Manual Statistics Made Learnable — A Learning Aid for Statistics Courses

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Increase site traffic, click-through rate and site stickiness. The HTML5 output offers the multi-device support, thereby, enabling learners to seamlessly move across devices as they complete a given course. However, the User Experience and interactions are largely aligned with the way learners consume content on desktops and laptops. As a result, this kind of a learning experience would work reasonably well on tablets. However, the approach has its limitations on smartphones on two counts:. In contrast to the adaptive mobile-friendly designs , responsive mobile-first designs should be used when the predominant consumption of content is expected to be on smartphones.

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More broadly, asking at what point new neurobiological knowledge is arising during ClSt and StLe investigations relies on largely distinct theoretical frameworks that revolve around null-hypothesis testing and statistical learning theory Figure 4. Both ClSt and StLe methods share the common goal of demonstrating relevance of a given effect in the data beyond the sample brain scans at hand. However, the attempt to show successful extrapolation of a statistical relationship at the general population is embedded in different mathematical contexts.

Knowledge generation in ClSt and StLe is hence rooted in different notions of statistical inference. Figure 4. Key concepts in classical statistics and statistical learning. Schematic with statistical notions that are relatively more associated with classical statistical methods left column or pattern-learning methods right column. As there is a smooth transition between the classical statistical toolkit and learning algorithms, some notions may be closely associated with both statistical cultures middle column.

The rationale behind hypothesis falsification is that one counterexample can reject a theory by deductive reasoning , while any quantity of evidence can not confirm a given theory by inductive reasoning Goodman, The investigator verbalizes two mutually exclusive hypotheses by domain-informed judgment. The alternative hypothesis should be conceived as the outcome intended by the investigator and to contradict the state of the art of the research topic.

The null hypothesis represents the devil's advocate argument that the investigator wants to reject i. If the null hypothesis can not be rejected which depends on power , then the test yields no conclusive result, rather than a null result Schmidt, In this way, classical hypothesis testing continuously replaces currently embraced hypotheses explaining a phenomenon in nature by better hypotheses with more empirical support in a Darwinian selection process.

Finally, Fisher, Neyman, and Pearson intended hypothesis testing as a marker for further investigation, rather than an off-the-shelf decision-making instrument Cohen, ; Nuzzo, In StLe instead, answers to how neurobiological conclusions can be drawn from a dataset at hand are provided by the Vapnik-Chervonenkis dimensions VC dimensions from statistical learning theory Vapnik, , The VC dimensions of a pattern-learning algorithm quantify the probability at which the distinction between the neural correlates underlying the face vs.

Such statistical approaches implement the inductive strategy to learn general principles i. Tenenbaum et al. VC dimensions are derived from the maximal number of different brain scans that can be correctly detected to belong to either the house condition or the face condition by a given model. The VC dimensions thus provide a theoretical guideline for the largest set of brain scan examples fed into a learning algorithm such that this model is able to guarantee zero classification errors.

As one of the most important results from statistical learning theory, in any intelligent learning system, the opportunity to derive abstract patterns in the world by reducing the discrepancy between prediction error from training data in-sample estimate and prediction error from independent test data out-of-sample estimate decreases with the higher model capacity and increases with the number of available training observations Vapnik and Kotz, ; Vapnik, In brain imaging, a learning algorithm is hence theoretically backed up to successfully predict outcomes in future brain scans with high probability if the choosen model ignores structure that is overly complicated, such as higher-order non-linearities between many brain voxels, and if the model is provided with a sufficient number of training brain scans.

Hence, VC dimensions provide explanations why increasing the number of considered brain voxels as input features i. Nevertheless, the VC dimensions provide justification that a certain learning model can be used to approximate that target function by fitting a model to a collection of input-output pairs. In short, VC dimensions is among the best frameworks to derive theoretical errors bounds for predictive models Abu-Mostafa et al. Further, some common invalidations of the ClSt and StLe statistical concern in neuroimaging studies performing classical inference is double dipping or circular analysis Kriegeskorte et al.

