voxelwise multivariate analysis of multimodality imaging

Voxelwise multivariate analysis of multimodality imaging

Melissa Naylor and Armin SchwartzmanBiostatistics

Harvard School of Public HealthDana-Farber Cancer Institute

June 2010

Outline• The Data• Standard Approach• Multiple Comparisons• The Problem• Multivariate Regression• Simulations• Analysis• Conclusions

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The Data: Three Image Modalities• deformation-based morphometry (DBM)

– determinant of the Jacobian matrix

• diffusion tensor images (DTI)– fractional anisotropy map after Eddy current and distortion correction

• perfusion weighted MRI (Perf)– partial volume corrected cerebral blood flow map

• 1) co-registered to subject’s T1 image• 2) mapped onto the atlas brain space using the deformation map from DBM• 3) divided by average perfusion in brain stem

All images were smoothed with a Gaussian kernel of variance 4, equivalent to SPM’s 10-12 mm FWHM kernel.

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Standard ApproachFor each modality, fit a univariate linear regression model.

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y = Xβ + �

y =

y1

y2...yn

is a vector of image values for a

single voxel.

X =

x11 x12 . . . x1p

x21 x22 . . . x2p... ... ...

xn1 xn2 . . . xnp

is a matrix of p

covariates for each subject.

β =

β1

β2...

βp

is a vector of coefficients for the

p covariates.

1

p covariates.

� =

�1

�2...�n

is a vector of errors.

y1y2...yn

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β1β2...βp

+

�1�2...�n

β̂ = (X �X)−1X �Y

2

vector of n image values for a single

voxel

matrix of p covariates for each subject.

vector of coefficients

for the p covariates

vector of n

errors

� =

�1�2...

�n


y1y2...yn

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β1β2...βp

+

�1�2...�n

B̂ = (X �X)−1X �Y

β̂ = (X �X)−1X �y

yi = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

yDBMJA = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

yDBMGA = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

3

Estimates of Beta

5

p covariates.

� =

�1

�2...�n


y1y2...yn

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β1β2...βp

+

�1�2...�n

β̂ = (X �X)−1X �Y



2

Multiple Comparisons• When multiple modalities are analyzed, one needs to adjust

for multiple comparisons.

• For three tests, each with α=0.05, P(at least one test significant) = 1- P(no significant results)

= 1- (1- 0.05)3

= 0.143

• One way to preserve α level for n tests is the Bonferroni correction:

threshold pvalues at 0.05/n or, equivalently, calculate adjusted pvalues

pnew = min(n*pold, 1) 6

The Problem

• Bonferroni tends to be overly conservative and therefore can lead to a loss in power.

• Taking into account correlation between the multiple outcomes can help increase power.

– Example: For three completely correlated outcomes, applying Bonferroni is equivalent to doing the same test three times with α = 0.05/n.

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• H0: β = 0 for all univariate models• HA: β ≠ 0 for at least one univariate model

• Fisher

• Stouffer

• BUT, these methods assume independence!

How else can we adjust for multiple comparisons?

9


yDTIFA = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

yDTIMD = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

yPerf = β1+Xageβ2+XI(male)β3+XI(alz)β4+�

p = P (χ22n ≥ −2

n�

i=1

ln pi)

yi ∼ N(β1 + XI(alz)β2 + Xnormβ3, 1)

3





p = P (χ22n ≥ −2

n�

i=1

ln pi)

p = 1− Φ

��ni=1 zi√n

�

3

zi = Φ−1(1− pi)


Cov(Y ) =

1 0.5 0.5 0.50.5 1 0.5 0.50.5 0.5 1 0.50.5 0.5 0.5 1

Cov(yj, yk) = 0.5

4

Multivariate Regression

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p covariates.

� =

�1

�2...�n


y = Xβ + �

matrix of image values for a single voxel.

y11 y12 . . . y1q

y21 y22 . . . y2q...

......

yn1 yn2 . . . ynq

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β11 β12 . . . β1q

β21 β22 . . . β2q... . . . . . . . . .

βp1 βp2 . . . βpq

+

�11 �12 . . . �1q

�21 �22 . . . �2q...

......

