% Random effects analyses for SPM'99 using a multi-level approach %_______________________________________________________________________ % % This SPM manual page attepts to give an overview of the issues and % implementation of random effects analyses in SPM. A full theoretical % treatment is beyond the scope of this "man" page. References are % provided below. % % See also: spm_getSPM.m (contrast evaluation) % spm_conman.m (contrast definition) % %======================================================================= % Overview: Fixed and random effects, the SPM multi-level approach %======================================================================= % % SPM uses a fixed effects model. That is, the parameters of the model % are assumed fixed, and of interest in their own right. The only % random quantity considered is the residual error, characterised by % the residual variance. % % In a model with repeated measurements on a number of subjects, the % residual error is within subject. If there is subject by response % interaction (as there almost certainly is), then inference from a % fixed effects analysis of the group average response is limited to % the particular subjects studied. That is, it is a case study. To % extend inference to the population from which the subjects were drawn % we have to acknowledge that it is not the average response of these % particular subjects that is of interest, but the average response for % the population from which they are drawn. That is, the response for % each subject is a random quantity, a random effect. To infer about % the population average effect we must account for the variance of % response from subject to subject, in addition to the scan to scan % (residual) variability. Thus, we must consider two components of % variance. % % In general mixed effects models (with both random and fixed effects) % are difficult to analyse. However, in functional neuroimaging the % models are often fairly simple, balanced, with the only random effect % of concern the subject by condition interaction: Further, the models % are usually separable into individual subject models (the standard % SPM99 designs enforce this). This enables a basic mixed effects % analysis to be easily be carried out in two stages, corresponding to % the two levels of variability in question. The first level is the % intra-subject level, the second the inter-subject level. % % With only one "scan" per subject, scan to scan residual variability % *is* between subject variance. For balanced designs, by summarising % the individual subject data with an appropriate measure, and then % assessing those measures across subject, a random effects analysis is % effected. Using the within-subject data to compute the summary % measure "scan" incorporates the within-subject error into the summary % scans, such that the between subject variability of these computed % summary "scans" consists of both within and between subject variance. % It's fairly easy for a balanced design to show that these are in % exactly the right ratio for a random effects assessment of the % overall (population) effect. % % This can be simply effected within SPM99: The individual subjects are % analysed at a within-subject level using balanced standard SPM % models. For each subject, the contrast of the parameter estimates of % interest is computed, and written out as a "contrast" image. (Note % that this is *not* the SPM, just the linear combination of the % parameter estimates.) Either individual single subject models or % subject-separable multi-subject models can be used, since the % parameters and therefore contrasts will be the same, all that is % required is that the design is balanced from a multi-subject % perspective (i.e. the individual subjects models and contrast weights % are the same). These contrast images surmise the response for each % subject, and are then used as input into SPM for a between-subjects % level analysis. This would usually involve a simple model: For % instance, a population mean effect could be assessed with a simple % one-sample t-test. A comparison of the mean responses of two % populations could be effected using a two- % % Global normalisation and grand mean scaling should have been handled % at the within-subject level, so there is no need for such things at % the between subject level. Similarly, the contrast images produced % are implicitly masked, so there is no need to (re)define the analysis % volume in the second level analysis. % % Although this second level (between subjects) analysis is a fixed % effects model, by using summary "scans", the effect is the population % effect (which *is* fixed). The residual error variance of this second % level model is the variance of the contrast images from subject to % subject, and consists of contributions from both the between and % within subject components of variance, in the correct proportions. It % can easily be shown that the resulting analysis is mathematically % identical to the appropriate random effects (strictly called mixed % effects) analysis of these data. % % %======================================================================= % The SPM99 approach %======================================================================= % % The basic procedure for a random effects analysis in SPM99 is as follows: % * First, fit the model for each subject % - you can model all the subjects together, % provided you use a subject separable model % - strictly speaking, the individual subject models should be % identical (i.e. this is a balanced design) % % * Define the effect of interest for each subject with a t-contrast. % - Each contrast will write a con_????.