RB-p
Schematic with Example Data
--------------------------------------------
| a | total
s | 1 2 3 4 |
----------+--------------------------+------
1 | 3 4 4 3 |
2 | 2 4 4 5 |
3 | 2 3 3 6 |
4 | 3 3 3 5 |
5 | 1 2 4 7 |
6 | 3 3 6 6 |
7 | 4 4 5 10 |
8 | 6 5 5 8 |
-------------------------------------+-------
mean | 3.0 3.5 4.25 6.25 | 4.25
sd | 1.5 0.93 1.04 2.12 | 1.88
var | 2.29 0.86 1.07 4.5 | 3.55
Or in abbreviated form:
Where S1 is the same group of 8 subjects.
Level a1
a2 a3 a4 Total
S1
n=8
S1
n=8
S1
n=8
S1
n=8
Mean 3.0
3.5 4.25 6.25 4.25
sd 1.51 0.93 1.04 2.12 1.88
Linear Model
Yij = μ + αj + πi + εij
μ = overall poulation mean
αj = the effect of treatment level j
πi = the effect of block i (in repeated measures subjects are the blocks)
εij = experimental error
Hypotheses
Assumptions
| 1. | Independence |
| 2. | Normality |
| 3. | Homogeneity of variance |
| 4. | No Non-additivity (no block*treatment interaction) |
| 5. | Compound symmetry in the variance-covariance matrix |
ANOVA Summary Table
| Source | SS | df | MS | F | p-value | Error | |
| 1 | Treatment | 49.00 | 3 | 16.33 | 11.63 | 0.0001 | [3] |
| 2 | Blocks (Subjects) | 31.50 | 7 | 4.50 | 3.20 | 0.0180 | [3] |
| 3 | Residual | 29.50 | 21 | 1.40 | |||
| Total | 110.00 | 31 |
Expected Mean Squares
Strength of Association
Using Stata: Method 1
input y a s x1 x2 x3 s1 s2 s3 s4 s5 s6 s7
3 1 1 1 1 1 1 1 1 1 1 1 1
2 1 2 1 1 1 -1 1 1 1 1 1 1
2 1 3 1 1 1 0 -2 1 1 1 1 1
3 1 4 1 1 1 0 0 -3 1 1 1 1
1 1 5 1 1 1 0 0 0 -4 1 1 1
3 1 6 1 1 1 0 0 0 0 -5 1 1
4 1 7 1 1 1 0 0 0 0 0 -6 1
6 1 8 1 1 1 0 0 0 0 0 0 -7
4 2 1 -1 1 1 1 1 1 1 1 1 1
4 2 2 -1 1 1 -1 1 1 1 1 1 1
3 2 3 -1 1 1 0 -2 1 1 1 1 1
3 2 4 -1 1 1 0 0 -3 1 1 1 1
2 2 5 -1 1 1 0 0 0 -4 1 1 1
3 2 6 -1 1 1 0 0 0 0 -5 1 1
4 2 7 -1 1 1 0 0 0 0 0 -6 1
5 2 8 -1 1 1 0 0 0 0 0 0 -7
4 3 1 0 -2 1 1 1 1 1 1 1 1
4 3 2 0 -2 1 -1 1 1 1 1 1 1
3 3 3 0 -2 1 0 -2 1 1 1 1 1
3 3 4 0 -2 1 0 0 -3 1 1 1 1
4 3 5 0 -2 1 0 0 0 -4 1 1 1
6 3 6 0 -2 1 0 0 0 0 -5 1 1
5 3 7 0 -2 1 0 0 0 0 0 -6 1
5 3 8 0 -2 1 0 0 0 0 0 0 -7
3 4 1 0 0 -3 1 1 1 1 1 1 1
5 4 2 0 0 -3 -1 1 1 1 1 1 1
6 4 3 0 0 -3 0 -2 1 1 1 1 1
5 4 4 0 0 -3 0 0 -3 1 1 1 1
7 4 5 0 0 -3 0 0 0 -4 1 1 1
6 4 6 0 0 -3 0 0 0 0 -5 1 1
10 4 7 0 0 -3 0 0 0 0 0 -6 1
8 4 8 0 0 -3 0 0 0 0 0 0 -7
end
tabstat y, by(a) stat(n mean sd var)
Summary for variables: y
by categories of: a
a | N mean sd variance
---------+----------------------------------------
1 | 8 3 1.511858 2.285714
2 | 8 3.5 .9258201 .8571429
3 | 8 4.25 1.035098 1.071429
4 | 8 6.25 2.12132 4.5
---------+----------------------------------------
Total | 32 4.25 1.883716 3.548387
--------------------------------------------------
histogram y, by(a) normal
anova y a s, repeated(a)
Number of obs = 32 R-squared = 0.7318
Root MSE = 1.18523 Adj R-squared = 0.6041
