Differences between revisions 8 and 41 (spanning 33 versions)

Longitudinal Statistics

This page describes ways of analyzing longitudinal data after processing it using the longitudinal stream in Freesurfer.

Longitudinal data are more complex than cross-sectional data, as repeated measures are correlated within each subject. The strength of this correlation will depend on the time separation between scans. In addition, extra care must be taken when the data exhibit significant between-subject variation in number of time points and between-scan intervals (imperfect timing). A statistical analysis should then consider these data features in order to obtain valid statistical inferences.

Freesurfer currently comes with (at least) three different frameworks for the analysis of longitudinal data:

Simplified repeated measures ANOVA (ignores correlation and timing of the measurement occasions)
Direct analysis of atrophy rates or percent changes (ignores correlation and single time points)
Linear mixed effects models <-- recommended (but more complex)

Simplified Repeated Measures ANOVA

This method can be used to check for differences between individual time points or compare time point differences across groups. For two time points it simplifies to a PairedAnalysis.

Advantages:

Included in mri_glmfit.
Does not assume any specific trend in the mean response over time and thus can capture complex trajectories.
Can make use of different multiple comparisons methods that come with mri_glmfit.

Disadvantages:

Does NOT consider the correlation among the repeated measures, and thus, there is a significant reduction in statistical power.
Does NOT consider the timing of the measurement occasions which may result in a further reduction in power.
Can only be applied to balanced data (all subject have their scans acquired at the same set of measurement occasions) with a small number of repeated measures (<=3).

For details see: RepeatedMeasuresAnova

Analysis of Rates or Percent Changes

To analyze, e.g. annualized percent change or atrophy rates for 2 or more time points, one can run a two stage model. This avoids dealing with the longitudinal correlation. The two stages are:

First, simplify the statistic to a single number for each subject (the difference of two time points, or the slope of the fitting line, or the annualized percent change, etc...).
Then analyze the obtained summary measure across subjects or groups with a standard GLM.

This model is quite simple and can be an option if all subjects have the same number of time points, approximately equally spaced. Linear fits into each subject data are often meaningful, as longitudinal change can be assumed to be almost linear within a short time frame in several applications.

Advantages:

Can deal with differently many and differently spaced time points (but does not model the difference in variability).
Works on ROI stat (e.g. aseg.stats or aparc.stats) and on cortical maps (e.g. thickness).
The second stage can be performed with QDEC (simple GUI) or directly with mri_glmfit.
The second stage analysis can make use of different multiple comparisons methods that come with mri_glmfit.
Scripts are available ( long_mris_slopes and long_stats_slopes ), no matlab needed.
For the simple case of two time points and when looking at simple differences this model simplifies to a paired analysis, but can additionally compute (symmetrized) percent changes.
Includes code for intersecting cortex labels (across time and across subjects) to make sure that all non-cortex vertices are excluded.

Disadvantages:

Does NOT model the correlation among the repeated measures, and thus, there is a significant reduction in statistical power.
Does NOT account for different certainty of within subject slopes depending on the number of time points and therefore it has the highest propensity to false positives (type I family wise error in the mass-univariate setting).
Difficult to model non-linear temporal behaviour.
Difficult to deal with time varying co-variates (slopes would need to be fit into those for each subject to reduce these to a single number).
Cannot include information from subjects with only a single time point and thus the results are likely to be biased and have less statistical power.

The linear mixed effects model overcomes these limitations and should be used if subjects have differently many time points (or for more complex modeling).

For details see: LongitudinalTwoStageModel

Linear Mixed Effects Model

A Linear Mixed Effects (LME) model is the most powerful and principled approach. We recommend this approach.

Advantages:

Works for both stats (univariate) and surface analysis (mass-univariate).
Can handle unequal timing and different number of time points across subjects (missing data).
Even subjects with only a single time point can be included into these models (make sure they also run through the longitudinal stream, available with version FS 5.2, to avoid a bias due to different image processing) .
Appropriately models the temporal correlation.
Can model different variances across measurement occasions.
Our mass-univariate method can deal very well with the spatial correlation among measurements on the cortex and is very fast by working with spatial regions in which the correlation structure is relative constant.
Can be used to model complex longitudinal behavior (e.g. quadratic, or piecewise linear trajectories) and time-varying covariates.
It seems to have become the consensus among statisticians that LME models are the right mechanism to study longitudinal data and they may be requested in journal publications by the reviewers.

