# The “Group” indicator in FSL

In FSL, when we create a design using the graphical interface in FEAT, or with the command Glm, we are given the opportunity to define, at the higher-level, the “Group” to which each observation belongs. When the design is saved, the information from this setting is stored in a text file named something as “design.grp”. This file, and thus the group setting, takes different roles depending whether the analysis is used in FEAT itself, in PALM, or in randomise.

What can be confusing sometimes is that, in all three cases, the “Group” indicator does not refer to experimental or observational group of any sort. Instead, it refers to variance groups (VG) in FEAT, to exchangeability blocks (EB) in randomise, and to either VG or EB in PALM, depending on whether the file is supplied with the options -vg or -eb.

In FEAT, unless there is reason to suspect (or assume) that the variances for different observations are not equal, all subjects should belong to group “1”. If variance groups are defined, then these are taken into account when the variances are estimated. This is only possible if the design matrix is “separable”, that is, it must be such that, if the observations are sorted by group, the design can be constructed by direct sum (i.e., block-diagonal concatenation) of the design matrices for each group separately. A design is not separable if any explanatory variable (EV) present in the model crosses the group borders (see figure below). Contrasts, however, can encompass variables that are defined across multiple VGs.

The variance groups not necessarily must match the experimental observational groups that may exist in the design (for example, in a comparison of patients and controls, the variance groups may be formed based on the sex of the subjects, or another discrete variable, as opposed to the diagnostic category). Moreover, the variance groups can be defined even if all variables in the model are continuous.

In randomise, the same “Group” setting can be supplied with the option -e design.grp, thus defining exchangeability blocks. Observations within a block can only be permuted with other observations within that same block. If the option --permuteBlocks is also supplied, then the EBs must be of the same size, and the blocks as a whole are instead then permuted. Randomise does not use the concept of variance group, and all observations are always members of the same single VG.

In PALM, using -eb design.grp has the same effect that -e design.grp has in randomise. Further using the option -whole is equivalent to using --permuteBlocks in randomise. It is also possible to use together -whole and -within, meaning that the blocks as a whole are shuffled, and further, observations within block are be shuffled. In PALM the file supplied with the option -eb can have multiple columns, indicating multi-level exchangeability blocks, which are useful in designs with more complex dependence between observations. Using -vg design.grp causes PALM to use the v– or G-statistic, which are replacements for the t– and F-statistics respectively for the cases of heterogeneous variances. Although VG and EB are not the same thing, and may not always match each other, the VGs can be defined from the EBs, as exchangeability implies that some observations must have same variance, otherwise permutations are not possible. The option -vg auto defines the variance groups from the EBs, even for quite complicated cases.

In both FEAT and PALM, defining VGs will only make a difference if such variance groups are not balanced, i.e., do not have the same number of observations, since heteroscedasticity (different variances) only matter in these cases. If the groups have the same size, all subjects can be allocated to a single VG (e.g., all “1”).

# Why the maximum statistic?

In brain imaging, each voxel (or vertex, or face, or edge) constitutes a single statistical test. Because thousands such voxels are present in an image, a single experiment results in thousands of statistical tests being performed. The p-value is the probability of finding a test statistic at least as large as the one observed in a given voxel, provided that no effect is present. A p-value of 0.05 indicates that, if an experiment is repeated 20 times and there are no effects, on average one of these repetitions will be considered significant. If thousands of tests are performed, the chance of obtaining a spuriously significant result in at least one voxel increases: if there are 1000 voxels, and at the same test level $\alpha = 0.05$, we expect, on average, to find 50 significant tests, even in the absence of any effect. This is known as the multiple testing problem. A review of the topic for brain imaging provided in Nichols and Hayasaka (2003) [see references at the end].

To take the multiple testing problem into account, either the test level ($\alpha$), or the p-values can be adjusted, such that instead of controlling the error rate at each individual test, the error rate is controlled for the whole set (family) of tests. Controlling such family-wise error rate (FWER) ensures that the chance of finding a significant result anywhere in the image is expected to be within a certain predefined level. For example, if there are 1000 voxels, and the FWER-adjusted test level is 0.05, we expect that, if the experiment is repeated for all the voxels 20 times, then on average in one of these repetitions there will be an error somewhere in the image. The adjustment of the p-values or of the test level is done using the distribution of the maximum statistic, something that most readers of this blog are certainly well aware of, as that permeates most of the imaging literature since the early 1990s.

Have you ever wondered why? What is so special about the distribution of the maximum that makes it useful to correct the error rate when there are multiple tests?

## Definitions first

Say we have a set of $V$ voxels in an image. For a given voxel $v$, $v \in \{1, \ldots, V\}$, with test statistic $t_v$, the probability that $t_v$ is larger than some cutoff $t$ is denoted by:

$\mathsf{P}(t_v > t) = 1 - F_v(t)$

where $F_v(t)$ is the cumulative distribution function (cdf) of the test statistic. If the cutoff $t$ is used to accept or reject a hypothesis, then we say that we have a false positive if an observed $t_v$ is larger than $t$ when there is no actual true effect. A false positive is also known as error type I (in this post, the only type of error discussed is of the type I).

For an image (or any other set of tests), if there is an error anywhere, we say that a family-wise error has occurred. We can therefore define a “family-wise null hypothesis” that there is no signal anywhere; to reject this hypothesis, it suffices to have a single, lonely voxel in which $t_v > t$. With many voxels, the chances of this happening increase, even if there no effect is present. We can, however, adjust our cuttoff $t$ to some other value $t_{\text{FWER}}$ so that the probability of rejecting such family-wise null hypothesis remains within a certain level, say $\alpha_{\text{FWER}}$.

## Union-intersection tests

The “family-wise null hypothesis” is effectively a joint null hypothesis that there is no effect anywhere. That is, it is an union-intersection test (UIT; Roy, 1953). This joint hypothesis is retained if all tests have statistics that are below the significance cutoff. What is the probability of this happening? From the above we know that $\mathsf{P}(t_v \leqslant t) = F_v(t)$. The probability of the same happening for all voxels simultaneously is, therefore, simply the product of such probabilities, assuming of course that the voxels are all independent:

$\mathsf{P}(\bigwedge_v t_v \leqslant t) = \prod_v \mathsf{P}(t_v \leqslant t) = \prod_v F_v(t)$

Thus, the probability that any voxel has a significant result, which would lead to the occurrence of a family-wise error, is $1-\prod_v F_v(t)$. If all voxels have identical distribution under the null, then the same is stated as $1- F_v(t)^V$.

## Distribution of the maximum

Consider the maximum of the set of $V$ voxels, that is, $M = \max{(t_v)}$. The random variable $M$ is only smaller or equal than some cutoff $t$ if all values $t_v$ are smaller or equal than $t$. If the voxels are independent, this enables us to derive the cdf of $M$:

$\mathsf{P}(M \leqslant t) = \prod_v \mathsf{P}(t_v \leqslant t) = \prod_v F_v(t)$.

Thus, the probability that $M$ is larger than some threshold $t$ is $1-\prod_v F_v(t)$. If all voxels have identical distribution under the null, then the same is stated as $1- F_v(t)^V$.

These results, lo and behold, are the same as those used for the UIT above, hence how the distribution of the maximum can be used to control the family-wise error rate (if the distribution of the maximum is computed via permutations, independence is not required).

## Closure

The above is not the only way in which we can see why the distribution of the maximum allows the control of the family-wise error rate. The work by Marcus, Peritz and Gabriel (1976) showed that, in the context of multiple testing, the null hypothesis for a particular test $v$ can be rejected provided that all possible joint (multivariate) tests done within the set and including $v$ are also significant, and doing so controls the family-wise error rate. For example, if there are four tests, $v \in \{1, 2, 3, 4\}$, the test in $v=1$ is considered significant if the joint tests using (1,2,3,4), (1,2,3), (1,2,4), (1,3,4), (1,2), (1,3), (1,4) and (1) are all significant (that is, all that include $v=1$). Such joint test can be quite much any valid test, including Hotelling’s $T^2$, MANOVA/MANCOVA, or NPC (Non-Parametric Combination), all of which are based on recomputing the test statistic from the original data, or others, based on the test statistics or p-values of each of the elementary $V$ tests, as in a meta-analysis.

