General Linear Model Framework Research Articles

Efficient modeling of neural activity using coupled renewal processes

Generalized Linear Models (GLMs) are stochastic models that have been successfully used to model neural activity of single cells and populations. Typically, this requires the spike train to be binned into a binary time series that is modeled as a Bernoulli-type GLM. Unless a loss in temporal precision is acceptable, the bin size has to be chosen on the order of milliseconds. This renders an application to long-term recordings infeasible for computational reasons. We propose a new, binning-free application of the GLM framework to model neural activity by noting that spike trains can equally be characterized by the sequence of continuous-valued inter-spike intervals (ISIs). It was shown that single-neuron spike trains can often be sufficiently modeled as a non-stationary renewal process. Here, we focus on the Gamma distribution for the inter-spike intervals as it allows the accommodation of a relative refractory period and includes the inhomogeneous Poisson process as a special case. We generalize the idea of a modulated Gamma process to the population level by including cross-couplings between the neurons in case of recordings from multiple neurons in parallel. In a Gamma-GLM, each observed inter-spike interval is modeled as a sample from a Gamma distribution whose mean can vary with time, e.g. through modulation by an external stimulus. Traditionally, ISIs are assumed to be independent. However, this renewal assumption can be relaxed by conditioning on the durations of previous intervals. An inter-neuron coupling is introduced by additionally conditioning on the times of spikes of the other neurons. The Gamma distribution and other canonical distributions for inter-spike intervals are part of the exponential family so that efficient maximum likelihood solutions for all model parameters can be obtained using the framework of Generalized Linear Models. Considering the ratio between bin sizes for time-discrete models and the typical length of cortical inter-spike intervals, the computational demand can be reduced by several orders of magnitude. Goodness-of-fit is assessed by adapting standard tests for point process models and evaluating spike prediction performance based on cross-validation. We demonstrate the versatility of the coupled Gamma model for estimating functional connectivity from multi-electrode recordings. With a data set obtained from multi-electrode recordings of the visual system of the awake monkey, we show that the test power of the Gamma-GLM for detecting functional connectivity links is comparable to the performance of a classical discrete-time GLM albeit requiring much less computational time and memory resources. Using the two models on the same data set, we are able to estimate the unknown, underlying connection density of the network. Finally, we discuss general limitations of coupled Gamma-GLMs on the type of interactions that can be faithfully represented and discuss similarities to other binning-free models, in particular Cox’s proportional hazards model.

Statistical Analysis of fMRI Time-Series: A Critical Review of the GLM Approach

Functional magnetic resonance imaging (fMRI) is one of the most widely used tools to study the neural underpinnings of human cognition. Standard analysis of fMRI data relies on a general linear model (GLM) approach to separate stimulus induced signals from noise. Crucially, this approach relies on a number of assumptions about the data which, for inferences to be valid, must be met. The current paper reviews the GLM approach to analysis of fMRI time-series, focusing in particular on the degree to which such data abides by the assumptions of the GLM framework, and on the methods that have been developed to correct for any violation of those assumptions. Rather than biasing estimates of effect size, the major consequence of non-conformity to the assumptions is to introduce bias into estimates of the variance, thus affecting test statistics, power, and false positive rates. Furthermore, this bias can have pervasive effects on both individual subject and group-level statistics, potentially yielding qualitatively different results across replications, especially after the thresholding procedures commonly used for inference-making.

Robust, unbiased general linear model estimation of phMRI signal amplitude in the presence of variation in the temporal response profile

To determine a simple yet robust method to generate parsimonious design matrices that accurately estimate the "pharmacological MRI" (phMRI) response amplitude in the presence of both confounding signals and variability in temporal profile. Variability in the temporal response profile of phMRI time series data is often observed. If not properly accounted for, this variation can result in inaccurate and unevenly biased signal amplitude estimates when modeled within a general linear model (GLM) framework. The approach uses a low-rank singular value decomposition (SVD) approximation to a set of vectors capturing anticipated variations of no interest around the signal model to generate additional regressors for the design matrix. The method is demonstrated for both plateau and bolus type phMRI response profiles in the presence of variation in signal onset and/or shape, and applied to an in vivo blood oxygenation level-dependent (BOLD) phMRI study of buprenorphine in healthy human subjects. In general, 2-3 additional regressors, capturing >75% of the anticipated variance, resulted in robust and unbiased signal amplitude estimates in the presence of substantial variability. This method provides a simple and flexible means to provide robust phMRI amplitude estimates within a GLM framework.

