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APTS CIS Preliminary Material

1 Introduction

The principal aim of these notes is to provide context and motivation for the APTS Computer Intensive Statistics Module, and to make the module as self-contained as is feasible. Statistics is a broad discipline and the APTS cohort naturally has a diverse range of backgrounds. If you have attended the earlier APTS modules this year, especially Statistical Computing and Statistical Inference, then you should be well prepared for this module.

Although it is likely that everyone attending the module will know everything they need to follow the lectures, there is one area which will require some preparation if it’s something with which you are not already comfortable. Indeed, there is one thing from which there is no escaping in computer intensive methods of any sort: implementation. It’s impossible to really understand the ideas which we’ll be discussing without experimenting with them yourself. As such, perhaps the most important prerequisite is competence with basic computer programming. In addition to the material described here, it is important that you are able to use the R (R Core Team 2013) programming language. If you haven’t already done so, then do please complete the R Programming Course for APTS Students before the start of the APTS week itself.

Two appendices are provided. You might very well already know everything in these appendices—if that’s the case, then great. If you have a less conventional statistical background and haven’t managed to attend the earlier APTS modules then you may find some parts of these notes less familiar in which case some references are provided, but rest assured that the module should be accessible to anyone who is pursuing a PhD in any aspect of statistics or applied probability (both interpreted broadly). Appendix A provides a compact summary of some statistical tasks which we will aim to address in this module. It is likely that anyone pursuing a PhD in statistics—especially anyone who has attended an APTS module on Statistical Inference—will be familiar with this material, and (re)reading the notes provided for that module would be good preparation for the present one, but it is also convenient to have a compact summary on hand. Appendix B summarises some basic notions of convergence of stochastic quantities. This is here only to ensure that everyone has had the opportunity to see these ideas before the week. If everything it contains is obvious to you then great; if it’s not then don’t panic, we will make limited use of these results and their friends to motivate and justify some of what we do in this module. But we won’t spend time proving them or focussing heavily on their technical consequences1. If you’re prepared to believe, in particular, that sample averages of collections of random variables might be expected to become close to the population average for large enough samples of sufficiently regular random variables then you can dispense with the detailed contents of Appendix B entirely. A few exercises are provided in each chapter for anyone who does want to revise these ideas or gain some exposure to them.

Before going any further, we need to establish what, exactly, computer intensive statistics actually is. Clearly, it is statistics and this must be borne in mind throughout: the computer is a tool which we are using to resolve statistical problems as well as we are able to. When we can dispense with complicated computational procedures without sacrificing accuracy or power then we should probably do so.

So what sorts of statistics are computer intensive? There are several situations in which we are likely to find ourself needing substantial computing power:

Big Data. A current buzzword, sometimes unhelpful, but it would be negligent not to mention in the current climate. Situations in which we have very large data sets necessarily involve substantial computational challenges: storing the data produced in fractions of a second at the large hadron collider at CERN is already a (computational) challenge; computing the sample moments of a set of \(10^{12}\) observations is not trivial. Doing anything interesting with such data is certainly computer intensive.

Big Models. If having a large and complicated data set is the characteristic of Big Data then the alternative position of having a large and complicated model can presumably be characterised as having big models. There are lots of subtly different situations which can be characterised as being difficult because of the complexity of the model. These include:

  • large, hierarchical Bayesian models with many levels of uncertainty and many latent variables;

  • latent variable models such as phylogenetic trees in which we have a parametric generative model for the process by which the tree was generated and observations of the leaves of the tree (the current generation of a population) but no measurements of the intermediate variables;

  • scientific models from which we can sample but which are so complicated that we can’t write down the likelihood—think of models in climate science, numerically solved differential equation models in physics and that sort of thing.

We will at least touch on all of these things at some point during the module. Roughly speaking, everything which we will discuss can be thought of as being computer intensive for one of two reasons: because we want to deal with so much data (in some sense) that doing anything with it is difficult or because what we want to do is intrinsically complicated.


These materials are closely based on those developed by previous module leaders, Dr Paul Jenkins (University of Warwick) and Prof. Adam Johansen (University of Warwick), in turn based on a course Prof. Johansen developed with Dr Ludger Evers (University of Glasgow).

2 Towards Computer Intensive Statistics

This chapter aims to introduce a few of the ideas which will be important in this module and to provide some pointers to directions which will be looked at during the week.