Such closed testing procedure (CTP) incurs an additional problem, though: the number of joint tests that needs to be done is $2^V-1$, which in imaging applications renders them unfeasible. However, there is one particular joint test that provides a direct algorithmic shortcut: using the $\max(t_v)$ as the test statistic for the joint test. The maximum across all $V$ tests is also the maximum for any subset of tests, such that these can be skipped altogether. This gives a vastly efficient algorithmic shortcut to a CTP, as shown by Westfall and Young (1993).

## Simple intuition

One does not need to chase the original papers cited above (although doing so cannot hurt). Broadly, the same can be concluded based solely on intuition: if the distribution of some test statistic that is not the distribution of the maximum within an image were used as the reference to compute the (FWER-adjusted) p-values at a given voxel $v$, then the probability of finding a voxel with a test statistic larger than $t_v$ anywhere could not be determined: there could always be some other voxel $v'$, with an even larger statistic (i.e., $t_{v'} > t_v$), but the probability of such happening would not be captured by the distribution of a non-maximum. Hence the chance of finding a significant voxel anywhere in the image under the null hypothesis (the very definition of FWER) would not be controlled. Using the absolute maximum eliminates this logical leakage.

## References

• Marcus R, Peritz E, Gabriel KR. On closed testing pocedures with special reference to ordered analysis of variance. Biometrika. 1976 Dec;63(3):655.
• Nichols T, Hayasaka S. Controlling the familywise error rate in functional neuroimaging: a comparative review. Stat Methods Med Res. 2003 Oct;12(5):419–46.
• Roy SN. On a heuristic method of test construction and its use in multivariate analysis. Ann Math Stat. 1953 Jun;24(2):220–38.
• Westfall PH, Young SS. Resampling-based multiple testing: examples and methods for p-value adjustment. New York, Wiley, 1993.

# Better statistics, faster

Permutation tests are more robust and help to make scientific results more reproducible by depending on fewer assumptions. However, they are computationally intensive as recomputing a model thousands of times can be slow. The purpose of this post is to briefly list some options available for speeding up permutation.

Firstly, no speed-ups may be needed: for small sample sizes, or low resolutions, or small regions of interest, a permutation test can run in a matter of minutes. For larger data, however, accelerations may be of use. One option is acceleration through parallel processing or GPUs (for example applications of the latter, see Eklund et al., 2012, Eklund et al., 2013 and Hernández et al., 2013; references below), though this does require specialised implementation. Another option is to reduce the computational burden by exploiting the properties of the statistics and their distributions. A menu of options includes:

• Do few permutations (shorthand name: fewperms). The results remain valid on average, although the p-values will have higher variability.
• Keep permuting until a fixed number of permutations with statistic larger than the unpermuted is found (a.k.a., negative binomial; shorthand name: negbin).
• Do a few permutations, then approximate the tail of the permutation distribution by fitting a generalised Pareto distribution to its tail (shorthand name: tail).
• Approximate the permutation distribution with a gamma distribution, using simple properties of the test statistic itself, amazingly not requiring any permutations at all (shorthand name: noperm).
• Do a few permutations, then approximate the full permutation distribution by fitting a gamma distribution (shorthand name: gamma).
• Run permutations on only a few voxels, then fill the missing ones using low-rank matrix completion theory (shorthand name: lowrank).

These strategies allow accelerations >100x, yielding nearly identical results as in the non-accelerated case. Some, such as tail approximation, are generic enough to be used nearly all the most common scenarios, including univariate and multivariate tests, spatial statistics, and for correction for multiple testing.

In addition to accelerating permutation tests, some of these strategies, such as tail and noperm, allow continuous p-values to be found, and refine the p-values far into the tail of the distribution, thus avoiding the usual discreteness of p-values, which can be a problem in some applications if too few permutations are done.

These methods are available in the tool PALM — Permutation Analysis of Linear Models — and the complete description, evaluation, and application to the re-analysis of a voxel-based morphometry study (Douaud et al., 2007) have been just published in Winkler et al., 2016 (for the Supplementary Material, click here). The paper includes a flow chart prescribing these various approaches for each case, reproduced below.

The hope is that these accelerations will facilitate the use of permutation tests and, if used in combination with hardware and/or software improvements, can further expedite computation leaving little reason not to use these tests.

## References

Contributed to this post: Tom Nichols, Ged Ridgway.

# Three HCP utilities

If you are working with data from the Human Connectome Project (HCP), perhaps these three small Octave/MATLAB utilities may be of some use:

• hcp2blocks.m: Takes the restricted file with information about kinship and zygosity and produces a multi-level exchangeability blocks file that can be used with PALM for permutation inference. It is fully described here.
• hcp2solar.m: Takes restricted and unrestricted files to produce a pedigree file that can be used with SOLAR for heritability and genome-wide association analyses.
• picktraits.m: Takes either restricted or unrestricted files, a list of traits and a list of subject IDs to produce tables with selected traits for the selected subjects. These can be used to, e.g., produce design matrices for subsequent analysis.

These functions need to parse relatively large CSV files, which is somewhat inefficient in MATLAB and Octave. Still, since these commands usually have to be executed only once for a particular analysis, a 1-2 minute wait seems acceptable.

## A fourth utility

For the HCP-S1200 release (March/2017), zygosity information is provided in the fields ZygositySR (self-reported zygosity) and ZygosityGT (zygosity determined by genetic methods for select subjects). If needed, these two fields can be merged into a new field named simply Zygosity. To do so, use a fourth utility, command mergezyg.

# Extreme value notes

Extreme values are useful to quantify the risk of catastrophic floods, and much more.

This is a brief set of notes with an introduction to extreme value theory. For reviews, see Leadbetter et al (1983) and David and Huser (2015) [references at the end]. Also of some (historical) interest might be the classical book by Gumbel (1958). Let $X_1, \dots, X_n$ be a sequence of independent and identically distributed variables with cumulative distribution function (cdf) $F(x)$ and let $M_n =\max(X_1,\dots,X_n)$ denote the maximum.

If $F(x)$ is known, the distribution of the maximum is:

$\begin{array}{lll} P(M_n \leqslant x) &=&P(X_1 \leqslant x, \dots, X_n \leqslant x) \\ &=& P(X_1 \leqslant x) \cdots P(X_n \leqslant x) = F^n(x). \end{array}$

The distribution function $F(x)$ might, however, not be known. If data are available, it can be estimated, although small errors on the estimation of $F(x)$ can lead to large errors concerning the extreme values. Instead, an asymptotic result is given by the extremal types theorem, also known as Fisher-Tippett-Gnedenko Theorem, First Theorem of Extreme Values, or extreme value trinity theorem (called under the last name by Picklands III, 1975).

But before that, let’s make a small variable change. Working with $M_n$ directly is problematic because as $n \rightarrow \infty$, $F^n(x) \rightarrow 0$. Redefining the problem as a function of $M_n^* = \frac{M_n-b_n}{a_n}$ renders treatment simpler. The theorem can be stated then as: If there exist sequences of constants $a_n \in \mathbb{R}_{+}$ and $b_n \in \mathbb{R}$ such that, as $n \rightarrow \infty$:

$P\left(M_{n}^{*} \leqslant x \right) \rightarrow G(x)$

then $G(x)$ belongs to one of three “domains of attraction”:

• Type I (Gumbel law): $\Lambda(x) = e^{-e^{-\frac{x-b}{a}}}$, for $x \in \mathbb{R}$ indicating that the distribution of $M_n$ has an exponential tail.
• Type II (Fréchet law): $\Phi(x) = \begin{cases} 0 & x\leqslant b \\ e^{-\left(\frac{x-b}{a}\right)^{-\alpha}} & x\; \textgreater\; b \end{cases}$ indicating that the distribution of $M_n$ has a heavy tail (including polynomial decay).
• Type III (Weibull law): $\Psi(x) = \begin{cases} e^{-\left( -\frac{x-b}{a}\right)^\alpha} & x\;\textless\; b \\ 1 & x\geqslant b \end{cases}$ indicating that the distribution of $M_n$ has a light tail with finite upper bound.

Note that in the above formulation, the Weibull is reversed so that the distribution has an upper bound, as opposed to a lower one as in the Weibull distribution. Also, the parameterisation is slightly different than the one usually adopted for the Weibull distribution.

These three families have parameters $a\; \textgreater\; 0$, $b$ and, for families II and III, $\alpha\; \textgreater\; 0$. To which of the three a particular $F(x)$ is attracted is determined by the behaviour of the tail of of the distribution for large $x$. Thus, we can infer about the asymptotic properties of the maximum while having only a limited knowledge of the properties of $F(x)$.

