Showing papers by "University of Chicago published in 1987"
TL;DR: In this article, the authors examined the relation between stock returns and stock market volatility and found that the expected market risk premium (the expected return on a stock portfolio minus the Treasury bill yield) is positively related to the predictable volatility of stock returns.
4,348 citations
TL;DR: If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution of parameters of interest.
Abstract: The idea of data augmentation arises naturally in missing value problems, as exemplified by the standard ways of filling in missing cells in balanced two-way tables. Thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze. This device is used to great advantage by the EM algorithm (Dempster, Laird, and Rubin 1977) in solving maximum likelihood problems. In situations when the likelihood cannot be approximated closely by the normal likelihood, maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements. From the Bayesian point of view, one must now calculate the posterior distribution of parameters of interest. If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution. It is the purpose of this article to explain how this can be done. The basic idea ...
4,020 citations
TL;DR: It is concluded that human coronary arteries enlarge in relation to plaque area and that functionally important lumen stenosis may be delayed until the lesion occupies 40 percent of the internal elastic lamina area.
Abstract: Whether human coronary arteries undergo compensatory enlargement in the presence of coronary disease has not been clarified. We studied histologic sections of the left main coronary artery in 136 hearts obtained at autopsy to determine whether atherosclerotic human coronary arteries enlarge in relation to plaque (lesion) area and to assess whether such enlargement preserves the cross-sectional area of the lumen. The area circumscribed by the internal elastic lamina (internal elastic lamina area) was taken as a measure of the area of the arterial lumen if no plaque had been present. The internal elastic lamina area correlated directly with the area of the lesion (r = 0.44, P less than 0.001), suggesting that coronary arteries enlarge as lesion area increases. Regression analysis yielded the following equation: Internal elastic lamina area = 9.26 + 0.88 (lesion area) + 0.026 (age) + 0.005 (heart weight). The correlation coefficient for the lesion area was significant (P less than 0.001), whereas the correlation coefficients for age and heart weight were not. The lumen area did not decrease in relation to the percentage of stenosis (lesion area/internal elastic lamina area X 100) for values between zero and 40 percent but did diminish markedly and in close relation to the percentage of stenosis for values above 40 percent (r = -0.73, P less than 0.001). We conclude that human coronary arteries enlarge in relation to plaque area and that functionally important lumen stenosis may be delayed until the lesion occupies 40 percent of the internal elastic lamina area. The preservation of a nearly normal lumen cross-sectional area despite the presence of a large plaque should be taken into account in evaluating atherosclerotic disease with use of coronary angiography.
3,631 citations
TL;DR: In this paper, the authors present a number of formal restrictions of this sort, investigate their behavior in specific examples, and relate these restrictions to Kohlberg and Mertens' notion of stability.
Abstract: Games in which one party conveys private information to a second through messages typically admit large numbers of sequential equilibria, as the second party may entertain a wealth of beliefs in response to out-of-equilibrium messages. By restricting those out-of equilibrium beliefs, one can sometimes eliminate many unintuitive equilibria. We present a number of formal restrictions of this sort, investigate their behavior in specific examples, and relate these restrictions to Kohlberg and Mertens` notion of stability.
3,290 citations
TL;DR: In this paper, the relation of the structure of an R set to its degree is discussed, and the infinite injury priority method is proposed to solve the problem of scaling and splitting R sets.
Abstract: TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4. Low degrees, atomless sets, and invariant degree classes. 5. Incompleteness and completeness for noninvariant properties. Chapter II. The structure, automorphisms, and elementary theory of the r.e. sets. 6. Basic facts and splitting theorems. 7. Hh-simple sets. 8. Major subsets and r-maximal sets. 9. Automorphisms of &. 10. The elementary theory of S. Chapter III. The structure of the r.e. degrees. 11. Basic facts. 12. The finite injury priority method. 13. The infinite injury priority method. 14. The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R.
1,932 citations
TL;DR: In this paper, the effects of short-sale constraints on the speed of adjustment (to private information) of security prices are modeled. But short-sellers do not bias prices upward, while non-prohibitive costs have the reverse effect.