There are a great many books on simulation-based computational statistics; some good examples are those of Voss (2013) which provides a gentle introduction to simulation-based computational methods, Robert and Casella (2004) which takes a rigorous but approachable look at the area from a mathematical/statistical perspective and Liu (2001) which many with backgrounds in the natural sciences find to be very accessible.

2.1 Simulation and the Monte Carlo Method

Simulation will occupy most of our time in this module. The idea of drawing random variables from specified distributions and using the resulting ensemble of realisations to approximate quantities of interest is intoxicatingly powerful for a method which, at least in principle, is very simple.

Methods based around simulation are typically termed Monte Carlo2 methods, following Metropolis and Ulam (1949). One characterization of such methods was given by Halton (1970):

Representing the solution of a problem as a parameter of a hypothetical population, and using a random sequence of numbers to construct a sample of the population, from which statistical estimates of the parameter can be obtained.

In some ways this is doing statistics backwards: rather than taking a real sample (of data) and attempting to infer parameters of an underlying population using analytical techniques, we devise a representation of a quantity of interest as a parameter of a hypothetical population and then obtain, artificially, a sample from that population before using the properties of this sample as a proxy for those of the population itself.

Perhaps an example might make it clear how simple this approach really is. You may well have seen this example before, but try to see it as a prototype for a general approach to approximate computation.

Example 2.1 (Computing \(\pi\) in the rain). Suppose we want to obtain an approximation of \(\pi\) using a simple experiment. Assume that we are able to produce “uniform rain” on the square extending to \(\pm1\) in two orthogonal directions, \([-1,1]^2 = [-1,1]\times[-1,1] = \{(x,y): x \in [-1,1], y \in [-1,1]\}\), such that the probability of a raindrop falling into a region \(\mathcal{R}\subseteq [-1,1]^2\) is proportional to the area of \(\mathcal{R}\), but independent of the position of \(\mathcal{R}\). It is easy to see that this is the case iff the two coordinates \(X,Y\) are independent realisations of uniform distributions on the interval \([-1,1]\) (in short \(X,Y \overset{\text{iid}}{\sim}{\textsf{U}{\left[-1,+1\right]}}\)).

Illustration of the estimation pi using uniform rain.

Figure 2.1: Illustration of the estimation \(\pi\) using uniform rain.

Now consider the probability that a raindrop falls into the unit circle (see Figure 2.1). It is \[{\mathbb{P}}(\textrm{drop within circle})=\frac{\textrm{area of the unit circle}}{\textrm{area of the square}}=\frac{\underset{\{x^2+y^2\leq 1\}}{\int\!\int}1\;dxdy}{\underset{\{-1\leq x,y\leq 1\}}{\int\!\int}1\;dxdy}=\frac{\pi}{2\cdot 2}=\frac{\pi}{4}.\] In other words, \[\pi = 4 \cdot {\mathbb{P}}(\textrm{drop within circle}),\] i.e. there is an expression for the desired quantity \(\pi\) as a function of a probability.

Of course we cannot compute \({\mathbb{P}}(\textrm{drop within circle})\) without knowing \(\pi\). However, we can estimate the probability using our raindrop experiment. If we observe \(n\) raindrops, then the number of raindrops \(M\) that fall inside the circle is a binomial random variable, \(M \sim {\textsf{Bin}\left( n,p \right)}\) with \(p={\mathbb{P}}(\textrm{drop within circle})\).

Thus, if we observe that \(m\) raindrops fall within the circle, we approximate \(p\) using its maximum-likelihood estimate \(\hat p= {m}/{n},\) and we can estimate \(\pi\) by \(\hat \pi =4\hat p =4 \cdot \frac{m}{n}.\) Assume we have observed, as in Figure 2.1, that 77 of the 100 raindrops were inside the circle. In this case, our estimate of \(\pi\) is \(\hat \pi = {4\times 77}/{100}=3.08\), which is clearly some way from the truth.

However the strong law of large numbers (Theorem B.2) guarantees that the estimator \(\hat\pi = 4M/n\) converges almost surely to \(\pi\). Figure 2.2 shows the estimate obtained after \(n\) iterations as a function of \(n\) for \(n=1,\dots,2000\). You can see that the estimate improves as \(n\) increases.