These three limiting cases are collectively termed extreme value distributions. Types II and III were identified by Fréchet (1927), whereas type I was found by Fisher and Tippett (1928). In his work, Fréchet used $M_n^* = \frac{M_n}{a_n}$, whereas Fisher and Tippett used $M_n^* = \frac{M_n-b_n}{a_n}$. Von Mises (1936) identified various sufficient conditions for convergence to each of these forms, and Gnedenko (1943) established a complete generalisation.

## Generalised extreme value distribution

As shown above, the rescaled maxima converge in distribution to one of three families. However, all are cases of a single family that can be represented as:

$G(x) = e^{-\left(1-\xi\left(\frac{x-\mu}{\sigma}\right)\right)^{\frac{1}{\xi}}}$

defined on the set $\left\{x:1-\xi\frac{x-\mu}{\sigma}\;\textgreater\;0\right\}$, with parameters $-\infty \;\textless \;\mu\;\textless\; \infty$ (location), $\sigma\;\textgreater\;0$ (scale), and $-\infty\;\textless\;\xi\;\textless\;\infty$ (shape). This is the generalised extreme value (GEV) family of distributions. If $\xi \rightarrow 0$, it converges to Gumbel (type I), whereas if $\xi < 0$ it corresponds to Fréchet (type II), and if $\xi\;\textgreater\;0$ it corresponds to Weibull (type III). Inference on $\xi$ allows choice of a particular family for a given problem.

## Generalised Pareto distribution

For $u\rightarrow\infty$, the limiting distribution of a random variable $Y=X-u$, conditional on $X \;\textgreater\; u$, is:

$H(y) = 1-\left(1-\frac{\xi y}{\tilde{\sigma}}\right)^{\frac{1}{\xi}}$

defined for $y \;\textgreater\; 0$ and $\left(1-\frac{\xi y}{\tilde{\sigma}}\right) \;\textgreater\; 0$. The two parameters are the $\xi$ (shape) and $\tilde{\sigma}$ (scale). The shape corresponds to the same parameter $\xi$ of the GEV, whereas the scale relates to the scale of the former as $\tilde{\sigma}=\sigma-\xi(u-\mu)$.

The above is sometimes called the Picklands-Baikema-de Haan theorem or the Second Theorem of Extreme Values, and it defines another family of distributions, known as generalised Pareto distribution (GPD). It generalises an exponential distribution with parameter $\frac{1}{\tilde{\sigma}}$ as $\xi \rightarrow 0$, an uniform distribution in the interval $\left[0, \tilde{\sigma}\right]$ when $\xi = 1$, and a Pareto distribution when $\xi \;\textgreater\; 0$.

## Parameter estimation

By restricting the attention to the most common case of $-\frac{1}{2}<\xi<\frac{1}{2}$, which represent distributions approximately exponential, parametters for the GPD can be estimated using at least three methods: maximum likelihood, moments, and probability-weighted moments. These are described in Hosking and Wallis (1987). For $\xi$ outside this interval, methods have been discussed elsewhere (Oliveira, 1984). The method of moments is probably the simplest, fastest and, according to Hosking and Wallis (1987) and Knijnenburg et al (2009), has good performance for the typical cases of $-\frac{1}{2}<\xi<\frac{1}{2}$.

For a set of extreme observations, let $\bar{x}$ and $s^2$ be respectively the sample mean and variance. The moment estimators of $\tilde{\sigma}$ and $\xi$ are $\hat{\tilde{\sigma}} = \frac{\bar{x}}{2}\left(\frac{\bar{x}^2}{s^2}+1\right)$ and $\hat{\xi}=\frac{1}{2}\left(\frac{\bar{x}^2}{s^2}-1\right)$.

The accuracy of these estimates can be tested with, e.g., the Anderson-Darling goodness-of-fit test (Anderson and Darling, 1952; Choulakian and Stephens, 2001), based on the fact that, if the modelling is accurate, the p-values for the distribution should be uniformly distributed.

## Availability

Statistics of extremes are used in PALM as a way to accelerate permutation tests. More details to follow soon.

## References

The figure at the top (flood) is in public domain.

# Non-Parametric Combination (NPC) for brain imaging

Have you ever had an analysis in which there was a large set of contrasts, all of interest, and you were worried about multiple testing? An eventual effect would be missed by a simple Bonferroni correction, but you did not know what else to do? Or did you have a set of different studies and you wished to obtain a style of meta-analytic result, indicating whether there would be evidence across all of them, without requiring the studies to be all consistently significant?

The Non-Parametric Combination (NPC) solves these issues. It is a way of performing joint inference on multiple data collected on the same experimental units (e.g., same subjects), all with minimal assumptions. The method was proposed originally by Pesarin (1990, 1992) [see references below], independently by Blair and Karninski (1993), and described extensively by Pesarin and Salmaso (2010). In this blog entry, the NPC is presented in brief, with emphasis on the modifications we introduce to render it feasible for brain imaging. The complete details are in our paper that has just been published in the journal Human Brain Mapping.

## NPC in a nutshell

The NPC consists of, in a first phase, testing each hypothesis separately using permutations that are performed synchronously across datasets; these tests are termed partial tests. The resulting statistics for each and every permutation are recorded, allowing an estimate of the complete empirical null distribution to be constructed for each one. In a second phase, the empirical p-values for each statistic are combined, for each permutation, into a joint statistic. As such a combined joint statistic is produced from the previous permutations, an estimate of its empirical distribution function is immediately known, and so is the p-value of the joint test. A flowchart of the original algorithm is shown below; click to see it side-by-side with the modified one (described below).

## A host of combining functions

The null hypothesis of the NPC is that null hypotheses for all partial tests are true, and the alternative hypothesis that any is false, which is the same null of a union-intersection test (UIT; Roy, 1953). The rejection region depends on how the combined statistic is produced. Various combining functions, which produce such combined statistics, can be considered, and some of the most well known are listed in the table below:

Method Statistic p-value
Tippett $\min \left(p_{k}\right)$ $1-\left(1-T\right)^{K}$
Fisher $-2 \sum_{k=1}^{K} \ln\left(p_{k}\right)$ $1-\chi^{2}\left(T;\;\nu=2K\right)$
Stouffer $\frac{1}{\sqrt{K}} \sum_{k=1}^{K} \Phi^{-1}\left(1-p_{k}\right)$ $1-\Phi\left(T;\;\mu=0,\;\sigma^2=1\right)$
Mudholkar–George $\frac{1}{\pi}\sqrt{\frac{3(5K+4)}{K(5K+2)}}\sum_{k=1}^{K} \ln\left(\frac{1-p_{k}}{p_{k}}\right)$ $1-t_{\text{cdf}}(T;\;\nu=5K+4)$

In the table, $K$ is the number of partial tests, and the remaining of the variables follow the usual notation (see the Table 1 in the paper for the complete description). Many of these combining functions were proposed over the years for applications such as meta-analyses, and many of them assume independence between the tests being combined, and will give incorrect p-values if such assumption is not met. In the NPC, lack of dependence is not a problem, even if these same functions are used: the synchronised permutations ensure that any dependence, if existing, is taken into account, and this is done so implicitly, with no need for explicit modelling.

The different combining functions lead to different rejection regions for the null hypothesis. For the four combining functions in the table above, the respective rejection regions are in the figure below.

The combining functions can be modified to allow combination of tests so as to favour hypotheses with concordant directions, or be modified for bi-directional tests. Click on the figure above for examples of these cases (again, see the paper for the complete details).

## Two problems, one solution

The multiple testing problem is well known in brain imaging: as an image comprises thousands of voxels/vertices/faces, correction is necessary. Bonferroni is in general too conservative, and various other approaches have been proposed, such as the random field theory. Permutation tests provide control over the familywise error rate (FWER) for the multiple tests across space, requiring only the assumption of exchangeability. This is all well known; see Nichols and Hayasaka (2003) and Winkler et al. (2014) for details.

However, another type of multiple testing is also common: analyses that test multiple hypotheses using the same model, multiple pairwise group comparisons, multiple and distinct models, studies using multiple modalities, that mix imaging and non-imaging data, that consider multiple processing pipelines, and even multiple multivariate analyses. All these common cases also need multiple testing correction. We call this multiple testing problem MTP-II, to discern it from the well known multiple testing problem across space, described above, which we term MTP-I.

