iMap4: An open source toolbox for the statistical fixation mapping of eye movement data with linear mixed modeling
Abstract
A major challenge in modern eye movement research is to statistically map where observers are looking, by isolating the significant differences between groups and conditions. As compared to the signals from contemporary neuroscience measures, such as magneto/electroencephalography and functional magnetic resonance imaging, eye movement data are sparser, with much larger variations in space across trials and participants. As a result, the implementation of a conventional linear modeling approach on twodimensional fixation distributions often returns unstable estimations and underpowered results, leaving this statistical problem unresolved (Liversedge, Gilchrist, & Everling, 2011). Here, we present a new version of the iMap toolbox (Caldara & Miellet, 2011) that tackles this issue by implementing a statistical framework comparable to those developed in stateoftheart neuroimaging dataprocessing toolboxes. iMap4 uses univariate, pixelwise linear mixed models on smoothed fixation data, with the flexibility of coding for multiple between and withinsubjects comparisons and performing all possible linear contrasts for the fixed effects (main effects, interactions, etc.). Importantly, we also introduced novel nonparametric tests based on resampling, to assess statistical significance. Finally, we validated this approach by using both experimental and Monte Carlo simulation data. iMap4 is a freely available MATLAB open source toolbox for the statistical fixation mapping of eye movement data, with a userfriendly interface providing straightforward, easytointerpret statistical graphical outputs. iMap4 matches the standards of robust statistical neuroimaging methods and represents an important step in the datadriven processing of eye movement fixation data, an important field of vision sciences.
Keywords
Eye movement analysis Statistical mapping Linear mixed modelsHuman beings constantly move the eyes to sample visual information of interest from the environment. Eye fixations deliver inputs with the highest resolution to the human visual cortex from the fovea, as well as blurry, lowspatialfrequency information from peripheral vision (Rayner, 1998). Thus, isolating statistically where and how long fixations are deployed to process visual information is of particular interest to behavioral researchers, psychologists, and neuroscientists. Moreover, fixation mapping has a wide range of practical applications in determining marketing strategies and the understanding of consumer behaviour (Duchowski, 2002).
Conventional eye movement data analyses rely on the estimation of probabilities of occurrence of fixations and saccades (or their characteristics, such as duration or length) within predefined regions of interest (ROIs), which are at best defined a priori—but often also defined a posteriori, on the basis of data exploration, which inflates the Type I error rate. Another issue with ROIs is of course that other important information not included in the ROI is discarded. In a continuous effort to circumvent the limitations of the ROI approach (for a detailed discussion on this point, see Caldara & Miellet, 2011), we previously developed an unbiased, datadriven approach to compute statistical fixation maps of eye movements: the iMap toolbox (Caldara & Miellet, 2011). From the very first version, the toolbox was developed as a MATLAB open source toolbox freely available for download online. The previous versions (1 and 2) made use of Gaussian smoothing and the random field theory as a statistical engine (Caldara & Miellet, 2011), which is one of the standard methods applied in statistical analyses for functional magnetic resonance imaging (fMRI) data (Penny, Friston, Ashburner, Kiebel, & Nichols, 2011). Version 3 introduced pixelwise t test and bootstrap clustering in order to generate selfcontained statistical maps (Miellet, Lao, & Caldara, 2014). However, all of the previous versions of iMap still suffered from a major limitation: They could only contrast two conditions at a time.
A major revision of the toolbox was necessary to enable the analysis of more complex experimental designs routinely used in the field. One of the most suitable and obvious statistical solutions to overcome this problem would be to implement a general linear model, a widespread approach in both behavioral and neuralimaging data analyses. In fact, many modern procedures for hypothesis testing, such as the t test, analysis of variance (ANOVA), regression, and so forth, belong to the family of general linear models. However, eye movement data are a sparse production of visual perceptual sampling. Unlike neuroimaging data, eye movement data contain many empty cells with little to no data points across the tested space (e.g., all of the pixels in an image). This caveat engenders a statistical problem when the same statistical inference procedure is applied on each pixel, regardless or whether or not its data are missing. To account for the sparseness and the high variation of spatial eye movement data, we developed a specific novel approach for smoothed fixation maps, which was inspired by the statistical framework implemented in diverse stateoftheart neuroimaging dataprocessing toolboxes: statistical parametric mapping (SPM; Penny et al., 2011), Fieldtrip (Oostenveld, Fries, Maris, & Schoffelen, 2011), and LIMO EEG (Pernet, Chauveau, Gaspar, & Rousselet, 2011). In the simplest case, users can apply a massive univariate, pixelwise linear mixed model (LMM) on the smoothed fixation data with the subject considered as a random effect, which offers the flexibility to code for multiple between and withinsubjects comparisons. Our approach allows users to perform all possible linear contrasts for the fixed effects (main effects, interactions, etc.) from the resulting model coefficients and the estimated covariance. Importantly, we also introduced a novel nonparametric statistical test based on resampling (permutation and bootstrap spatial clustering) to assess the statistical significance of the linear contrasts (Pernet, Latinus, Nichols, & Rousselet, 2015; Winkler, Ridgway, Webster, Smith, & Nichols, 2014).