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 80.50 10 8.05 5.73 0.0004
|
a | 49.00 3 16.3333333 11.63 0.0001
s | 31.50 7 4.50 3.20 0.0180
|
Residual | 29.50 21 1.4047619
-----------+----------------------------------------------------
Total | 110.00 31 3.5483871
Between-subjects error term: s
Levels: 8 (7 df)
Lowest b.s.e. variable: s
Repeated variable: a
Huynh-Feldt epsilon = 0.8343
Greenhouse-Geisser epsilon = 0.6195
Box's conservative epsilon = 0.3333
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
a | 3 11.63 0.0001 0.0003 0.0015 0.0113
Residual | 21
-----------+----------------------------------------------------
mat list e(Srep)
symmetric e(Srep)[4,4]
c1 c2 c3 c4
r1 2.2857143
r2 1.1428571 .85714286
r3 .71428571 .28571429 1.0714286
r4 1.2857143 .28571429 .92857143 4.5
omega2 a
omega squared for a = 0.4991
effect size = 0.9981
nonadd y a s
Tukey's test of nonadditivity for randomized block designs
Ho: model is additive (no block*treatment interaction)
F (1,20) = 1.2795813 Pr > F: .27135918
quietly anova y a s /* quietly rerun anova to get correct mse */
fhcomp a
Fisher-Hayter pairwise comparisons for variable a
studentized range critical value(.05, 3, 21) = 3.5647267
mean critical
grp vs grp group means dif dif
-------------------------------------------------------
1 vs 2 3.0000 3.5000 0.5000 1.4938
1 vs 3 3.0000 4.2500 1.2500 1.4938
1 vs 4 3.0000 6.2500 3.2500* 1.4938
2 vs 3 3.5000 4.2500 0.7500 1.4938
2 vs 4 3.5000 6.2500 2.7500* 1.4938
3 vs 4 4.2500 6.2500 2.0000* 1.4938
tukeyhsd a
Tukey HSD pairwise comparisons for variable a
studentized range critical value(.05, 4, 21) = 3.9419483
uses harmonica mean sample size = 8.000
mean critical
grp vs grp group means dif dif
-------------------------------------------------------
1 vs 2 3.0000 3.5000 0.5000 1.6518
1 vs 3 3.0000 4.2500 1.2500 1.6518
1 vs 4 3.0000 6.2500 3.2500* 1.6518
2 vs 3 3.5000 4.2500 0.7500 1.6518
2 vs 4 3.5000 6.2500 2.7500* 1.6518
3 vs 4 4.2500 6.2500 2.0000* 1.6518
regress y x1 x2 x3 s1 s2 s3 s4 s5 s6 s7
Source | SS df MS Number of obs = 32
-------------+------------------------------ F( 10, 21) = 5.73
Model | 80.50 10 8.05 Prob > F = 0.0004
Residual | 29.50 21 1.4047619 R-squared = 0.7318
-------------+------------------------------ Adj R-squared = 0.6041
Total | 110.00 31 3.5483871 Root MSE = 1.1852
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | -.25 .2963066 -0.84 0.408 -.8662034 .3662034
x2 | -.3333333 .1710727 -1.95 0.065 -.6890985 .0224318
x3 | -.6666667 .1209667 -5.51 0.000 -.9182306 -.4151027
s1 | -.125 .4190409 -0.30 0.768 -.9964432 .7464432
s2 | .0416667 .2419334 0.17 0.865 -.4614613 .5447946
s3 | .0208333 .1710727 0.12 0.904 -.3349318 .3765985
s4 | .0125 .1325124 0.09 0.926 -.2630745 .2880745
s5 | -.1583333 .1081959 -1.46 0.158 -.383339 .0666723
