In lifetesting, medical follow-up, and other fields the observation of the time of occurrence of the event of interest (called a death) may be prevented for some of the items of the sample by the previous occurrence of some other event (called a loss). Losses may be either accidental or controlled, the latter resulting from a decision to terminate certain observations. In either case it is usually assumed in this paper that the lifetime (age at death) is independent of the potential loss time; in practice this assumption deserves careful scrutiny. Despite the resulting incompleteness of the data, it is desired to estimate the proportion P(t) of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t). The observation for each item of a suitable initial event, marking the beginning of its lifetime, is presupposed. For random samples of size N the product-limit (PL) estimate can be defined as follows: L...

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Nonparametric Estimation from Incomplete Observations

formalization, and abstraction in biology, so too does mathematics’ relevance to the field (Fagerström et al. 1996). Mathematics is particularly important for analyzing and characterizing random variation of, for example, size and weight of individuals in populations, their sensitivity to chemicals, and time-to-event cases, such as the amount of time an individual needs to recover from illness. The frequency distribution of such data is a major factor determining the type of statistical analysis that can be validly carried out on any data set. Many widely used statistical methods, such as ANOVA (analysis of variance) and regression analysis, require that the data be normally distributed, but only rarely is the frequency distribution of data tested when these techniques are used. The Gaussian (normal) distribution is most often assumed to describe the random variation that occurs in the data from many scientific disciplines; the well-known bell-shaped curve can easily be characterized and described by two values: the arithmetic mean ̄x and the standard deviation s, so that data sets are commonly described by the expression x̄ ± s. A historical example of a normal distribution is that of chest measurements of Scottish soldiers made by Quetelet, Belgian founder of modern social statistics (Swoboda 1974). In addition, such disparate phenomena as milk production by cows and random deviations from target values in industrial processes fit a normal distribution. However, many measurements show a more or less skewed distribution. Skewed distributions are particularly common when mean values are low, variances large, and values cannot be negative, as is the case, for example, with species abundance, lengths of latent periods of infectious diseases, and distribution of mineral resources in the Earth’s crust. Such skewed distributions often closely fit the log-normal distribution (Aitchison and Brown 1957, Crow and Shimizu 1988, Lee 1992, Johnson et al. 1994, Sachs 1997). Examples fitting the normal distribution, which is symmetrical, and the lognormal distribution, which is skewed, are given in Figure 1. Note that body height fits both distributions. Often, biological mechanisms induce log-normal distributions (Koch 1966), as when, for instance, exponential growth is combined with further symmetrical variation: With a mean concentration of, say, 106 bacteria, one cell division more— or less—will lead to 2 × 106—or 5 × 105—cells.Thus, the range will be asymmetrical—to be precise, multiplied or divided by 2 around the mean. The skewed size distribution may be why “exceptionally”big fruit are reported in journals year after year in autumn. Such exceptions, however, may well be the rule: Inheritance of fruit and flower size has long been known to fit the log-normal distribution (Groth 1914, Powers 1936, Sinnot 1937). What is the difference between normal and log-normal variability? Both forms of variability are based on a variety of forces acting independently of one another. A major difference, however, is that the effects can be additive or multiplicative, thus leading to normal or log-normal distributions, respectively.

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Log-normal Distributions across the Sciences: Keys and Clues

Susceptibility to systemic meningococcal disease is related to a selective deficiency of humoral antibodies to pathogenic strains of meningococci. In a study of the age-specific incidence of meningococcal meningitis in the United States, it was found that the proportion of individuals with serum bactericidal activity to meningococci of serogroups A, B, and C was reciprocally related to the incidence of disease. The prevalence of bactericidal activity was highest at birth and among adults, and lowest in infants between 6 and 24 months of age. Sera from 51 of 54 prospective cases of meningococcal disease among military recruits were deficient in antibodies to homologous and heterologous strains of pathogenic meningococci as determined by serum bactericidal activity and indirect immunofluorescence. Such sera, however, could support the bactericidal activity of purified human gamma globulin (Cohn fraction II), and such individuals could respond immunologically to infection with meningococci. The implication is that susceptible persons are deficient in antimeningococcal antibodies because they have not received significant exposure to meningococcal antigens in the past. The fate of individuals who lack bactericidal antibodies to pathogenic meningococci was determined during an outbreak of group C meningitis among military recruits. The incidence of disease was found to be primarily associated with the incidence of exposure of susceptibles to the pathogenic strains. Whereas 81.5% of the presumed susceptibles acquired a meningococcal strain, only 24.1% acquired an organism similar to the prevalent disease-producing strains. Of the exposed susceptibles, 38.5% developed systemic meningococcal disease.

/pdf/human-immunity-to-the-meningococcus-i-the-role-of-humoral-1namleo7vv.pdf

Human immunity to the meningococcus i. the role of humoral antibodies

Contrasting treatment‐specific survival using double‐robust estimators

Results of the present study suggest that natural immunity to meningococcal disease is initiated, reinforced, and broadened by intermittent carriage of different strains of meningococci throughout life. In young adults, carriage of meningococci in the nasopharynx is an efficient process of immune sensitization. 92% of carriers of serogroup B, C, or Bo meningococci were found to develop increased titers of serum bactericidal activity to their own meningococcal isolate, and 87% developed bactericidal activity to heterologous strains of pathogenic meningococci. The rise in bactericidal titer occurred within 2 wk of onset of the carrier state, and was accompanied by an increase in titer of specific IgG, IgM, and IgA antibodies to meningococci. In early childhood, when few children have antibodies to pathogenic meningococci, active immunization seems to occur as a result of carriage of atypical, nonpathogenic strains. Immunity to systemic meningococcal infection among infants in the neonatal period is associated with the passive transfer of IgG antibodies from mother to fetus. The antigenic determinants which initiate the immune response to meningococci include the group-specific C polysaccharide, cross-reactive antigens, and type-specific antigens.

https://rupress.org/jem/article-pdf/129/6/1327/1392586/1327.pdf

Human immunity to the meningococcus. II. Development of natural immunity.

The incubation period of poliomyelitis.

IN laboratory animals where time of infection and dose of bacilli can be controlled there is substantial agreement that primary infection with tuberculosis, if not overwhelming, results in a measurably increased resistance to subsequent inoculations with tubercle bacilli.1' 2, 3 In man the picture is less clear. Although the typical parenchymal infiltration of primary infection, followed by resolution, fibrosis and calcification, is not infrequently observed roentgenologically, the exact date of reinfection and the duration of exposure are rarely known, and the dosage of bacilli is wholly problematical. Furthermore, it has not been demonstrated how far the results of inoculation experiments with tuberculosis in laboratory animals can be applied to natural infection in man. From a public health standpoint the relationship between primary infection with tuberculosis and the development of subsequent progressive disease has become a problem of major importance. Is a primary infection acquired during childhood a relative safeguard against subsequent exposure? or does the sensitization acquired through such infection result in increased susceptibility to destructive lesions? Much evidence might be cited to

https://ajph.aphapublications.org/doi/pdfplus/10.2105/AJPH.29.12.1318

Philip E. Sartwell

Papers

The incubation period of poliomyelitis.

Development of Tuberculosis in Infected Children.

Observations on an outbreak of infectious hepatitis in Baltimore during 1951.

Influence of the dynamic character of chronic disease on the interpretation of morbidity rates.

Epidemiological Notes on Meningococcal Meningitis in the Army