Disadvantages:

More complicated use (e.g. requires distinguishing mixed effects from fixed effects ...).
Currently, our implementation is in Matlab only.
Currently only offers FDR for multiple comparisons correction.

For details see: LinearMixedEffectsModels

MartinReuter

-  ⇤ ← Revision 8 as of 2012-11-09 18:41:44 → 
  Size: 17548
  Editor: jbernal
  Comment:
+   ← Revision 41 as of 2015-11-09 15:12:19 → ⇥
  Size: 6338
  Editor: AllisonMoreau
  Comment: fixing typo
-Deletions are marked like this.
+Additions are marked like this.
 Line 1:
-#acl LcnGroup:read,write,delete,revert All:
+#acl LcnGroup:read,write,delete,revert All:read
 Line 3:
-LME Matlab tools. Author: Jorge Luis Bernal Rusiel, 2012. jbernal@nmr.mgh.harvard.edu or jbernal0019@yahoo.es
+This page describes ways of analyzing longitudinal data after processing it using the [[LongitudinalProcessing|longitudinal stream]] in Freesurfer.
 Line 5:
-If you use these tools in your analysis please cite:
+Longitudinal data are more complex than cross-sectional data, as repeated measures are correlated within each subject. The strength of this correlation will depend on the time separation between scans. In addition, extra care must be taken when the data exhibit significant between-subject variation in number of time points and between-scan intervals (imperfect timing).  A statistical analysis should then consider these data features in order to obtain valid statistical inferences.
 Line 7:
-Bernal-Rusiel J.L., Greve D.N., Reuter M., Fischl B., Sabuncu M.R., 2012. Statistical Analysis of Longitudinal Neuroimage Data with Linear Mixed Effects Models, NeuroImage, doi:10.1016/j.neuroimage.2012.10.065.
+Freesurfer currently comes with (at least) three different frameworks for the analysis of longitudinal data:
 Line 9:
-These Matlab tools are freely distributed and intended to help neuroimaging researchers when analysing longitudinal neuroimaging (LNI) data. The statistical analysis of such type of data is arguable more challenging than the cross-sectional or time series data traditionally encountered in the neuroimaging field. This is because the timing associated with the measurement occasions and the underlying biological process under study are not usually under full experimental control.
+. Simplified [[RepeatedMeasuresAnova|repeated measures ANOVA]] (ignores correlation and timing of the measurement occasions)
 1. [[LongitudinalTwoStageModel|Direct analysis of atrophy]] rates or percent changes (ignores correlation and single time points)
 1. [[LinearMixedEffectsModels|Linear mixed effects models]] <-- '''recommended''' (but more complex)
-Line 11:
+Line 13:
-There are two aspects of longitudinal data that require correct modeling: The mean response over time and the covariance among repeated measures on the same individual. I hope these tools can serve for such modeling purpose as they provide functionality for exploratory data visualization, model specification, model selection, parameter estimation, inference and power analysis including sample size estimation. They are specially targeted to be used with Freesurfer's data but can be used with any other data as long as they are loaded into Matlab and put into the appropriate format. Here are some recommendations about how to use these tools.
+----
-Line 13:
+Line 15:
-<<TableOfContents>>
+== Simplified Repeated Measures ANOVA ==
This method can be used to check for differences between individual time points or compare time point differences across groups. For two time points it simplifies to a PairedAnalysis.
-Line 15:
+Line 18:
-== Preparing your data ==