This occurs when, for instance, first correlating a behavioral measure with brain activity and then using the identified subset of brain voxels for a second correlation analysis with that same behavioral measurement Lieberman et al. In this scenario, voxels are submitted to two statistical tests with the same goal in a nested, non-independent fashion 5 Freedman, This corrupts the validity of the null hypothesis on which the reported test results conditionally depend.

Importantly, this case of repeating a same statistical estimation with iteratively pruned data selections on the training data split is a valid routine in the StLe framework, such as in recursive feature extraction Guyon et al. However, double-dipping or circular analysis in ClSt applications to neuroimaging data have an analog in StLe analyses aiming at out-of-sample generalization: data-snooping or peeking Pereira et al.

This can occur, for instance, when performing simple e. Data-snooping can lead to overly optimistic cross-validation estimates and a trained learning algorithm that fails on fresh data drawn from the same distribution Abu-Mostafa et al. Rather than a corrupted null hypothesis, it is the error bounds of the VC dimensions that are loosened and, ultimately, invalidated because information from the concealed test set influences model selection on the training set.

In sum, statistical inference in ClSt is drawn by using the entire data at hand to formally test for theoretically guaranteed extrapolation of an effect to the general population. In stark contrast, inferential conclusions in StLe are typically drawn by fitting a model on a larger part of the data at hand i. As such, ClSt has a focus on in-sample estimates and explained-variance metrics that measure some form of goodness of fit, while StLe has a focus on out-of-sample estimates and prediction accuracy. Vignette: After isolating the neural correlates underlying face processing, the neuroimaging investigator wants to examine their relevance in psychiatric disease.

In addition to the 40 healthy participants, 40 patients diagnosed with schizophrenia are recruited and administered the same experimental paradigm and set of face and house pictures.

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In this clinical fMRI study on group differences, the investigator wants to explore possible imaging-derived markers that index deficits in social-affective processing in patients carrying a diagnosis of schizophrenia. Question: Can metrics of statistical relevance from ClSt and StLe be combined to corroborate a given candidate biomarker? Many neuroscientists have thus adopted a natural habit of assessing the quality of statistical relationships by means of p -values, effect sizes, confidence intervals, and statistical power.

These are ubiquitously taught and used at many universities, although they are not the only coherent set of statistical diagnostics Figure 5. These outcome metrics from ClSt may for instance be less familiar to some scientists with a background in computer science, physics, engineering, or philosophy. As an equally legitimate and internally coherent, yet less widely known diagnostic toolkit from the StLe community, prediction accuracy, precision, recall, confusion matrices, F1 score, and learning curves can also be used to measure the relevance of statistical relationships Abu-Mostafa et al.

Figure 5. Key differences between measuring outcomes in classical statistics and statistical learning. Ten intuitions on quantifying statistical modeling outcomes that tend to be relatively more true for classical statistical methods blue or pattern-learning methods red. ClSt typically yields point estimates and interval estimates e. In many cases, classical inference is a judgment about an entire data sample, whereas a trained predictive model can obtain quantitative answers from a single data point. On a general basis, applications of ClSt and StLe methods may not judge findings on identical grounds Breiman, ; Shmueli, ; Lo et al.

There is an often-overlooked misconception that models with high explanatory performance do necessarily exhibit high predictive performance Wu et al. For instance, brain voxels in ventral visual stream found to well explain the difference between face processing in healthy and schizophrenic participants based on an ANOVA may not in all cases be the best brain features to train a support vector machine to predict this group effect in new participants. An important outcome measure in ClSt is the quantified significance associated with a statistical relationship between few variables given a pre-specified model.

ClSt tends to test for a particular structure in the brain data based on analytical guarantees , in form of as mathematical convergence theorems about approximating the population properties with increasing sample size. The outcome measure for StLe is the quantified generalization of patterns between many variables or, more generally, the robustness of special structure in the data Hastie et al. In the neuroimaging literature, reports of statistical outcomes have previously been noted to confuse diagnostic measures from classical statistics and statistical learning Friston, For neuroscientists adopting a ClSt culture computing p -values takes a central position.