{img,hdr} analyze % image containing the contrast of the parameter estimates % % * Armed with one contrast image per subject, proceed to the second % (between subject) level, feeding the contrast images back into SPM, % using simple models (usually "Basic stats" models) as appropriate. % % Since you can do this with any one dimensional contrast that would % produce an SPM{t}, you can look at regression slopes, interactions % and so on, in addition to simple categorical comparisons. % % ---------------- % % * Sphericity: % % You should only take a single contrast per subject forward to a % second level analysis. Taking more assumes that the repeat contrasts % within subject are independent: implicitly implying that the variance % is "spherical". It is fairly rare that repeated measures data is % spherical, so it's safest to just put in one contrast per subject. % The degrees of freedom should then be just less than the number of % subjects, which is usually quite low. % % ---------------- % % * Implicit masking: % % SPM computes statistics only at voxels for which there is a full data % set after applying and implicit, explicit or (analysis) threshold % masking. This set of voxels constitutes the analysis volume for these % data, usually chosen to be the intracerebral volume. (For PET/SPECT & % basic designs, see spm_spm_ui.m for definitions of these masks and % thresholds, and the options available. For fMRI, explicit masking is % currently unavailable, and the "analysis threshold" is hard coded at % the global mean (after grand mean scaling), which usually includes % all intracerebral voxels.) % % ( Note that this refers to the masks and thresholds used to define ) % ( the volume for analysis when setting up a design, not the ) % ( thresholds and masks used when examining SPM's in the results ) % ( section. The contrast and SPM{t} images (or ESS & SPM{F} images for ) % ( an F-contrast - see spm_getSPM.m) are written by the results ) % ( section once a new contrast is defined, but before any height or ) % ( extent thresholding is applied. Only the write-filtered button in ) % ( the results interface writes out height and extent filtered SPM{t} ) % % For a given statistical analysis, the output images (parameter images % - beta_????.{img,hdr}, variance image - ResMS.{img,hdr}, contrast % images - (con_????.{img,hdr}), ESS images - ess_????.{img,hdr}, and % SPM images - spm{T,F}_????.{img,hdr}) are written as floating point % images, with voxels outwith the analysis set to NaN - not a number. % These images are implicitly masked, since SPM will omit any voxel % with value NaN in any image. % % Thus, when entering contrast images into a second level analysis, % there is no need to explicitly define the volume for analysis since % the contrast images are already implicitly masked. The analysis % volume will be the intersection of the analysis volumes for the % individual subjects. % % ---------------- % % * Global normalisation and grand mean scaling: % % Global normalisation and grand mean scaling should have been handled % at the within-subject level, so there is no need for such things at % the between subject level. % % ---------------- % % * Only use contrast images as input for higher level analyses: % % It's contrast images from a first level (individual subject say) % analysis that can be re-entered into SPM to effect an analysis at a % higher level (across subjects say). In SPM99 these are named % con_????.img. % % The statistic (SPM) images (SPMt_????.img & SPMF_????.img) should % *not* be entered into a second level analysis if you want to effect a % random effects analysis. This would basically be assessing the % significance (across subjects) of the individual subjects % significance! (Rather than the significance (across subjects) of the % response. % % It's possible that this confusion may arise because you enter % contrast weights to get a SPM{t}. However, the term contrast itself % refers only to the weighted sum of the parameters, whose estimates % given in the contrast images only form the numerator of the SPM{t}. % (The SPM{t} is formed by dividing the contrast image by a suitable % estimate of the standard error.) % % The contrast surmises the effect, the SPM{t} the evidence for the % effect in comparison to the residual variance for the model under % consideration. % % Contrast images should be used because they are guaranteed to be % estimable whatever the design. In general the parameter images % (beta_????.img) are not estimable: The "parameter estimability" bar % on SPM printouts tells you which parameters are uniquely estimable % for this model. A contrast with a single "1" picking out these % estimable parameters would be a valid contrast, and the contrast % image would be the same as the parameter image. % % ---------------- % % * Miscellaneous notes: % % For er-fMRI, you would probably need to fix the slice timing of 2D % multi-slice acquisitions by temporal interpolation. Your model would % have to use a canonical response (rather than a set of basis % functions), such that the effect of interest can be extracted with a % single contrast. You then put these contrast images into the between % subject level analysis... % % Note that this only works for one-dimensional contrasts, i.e. a % t-contrast, with contrast weights a single vector. For