In the next section, we briefly describe the key concepts of the LMM approach. We then introduce the novel nonparametric statistical approach on the fixed effects that we implemented in iMap4, which uses a resampling procedure and spatial clustering. We also report a validation of the proposed resampling procedures, and illustrate how iMap4 can be used, with both a subset of data from a previous study and computersimulated data. Finally, we give an overview of future development and discuss technical insights on eye fixation mapping.
Linear mixed models
In this part, we outline the key elements and concepts of LMMs in comparison with general linear models (GLM) and hierarchical linear models (HLM). Mixed models represent a complex topic, and the discussion of many underlying mathematical details goes beyond the scope of this article. For general, thoughtful introductions to mixed models, users of the toolbox should refer to Raudenbush and Bryk (2002) and McCulloch, Searle, and Neuhaus (2011). Users may also wish to consult the documentation and help files of the LinearMixedModel class in the MATLAB Statistics Toolbox for details about parameter estimation and the available methods (www.mathworks.com/help/stats/linearmixedmodelclass.html).
The GLM Y = X β + ε could be easily extended into a generalized form with ε ~ Ν(0, σ ^{2} V) where V is some known positive definite matrix. Moreover, if a more specific structure of the error ε is available, the GLM (Eq. 2), which has one randomeffect term (the error ε), could be further extended into a mixed model. Mixed models include additional randomeffect terms that can represent the clusters or classes. In a typical neuroimaging study, this could be the subjects or groups. In the following example, we consider a simplified case in which only the subject is considered as the additional random effect. This type of model is one of the most widely used models in both fMRI and electroencephalography (EEG).
HLMs are specific cases of LMMs. In a mixed model, factors are not necessarily hierarchical. Moreover, crossed factors between fixed effects and random effects are much easier to model in mixed than in hierarchical models. In additions, the fixed effects and random effects are estimated simultaneously in mixed models, which is not always the case in hierarchical models.
Parameter estimation in mixed models is much more complicated than in GLM or HLM. Assuming that the model in Eq. 13 has the error covariance matrix R: \( \mathrm{v}\mathrm{a}\mathrm{r}\left(\mathbf{Y}\boldsymbol{b}\right)=\boldsymbol{R} \), this model is equivalent to \( y\sim \mathbf{N}\left(\boldsymbol{X}\boldsymbol{\beta },\;\boldsymbol{V}\right),\;\boldsymbol{V}=\boldsymbol{Z}\boldsymbol{D}{\boldsymbol{Z}}^{\boldsymbol{T}}+\boldsymbol{R} \). The estimation of the fixed effects β requires prior knowledge of V, which is usually unavailable. In practice, the variance component V is commonly replaced by an estimation \( \widehat{\boldsymbol{V}} \) based on one of several approaches, such as ANOVA, maximum likelihood (ML) estimation, restricted maximum likelihood (ReML) estimation, or Monte Carlo approximation (McCulloch, Searle, & Neuhaus, 2011; Pinheiro & Bates, 2000). In general, the modelfitting procedure of LMM is implemented in major statistical packages (e.g., R and Stata) by solving Henderson’s mixed model equation. iMap4 calls the MATLAB class LinearMixedModel from Statistics Toolbox (versions R2013b or above) to estimate the coefficients (fixed effect β and random effect b) and the covariance matrix V with various options (key concepts with regard to parameter estimations can be found in the MATLAB documentation: www.mathworks.com/help/stats/estimatingparametersinlinearmixedeffectsmodels.html). In brief, model coefficients are estimated by ML or ReML, and the pattern of the covariance matrix of the random effects (D) could take the form of a full covariance matrix, a diagonal covariance matrix, or other symmetry structure.
Statistical inferences in LMM are also much more complex than in a GLM. In a balanced design, or with the variance component V known, hypothesis testing of the fixed effect follows Eq. 8 or 11 as an exact test. However, in an unbalanced design with random effects, no exact F statistics are available, since biases in the estimation usually result in an unknown distribution of F (KheradPajouh & Renaud, 2015). Although F and t values are available as approximate tests in most statistical packages, Baayen, Davidson, and Bates (2008) discouraged the usage of t or F statistics, and especially report of the p value, in mixed models. Other approaches have also been proposed. For example, likelihood ratio tests could be performed to test composite hypotheses by comparing the desired model with the reduced model. However, there are many constraints on the application of likelihood ratio tests (e.g., the method of model fitting and selection of the reduced model). Moreover, running multiple copies of similar LMMs is computationally expensive, especially in the context of pixelwise testing, such as in iMap4.