There are two types of analyses that can be done: univariate and mass-univariate. The first step is to load your data into Matlab. If you are working with Freesurfer then univariate data (eg. Hippocampus' volume) can be loaded using Qdec tables. There are, under the Qdec directory, some simple example scripts for reading and writing Freesurfer's Qdec tables.
+'''Advantages:'''
-Line 18:
+Line 20:
-In order to read mass-univariate data you should use the following scripts:
+ * Included in mri_glmfit.
 * Does not assume any specific trend in the mean response over time and thus can capture complex trajectories.
 * Can make use of different multiple comparisons methods that come with mri_glmfit.
-Line 20:
+Line 24:
-''fs_read_label.m ''
+'''Disadvantages:'''
-Line 22:
+Line 26:
-''fs_read_surf.m ''
+ * Does NOT consider the correlation among the repeated measures, and thus, there is a significant reduction in statistical power.
 * Does NOT consider the timing of the measurement occasions which may result in a further reduction in power.
 * Can only be applied to balanced data (all subject have their scans acquired at the same set of measurement occasions) with a small number of repeated measures (<=3).
-Line 24:
+Line 30:
-''fs_read_Y.m ''
+For details see: RepeatedMeasuresAnova
-Line 26:
+Line 32:
-The last two depend on Freesurfer's scripts so you need to have installed Freesurfer software package and included the Freesurfer's matlab subdirectory in your Matlab's search path.
+----
-Line 28:
+Line 34:
-Previously, the mass-univariate data can be generated in Freesurfer by running variants of the following commands:
+== Analysis of Rates or Percent Changes ==
To analyze, e.g. annualized percent change or atrophy rates for 2 or more time points, one can run a two stage model. This avoids dealing with the longitudinal correlation. The two stages are:
-Line 30:
+Line 37:
-'''mris_preproc''' --qdec qdec.table.dat --target study_average --hemi lh --meas thickness --out lh.thickness.mgh (assembles your thickness data into a single lh.thickness.mgh file)
+. First, simplify the statistic to a single number for each subject (the difference of two time points, or the slope of the fitting line, or the annualized percent change, etc...).
 1. Then analyze the obtained summary measure across subjects or groups with a standard GLM.
-Line 32:
+Line 40:
-'''mri_surf2surf''' --hemi lh --s study_average --sval lh.thickness.mgh --tval lh.thickness_sm10.mgh --fwhm-trg 10 --cortex  --noreshape (smooths the cortical thickness maps with FWHM=10 mm. Note the --cortex and --noreshape options)
+This model is quite simple and can be an option if all subjects have the same number of time points, approximately equally spaced. Linear fits into each subject data are often meaningful, as longitudinal change can be assumed to be almost linear within a short time frame in several applications.
-Line 34:
+Line 42:
-Then you can load the cortical thickness data lh.thickness_sm10.mgh into Matlab using fs_read_Y.m
+'''Advantages:'''
-Line 36:
+Line 44:
-eg.
+ * Can deal with differently many and differently spaced time points (but does not model the difference in variability).
 * Works on ROI stat (e.g. aseg.stats or aparc.stats) and on cortical maps (e.g. thickness).
 * The second stage can be performed with QDEC (simple GUI) or directly with mri_glmfit.
 * The second stage analysis can make use of different multiple comparisons methods that come with mri_glmfit.
 * Scripts are available ( long_mris_slopes and long_stats_slopes ), no matlab needed.
 * For the simple case of two time points and when looking at simple differences this model simplifies to a paired analysis, but can additionally compute (symmetrized) percent changes.