s6 | -.2916667 .0914422 -3.19 0.004 -.4818312 -.1015022
s7 | -.25 .0791913 -3.16 0.005 -.4146873 -.0853127
_cons | 4.25 .2095204 20.28 0.000 3.814278 4.685722
------------------------------------------------------------------------------
test x1 x2 x3
( 1) x1 = 0.0
( 2) x2 = 0.0
( 3) x3 = 0.0
F( 3, 21) = 11.63
Prob > F = 0.0001
test s1 s2 s3 s4 s5 s6 s7
( 1) s1 = 0.0
( 2) s2 = 0.0
( 3) s3 = 0.0
( 4) s4 = 0.0
( 5) s5 = 0.0
( 6) s6 = 0.0
( 7) s7 = 0.0
F( 7, 21) = 3.20
Prob > F = 0.0180
xi3: regress y r.a r.s
r.a _Ia_1-4 (naturally coded; _Ia_1 omitted)
r.s _Is_1-8 (naturally coded; _Is_1 omitted)
Source | SS df MS Number of obs = 32
-------------+------------------------------ F( 10, 21) = 5.73
Model | 80.50 10 8.05 Prob > F = 0.0004
Residual | 29.50 21 1.4047619 R-squared = 0.7318
-------------+------------------------------ Adj R-squared = 0.6041
Total | 110.00 31 3.5483871 Root MSE = 1.1852
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Ia_2 | .5 .5926133 0.84 0.408 -.7324067 1.732407
_Ia_3 | 1 .5132181 1.95 0.065 -.0672955 2.067296
_Ia_4 | 2.666667 .4838667 5.51 0.000 1.660411 3.672923
_Is_2 | .25 .8380817 0.30 0.768 -1.492886 1.992886
_Is_3 | -.125 .7258001 -0.17 0.865 -1.634384 1.384384
_Is_4 | -.0833333 .6842909 -0.12 0.904 -1.506394 1.339727
_Is_5 | -.0625 .6625618 -0.09 0.926 -1.440373 1.315373
_Is_6 | .95 .6491753 1.46 0.158 -.4000339 2.300034
_Is_7 | 2.041667 .6400955 3.19 0.004 .7105153 3.372818
_Is_8 | 2 .6335302 3.16 0.005 .6825018 3.317498
_cons | 4.25 .2095204 20.28 0.000 3.814278 4.685722
------------------------------------------------------------------------------
describe _Ia_2-_Is_8
storage display value
variable name type format label variable label
-------------------------------------------------------------------------------
_Ia_2 double %10.0g a(2 vs. 1)
_Ia_3 double %10.0g a(3 vs. 2-)
_Ia_4 double %10.0g a(4 vs. 3-)
_Is_2 double %10.0g s(2 vs. 1)
_Is_3 double %10.0g s(3 vs. 2-)
_Is_4 double %10.0g s(4 vs. 3-)
_Is_5 double %10.0g s(5 vs. 4-)
_Is_6 double %10.0g s(6 vs. 5-)
_Is_7 double %10.0g s(7 vs. 6-)
_Is_8 double %10.0g s(8 vs. 7-)
test _Ia_2 _Ia_3 _Ia_4
( 1) _Ia_2 = 0.0
( 2) _Ia_3 = 0.0
( 3) _Ia_4 = 0.0
F( 3, 21) = 11.63
Prob > F = 0.0001
test _Is_2 _Is_3 _Is_4 _Is_5 _Is_6 _Is_7 _Is_8
( 1) _Is_2 = 0.0
( 2) _Is_3 = 0.0
( 3) _Is_4 = 0.0
( 4) _Is_5 = 0.0
( 5) _Is_6 = 0.0
( 6) _Is_7 = 0.0
( 7) _Is_8 = 0.0
F( 7, 21) = 3.20
Prob > F = 0.0180Using Stata: Method 2
Randomized block data often come in a wide form, in which, each of the repeated measures is a separate variable. Before you can alanlyze these data using the Stata anova command you need to reshape the data into a long form. While the data are still wide you can compute the correlation and covariance matrices using the corr command.