One of the many combining functions possible with NPC, the one proposed by Tippett (1931), has a further property that makes it remarkably interesting. The Tippett function uses the smallest p-value across partial tests as its test statistic. Alternatively, if all statistics are comparable, it can be formulated in terms of the maximum statistic. It turns out that the distribution of the maximum statistic across a set of tests is also the distribution that can be used in a closed testing procedure (Marcus et al., 1976) to correct for the familywise error rate (FWER) using resampling methods, such as permutation. In the context of joint inference, FWER-correction can also be seen as an UIT. Thus, NPC offers a link between combination of multiple tests, and correction for multiple tests, in both cases regardless of any dependence between such tests.

This means that the MTP-II, for which correction in the parametric realm is either non-existing or fiendishly difficult, can be accommodated easily. It requires no explicit modelling of the dependence between the tests, and the resulting error rates are controlled exactly at the test level, adding rigour to what otherwise could lead to an excess of false positives without correction, or be overly conservative if a naïve correction such as Bonferroni were attempted.

## Modifying for imaging applications

As originally proposed, in practice NPC cannot be used in brain imaging. As the statistics for all partial tests for all permutations need to be recorded, an enormous amount of space for data storage is necessary. Even if storage space were not a problem, the discreteness of the p-values for the partial tests is problematic when correcting for multiple testing, because with thousands of tests in an image, ties are likely to occur, further causing ties among the combined statistics. If too many tests across an image share the same most extreme statistic, correction for the MTP-I, while still valid, becomes less powerful (Westfall and Young, 1993; Pantazis et al., 2005). The most obvious workaround — run an ever larger number of permutations to break the ties — may not be possible for small sample sizes, or when possible, requires correspondingly larger data storage.

The solution is loosely based on the direct combination of the test statistics, by converting the test statistics of the partial tests to values that behave as p-values, using the asymptotic distribution of the statistics for the partial tests. We call these as “u-values”, in order to emphasise that they are not meant to be read or interpreted as p-values, but rather as transitional values that allow combinations that otherwise would not be possible.

For spatial statistics, the asymptotic distribution of the combined statistic is used, this time to produce a z-score, which can be subjected to the computation of cluster extent, cluster mass, and/or threshold-free cluster enhancement (TFCE; Smith and Nichols, 2009). A flow chart of the modified algorithm is shown below. Click to see it side-by-side with the original.

## More power, fewer assumptions

One of the most remarkable features of NPC is that the synchronised permutations implicitly account for the dependence structure among the partial tests. This means that even combining methods originally derived under the assumption of independence can be used when such independence is untenable. As the p-values are assessed via permutations, distributional restrictions are likewise not necessary, liberating NPC from most assumptions that thwart parametric methods in general. This renders NPC a good alternative to classical multivariate tests, such as MANOVA, MANCOVA, and Hotelling’s T2 tests: each of the response variables can be seen as an univariate partial test in the context of the combination, but without the assumptions that are embodied in these old multivariate tests.

As if all the above were not already sufficient, NPC is also more powerful than such classical multivariate tests. This refers to its finite sample consistency property, that is, even with fixed sample size, as the number of modalities being combined increases, the power of the test also increases. The power of classical multivariate tests, however, increases up to a certain point, then begins to decrease, eventually reaching zero when the number of combining variables match the sample size.

The figure below summarises the analysis of a subset of the subjects of a published FMRI study (Brooks et al, 2005) in which painful stimulation was applied to the face, hand, and foot of 12 subjects. Using permutation tests separately, no results could be identified for any of the three types of stimulation. A simple multivariate test, the Hotelling’s T2 test, even assessed using permutations, did not reveal any effect of stimulation either. The NPC results, however, suggest involvement of large portions of the anterior insula and secondary somatosensory cortex. The Fisher, Stouffer and Mudholkar–George combining functions were particularly successful in recovering a small area of activity in the midbrain and periaqueductal gray area, which would be expected from previous studies on pain, but that could not be located from the original, non-combined data.

Detailed assessment of power, using variable number of modalities, and of modalities containing signal, is shown in the paper.

## Combinations or conjunctions?

Combination, as done via NPC, is different than conjunctions (Nichols et al., 2005) in the following: in the combination, one seeks for aggregate significance across partial tests, without the need that any individual study is necessarily significant. In the conjunction, it is necessary that all of them, with no exception, is significant. As indicated above, the NPC forms an union-intersection test (UIT; Roy, 1953), whereas the conjunctions form an intersection-union test (IUT; Berger, 1982). The former can be said to be significant if any (or an aggregate) of the partial tests is significant, whereas the latter is significant if all the partial tests are.

## Availability

The NPC, with the modifications for brain imaging, is available in the tool PALM — Permutation Analysis of Linear Models. It runs in either Matlab or Octave, and is free (GPL).

## References

Contributed to this post: Tom Nichols.

# Permutation tests in the Human Connectome Project

Permutation tests are known to be superior to parametric tests: they are based on only few assumptions, essentially that the data are exchangeable, and allow the correction for the multiplicity of tests and the use of various non-standard statistics. The exchangeability assumption allows data to be permuted whenever their joint distribution remains unaltered. Usually this means that each observation needs to be independent from the others.

In many studies, however, there are repeated measurements on the same subjects, which violates exchangeability: clearly, the various measurements obtained from a given subject are not independent from each other. In the Human Connectome Project (HCP) (Van Essen et al, 2012; 2013; see references at the end), subjects are sampled along with their siblings (most of them are twins), such that independence cannot be guaranteed either.

In Winkler et al. (2014), certain structured types of non-independence in brain imaging were addressed through the definition of exchangeability blocks (EBs). Observations within EB can be shuffled freely or, alternatively, the EBs themselves can be shuffled as a whole. This allows various designs that otherwise could not be assessed through permutations.

The same idea can be generalised for blocks that are nested within other blocks, in a multi-level fashion. In the paper Multi-level Block Permutation (Winkler et al., 2015) we presented a method that allows blocks to be shuffled a whole, and inside them, sub-blocks are further allowed to be shuffled, in a recursive process. The method is flexible enough to accommodate permutations, sign-flippings (sometimes also called “wild bootstrap”), and permutations together with sign-flippings.

In particular, this permutation scheme allows the data of the HCP to be analysed via permutations: subjects are allowed to be shuffled with their siblings while keeping the joint distribution intra-sibship maintained. Then each sibship is allowed to be shuffled with others of the same type.

In the paper we show that the error type I is controlled at the nominal level, and the power is just marginally smaller than that would be obtained by permuting freely if free permutation were allowed. The more complex the block structure, the larger the reductions in power, although with large sample sizes, the difference is barely noticeable.

Importantly, simply ignoring family structure in designs as this causes the error rates not to be controlled, with excess false positives, and invalid results. We show in the paper examples of false positives that can arise, even after correction for multiple testing, when testing associations between cortical thickness, cortical area, and measures of body size as height, weight, and body-mass index, all of them highly heritable. Such false positives can be avoided with permutation tests that respect the family structure.

The figure at the top shows how the subjects of the HCP (terminal dots, shown in white colour) can be shuffled or not, while respecting the family structure. Blue dots indicate branches that can be permuted, whereas red dots indicate branches that cannot (see the main paper for details). This diagram includes 232 subjects of an early public release of HCP data. The tree on the left considers dizygotic twins as a category on their own, i.e., that cannot be shuffled with ordinary siblings, whereas the tree on the right considers dizygotic twins as ordinary siblings.

The first applied study using our strategy has just appeared. The method is implemented in the freely available package PALM — Permutation Analysis of Linear Models, and a set of practical steps to use it with actual HCP data is available here.

# Another look into Pillai’s trace

In a previous post, all commonly used univariate and multivariate test statistics used with the general linear model (GLM) were presented. Here an alternative formulation for one of these statistics, the Pillai’s trace (Pillai, 1955, references at the end), commonly used in MANOVA and MANCOVA tests, is presented.

