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Introduction |
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Most of the material and notations are
from Doug WardÕs manuals for the programs 3dttest, 3dANOVA, 3dANOVA2, 3dANOVA3,
and 3dRegAna, and from Gang ChenÕs manual for 3dANOVA4 |
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Documentation available with the AFNI
distribution |
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Lots of stuff (theory, examples)
therein |
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Software and documentation files are
based on these books: |
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Applied Linear Statistical Models by
Neter, Wasserman, and Kutner (4th edition) |
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Applied Regression Analysis by Draper
and Smith (3rd edition) |
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General steps |
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Smoothing (3dmerge -1blur_fwhm) |
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Normalization (3dcalc) |
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Deconvolution/Regression (3dDeconvolve) |
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Co-registration of individual analyses
to common ÒspaceÓ (adwarp -dxyz) |
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Group analysis (3dttest, 3dANOVA, É) |
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Post-analysis (AlphaSim, conjunction
analyses, É) |
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Interpretation and Thinking |
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Data Preparation: Spatial Smoothing |
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Spatial variability of both FMRI
activation and the Talairach transform (the common space) can result in
little or no overlap of function between subjects. |
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Data smoothing is used to reduce this
problem. |
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Leads to loss of spatial resolution,
but that is a price to be paid with the Talairach transform (or any current
technique that does inter-subject anatomical alignments) |
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In principle, smoothing should be done
on time series data, before data fitting (i.e., before 3dDeconvolve or 3dNLfim,
etc.) |
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Otherwise one has to decide on how to
smooth statistical parameters. |
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In statistical data sets, each voxel
has a multitude of different parameters associated with it like a regression
coefficient, t-statistic, F-statistic, etc. |
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Combining some statistical parameters
across voxels would result in parameters with unknown distributions |
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It is OK to blur percent signal change
values that come out of the regression analysis, since these numbers depend
linearly on the input data (unlike the F- and t-statistics) |
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Blurring in 3D is done using 3dmerge
with the -1blur_fwhm option |
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Blurring on the surface is done with
program SurfSmooth |
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Data Preparation: Parameter
Normalization |
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Parameters quantifying activation must
be normalized before group comparisons. |
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FMRI signal amplitude varies for
different subjects, runs, scanning sessions, regressors, image reconstruction
software, modeling strategies, etc. |
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Amplitude measures (regression
coefficients) can be turned to percent signal change from baseline (do it
before the individual analysis in 3dDeconvolve). |
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Equations to use with 3dcalc to
calculate percent signal change |
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100 bi / b0
(basic formula) |
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100 bi / b0 * c (mask out the outside of the brain) |
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bi = coefficient for regressor i (output from 3dDeconvolve) |
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b0 = baseline estimate
(output from 3dTstat -mean) |
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c
= threshold value generated from running 3dAutomask -dilate |
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This will be included into 3dDeconvolve
in a future release |
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Other normalization methods, such as z-score
transformations of statistics, can also be used. |
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Data Preparation: Co-Registration (AKA
ÒSpatial NormalizationÓ) |
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Group analyses are performed on a
voxel-by-voxel basis |
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All data sets used in the analysis must
be aligned and defined over the same spatial domain. |
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Talairach domain for volumetric data |
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Landmarks for the transform are set on
high-res. anatomical data using AFNI |
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Functional data volumes are then
transformed using AFNI interactively or adwarp from command line (use option -dxyz
with about the same resolution as EPI data Ñ do not use the default 1 mm
resolution!) |
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Standard meshes and spherical
coordinate system for surface data |
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Surface models of the cortical surface
are warped to match a template surface using Caret/SureFit (http://brainmap.wustl.edu)
or FreeSurfer (http://surfer.nmr.mgh.harvard.edu) |
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Standard-mesh surface models are then
created with SUMA (http://afni.nimh.nih.gov/ssc/ziad/SUMA) to allow for
node-based group analysis using AFNIÕs programs |