Besides the practical problem of statistical inferences with LMM, another main challenge in the application of LMM to spatial eye movement data is the Type I error from multiple comparisons. To resolve these issues, we adopted resampling techniques for nullhypothesis statistical testing, as is suggested in neuroimaging analysis with GLM or HLM (Pernet et al., 2015; Winkler et al., 2014). Nonparametric statistics using Monte Carlo simulation are ideal for both parameter estimation and hypothesis testing (Baayen et al., 2008; KheradPajouh & Renaud, 2015). In iMap4, we adapted a simplified version of the permutation test suggested by Winkler et al. (2014) and a bootstrap clustering method similar to the one applied in LIMO EEG (Pernet et al., 2011). Details of the proposed algorithm and preliminary validation result are described in the following section.
Pixelwise modeling and spatial clustering
For s ∈ D of the search space.
This step is essential to account for the spatial uncertainty of eye movement recordings (both mechanical and physiological) and the sparseness of the fixation locations. The Gaussian kernel could also be replaced by other 2D spatial filters to best suit the research question.
The resulting smoothed fixation map is a 3D matrix. The last two dimensions of the fixation matrix are the sizes of the stimuli or search space. The information of each entry in the first dimension is stored in a predictor table, which is generated from the experiment condition matrix. Each experiment condition can be coded at the singletrial level in the predictor table, or as one entry by taking the average map across trials.
In addition, iMap4 provides a robust estimation option by applying Winsorization in order to limit extreme values in the smoothed fixation matrix. The goal here is to reduce the effect of any potential outliers. Additional options include: spatial normalization (zscored map or probability map), spatial downsampling (linear transformation using imresize in MATLAB) to optimize computing speed, and mask creation to exclude irrelevant pixels.
The resulting 3D fixation matrix is then modeled in a LMM as the response variable. The results are saved as a MATLAB structure (LMMmap, as in the examples below). The fields of LMMmap are nearly identical to the output from the LinearMixedModel class. For each modeled pixel, iMap4 saves the model criterion, variances explained, error sum of squares, coefficient estimates, and their covariance matrix for both fixed and random effects, and the ANOVA results for the LMM. Additional modeling specifications, as well as other model parameters, including the LMM’s formula, design matrix for fixed and random effect, and residual degrees of freedom, are also saved in LMMmap. Linear contrasts and other analyses based on variance or covariance can be performed afterward from the modelfitting information. Any other computation on the LinearMixedModel output can also be replicated with LMMmap.
One of the crucial assumptions of pixelwise modeling is that all pixels are independent and identically distributed. Of course, this assumption is never satisfied, neither before nor after smoothing. To ensure valid inferences on activity patterns in a large 2D pixel space, we applied nonparametric statistics to resolve the biases in parameter estimation and problems arising from multiple comparisons. We developed two resamplingbased statistical hypothesistesting methods for the fixedeffect coefficients: a universal permutation test and a universal bootstrap clustering test.
For s ∈ D of the search space.
For any permutation test, iMap4 performs the following algorithms on Y _{ fixed } for each pixel.
Algorithm 1

Performs a linear transformation on the design matrix X to get a new design matrix M so that the partitioning of M = [M1, M2]. Then iMap4 computes the new coefficients by projecting Y _{ fixed } to the pseudoinverse of M. The design matrix M is created so that the original hypothesis testing is equivalent to the hypothesis regarding the M1 coefficients. The matrix transformation and partition are the same as the algorithm described in Winkler et al. (2014, Appx. A).

Computes the residuals related to the hypothesis by subtracting the variance accounted for by M2 from Y _{ fixed }, to get Y _{ rr }.

Fits Y _{ rr } to M by solving Y _{ rr } = M β _{ m } + ε, and gets the statistical value F _{ rr } of M1 according to Eqs. 10 and 11. Note that to replicate the original hypothesis testing on the fixed effect, the new contrast c’ is just used to partition M into M1 and M2.

Permutes the rows of the design matrix M to obtain the new design matrix M ^{ * }.

Fits Y _{ rr } to M ^{ * } and gets the F _{ rr }* of M1 ^{ * }.
 Repeats the previous two steps a large number of times (k resamplings/repetitions), and the p value is then defined as in Eq. 16. Importantly, the familywise error rate (FWER) corrected p value is computed by comparing the largest F _{ rr }* across all tested pixels in one resampling with the original F _{ rr }:$$ p=\frac{\left(\#\ {{\mathbf{F}}_{\boldsymbol{rr}}}^{*}\ge\ {\mathbf{F}}_{\boldsymbol{rr}}\right)}{k}. $$(16)
Algorithm 1 is a simplified version of Winkler et al. (2014, Algorithm 1): The resampling table includes permutation but not signflipping, which assumes the errors to be independent and symmetric. Thus, the underlying assumptions are stronger than with classical permutations, which require only exchangeable errors (Winkler et al., 2014).