The p-value denotes the probability of observing a result at least as extreme as a test statistic, assuming the null hypothesis is true. Under the condition of sufficiently high power cf. Counterintuitively, it is not an immediate judgment on the alternative hypothesis H 1 preferred by the investigator Cohen, ; Anderson et al. P -values do also not qualify the possibility of replication. It is another important caveat that a finding in the brain becomes more statistically significant i.

The essentially binary p -value i. The effect size allows the identification of marginal effects that pass the statistical significance threshold but are not practically relevant in the real world. The p -value is a deductive inferential measure, whereas the effect size is a descriptive measure that follows neither inductive nor deductive reasoning. The normalized effect size can be viewed as the strength of a statistical relationship—how much H 0 deviates from H 1 , or the likely presence of an effect in the general population Chow, ; Ferguson, ; Kelley and Preacher, This diagnostic measure is often unit-free, sample-size independent, and typically standardized.

As a property of the actual statistical test, the effect size can be essential to report for biological understanding, but has different names and takes various forms, such as rho in Pearson correlation, eta 2 in explained variances, and Cohen's d in differences between group averages. Additionally, the certainty of a point estimate i. These variability diagnostics indicate a range of values between which the true value will fall a given proportion of the time Estes, ; Nickerson, ; Cumming, The tighter the confidence interval, the smaller the variance of the point estimate of the population parameter in each drawn sample.

The estimation of confidence intervals is influenced by sample size and population variability. Confidence intervals may be asymmetrical ignored by Gaussianity assumptions; Efron, , can be reported for different statistics and with different percentage borders. Notably, they can be used as a viable surrogate for formal tests of statistical significance in many scenarios Cumming, Some confidence intervals can be computed in various data scenarios and statistical regimes, whereas the power may be especially meaningful within the culture of classical hypothesis testing Cohen, , ; Oakes, To estimate power the investigator needs to specify the true effect size and variance under H 1.

The ClSt-minded investigator can then estimate the probability for rejecting null hypotheses that should be rejected, at the given threshold alpha and given that H 1 is true. A high power thus ensures that statistically significant and non-significant tests indeed reflect a property of the population Chow, Intuitively, a small confidence interval around a relevant effect suggests high statistical power.

False negatives i. Ioannidis, Concretely, an underpowered investigation means that the investigator is less likely to be able to distinguish between H 0 and H 1 at the specified significance threshold alpha. Power calculations depend on several factors, including significance threshold alpha, the effect size in the population, variation in the population, sample size n , and experimental design Cohen, While neuroimaging studies based on classical statistical inference ubiquitously report p -values and confidence intervals, there have however been few reports of effect size in the neuroimaging literature Kriegeskorte et al.

Effect sizes are however necessary to compute power estimates. This explains the even rarer occurrence of power calculations in the neuroimaging literature Yarkoni and Braver, ; but see Poldrack et al. Given the importance of p -values and effect sizes, the goal of computing both these useful statistics, such as for group differences in the neural processing of face stimuli, can be achieved based on two independent samples of these experimental data especially if some selection process has been used. One sample would be used to perform statistical inference on the neural activity change yielding a p -value and one sample to obtain unbiased effect sizes.

Further, it has been previously emphasized Friston, that p -values and effect sizes reflect in-sample estimates in a retrospective inference regime ClSt. These metrics find an analog in out-of-sample estimates issued from cross-validation in a prospective prediction regime StLe. Instead, classification accuracy on fresh data is a frequently reported performance metric in neuroimaging studies using learning algorithms.

The classification accuracy is a simple summary statistic that captures the fraction of correct prediction instances among all performed applications of a fitted model. Basing interpretation on accuracy alone can be an insufficient diagnostic because it is frequently influenced by the number of samples, the local characteristics of hemodynamic responses, efficiency of experimental design, data folding into train and test sets, and differences in the feature number p Haynes, A potentially under-exploited data-driven tool in this context is bootstrapping.