Thalamic metabolism and neurological outcome after traumatic brain injury. A voxel-based morphometric FDG-PET study

Objective To study the relationship between thalamic metabolism and neurological outcome in patients who had sustained a traumatic brain injury (TBI). Methods Nineteen patients who had sustained a severe TBI and ten control subjects were included in this study. Six of the 19 patients had a low level of consciousness (vegetative state or minimally conscious state), while thirteen showed normal consciousness. All patients underwent an 18F-FDG PET, 459.4 ± 470.9 days after the TBI. The FDG-PET images were normalised in intensity, with a metabolic template being created from data derived from all subjects. The thalamic trace was generated automatically with a mask of the region of interest to evaluate its metabolism. A comparison between the two groups was carried out by a two sample voxel-based T-test, under the General Linear Model (GLM) framework. Results Patients with low consciousness had lower thalamic metabolism (MNI-Talairach coordinates: 12, —24, 18; T=4.1) than patients with adequate awareness (14, —28, 6; T=5.5). Control subjects showed the greatest thalamic metabolism compared to both patients groups. These differences in metabolism were more pronounced in the internal regions of the thalamus. Conclusions The applied method may be a useful ancillary tool to assess neurological outcomes after a TBI, since it permits an objective quantitative assessment of metabolic function for groups of subjects. Our results confirm the vulnerability of the thalamus to suffering the effects of the acceleration-deceleration forces generated during a TBI. It is hypothesized that patients with low thalamic metabolism represent a subset of subjects highly vulnerable to neurological and functional disability after TBI.

Metabolismo talámico y situación neurológica tras un traumatismo craneoencefálico. Estudio mediante PET-FDG y morfometría basada en vóxel

ObjectiveTo study the relationship between thalamic metabolism and neurological outcome in patients who had sustained a traumatic brain injury (TBI). MethodsNineteen patients who had sustained a severe TBI and ten control subjects were included in this study. Six of the 19 patients had a low level of consciousness (vegetative state or minimally conscious state), while thirteen showed normal consciousness. All patients underwent a PET with 18F-FDG, 459.4 ± 470.9 days after the TBI. The FDG-PET images were normalized in intensity, with a metabolic template being created from data derived from all subjects. The thalamic trace was generated automatically with a mask of the region of interest in order to evaluate its metabolism. A comparison between the two groups was carried out by a two sample voxel-based T-test, under the General Linear Model (GLM) framework. ResultsPatients with low consciousness had lower thalamic metabolism (MNI-Talairach coordinates: 12, -24, 18; T=4.1) than patients with adequate awareness (14, -28, 6; T=5.5). Control subjects showed the greatest thalamic metabolism compared to both patients groups. These differences in metabolism were more pronounced in the internal regions of the thalamus. ConclusionsThe applied method may be a useful ancillary tool to assess neurological outcomes after a TBI, since it permits an objective quantitative assessment of metabolic function for groups of subjects. Our results confirm the vulnerability of the thalamus to suffering the effects of the acceleration-deceleration forces generated during a TBI. It is hypothesized that patients with low thalamic metabolism represent a subset of subjects highly vulnerable to neurological and functional disability after TBI.

Population decoding of motor cortical activity using a generalized linear model with hidden states

Generalized linear models (GLMs) have been developed for modeling and decoding population neuronal spiking activity in the motor cortex. These models provide reasonable characterizations between neural activity and motor behavior. However, they lack a description of movement-related terms which are not observed directly in these experiments, such as muscular activation, the subject’s level of attention, and other internal or external states. Here we propose to include a multi-dimensional hidden state to address these states in a GLM framework where the spike count at each time is described as a function of the hand state (position, velocity, and acceleration), truncated spike history, and the hidden state. The model can be identified by an Expectation–Maximization algorithm. We tested this new method in two datasets where spikes were simultaneously recorded using a multi-electrode array in the primary motor cortex of two monkeys. It was found that this method significantly improves the model-fitting over the classical GLM, for hidden dimensions varying from 1 to 4. This method also provides more accurate decoding of hand state (reducing the mean square error by up to 29% in some cases), while retaining real-time computational efficiency. These improvements on representation and decoding over the classical GLM model suggest that this new approach could contribute as a useful tool to motor cortical decoding and prosthetic applications.