Importantly, this test is exact only under a balanced design with no missing values and only subjects as a random effect. As was previously shown in KheradPajouh and Renaud (2015), a general and exact permutation approach for mixedmodel designs should be performed on modified residuals that have up to secondmoment exchangeability. This is done to satisfy the important assumptions for repeated measures ANOVA: normality and the sphericity of errors. However, there are strict requirements to achieve this goal: careful transformation and partition of both the fixed and randomeffects design matrices, and removal of the random effects related to M2 (KheradPajouh & Renaud, 2015). In iMap4, we perform an approximation version by removing all random effects to increase the efficiency and speed of the huge amount of resampling computation in our pixelwise modeling algorithm. Validation and simulation data set indeed showed that the sensitivity and the false alarm rate of the proposed algorithm were not compromised.
Algorithm 2

For each unique categorical variable, iMap4 removes the conditional expectations from Y _{ fixed } for each pixel. A random shuffling is then performed on the centered data to acquire Y _{ c } , so that any potential covariance is also disrupted. This is done to construct the true empirical nullhypothesis distribution in which all elements and their linear combinations in Y _{ c } have expected values equal to 0.

Randomly draws with replacement from {X, Z, Y _{ c }} equal numbers of subjects {X*, Z*, Y _{ c }*}.

Fits Y _{ c }* to X^{*} by solving Y _{ c }* = X*β* + ε. For a given hypothesis or linear contrast c (as in Eq. 9), iMap4 computes the statistics value F* according to Eqs. 10 and 11, and their parametric p value under the GLM framework.

Thresholds the statistical maps F^{*} at p*≤.05 and records the desired maximum cluster characteristics across all significant clusters. The cluster characteristics considered are cluster mass (summed F value within a cluster), cluster extent (size of the cluster), and cluster density (mean F value).

The previous three steps are repeated a large number of times, to get the cluster characteristic distribution under the null hypothesis.

Thresholds the original statistical map F at \( p\le .05 \) and compares the selected cluster characteristic with the value of the null distribution corresponding to the 95th percentile. Any cluster with the chosen characteristic larger than this threshold is considered significant.
The bootstrap clustering approach is identical to the bootstrap procedure described by Pernet et al. (2011; Pernet et al., 2015) if only a subject intercept is considered as the random effect. In addition, Algorithm 2 extents the philosophy and approach presented by Pernet et al. (2011; Pernet et al., 2015) to nonhierarchical mixedeffect models.
It is worth noting that we implemented in iMap4 a highperformance algorithm to minimize the computational demands of the large amount of resampling. The model fitting in both resampling approaches makes use of ordinary least squares. The inversion of the covariance matrices (required for Eq. 11) is computed on the upper triangular factor of the Cholesky decomposition. Calculation of the quartic form (as in Eq. 11) for all pixels is optimized by constructing a sparse matrix of the inverse of the covariance matrix. More details of these algrebraic simplifications can be found in the imapLMMresample function in iMap4.
Other multiplecomparison correction methods, such as Bonferroni correction, false discovery rate, or random field theory (RFT), could also be applied. A thresholdfree cluster enhancement algorithm could also be applied on the statistical (Fvalue) maps as an option after the permutation and bootstrap clustering procedures (Smith & Nichols, 2009).
We performed a validation study to assess the Type I error rate when applying the permutation and bootstrap clustering approach for hypothesis testing. We used a balanced repeated measures ANOVA design with a twolevel betweengroup factor and a threelevel withingroup factor. A total population of 134 observers (67 in each group) was drawn from previous faceviewing eye movement studies. We centered the cell means for the whole dataset to obtain the validation dataset under the null hypothesis (similar to Step 1 in Algorithm 2). Thus, we used real data to warrant realistic distributions and centered them to ensure that H0 was confirmed. Any significant output from iMap4 performed on this dataset would be considered as a false alarm (Type I error).
Graphical user interface (GUI) and command line handling
iMap4 runs on MATLAB 2013b and above, since it requires some essential functions and classes from the Image Processing Toolbox and Statistics Toolbox in these versions. iMap4 will execute in parallel on multicores or distributed workers, when available.
Applications to real and simulation data
In the following examples, we illustrate iMap4’s flexibility and power with two real data sets and a computer simulation. All material and codes presented here are available in the iMap4 installation package.
Example 1
We consider first a subset of participants from Bovet, Lao, Bartholomée, Caldara, & Raymond, (2016), as a demonstration of the analysis procedure in iMap4. A stepbystep demonstration is available in the user guidebook and example code.
In short, the dataset consists of eye movement data from 20 male observers during a gazecontingent study. Observers viewed computerrendered female bodies in different conditions and performed a behavioral task (i.e., subjective ratings of bodily attractiveness). This was a withinsubjects design with two experimental manipulations: the viewing condition (three levels: 2° spotlight, 4° spotlight, or natural viewing) and body orientation (two levels: front view or back view). The aim of the study was to evaluate the use of visual information for bodily attractiveness evaluation in the male observers. Other details of the experiment can be found in the article.
Fixation durations were projected into the twodimensional space according to their coordinates at the singletrial level. The fixation duration maps were first smoothed at 1° of visual angle. We used the “estimated” option by taking the expected values across trials within the same condition independently for each observer. To reduce the computation time, we downsampled the fixation map to 256*205 pixels and applied a mask to only model the pixels with average durations longer than half of the minimum fixation duration input.