1,866 citations
TL;DR: Sequencing of cloned human apolipoprotein(a) complementary DMA shows that it is very similar to human plasminogen.
Abstract: Lipoprotein(a) is an LDL-like lipoprotein whose concentration in plasma is correlated with atherosclerosis. The characteristic protein component of lipoprotein(a) is apolipoprotein(a) which is disulphide-linked to apolipoprotein B-100. Sequencing of cloned human apolipoprotein(a) complementary DNA shows that it is very similar to human plasminogen. It contains a serine protease domain and two types of plasminogen-like kringle domains, one of which is present in 37 copies.
1,824 citations
TL;DR: The authors showed that the positive test strategy can be a very good heuristic for determining the truth or falsity of a hypothesis under realistic conditions, but it can also lead to systematic errors or inefficiencies.
Abstract: Strategies for hypothesis testing in scientific investigation and everyday reasoning have interested both psychologists and philosophers. A number of these scholars stress the importance of disconfir. marion in reasoning and suggest that people are instead prone to a general deleterious "confirmation bias" In particula~ it is suggested that people tend to test those cases that have the best chance of verifying current beliefs rather than those that have the best chance of falsifying them. We show, howeve~ that many phenomena labeled "confirmation bias" are better understood in terms of a general positive test strate~. With this strategy, there is a tendency to test cases that are expected (or known) to have the property of interest rather than those expected (or known) to lack that property. This strategy is not equivalent to confirmation bias in the first sense; we show that the positive test strategy can be a very good heuristic for determining the truth or falsity of a hypothesis under realistic conditions~ It can, howeve~ lead to systematic errors or inefficiencies. The appropriateness of human hypotheses-testing strategies and prescriptions about optimal strategies must he understood in terms of the interaction between the strategy and the task at hand.
1,811 citations
TL;DR: In this paper, a method for producing Latin hypercube samples when the components of the input variables are statistically dependent is described, and the estimate is also shown to be asymptotically normal.
Abstract: Latin hypercube sampling (McKay, Conover, and Beckman 1979) is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables. The asymptotic variance of such an estimate is obtained. The estimate is also shown to be asymptotically normal. Asymptotically, the variance is less than that obtained using simple random sampling, with the degree of variance reduction depending on the degree of additivity in the function being integrated. A method for producing Latin hypercube samples when the components of the input variables are statistically dependent is also described. These techniques are applied to a simulation of the performance of a printer actuator.
1,750 citations
TL;DR: This article proposed a two-stage hierarchical linear model (HLM) to study the structure of individual growth and estimate important statistical and psychometric properties of collections of growth trajectories, discovering correlates of change factors that influence the rate at which individuals develop; and testing hypotheses about the effects of on or more experimental or quasi-experimental treatments on growth curves.
Abstract: Developments over the past 10 years in the statistical theory of hierarchical linear models (HLMs) now enable an integrated approach for (a) studying the structure of individual growth and estimating important statistical and psychometric properties of collections of growth trajectories; (b) discovering correlates of change factors that influence the rate at which individuals develop; and (c) testing hypotheses about the effects of on or more experimental or quasi-experimental treatments on growth curves. The approach is based on a two-stage hierarchical model. An example based on Head Start data illustrated key analytic uses of HLMs; (a) describing the structure of the mean growth trajectory; (b) estimating the extent and character of individual variation around mean growth; (c) assessing the reliability of measures for studying both status and change; (d) estimating the correlation between subjects entry status and rates of growth; (e) estimating correlates of both status and change; (f) assessing the adequacy of between-subjects models by estimating reduction in unexplained parameter variance (reduction in uncertainty about the individual growth parameters as distinguished from errors in their estimation); and (g) predicting future individual growth. HLMs can be applied in experimental and quasi-experimental settings. The HLM approach requires multi-time point data. The special strengths of HLMs in individual prediction are remarkable. The study of growth curves using HLMs requires special care to distributional assumptions covariance assumptions and the metric of measurement. HLMs seem broadly applicable to the study of change and are likely to extend substantially the empirical research on change. To the extent that HLMs enrich the class of testable hypotheses about the structure of growth it may also encourage a broadened discussion about the nature of change itself.