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Once data is aligned, analysis is
carried out voxel-by-voxel or node-by-node |
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The percent signal change from each
subject in each task/stimulus state are usually the numbers that will be
compared and contrasted |
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Resulting statistics (voxel-wise or
node-wise) can then be displayed in AFNI and/or SUMA |
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Overview of Statistical Testing of
Group Datasets with AFNI programs |
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Parametric Tests: |
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Assume data are normally distributed
(Gaussian) |
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3dttest (paired, unpaired) |
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3dANOVA (or 3dANOVA2 or 3dANOVA3) |
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3dRegAna (regression, unbalanced ANOVA,
ANCOVA) |
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3dANOVA4 = Matlab script for one-,
two-, three- and four-way ANOVA |
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Non-parametric analyses: |
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No assumption of normality |
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Tends to be less sensitive to outliers
(more robust) |
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3dWilcoxon (~t-test paired) |
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3dMannWhitney (~t-test unpaired) |
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3dKruskalWallis (~3dANOVA) |
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3dFriedman (~3dANOVA2) |
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Permutation test |
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Less sensitive and less flexible than
parametric tests |
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In practice, seems to make little
difference |
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Probably because number of datasets and
subjects is usually small (hard to tell if data is non-Gaussian when only
have a few sample points) |
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t-Tests [starting easy, but contains most of the ideas] |
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Program 3dttest |
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Used to test if the mean of a set of
values is significantly different from a constant (usually 0) or the
mean of another set of values. |
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Assumptions |
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Values in each set are normally
distributed |
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Equal variance in both sets |
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Values in each set are independent Þ unpaired t-test |
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Values in each set are dependent Þ paired t-test |
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Example: 20 subjects are tested for the
effects of 2 drugs A and B |
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Case 1: 10 subjects were given drug A
and the other 10 subjects given drug B. |
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Unpaired t-test is used to test: mA
= mB? (mean response is different?) |
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Equivalent to one-way ANOVA with
between-subjects design of equal sample size Þ can also run 3dANOVA
(treating subjects as multiple measurements) |
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Case 2: 20 subjects were given both
drugs at different times. |
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Paired t-test is used to test: mA
= mB? |
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Case 3: 20 subjects were given drug A. |
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t-test is used to test if drug effect
is significant at group level: mA = 0? |
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1-Way ANOVA |
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Program 3dANOVA |
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Determine whether treatments (levels)
of a single factor (independent parameter) has an effect on the measured
response (dependent parameter, like FMRI percent signal change due to some
stimulus). |
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Examples of factor: subject type, task
type, task difficulty, drug type, drug dosage, etc. |
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Within a factor are levels: different
sub-categorizations |
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Example: factor=subject type; level
1=normals, level 2=patients with mild symptoms, level 3=patients with severe
symptoms |
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The various AFNI ANOVA programs differ
in the number of factors they allow: 3dANOVA allows 1 factor, comprising up to 100 levels |
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Assumptions |
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Values are normally distributed |
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No assumptions about relationship
between dependent and independent variables (e.g., not necessarily linear) |
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Independent variables are qualitative |
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Can also use 3dttest if there are only
two levels |
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The 1-way 3dANOVA analysis is a
generalization to multiple levels of an unpaired 3dttest (for generalization
of paired, wait for 3dANOVA2) |
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Example: r different types of subjects
performed the same task in the scanner |
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Null Hypothesis: H0 : m1 = m2 = É = mr |
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i.e., subject type has
no effect on mean signal in this voxel |
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Alternative Hypothesis: Ha
: not all mi are equal |
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i.e., at least one