Notice that the mean fixation duration for each condition and subject were treated as random effects to control for the variation across individuals. The parameters were fitted with restricted maximum likelihood estimation (ReML).
Example 2
As a second demonstration, we reanalyzed the full dataset from one of our previous studies, Miellet, He, Zhou, Lao, and Caldara (2012).
Previous studies testing Western Caucasian (WC) and East Asian (EA) observers had shown that people deploy different eye movement strategies during free viewing of faces. WC observers fixate systematically toward the eyes and mouth, following a triangular pattern, whereas EA observers predominantly fixated at the center of the face (Blais, Jack, Scheepers, Fiset, & Caldara, 2008; Caldara, Zhou, & Miellet, 2010). Moreover, human observers can flexibly adjust their eye movement strategies to adapt to environmental constraints, as has been shown using different gazecontingent paradigms (Caldara, Zhou, & Miellet, 2010; Miellet et al., 2012). In our 2012 study, we tested two groups of observers in a face task in which their foveal vision was restricted by a blind spot. This was a mixed design with the culture of the observers as the betweensubjects factor (WCs or EAs) and the blind spot size as the withinsubjects factor (four level: natural viewing, 2° blindspot, 5° blindspot, or 8° blindspot). For more details of the experiment, please refer to Miellet et al. (2012).
Only the subject predictor was treated as a random effect, and the model was fitted using ML.
Example 3
 In a 4*4 grid, we introduced a different linear relationship in each cell between fixation number and subjective rating. Figure 7a shows the linear relationships we introduced for one subject. We varied the slope and the strength of the linear association. The correlation was strongest on the top row (r = .9), and there was no correlation on the bottom row (r = 0). The slope varied among [1, 0.4, –0.2, –0.8] across the columns. Note that each dot on a scatterplot represents one trial, and the dots with the same rating (value on the xaxis) across subplots belong to the same trial. The resulting matrices after this step were a onedimensional array Rating and a twodimensional matrix P (matrix size: 16 * number of trials)

The spatial locations of fixations were generated using linear Gaussian random fields. For each trial, we created a Gaussian mixture model gm using the gmdistribution class in MATLAB. The Gaussian mixture model gm contains 16 (4*4) 2D Gaussian distribution components. The center of each component aligned with the center of each grid, and the covariance was an identity matrix with 1° of visual angle on the diagonal. Crucially, the mixing proportion of each component was decided by the column of the specific trial in P. A number of random fixations were then generated from this Gaussian mixture model gm. See Fig. 7b for a realization of one random trial for one subject.
The significant regression coefficients of Rating are shown in Fig. 7d. iMap4 accurately rejected the null hypothesis for most conditions when there was a significant relationship. For the most robust effect (r = .9), iMap4 accurately estimated the coefficients. It also correctly reported a null result for r = 0. Moreover, iMap4 did not report any significant effect for the weakest relationship (slope = –0.2, r = .3), due to the lack of power. Indeed, further simulations showed that increasing the numbers of fixations, trials, or subjects would lead to significance.
Discussion and future developments
In the present article, we have reported a major update of iMap, a toolbox for statistical fixation mapping of eye movement data. While keeping unchanged the general datadriven philosophy of iMap, we significantly improved the underlying statistical engine, by incorporating pixelwise LMMs and a variety of robust nonparametric statistics. Crucially, the new analysis pipeline allows for the testing of complex designs while controlling for a wide range of random factors. We also implemented a full GUI to make this approach more accessible to MATLAB beginners. Examples from empirical and computersimulated datasets showed that this approach has a slightly conservative FWER under H0, while remaining highly sensitive to actual effects (e.g., Fig. 6d). The present method represents a significant advance in eye movement data analysis, particularly for analyzing experimental designs using normalized visual stimuli. In fact, iMap4 uses a statistical inference method similar to those in fMRI and magnetoencephalography/EEG analysis. The interpretation of the statistical maps is simply done by looking at which stimulus features/pixels relate to the significant areas (after multiplecomparison correction). This procedure is similar to the interpretation of fMRI results: after a significant region is revealed, we can use its spatial coordinates to check in which part of the cortex the region activated above chance level is located.
As a powerful statistical tool, LMMs are gaining popularity in psychological research and have previously been applied in eye movement studies (e.g., Kliegl, Masson, & Richter, 2010). Similarly, particular cases of LMM, such as HLM or twolevel models, are now standard dataprocessing approaches in neuroimaging studies. As a general version of HLMs, LMMs are much more flexible and powerful than other multilevel models. Most importantly, an exact same LMM could be applied to behavior, eye movement, and neuroimaging data, bridging these different measures to allow drawing more direct and complete conclusions.