The archetypical example of computer-intensive statistical method enables population-level inference of unknown distributions largely independent of model complexity by repeated random draws from the neuroimaging data sample at hand Efron, ; Efron and Tibshirani, This opportunity to equip various point estimates by an interval estimate of certainty e. Besides providing confidence intervals, bootstrapping can also perform non-parametric null hypothesis testing. This may be one of few examples of a direct connection between ClSt and StLe methodology.

Alternatively, binomial tests have been used to obtain a p -value estimate of statistical significance from accuracies and other performance scores Pereira et al. It has frequently been employed to reject the null hypothesis that two categories occur equally often. There are however increasing concerns about the validity of this approach if statistical independence between the performance estimates e.

Yet another option to derive p -values from classification performances of two groups is label permutation based on non-parametric resampling procedures Nichols and Holmes, ; Golland and Fischl, This algorithmic significance-testing tool can serve to reject the null hypothesis that the neuroimaging data do not contain relevant information about the group labels in many complex data analysis settings. The neuroscientist who adopted a StLe culture is in the habit of corroborating prediction accuracies using cross-validation: the de facto standard to obtain an unbiased estimate of a model's capacity to generalize beyond the brain scans at hand Hastie et al.

Model assessment is commonly done by training on a bigger subset of the available data i. Cross-validation typically divides the sample into data splits such that the class label i. The pairs of model-predicted label and the corresponding true label for each data point i. Accuracy and the other performance metrics are often computed separately on the training set and the test set.

Additionally, the measures from training and testing can be expressed by their inverse e. The classification accuracy can be further decomposed into group-wise metrics based on the so-called confusion matrix , the juxtaposition of the true and predicted group memberships. The precision measures Table 1 how many of the labels predicted from brain scans are correct, that is, how many participants predicted to belong to a certain class really belong to that class. Put differently, among the participants predicted to suffer from schizophrenia, how many have really been diagnosed with that disease?

On the other hand, the recall measures how many labels are correctly predicted, that is, how many members of a class were predicted to really belong to that class. Hence, among the participants known to be affected by schizophrenia, how many were actually detected as such? Neither accuracy, precision, or recall allow injecting subjective importance into the evaluation process of the learning algorithm.

This disadvantage is addressed by the F beta score : a weighted combination of the precision and recall prediction scores. Concretely, the F 1 score would equally weigh precision and recall of class predictions, while the F 0. Moreover, applications of recall, precision, and F beta scores have been noted to ignore the true negative cases as well as to be highly susceptible to estimator bias Powers, Needless to say, no single outcome metric can be equally optimal in all contexts.

Extending from the setting of healthy-diseased classification to the multi-class setting e. Rather than reporting mere better-than-chance findings in StLe analyses, it becomes more important to evaluate the F 1 , precision and recall scores for each class to be predicted in the brain scans e. In fact, sensitivity equates with recall. Specificity does however not equate with precision. Again, Type I and II errors are related to the entirety of data points in a ClSt regime and prediction is only evaluated on a test data split of the sample in an StLe regime.

Finally, StLe-minded investigators use learning curves Abu-Mostafa et al. For increasingly bigger subsets of the training set, a classification algorithm is trained on that current share of the training set and then evaluated for accuracy on the always-same test set. Across subset instances, simple models display relatively high in-sample error because they can not approximate the target function very well underfitting but exhibit good generalization to unseen data with relatively low out-of-sample error.

Yet, complex models display relatively low in-sample error because they adapt too well to the data overfitting with difficulty to extrapolate to newly sampled data with high out-of-sample error. Put differently, a big gap between high in-sample and low out-of-sample performance is typically observed for high-variance models, such as artificial neural network algorithms or random forests.

These performance metrics from different data splits often converge for high-bias models, such as linear support vector machines and logistic regression. In sum, the ClSt and StLe communities rely on diagnostic metrics that are largely incongruent and may therefore not lend themselves for direct comparison in all practical analysis settings. Vignette: The investigator is interested in potential differences in brain volume that are associated with an individual's age continuous target variable. This L1-penalized residual-sum-of-squares regression performs automatic variable selection i.

Assessing generalization performance of different sparse models using 5-fold cross-validation yields the non-zero coefficients for few brain voxels whose volumetric information is most predictive of an individual's age. Question: How can the investigator perform classical inference to know which of the gray-matter voxels selected to be predictive for biological age are statistically significant?

