -9.987433 -4.52156
_IfemXpro_~2 | -1.137957 2.954299 -0.39 0.701 -6.964625 4.68871
_IfemXpro_~3 | 5.598712 2.771365 2.02 0.045 .1328381 11.06459
_cons | 51.23086 .6286312 81.50 0.000 49.99103 52.47068
-------------+----------------------------------------------------------------
math |
_Ifemale_1 | -.647462 1.266268 -0.51 0.610 -3.144881 1.849957
_Iprog_2 | 6.737988 1.48773 4.53 0.000 3.803786 9.67219
_Iprog_3 | -6.94521 1.395608 -4.98 0.000 -9.697723 -4.192698
_IfemXpro_~2 | -.3983903 2.97546 -0.13 0.894 -6.266794 5.470014
_IfemXpro_~3 | -.3983721 2.791217 -0.14 0.887 -5.903398 5.106654
_cons | 51.08666 .633134 80.69 0.000 49.83795 52.33537
------------------------------------------------------------------------------
mvtest _Ifemale_1
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_Ifemale_1" Effect(s)
S=1 M=.5 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.82380984 13.6878 3 192.0000 0.0000
Pillai's Trace 0.17619016 13.6878 3 192.0000 0.0000
Hotelling-Lawley Trace 0.21387237 13.6878 3 192.0000 0.0000
mvtest _Iprog_2 _Iprog_3
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_Iprog_2 _Iprog_3" Effect(s)
S=2 M=0 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.73051851 10.8797 6 384.0000 0.0000
Pillai's Trace 0.27116893 10.0907 6 386.0000 0.0000
Hotelling-Lawley Trace 0.36658079 11.6695 6 382.0000 0.0000
mvtest _IfemXpro_1_2 _IfemXpro_1_3
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_IfemXpro_1_2 _IfemXpro_1_3" Effect(s)
S=2 M=0 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.93205692 2.2916 6 384.0000 0.0347
Pillai's Trace 0.06913323 2.3034 6 386.0000 0.0338
Hotelling-Lawley Trace 0.07161894 2.2799 6 382.0000 0.0356
Now to follow up on the interaction effect using mvreg, xi3 and mvtest by
doing some testd of simple main effects.
xi3: mvreg read write math = a.female@g.prog
a.female _Ifemale_0-1 (naturally coded; _Ifemale_1 omitted)
g.prog _Iprog_1-3 (naturally coded; _Iprog_1 omitted)
Equation Obs Parms RMSE "R-sq" F P
----------------------------------------------------------------------
read 200 6 9.301994 0.1976 9.553455 0.0000
write 200 6 8.263856 0.2590 13.56062 0.0000
math 200 6 8.32305 0.2306 11.62587 0.0000
------------------------------------------------------------------------------
| Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
read |
_Iprog_2 | 6.217423 1.662715 3.74 0.000 2.938105 9.496742
_Iprog_3 | -3.795937 1.916533 -1.98 0.049 -7.575852 -.0160219
_Ife0Wpr1 | 5.994048 2.779502 2.16 0.032 .5121253 11.47597
_Ife0Wpr2 | .2076302 1.825609 0.11 0.910 -3.392959 3.80822
_Ife0Wpr3 | -1.014493 2.639461 -0.38 0.701 -6.220216 4.191231
_cons | 50.76252 .7076023 71.74 0.000 49.36694 52.1581
-------------+----------------------------------------------------------------
write |
_Iprog_2 | 4.905186 1.477149 3.32 0.001 1.991852 7.818519
_Iprog_3 | -4.801904 1.70264 -2.82 0.005 -8.159965 -1.443842
_Ife0Wpr1 | -4.107143 2.469299 -1.66 0.098 -8.977261 .7629757
_Ife0Wpr2 | -2.969186 1.621864 -1.83 0.069 -6.167935 .2295642
_Ife0Wpr3 | -9.136876 2.344887 -3.90 0.000 -13.76162 -4.512131
_cons | 51.23086 .6286312 81.50 0.000 49.99103 52.47068
-------------+----------------------------------------------------------------
math |
_Iprog_2 | 6.737988 1.48773 4.53 0.000 3.803786 9.67219
_Iprog_3 | -3.576216 1.714836 -2.09 0.038 -6.958332 -.1941007
_Ife0Wpr1 | .3154762 2.486987 0.13 0.899 -4.589527 5.220479
_Ife0Wpr2 | .7138665 1.633481 0.44 0.663 -2.507796 3.935529
_Ife0Wpr3 | .9130435 2.361683 0.39 0.699 -3.744828 5.570915
_cons | 51.08666 .633134 80.69 0.000 49.83795 52.33537
------------------------------------------------------------------------------
describe _Ife0Wpr1 _Ife0Wpr2 _Ife0Wpr3
storage display value
variable name type format label variable label
-------------------------------------------------------------------------------
_Ife0Wpr1 double %10.0g female(0 vs. 1) @ prog==1
_Ife0Wpr2 double %10.0g female(0 vs. 1) @ prog==2
_Ife0Wpr3 double %10.0g female(0 vs. 1) @ prog==3
mvtest _Ife0Wpr1
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_Ife0Wpr1" Effect(s)
S=1 M=.5 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.92126494 5.4697 3 192.0000 0.0013
Pillai's Trace 0.07873506 5.4697 3 192.0000 0.0013
Hotelling-Lawley Trace 0.08546408 5.4697 3 192.0000 0.0013
mvtest _Ife0Wpr2
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_Ife0Wpr2" Effect(s)
S=1 M=.5 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.96490405 2.3278 3 192.0000 0.0759
Pillai's Trace 0.03509595 2.3278 3 192.0000 0.0759
Hotelling-Lawley Trace 0.03637247 2.3278 3 192.0000 0.0759
mvtest _Ife0Wpr3
MULTIVARIATE TESTS OF SIGNIFICANCE
Multivariate Test Criteria and Exact F Statistics for
the Hypothesis of no Overall "_Ife0Wpr3" Effect(s)
S=1 M=.5 N=95
Test Value F Num DF Den DF Pr > F
Wilks' Lambda 0.88168270 8.5885 3 192.0000 0.0000
Pillai's Trace 0.11831730 8.5885 3 192.0000 0.0000
Hotelling-Lawley Trace 0.13419488 8.5885 3 192.0000 0.0000
Multivariate Course Page
Phil Ender, 10nov05, 3feb03; 20may02; 29jan98