Also Know as Hierarchical Designs
Compare these Three Designs
Crossed, nested, and confounded.
a1 | a2 | |
b1 | s1 | s5 |
b2 | s2 | s6 |
b3 | s3 | s7 |
b4 | s4 | s8 |
a1 | a2 | |
b1 | s1 | |
b2 | s2 | |
b3 | s3 | |
b4 | s4 |
a1 | a2 | |
b1 | s1 | |
b2 | s2 |
Linear Model
Yijk = μ + αj + βk(j) + εi(jk)
Expected Mean Squares
E(MSA) = σ2ε + nσ2β + nqσ2α
E(MSB(A)) = σ2ε + nσ2β
E(MSresid) = σ2ε
ANOVA Summary Table for CRH-pq(A) where A is a Fixed Variable
Source Errorterm df 1 A [2] p-1 2 B(A) [3] p(q(j)-1) 3 Residual pq(j)(n-1)
Example CRH-2,8(A)
a1b1 3 6 3 3 a1b2 1 2 2 2 a1b3 5 6 5 6 a1b4 2 3 4 3 a2b5 7 8 7 6 a2b6 4 5 4 3 a2b7 7 8 9 8 a2b8 10 10 9 11
Using Stata
input a b y x1 x2 x3 x4 x5 x6 x7 1 1 3 1 1 1 1 0 0 0 1 1 6 1 1 1 1 0 0 0 1 1 3 1 1 1 1 0 0 0 1 1 3 1 1 1 1 0 0 0 1 2 1 1 -1 1 1 0 0 0 1 2 2 1 -1 1 1 0 0 0 1 2 2 1 -1 1 1 0 0 0 1 2 2 1 -1 1 1 0 0 0 1 3 5 1 0 -2 1 0 0 0 1 3 6 1 0 -2 1 0 0 0 1 3 5 1 0 -2 1 0 0 0 1 3 6 1 0 -2 1 0 0 0 1 4 2 1 0 0 -3 0 0 0 1 4 3 1 0 0 -3 0 0 0 1 4 4 1 0 0 -3 0 0 0 1 4 3 1 0 0 -3 0 0 0 2 5 7 -1 0 0 0 1 1 1 2 5 8 -1 0 0 0 1 1 1 2 5 7 -1 0 0 0 1 1 1 2 5 6 -1 0 0 0 1 1 1 2 6 4 -1 0 0 0 -1 1 1 2 6 5 -1 0 0 0 -1 1 1 2 6 4 -1 0 0 0 -1 1 1 2 6 3 -1 0 0 0 -1 1 1 2 7 7 -1 0 0 0 0 -2 1 2 7 8 -1 0 0 0 0 -2 1 2 7 9 -1 0 0 0 0 -2 1 2 7 8 -1 0 0 0 0 -2 1 2 8 10 -1 0 0 0 0 0 -3 2 8 10 -1 0 0 0 0 0 -3 2 8 9 -1 0 0 0 0 0 -3 2 8 11 -1 0 0 0 0 0 -3 end table b a,contents(freq mean y sd y) ----------+------------------- | a b | 1 2 ----------+------------------- 1 | 4 | 3.75 | 1.5 | 2 | 4 | 1.75 | .5 | 3 | 4 | 5.5 | .5773503 | 4 | 4 | 3 | .8164966 | 5 | 4 | 7 | .8164966 | 6 | 4 | 4 | .8164966 | 7 | 4 | 8 | .8164966 | 8 | 4 | 10 | .8164966 ----------+------------------- histogram y, by(a b) normal anova y a / b|a / Number of obs = 32 R-squared = 0.9214 Root MSE = .877971 Adj R-squared = 0.8985 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 217.00 7 31.00 40.22 0.0000 | a | 112.50 1 112.50 6.46 0.0440 b|a | 104.50 6 17.4166667 -----------+---------------------------------------------------- b|a | 104.50 6 17.4166667 22.59 0.0000 | Residual | 18.50 24 .770833333 -----------+---------------------------------------------------- Total | 235.50 31 7.59677419 regress y x1 x2 x3 x4 x5 x6 x7 Source | SS df MS Number of obs = 32 ---------+------------------------------ F( 7, 24) = 40.22 Model | 217.00 7 31.00 Prob > F = 0.0000 Residual | 18.50 24 .770833333 R-squared = 0.9214 ---------+------------------------------ Adj R-squared = 0.8985 Total | 235.50 31 7.59677419 Root MSE = .87797 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x1 | -1.875 .1552048 -12.08 0.000 -2.195327 -1.554673 x2 | 1 .3104097 3.22 0.004 .3593459 1.640654 x3 | -.9166667 .1792151 -5.11 0.000 -1.286548 -.5467849 x4 | .1666667 .1267242 1.32 0.201 -.0948793 .4282126 x5 | 1.5 .3104097 4.83 0.000 .8593459 2.140654 x6 | -.8333333 .1792151 -4.65 0.000 -1.203215 -.4634515 x7 | -.9166667 .1267242 -7.23 0.000 -1.178213 -.6551207 _cons | 5.375 .1552048 34.63 0.000 5.054673 5.695327 ------------------------------------------------------------------------------ test2 x1 / x2 x3 x4 x5 x6 x7 /* available from ATS */ Testing: x1 Error term: x2 x3 x4 x5 x6 x7 F( 1, 6) = 6.46 Prob > F = 0.0440 test x2 x3 x4 x5 x6 x7 ( 1) x2 = 0.0 ( 2) x3 = 0.0 ( 3) x4 = 0.0 ( 4) x5 = 0.0 ( 5) x6 = 0.0 ( 6) x7 = 0.0 F( 6, 24) = 22.59 Prob > F = 0.0000Multilevel Model Using xtmixed
It is also possible to analyze these data using a multilevel model approach equivalent to using proc mixed in SAS or using HLM. We will run this as a random intercept restricted maximum likelihood model.