input s y1 y2 y3 y4
1 3 4 4 3
2 2 4 4 5
3 2 3 3 6
4 3 3 3 5
5 1 2 4 7
6 3 3 6 6
7 4 4 5 10
8 6 5 5 8
end
corr y1 y2 y3 y4, cov /* note: covariance matrix same as e(Srep) */
(obs=8)
| y1 y2 y3 y4
---------+------------------------------------
y1 | 2.28571
y2 | 1.14286 .857143
y3 | .714286 .285714 1.07143
y4 | 1.28571 .285714 .928571 4.5
corr y1 y2 y3 y4
(obs=8)
| y1 y2 y3 y4
-------------+------------------------------------
y1 | 1.0000
y2 | 0.8165 1.0000
y3 | 0.4564 0.2981 1.0000
y4 | 0.4009 0.1455 0.4229 1.0000
reshape long y, i(s) j(a)
(note: j = 1 2 3 4)
Data wide -> long
-----------------------------------------------------------------------------
Number of obs. 8 -> 32
Number of variables 5 -> 3
j variable (4 values) -> a
xij variables:
y1 y2 ... y4 -> y
-----------------------------------------------------------------------------
list
[output omitted]
anova y a s, repeated(a)
Number of obs = 32 R-squared = 0.7318
Root MSE = 1.18523 Adj R-squared = 0.6041
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 80.50 10 8.05 5.73 0.0004
|
a | 49.00 3 16.3333333 11.63 0.0001
s | 31.50 7 4.50 3.20 0.0180
|
Residual | 29.50 21 1.4047619
-----------+----------------------------------------------------
Total | 110.00 31 3.5483871
Between-subjects error term: s
Levels: 8 (7 df)
Lowest b.s.e. variable: s
Repeated variable: a
Huynh-Feldt epsilon = 0.8343
Greenhouse-Geisser epsilon = 0.6195
Box's conservative epsilon = 0.3333
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
a | 3 11.63 0.0001 0.0003 0.0015 0.0113
Residual | 21
-----------+----------------------------------------------------
Nonadditivity
The linear model that is nonadditive would look as follows;
Yij = μ + αj + πi + απij + εij
One obvious problem is that απij & εij cannot exist as separate entities. The expected mean squares in the nonadditive model are as follows:
E(MSA) σ2ε + nσ2α
E(MSBlk) σ2ε + pσ2π
E(MSRes) σ2ε + σ2απ
Using the MSRes as the error term, in a nonadditive model, in the F-ratios for treatment and blocks tends to negatively bias the F-ratio. Tukey (1949) came up with a relatively simple test of additivity. This test examines a one degree of freedom linear*linear interaction to test for the presence of nonadditivity. The Tukey test can be run using the Stata nonadd command from ATS.
Using Stata to Obtain Tukey's Test of Additivity
use http://www.gseis.ucla.edu/courses/data/rb4new
anova y a s, repeated(a)
Number of obs = 32 R-squared = 0.7318
Root MSE = 1.18523 Adj R-squared = 0.6041
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 80.50 10 8.05 5.73 0.0004
|
a | 49.00 3 16.3333333 11.63 0.0001
s | 31.50 7 4.50 3.20 0.0180
|
Residual | 29.50 21 1.4047619
-----------+----------------------------------------------------
Total | 110.00 31 3.5483871
Between-subjects error term: s
Levels: 8 (7 df)
Lowest b.s.e. variable: s
Repeated variable: a
Huynh-Feldt epsilon = 0.8343
Greenhouse-Geisser epsilon = 0.6195
Box's conservative epsilon = 0.3333
------------ Prob > F ------------
Source | df F Regular H-F G-G Box
-----------+----------------------------------------------------
a | 3 11.63 0.0001 0.0003 0.0015 0.0113
Residual | 21
-----------+----------------------------------------------------
nonadd y a s
Tukey's test of nonadditivity for randomized block designs
Ho: model is additive (no block*treatment interaction)
F (1,20) = 1.2795813 Pr > F: .27135918Blocking
Blocking can be used to hold a nuisance variable constant. In testing math performance, one could block on reading ability by matching subjects on reading test scores. Then each subject within a block is randomly assigned to one of several treatment groups.
Carrying matching to an extreme occurs when the subjects themselves become the block. This is a repeated measures type design.
Counter Balancing
When a randomized block design is used to analyze repeated measures type data, care should be taken to counter balance the order of the treatment so as to eliminate the possibility of order effects in final results. Counter balancing cannot, of course, be done when time is the repeated factor.
Variance-Covariance Matrices
The variance-covariance matrix for the repeated effect can take on any several different forms. Here are some examples:
Independence - 1 parameter
σ2 0 σ2 0 0 σ2
Compound Symmetry - 2 parameters
σ2 σ1 σ2 σ1 σ1 σ2
Autoregressive - 3 parameters
σ2 σρ σ2 σρ2 σρ σ2
Unstructured - 6 parameters
σ12 σ12 σ22 σ13 σ23 σ32
The independence covariance structure is typically found in between subject designs with random sampling and random assignment. When the assumption of independence is met the covariance matrix will have zeros or near zeros in the off diagonals. When the assumption of homogeneity of variance is met the diagonal values will all be equal or nearly equal.
