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The regression equation describes two characteristics of the regression line: its slope and its Y intercept buy provigil 100mg with mastercard sleep aid 100 mg. The slope is a number that indicates how slanted the regression line is and the direc- tion in which it slants 100mg provigil visa insomnia kills. When no relationship is present, the regression line is horizontal, such as line A, and the slope is zero. A positive linear relationship produces regression lines such as B and C; each of these has a slope that is a positive number. A negative linear relationship, such as line D, yields a slope that is a negative number. The Y intercept is the value of Y at the point where the regression line intercepts, or crosses, the Y axis. If we extended line C, it would intercept the Y axis at a point below the X axis, so its Y intercept is a negative Y score. Because line D reflects a negative relationship, its Y intercept is the relatively high Y score of 9. When there is no relationship, the regression line is flat and every Y¿ equals the Y intercept. The regression equation works like this: The slope indicates the direction in which the Ys change as X increases and the rate at which they change. The Y intercept indicates the starting point from which the Y scores begin to change. Thus, together, the slope and intercept describe how, starting at a particular Y score, the Y scores tend to change by a specific amount as the X scores increase. As an example, say that we have developed a test to identify (predict) those indi- viduals who will be good or bad workers at a factory that makes “widgets. The predictor (X) variable is participants’ scores on the widget test, and the criterion (Y) variable is the number of widgets they produced. This is a very strong, positive linear relationship, and so the test will be what researchers call “a good predictor” of widget-making. The numerator of the formula for b is the same as the numerator in the formula for r, and the denominator of the formula for b is the left-hand quantity in the denominator of the formula for r. This positive slope indicates a positive relationship, which fits with the positive r of 1. Had the rela- tionship been negative, the formula would have produced a negative number here. Computing the Y Intercept The formula for the Y intercept of the linear regression line is a 5 Y 2 1b21X2 First, multiply the mean of all X scores times the slope of the regression line. Describing the Linear Regression Equation Once you have computed the Y intercept and the slope, rewrite the regression equation, substituting the computed values for a and b. Plotting the Regression Line We use the finished regression equation to plot our linear regression line. To draw a line, we need at least two data points, so choose a low and high X score, insert each into the regression equation, and compute the Y¿ for that X. Therefore, we also use the finished regression equation to predict anyone’s Y score if we know their X score. In fact, computing any Y¿ using the equation is the equivalent of going to the graph and traveling vertically from the X score up to the regression line and then left to the value of Y¿ on the Y axis. We can compute Y¿ for any value of X that falls within the range of Xs in our data, even if it’s a score not found in the original sample: No one scored an X of 1. Our regression equa- tion is based only on widget test scores between 1 and 4, so we shouldn’t predict a Y for an X of, for example, 6. This is because we can’t be sure what the nature of the relationship is at 6—maybe it’s curvilinear or has a steeper slope. Putting all of this together, the preceding computations are summarized in Table 8. Substitute the values of a and b into the formula for the regression equation: Y¿ 5 1b21X2 1 a 5.
There was very considerable diversity among Greek medical people purchase provigil 100mg with visa insomnia jjcc, not only between the ‘rational’ buy provigil 100 mg visa sleep aid blood pressure, Introduction 19 philosophically inspired medicine that we find in the Hippocratic writings on the one hand and what is sometimes called the ‘folk medicine’ practised by drugsellers, rootcutters and suchlike on the other, but even among more intellectual, elite physicians themselves. One of the crucial points on which they were divided was precisely the ‘philosophical’ nature of medicine – the question of to what extent medicine should be built on the foundation of a comprehensive theory of nature, the world and the universe. It is interesting in this connection that one of the first attestations of the word philosophia in Greek literature occurs in a medical context – the Hippocratic work On Ancient Medicine – where it is suggested that this is not an area with which medicine should engage itself too much. It is clear from the context that what the author has in mind is approaches to medicine that take as their point of departure a general theory about ‘nature’ (phusis), more in particular theories that reduce all physical phenomena to unproven ‘postu- lates’ (hupotheseis), such as the elementary qualities hot, cold, dry and wet – theories which the author associates with the practice of Empedocles, who reduced natural phenomena to the interaction and combinations of the four elements earth, fire, water and air. The polemical tone of the treatise suggests that such ‘philosophical’ approaches to medicine were be- coming rather popular, and this is borne out by the extant evidence such as that provided by the Hippocratic treatises mentioned above. There were a number of medical authors for whom what we call ‘philosophy’ would not have been an inappropriate term to describe their projects – regardless of whether or not they knew and used the term. To this group certainly belongs the author of the treatise which is de- servedly one of the most famous writings in the Hippocratic Corpus, On the Sacred Disease. As I alluded to above, this work has long been read as the paradigm of Greek fifth-century