F-contrasts % (which you would use for a two-sided t-test, or to test any overall % effect for a set of basis functions) you need a multivariate second % level analysis. This should be possible with the Multivariate % toolbox... % % %======================================================================= % References %======================================================================= % % A paper is currently in preparation. However, the basic gist is % reported in the abstract: % Holmes AP, Friston KJ (1998) % "Generalisability, Random Effects & Population Inference" % NeuroImage 7(4-2/3):S754 % The corresponding poster (HBM'98) is available from % http://www.fil.ion.ucl.ac.uk/spm/RFXposter.pdf % % ---------------- % % This basic approach using summary measures is pretty standard. A good % introductory paper (albeit for clinical trials) is: % Frisson L & Pocock SJ (1992) % "Repeated measures in clinical trials: Analysis using mean % summary statistics and its implications for design" % Statistics in Medicine 11:1685-1704 % % ---------------- % % The basic concepts of random effects are expounded in all but the % most basic statistics books on "Design of Experiments". Other key % phrases to look out for are "variance components", "mixed effects % models", "repeated measures", "multi-level modeling" & "hierarchical % modeling". I've found the first chapter of Searle, Casella & % McCulloch's 1992 book "Variance Components" (Wiley, London) a good % description accessible to non-statisticians. % % ---------------- % % Roger Woods (1996) paper "Modeling for intergroup comparisons of % imaging data" (Neuroimage, 4(3/3):S84-94) also provides a readable % introduction to the concepts, with a neuroimaging slant. % % ---------------- % % A final reference of interest: Keith Worsley et al's seminal 1991 % paper for addressing the multiple comparisons problem also proposed % the use (for PET), of a two-sample t-statistic computed from average % condition images! (Although they used a variance estimate pooled over % the entire intracerebral volume rather than the voxel level variance % estimate used in SPM.) This is a repeated measures paired t-test % approach, identical to the current SPM approach, although (possibly) % primarily motivated by the substantial data reduction the use of % summary images confers. % % Worsley KJ, Marrett S, Neelin P, Evans AC (1992) % "A three-dimensional statistical analysis for CBF % activation studies in human brain" % Journal of Cerebral Blood Flow and Metabolism, 12:900-918 % % %======================================================================= % The SPM96 approach reprised (for historical interest) %======================================================================= % % With SPM96, basic random effects analyses for simple categorical % compariosns were retrospectively implemented in a limited way using % the "Random Effects Kit": % http://www.fil.ion.ucl.ac.uk/spm/spm96.html#RFX96 This allowed random % effects analyses of simple categorical comparisons by computing % adjusted mean condition images. With two conditions, this resulted in % two images per subject, which were then assessed at the subject level % using a paired t-test. % % The adjusted condition images obtained from the SPM96 AdjMean/fMRI % are the parameter estimates for a very simple model using box-cars, % optionally convolved with a synthetic haemodynamic response function, % and optionally including global intensity normalisation. Although % these adjusted mean condition images themselves may not be uniquely % estimated, the difference between any two (usually) is (for the % limited models of the AdjMean/fMRI module). I.e. The difference % between two of these adjusted mean condition images is the same as a % contrast image for a contrast with weights of the form % [...+1...-1...] which would contrast the two conditions. % % By putting the pairs of adjusted mean condition images for each % subject into SPM96's PETstats "Multi-Subject: Conditions only" design % (with no global normalisation), you're just doing a paired t-test: % This SPM model used this way is equivalent to doing a t-test on the % inter-condition intra-subject differences. I.e., The SPM96 way is % implicitly putting [...+1...-1...] type contrasts into the second % level analysis, by virtue of using a paired t-test on the adjusted % mean condition images, which are just parameter estimate images. % % The reason for adopting this convoluted scenario was that SPM96 had % no implicit or explicit masking - the only way to specify the volume % for analysis in SPM96 is via an analysis threshold (expressed as a % proportion of the global mean for each scan). For such an analysis % threshold to be able to limit the analysis to intracerebral voxels % (say), the input images clearly must exhibit some structure, which % contrast images (or difference images) do not. (This was frequently % mistakenly presented as a deficiency in SPM96's handling of negative % data.) Adjusted condition images had the advantage of containing % structure, and yet when assessed in a pairwise analysis effected the % appropriate analysis. % % ---------------- % % The AdjMean functions of the SPM96 random effects kit are included in % SPM99, but we recommend using the new main SPM stats routines to % define an appropriate model and extract an appropriate contrast image % as summary image. %_______________________________________________________________________ % @(#)spm_RandFX.man 2.2 Andrew Holmes 00/01/25