Estimate of pi together with approximate confidence intervals resulting from the raindrop experiment. Notice the jagged evolution of the confidence interval estimate: this is something which must be borne in mind when using simulation, if we use uncertainty estimates based upon estimated values then the quality of the estimate will determine also the quality of the *uncertainty* estimate. We can dramatically underestimate our own uncertainty if we are not careful.

Figure 2.2: Estimate of \(\pi\) together with approximate confidence intervals resulting from the raindrop experiment. Notice the jagged evolution of the confidence interval estimate: this is something which must be borne in mind when using simulation, if we use uncertainty estimates based upon estimated values then the quality of the estimate will determine also the quality of the uncertainty estimate. We can dramatically underestimate our own uncertainty if we are not careful.

We can assess the quality of our estimate by computing a confidence interval for \(\pi\). As we have \(X\sim {\textsf{Bin}\left( 100,p \right)}\), we can obtain a 95% confidence interval for \(p\) using a Normal approximation: \[\left[ 0.77 - 1.96 \cdot \sqrt{\frac{0.77\cdot (1-0.77)}{100}}, \; 0.77 + 1.96 \cdot \sqrt{\frac{0.77\cdot (1-0.77)}{100}}\right ] = [ 0.6875, \; 0.8525 ],\] As our estimate of \(\pi\) is four times the estimate of \(p\), we now also have a confidence interval for \(\pi\) which is simply \([2.750, \; 3.410]\).

In more general terms, let \(\hat \pi_n=4\hat p_n\) denote the estimate after having observed \(n\) raindrops. A \((1-2\alpha)\) confidence interval for \(\pi\) is \[\left[\hat \pi_n - z_{1-\alpha} \sqrt{\frac{\hat\pi_n(4-\hat\pi_n)}{n}}, \hat \pi_n + z_{1-\alpha} \sqrt{\frac{\hat\pi_n(4-\hat\pi_n)}{n}}\right].\]

Recall the main steps of this process:

  • We have written the quantity of interest (in our case \(\pi\)) as an expectation (a probability is a special case of an expectation as \({\mathbb{P}}(A)={\mathbb{E}_{}\left[\mathbb{I}_A\right]}\) where \(\mathbb{I}_A(x)\) is the indicator function which takes value \(1\) if \(x\in A\) and 0 otherwise).

  • We have replaced this algebraic representation of the quantity of interest with a sample approximation. The strong law of large numbers guaranteed that the sample approximation converges to the algebraic representation, and thus to the quantity of interest. Furthermore, the central limit theorem (Theorem B.3) allows us to assess the speed of convergence.

Of course we’d never use a method like this one to estimate \(\pi\); there are much faster ways of getting good estimates. Indeed, the rate of convergence here illustrates that these methods can be really computationally intensive. One major advantage of such methods, as we shall see, is that the rate of convergence of Monte Carlo estimates of expectations is independent of the dimension of the space on which the integral is defined, unlike more traditional approaches to numerical integration, and it is for hard problems in which other methods fail to produce any meaningful solution that simulation-based strategies are most useful.

2.2 Bootstrap Methods

Bootstrap methods are based around the, at first rather fanciful, idea that if we sample \(n\) times with replacement from a simple random sample of size \(n\) then the relationship between our resampled set of \(n\) points and the empirical distribution of the original sample is the same as the relationship between the original sample and its true distribution. This can be justified, at least asymptotically under regularity conditions via the Glivenko–Cantelli theorem (Theorem B.4) and its relatives. Sampling with replacement from a sample is equivalent to sampling from the distribution with its empirical distribution function.

There are many applications of bootstrap techniques, and many variations on the basic idea. In broad terms, the approach can be most useful when we want to construct confidence intervals without being able to compute the distribution of the test statistic. They can allow us:

  • to mitigate certain types of bias; and

  • to obtain similar results to those yielded by higher order asymptotic theory without doing the analysis that would be required to obtain them.

The paper of Efron (1979) is a reasonable first reference; a number of books on the subject have been written, including Efron (1982). If you already have Wasserman (2004) then you may find the informal introduction to these techniques presented in Chapter 8 helpful.

Bootstrap methods—and their strengths and limitations—will be discussed in the lectures.