rationalism. And it is certainly true that this author, in claiming that epilepsy ‘has a nature’, is doing something very similar to what the Presocratics did in inquiring into the ‘nature’ (phusis) of things, namely their origin, source of growth and identi- fying structure – be they earthquakes and solar eclipses, or bodily processes and changes, illnesses, conditions, affections, symptoms, or substances like foods, drinks, drugs and poisons and the effects they produced on the bod- ies of human beings. And just like Ionian philosophers such as Anaximenes and Anaxagoras in their explanations of earthquakes, solar eclipses, thun- derstorms and other marvellous phenomena, he produces a ‘natural’ ex- planation for a phenomenon – in his case ‘the so-called sacred disease’, epilepsy – that used to be seen as the manifestation of immediate divine agency. Epilepsy, the author argues, like all other diseases (and, one may 20 Medicine and Philosophy in Classical Antiquity add, like all other phenomena), has its own ‘nature’, its peculiar determined, normal, stable and self-contained identity. Knowledge of this identity, and of the regularity that results from this, will allow one to recognise and understand individual instances of the phenomenon, to predict its future occurrence and by medical intervention influence it or even prevent it from happening or spreading. And in making this claim, the author polemicises against people whom he calls ‘magicians, quacks, charlatans’, who regard the disease as a form of whimsical, unpredictable divine intervention or even demonic possession and whose therapeutic practice is determined by magical beliefs and procedures. He describes the development of the disease from its earliest, prenatal and indeed ancestral stages; he identifies the brain as the seat of consciousness (a theme I shall examine in greater depth in chapter 4) and as the primary organ affected by the disease; he discusses a whole range of additional factors, both internal and external, that set the disease in motion or influence its actual development; and he gives a vivid account of the various stages of an epileptic seizure, relating each of the symptoms to a particular underlying physiological cause. Yet for all the emphasis on the naturalness and ‘rationality’ of his ap- proach, we shall see in chapter 1 that the author rules out neither the divinity of the diseases nor the possibility of divine intervention as such. He is distinguishing between an appropriate appeal to the gods for purifi- cation from the ‘pollution’ (miasma) of moral transgressions (hamartemata¯ ) that has disturbed the relationship between man and the gods, and an in- appropriate appeal to the gods for the purification of the alleged pollution of the ‘so-called sacred disease’. This is inappropriate, he says, for diseases are not sent by a god – to say so would be blasphemous, he insists – they are natural phenomena which can be cured by natural means, and they do not constitute a pollution in the religious sense. The text has often been read as if the author ruled out divine ‘intervention’ as such. But in fact, there is no evidence that he does – indeed, he does not even rule out that gods may cure diseases, if approached in the proper way and on the basis of appropriate premises. Such negative readings of the text attributing to the author the ruling out of all forms of divine intervention have presumably been inspired by a wishful belief among interpreters to ‘rationalise’ or ‘secularise’ Hippo- cratic medicine – a belief possibly inspired by the desire to see Hippocratic medicine as the forerunner of modern biomedicine, and which can be par- alleled with interpretative tendencies to ‘demythologise’ philosophers such as Parmenides, Pythagoras and Empedocles to make them fit our concept of ‘philosophy’ more comfortably. Yet recently, there has been a renewed appreciation of the ‘mythical’ or ‘religious’ aspects of early Greek thought, Introduction 21 and a readiness to take documents such as the Dervenyi papyrus, the in- troduction of Parmenides’ poem and the Purifications of Empedocles more seriously. Similar ‘paradigm’ shifts have taken place in the study of Hip- pocratic medicine, and there is now a much greater willingness among interpreters25 to accept the religious and ‘rational’ elements as coexistent and – at least in their authors’ conception – compatible. The question is not so much to disengage from their mythical context those elements which we, or some of us, regard as philosophically interesting from a contemporary perspective, but rather to try to see how those elements fit into that context. Within this approach, the author of On the Sacred Disease can be regarded as an exponent of a modified or ‘purified’ position on traditional religious beliefs without abandoning those beliefs altogether and, as such, he can be said to have contributed also to the development of Greek religious or the- ological thought; for his arguments closely resemble those found in Plato’s ‘outlines of theology’ in the second book of the Republic, or, as I said above, Aristotle’s arguments against the traditional belief that dreams are sent by the gods in his On Divination in Sleep (see also chapter 6). One further, paradoxical aspect of On the Sacred Disease and its alleged ‘rationality’ is worth mentioning here. On the one hand, it is probably the best known of all the Hippocratic writings after the Oath, and its author (who is widely agreed to be also the author of Airs, Waters, Places) has often been regarded as one of the most plausible candidates for being identical with the historical Hippocrates. On the other hand, this is a fairly recent development, which stands in marked contrast to the rather marginal position the treatise occupied in ancient perceptions of Hippocrates. It hardly figures in ancient lists of Hippocratic writings, and it is particularly striking for its almost complete absence from Galen’s references to the Hippocratic Corpus.
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