As with linear and logistic regressions, generalized linear models can be fit to multilevel structures by including coefficients for group indicators and then adding group-level models. We illustrate in this chapter with three examples from our recent applied research: an overdispersed Poisson model for police stops, a multinomial logistic model for storable voting, and an overdispersed Poisson model for social networks.

Overdispersed Poisson regression: police stops and ethnicity

We return to the New York City police example introduced in Sections 1.2 and 6.2, where we formulated the problem as an overdispersed Poisson regression, and here we generalize to a multilevel model. In order to compare ethnic groups while controlling for precinct-level variation, we perform multilevel analyses using the city's 75 precincts. Allowing precinct-level effects is consistent with theories of policing such as the “broken windows” model that emphasize local, neighborhood-level strategies. Because it is possible that the patterns are systematically different in neighborhoods with different ethnic compositions, we divide the precincts into three categories in terms of their black population: precincts that were less than 10% black, 10%–40% black, and more than 40% black. We also account for variation in stop rates between the precincts within each group. Each of the three categories represents roughly one-third of the precincts in the city, and we perform separate analyses for each set.

Overdispersion as a variance component

As discussed in Chapter 6, data that are fit by a generalized linear model are overdispersed if the data-level variance is higher than would be predicted by the model.

Whenever we represent inferences for a parameter using a point estimate and standard error, we are performing a data reduction. If the estimate is normally distributed, this summary discards no information because the normal distribution is completely defined by its mean and variance. But in other cases it can be useful to represent the uncertainty in the parameter estimation by a set of random simulations that represent possible values of the parameter vector (with more likely values being more likely to appear in the simulation). By simulation, then, we mean summarizing inferences by random numbers rather than by point estimates and standard errors.

Simulation of probability models

In this section we introduce simulation for two simple probability models. The rest of the chapter discusses how to use simulations to summarize and understand regressions and generalize linear models, and the next chapter applies simulation to model checking and validation. Simulation is important in itself and also prepares for multilevel models, which we fit using simulation-based inference, as described in Part 2B.

A simple example of discrete predictive simulation

How many girls in 400 births? The probability that a baby is a girl or boy is 48.8% or 51.2%, respectively. Suppose that 400 babies are born in a hospital in a given year. How many will be girls?

It is not always appropriate to fit a classical linear regression model using data in their raw form. As we discuss in Sections 4.1 and 4.4, linear and logarithmic transformations can sometimes help in the interpretation of the model. Nonlinear transformations of the data are sometimes necessary to more closely satisfy additivity and linearity assumptions, which in turn should improve the fit and predictive power of the model. Section 4.5 presents some other univariate transformations that are occasionally useful. We have already discussed interactions in Section 3.3, and in Section 4.6 we consider other techniques for combining input variables.

Linear transformations

Linear transformations do not affect the fit of a classical regression model, and they do not affect predictions: the changes in the inputs and the coefficients cancel in forming the predicted value Xβ. However, well-chosen linear transformation can improve interpretability of coefficients and make a fitted model easier to understand. We saw in Chapter 3 how linear transformations can help with the interpretation of the intercept; this section provides examples involving the interpretation of the other coefficients in the model.

Scaling of predictors and regression coefficients. The regression coefficient βj represents the average difference in y comparing units that differ by 1 unit on the jth predictor and are otherwise identical. In some cases, though, a difference of 1 unit on the x-scale is not the most relevant comparison.

Not only in the commercial world but in the realm of ideas as well, our age is holding a veritable clearance sale. Everything is had so dirt cheap that it is doubtful whether in the end anyone will bid. Every speculative score-keeper who conscientiously keeps account of the momentous march of modern philosophy, every lecturer, tutor, student, every outsider and insider in philosophy does not stop at doubting everything but goes further. Perhaps it would be inappropriate and untimely to ask them where they are actually going, but it is surely polite and modest to take it for granted that they have doubted everything, since otherwise it would certainly be peculiar to say that they went further. All of them then have made this preliminary movement, and presumably so easily that they do not find it necessary to drop a hint about how, for not even the one who anxiously and worriedly sought a little enlightenment found so much as an instructive tip or a little dietary prescription on how to conduct oneself under this enormous task. “But Descartes has done it, hasn't he?” Descartes, a venerable, humble, honest thinker whose writings surely no one can read without the deepest emotion, has done what he has said and said what he has done. Alas! Alas! Alas! That is a great rarity in our age! As he himself reiterates often enough, Descartes did not doubt with respect to faith.