Generalized linear and additive models for psychophysical data

What do such diverse paradigms as classification images, difference scaling and additive conjoint measurement have in common? We introduce a general framework that permits modeling and evaluating experiments covering a broad range of psychophysical tasks. Psychophysical data are considered within a signal detection model in which a decision variable, d, which is some function, f, of the stimulus conditions, S, is related to the expected probability of response, E[P], through a psychometric function, G: E[P] = G(f(d(S))). In many cases, the function f is linear, in which case the model reduces to E[P] = G(Xb), where X is a design matrix describing the stimulus configuration and b a vector of weights indicating how the observer combines stimulus information in the decision variable. By inverting the psychometric function, we obtain a Generalized Linear Model (GLM). We demonstrate how this model, which has previously been applied to calculation of signal detection theory parameters and fitting the psychometric function, is extended to provide maximum likelihood solutions for three tasks: classification image estimation, difference scaling and additive conjoint measurement. Within the GLM framework, nested hypotheses are easily set-up in a manner resembling classical analysis of variance. In addition, the GLM is easily extended to fitting and evaluating more flexible (nonparametric) models involving arbitrary smooth functions of the stimulus. In particular, this approach permits a principled approach to fitting smooth classification images.

Entropy coefficient of determination for generalized linear models

The objective of the present paper is to propose a predictive power measure for generalized linear models (GLMs). First, basic predictive power measures for GLMs are compared with respect to some desirable properties. We propose a generalized coefficient of determination for GLMs, which is referred to as the entropy coefficient of determination (ECD). The advantage of the measure is discussed in the GLM framework. Second, the asymptotic properties of the maximum likelihood estimator of ECD are discussed. Third, ECD is applied to GLMs with polytomous response variables. Finally, discussions and a conclusion to this study are provided.

Improved characterization of BOLD responses for evoked sensory stimuli

Pain and somatosensory processing involves an interaction of multiple neuronal networks. One result of these complex interactions is the presence of differential responses across brain regions that may be incompletely modeled by a straightforward application of standard general linear model (GLM) approaches based solely on the applied stimulus. We examined temporal blood oxygenation-level dependent (BOLD) signatures elicited by two stimulation paradigms (brush and heat) providing innocuous and noxious stimuli. Data were acquired from 32 healthy male subjects (2 independent cohorts). Regional time courses and model-free analyses of the first cohort revealed distinct temporal features of the BOLD responses elicited during noxious versus innocuous stimulation. Specifically, a biphasic (dual peak) BOLD signal was observed in response to heat but much less so in response to brush stimuli. This signal was characterized by a stimulus-locked response along with a second peak delayed by ∼12.5 s. A cross-validation error analysis determined a modified design matrix comprising two explanatory variables (EVs) as a parsimonious means to model the biphasic responses within a GLM framework. One EV was directly derived from the stimulation paradigm (EV1), while the second EV (EV2) was EV1 shifted by 12.5 s. The 2EV GLM analysis enabled a more detailed characterization of the elicited BOLD responses, particularly during pain processing. This was confirmed by application of the model to a second, independent cohort[AU1]. Furthermore, the delayed component of the biphasic response was strongly associated with the noxious heat stimuli, suggesting that this may represent a sensitive fMRI link of pain processing.