We begin with a multivariate general linear model expressed as:

$\mathbf{Y} = \mathbf{M} \boldsymbol{\psi} + \boldsymbol{\epsilon}$

where $\mathbf{Y}$ is the $N \times K$ full rank matrix of observed data, with $N$ observations of $K$ distinct (possibly non-independent) variables, $\mathbf{M}$ is the full-rank $N \times R$ design matrix that includes explanatory variables (i.e., effects of interest and possibly nuisance effects), $\boldsymbol{\psi}$ is the $R \times K$ vector of $R$ regression coefficients, and $\boldsymbol{\epsilon}$ is the $N \times K$ vector of random errors. Estimates for the regression coefficients can be computed as $\boldsymbol{\hat{\psi}} = \mathbf{M}^{+}\mathbf{Y}$, where the superscript ($^{+}$) denotes a pseudo-inverse.

## The null hypothesis, and a simplification

One is generally interested in testing the null hypothesis that a contrast of regression coefficients is equal to zero, i.e., $\mathcal{H}_{0} : \mathbf{C}'\boldsymbol{\psi}\mathbf{D} = \boldsymbol{0}$, where $\mathbf{C}$ is a $R \times S$ full-rank matrix of $S$ contrasts of coefficients on the regressors encoded in $\mathbf{X}$, $1 \leqslant S \leqslant R$ and $\mathbf{D}$ is a $K \times Q$ full-rank matrix of $Q$ contrasts of coefficients on the dependent, response variables in $\mathbf{Y}$, $1 \leqslant Q \leqslant K$; if $K=1$ or $Q=1$, the model is univariate. Once the hypothesis has been established, $\mathbf{Y}$ can be equivalently redefined as $\mathbf{Y}\mathbf{D}$, such that the contrast $\mathbf{D}$ can be omitted for simplicity, and the null hypothesis stated as $\mathcal{H}_{0} : \mathbf{C}'\boldsymbol{\psi} = \boldsymbol{0}$.

## Model partitioning

It is useful to consider a transformation of the model into a partitioned one:

$\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\boldsymbol{\gamma} + \boldsymbol{\epsilon}$

where $\mathbf{X}$ is the matrix with regressors of interest, $\mathbf{Z}$ is the matrix with nuisance regressors, and $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$ are respectively the vectors of regression coefficients. From this model we can also define the projection (hat) matrices $\mathbf{H}_{\mathbf{X}}=\mathbf{X}\mathbf{X}^{+}$ and $\mathbf{H}_{\mathbf{Z}}=\mathbf{Z}\mathbf{Z}^{+}$ due to tue regressors of interest and nuisance, respectively, and the residual-forming matrices $\mathbf{R}_{\mathbf{X}}=\mathbf{I}-\mathbf{H}_{\mathbf{X}}$ and $\mathbf{R}_{\mathbf{Z}}=\mathbf{I}-\mathbf{H}_{\mathbf{Z}}$.

Such partitioning is not unique, and schemes can be as simple as separating apart the columns of $\mathbf{M}$ as $\left[ \mathbf{X} \; \mathbf{Z} \right]$, with $\boldsymbol{\psi} = \left[ \boldsymbol{\beta}' \; \boldsymbol{\gamma}' \right]'$. More involved strategies can, however, be devised to obtain some practical benefits. One such partitioning is to define $\mathbf{X} = \mathbf{M} \mathbf{Q} \mathbf{C} \left(\mathbf{C}'\mathbf{Q}\mathbf{C}\right)^{-1}$ and
$\mathbf{Z} = \mathbf{M} \mathbf{Q} \mathbf{C}_v \left(\mathbf{C}_v'\mathbf{Q}\mathbf{C}_v\right)^{-1}$, where $\mathbf{Q}=(\mathbf{M}'\mathbf{M})^{-1}$, $\mathbf{C}_v=\mathbf{C}_u-\mathbf{C}(\mathbf{C}'\mathbf{Q}\mathbf{C})^{-1}\mathbf{C}'\mathbf{Q}\mathbf{C}_u$, and $\mathbf{C}_u$ has $r-\mathsf{rank}\left(\mathbf{C}\right)$ columns that span the null space of $\mathbf{C}$, such that $[\mathbf{C} \; \mathbf{C}_u]$ is a $r \times r$ invertible, full-rank matrix (Smith et al, 2007). This partitioning has a number of features: $\boldsymbol{\hat{\beta}} = \mathbf{C}'\boldsymbol{\hat{\psi}}$, $\widehat{\mathsf{Cov}}(\boldsymbol{\hat{\beta}}) = \mathbf{C}'\widehat{\mathsf{Cov}}(\boldsymbol{\hat{\psi}})\mathbf{C}$, i.e., estimates and variances of $\boldsymbol{\beta}$ for inference on the partitioned model correspond exactly to the same inference on the original model, $\mathbf{X}$ is orthogonal to $\mathbf{Z}$, and $\mathsf{span}(\mathbf{X}) \cup \mathsf{span}(\mathbf{Z}) = \mathsf{span}(\mathbf{M})$, i.e., the partitioned model spans the same space as the original.

Another partitioning scheme, derived by Ridgway (2009), defines $\mathbf{X}=\mathbf{M}(\mathbf{C}^+)'$ and $\mathbf{Z}=\mathbf{M}-\mathbf{M}\mathbf{C}\mathbf{C}^{+}$. As with the previous strategy, the parameters of interest in the partitioned model are equal to the contrast of the original parameters. A full column rank nuisance partition can be obtained from the singular value decomposition (SVD) of $\mathbf{Z}$, which will also provide orthonormal columns for the nuisance partition. Orthogonality between regressors of interest and nuisance can be obtained by redefining the regressors of interest as $\mathbf{R}_{\mathbf{Z}}\mathbf{X}$.

## The usual multivariate statistics

For the multivariate statistics, define generically:

$\mathbf{H}=(\mathbf{C}'\boldsymbol{\hat{\psi}}\mathbf{D})' \left(\mathbf{C}'(\mathbf{M}'\mathbf{M})^{-1}\mathbf{C} \right)^{-1} (\mathbf{C}'\boldsymbol{\hat{\psi}}\mathbf{D})$

as the sums of products explained by the model (hypothesis) and:

$\mathbf{E} = (\boldsymbol{\hat{\epsilon}}\mathbf{D})'(\boldsymbol{\hat{\epsilon}}\mathbf{D})$

as the sums of the products of the residuals, i.e., that remain unexplained. With the simplification to the original model that redefined $\mathbf{Y}$ as $\mathbf{Y}\mathbf{D}$, the $\mathbf{D}$ can be dropped, so that we have $\mathbf{H}=(\mathbf{C}'\boldsymbol{\hat{\psi}})' \left(\mathbf{C}'(\mathbf{M}'\mathbf{M})^{-1}\mathbf{C} \right)^{-1} (\mathbf{C}'\boldsymbol{\hat{\psi}})$ and $\mathbf{E} = \boldsymbol{\hat{\epsilon}}'\boldsymbol{\hat{\epsilon}}$. The various well-known multivariate statistics (see this earlier blog entry) can be written as a function of $\mathbf{H}$ and $\mathbf{E}$. Pillai’s trace is:

$T=\mathsf{trace}\left(\mathbf{H}(\mathbf{H}+\mathbf{E})^{-1}\right)$

## More simplifications

With the partitioning, other simplifications are possible:

$\mathbf{H}=\boldsymbol{\hat{\beta}}' (\mathbf{X}'\mathbf{X})\boldsymbol{\hat{\beta}} = ( \mathbf{X}\boldsymbol{\hat{\beta}})'(\mathbf{X}\boldsymbol{\hat{\beta}})$

Recalling that $\mathbf{X}'\mathbf{Z}=\mathbf{0}$, and defining $\tilde{\mathbf{Y}}=\mathbf{R}_{\mathbf{Z}}\mathbf{Y}$, we have:

$\mathbf{H} = (\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}})'(\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}) = \tilde{\mathbf{Y}}'\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}$

The unexplained sums of products can be written in a similar manner:

$\mathbf{E} = (\mathbf{R}_{\mathbf{X}}\tilde{\mathbf{Y}})'(\mathbf{R}_{\mathbf{X}}\tilde{\mathbf{Y}}) = \tilde{\mathbf{Y}}'\mathbf{R}_{\mathbf{X}}\tilde{\mathbf{Y}}$

The term $\mathbf{H}+\mathbf{E}$ in the Pillai’s trace can therefore be rewritten as:

$\mathbf{H}+\mathbf{E}= \tilde{\mathbf{Y}}'(\mathbf{H}_{\mathbf{X}} + \mathbf{R}_{\mathbf{X}})\tilde{\mathbf{Y}} = \tilde{\mathbf{Y}}'\tilde{\mathbf{Y}}$