subject type had a different mean FMRI signal |
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3dANOVA is effectively a generalization
of the unpaired t-test to multiple columns of data (a further refinement will
be introduced with 3dANOVA3) |
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As such, 3dANOVA is probably not
appropriate when comparing results of different tasks on the same subjects
(need a generalization of the paired t-test: 3dANOVA2) |
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Difference from doing unpaired t-tests
on pairs of columns: variance estimates are all pooled together, increasing
the denominatorial degrees of freedom |
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Assumption is that data fluctuations in
each column have same variance |
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ANOVA: Which levels had an effect or
were different from one another? |
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Usually, just knowing that there is a main
effect (some of the means are different, but no information about which ones)
isnÕt enough, so there is a number of options to let you look for more detail |
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Which treatment means (mi )
are 0 ? |
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e.g., is the response of subjects in
level #3 different from 0 ? |
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t-statistic with option -mean in 3dANOVA |
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Similar to using 3dttest -base1 0 (single sample test) to test only the
data from those subjects |
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Which treatment means are different
from each other ? |
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e.g., is the response of subjects in
level #3 different from those in level #2 ? |
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t-statistic with option -diff in 3dANOVA |
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Similar to using 3dttest (unpaired)
between the data from these sets of subjects |
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Which linear combination of means (contrasts)
are 0 ? |
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e.g., is the average response of
subjects in level #1 different from the combined average of subjects in
levels #2 and #3 ? |
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t-statistic with option -contr in 3dANOVA |
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2-Way ANOVA: test for effects of two
independent factors on measurements |
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This is a fully crossed analysis: all
combinations of factor levels are measured |
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In particular, if one factor is
ÒsubjectÓ, then all subjects are tested in all levels of the other factor |
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Program is limited to balanced designs:
Must have same number of measurements in each ÒcellÓ (combinations of factor
levels) |
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Example: Stimulus type for factor A and
subject for factor B |
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Each subject is a level within factor B
(1 measurement per cell) |
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This is a fixed effect « random effect model = Òmixed effectÓ model |
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Example: Stimulus type for factor A and
drug treatment for factor B |
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Each subject is an independent
measurement for both factors, all levels |
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This is a fixed effect « fixed effect model |
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If you also want to treat subject as a
separate factor, need 3dANOVA3 |
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Example: Stimulus type for factor A,
stimulus day for factor B |
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With one fixed subject, for a
longitudinal study (e.g., training between scan days) |
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This also is a fixed effect « fixed effect model |
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Again, multiple subjects could be
treated as independent measurements in 3dANOVA2, or as a third factor in 3dANOVA3 |
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Choose between two types of analysis
for each factor: fixed and random effects |
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Fixed effects factor = differences
between levels in this factor are modeled as deterministic differences in the
mean measurements (as in 3dANOVA and 3dttest) |
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Useful for most categories under the
experimenterÕs control or observation |
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Allows same type of statistics as 3dANOVA: |
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factor main effect (are all the mean
activations of each level in this factor the same?) |
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differences between level pairs (e.g.,
level #2 same as #3?) |
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more complex contrasts (e.g., average
of levels #1 and #2 same as level #3?) |
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If both factors are modeled as fixed
effects with multiple measurements (e.g., subjects): |
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Can also test for interaction between
the factors |
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Are there any combinations of factor
levels whose means Òstick outÓ [e.g., mean of cell #(A1,B2)
differs from (#A1 mean)+(#B2 mean)]? |
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Example: A=stimulus type, B=drug type;
then cell #(A1,B2) is FMRI response (in each voxel) to
stimulus #1 and drug #2 |
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Interaction test would determine if any
individual combination of drug type and stimulus type was abnormal |
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e.g., if stimulus #1 averages a high