This is an important concern because most statistical methods currently applied to large datasets perform some explicit or implicit form of variable selection Jenatton et al. There are even many different forms of preliminary selection of variables before performing significance tests on them. Beyond neuroscience, generalization-approved statistical learning models are routinely solving a diverse set of real-world challenges.

This includes algorithmic trading in financial markets, fraud detection in credit card transactions, real-time speech translation, SPAM filtering for e-mails, face recognition in digital cameras, and piloting self-driving cars Jordan and Mitchell, ; LeCun et al. In all these examples, statistical learning algorithms successfully generalize to unseen, later acquired data and thus tackle the problem heuristically without classical significance test on specific variables or for overall model performance.

Second, the LASSO has been introduced as an elegant solution to the combinatorial problem of what subset of gray-matter voxels is sufficient for predicting an individual's age by automatic variable selection Tibshirani, Computing voxel-wise p -values would recast this high-dimensional pattern-learning setting i. Yet, recasting into the mass-univariate setting would ignore the sophisticated selection process that led to the predictive model with a reduced number of variables Wu et al. Put differently, variable selection via the LASSO is itself a stochastic process that is however not accounted for by the theoretical guarantees of classical inference for statistical significance Berk et al.

Put in yet another way, data-driven model selection is corrupting the null hypothesis of classical statistical inference because the sampling distribution of the parameter estimates is altered. Third, the portrayed conflict between more exploratory model selection by cross-validation StLe and more confirmatory classical inference ClSt is currently at the frontier of statistical development Loftus, ; Taylor and Tibshirani, New methods for so-called post-selection inference or selective inference allow computing p -values for a set of features that have previously been chosen to be meaningful predictors by some criterion, one example being sparsity-incuding prediction algorithms such as LASSO.

According to the theory of ClSt, the statistical model is to be chosen before visiting the data. Classical statistical tests and confidence intervals therefore become invalidated and the p -values become optimistically biased Berk et al. Consequently, the association between a predictor and the target variable must be even stronger to certify the same level of significance. As an ordinary null hypothesis can hardly be adopted in this adaptive testing setting, conceptual extension is also prompted on the level of ClSt theory itself Hastie et al.

For instance, closed-form solutions to adjusted classical inference after variable selection already exist for principal component analysis Choi et al. Moreover, a simple alternative to formally account for preceding model selection is data splitting Cox, ; Wasserman and Roeder, ; Fithian et al. In this procedure, the variable selection procedure is computed on one data split and p -values are computed on the remaining second data split.

However, such data splitting is not always possible and will incur power losses. In sum, in many analysis settings, the same data should typically not be used to first apply supervised learning algorithms for automatic selection of the most predictive variables and to then test for statistical significance of the variables already found to be most predictive based on these data points. The recent developments for post-selection inference can be viewed as an attempt to reconcile certain aspects of how the StLe and ClSt paradigms draw conclusions from data.

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Vignette: The investigator is interested in potential brain structure differences that are associated with an individual's gender categorical target variable in the voxel-based morphometry data of the 1,subject HCP release Human Connectome Project; Van Essen et al. To this end, ANOVA univariate test for statistical significance belonging to ClSt is initially used to obtain a ranking of the most relevant 10, features from the gray matter.

Question: Is an analysis pipeline with univariate classical inference and subsequent high-dimensional prediction valid if both steps rely on gender as the target variables? The implications of feature engineering procedures applied before training a learning algorithm is a frequent concern and can require subtle answers Guyon and Elisseeff, ; Kriegeskorte et al. In most applications of predictive models the large majority of brain voxels will not be very informative Brodersen et al. The described scenario of dimensionality reduction by feature selection to focus prediction is clearly allowed under the condition that the ANOVA is not computed on the entire data sample.

Rather, the initial identification of voxels explaining most variance between the male and female individuals should be computed only on the training set in each cross-validation fold. In the training set and test set of each fold the same identified candidate voxels are then regrouped into a feature space that is fed into the support vector machine algorithm. This ensures an identical feature space for model training and model testing but its construction only depends on structural brain scans from the training set. Generally, voxel preprocessing performed before model training is authorized if the feature space construction is not influenced by properties of the concealed test set.