Using the property that the trace of a product is invariant to a circular permutation of the factors, Pillai’s statistic can then be written as:

$\begin{array}{ccl} T&=&\mathsf{trace}\left(\tilde{\mathbf{Y}}'\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}\left(\tilde{\mathbf{Y}}'\tilde{\mathbf{Y}}\right)^{+}\right)\\ {}&=&\mathsf{trace}\left(\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}\left(\tilde{\mathbf{Y}}'\tilde{\mathbf{Y}}\right)^{+}\tilde{\mathbf{Y}}'\right)\\ {}&=&\mathsf{trace}\left(\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}\tilde{\mathbf{Y}}^{+}\right)\\ \end{array}$

## The final, alternative form

Using sigular value decomposition we have $\tilde{\mathbf{Y}} = \mathbf{U}\mathbf{S}\mathbf{V}'$ and $\tilde{\mathbf{Y}}^{+} = \mathbf{V}\mathbf{S}^{+}\mathbf{U}'$, where $\mathbf{U}$ contains only the columns that correspond to non-zero eigenvalues. Thus, the above can be rewritten as:

$\begin{array}{ccl} T&=&\mathsf{trace}\left(\mathbf{H}_{\mathbf{X}} \mathbf{U}\mathbf{S}\mathbf{V}' \mathbf{V}\mathbf{S}^{+}\mathbf{U}' \right)\\ {}&=&\mathsf{trace}\left(\mathbf{H}_{\mathbf{X}} \mathbf{U}\mathbf{U}' \right)\\ \end{array}$

The SVD transformation is useful for languages or libraries that offer a fast implementation. Otherwise, using a pseudoinverse yields the same result, perhaps only slightly slower. In this case, $T=\mathsf{trace}\left(\mathbf{H}_{\mathbf{X}}\tilde{\mathbf{Y}}\tilde{\mathbf{Y}}^{+}\right)$.

## Importance

If we define $\mathbf{A}\equiv\mathbf{H}_{\mathbf{X}}$ and $\mathbf{W}\equiv\mathbf{U}\mathbf{U}'$ (or $\mathbf{W}\equiv\tilde{\mathbf{Y}}\tilde{\mathbf{Y}}^{+}$), then $T=\mathsf{trace}\left(\mathbf{A}\mathbf{W}\right)$. The first three moments of the permutation distribution of statistics that can be written in this form can be computed analytically once $\mathbf{A}$ and $\mathbf{W}$ are known. With the first three moments, a gamma distribution (Pearson type III) can be fit, thus allowing p-values to be computed without resorting to the usual beta approximation to Pillai’s trace, nor using permutations, yet with results that are not based on the assumption of normality (Mardia, 1971; Kazi-Aoual, 1995; Minas and Montana, 2014).

## Availability

This simplification is available in PALM, for use with imaging and non-imaging data, using Pillai’s trace itself, or a modification that allows inference on univariate statistics. As of today, this option is not yet documented, but should become openly available soon.

## References

Update: 20.Jan.2016: A slight simplification was applied to the formulas above so as to make them more elegant and remove some redundancy. The result is the same.

# The lady tasting tea experiment

Can you tell?

The now famous story is that in an otherwise unremarkable summer afternoon in Cambridge in the 1920’s, a group of friends eventually discussed about the claims made by one of the presents about her abilities on discriminating whether milk was poured first or last when preparing a cup of tea with milk. One of the presents was Ronald Fisher, and the story, along with a detailed description of how to conduct a simple experiment to test the claimed ability, and how to obtain an exact solution, was presented at length in the Chapter 2 of his book The Design of Experiments, a few lines of which are quoted below:

A lady declares that by tasting a cup of tea made with milk she can discriminate whether the milk or the tea infusion was first added to the cup. We will consider the problem of designing an experiment by means of which this assertion can be tested. […] [It] consists in mixing eight cups of tea, four in one way and four in the other, and presenting them to the subject for judgment in a random order. The subject has been told in advance of that the test will consist, namely, that she will be asked to taste eight cups, that these shall be four of each kind […]. — Fisher, 1935.

There are $\frac{8!}{4!4!}=70$ distinct possible orderings of these cups, and by telling the subject in advance that there are four cups of each type, this guarantees that the answer will include four of each.

The lady in question eventually answered correctly six out of the eight trials. The results can be assembled in a 2 by 2 contingency table:

True order: Total (margin)
Tea first Milk first
Lady’s Guesses: Tea first $a=3$ $b=1$ $a+b=4$
Milk first $c=1$ $d=3$ $c+d=4$
Total (margin) $a+c=4$ $b+d=4$ $n=8$

With these results, what should be concluded about the ability of the lady in discriminating whether milk or tea was poured first? It is not possible to prove that she would never be wrong, because if a sufficiently large number of cups of tea were offered, a single failure would disprove such hypothesis. However, a test that she is never right can be disproven, with a certain margin of uncertainty, given the number of cups offered.

## Solution using Fisher’s exact method

Fisher presented an exact solution for this experiment. It is exact in the sense that it allows an exact probability to be assigned to each of the possible outcomes. The probability can be calculated as:

$P_{\text{Fisher}} = \dfrac{(a+b)!(c+d)!(a+c)!(b+d)!}{n!a!b!c!d!}$

For the particular configuration of the contingency table above, the probability is $\frac{16}{70} = 0.22857$. This is not the final result, though: what matters to disprove the hypothesis that she is not able to discriminate is how likely it would be for her to find a result at least as extreme as the one observed. In this case, there is one case that is more extreme, which would be the one in which she would have made correct guesses for all the 8 cups, in which case the values in the contingency table above would have been $a = 4$, $b = 1$, $c = 1$, and $d = 4$, with a probability computed with the same formula as $\frac{1}{70} = 0.01429$. Adding these two probabilities together yield $\frac{16+1}{70}=0.24286$.

Thus, if the lady is not able to discriminate whether tea or milk was poured first, the chance of observing a result at least as favourable towards her claim would be 0.24286, i.e., about 24%.

If from the outset we were willing to consider a significance level 0.05 (5%) as an informal rule to disprove the null hypothesis, we would have considered the p-value = 0.24286 as non-significant. This p-value is exact, a point that will become more clear below, in the section about permutation tests.

## Using the hypergeometric distribution directly

The above process can become laborious for experiments with larger number of trials. An alternative, but equivalent solution, is to appeal directly to the hypergeometric distribution. The probability mass function (pmf) of this distribution can be written as a function the parameters of the contingency table as:

$P(X=a) = \dfrac{\binom{a+b}{a}\binom{c+d}{c}}{\binom{n}{a+c}}$

The pmf is equivalent to Fisher’s exact formula to compute the probability of a particular configuration. The cumulative density function, which is conditional on the margins being fixed, is:

$P(X \geqslant a) = \sum_{j=a}^{J}\dfrac{\binom{a+b}{j}\binom{c+d}{a+c-j}}{\binom{n}{a+c}} = \sum_{j=a}^{J}\dfrac{(a+b)!(c+d)!(a+c)!(b+d)!}{n!j!(a+b-j)!(a+c-j)!(d-a+j)!}$

where $J=\min(a+c,a+b)$. Computing from the above (details omitted), yield the same value as using Fisher’s presentation, that is, the p-value is (exactly) 0.24286.

## Solution using Pearson’s $\chi^2$ method

Much earlier than the tea situation described above, Karl Pearson had already considered the problem of inference in contingency tables, having proposed a test based on a $\chi^2$ statistic.

$\chi^2 = \sum_{i=1}^{R}\sum_{j=1}^{C}\dfrac{(O_{ij}-E_{ij})^2}{E_{ij}}$

where $O_{ij}$ is the observed value for the element in the position $(i,j)$ in the table, $R$ and $C$ are respectively the number of rows and columns, and $E_{ij}$ is the expected value for these elements if the null hypothesis is true. The values $E_{ij}$ can be computed as the product of the marginals for row $i$ and column $j$, divided by the overall number of observations $n$. A simpler, equivalent formula is given by:

$\chi^2 = \dfrac{n(ad - bc)^2}{(a+b)(c+d)(a+c)(b+d)}$

A p-value can be computed from the $\chi^2$ distribution with degrees of freedom $\nu=(R-1)(C-1)$.