response, and drug #2 averages no effect on response, but when together,
value in cell #(A1,B2) averages small |
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i.e., Effect of one factor (stimulus)
depends on level of other factor (drug) |
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no interaction means the effects of the
factors are always just additive |
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Inter-factor contrasts can then be used
to test individual combinations of cells to determine which cell(s) the
interaction comes from |
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Random effects factor = differences
between levels in this factor are modeled as random fluctuations |
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Useful for categories not under
experimenterÕs control or observation |
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In FMRI, is especially useful for
subjects; a good rule is |
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treat subjects as a separate random
effects factor rather than |
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as multiple independent measurements
inside fixed-effect factors |
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In such a case, usually have 1
measurement per cell (each cell is the combination of a level from the other
factor with 1 subject) |
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This is sometimes called a Òrepeated
measures ANOVAÓ, when we have multiple measurements on each subject (in this
case, across different stimulus classes) |
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Treating subjects as a random factor in
a fully crossed analysis is a generalization of the paired t-test |
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intra-subject and inter-subject data
variations are modeled separately |
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which can let you detect small
intra-subject changes due to the fixed-effect factors that might otherwise be
overwhelmed by larger inter-subject fluctuations |
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Main effect for a random effects factor
tests if fluctuations among levels in this factor have additional variance
above that from the other random fluctuations in the data |
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e.g., Are inter-subject fluctuations
bigger than intra-subject fluctuations? |
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Not usually very interesting when
random factor = subject |
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It is hard to think of a good FMRI
example where both factors would be random |
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3dANOVA2: Usually have 1 fixed factor
and 1 random factor = mixed effects analysis |
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3dANOVA2: A test case |
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Michael S. Beauchamp, Kathryn E. Lee,
James V. Haxby, and Alex Martin, fMRI Responses to Video and Point-Light
Displays of Moving Humans and Manipulable Objects, Journal of Cognitive
Neuroscience, 15: 991-1001
(2003). |
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Purpose is to study the organization of
brain responses to different types of complex visual motion (the 4 levels
within factor A) from 9 subjects (the levels within factor B) |
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Data from 3 of the subjects, and
scripts to process it with AFNI programs, are available in AFNI HowTo #5
(hands-on) |
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Available for download at the AFNI web
site: http://afni.nimh.nih.gov/afni/doc/howto/ |
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If you want all the data, it is at the
FMRI Data Center at Dartmouth: http://www.fmridc.org |
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Or at least, it should be (but they
havenÕt posted it yet for some reason) |
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Data Processing Outline |
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Image registration with 3dvolreg |
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Images smoothed (4 mm FWHM) with 3dmerge |
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IRF for each of the 4 stimuli were
obtained using 3dDeconvolve |
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Regressor coefficients (IRFs) were
normalized to percent signal change (using 3dcalc) |
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An average activation measure was
obtained by averaging IRF amplitude from the 4th through the 10th second of
the response (using 3dTstat) |
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Capturing the positive
blood-oxygenation level dependent response but not any post-stimulus
undershoot |
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These activation measures will be the
measurements in the ANOVA table |
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After each subjectÕs results are warped
to Talairach coordinates, using adwarp program |
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3dANOVA2 was carried out with: |
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Factor A, fixed effects: levels = HM, TM, HP, TP (4
types of stimuli) |
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Factor B, random effects: levels = 9 subjects |
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1 measurement per cell |
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3dANOVA2 script |
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3dANOVA2 -type 3 -alevels 4
-blevels 9 \ |
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-dset 1 1 ED+tlrc'[0]' -dset 2 1
ED+tlrc'[1]' \ |
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-dset 3 1 ED+tlrc'[2]' -dset 4 1
ED+tlrc'[3]' \
-dset 1 2 EE+tlrc'[0]' -dset 2 2 EE+tlrc'[1]' \ |
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-dset 3 2 EE+tlrc'[2]' -dset 4 2
EE+tlrc'[3]' \ |
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É É |
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-dset 1 9 FN+tlrc'[0]' -dset 2 9
FN+tlrc'[1]' \ |
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-dset 3 9 FN+tlrc'[2]' -dset 4 9
FN+tlrc'[3]' \ |