1,416 citations
TL;DR: It is shown that neurons have a number of different types of calcium channels, each with their own unique properties and pharmacology, and these calcium channels may be important in the control of different aspects of nerve activity.
Abstract: Recent investigations have demonstrated that neurons have a number of different types of calcium channels, each with their own unique properties and pharmacology. These calcium channels may be important in the control of different aspects of nerve activity. Some of the possibilities can now be discussed.
TL;DR: In this paper, a model of intertemporal choice that incorporates "savoring" and "dread" is presented, which explains why an individual with positive time preference may delay desirable outcomes or get unpleasant outcomes over with quickly.
Abstract: This paper presents a model of intertemporal choice that incorporates "savoring" and "dread"-i.e., utility from anticipat ion of delayed consumption. The model explains why an individual with positive time preference may delay desirable outcomes or get unpleas ant outcomes over with quickly, contrary to the prediction of convent ional formulations of intertemporal choice. Implications of savoring and dread for savings behavior, empirical estimation of discount rate s, and public policy efforts to combat myopic behavior are explored. The model provides an explanation for common violations of the indepe ndence axiom as applied to intertemporal choice. Copyright 1987 by Royal Economic Society.
TL;DR: Cognitive research can help in identifying situations that inhibit or facilitate recall and can reveal inferences that affect the accuracy of respondents' answers.
Abstract: Survey questions often probe respondents for quantitative facts about events in their past: "During the last 2 weeks, on days when you drank liquor, about how many drinks did you have?" "During the past 12 months, how many visits did you make to a dentist?" "When did you last work at a full-time job?" are all examples from national surveys. Although questions like these make an implicit demand to remember and enumerate specific autobiographical episodes, respondents frequently have trouble complying because of limits on their ability to recall. In these situations, respondents resort to inferences that use partial information from memory to construct a numeric answer. Results from cognitive psychology can be useful in understanding and investigating these phenomena. In particular, cognitive research can help in identifying situations that inhibit or facilitate recall and can reveal inferences that affect the accuracy of respondents' answers.
TL;DR: In this paper, a probe molecule coumarin 153 (Cu153) and picosecond spectroscopic techniques were used to examine the solvation dynamics in polar liquids and showed that the frequency of the electronic spectrum of this probe provides a convenient measure of solvation energetics.
Abstract: Solvation dynamics in polar liquids have been examined using the probe molecule coumarin 153 (Cu153) and picosecond spectroscopic techniques. Steady‐state absorption and fluorescence spectra of Cu153 as a function of solvent show that the frequency of the electronic spectrum of this probe provides a convenient measure of solvation energetics. Both nonspecific dipolar and to a smaller degree H‐bonding solute–solvent interactions are involved. Time‐correlated single photon counting was used to observe time‐dependent shifts of the fluorescence spectrum of Cu153 in a variety of alcohols, propylene carbonate, and N‐methylpropionamide solvents as a function of temperature. These time‐dependent spectral shifts provide a direct measure of the time dependence of the solvation process. Theoretical models that treat the solvent as a dielectric continuum do not adequately account for the observed solvation dynamics. In the solvents studied, such theories predict a single exponential shift of the fluorescence spectrum...
TL;DR: In this article, the authors derived Yukawa couplings in the effective field theory for fermionic strings on orbifolds and applied them to the scattering of four twisted string states, which allowed the extraction of operator product coefficients of conformal twist fields.
01 Nov 1987
TL;DR: A description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory, focuses upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularity; and f, which describes how densely they are distributed.
Abstract: We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f , which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f ( α ). We apply this formation to the 2 ∞ cycle of period doubling, to the devil's staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.
Book•
01 Jan 1987TL;DR: Richards's book is now the obvious introduction to the history of ideas about mind and behavior in the nineteenth century as mentioned in this paper, and not since the publication of Michael Ghiselin's "The Triumph of the Darwinian Method" has there been such an ambitious, challenging, and methodologically self-conscious interpretation of the rise and development and evolutionary theories and Darwin's role therein.