In the present scenario, the Vapnik-Chervonenkis bounds of the cross-validation estimator are therefore not loosened or invalidated if class labels have been exploited for feature selection or depending on whether the feature selection procedure is univariate or multivariate Abu-Mostafa et al. Put differently, the cross-validation procedure simply evaluates the entire prediction process including the automatized and potentially nested dimensionality reduction approaches. In sum, in an StLe regime, using class information during feature preprocessing for a cross-validated supervised estimator is not an instance of data-snooping or peeking if done exclusively on the training set Abu-Mostafa et al.

At the core of this explanation is the goal of cross-validation to yield out-of-sample estimates. In stark contrast, remember that null-hypothesis testing yields in-sample estimates as it needs all available data points to take its decision. Using the class labels for a variable selection step just before null-hypothesis testing on a same data sample would invalidate the null hypothesis Kriegeskorte et al. Consequently, in a ClSt regime, using class information to select variables before null-hypothesis testing will incur an instance of double-dipping or circular analysis.

This also occurs when, for instance, first correlating a behavioral measure with brain activity and then using the identified subset of brain voxels for a second correlation analysis with that same behavioral measurement Lieberman et al. In this scenario, voxels are submitted to two statistical tests with the same goal in a nested, non-independent fashion Freedman, Regarding interpretation of the results, the classifier will miss some brain voxels that only carry relevant information when considered in voxel ensembles.

Univariate feature selection in high-dimensional brain scans may therefore systematically encourage model selection i. Concretely, in the discussed scenario the classifier learns complex patterns between voxels that were previously chosen to be individually important. Remember also that variables that have a statistically significant association with a target variable do not necessarily have good generalization performance , and vice versa Shmueli, ; Lo et al.

On the upside, it is frequently observed that the combination of whole-brain univariate feature selection and linear classification is among the best approaches if the primary goal is maximizing prediction performance as opposed to maximizing interpretability. This allows recasting the StLe regime into a ClSt regime in order to fit a GLM and perform classical statistical tests instead of training a predictive classification algorithm Brodersen et al.

In sum, in many analysis settings, prediction algorithms can be trained after choosing the input variables most significantly associated with an explanatory target variable if the initial classical inference p -values is performed only in the training set and the ensuing evaluation of algorithm generalization prediction performance is performed on the independent test set. Vignette: Each functionally specialized region in the human brain probably has a unique set of long-range connections Passingham et al. This notion has prompted connectivity-based parcellation methods in neuroimaging that segregate an ROI can be locally circumscribed or brain global; Eickhoff et al.

The whole-brain connectivity for each ROI voxel is computed and the voxel-wise connectional fingerprints are submitted to a clustering algorithm i. The investigator wants to apply connectivity-based parcellation to the fusiform gyrus to segregate this ROI into cortical modules that exhibit similar connectivity patterns with the rest of the brain and are, thus potentially, functionally distinct. That is, voxels within the same cluster in the ROI will have more similar whole-brain connectivity properties than voxels from different clusters in the fusiform gyrus.

Question: Is it possible to decide whether the obtained brain clusters are statistically significant? In essence, the aim of connectivity-guided brain parcellation is to find useful, simplified structure by imposing circumscribed compartments on brain topography Yeo et al. This is typically achieved by using k-means, hierarchical, Ward, or spectral clustering algorithms Thirion et al.

Putting on the ClSt hat, an ROI clustering result would be deemed statistically significant if the obtained data are incompatible with the null hypothesis that the investigator seeks to reject Everitt, ; Halkidi et al. Choosing a test statistic for clustering solutions to obtain p -values is difficult Vogelstein et al. Put differently, for classical inference based on statistical hypothesis testing one may need to pick an arbitrary null hypothesis to falsify.