Compound symmetry is often found in repeated measures designs which are not time dependent. If the order of the responses does not matter than the covariances are exchangable. The term exchangable is synonymous with compound symmetry. You may also hear the terms spherical or circular which are mathematical measures equating to compound symmetry.
If the response variable is measured at different time points an autoregressive covariance structure often results, in which, observations more closely associated in time are more strongly correlated than observations farther apart in time.
Finally, when there does not seem to be any discernable pattern in the covariance matrix the matrix is said to be unstructured. The multivariate option below makes use of an unstructured covariance matrix.
The Compound Symmetry Assumption Concerning Variance-Covariance Matrix
Generalizing on the concept of homogeneity of variance is the idea of homogeneity of the variance-covariance matrix. With compound symmetry the diagonal values of the variance-covariance matrix (variances) are approximately equal. Further, the off-diagonal values (covariances) are different from the diagonal values but are approximately equal to one another.
In order for the probabilities associated with the F-ratio to be correct the variance-covariance matrix needs to have compound symmetry. When the variance-covariance matrix has compound symmetry then the p-values associated with the regular F-ratio are used.
When the variance-covariance matrix does not have compound symmetry it is necessary to use the p-values associated with one of the conservative F procedures: Huynh-Feldt, Greenhouse-Geisser, or Box's conservative F. Here is the recommended 3-step procedure:
The 3-step procedure is recommended for even small deviations from compound symmetry.
Missing Data
Randomized block designs do not allow for missing data. You need to have an observation in each treatment level for each block. It is possible, but not always desirable, to estimate a relatively small number of missing observations.
The Multivariate Option
It is possible to analyze the data using multivariate analysis of variance as an alternative to the univariate randomized block analysis. The multivariate solution does not have assumptions concerning compound symmetry or nonadditivity. In fact, multivariate repeated measures makes no assumptions concerning the structure of the covariance matrix. It does require that the data be multivariate normal.
input s y1 y2 y3 y4
1 3 4 4 3
2 2 4 4 5
3 2 3 3 6
4 3 3 3 5
5 1 2 4 7
6 3 3 6 6
7 4 4 5 10
8 6 5 5 8
end
corr y1 y2 y3 y4, cov
(obs=8)
| y1 y2 y3 y4
---------+------------------------------------
y1 | 2.28571
y2 | 1.14286 .857143
y3 | .714286 .285714 1.07143
y4 | 1.28571 .285714 .928571 4.5
corr y1 y2 y3 y4
(obs=8)
| y1 y2 y3 y4
-------------+------------------------------------
y1 | 1.0000
y2 | 0.8165 1.0000
y3 | 0.4564 0.2981 1.0000
y4 | 0.4009 0.1455 0.4229 1.0000Next, we trick manova into running a model that has just within-subjects effects by creating our own constant, called cons, and running the model with the nonconstant option. We will also create contrasts among the dependent variables, using an effect coding.
generate con = 1
manova y1 y2 y3 y4 = con, noconstant
Number of obs = 8
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
-----------+--------------------------------------------------
con | W 0.0196 1 4.0 4.0 49.92 0.0011 e
| P 0.9804 4.0 4.0 49.92 0.0011 e
| L 49.9217 4.0 4.0 49.92 0.0011 e
| R 49.9217 4.0 4.0 49.92 0.0011 e
|--------------------------------------------------
Residual | 7
-----------+--------------------------------------------------
Total | 8
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
mat ycomp = (1,0,0,-1\0,1,0,-1\0,0,1,-1)
manovatest con, ytrans(ycomp)
Transformations of the dependent variables
(1) y1 - y4
(2) y2 - y4
(3) y3 - y4
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
-----------+--------------------------------------------------
con | W 0.2458 1 3.0 5.0 5.11 0.0554 e
| P 0.7542 3.0 5.0 5.11 0.0554 e
| L 3.0682 3.0 5.0 5.11 0.0554 e
| R 3.0682 3.0 5.0 5.11 0.0554 e
|--------------------------------------------------
Residual | 7
---------------------------------------------------------
In this example, the multivariate test is not significant, this result conflicts with
the conservative univariate F-tests, which were significant with p < .01. The multivariate
F has 3 and 5 degrees of freedom while the univariate F had 3 and 21. The difference can be
attributed to the greater number of parameters (4 variances and 6 covariances) that are
estimated in the multivariate case. For larger datasets, you would expect the multivariate test
to be at least as powerful as the univariate test.
Linear Statistical Models Course
Phil Ender, 25apr06, 12Feb98