Abstract: With insight and wit, Robert J. Richards focuses on the development of evolutionary theories of mind and behavior from their first distinct appearance in the eighteenth century to their controversial state today. Particularly important in the nineteenth century were Charles Darwin's ideas about instinct, reason, and morality, which Richards considers against the background of Darwin's personality, training, scientific and cultural concerns, and intellectual community. Many critics have argued that the Darwinian revolution stripped nature of moral purpose and ethically neutered the human animal. Richards contends, however, that Darwin, Herbert Spencer, and their disciples attempted to reanimate moral life, believing that the evolutionary process gave heart to unselfish, altruistic behavior. "Richards's book is now the obvious introduction to the history of ideas about mind and behavior in the nineteenth century." Mark Ridley, Times Literary Supplement "Not since the publication of Michael Ghiselin's "The Triumph of the Darwinian Method "has there been such an ambitious, challenging, and methodologically self-conscious interpretation of the rise and development and evolutionary theories and Darwin's role therein." John C. Greene, "Science " "His book . . . triumphantly achieves the goal of all great scholarship: it not only informs us, but shows us why becoming thus informed is essential to understanding our own issues and projects." Daniel C. Dennett, "Philosophy of Science ""
TL;DR: The ascending pathway for visceral sensory information appears to be viscerotopically organized at all levels of the neuraxis, including the insular cortex.
Abstract: The functional organization of the insular cortex was studied by recording neuronal responses to visceral sensory stimuli. Horseradish peroxidase (HRP) was then iontophoresed at the recording sites to identify afferents from the ventrobasal thalamus to specific visceroceptive sites in the insular cortex. The relationship of the ventrobasal thalamus to the insular cortex and to brainstem relay nuclei for the ascending visceral projections was then examined by using the axonal transport of HRP, wheat germ agglutinin conjugated to HRP (WGA-HRP), and fluorescent dyes. Of a total of 55 neurons that were tested for responses to visceral sensory stimuli, 33 units responded to at least one visceral sensory modality: 6 received gastric mechanoreceptor input, 8 responded to taste inputs, 13 were activated by arterial chemoreceptors and/or showed respiratory related activity, and 6 responded to cardiovascular baroreceptor stimulation. On the basis of its cytoarchitecture and connections with the thalamus, the insular cortex was divided into a dorsal granular area, an intermediate dysgranular region, and a ventral agranular strip. Taste-responsive neurons were located anteriorly, primarily in the dysgranular region, whereas unit responses to general visceral modalities were distributed dorsally and posteriorly in the granular insular cortex. Gastric mechanoreceptor-responsive units were situated more dorsally and anteriorly in the granular insular cortex, while cardiopulmonary inputs were located more ventrally and posteriorly. Injections of HRP into the gustatory insular cortex resulted in retrograde labeling of neurons in the parvicellular part of the ventroposterior medial thalamic nucleus (VPMpc). Injections into the general visceral insular cortex retrogradely labeled neurons lateral to VPMpc in the ventroposterior lateral parvicellular thalamic nucleus (VPLpc). Injections of HRP, WGA-HRP, and fluorescent dyes into VPMpc and VPLpc verified that their projection to the insular cortex is topographically organized. In the same experiments, retrogradely labeled neurons in the parabrachial nucleus identified the likely subnuclei within this nucleus for relay of visceral sensory information to the thalamus. Injections of WGA-HRP into the parabrachial nucleus demonstrated that its projection to the ventrobasal thalamus is also topographically organized. These results demonstrate the relationship of general visceral and special visceral (taste) representations in the insular cortex. The ascending pathway for visceral sensory information appears to be viscerotopically organized at all levels of the neuraxis, including the insular cortex.
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TL;DR: The authors describes the sources of this deficiency and indicates just what elements have come to be missing from home and community These are conceptualized generally as social capital which is now in short supply for children and youth and a general approach to increasing the social capital available to the next generation is indicated.