2.3 Markov chains and Monte Carlo

Markov chains are objects with which you will become familiar during the APTS week, if you are not already. Roughly speaking, a Markov chain is a stochastic process for which the distribution of its future states is independent of its past given its current value. A more formal coverage of Markov chains was provided by the Applied Stochastic Processes module which took place in the previous APTS week.

The ergodic hypothesis was originally the work of Boltzmann and was intended to provide a characterisation of the long term average behaviour of a thermodynamic system. It said, approximately, that the time averaged behaviour of the microscopic configuration of a system was the same as the instantaneous average over a hypothetical ensemble of systems prepared in a particular way. In modern terms we would think of that hypothetical ensemble as a way of describing a probability distribution and we could think of the ergodic hypothesis as telling us (assuming it to be true) that the long-time-average behaviour of the stochastic process describing the evolution of the system coincided with an expectation with respect to that probability distribution.

The previous paragraph appears to be hinting at something like a law of large numbers and, indeed, that’s what a modern ergodic theorem describes. From a simulation point of view this is a tremendously powerful concept and there is an enormous literature on Markov chain Monte Carlo methods in which the trajectories of Markov chains are simulated for a long period of time and averages over these trajectories are calculated as a proxy for the expectation with respect to a particular distribution. A major contributing factor to the popularity and pre-eminence of these methods amongst computational statistics is that there exist recipes for the construction of Markov chains whose ergodic averages coincide with expectations with respect to essentially any distribution of interest, particularly the Metropolis (Metropolis et al. 1953) algorithm and its variants such as the Metropolis–Hastings algorithm (Hastings 1970) and the reversible jump algorithm (Green 1995). For some well-behaved statistical problems, the so-called Gibbs sampler (Geman and Geman 1984) provides what may seem an even more automatic approach, but it does come at the cost of some additional human calculation.

During the course of the APTS week we’ll see how to construct Markov chains to answer particular questions of interest and touch on all of the algorithms mentioned above.

There are vast numbers of resources which provide information on Markov chain Monte Carlo methods. I find that Robert and Casella (2004) covers this particular sub-field of Monte Carlo very well, but many other resources exist. For many years Gilks, Richardson, and Spiegelhalter (1996) has been something of a canonical reference, but has perhaps aged a little since its publication; perhaps less venerable but undoubtedly more up to date is Brooks et al. (2011).

2.4 Big Data is a Big Problem

Here’s the description of a dataset from a piece of work in which one of us was involved (Chan, Jenkins, and Song 2012):

We applied our method to samples from two populations of D. melanogaster (fruit flies): Raleigh, USA (RAL) and Gikongoro, Rwanda (RG). The RAL dataset consisted of the genomes of 37 inbred lines sequenced at a coverage of \(\geq 10\times\) by the Drosophila Population Genomics Project. The RG dataset comprised 22 genomes from haploid embryos sequenced at a coverage of \(\geq 25\times\) by the Drosophila Population Genomics Project 2…we were able to designate the ancestral allele in 1,755,040 of 2,475,674 high quality (quality score \(Q \geq 30\)) SNPs (single nucleotide polymorphisms) in the RAL sample (70.9%), and 2,213,312 out of 3,134,295 high quality SNPs in the RG sample (70.6%).

Each genome is summarised by its positions that differ from some other members of the sample (“SNPs”), so the data can be summarised by tables of size \(37 \times 2,475,674\) (RAL) and \(22 \times 3,134,295\) (RG). By the standards of modern statistics, these aren’t particularly large data sets, yet managing, storing and performing inference for data sets on this scale required considerable thought and effort.

The need for tools which can deal efficiently with data sets of this magnitude (and, indeed, much larger ones) is one of the reasons that computer intensive statistics is important. It would not be feasible to compute the sample mean of a data set with a quarter of a million elements without making use of a computer; of course, performing meaningful inference typically requires much more sophisticated computations than that.

A question to think about: from your personal perspective: how big is a large data set and how big is an enormous data set?

2.5 Warm-Up Exercises

Exercise 2.1 (Preliminary Simulation). Familiarise yourself with the support provided by R for simple simulation tasks. In particular:

  1. Generate a large sample of \({\textsf{N}\left( 3,7 \right)}\) random variables (see rnorm); plot a histogram (hist with sensibly-chosen bins) of your sample and overlay a normal density upon it (see dnorm, lines).