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-amean 1 TM -amean 2 HM -amean 3 TP
-amean 4 HP \ |
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-acontr 1 1 1 1 AllAct
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-acontr -1 1 -1 1 HvsT \
-acontr 1 1 -1 -1 MvsP \
-acontr 0 1 0 -1 HMvsHP \
-acontr 1 0 -1 0 TMvsTP
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-acontr 0 0 -1 1 HPvsTP \
-acontr -1 1 0 0 HMvsTM
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-acontr 1 -1 -1 1 Inter
\ |
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-fa StimEffect \ |
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-bucket AvgANOVA |
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3dANOVA2: specifying which statistics
to output |
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3dANOVA2 -type 3 -alevels 4 -blevels 9 ÉÉ \ |
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-amean 1 TM -amean 2 HM -amean 3 TP -amean 4 HP \ |
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-acontr 1 1 1 1 AllAct \
-acontr -1 1 -1 1 HvsT
\
-acontr 1 1 -1 -1
MvsP \
-acontr 0 1 0 -1 HMvsHP
\
-acontr 1 0 -1 0 TMvsTP \
-acontr 0 0 -1 1 HPvsTP
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-acontr -1 1 0 0 HMvsTM \
-acontr 1 -1 -1 1 Inter \ |
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-fa StimEffect \
-bucket AvgANOVA |
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-amean 1 TM: estimate mean of factor A, level 1 and label it TM in the
output dataset |
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-acontr : specifies contrast matrix and label in output dataset |
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1 1 1 1: all of factor A's levels summed = 0? |
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-1 1 -1 1: contrast between human and tools (HM
+ HP) Ð (TM + TP) |
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1 1 -1 -1:
contrast between motion and points (HM + TM) Ð (HP + TP) |
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0 1 0 -1: contrast between human motion and
points (HM Ð HP) |
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É É |
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-fa StimEffect: F-statistic for main effect of factor
A (any differences among stimuli?) |
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-bucket AvgANOVA: prefix of output
dataset containing statistical results |
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3dANOVA2: viewing results |
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Main effect: Regions showing presence
of differences in activation due to changes in stimulus type (which
differences must be determined via later contrasts) |
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view StimEffect sub-bricks for function
and threshold (F-stat = 15, p =10-5) |
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Factor Means: Activation in response to
each category |
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view TM, HM, etc. sub-bricks (t-stat =
10.6, p = 10-10) |
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all categories appear to activate same
areas |
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Choose AllAct sub-bricks for finding
regions activated by at least one of the stimuli |
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this region of activation is often used
to select an ROI which is examined for subtler effects |
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Choose HvsT (human versus tools)
sub-bricks |
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note small range of t-values (subtler
effects, if any) |
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lower t-stat threshold to 4, p ~ 5x10-4 |
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might want to restrict hypothesis
testing to region activated by stimuli |
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Look for interactions that might
complicate your fairy tale (AKA hypothesis) |
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view the Inter sub-bricks to determine
if some areas for which the contrast (TM+HP)Ð(HM+TP) is significant |
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Hopefully youÕll find few/none, or be
prepared to explain such activations |
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3-Way ANOVA: 3dANOVA3 (again, balanced
designs only) |
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Read the manual first and understand
what options are available |
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It is important to understand 2-way
ANOVA before moving up to the big time show! |
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Has several fixed effects and random
effects combinations |
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Has new concept: nested design (vs.
fully crossed design) |
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Nested design is for use when you have
2 fixed effects factors and 1 random effects factor where the subjects for
the random effects factor depend on one of the fixed effect factors; example: |
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factor A = subject type; level
#1=normal, #2=genotype Q, #3=genotype R |
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factor B = stimulus type; levels
#1Ð4=different types of videos |
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factor C = subject; levels #1Ð10 = 30
different subjects, 10 in each of the factor A levels; C is ÒnestedÓ inside A |
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Nested design is a mixture of unpaired
and paired tests |
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Will be like ÒpairedÓ for tests across
stimulus type (factor B levels) |
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Will be like ÒunpairedÓ across subject
types (factor A levels) |
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Fully crossed design is when the
subjects are common across the other factors |
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As was said before, un-nested design is
a generalization of paired t-test |
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Treating the subjects correctly is a
crucially important decision |