Abstract: Families at all economic levels are becoming increasingly ill-equipped to provide the setting that schools are designed to complement and augment in preparing the next generation This paper describes the sources of this deficiency and indicates just what elements have come to be missing from home and community These are conceptualized generally as social capital which is now in short supply for children and youth A general approach to increasing the social capital available to the next generation is indicated
TL;DR: In this paper, it was shown that part of all of the C-delta is primary, not shock-produced, diamond formed by stellar condensation as a metastable phase.
Abstract: Primitive meteorites contain up to 400 ppm of a very fine-grained type of carbon tentatively called C-delta. Evidence is presented here that part of all of the C-delta is primary, not shock-produced, diamond formed by stellar condensation as a metastable phase. It appears that interstellar dust contains diamond.
TL;DR: This paper found that the content of network television news accounts for a high proportion of aggregate changes (from one survey to another) in U.S. citizens' policy preferences, while special interest groups tend to have a negative impact.
Abstract: Democratic theory must pay attention to what influences public opinion. In this study the content of network television news is shown to account for a high proportion of aggregate changes (from one survey to another) in U.S. citizens' policy preferences. Different news sources have different effects. News commentators (perhaps reflecting elite or national consensus or media biases) have a very strong positive impact, as do experts. Popular presidents tend to have positive effects, while unpopular presidents do not. In contrast, special interest groups tend to have a negative impact.
TL;DR: The origins of the cholinergic and other afferents of several thalamic nuclei were investigated in the rat by using the retrograde transport of wheat germ agglutinin conjugated‐horseradish peroxidase in combination with the immunohistochemical localization of choline acetyltransferase immunoreactivity.
Abstract: The origins of the cholinergic and other afferents of several thalamic nuclei were investigated in the rat by using the retrograde transport of wheat germ agglutinin conjugated-horseradish peroxidase in combination with the immunohistochemical localization of choline acetyltransferase immunoreactivity. Small injections placed into the reticular, ventral, laterodorsal, lateroposterior, posterior, mediodorsal, geniculate, and intralaminar nuclei resulted in several distinct patterns of retrograde labelling. As expected, the appropriate specific sensory and motor-related subcortical structures were retrogradely labelled after injections into the principal thalamic nuclei. In addition, other basal forebrain and brainstem structures were also labelled, with their distribution dependent on the site of injection. A large percentage of these latter projections was cholinergic. In the brainstem, the cholinergic pedunculopontine tegmental nucleus was retrogradely labelled after all thalamic injections, suggesting that it provides a widespread innervation to the thalamus. Neurons of the cholinergic laterodorsal tegmental nucleus were retrogradely labelled after injections into the anterior, laterodorsal, central medial, and mediodorsal nuclei, suggesting that it provides a projection to limbic components of the thalamus. Significant basal forebrain labelling occurred only with injections into the reticular and mediodorsal nuclei. Only injections into the reticular nucleus resulted in retrograde labelling of the cholinergic neurons in the nucleus basalis of Meynert. The results provide evidence for an organized system of thalamic afferents arising from cholinergic and noncholinergic structures in the brainstem and basal forebrain. The brainstem structures, especially the cholinergic pedunculopontine tegmental nucleus, appear to project directly to principal thalamic nuclei, thereby providing a possible anatomical substrate for mediating the well-known facilitory effects of brainstem stimulation upon thalamocortical transmission.
TL;DR: The adaptive response of the arterial wall and intimal thickening under conditions of increased flow in an atherogenic model was studied in six cynomolgus monkeys fed a diet containing 2% cholesterol and 25% peanut oil.
TL;DR: This article investigated the role of programming knowledge in program comprehension and the nature of mental representations of programs; specifically, whether procedural (control flow) or functional (goal hierarchy) relations dominate programmers' mental representations.
TL;DR: A detailed cytoarchitectonic delineation of the mesopontine tegmentum, including the PPTn, was performed employing tissue stained for Nissl substance and the relationships between neuronal elements staining for ChAT and tyrosine hydroxylase were investigated on tissue stained immunohistochemically.