 * Includes code for intersecting cortex labels (across time and across subjects) to make sure that all non-cortex vertices are excluded.
-Line 38:
+Line 52:
-''[Y,mri] = fs_read_Y('lh.thickness_sm10.mgh'); ''
+'''Disadvantages:'''
-Line 40:
+Line 54:
-You should also read the spherical surface (lh.sphere) and cortex label (lh.cortex.label) of study_average.
+ * Does NOT model the correlation among the repeated measures, and thus, there is a significant reduction in statistical power.
 * Does NOT account for different certainty of within subject slopes depending on the number of time points and therefore it has the highest  propensity to false positives (type I family wise error in the mass-univariate setting).
 * Difficult to model non-linear temporal behaviour.
 * Difficult to deal with time varying co-variates (slopes would need to be fit into those for each subject to reduce these to a single number).
 * Cannot include information from subjects with only a single time point and thus the results are likely to be biased and have less statistical power.
-Line 42:
+Line 60:
-eg.
+The linear mixed effects model overcomes these limitations and should be used if subjects have differently many time points (or for more complex modeling).
-Line 44:
+Line 62:
-''lhsphere = fs_read_surf('$FsDir/freesurfer/subjects/fsaverage/surf/lh.sphere'); ''
+For details see: LongitudinalTwoStageModel
-Line 46:
+Line 64:
-''lhcortex = fs_read_label('$FsDir/freesurfer/subjects/fsaverage/label/lh.cortex.label'); ''
+----
-Line 48:
+Line 66:
-Once you have your data in Matlab you need to build your design matrix. For computational efficiency reasons, these tools require the data ordered according to time for each individual (that is, your design matrix needs to have all the repeated assessments for the first subject, then all for the second and so on). You can use the script:
+== Linear Mixed Effects Model ==
A Linear Mixed Effects (LME) model is the most powerful and principled approach. We recommend this approach.
-Line 50:
+Line 69:
-''sortData ''
+'''Advantages:'''
-Line 52:
+Line 71:
-For example, if you have your covariates in a Qdec table then you can use the following code
+ * Works for both stats (univariate) and surface analysis (mass-univariate).
 * Can handle unequal timing and different number of time points across subjects (missing data).
 * Even subjects with only a single time point can be included into these models (make sure they also run through the longitudinal stream, available with version FS 5.2, to avoid a bias due to different image processing) .
 * Appropriately models the temporal correlation.
 * Can model different variances across measurement occasions.
 * Our mass-univariate method can deal very well with the spatial correlation among measurements on the cortex and is very fast by working with spatial regions in which the correlation structure is relative constant.
 * Can be used to model complex longitudinal behavior (e.g. quadratic, or piecewise linear trajectories) and time-varying covariates.
 * It seems to have become the consensus among statisticians that LME models are the right mechanism to study longitudinal data and they may be requested in journal publications by the reviewers.
-Line 54:
+Line 80:
-''Qdec = fReadQdec('qdec.table.dat'); ''
+'''Disadvantages:'''
-Line 56:
+Line 82:
-''Qdec = rmQdecCol(Qdec,1); ''
+ * More complicated use (e.g. requires distinguishing mixed effects from fixed effects ...).
 * Currently, our implementation is in Matlab only.
 * Currently only offers FDR for multiple comparisons correction.
-Line 58:
+Line 86:
-''sids = Qdec(:,1); ''
+For details see: LinearMixedEffectsModels
-Line 60:
+Line 88:
-''Qdec = rmQdecCol(Qdec,1); ''