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Unlike 3dANOVA2, 3dANOVA3 does not
currently allow for arbitrary contrasts between random cells in different
factors/different levels |
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4-Way ANOVA: ready to rock-n-roll (for
the daring and intrepid) |
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Interactive Matlab script |
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Can run both crossed and nested (i.e., subject
nested into gender) design |
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Heavy duty computation + Matlab: expect
to take 10s of minutes to hours |
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Same script can also do ANOVA, ANOVA2,
and ANOVA3 analyses |
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Includes contrast tests across all
factors |
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At present, must have a balanced design
with no missing data |
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equal number of entries in each cell |
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can be a problem when studying patients
(e.g., hard to find some genotypes) |
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Working now to implement more options,
such as |
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ANCOVA (ANOVA plus regression with
continuous covariates; e.g., age) |
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Recent news: now working!! |
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unbalanced designs (uneven numbers of
entries in cells, or levels in factors) |
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Recent news: now working for some
cases! |
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missing data (e.g., some subjects
couldnÕt perform certain tasks) |
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Goal: be a user-friendly alternative to
running 3dRegAna for most complicated analyses of group datasets |
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Goal: once program is stabilized,
re-write in C for speed and independence from the commercial product Matlab |
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In a mixed effects model, ANOVA cannot
deal with unequal variances in the random factor between different levels of
a fixed factor |
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Example: 2-way layout, factor
A=stimulus type (fixed effect), factor B=subject (random effect) |
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As seen earlier, ANOVA can detect differences in means
between levels in A (different stimuli) |
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But if the measurements from different
stimuli also have significantly different variances (e.g., more attentional
wandering in one task vs. another), then the ANOVA model for the signal is
wrong |
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In general, this ÒheteroscedasticityÓ
problem is a difficult one, even in a 2-sample t-test; there is no exact F-
or t-statistic to test when the means and the variances might differ
simultaneously |
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Although ANOVA does allow somewhat for
intra-subject correlations in measurements, it is not fully general |
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Example: 2-way layout as above, 3
stimulus types in factor A; general correlation matrix between the 3
different types of responses is
but ANOVA only properly deals with the case r12=r13=r23
(recall we are assuming subject effects are random; this is the
correlation matrix for the intra-subject random responses). |
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Possible solution: general
linear-quadratic minimum variance mixed effects modeling |
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A statistical theory not yet much
applied to FMRI data (but it will be, someday) |
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Questions of sample size (number of
subjects needed) will surely arise |
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Conjunction Junction: WhatÕs Your
Function? |
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The program 3dcalc is a general purpose
program for performing logic and arithmetic calculations |
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Command line is of the format |
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3dcalc -a Dset1 -b Dset2 ... -expr
Ò(a * b ...)Ó |
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Some expressions can be used to select voxels with values v
meeting certain criteria: |
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Find voxels where v ³ th and mark them with value=1 |
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expression = step (v Ð th) (result is 1 or 0) |
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In a range of values: thmin ²
v ² thmax |
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expression = step (v Ð thmin) * step (thmax
- v) |
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Exact value: v = n |
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expression = equals(v
Ð n) |
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Create masks to apply to functional datasets |
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Two values both above threshold (e.g.,
active in both tasks; ÒconjunctionÓ) |
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expression = step(v-A)*step(w-B) |
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Regression Analysis: 3dRegAna |
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Simple linear regression: |
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Y = b0 + b1X1,+
e |
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where Y represents the FMRI measurement
(i.e., percent signal change) and X is the independent variable (i.e., drug
dose) |
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Multiple linear regression: |
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Y = b0 + b1X1
+ b2X2 + b3X3 + É+ e |
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Regression with qualitative and
quantitative variables (ANCOVA) |
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i.e., drug dose (5mg, 12mg, 23mg, etc.)