Under the null, we can expect a value equal to 2 in each of the 2 cells, that is, the lady would for each cup have a 50:50 chance of answering correctly. For the original tea tasting experiment, Pearson’s method give quite inaccurate results: $\chi^2=2$, which corresponds to a p-value of 0.07865. However, it is well known that this method is inaccurate if cells in the table have too small quantities, usually below 5 or 6.

## Improvement using Yates’ continuity correction

To solve the above well-known issue with small quantities, Yates (1934) proposed a correction, such that the test statistic becomes:

$\chi^2 = \sum_{i,j}\dfrac{\left(|O_{ij}-E_{ij}|-\frac{1}{2}\right)^2}{E_{ij}}$

Applying this correction to the original tea experiment gives $\chi^2=0.5$, and a p-value of 0.23975, which is very similar to the one given by the Fisher method. Note again that this approach, like the $\chi^2$ test, predates Fisher’s exact test.

## Equivalence of Fisher’s exact test and permutation tests

The method proposed by Fisher corresponds to a permutation test. Let $\mathbf{x}$ be a column vector containing binary indicators for whether milk was truly poured first. Let $\mathbf{y}$ be a column vector containing binary indicators for whether the lady answered that milk was poured first. The general linear model (GLM) can be used as usual, such that $\mathbf{y}=\mathbf{x}\beta + \boldsymbol{\epsilon}$, where $\beta$ is a regression coefficient, and $\boldsymbol{\epsilon}$ are the residuals.

Under the null hypothesis that the lady cannot discriminate, the binary values in $\mathbf{y}$ can be permuted randomly. There are $\binom{8}{4}=70$ possible unique rearrangements. Out of these, in 17, there are 6 or more (out of 8) correct answers matching the values in $\mathbf{x}$, which gives a p-value 17/70 = 0.24286.

Note that the strategy using the GLM can be used even if both variables $\mathbf{x}$ and $\mathbf{y}$ are binary, as in the example of the tea tasting, even if the residuals are not normally distributed (permutation tests do not depend on distributional assumptions), and even considering that values in $\beta$ can lead to non-sensical predictions in $\mathbf{y}$, as prediction is not the objective of this analysis.

## Why not a binomial test?

The binomial test could be considered if the lady did not know in advance that there were 4 cups of each mixture order. Since she knew (so the two margins of the table are fixed), each cup was not independent from each other, and her possible answers had to be constrained by answers previously given. The binomial test assumes independence, thus, is not an option for this analysis.

## Relevance

Using this simple experiment, Fisher established most of the fundamental principles for hypothesis testing, which contributed to major advances across biological and physical sciences. A careful read of the original text shows a precise use of terms, in a concise and unambiguous presentation, in contrast with many later and more verbose textbooks that eventually hid from readers most of the fundamental principles.

## References

The photograph at the top (tea with milk) is in public domain.

# Variance components in genetic analyses

Pedigree-based analyses allow investigation of genetic and environmental influences on anatomy, physiology, and behaviour.

Methods based on components of variance have been used extensively to assess genetic influences and identify loci associated with various traits quantifying aspects of anatomy, physiology, and behaviour, in both normal and pathological conditions. In an earlier post, indices of genetic resemblance between relatives were presented, and in the last post, the kinship matrix was defined. In this post, these topics are used to present a basic model that allows partitioning of the phenotypic variance into sources of variation that can be ascribed to genetic, environmental, and other factors.

## A simple model

Consider the model:

$\mathbf{Y} = \mathbf{X}\mathbf{B} + \boldsymbol{\Upsilon}$

where, for $S$ subjects, $T$ traits, $P$ covariates and $K$ variance components, $\mathbf{Y}_{S \times T}$ are the observed trait values for each subject, $\mathbf{X}_{S \times P}$ is a matrix of covariates, $\mathbf{B}_{P \times T}$ is a matrix of unknown covariates’ weights, and $\boldsymbol{\Upsilon}_{S \times T}$ are the residuals after the covariates have been taken into account.

The elements of each column $t$ of $\boldsymbol{\Upsilon}$ are assumed to follow a multivariate normal distribution $\mathcal{N}\left(0;\mathbf{S}\right)$, where $\mathbf{S}$ is the between-subject covariance matrix. The elements of each row $s$ of $\boldsymbol{\Upsilon}$ are assumed to follow a normal distribution $\mathcal{N}\left(0;\mathbf{R}\right)$, where $\mathbf{R}$ is the between-trait covariance matrix. Both $\mathbf{R}$ and $\mathbf{S}$ are seen as the sum of $K$ variance components, i.e. $\mathbf{R} = \sum_{k} \mathbf{R}_{k}$ and $\mathbf{S} = \sum_{k} \mathbf{S}_{k}$. For a discussion on these equalities, see Eisenhart (1947) [see references at the end].

## An equivalent model

The same model can be written in an alternative way. Let $\mathbf{y}_{S \cdot T \times 1}$ be the stacked vector of traits, $\mathbf{\tilde{X}}_{S \cdot T \times P \cdot T} = \mathbf{X} \otimes \mathbf{I}_{T \times T}$ is the matrix of covariates, $\boldsymbol{\beta}_{P \cdot T \times 1}$ the vector with the covariates’ weights, $\boldsymbol{\upsilon}_{S \cdot T \times 1}$ the residuals after the covariates have been taken into account, and $\otimes$ represent the Kronecker product. The model can then be written as:

$\mathbf{y} = \mathbf{\tilde{X}}\boldsymbol{\beta} + \boldsymbol{\upsilon}$

The stacked residuals $\boldsymbol{\upsilon}$ is assumed to follow a multivariate normal distribution $\mathcal{N}\left(0;\boldsymbol{\Omega}\right)$, where $\boldsymbol{\Omega}$ can be seen as the sum of $K$ variance components:

$\boldsymbol{\Omega} = \sum_{k} \mathbf{R}_k \otimes \mathbf{S}_k$

The $\boldsymbol{\Omega}$ here is the same as in Almasy and Blangero (1998). $\mathbf{S}_k$ can be modelled as correlation matrices. The associated scalars are absorbed into the (to be estimated) $\mathbf{R}_k$. $\mathbf{R}$ is the phenotypic covariance matrix between the residuals of the traits:

$\mathbf{R} = \left[ \begin{array}{ccc} \mathsf{Var}(\upsilon_1) & \cdots & \mathsf{Cov}(\upsilon_1,\upsilon_T) \\ \vdots & \ddots & \vdots \\ \mathsf{Cov}(\upsilon_T,\upsilon_1) & \cdots & \mathsf{Var}(\upsilon_T) \end{array}\right]$

whereas each $\mathbf{R}_k$ are the share of these covariances attributable to the $k$-th component:

$\mathbf{R}_k = \left[ \begin{array}{ccccc} \mathsf{Var}_k(\upsilon_1) & \cdots & \mathsf{Cov}_k(\upsilon_1,\upsilon_T) \\ \vdots & \ddots & \vdots \\ \mathsf{Cov}_k(\upsilon_T,\upsilon_1) & \cdots & \mathsf{Var}_k(\upsilon_T) \end{array}\right]$

$\mathbf{R}$ can be converted to a between-trait phenotypic correlation matrix $\mathbf{\mathring{R}}$ as:

$\mathbf{\mathring{R}} = \left[ \begin{array}{ccc} \frac{\mathsf{Var}(\upsilon_1)}{\mathsf{Var}(\upsilon_1)} & \cdots & \frac{\mathsf{Cov}(\upsilon_1,\upsilon_T)}{\left(\mathsf{Var}(\upsilon_1)\mathsf{Var}(\upsilon_T)\right)^{1/2}} \\ \vdots & \ddots & \vdots \\ \frac{\mathsf{Cov}(\upsilon_1,\upsilon_T)}{\left(\mathsf{Var}(\upsilon_1)\mathsf{Var}(\upsilon_T)\right)^{1/2}} & \cdots & \frac{\mathsf{Var}(\upsilon_T)}{\mathsf{Var}(\upsilon_T)} \end{array}\right]$

The above phenotypic correlation matrix has unit diagonal and can still be fractioned into their $K$ components:

$\mathbf{\mathring{R}}_k = \left[ \begin{array}{ccc} \frac{\mathsf{Var}_k(\upsilon_1)}{\mathsf{Var}(\upsilon_1)} & \cdots & \frac{\mathsf{Cov}_k(\upsilon_1,\upsilon_T)}{\left(\mathsf{Var}(\upsilon_1)\mathsf{Var}(\upsilon_T)\right)^{1/2}} \\ \vdots & \ddots & \vdots \\ \frac{\mathsf{Cov}_k(\upsilon_T,\upsilon_1)}{\left(\mathsf{Var}(\upsilon_T)\mathsf{Var}(\upsilon_1)\right)^{1/2}} & \cdots & \frac{\mathsf{Var}_k(\upsilon_T)}{\mathsf{Var}(\upsilon_T)} \end{array}\right]$

The relationship $\mathbf{\mathring{R}} = \sum_k \mathbf{\mathring{R}}_k$ holds. The diagonal elements of $\mathbf{\mathring{R}}_k$ may receive particular names, e.g., heritability, environmentability, dominance effects, shared enviromental effects, etc., depending on what is modelled in the corresponding $\mathbf{S}_k$. However, the off-diagonal elements of $\mathbf{\mathring{R}}_k$ are not the $\rho_k$ that correspond, e.g. to the genetic or environmental correlation. These off-diagonal elements are instead the signed $\text{\textsc{erv}}$ when $\mathbf{S}_k=2\cdot\boldsymbol{\Phi}$, or their $\text{\textsc{erv}}_k$-equivalent for other variance components (see below). In this particular case, they can also be called “bivariate heritabilities” (Falconer and MacKay, 1996). A matrix $\mathbf{\breve{R}}_k$ that contains these correlations $\rho_k$, which are the fraction of the variance attributable to the $k$-th component that is shared between pairs of traits is given by:

$\mathbf{\breve{R}}_k = \left[ \begin{array}{ccc} \frac{\mathsf{Var}_k(\upsilon_1)}{\mathsf{Var}_k(\upsilon_1)} & \cdots & \frac{\mathsf{Cov}_k(\upsilon_1,\upsilon_T)}{\left(\mathsf{Var}_k(\upsilon_1)\mathsf{Var}_k(\upsilon_T)\right)^{1/2}} \\ \vdots & \ddots & \vdots \\ \frac{\mathsf{Cov}_k(\upsilon_T,\upsilon_1)}{\left(\mathsf{Var}_k(\upsilon_T)\mathsf{Var}_k(\upsilon_1)\right)^{1/2}} & \cdots & \frac{\mathsf{Var}_k(\upsilon_T)}{\mathsf{Var}_k(\upsilon_T)} \end{array}\right]$

As for the phenotypic correlation matrix, each $\mathbf{\breve{R}}_k$ has unit diagonal.

## The most common case

A particular case is when $\mathbf{S}_1 = 2\cdot\boldsymbol{\Phi}$, the coefficient of familial relationship between subjects, and $\mathbf{S}_2 = \mathbf{I}_{S \times S}$. In this case, the $T$ diagonal elements of $\mathbf{\mathring{R}}_1$ represent the heritability ($h_t^2$) for each trait $t$. The diagonal of $\mathbf{\mathring{R}}_2$ contains $1-h_t^2$, the environmentability. The off-diagonal elements of $\mathbf{\mathring{R}}_1$ contain the signed $\text{\textsc{erv}}$ (see below). The genetic correlations, $\rho_g$ are the off-diagonal elements of $\mathbf{\breve{R}}_1$, whereas the off-diagonal elements of $\mathbf{\breve{R}}_2$ are $\rho_e$, the environmental correlations between traits. In this particular case, the components of $\mathbf{R}$, i.e., $\mathbf{R}_k$ are equivalent to $\mathbf{G}$ and $\mathbf{E}$ covariance matrices as in Almasy et al (1997).

## Relationship with the ERV

To see how the off-diagonal elements of $\mathbf{\mathring{R}}_k$ are the signed Endophenotypic Ranking Values for each of the $k$-th variance component, $\text{\textsc{erv}}_k$ (Glahn et al, 2011), note that for a pair of traits $i$ and $j$:

$\mathring{R}_{kij} = \frac{\mathsf{Cov}_k(\upsilon_i,\upsilon_j)}{\left(\mathsf{Var}(\upsilon_i)\mathsf{Var}(\upsilon_j)\right)^{1/2}}$

Multiplying both numerator and denominator by $\left(\mathsf{Var}_k(\upsilon_i)\mathsf{Var}_k(\upsilon_j)\right)^{1/2}$ and rearranging the terms gives:

$\mathring{R}_{kij} = \frac{\mathsf{Cov}_k(\upsilon_i,\upsilon_j)}{\left(\mathsf{Var}_k(\upsilon_i)\mathsf{Var}_k(\upsilon_j)\right)^{1/2}} \left(\frac{\mathsf{Var}_k(\upsilon_i)}{\mathsf{Var}(\upsilon_i)}\frac{\mathsf{Var}_k(\upsilon_j)}{\mathsf{Var}(\upsilon_j)}\right)^{1/2}$

When $\mathbf{S}_k = 2\cdot\boldsymbol{\Phi}$, the above reduces to $\mathring{R}_{kij} = \rho_k \sqrt{h^2_i h^2_j}$, which is the signed version of $\text{\textsc{erv}}=\left|\rho_g\sqrt{h_i^2h_j^2}\right|$ when $k$ is the genetic component.

## Positive-definiteness

$\mathbf{R}$ and $\mathbf{R}_k$ are covariance matrices and so, are positive-definite, whereas the correlation matrices $\mathbf{\mathring{R}}$, $\mathbf{\mathring{R}}_k$ and $\mathbf{\breve{R}}_k$ are positive-semidefinite. A hybrid matrix that does not have to be positive-definite or semidefinite is:

$\mathbf{\check{R}}_k = \mathbf{I} \odot \mathbf{\mathring{R}}_k + \left(\mathbf{J}-\mathbf{I}\right) \odot \mathbf{\breve{R}}_k$

where $\mathbf{J}$ is a matrix of ones, $\mathbf{I}$ is the identity, both of size $T \times T$, and $\odot$ is the Hadamard product. An example of such matrix of practical use is to show concisely the heritabilities for each trait in the diagonal and the genetic correlations in the off-diagonal.

## Cauchy-Schwarz

Algorithmic advantages can be obtained from the positive-definiteness of $\mathbf{\mathring{R}}_k$. The Cauchy–Schwarz theorem (Cauchy, 1821; Schwarz, 1888) states that:

$\mathring{R}_{kij} \leqslant \sqrt{\mathring{R}_{kii}\mathring{R}_{kjj}}$

Hence, the bounds for the off-diagonal elements can be known from the diagonal elements, which, by their turn, are estimated in a simpler, univariate model.

The Cauchy-Schwarz inequality imposes limits on the off-diagonal values of the matrix that contains the genetic covariances (or bivariate heritabilities).

## Parameter estimation

Under the multivariate normal assumption, the parameters can be estimated maximising the following loglikelihood function:

$\mathcal{L}\left(\mathbf{R}_k,\boldsymbol{\beta}\Big|\mathbf{y},\mathbf{\tilde{X}}\right) = -\frac{1}{2} \left(N \ln 2\pi + \ln \left|\boldsymbol{\Omega}\right| + \left(\mathbf{y}-\mathbf{\tilde{X}}\boldsymbol{\beta}\right)'\boldsymbol{\Omega}\left(\mathbf{y}-\mathbf{\tilde{X}}\boldsymbol{\beta}\right)\right)$

where $N=S \cdot T$ is the number of observations on the stacked vector $\mathbf{y}$. Unbiased estimates for $\boldsymbol{\beta}$, although inefficient and inappropriate for hypothesis testing, can be obtained with ordinary least squares (OLS).

## Parametric inference

Hypothesis testing can be performed with the likelihood ratio test (LRT), i.e., the test statistic is produced by subtracting from the loglikelihood of the model in which all the parameters are free to vary ($\mathcal{L}_1$), the loglikelihood of a model in which the parameters being tested are constrained to zero, the null model ($\mathcal{L}_0$). The statistic is given by $\lambda = 2\left(\mathcal{L}_1-\mathcal{L}_0\right)$ (Wilks, 1938), which here is asymptotically distributed as a 50:50 mixture of a $\chi^2_0$ and $\chi^2_{\text{df}}$ distributions, where df is the number of parameters being tested and free to vary in the unconstrained model (Self and Liang, 1987). From this distribution the p-values can be obtained.

## References

The photograph at the top (elephants) is by Anja Osenberg and was generously released into public domain.