Fear and Trembling, written when the author was only thirty years old, is in all likelihood Søren Kierkegaard's most-read book. This would not have surprised Kierkegaard, who wrote prophetically in his journal that “once I am dead, Fear and Trembling alone will be enough for an imperishable name as an author. Then it will [be] read, translated into foreign languages as well.” In one sense the book is not difficult to read. It is often assigned in introductory university classes, for it is the kind of book that a novice in philosophy can pick up and read with interest and profit – stimulating questions about ethics and God, faith and reason, experience and imagination. However, in another sense the book is profoundly difficult, the kind of book that can be baffling to the scholar who has read it many times and studied it for years – giving rise to a bewildering variety of conflicting interpretations.

Many of these interpretations have focused on the book's relation to Kierkegaard's own life, and in particular on the widely known story of Kierkegaard's broken engagement to Regine Olsen. There is little doubt that part of Kierkegaard's own motivation for writing Fear and Trembling was to present a disguised explanation to Regine of his true reasons for breaking off the engagement. However, it is just as certain that the philosophical importance of the book does not depend on these personal and biographical points; the book can be read and has been read with profit by those with no knowledge of Kierkegaard's own life.

On one occasion when the price of spices in Holland became somewhat slack, the merchants let a few loads be dumped at sea in order to drive up the price. This was a pardonable, perhaps a necessary stratagem. Do we need something similar in the world of spirit? Are we so sure of having attained the highest that there is nothing left to do except piously to delude ourselves that we have not come so far in order still to have something with which to fill the time? Does the present generation need such a self-deception? Should a virtuosity in this be cultivated in it, or is it not rather sufficiently perfected in the art of self-deception? Or is what it needs not rather an honest earnestness that fearlessly and incorruptibly calls attention to the tasks, an honest earnestness that lovingly preserves the tasks, that does not make people anxiously want to rush precipitously to the highest but keeps the tasks young, beautiful, delightful to look upon, and inviting to all, yet also difficult and inspiring for the noble-minded (for the noble nature is inspired only by the difficult)? Whatever one generation learns from another, no generation learns the genuinely human from a previous one. In this respect, every generation begins primitively, has no other task than each previous generation, and advances no further, provided the previous generation has not betrayed the task and deceived itself.

If there were no eternal consciousness in a human being, if underlying everything there were only a wild, fermenting force writhing in dark passions that produced everything great and insignificant, if a bottomless, insatiable emptiness lurked beneath everything, what would life be then but despair? If such were the case, if there were no sacred bond that tied humankind together, if one generation after another rose like leaves in the forest, if one generation succeeded another like the singing of birds in the forest, if the human race passed through the world as a ship through the sea, as the wind through the desert, a thoughtless and futile activity, if an eternal oblivion always hungrily lay in wait for its prey and there were no power strong enough to snatch it away – then how empty and hopeless life would be! But that is why it is not so, and as God created man and woman, so he fashioned the hero and the poet or orator. The latter can do nothing that the former does, he can only admire, love, and rejoice in the hero. Yet he too is happy, no less than the former, for the hero is so to speak his better nature with which he is infatuated yet delighted that it is after all not himself, that his love can be admiration.

A Preliminary Outpouring from the Heart

An old adage drawn from the external and visible world says: “Only the one who works gets the bread.” Oddly enough, the adage does not apply in the world where it is most at home, for the external world is subject to the law of imperfection, and here it happens again and again that the one who does not work also gets the bread, and the one who sleeps gets it more abundantly than the one who works. In the external world everything belongs to the possessor; it toils slavishly under the law of indifference, and the genie of the ring obeys whoever has the ring, whether he is a Noureddin or an Aladdin, and whoever has the world's treasures has them no matter how he got them. In the world of spirit it is otherwise. Here an eternal divine order prevails, here it does not rain on both the just and the unjust, here the sun does not shine on both good and evil, here it holds true that only the one who works gets the bread, only the one who was in anxiety finds rest, only the one who descends into the underworld rescues the beloved, only the one who draws the knife gets Isaac. The one who will not work does not get the bread but is deceived, just as the gods deceived Orpheus with an airy apparition instead of the beloved, deceived him because he was sentimental, not courageous, deceived him because he was a lute player, not a man.