�n1 �n2 . . . �nq

2

matrix of nq image values for a single

voxel

matrix of p covariates for each subject.

vector of coefficients for

the pq covariates

vector of nq errors

p covariates.

� =

�1

�2...�n


y = Xβ + �

Y = XB + �


y11 y12 . . . y1q

y21 y22 . . . y2q...

......

yn1 yn2 . . . ynq

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β11 β12 . . . β1q

β21 β22 . . . β2q... . . . . . . . . .


+

�11 �12 . . . �1q

�21 �22 . . . �2q...

......

�n1 �n2 . . . �nq

2

� =

�1�2...

�n


y1y2...yn

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β1β2...βp

+

�1�2...�n

B̂ = (X �X)−1X �Y

β̂ = (X �X)−1X �y




3

More specifically...

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p covariates.

� =

�1

�2...�n


y = Xβ + �

Y = XB + �


y11 y12 . . . y1q

y21 y22 . . . y2q...

......

yn1 yn2 . . . ynq

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β11 β12 . . . β1q

β21 β22 . . . β2q... . . . . . . . . .


+

�11 �12 . . . �1q

�21 �22 . . . �2q...

......

�n1 �n2 . . . �nq

2

� =

�1

�2...�n


y = Xβ + �

Y = XB + �


y11 y12 . . . y1q

y21 y22 . . . y2q...

......

yn1 yn2 . . . ynq

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β11 β12 . . . β1q

β21 β22 . . . β2q... . . . . . . . . .


+

�11 �12 . . . �1q

�21 �22 . . . �2q...

......

�n1 �n2 . . . �nq

more specific

y1DBM y1FA y1Perf

y2DBM y2FA y2Perf...

......

y84DBM y84FA y84Perf

=

2

1 x1age x1sex x1AD

1 x2age x2sex x2AD...

......

1 x84age x84sex x84AD

βint,DBM βint,FA βint,Perf

βage,DBM βage,FA βage,Perf

βsex,DBM βsex,FA βsex,Perf

βAD,DBM βAD,FA βAD,Perf

+

�1DBM �1FA �1Perf

�2DBM �2FA �2Perf...

......

�84DBM �84FA �84Perf

� =

�1�2...

�n


y1y2...yn

=

x11 x12 . . . x1p

x21 x22 . . . x2p...

......

xn1 xn2 . . . xnp

β1β2...βp

+

�1�2...�n

B̂ = (X �X)−1X �Y

β̂ = (X �X)−1X �y

yi = β1 + Xageβ2 + XI(male)β3 + XI(alz)β4 + �

yDBMJA = β1 + Xageβ2 + XI(male)β3 + XI(alz)β4 + �

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Simulations: 4 Independent Outcomes

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β3 = (2 2 2 0)

Simulated data:

Analysis: H0: β3 = 0 for all outcomesHA: β3 ≠ 0 for at least one outcome

β3 = (2 2 0 0)

β3 = (0 0 0 0) β3 = (2 0 0 0)




p = P (χ22n ≥ −2

n�

i=1

ln pi)


3

Simulations: 4 Correlated Outcomes

13

β3 = (2 2 2 0)

Simulated data:

Analysis: H0: β3 = 0 for all outcomesHA: β3 ≠ 0 for at least one outcome

β3 = (2 2 0 0)

β3 = (0 0 0 0) β3 = (2 0 0 0)

Cov(Y ) =

1 0.5 0.5 0.50.5 1 0.5 0.50.5 0.5 1 0.50.5 0.5 0.5 1

Cov(yj, yk) = 0.5

4




p = P (χ22n ≥ −2

n�

i=1

ln pi)


3

Analysis: -log10 p-values

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Multiple Comparisons Across Voxels

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We set FDR at 0.10 and used the empirical null to obtain a p-value threshold of 6.2654 x 10-5.

Analysis: -log10 p-values thresholded

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Analysis: standardized β estimates

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DBM FA Perfusion

Note: The color bars are not necessarily equal for these images.

Conclusions• Voxelwise multivariate regression is a powerful tool for

analyzing multiple imaging modalities.

• By estimating the covariance structure, the multivariate model is able to account for multiple comparisons without assuming independence.

• The estimated coefficients in the model provide an easy way to examine correlated changes in the brain.

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voxelwise multivariate analysis of multimodality imaging

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