is quantitative while drug type (Nicotine, THC, Cocaine) or age group (young
vs. old) or genotype is qualitative, and usually called dummy (or indicator)
variable |
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ANOVA with unequal sample sizes (with
indicator variables) |
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Polynomial regression: |
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Y = b0 + b1X1
+ b2X12 + É + e |
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Linear regression: model is a linear
function of its unknowns bi
, NOT its independent variables Xi |
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Not for fitting time series, use 3dDeconvolve
(or 3dNLfim) instead |
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F-test for Lack of Fit (lof) |
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If multiple measurements are available
(and they should be), a Lack Of Fit (lof) test is first carried out. |
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Hypothesis: |
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H0: E(Y) = b0
+ b1X1 + b2X2 + É,+ bp-1Xp-1 |
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Ha: E(Y) b0
+ b1X1 + b2X2 + É,+ bp-1Xp-1 |
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Hypothesis is tested by comparing the
variance of the modelÕs lack of fit to the measurement variance at each point
(pure error). |
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If Flof is significant then model is
inadequate. STOP HERE. |
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Reconsider independent variables, try
again. |
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If Flof is insignificant then model appears
adequate, so far. |
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It is important to test for the lack of
fit: |
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The remainder of the analysis assumes
an adequate model is used |
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You will not be visually inspecting the
goodness of the fit for thousands of voxels! |
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Test for Significance of Linear
Regression |
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This is done by testing whether
additional parameters significantly improve the fit |
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For simple case |
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Y = b0 + b1X1
+ e |
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H0: b1 = 0 |
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H1: b1 0 |
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For general case |
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Y = b0 + b1X1
+ b2X2 +
É + bq-1Xq-1 + bqXq + É + bp-1Xp-1 +
e |
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H0: bq = bq+1
= ... = bp-1 = 0 |
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Ha: bk 0, for
some k, q ² k ² p-1 |
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Freg is the F-statistic for determining
if the Full model significantly improved on the reduced model |
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NOTE: This F-statistic is assumed to
have a central F-distribution. This is not the case when there is a lack of
fit |
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3dRegAna: Other statistics |
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How well does model fit data? |
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R2 (coefficient of multiple
determination) is the proportion of the variance in the data accounted for by
the model 0 ² R2 ² 1. |
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i.e., if R2 = 0.26 then 26%
of the dataÕs variation about their mean is accounted for by the model. So
this might indicate the model, even if significant, might not be that useful
(depends on what use you have in mind) |
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Having said that, you should consider R2
relative to the maximum it can achieve given the pure error which cannot be
modeled. [cf. Draper & Smith, chapter 2]. |
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Are individual parameters bk
significant? |
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t-statistic is calculated for each
parameter |
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helps identify parameters that can be
discarded to simplify the model |
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R2 and t-statistic are
computed for full (not reduced) model |
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3dRegAna: Qualitative Variables
(ANCOVA) |
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Qualitative variables can also be used |
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i.e., WeÕre modeling the response
amplitude to a stimulus of varying contrast when subjects are either young,
middle-aged or old. |
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X1 represents the stimulus
contrast (quantitative): continuous covariate |
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Create indicator variables X2
and X3 to represent age: |
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X2 = 1 if subject is
middle-aged |
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= 0 otherwise |
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X3 = 1 if subject is old
(i.e., at least 1 year older than Bob Cox) |
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= 0 otherwise |
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Full Model (no interactions between age
and contrast) |
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Y = b0 + b1X1
+ b2X2 + b3X3 + e |
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E(Y) = b0 + b1X1
for young
subjects |
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E(Y) = ( b0 + b2 )
+ b1X1 for middle-aged subjects |
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E(Y) = ( b0 + b3 )
+ b1X1 for old subjects |
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Full Model (with interactions between
age and contrast) |
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Y = b0 + b1X1
+ b2X2 + b3X3 + b4X2X1
+ b5X3X1 + e |
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E(Y) = b0 + b1X1
for young
subjects |
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E(Y) = ( b0 + b2 ) + ( b1 + b4
)X1 for middle-aged subjects |
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E(Y) = ( b0 + b3 )
+ ( b1 + b5 )X1 for old subjects |
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Will be easier to run analysis in
Matlab script for 3dANOVA4, when ready! |
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3dRegAna: ANOVA with unequal samples |
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3dANOVA2 and 3dANOVA3 do not allow for
unequal samples in each combination of factor levels |
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Can use 3dRegAna to look for main
effects and interactions |
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The analysis method involves the use of
indicator variables so it is practical for small for small number (~3) of
factor levels |
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Details are in the 3dRegAna manual |
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method is significantly more
complicated than running ANOVA; you must understand the math |
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avoid this, if you can, especially if
you have more than 4 factor levels or more than 2 factors |
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Interactions hard to interpret, and
contrast tests unavailable |
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Will be easier to run analysis in
Matlab script for 3dANOVA4, when ready! |