There was once a man who as a child had heard that beautiful story about how God tested Abraham and how he withstood the test, kept the faith, and received a son a second time contrary to expectation. When the man became older, he read the same story with even greater admiration, for life had separated what had been united in the child's pious simplicity. Indeed, the older he became, the more often his thoughts turned to that story; his enthusiasm became stronger and stronger, and yet he could understand the story less and less. Finally he forgot everything else because of it; his soul had only one wish, to see Abraham, one longing, to have been a witness to that event. His desire was not to see the beautiful regions of the Far East, not the earthly splendor of the Promised Land, not that god-fearing married couple whose old age God had blessed, not the venerable figure of the aged patriarch, not the vigorous youth of Isaac bestowed by God – it would not have mattered to him if the same thing had taken place on a barren heath. His longing was to accompany them on the three day journey when Abraham rode with sorrow before him and Isaac by his side. His wish was to be present at the hour when Abraham lifted up his eyes and saw Mount Moriah in the distance, the hour he left the asses behind and went up the mountain alone with Isaac, for what engrossed him was not the artistic weave of the imagination but the shudder of the thought.

The Hellenistic period as conventionally understood is framed by two military conquests: the first, the Macedonian invasion of the Persian Empire under Alexander, rapid and deliberate (334–323 bc), the second, the Roman takeover of much of the Hellenistic world, hardly deliberate but a long-drawn process, which started in the late third century bc but was not complete till 30 bc with the overthrow of the Ptolemaic dynasty after the battle of Actium. War in this period was a constant presence which shaped the history of the times in many ways. Conquest and empire are the leading themes: they have had a long and varied life from antiquity to the present day, and are unlikely to lose their relevance in the foreseeable future.

Hellenistic studies are at present in a thriving condition, as a glance at any bibliography will show. When the first edition of this book was written only one general survey of the Hellenistic age in English was available (Tarn and Griffith (1953)), but since then they have multiplied and there has been a profusion of specialist studies. What George Grote once wrote in the Preface to his great History of Greece (12 volumes, first published 1846–56) now seems an outdated curiosity: ‘After the generation of Alexander, the political action of Greece becomes cramped and degraded – no longer interesting to the reader, or operative on the destinies of the world […] As a whole, the period between 300 bc and the absorption of Greece by the Romans is of no interest in itself, and is only so far of value as it helps us to understand the preceding centuries.'

Alexander's legacy to the world was a mess. His sudden death in Babylon in June 323 left Macedon in an unprecedented situation: there was no designated successor capable of taking over, and Macedon did not have the constitutional machinery to handle the emergency. The result was an open-ended power struggle between his leading followers which never came to a final conclusion. Within a generation three major monarchies established themselves in a lasting way – the Ptolemies in Egypt (chapter 7), the Seleucids in Asia (chapter 5), the Antigonids in Macedon (chapter 3) – but there was always room for newcomers, such as the Attalids of Pergamum (chapter 6), the Greek rulers in Bactria, and non-Greek monarchies, such as the dynasties in Bithynia, Pontus, and further afield Parthia and India.

The surviving literary sources place the leading Macedonians at the centre of the story. They evoke conflicting assessments. ‘All those who were associated with Philip, and afterwards with Alexander, showed themselves to be truly royal in their magnanimity, their self-control and their daring … After Alexander's death, when they became rivals for the possession of an empire which covered much of the world, they filled many history books with the glory of their achievements’ (Polybius VIII.10). Contrast Plutarch (Pyrrhus 12): ‘They are perpetually at war, because for them plotting and being envious of each other is second nature, and they use the words war and peace just like current coin, to serve their present needs, but in defiance of justice. Indeed they are really better men when they go to war openly than when they conceal under the names of justice and friendship those periods when they are at leisure and abstrain from acts of wrongdoing.’