xtmixed y i.a || b: , var /* reml - restricted maximum likelihood model */ Performing EM optimization: Performing gradient-based optimization: Iteration 0: log restricted-likelihood = -50.78963 Iteration 1: log restricted-likelihood = -50.78963 Computing standard errors: Mixed-effects REML regression Number of obs = 32 Group variable: b Number of groups = 8 Obs per group: min = 4 avg = 4.0 max = 4 Wald chi2(1) = 6.46 Log restricted-likelihood = -50.78963 Prob > chi2 = 0.0110 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 2.a | 3.75 1.475495 2.54 0.011 .8580829 6.641917 _cons | 3.5 1.043333 3.35 0.001 1.455106 5.544894 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ b: Identity | var(_cons) | 4.161463 2.514498 1.273271 13.60101 -----------------------------+------------------------------------------------ var(Residual) | .7708331 .2225203 .4377636 1.357316 ------------------------------------------------------------------------------ LR test vs. linear regression: chibar2(01) = 31.43 Prob >= chibar2 = 0.0000 test 2.a ( 1) [y]2.a = 0 chi2( 1) = 6.46 Prob > chi2 = 0.0110 anovalator a, main fratio anovalator main-effect for a chi2(1) = 6.4593237 p-value = .01103716 scaled as F-ratio = 6.4593237Examples of additional nested models
CRH-pq(A)r(A*B)
Linear Model
Yijkl = μ + αj + βk(j) + γl(jk) + εi(jkl)
Schematic
a1 | a2 | |
b1 c1 | s1 | |
b1 c2 | s2 | |
b2 c3 | s3 | |
b2 c4 | s4 | |
b3 c5 | s5 | |
b3 c6 | s6 | |
b4 c7 | s7 | |
b4 c8 | s8 |
Anova Summary Table for CRH-pq(A)r(A*B) where A is a Fixed Variable Source Errorterm df 1 A [2] p-1 2 B(A) [3] p(q(j)-1) 3 C(A*B) [4] pq(j)(r(jk)-1) 4 Residual pq(j)r(jk)(n-1)
CRPH-pq(A)r
Linear Model
Yijkl = μ + αj + βk(j) + γl + αγjl + βγk(j)l + εi(jkl)
Schematic
a1 c1 | a1 c2 | a2 c1 | a2 c2 | |
b1 | s1 | s3 | ||
b2 | s2 | s4 | ||
b3 | s5 | s7 | ||
b4 | s6 | s8 |
Anova Summary Table for CRPH-pq(A)r where A & C are Fixed Variables Source Errorterm df 1 A [2] p-1 2 B(A) [6] p(q(j)-1) 3 C [5] r-1 4 A*C [5] (p-1)(r-1) 5 B(A)*C [6] p(q(j)-1)(r-1) 6 Residual pq(j)r(n-1)CRPH-pq(A)r(A)
Linear Model
Yijkl = μ + αj + βk(j) + γl(j) + βγk(j)l(j) + εi(jkl)
Schematic
a1 | a2 | |
b1 c1 | s1 | |
b1 c2 | s2 | |
b2 c1 | s3 | |
b2 c2 | s4 | |
b3 c3 | s5 | |
b3 c4 | s6 | |
b4 c3 | s7 | |
b4 c4 | s8 |
Anova Summary Table for CRPH-pq(A)r(A) where A & C are Fixed Variables Source Errorterm df 1 A [2] p-1 2 B(A) [5] p(q(j)-1) 3 C(A) [4] p(r(j)-1) 4 B(A)*C(A) [5] p(q(j)-1)(r(j)-1) 5 Residual pq(j)r(j)(n-1)CRPH-pqr(A*B)
Linear Model
Yijkl = μ + αj + βk + γl(jk) + αβjk + εi(jkl)
Schematic
a1 b1 | a1 b2 | a2 b1 | a2 b2 | |
c1 | s1 | |||
c2 | s2 | |||
c3 | s3 | |||
c4 | s4 | |||
c5 | s5 | |||
c6 | s6 | |||
c7 | s7 | |||
c8 | s8 |
Anova Summary Table for CRPH-pqr(A*B) where A & B are Fixed Variables Source Errorterm df 1 A [3] p-1 2 B [3] q-1 3 C(A*B) [5] pq(r(jk)-1) 4 A*B [5] (p-1)(q-1) 5 Residual pqr(jk)(n-1)
Linear Statistical Models Course
Phil Ender, 17sep10, 14may06, 9may00