Inference For High-Dimensional Repeated Measures and Split-Plot-Designs

Split-Plot-Designs

Paavo Sattler and Markus Pauly

Ulm University Institute of Statistics [email protected]

EMS 2017 Helsinki

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Introduction and Motivation

Motivating Example  Repeated Measures

Sleep-Laboratory Trial, Jordan et al. (2004)

I

Figure 1 Prostaglandin-D-synthase (β-trace) levels in 10 healthy

male subjects in a sleep-lab trial under dierent sleep conditions

(normal sleep / sleep deprivation / recovery sleep / REM sleep

deprivation) at the time points 24h, 4h, 8h, 12h, 16h, and 20h for 4

consecutive nights and days

Motivating Example  Repeated Measures

Sleep-Laboratory Trial, Jordan et al. (2004)

I

Figure 2 Prostaglandin-D-synthase (β-trace) levels in 10 healthy female subjects in a sleep-lab trial under dierent sleep conditions (normal sleep / sleep deprivation / recovery sleep / REM sleep deprivation) at the time points 24h, 4h, 8h, 12h, 16h, and 20h for 4 consecutive nights and days

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Introduction and Motivation

Observations

a = 2 groups

n ₁ = n ₂ = 10 sample sizes

d = 24 time points (dimension) for each observation

⇒ High-dimensional setting

⇒ Classical multivariate procedures (as Hotelling's T ² ) not applicable Factorial structure on repeated measures:

I

Factor I: 4 interventions (normal sleep / sleep deprivation / recovery / REM sleep deprivation)

I

Factor T: 6 measurements per day (at 24h, 4h, 8h, 12h, 16h, and 20h) Some questions of interest (null hypotheses)

I

H

( G): No gender eect

I

H

( T): No time eect

I

H

( GT): No interaction eect between gender and time

Split-Plot-Designs

Split-Plot-Design Model

I

X

∼ N

(µ

, Σ

), µ

∈ R

, Σ

> 0, independent random vectors

I

i = 1, . . . , a independent groups (e.g., a = 2) with

I

k = 1, . . . , n

independent subjects/units in group i

I

N = P

i =1

n

total sample size

I

Note: Heteroscedasticity allowed: Σ

6= Σ

for all i 6= j.

General linear Hypotheses

I

Write µ = (µ

, . . . , µ

)

∈ R

F

H

(T) : Tµ = 0, T = T

⊗ T

: proper projection contrast matrix

I

Some special cases

F

T

= P

⊗

J

- No group eect

F

T

= P

⊗ P

- No interaction eect time×group

J d := 1 d ×d P d := I d − 1/d · J d

1

T = H

(HH

)

H unique for contrast matrix H with T

= T and T

= T.

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

High-Dimensional Split-Plot-Designs

Test Statistics and Asymptotics

Test Statistic based on

Q _N = N · X ^> TX, where

X = (X ^> ₁ , ..., X ^> a ) ^> denotes the vector of pooled group mean vectors Asymptotic Frameworks

(i) a xed and min(d, n

, . . . , n

) → ∞, (ii) d xed and min(a, n

, . . . , n

) → ∞, (iii) or even min(a, d, n

, . . . , n

) → ∞.

Surprising result

W f _N = Q _N − E _H

(Q _N ) pVar H

(Q N )

behaves similar under all frameworks, if H 0 is fullled.

Test Statistics and Asymptotics

Under H ₀ it holds

Q _N = ^d

ad

X

`= 1

λ _{`</sub> C <sub>`} E (Q _N ) = tr(TV N ) Var(Q N ) = 2 tr TV N ) ²

with C ` i .i .d .

∼ χ ² ₁ and λ ` are the eigenvalues of TV N T, where V N := L a

i = 1 N n

Σ _i .

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

High-Dimensional Split-Plot-Designs

Test Statistics and Asymptotics

Theorem

Under each of the asymptotic frameworks (i)-(iii) it holds under H ₀ ( T) : Tµ = 0

(a) W f _N −→ U ∼ N( ^d 0, 1) if β ₁ → 0, (b) W f _N −→ C ∼ ^{d χ √}

^{− 1}

2 if β ₁ → 1, where β ₁ = λ _max /

q P ad

`= 1 λ <sup>2</sup> <sub>`</sub> is the largest standardized EV of TV N T, where V N := L a

i = 1 N

n

Σ _i .

Challenges

I

a, d and all n

have inuence on the EV and β

I

Ratio-consistent estimation of E

(Q

) , Var

(Q

) and other traces of certain powers are complicated.

I

In particular, the asymptotic frameworks (i)-(iii) have to be taken into account.