  2. Use sample to simulate the rolling of 1,000 dice; use your sample to estimate the average value obtained when rolling a standard die. Show how the estimate obtained using \(n\) dice behaves for \(n \in [0, 1000]\) using a simple plot.

Exercise 2.2 (Towards Bootstrap Methods). Imagine that you have a sample of 1,000 values from a population of unknown distribution (for our purposes, you can obtain such a sample using rnorm as in the previous question and pretending that the distribution is unknown).

  1. Write code to repeat the following 1,000 times:

    1. Sample 1,000 times with replacement from the original sample to obtain 1,000 resampled sets of values.

    2. Compute the sample mean of your resampled set.

  2. You now have 1,000 sample means for resampled subsets of your data. Find the 5th and 95th percentile of this collection of resampled means.

  3. How does this compare with a standard 90% confidence interval for the mean of a sample of size 1,000 from your chosen distribution?

  4. Repeat the above using the median rather than mean.

  5. Why might this be a useful technique? Note that we haven’t done anything to justify the approach yet.

Exercise 2.3 (Transformation Methods). The Box–Muller method transforms a pair of uniformly-distributed random variables to obtain a pair of independent standard normal random variates. If \[U_1, U_2 \overset{\text{iid}}{\sim}{\textsf{U}{\left[0,1\right]}},\] and \[\begin{aligned} X_1=&\sqrt{-2\log(U_1)} \cdot \cos(2\pi U_2), \\ X_2=&\sqrt{-2\log(U_1)} \cdot \sin(2\pi U_2), \end{aligned}\] then \(X_1,X_2 \overset{\text{iid}}{\sim}{\textsf{N}\left( 0,1 \right)}\).

  1. Write a function which takes as arguments two vectors \((\mathbf{U}_1,\mathbf{U}_2)\) of uniformly distributed random variables, and returns the two vectors \((\mathbf{X}_1,\mathbf{X}_2)\) obtained by applying the Box–Muller transform elementwise.

  2. The R function runif provides access to a ‘pseudo-random number generator’ (PRNG), which we’ll discuss further during the module. Generate 10,000 \({\textsf{U}{\left[0,1\right]}}\) random variables using this function, and transform this vector to two vectors, each of 5,000 normal random variates.

  3. Check that the result from (b) is plausibly distributed as pairs of independent, standard normal random variables, by creating a scatter plot of your data.

Exercise 2.4 (Simulating Markov chains). Consider a simple board game in which players take it in turns to move around a circular board in which an annulus is divided into 40 segments. Players move by rolling two standard dice and moving their playing piece, clockwise around the annulus, the number of spaces indicated. For simplicity we’ll neglect the game’s other features.

  1. All players begin in a common space. Write R code to simulate a player’s position after three moves of the game and repeat this 1,000 or so times. Plot a histogram to show the distribution of player positions after three moves. Is this consistent with your expectations?

  2. Now modify your code to simulate the sequence of spaces occupied by a player over their first 10,000 moves. Plot a histogram to show the occupancy of each of the forty spaces during these 10,000 moves. Are there any interesting features?

  3. If a player’s score increases by 1 every time they land on the first space (the starting space), 2 every time they land in the space after that and so on up to 40 for landing in the space immediately before that space then approximately what would be the long-run average number of points per move (use your simulation to obtain an approximate answer).


A Inference and Estimators

Computer intensive statistics is still statistics, and it is important not to lose sight of that fact. It’s necessary to focus on the core ideas of computer intensive statistics in this module due to the limited time available, but it is important to remember why we want to address the problems we consider.

One of the major tasks of computer intensive statistics is to provide (approximate) estimates in situations in which the estimators of interest are analytically intractable. This chapter provides a short reminder of some of the estimators of widespread use in statistics that will feature in this module.

Most if not all of this material was covered in much greater depth in the Statistical Inference module.

A.1 Inference and Some Common Estimators

Computationally intensive methods are extremely prevalent in Bayesian statistics and people often make the mistake of thinking the two are synonymous. This is not the case. We will see in this module that computer intensive methods can be very useful in other areas of statistics, including likelihood-based inference. With this in mind, it is useful to recall a number of common estimation tasks we might want to carry out. It is very likely that you’ve come across all of these things before; the particular character of our interest is that we shall seek to find approximate solutions to these estimation problems in settings in which they are not analytically tractable (either because the computation is apparently not possible even in abstract terms or because carrying it out would take many times the age of the universe).