Chronic Recurrent Multifocal Osteomyelitis: Comparison of Whole-Body MR Imaging with Radiography and Correlation with Clinical and Laboratory Data

To describe whole-body magnetic resonance (MR) imaging appearance of chronic recurrent multifocal osteomyelitis (CRMO) and assess the role of MR imaging versus radiography in diagnosis of disease and correlation with clinical findings and laboratory data. Institutional review board approved this retrospective HIPAA-compliant study; informed consent was waived. T1-weighted, short inversion time inversion-recovery, and contrast material-enhanced T1-weighted whole-body MR imaging was performed and two-plane radiographs, clinical findings, and laboratory data were reviewed in 13 children (median age, 13 years) with CRMO. Lesion depiction, location, and characterization and extraskeletal abnormalities were evaluated. MR imaging findings were compared with clinical and laboratory data and radiographic results. Data analysis was performed, and diagnostic performance statistics of radiography, physical examination results, and serum inflammatory markers were calculated. General multilevel linear modeling framework was used. Odds ratios were calculated to estimate effect of age, erythrocyte sedimentation rate (ESR), and C-reactive protein (CRP) level on reliabilities. Associations of ESR and CRP level with total number of lesions were assessed (chi(2) test). MR imaging depicted 101 ill-defined edemalike osseous lesions. Most frequent anatomic sites were distal femur (21%, 21 of 101), proximal tibia (17%, 17 of 101), and distal tibia and fibula (14% each, 14 of 101). In tubular bones (70 anatomic sites), metaphysis (86%, 60 of 70) and epiphysis (67%, 47 of 70) were involved. Contiguous physeal relationship (89%, 66 of 74), periosteal reaction (48%, 48 of 101), and symmetric involvement (85%, 11 of 13) were present. MR imaging demonstrated multifocality in all patients. There were no extraskeletal abnormalities and no relationship between serum inflammatory markers and number of symptomatic anatomic sites (P = .472). Sensitivity for radiography was 0.13 (70 of 119); physical examination, 0.31 (52 of 299); and serum inflammatory markers, 0.15 (two of 13). Whole-body MR imaging is useful for detection of CRMO, particularly in indeterminate cases, because it is more likely to show abnormalities.

A semiparametric approach to hidden Markov models under longitudinal observations

We propose a hidden Markov model for longitudinal count data where sources of unobserved heterogeneity arise, making data overdispersed. The observed process, conditionally on the hidden states, is assumed to follow an inhomogeneous Poisson kernel, where the unobserved heterogeneity is modeled in a generalized linear model (GLM) framework by adding individual-specific random effects in the link function. Due to the complexity of the likelihood within the GLM framework, model parameters may be estimated by numerical maximization of the log-likelihood function or by simulation methods; we propose a more flexible approach based on the Expectation Maximization (EM) algorithm. Parameter estimation is carried out using a non-parametric maximum likelihood (NPML) approach in a finite mixture context. Simulation results and two empirical examples are provided.

A novel ANCOVA design for analysis of MEG data with application to a visual attention study

Statistical inference from MEG-based distributed activation maps is well suited to the general linear modeling framework, a standard approach to the analysis of fMRI and PET neuroimaging studies. However, there are important differences from the other neuroimaging modalities related to how observations are created and fitted in GLM models, as well as how subsequent statistical inference is performed. In this paper, we demonstrate how MEG oscillatory components can be analyzed in this framework based on a custom ANCOVA model that takes into account baseline and inter-hemispheric effects, rather than a simpler ANOVA design. We present the methodology using as an example an MEG study of visual spatial attention, since the model design depends on the specific experiment and neuroscience hypotheses being tested. However, the techniques presented here can be readily adapted to accommodate other experimental paradigms. We create statistics that estimate the temporal evolution of attention effects on alpha power in several cortical regions. We present evidence for direction-specific attention effects on alpha activity in occipital and parietal regions and demonstrate the sub-second timing of these effects in each region. The results support a mechanism for anticipatory attentional deployment that dynamically modulates the local alpha synchrony in a network of parietal control and occipital sensory regions.

GLUMIP 2.0: SAS/IML Software for Planning Internal Pilots.

