Group Analysis with FourWay ANOVA in AFNI

1
Group Analysis with Four-Way ANOVA in AFNI
Gang Chen, Ziad S. Saad, Robert W. Cox Scientific
and Statistical Computing Core, National
Institute of Mental Health National Institutes of
Health, Department of Health and Human Services,
USA
Introduction
We start with a prototype of four-way ANOVA with
a basic design of AXBXCXD with all factors fixed.
The corresponding cell means model leads to a
general linear model y X? ?where y is an n
? 1 vector of the observation values, X is the
cell means design n ? m matrix, ? is the m ? 1
regression coefficient vector, and ? the random
error n ? 1 vector. This leads to solving the
normal equations for ordinary least squares
estimationX'X? X'yDue to the coding with
dummy variables, design matrix X is rank deficit
as rank(X)ltm. In the meantime, a constraints
matrix C is defined based on all the factors and
their various interactions. The numerical
calculations are done through the following basic
steps (1) QR decomposition of constraints
matrix CCEc QcRcEc is a permutation matrix
so that ?diag(Rc)? is decreasing.(2) Projection
of the design matrix X into the null space Qc0 of
the constraints matrix C, which is composed of
those rows of Qc corresponding to the diagonal
zeros in RcXp XQc0 (3) QR decomposition of
XpXpEd QdRdAgain Ed is a permutation matrix
so that ?diag(Rd)? is decreasing.(4) The
degrees of freedom (df) and sum of squares (SS)
df rank(Qd)SS y2 Qd' y2 The above
steps apply to computing random error and all
ANOVA terms (main effects and interactions) as
well. Other design types are also based on this
basic algorithm. As a demonstration, we assume a
four-way ANOVA with a design of B?C ? D(A), and
among the four factors, A, B, and C are fixed
while D is random and nested within A. Following
the rules of thumb for writing the ANOVA table
(1, 2), we have an ANOVA table with all available
variation sources and their corresponding F
statistics. Various contrasts with their t
statistics are constructed in the same fashion
with relevant variance estimates.
Four-way ANOVA with an unbalanced design (unequal
sample size) and with covariates (ANCOVA) are
currently under development. The package for
four-way ANOVA can be downloaded from the AFNI
website http//afni.nimh.nih.gov/sscc/gangc
Software Implementation
Experimental designs with FMRI are increasingly
requiring more factors in group analysis, thus
impelling the creation of a four-way ANOVA
program in AFNI. With potential expansion to a
program capable of running ANCOVA and unbalanced
designs, the four-way ANOVA for AFNI datasets is
currently implemented in Matlab by converting
factors into dummy variables. QR decomposition is
used to solve the normal equations of the general
linear system. Five design types (fixed/random
and crossed/nested) are embedded in the program,
allowing for the user to analyze most typical
experiments. We present a streamlined way of
running ANOVA in which information requested for
the four-way analysis is straightforward and
saved for the users records. Runtime for a
typical four-way ANOVA is usually about half an
hour.
Sample Dialog Questions and Answers How many
factors? 4 Choose design type (0, 1, 2, 3, 4, 5,
...) 2 How many slices along the Z axis?
40 Label for No. 1 factor MD How many levels
does factor A (MD) have? 2 Label for No. 1 level
of factor A (MD) is VI1 Label for No. 2 level of
factor A (MD) is AU Label for No. 4 factor
SJ How many levels does factor D (SJ) have?
12 Label for No. 1 level of factor D (SJ) is
S1 There should be totally 96 input files.
Correct? (1 - Yes 0 - No) 1 (1) factor
combination factor A (MD) at level 2 (VI1)
factor B (FB) at level 2 (NW) factor C (CG)
at level 1 (AN) factor D (SJ) at level 12 (S1)
is ss15.a_sound.irf.meantlrc.BRIK How many
2nd-roder contrasts? (0 if none) 7 Label for 2nd
order contrast No. 1 is vis_avt How many terms
are involved? 2 Factor index for No. 1 term is
(e.g., 0120) 1010 Corresponding coefficient
(i.e., 1 or -1) 1 Factor index for No. 2 term is
(e.g., 0120) 1020 Corresponding coefficient
(i.e., 1 or -1) -1 Running ANOVA on
slice 1... done in 20.748358 seconds

Four-Way ANOVA Table (BF?CF?DR(AF))
Theory and Numerical Considerations
Group analysis is a critical stage in FMRI
analysis when the investigator makes some
generalization about the conditions/stimuli or
their comparisons from single subject to
population level. Such a step usually involves
the analysis of variance (ANOVA) with various
categorizations of stimulus by treating subjects
as a random factor. Previously one-, two-, and
three-way ANOVAs were implemented in AFNI in C as
three separate programs by calculating various
sums of squares and t/F statistics. Until
recently, these programs met the needs of the
users. However, contrast tests among second-order
and above terms were not available in three-way
ANOVA due to its complications in computation.
More importantly, as investigations get more
complicated and refined, higher numbers of
stimulus categorization are involved in the
analysis at group level, and thus a four-way
ANOVA in AFNI became highly desirable. Other
than the numbers of stimulus categorization,
concomitant variables (covariates) and unbalanced
design or missing data are very typically
encountered in FMRI group analysis. With these
considerations in mind, a general linear model
approach was adopted by coding factor levels into
values of dummy variables. Numerical computation
is not done through indexing terms as in previous
ANOVA programs instead the QR decomposition of
the design matrix is used to project each term
onto its corresponding subspace, and to obtain
various sums of squares for all possible terms.
Five Design Types of Four-Way ANOVA

• Neter, J., Kutner, M. H., Nachtsheim, C. J., and
  Wasserman, W. (1996), Allied Linear Statistical
  Models, Fourth Edition, McGraw-Hill.
• Keppel, G., and Wickens, T. (2004), Design and
  Analysis. A Research Handbook (4th Ed.), Prentice
  Hall.