''M = Qdec2num(Qdec); ''

''[M,Y,ni] = sortData(M,2,Y,sids); ''

== Model specification ==
Analysis of longitudinal data should always start by simple graphical displays of the data. A natural way is through the use of time plots. A time plot is a scatter-plot with the responses on the vertical axis and the measurement times on the horizontal axis. In general it is usually more informative to display a time plot of the smoothed mean response over time. This graphical display can be quite enlightening and can provide the basis for choosing an appropriate model for the analysis of change over time. The toolbox provides the following two functions:

''lme_timePlot ''

''lme_lowessPlot ''

In the mass-univariate case it is more difficult to determine the trend in the mean response over time since it may vary across different regions of the cortex. You can plot the trends in different regions (eg.from the Freesurfer's parcelation) and select the most complex trend for your model (eg. quadratic in time over linear in time, etc...). We recommend to order the columns of your design matrix in the following way: First, the intercept term (which is a column of ones); second, the time covariate; third, any time-varying covariates (eg. time^2); fourth, the group covariates of interest (eg. a binary variable indicating whether the subject is an Alzheimer's patient or not) and their interactions with the time-varying covariates; finally any other nuisance time-invariant covariate (eg. gender). See bellow in section 7 for an example of a model for the design matrix.

== Parameter estimation ==
A very flexible and versatile approach for analyzing longitudinal continuous data is the linear mixed effects (LME) regression paradigm. This paradigm can provide parsimonious models for both the trend in the mean response over time and the covariance among repeated measures on the same individual. The toolbox provides three types of tools for fitting LME models:

a)Univariate tools

''lme_fit_FS ''

''lme_fit_EM ''

''lme_fit_NR ''

b)Mass-univariate tools

''lme_mass_fit_vw ''

''lme_mass_fit ''

c)Novel mass-univariate tools (spatiotemporal models) Spatiotemporal models are more powerful to detect effects in your data and usually require less computation time than traditional vertex-wise mass-univariate tools. Here you should first compute initial covariance component estimates using:

''lme_mass_fit_init  '' or''   lme_mass_fit_EMinit ''

these estimates can then be used to segment the brain into homogeneous regions with similar random effects covariance parameters by

''lme_mass_!RgGrow''

The spatiotemporal model can then be fitted by using

''lme_mass_fit_Rgw ''

together with the previous segmentation and initial covariance estimates.

== Model selection ==
Here a likelihood ratio test can be used to compare a model with q random effects against a model with q+1 random effects using

a)Univariate

''lme_LR ''

b)Mass-univariate

''lme_mass_LR'' (a correction for multiple comparisons is then required)

== Inference ==
a)Univariate

''lme_F ''

b)Mass-univariate

''lme_mass_F'' (a correction for multiple comparisons is then required)

At this time the only tools provided for multiple comparisons correction are

''lme_mass_FDR'' (Original Benjamini and Hochberg FDR procedure)

''lme_mass_FDR2'' (A powerful two-stage FDR procedure)

== Power analysis ==
Only univariate tools are provided:

''lme_plannedPower ''

''lme_plannedSampleSize ''

''lme_realizedPower ''

== Example data analyses ==
Here are two example analyses using ADNI data processed with Freesurfer

a)Univariate

In this case the interest was in determining the differences in mean hippocampal volume change over time among four groups of individuals: healthy controls (HC), stable mild cognitive impairment patients (sMCI), MCI patients who converted to Alzheimer's disease (cMCI) during the follow-up period and patients with Alzheimer's disease (AD) at baseline. A Qdec table file qdec.table.dat was used to process in Freesurfer all time point scans from all subjects included in the study. It contains the following columns:

1) Freesurfer's fsid (fsid)

2) subject ID (sID, in this case an integer between one and the total number of subjects in the study)

3) time from baseline (time)

4) group (HC=1, sMCI=2, cMCI=3, AD=4)

5) Apoe4 (E4:carriers=1, non-carriers=0)

6) Gender (female=1, male=0)

7) Age at baseline (!BslAge)

8) Education (in years).

The qdec.table.dat file was then used to collect volumetric data for all subjects within the Freesurfer's Qdec interface. This resulted in a new Qdec table which had the previous columns plus three new columns:

9) left Hippocampus volume

10) right Hippocampus volume

11) total intracranial volume (ICV).

The data was loaded into Matlab using:

''Qdec = fReadQdec('qdec.table.dat'); ''

''Qdec = rmQdecCol(Qdec,1); ''

''sID = Qdec(:,1); ''

''Qdec = rmQdecCol(Qdec,1); ''

''M = Qdec2num(Qdec); ''

''Y = M(:,7:9); ''

''M = M(:,1:6); ''

All the data was ordered according to time for each subject using

''[M,Y,ni] = sortData(M,1,Y,sID); ''

where M,Y,ni are now the ordered design matrix, the ordered data and the vector with the number of repeated measures for each subject respectively. The mean response trends over time were then plotted

''lme_lowessPlot(M(:,1),Y(:,1)+Y(:,2),0.70,M(:,2)); ''

These plots revealed a linear trajectory over time for the four groups. So, the following LME model was proposed:

Y_ij = B_1 + B_2*t_ij + B_3*sMCI_i + B_4*sMCI_i*t_ij + B_5*cMCI_i + B_6*cMCI_i*t_ij + B_7*AD_i + B_8*AD_i*t_ij + B_9*E4_i + B_10*E4_i*t_ij + B_11*Gender_i + B_12*!BslAge_i + B_13*Education_i + B_14*ICV_i + b_1i + b_2i + e_ij

Thus, our design matrix X was built from matrix M to represent the previous model. The design matrix X then had 14 columns:

1) intercept (all ones)

2) time (t_ij)

3) one for sMCI and zero otherwise (sMCI_i)

4) colum 3) .* time

5) one for cMCI and zero otherwise (cMCI_i)

6) colum 5) .* time

7) one for AD and zero otherwise (AD_i)

8) colum 7) .* time

9) E4

10) E4 .* time

11) Gender

12) !BslAge

13) Education

14) ICV (converted to liters).