A.1.1 Maximum Likelihood Estimates

Given a generative model for our data, \({\boldsymbol{X}}\sim f(\cdot\mid {\boldsymbol{\theta}})\), the likelihood is the function \(L({\boldsymbol{\theta}};{\boldsymbol{x}})\) viewed as a function of the parameter vector, \({\boldsymbol{\theta}}\), with the observed data, \({\boldsymbol{x}}\), treated as fixed.

The maximum likelihood estimator is: \[\label{eq:mle} \hat{\boldsymbol{\theta}}_{\text{MLE}} := \mathop{\text{arg max}}_\theta L({\boldsymbol{\theta}};{\boldsymbol{x}}).\]

It is common to work with the logarithm of the likelihood, \(\ell({\boldsymbol{\theta}};{\boldsymbol{x}}) = \log L({\boldsymbol{\theta}};{\boldsymbol{x}})\) for numerical reasons. In particular, if \({\boldsymbol{x}}= (x_1,\dots,x_n)\) and \(X_i \overset{\text{iid}}{\sim}f(\cdot|{\boldsymbol{\theta}})\) then: \[\begin{aligned} L({\boldsymbol{\theta}};{\boldsymbol{x}}) =& \prod_{i=1}^n f(x_i\mid {\boldsymbol{\theta}}), & \ell({\boldsymbol{\theta}};{\boldsymbol{x}}) =& \sum_{i=1}^n \log f(x_i\mid {\boldsymbol{\theta}}),\end{aligned}\] and it is typically much easier to work with \(\ell\) than \(L\). By the strict monotonicity of the logarithm (and non-negativity of the likelihood) it is clear that: \[\mathop{\text{arg max}}_\theta L({\boldsymbol{\theta}};{\boldsymbol{x}}) = \mathop{\text{arg max}}_\theta \ell({\boldsymbol{\theta}};{\boldsymbol{x}}).\]

For complex models, obtaining this maximiser analytically can be impossible; we’ll see that there are (at least approximate) computational solutions to this problem during this module.

A.1.2 Confidence Intervals

A confidence interval is a random set which will contain the true value of the parameter with a specified probability with respect to replication of the sampling experiment.

If we are interested in a (real-valued) parameter \(\theta\) and a density \(f({\boldsymbol{x}};\theta)\) describing a data-generating process then we seek random variables \(L({\boldsymbol{X}})\) and \(U({\boldsymbol{X}})\) such that \({\mathbb{P}}(L({\boldsymbol{X}}) \leq \theta \leq U({\boldsymbol{X}})) = 1 - \alpha\) for some confidence level \(\alpha\). Note that \(\theta\) is not treated as random here, but as fixed and unknown; it is \(L({\boldsymbol{X}})\) and \(U({\boldsymbol{X}})\) that are random and the probability is with respect to their distribution under repeated realisations of the experiment which realises the random variable \({\boldsymbol{X}}\) describing the data.

A level \(\alpha\) confidence interval for \(\theta\), then, is a random interval \([L({\boldsymbol{X}}),U({\boldsymbol{X}})]\) which would contain the true parameter a proportion (approximately / exactly) \(\alpha\) of the time if we carried out the whole procedure (a very large number of / infinitely many) times.

A.1.3 Hypothesis Tests

Closely related to the notion of a confidence interval is the hypothesis test. In order to distinguish between two possible explanations for observed data, a default scenario \(H_0\) termed the null hypothesis and an alternative \(H_1\), this procedure seeks to reject \(H_0\) when the data is in an appropriate sense unlikely to have arisen if \(H_0\) is true and not to reject \(H_0\) otherwise.

More precisely, given a test statistic \(T({\boldsymbol{X}})\), we seek a set of values \(C_\alpha\) such that \[\begin{aligned} {\mathbb{P}}(T({\boldsymbol{X}}) \in C_\alpha\mid H_0 \textrm{ is true}) &= \alpha, & {\mathbb{P}}(T({\boldsymbol{X}}) \in C_\alpha\mid H_1 \textrm{ is true}) > \alpha.\end{aligned}\] Often \(C_\alpha\) is obtained as the complement of an interval of values which are likely under the null hypothesis (viewed relatively, contrasting with the plausibility of those values under the particular alternative hypothesis). See Figure A.1 for an illustration.