I

Large numbers are required

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Better Approximation

Dierent Approximation

Idea of Pauly et al. (2015):

Approximation with sequence of distributions K f such that both cases are covered, i.e.

K _f = χ ² _f − f

√ 2f

−→ d

( N( 0, 1) if f → ∞

χ √

− 1

2 if f → 1

f P = tr ³ ( TV N ) ²
/ tr ² ( TV N ) ³

(even ts skewness) Advantage: τ P = 1/f P → 0 ⇐⇒ β ₁ → 0 and

τ _P → 1 ⇐⇒ β ₁ → 1.

Dierent Approximation

Idea of Pauly et al. (2015):

Approximation with sequence of distributions K f such that both cases are covered, i.e.

K _f = χ ² _f − f

√ 2f

−→ d

( N( 0, 1) if f → ∞

χ √

− 1

2 if f → 1

f P = tr ³ ( TV N ) ²
/ tr ² ( TV N ) ³

(even ts skewness) Advantage: τ P = 1/f P → 0 ⇐⇒ β ₁ → 0 and

τ P → 1 ⇐⇒ β ₁ → 1.

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Better Approximation

Proposed Estimators

I

Estimators of symmetrized U-statistics-type

⇒ Leads to ratio-consistent estimators for all frameworks (i)-(iii) such that

W

= Q

− b E

(Q

) q

Var d

(Q

)

fullls W

− f W

= o

( 1) under (a) and (b).

Proposed Estimators for τ P

Let be

Z (`

,`

,...,`

) :=

r N

n ₁ X _1,`

− X _1,`

>

, . . . , r N

n a X a,`

− X a,`

>

! >

TZ (`

,`

,...,`

) ∼ N _ad ( 0 ad , 2TV N T)

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Better Approximation

Proposed Estimators for τ P

and

Λ ₁ (` <sub>1,1</sub> , . . . , ` _6,a ) = Z (`

,`

,...,`

,`

) > TZ (`

,`

,...,`

,`

) , Λ ₂ (` <sub>1,1</sub> , . . . , ` _6,a ) = Z (`

,`

,...,`

,`

) > TZ (`

,`

,...,`

,`

) , Λ ₃ (` <sub>1,1</sub> , . . . , ` _6,a ) = Z (`

,`

,...,`

,`

) > TZ (`

,`

,...,`

,`

) ,

E

3

Y

m= 1

Λ m (` <sub>1,1</sub> , . . . , ` _6,a )

!

= 8 tr 

( TV N ) ³



Proposed Estimators for τ P

C ₁ =

n

P

`<sub>1,1</sub>,...,`_6,1=1

`<sub>1,1</sub>6=...6=`_6,1

· · · P ⁿ

`<sub>1,a</sub>,...,`_6,a=1

`<sub>1,a</sub>6=...6=`_6,a

Λ

(`

,...,`

)·Λ

(`

,...,`

)·Λ

(`

,...,`

) 8· Q

i =1 ni !

(

)

I

E (C

) = tr 

( TV

)



I

Ratio-consistent in framework (i)-(iii) if p > 1 exists with min(n

, ..., n

) = O(a

) .

I

Similar but even more complicated we can nd a ratio-consistent estimator C

in framework (i)-(iii).

I

Q

(ni−6)!

summations

⇒ Subsampling version C

, C

of this estimators should be used.

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Better Approximation

Dierent Approximation

With other estimators results in ratio-consistent estimators τ b _p for τ P

and bf P = 1/b τ _P . Theorem

In framework (i)-(iii) it holds under H 0 ( T) : Tµ = 0:

χ ²

f b

− b f _P q

2bf P

−→ d

( N( 0, 1) if τ P → 0

χ √

− 1

2 if τ P → 1

Faster convergence

No decision between both asymptotic distributions

Moreover: Behaviour of β ₁ checkable via b τ p

Simulations for β 1 → 0  in case of α = 5%

n ₁ = 20 and n ₂ = 30 runs=10000

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Summary

Simulations for β 1 → 1  in case of α = 5%

Conclusion

Asymptotic test for framework (i)-(iii) Estimators for framework (i)-(iii)

Multivariate normal distribution is the main requirement Extension of existing approaches require β ₁ → 0, like ? ² Even if lim β ₁ → ρ ∈ ( 0, 1) we had some results

In the sleep trial H ₀ ( T) can be rejected

2

Chen, S. X. and Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist., 38(2):808-835.

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017

Summary

References

Chen, S. X. and Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist., 38(2):808835.

Pauly, M., Ellenberger, D., and Brunner, E. (2015). Analysis of high-dimensional

one group repeated measures designs. Statistics, 49:12431261.

Finally

Thanks for your attention!

Paavo Sattler (Ulm) High-Dimensional Split-Plot-Designs July 2017