The parameters of this model were estimated using

''total_hipp_vol_stats = lme_fit_FS(X,[1 2],Y(:,1)+Y(:,2),ni); ''

Note that this model contains two random effects corresponding to the intercept and slope terms. This model can be compared with the model with only a single random effect corresponding to the intercept:

''total_hipp_vol_stats_1RF = lme_fit_FS(X,[1],Y(:,1)+Y(:,2),ni); ''

using the likelihood ratio test:

''lr = lme_LR(total_hipp_vol_stats.lreml,total_hipp_vol_stats_1RF.lreml,1); ''

The previous test was significant (pval<0.0001) indicating that the model with two random effects is significantly better than the model with a single random effect. To test if there was any difference in the rate of change over time among the four groups the following contrast was used

''C = [zeros(3,3) [1 0 0 0 0; -1 0 1 0 0; 0 0 -1 0 1] zeros(3,6)]; ''

and the F-test

''F_C = lme_F(total_hipp_vol_stats,C); ''

The p-value of the test was in ''F_C.pval''.

b)Mass-univariate

In this case the interest was in determining regionally specific differences in cortical thickness atrophy rate over time among the previous four groups of individuals: HC, sMCI, cMCI and AD. Thus we had the same Qdec table of the previous example. The following Freesurfer's comands were used to generate the spatial cortical thickness data:

'''mris_preproc''' --qdec qdec.table.dat --target fsaverage --hemi lh --meas thickness --out lh.thickness.mgh

'''mri_surf2surf''' --hemi lh --s fsaverage --sval lh.thickness.mgh --tval lh.thickness_sm10.mgh --fwhm-trg 10 --cortex  --noreshape

Then the spatially smoothed data lh.thickness_sm10.mgh were loaded into Matlab using

''[Y,mri] = fs_read_Y('lh.thickness_sm10.mgh'); ''

As in the previous example, the data were ordered according to time for each individual:

''Qdec = fReadQdec('qdec.table.dat'); ''

''Qdec = rmQdecCol(Qdec,1); ''

''sID = Qdec(:,1); ''

''Qdec = rmQdecCol(Qdec,1); ''

''M = Qdec2num(Qdec); ''

''[M,Y,ni] = sortData(M,1,Y,sID); ''

The fsaverage's spherical surface (lh.sphere) and cortex label (lh.cortex.label) were read with:

''lhsphere = fs_read_surf('$FsDir/freesurfer/subjects/fsaverage/surf/lh.sphere'); ''

''lhcortex = fs_read_label('$FsDir/freesurfer/subjects/fsaverage/label/lh.cortex.label'); ''

A priori, we expected a linear model of cortical thickness atrophy over time. However, in order to account for any possible consistent non-linearity across vertices we carried out a model selection procedure starting with a quadratic model over time. So, the following LME model was initially proposed:

Y_ij = B_1 + B_2*t_ij + B_3*t_ij*t_ij + B_4*sMCI_i + B_5*sMCI_i*t_ij + B_6*sMCI_i*t_ij*t_ij + B_7*cMCI_i + B_8*cMCI_i*t_ij + B_9*cMCI_i*t_ij*t_ij + B_10*AD_i + B_11*AD_i*t_ij + B_12*AD_i*t_ij*t_ij + B_13*E4_i + B_14*E4_i*t_ij + B_15*Gender_i + B_16*!BslAge_i + B_17*Education_i + b_1i + b_2i + e_ij

Thus, our design matrix X was built from matrix M to represent the previous model.Then initial vertex-wise temporal covariance estimates were computed with:

''[lhTh0,lhRe] = lme_mass_fit_EMinit(X,[1 2],Y,ni,lhcortex,3); ''

These covariance estimates were segmented into homogeneous regions using:

''[lhRgs,lhRgMeans] = lme_mass_!RgGrow(lhsphere,lhRe,lhTh0,lhcortex,2,95); ''