However, there are both theoretical and practical challenges in using LMM for the statistical spatial mapping of fixation data. First, the fixation locations are too sparse to directly apply pixelwise modeling. Similarly to previous versions of iMap, we used spatial smoothing of the fixation locations, a preprocessing step necessary to account for the measurement error of eyetrackers and the imprecision of the physiological system (i.e., the human eye). The second issue is selecting the appropriate hypothesis testing for LMM and the multiplecomparison problems caused by modeling massive number of pixels in nonbalanced designs. We addressed this issue by applying nonparametric statistics based on resampling and spatial clustering. Another important challenge is the constraint of computational resources. Parameter estimations using LMM, the pixelwise modeling approach, and resampling techniques are very computationally demanding and timeconsuming. To produce a useful but also usable tool, we adapted many advanced and novel algorithms, such as parallel computing. Preprocessing options such as downsampling and applying a mask also significantly decrease the computational time of iMap4.
The comparison among ROIs/areas of interest, iMap 2.0, and the current version
In classical eye movement data analyses, particularly those considering fixation locations, the main challenge for statstially identifying the regions that have been fixated above chance level lies in the fact that we are facing a highdimensional data space. Mathematically, each pixel represents one dimension that could be potentially important. However, it is trivial to say that many of these dimensions are redundant and could be reduced to a particular set of representations or features. In other words, eye fixation data points are embedded in a highdimensional pixel space, but they actually occupy only a subspace with much lower dimensionality (Belkin & Niyogi, 2003). Indeed, in similar highdimensional datasets, a lowdimensional structure is often assumed and is naturally the main focus for investigation. Thus, by arbitrarily choosing one or multiple ROIs, one can represent the highdimensional dataset as a lowdimensional manifold. The fixation map thus projects into this manifold, and all the pixels within the same ROI are then considered as being in the same dimension. In this case, each ROI represents one feature. Such a method is comparable to early neural network and many other linear dimension reduction methods in the machinelearning literature with handcoded features (LeCun, Haffner, Bottou, & Bengio, 1999; Sorzano, Vargas, & Montano, 2014).
The early versions of iMap (1 and 2) adopted a similar logic, but relied on RFT to isolate datadriven features. Therefore, the fixation bias in each pixel was projected into a lowerdimensional subspace, resulting in fixation clusters. Secondlevel statistics were then computed at the cluster level instead of the pixel level to perform statistical inference (Miellet, Lao, & Caldara, 2014).
From iMap 3 onward, we took a very different approach. We used spatial clustering and multiplecomparison correction to avoid the use of secondlevel statistics to perform statistical inference. In iMap4, the fixation bias is similarly modeled on each pixel using a flexible yet powerful statistical model: the LMM. The LMM, in combination with nonparametric statistics and a spatial clustering algorithm, directly isolates the significant pixels. As a result, the iMap4 outputs can be interpreted intuitively and straightforwardly at the map level (i.e., by visualizing the areas reaching significance from the tested hypothesis).
Parameter settings and statistical choices
Our aim was and still is the development of a datadriven and fully automatized analysis tool. However, even in iMap4 some of the parameters in the analysis rely on a user’s expertise and subjective choices, which thus should be considered carefully before use. These parameters include the kernel size for the smoothing procedure, the spatial downsampling and masking, the spatial normalization, and the choice of statistics.
The rationale for the determining the kernel size for the smoothing procedure has been previously discussed (Caldara & Miellet, 2011), and the majority of the arguments we put forward in this previous article still hold true. Here, we would remind users that the spatial smoothing procedure mainly resolves the sparseness of fixation data. It also partially accounts for the spatial covariance, which is ignored in univariate pixelwise modeling. Finally, it accounts for the recording errors from eyetrackers, such as drift during the calibration, pupilsize variations, and so forth.
We also recommend that users perform downsampling and apply a mask before modeling their data. This step is important to reducing computational demands (time, memory, etc.). In general, we recommend that the downsampling factor not be bigger than half of the smoothing kernel size. In other words, if the FWHM of the Gaussian kernel is 10 pixels, the rescale factor should be less than 5. We are currently running further simulations and validations to investigate the best parameters under different settings, and hopefully will provide a statistical datadriven solution for this choice in future updates.
Spatial normalization (via a zscored or probability map) is available as an option in iMap4. Spatial normalization used to be a standard preprocessing procedure in previous versions of iMap. However, the hypotheses tested on raw fixation duration/number maps are fundamentally different from their spatially normalized versions. Importantly, after spatial normalization, the interpretation of results should be drawn on a spatially relative bias instead of on the absolute differences. Of course, if the viewing duration in each trial is constant within an experiment, spatial normalization will not make any difference.
For iMap4 we developed two main, nonparametric statistics based on resampling techniques. It is worth noting that different applicability comes with the choice of permutation tests versus bootstrap spatialclustering tests. In our own experience during empirical and simulation studies, permutation tests are more sensitive for studies with small sample sizes; the bootstrapclustering approach usually gives more homogeneous results but is biased toward bigger clusters. We suggest that users adopt a “wisdom of crowds” approach and look at the agreement among different approaches before concluding on the data analysis (Marbach et al., 2012). Nonconvergent results should be interpreted carefully.