It follows that neither the ClSt notions of effect size and power do seem to apply in the case of brain parcellation also a frequent question by paper reviewers. Instead of classical inference to formally test for a particular structure in the clustering results, the investigator actually needs to resort to exploratory approaches that discover and assess structure in the neuroimaging data Tukey, ; Efron and Tibshirani, ; Hastie et al.

Although statistical methods span a continuum between the two poles of ClSt and StLe, finding a clustering model with the highest fit in the sense of explaining the regional connectivity differences at hand is perhaps more naturally situated in the StLe community. Putting on the StLe hat, the investigator realizes that the problem of brain parcellation constitutes an unsupervised learning setting without any target variable y to predict e.

In clustering analysis, there are many possible transformations, projections, and compressions of X but there is usually no unique criterion of optimality that clearly suggests itself. Evaluating the adequacy of clustering results is therefore conventionally addressed by applying different cluster validity criteria Thirion et al. These heuristic metrics are useful and necessary because clustering algorithms will always find some subregions in the investigator's ROI, that is, find relevant structure with respect to the particular optimization objective of the clustering algorithm whether such structure truly exists in nature or not.

The various clustering validity criteria, possibly based on information theory, topology, or consistency Eickhoff et al. Given that the notions of optimality are not coherent with each other Shalev-Shwartz and Ben-David, ; Thirion et al. Evidently, the discovered set of connectivity-derived clusters only represent hints to candidate brain modules.

Nevertheless, such clustering solutions provide important means to narrow down high-dimensional neuroimaging data. Preliminary clustering results broaden the space of research hypotheses that the investigator can articulate. For instance, unexpected discovery of a candidate brain region cf. Mars et al. Brain parcellation can thus be viewed as an exploratory unsupervised method outlining relevant structure in neuroimaging data that can subsequently be tested as research hypotheses in targeted future neuroimaging studies on classical inference or out-of-sample generalization.

In sum, in most analysis settings, quantifying the importance of clustering solutions is inherently ill-posed because, without an explanatory target variable, many different low-dimensional reexpressions of high-dimensional input data can be useful. Choosing the right variant among the possible dimensionality reductions by clustering algorithms alone can typically not be done based on extrapolation metrics from ClSt p -values, effect size, power or StLe out-of-sample prediction performance, learning curves.

A novel scientific fact about the brain is only valid in the context of the complexity restrictions that have been imposed on the studied phenomenon during the investigation Box, Tools of the imaging neuroscientist's statistical arsenal can be placed on a continuum between classical inference by hypothesis falsification and increasingly used out-of-sample generalization by extrapolating complex patterns to independent data Efron and Hastie, While null-hypothesis testing has been dominating academic milieus in the empirical sciences and statistics departments for several decades, statistical learning methods are perhaps still more prevalent in data-intensive industries Breiman, ; Vanderplas, ; Henke et al.

This sociological segregation may contribute to the existing confusion about the mutual relationship between the ClSt and StLe camps in application domains such as imaging neuroscience. Despite the incongruent historical trajectories and theoretical foundations, both statistical cultures aim at inferential conclusions by extracting new knowledge from data using mathematical models Friston et al.

However, an observed effect in the brain with a statistically significant p -value does not in all cases generalize to future brain recordings Shmueli, ; Arbabshirani et al. Conversely, a neurobiological effect that can be successfully captured by a learning algorithm as evidenced by out-of-sample generalization does not invariably entail a significant p -value when submitted to null-hypothesis testing. The distributional properties of brain data important for high statistical significance and for high prediction accuracy are not identical Efron, ; Lo et al.

The goal and permissible conclusions of a neuroscientific investigation are therefore conditioned by the adopted statistical framework cf. Feyerabend, Awareness of the prediction-inference distinction will be criticial to keep pace with the increasing information detail of neuroimaging data repositories Eickhoff et al.

Ultimately, statistical inference is not a uniquely defined concept. The author confirms being the sole contributor of this work and approved it for publication. The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. In the optimization setting of finite spaces, all algorithms searching an extremum perform identically when averaged across possible cost functions. Abu-Mostafa, Y. Learning from Data. Google Scholar. Altman, D. Statistics notes: diagnostic tests 2: predictive values.

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