Here both lhTh0 and lhRgMeans maps were overlaid onto lhsphere and visually compared each other to ensure that they were similar enough (the essential spatial organization of the initial covariance estimates was not lost after the segmentation procedure):

''surf.faces =  lhsphere.tri; ''

''surf.vertices =  lhsphere.coord'; ''

''figure; p1 = patch(surf); ''

''set(p1,'facecolor','interp','edgecolor','none','facevertexcdata',Th0(1,:)'); ''

''figure; p2 = patch(surf); set(p2,'facecolor','interp','edgecolor','none','facevertexcdata',lhRgMeans(1,:)'); ''

''The spatiotemporal LME model was fitted using: ''

''lhstats = lme_mass_fit_Rgw(X,[1 2],Y,ni,lhTh0,lhRgs,lhsphere); ''

Note that there is no need to pass the lhcortex label into the previous estimation function since non-cortex vertices (codified as 0 in lhRgs segmentation vector) are not automatically considered for estimation. If it is desired to compare the previous LME model with the one with a single random effect (or the one with three random effects) the same segmentation must be used to estimate the parameters for both models. Here, in order to ensure validity of both models, the segmentation obtained for the most complex model must be used. This is because the same permisibility values for the segmentation procedure result in less and larger regions for the model with lesser number of random effects and using different segmentations for different models will invalid the subsequent likelihood ratio test. For instance, to compare the previous LME model with the one with a single random effect for the intercept term, the following steps can be followed:

i) Fit the model with one random effect using the segmentation obtained from the previous model:

''lhTh0_1RF = lme_mass_fit_EMinit(X,[1],Y,ni,lhcortex,3); ''

''lhstats_1RF = lme_mass_fit_Rgw(X,[1],Y,ni,lhTh0_1RF,lhRgs,lhsphere); ''

ii) Apply the likelihood ratio test:

''LR_pval = lme_mass_LR(lh_stats,lhstats_1RF,1); ''

iii) Correct for multiple comparisons:

''dvtx = lme_mass_FDR2(LR_pval,ones(1,length(LR_pval)),lhcortex,0.05,1); ''

In our case most cortex vertices survived the above correction (''length(dvtx)'' > 80% of ''length(lhcortex)'') and therefore we assumed that the model with two random effects was significantly better than the model with a single random effect. We then tested the null hypothesis of no group differences in the quadratic term:

''CM.C = [zeros(3,5) [1 0 0 0 0 0 0;-1 0 0 1 0 0 0;0 0 0 -1 0 0 1] zeros(3,5)];''

''F_lhstats = lme_mass_F(lhstats,CM); ''

and multiple comparisons were corrected with:

''dvtx = lme_mass_FDR2(F_lhstats.pval,F_lhstats.sgn,lhcortex,0.05,0); ''

As dvtx was empty, which means that no vertex survived the multiple comparison correction, then all quadratic terms were removed from the model. Thus a linear model over time with two random effects, as in the univariate case, was then fitted using the above functions (lme_mass_fit_EMinit, lme_mass_!RgGrow and lme_mass_fit_Rgw) and the null hypothesis of no group differences in the rate of change over time among the four groups was tested with the following contrast:

''CM.C = [zeros(3,3) [1 0 0 0 0; -1 0 1 0 0; 0 0 -1 0 1] zeros(3,5)]; ''

''F_lhstats = lme_mass_F(lhstats,CM); ''

The Freesurfer significance map was then written to disc for visualization and post-processing in Freesurfer (eg. using tksurfer tool):

''fs_write_fstats(F_lhstats,mri,'sig.mgh','sig'); ''

Alternatively, a correction for multiple comparisons can be carried out directly here with

''[detvtx,sided_pval,pth] = lme_mass_FDR2(F_lhstats.pval,F_lhstats.sgn,lhcortex,0.05,0); ''

then the sided p-values can be saved with

''fs_write_Y(sided_pval,mri,'spval.mgh'); ''

and finally the Freesurfer's '''mri_surfcluster''' cmd can be used with spval.mgh, lhcortex and pth to generate the set of detected clusters and their anatomical coordinates.
+----
MartinReuter