Abstract: The pedunculopontine tegmental nucleus (PPTn) was originally defined on cytoarchitectonic grounds in humans. We have employed cytoarchitectonic, cytochemical, and connectional criteria to define a homologous cell group in the rat. A detailed cytoarchitectonic delineation of the mesopontine tegmentum, including the PPTn, was performed employing tissue stained for Nissl substance. Choline acetyltransferase (ChAT) immunostained tissue was then analyzed in order to investigate the relationship of cholinergic perikarya, dendritic arborizations, and axonal trajectories within this cytoarchitectonic scheme. To confirm some of our cytoarchitectonic delineations, the relationships between neuronal elements staining for ChAT and tyrosine hydroxylase were investigated on tissue stained immunohistochemically for the simultaneous demonstration of these two enzymes.
The PPTn consists of large, multipolar neurons, all of which stain immunohistochemically for ChAT. It is present within cross-sections that also include the A-6 through A-9 catecholamine cell groups and is traversed by catecholaminergic axons within the dorsal tegmental bundle and central tegmental tract. The dendrites of PPTn neurons respect several nuclear boundaries and are oriented perpendicularly to several well-defined fiber tracts. Cholinergic axons ascend from the mesopontine tegmentum through the dorsal tegmental bundle and a more lateral dorsal ascending pathway. A portion of the latter terminates within the lateral geniculate nucleus.
It has been widely believed that the PPTn is reciprocally connected with several extrapyramidal structures, including the globus pallidus and substantia nigra pars reticulata. Therefore, the relationships of pallidotegmental and nigrotegmental pathways to the PPTn were investigated employing the anterograde autoradiographic methodology. The reciprocity of tegmental connections with the substantia nigra and entopeduncular nucleus was investigated employing combined WGA-HRP injections and ChAT immunohistochemistry.
The pallido- and nigrotegmental terminal fields did not coincide with the PPTn, but, rather, were located just medial and dorsomedial to it (the midbrain extrapyramidal area). The midbrain extrapyramidal area, but not the PPTn, was reciprocally connected with the substantia nigra and entope-duncular nucleus. We discuss these results in light of other cytoarchitec-tonic, cytochemical, connectional, and physiologic studies of the functional anatomy of the mesopontine tegmentum.
TL;DR: The electronic properties of a tight-binding model which possesses two types of hopping matrix element arranged in a Fibonacci sequence are studied and the fractal dimensions f(ae) represents the global scaling properties of the Cantor-set spectrum.
Abstract: The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show ``critical'' (or ``exotic'') behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ${\ensuremath{\alpha}}_{E}$ which takes a value in the range [${\ensuremath{\alpha}}_{E}^{\mathrm{min}}$,${\ensuremath{\alpha}}_{E}^{\mathrm{max}}$]. The fractal dimensions f(${\ensuremath{\alpha}}_{E}$) of these singularities in the Cantor set are calculated. This function f(${\ensuremath{\alpha}}_{E}$) represents the global scaling properties of the Cantor-set spectrum.
TL;DR: In this paper, the effects of large (block) transactions on the prices of common stocks traded on the New York Stock Exchange were examined. And the results suggest that price effects are predominantly temporary for seller-initiated transactions and permanent for buyer-inotiated transactions, while the price effects vary cross-sectionally according to the size of the block.
TL;DR: It is demonstrated that the major EBV outer membrane glycoprotein, gp350/220, is a highly specific ligand for CR2, the type 2 complement receptor of the EBV receptor.
TL;DR: Analysis indicates that two components govern the spectral lens density function, with one increasing gradually during life, and the average lens density increases linearly at 400 nm between the ages of 20 and 60 and above.
Abstract: The optical density of the human lens changes during life. Literature concerning both the spectral density function and the rate of such changes is reviewed. Analysis indicates that two components govern the spectral lens density function, with one increasing gradually during life. The average lens density increases linearly at 400 nm by 0.12 density unit per decade between the ages of 20 and 60 and by 0.40 density unit per decade above age 60. A tabulation of the two components of the average 32-yr old lens is given, as are equations to derive the average spectral lens density functions for observers aged 20–80.