Internal pilot designs involve conducting interim power analysis (without interim data analysis) to modify the final sample size. Recently developed techniques have been described to avoid the type I error rate inflation inherent to unadjusted hypothesis tests, while still providing the advantages of an internal pilot design. We present GLUMIP 2.0, the latest version of our free SAS/IML software for planning internal pilot studies in the general linear univariate model (GLUM) framework. The new analytic forms incorporated into the updated software solve many problems inherent to current internal pilot techniques for linear models with Gaussian errors. Hence, the GLUMIP 2.0 software makes it easy to perform exact power analysis for internal pilots under the GLUM framework with independent Gaussian errors and fixed predictors.

Comparison of the power of haplotype-based versus single- and multilocus association methods for gene × environment (gene × sex) interactions and application to gene × smoking and gene × sex interactions in rheumatoid arthritis

Accounting for interactions with environmental factors in association studies may improve the power to detect genetic effects and may help identifying important environmental effect modifiers. The power of unphased genotype-versus haplotype-based methods in regions with high linkage disequilibrium (LD), as measured by D', for analyzing gene x environment (gene x sex) interactions was compared using the Genetic Analysis Workshop 15 (GAW15) simulated data on rheumatoid arthritis with prior knowledge of the answers. Stepwise and regular conditional logistic regression (CLR) was performed using a matched case-control sample for a HLA region interacting with sex. Haplotype-based analyses were performed using a haplotype-sharing-based Mantel statistic and a test for haplotype-trait association in a general linear model framework. A step-down minP algorithm was applied to derive adjusted p-values and to allow for power comparisons. These methods were also applied to the GAW15 real data set for PTPN22.For markers in strong LD, stepwise CLR performed poorly because of the correlation/collinearity between the predictors in the model. The power was high for detecting genetic main effects using simple CLR models and haplotype-based methods and for detecting joint effects using CLR and Mantel statistics. Only the haplotype-trait association test had high power to detect the gene x sex interaction.In the PTPN22 region with markers characterized by strong LD, all methods indicated a significant genotype x sex interaction in a sample of about 1000 subjects. The previously reported R620W single-nucleotide polymorphism was identified using logistic regression, but the haplotype-based methods did not provide any precise location information.

Accounting for variability in individual hierarchical clinical trial data

Meta-analytical approaches have been extensively used to analyze medical data. In most cases, the data come from different studies or independent trials with similar characteristics. However, these methods can be applied in a broader sense. In this paper, we show how existing meta-analytic techniques can also be used as well when dealing with parameters estimated from individual hierarchical data. Specifically, we propose to apply statistical methods that account for the variances (and possibly covariances) of such measures. The estimated parameters together with their estimated variances can be incorporated into a general linear mixed model framework. We illustrate the methodology by using data from a first-in-man study and a simulated data set. The analysis was implemented with the SAS procedure MIXED and example code is offered.

Study-level wavelet cluster analysis and data-driven signal models in pharmacological MRI

In pharmacological MRI (phMRI) studies tracking signal changes following the acute administration of a compound, the spatiotemporal pattern of response is often unknown a priori. Moreover, when analysed within a general linear model (GLM) framework, the experimental paradigm of a single injection point under-informs the construction of an appropriate signal model, and information from pharmacokinetics or ancillary in vivo studies may be unavailable or insufficient to accurately describe the dynamic signal changes observed following injection of the drug. Here, we extend the application of a data-driven clustering algorithm, wavelet cluster analysis (WCA), to phMRI data from one or more groups of subjects in a study. A WCA decomposition of spatially concatenated time series’ provides a compact overview of spatiotemporal response patterns across cohorts, highlighting typical temporal signatures, brain regions implicated in the response and inter-subject variability. Further, we demonstrate the use of regressors based on selected temporal components as suitable signal models in GLM-based analyses, resulting in a close fit to dynamic phMRI signal changes. This approach is illustrated with simulated data and two representative in vivo phMRI studies in the rat (nicotine and apomorphine challenges).