An alternative to pixelwise approaches
In recent years, other frameworks have been also developed to model eyetracking data (Boccignone, 2015). One such approach is the aforementioned Poisson point process model (Barthelmé et al., 2013). It is a wellestablished statistical model when the point (fixation) occurrence is the main concern. Under some transformation, the Poisson point processes model of fixation occurrences could be expressed and modeled as a logistic regression, making it straightforward to apply using conventional statistical software (Barthelmé & Chopin, 2015). For example, Nuthmann and Einhäuser (2015) made use of logistic mixed models to determine the influence of low and high visual properties in scene images on eye movements. Moreover, smooth effect and spatial covariants could be captured by applying regression splines in a generalized additive model, as demonstrated in Barthelmé and Chopin (2015).
Importantly, the point process model addresses different questions than does iMap. It is most appropriate when the effect of spatial location is considered irrelevant, a nuisance effect, or a fixed intercept (see, e.g., Barthelmé & Chopin, 2015; Nuthmann & Einhäuser, 2015). As a comparison, in iMap the parameters of interest are location specific, varying from pixel to pixel. In other words, the differences or effects among different conditions are locationspecific, forming a complex pattern in two dimensions. These high dimension effects are more natural and easy to model using a pixelwise model, as in iMap4.
Conclusion and future development
In conclusion, we have presented an advanced eye movement analysis approach using LMMs and nonparametric statistics: iMap4. This method is implemented in MATLAB with a userfriendly interface. We aimed to provide a framework for analyzing spatial eye movement data with the most sophisticated statistical modeling to date. The procedure described in the present article currently represents our best attempt to conform with the conventional nullhypothesis testing, while providing options for robust statistics. We currently are still working on many improvements, including functions to compare different fitted models, statistics on the randomeffect coefficients, and replacing LMMs with generalized LMMs for modeling fixation numbers (Bolker, Brooks, Clark, Geange, Poulsen, Stevens, & White, 2009). In the future, we will also switch our focus to Bayesian statistics and the generative model (such as the Gaussian process) in an effort to develop a unified model of statistical inference for eye movement data (Jaynes & Bretthorst, 2003).
Notes
Author note
The authors declare no competing financial interests. This study was supported by the Swiss National Science Foundation (Grant No. 100014_138627). awarded to R.C. The authors would like to thank Dr Simon Barthelmé and an anonymous reviewer for their helpful comments that contributed to improving the final version of the article.
References
 Baayen, R. H., Davidson, D. J., & Bates, D. M. (2008). Mixedeffects modeling with crossed random effects for subjects and items. Journal of Memory and Language, 59, 390–412. doi: 10.1016/j.jml.2007.12.005 CrossRefGoogle Scholar
 Barthelmé, S., & Chopin, N. (2015). The Poisson transform for unnormalised statistical models. Statistics and Computing, 25, 767–780.CrossRefGoogle Scholar
 Barthelmé, S., Trukenbrod, H., Engbert, R., & Wichmann, F. (2013). Modeling fixation locations using spatial point processes. Journal of Vision, 13(12), 1. doi: 10.1167/13.12.17 CrossRefPubMedGoogle Scholar
 Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15, 1373–1396.CrossRefGoogle Scholar
 Blais, C., Jack, R. E., Scheepers, C., Fiset, D., & Caldara, R. (2008). Culture shapes how we look at faces. PLoS ONE, 3, e3022.CrossRefPubMedPubMedCentralGoogle Scholar
 Boccignone, G. (2015). Advanced statistical methods for eye movement analysis and modeling: A gentle introduction. arXiv preprint. arXiv:1506.07194.Google Scholar
 Bolker, B. M., Brooks, M. E., Clark, C. J., Geange, S. W., Poulsen, J. R., Stevens, M. H. H., & White, J. S. (2009). Generalized linear mixed models: A practical guide for ecology and evolution. Trends in Ecology and Evolution, 24, 127–135. doi: 10.1016/j.tree.2008.10.008 CrossRefPubMedGoogle Scholar
 Bovet, J., Lao, J., Bartholomée, O., Caldara, R., & Raymond, M. (2016). Mapping females’ bodily features of attractiveness. Scientific Reports, 6, 18551. doi: 10.1038/srep18551
 Caldara, R., & Miellet, S. (2011). iMap: A novel method for statistical fixation mapping of eye movement data. Behavior Research Methods, 43, 864–878. doi: 10.3758/s134280110092x CrossRefPubMedGoogle Scholar