Regression-Based Association Analysis with Clustered Haplotypes through Use of Genotypes

Haplotype-based association analysis has been recognized as a tool with high resolution and potentially great power for identifying modest etiological effects of genes. However, in practice, its efficacy has not been as successfully reproduced as expected in theory. One primary cause is that such analysis tends to require a large number of parameters to capture the abundant haplotype varieties, and many of those are expended on rare haplotypes for which studies would have insufficient power to detect association even if it existed. To concentrate statistical power on more-relevant inferences, in this study, we developed a regression-based approach using clustered haplotypes to assess haplotype-phenotype association. Specifically, we generalized the probabilistic clustering methods of Tzeng to the generalized linear model (GLM) framework established by Schaid et al. The proposed method uses unphased genotypes and incorporates both phase uncertainty and clustering uncertainty. Its GLM framework allows adjustment of covariates and can model qualitative and quantitative traits. It can also evaluate the overall haplotype association or the individual haplotype effects. We applied the proposed approach to study the association between hypertriglyceridemia and the apolipoprotein A5 gene. Through simulation studies, we assessed the performance of the proposed approach and demonstrate its validity and power in testing for haplotype-trait association.

Analysis of cod catch data from Icelandic groundfish surveys using generalized linear models

Abstract Catch data from the Icelandic groundfish surveys are analyzed using generalized linear models (GLM). The main goal is to test the effects of environmental variables on the expected cod catch and to distinguish between the gamma and log-normal distributions for the error structure. Only positive catch data are included in this work, i.e. only the positive part of a delta–gamma or delta–log-normal distribution is examined. The distributions are compared via a Kolmogorov–Smirnov goodness of fit test. Polynomials are used to describe the relationship between each environmental variable and the cod catch and their effects are tested within the GLM framework (a continuous model). Finally, an attempt is made to locate temperature fronts in the ocean by estimating the temperature gradient vector at each data point. The effect of the size of this gradient vector is then tested within in the GLM framework. A stratification model with only spatial and time effects explains 80% of the variation but that comes with a high cost of degrees of freedom. Most of the tested effects are found to be significant but the continuous model only captures 45% of the total variation. The size of the temperature gradient vector is found to be statistically significant though only a small portion of the variation in the data is explained by this term.

Optimal spatial regularisation of autocorrelation estimates in fMRI analysis

In the General Linear Model (GLM) framework for the statistical analysis of fMRI data, the problem of temporal autocorrelations in the residual signal (after regression) has been frequently addressed in the open literature. There exist various methods for correcting the ensuing bias in the statistical testing, among which the prewhitening strategy, which uses a prewhitening matrix for rendering the residual signal white (i.e., without temporal autocorrelations). This correction is only exact when the autocorrelation structure of the noise-generating process is accurately known, and the estimates derived from the fMRI data are too noisy to be used for correction. Recently, Worsley and co-workers proposed to spatially smooth the noisy autocorrelation estimates, effectively reducing their variance and allowing for a better correction. In this article, a systematic study into the effect of the smoothing kernel width is performed and a method is introduced for choosing this bandwidth in an "optimal" manner. Several aspects of the prewhitening strategy are investigated, namely the choice of the autocorrelation estimate (biased or unbiased), the accuracy of the estimates, the degree of spatial regularisation and the order of the autoregressive model used for characterising the noise. The proposed method is extensively evaluated on both synthetic and real fMRI data.

Experimentation with Randomization Restrictions: Targeting Practical Implementation

AbstractRandomization, one of the fundamental principles of statistically designed experiments, is not always easy to implement in practice. In fact, many industrial settings involve factors of interest that are difficult to change, requiring some modification to a completely randomized test sequence. Practical issues associated with restricted randomization commonly arise regarding design efficiencies, design requirements for error estimation, overall ease of software‐assisted analysis, required replication, tests for nonlinearity, and sequential testing plans. The purpose of this paper is to deal directly with these roadblocks to efficient and effective experimentation under randomization restrictions. The general approach involves modifying standard split‐plot designs with appropriate augmentation followed by a two‐stage model building method using a standard general linear model framework. An alternative procedure for analyzing two‐level factorial split‐plot designs is provided for commonly occurring situations involving more than one hard‐to‐change factor. The proposed methods were developed and implemented in conducting experiments for wind tunnel applications. These tests serve as case studies for the paper. We also suggest possible methods for limiting the replication required for estimating whole plot error. Copyright © 2004 John Wiley & Sons, Ltd.