 Caldara, R., Zhou, X., & Miellet, S. (2010). Putting culture under the “spotlight” reveals universal information use for face recognition. PLoS ONE, 5, e9708. doi: 10.1371/journal.pone.0009708 CrossRefPubMedPubMedCentralGoogle Scholar
 Christensen, R. (2011). Plane answers to complex questions: The theory of linear models. Berlin: Springer.CrossRefGoogle Scholar
 Duchowski, A. T. (2002). A breadthfirst survey of eyetracking applications. Behavior Research Methods, Instruments, & Computers, 34, 455–470.CrossRefGoogle Scholar
 Friston, K. J., Stephan, K. E., Lund, T. E., Morcom, A., & Kiebel, S. (2005). Mixedeffects and fMRI studies. NeuroImage, 24, 244–252.CrossRefPubMedGoogle Scholar
 Jaynes, E. T., & Bretthorst, G. L. (2003). Probability theory: The logic of science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
 KheradPajouh, S., & Renaud, O. (2010). An exact permutation method for testing any effect in balanced and unbalanced fixed effect ANOVA. Computational Statistics and Data Analysis, 54, 1881–1893.CrossRefGoogle Scholar
 KheradPajouh, S., & Renaud, O. (2015). A general permutation approach for analyzing repeated measures ANOVA and mixedmodel designs. Statistical Papers, 56, 947–967. doi: 10.1007/s0036201406173 CrossRefGoogle Scholar
 Kliegl, R., Masson, M. E., & Richter, E. M. (2010). A linear mixed model analysis of masked repetition priming. Visual Cognition, 18, 655–681.CrossRefGoogle Scholar
 LeCun, Y., Haffner, P., Bottou, L., & Bengio, Y. (1999). Object recognition with gradientbased learning Shape, contour and grouping in computer vision (pp. 319–345). New York: Springer.CrossRefGoogle Scholar
 Liversedge, S., Gilchrist, I., & Everling, S. (2011). The Oxford handbook of eye movements. Oxford: Oxford University Press.CrossRefGoogle Scholar
 Marbach, D., Costello, J. C., Küffner, R., Vega, N. M., Prill, R. J., Camacho, D. M., . . . Stolovitzky, G. (2012). Wisdom of crowds for robust gene network inference. Nature Methods, 9, 796–804. doi: 10.1038/nmeth.2016
 McCulloch, C. E., Searle, S. R., & Neuhaus, J. M. (2011). Generalized, linear, and mixed models. New York: Wiley.Google Scholar
 Miellet, S., He, L., Zhou, X., Lao, J., & Caldara, R. (2012). When East meets West: Gazecontingent blindspots abolish cultural diversity in eye movements for faces. Journal of Eye Movement Research, 5(5), 1–12.Google Scholar
 Miellet, S., Lao, J., & Caldara, R. (2014). An appropriate use of iMap produces correct statistical results: A reply to McManus (2013) “iMAP and iMAP2 produce erroneous statistical maps of eyemovement differences.”. Perception, 43, 451–457.CrossRefPubMedGoogle Scholar
 Nuthmann, A., & Einhäuser, W. (2015). A new approach to modeling the influence of image features on fixation selection in scenes. Annals of the New York Academy of Sciences, 1339, 82–96.CrossRefPubMedPubMedCentralGoogle Scholar
 Oostenveld, R., Fries, P., Maris, E., & Schoffelen, J.M. (2011). FieldTrip: Open source software for advanced analysis of MEG, EEG, and invasive electrophysiological data. Computational Intelligence and Neuroscience, 2011, 9.CrossRefGoogle Scholar
 Penny, W. D., Friston, K. J., Ashburner, J. T., Kiebel, S. J., & Nichols, T. E. (2011). Statistical parametric mapping: The analysis of functional brain images: The analysis of functional brain images. San Diego: Academic Press.Google Scholar
 Pernet, C. R., Chauveau, N., Gaspar, C., & Rousselet, G. A. (2011). LIMO EEG: A toolbox for hierarchical LInear MOdeling of ElectroEncephaloGraphic data. Computational Intelligence and Neuroscience, 2011, 11.CrossRefGoogle Scholar
 Pernet, C. R., Latinus, M., Nichols, T. E., & Rousselet, G. A. (2015). Clusterbased computational methods for mass univariate analyses of eventrelated brain potentials/fields: A simulation study. Journal of Neuroscience Methods, 250, 85–93. doi: 10.1016/j.jneumeth.2014.08.003 CrossRefPubMedPubMedCentralGoogle Scholar
 Pinheiro, J. C., & Bates, D. M. (2000). Mixedeffects models in S and SPLUS. Berlin: Springer.CrossRefGoogle Scholar
 Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods. Thousand Oaks: Sage.Google Scholar
 Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological Bulletin, 124, 372–422. doi: 10.1037/00332909.124.3.372 CrossRefPubMedGoogle Scholar
 Smith, S. M., & Nichols, T. E. (2009). Thresholdfree cluster enhancement: Addressing problems of smoothing, threshold dependence and localisation in cluster inference. NeuroImage, 44, 83–98.CrossRefPubMedGoogle Scholar
 Sorzano, C. O. S., Vargas, J., & Montano, A. P. (2014). A survey of dimensionality reduction techniques. arXiv preprint. arXiv:1403.2877.Google Scholar
 Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., & Nichols, T. E. (2014). Permutation inference for the general linear model. NeuroImage, 92, 381–397.CrossRefPubMedPubMedCentralGoogle Scholar