Sensitivity analysis in principal component analysis:influence on the subspace spanned by principal components.

The problem of detecting influential observations in principalcomponent analysis was discussed by several authors. Radhakrishnan and kshirsagar ( 1981 ), Critchley ( 1985 ), jolliffe ( 1986 )among others discussed this topicby using the influence functions I(X;θ_s)and I(X;V_s)of eigenvalues and eigenvectors, which wwere derived under the assumption that the eigenvalues of interest were simple. In this paper we propose the influence functionsI(X;∑^q _s=1θ_sV_sV_s ^T)and I(x;∑^q _s=1V_sV_s ^t)(q<p;p:number of variables) to investigate the influence onthe subspace spanned by principal components. These influence functions are applicable not only to the case where the edigenvalues of interst are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

Influence functions related to eigenvalue problems which appear in multivariate analysis

Tanaka(1988) derived two influence functions related to an ordinary eigenvalue problem (A–λ_s I)v_s = 0 of a real symmetric matrix A and used them for sensitivity analysis in principal component analysis. One of these influence functions was used to develop sensitivity analysis in factor analysis (see e.g. Tanaka and Odaka, 1988a). The present paper derives some additional influence functions related to an ordinary eigenvalue problem and also several influence functions related to a generalized eigenvalue problem (A–θ_s A)u_s = 0, where A and B are real symmetric and real symmetric positive definite matrices, respectively. These influence functions are applicable not only to the case where the eigenvalues of interest are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

Sensitivity analysis in multivariate methods: Decomposition of an arbitrary influence into a finite number of components

The influence function of the covariance matrix is decomposed into a finite number of components. This decomposition provides a useful tool to develop efficient methods for computing empirical influence curves related to various multivariate methods. It can also be used to characterize multivariate methods from the sensitivity perspective. A numerical example is given to demonstrate efficient computing and to characterize some procedures of exploratory factor analysis.     

Coincidence of two failure rate models

In considering an accelerated life test model, Bhattacharyya and Soejoeti (1989) proposed the tampered failure rate (TFR) model, and they gave a necessary and sufficient condition for the model to coincide with the tampered random variable (TRV) model due to DeGroot and Goel (1979). The main purpose of this paper is to show that their claim on the necessity is not true, in a stronger sense than theirs.     

An axiomatization of the Kalai-Smorodinsky solution when the feasible sets can be finite

We axiomatize the Kalai-Smorodinsky solution (1975) in the Nash bargaining problems if the feasible sets can be finite. We show that the Kalai-Smorodinsky solution is the unique solution satisfying Continuity (in the Hausdorff topology endowed with payoffs space), Independence (which is weaker than Nash's one and essentially equivalent to Roth (1977)'s one), Symmetry, Invariance (both of which are the same as in Kalai and Smorodinsky), and Monotonicity (which reduces to a little bit weaker version of the original if the feasible sets are convex). Received: 4 November 1999/Accepted: 6 June 2001     

Heritability estimates of life history traits in small white butterflyPieris rapae crucivora

Summary Heritabilities and genetic correlations of life history characters (pupal weight, age-specific fecundities, and egg weight) of small white butterflyPieris rapae crucivora are estimated by a quantitative genetic method (sib analysis). The results indicate moderate or high heritabilities and a largely negative genetic correaltion in age-specific fecundities.     

The evaluation and selection of adequate causal models: a compensatory education example

Procedures for ascertaining relative model adequacy in latent variable structural relations models are discussed. Under diverse methods of estimation, this determination may be assessed using the chi square goodness of fit statistic, incremental fit indices for covariance structure models, and latent variable coefficients of determination. An example from evaluation research is taken (cf. Magidson, 1977; Bentler & Woodward, 1978). Numerical sensitivity of parameter estimates under alternative model specifications is demonstrated. Interpretive implications based on these procedures are discussed in terms of parameter sensitivity to alternative model specifications.     

Benchmark Duration of Work Hours for Development of Fatigue Symptoms in Japanese Workers with Adjustment for Job‐Related Stress

The objective of this study was to calculate benchmark durations and lower 95% confidence limits for benchmark durations of working hours associated with subjective fatigue symptoms by applying the benchmark dose approach while adjusting for job‐related stress using multiple logistic regression analyses. A self‐administered questionnaire was completed by 3,069 male and 412 female daytime workers (age 18–67 years) in a Japanese steel company. The eight dependent variables in the Cumulative Fatigue Symptoms Index were decreased vitality, general fatigue, physical disorders, irritability, decreased willingness to work, anxiety, depressive feelings, and chronic tiredness. Independent variables were daily working hours, four subscales (job demand, job control, interpersonal relationship, and job suitability) of the Brief Job Stress Questionnaire, and other potential covariates. Using significant parameters for working hours and those for other covariates, the benchmark durations of working hours were calculated for the corresponding Index property. Benchmark response was set at 5% or 10%. Assuming a condition of worst job stress, the benchmark duration/lower 95% confidence limit for benchmark duration of working hours per day with a benchmark response of 5% or 10% were 10.0/9.4 or 11.7/10.7 (irritability) and 9.2/8.9 or 10.4/9.8 (chronic tiredness) in men and 8.9/8.4 or 9.8/8.9 (chronic tiredness) in women. The threshold amounts of working hours for fatigue symptoms under the worst job‐related stress were very close to the standard daily working hours in Japan. The results strongly suggest that special attention should be paid to employees whose working hours exceed threshold amounts based on individual levels of job‐related stress.     

Decrease in respiratory rate in a wolf spider,Pardosa astrigera (L. Koch), under starvation

Summary Effects of starvation on the suryival period and the respiratory rate in adults of a wolf spider,Pardosa astrigera (L. Koch), were investigated. The spiders used were divided into four groups: well-fed, starved and two limited food groups; in the latter two, each spider was supplied with one leafhopper every second or third day. Adult males and females ofP. astrigera could survive for a long time; 28.8±2.7 days and 54.4±18.9 days, respectively, without any food. The longevities shown here were 73.8% for males and 78.6% for females of those of well-fed spiders, indicating thatP. astrigera adults have a strong tolerance to starvation. The respiratory rate of well-fed adults showed no tendency to increase or decrease with their aging; the mean respiratory rates were 4.86×10⁻⁴ mg CO₂/mg f.w. (fresh body weight)/hr for males and 3.80×10⁻⁴ mg CO₂/mg f.w./hr for females. The respiratory rates of starved spiders increased during the first two days of starvation but decreased markedly from the third to the twelfth day, and thereafter retained an almost constant level for each sex. The mean respiratory rates after the twelfth day of starvation were 2.49×10⁻⁴ mg CO₂/mg f.w./hr for males and 2.76×10⁻⁴ mg CO₂/mg f.w./hr for females; these values were respectively 48.4% and 63.0% of those prior to starvation. The fresh body weight of starved spiders decreased linearly with time but the rate was small. The respiratory rates of the limited food groups tended to decline with time and thereby their weight losses were minimized. The decrease in the respiratory rate under starvation was considered not to be due to spider exhaustion or senescence but due to an intrinsic change in behaviour and/or metabolism, because when the spiders were supplied with ample food for five days after starvation, the respiratory rate and the body weight rapidly recovered to near the levels prior to starvation. It is suggested that starved spiders use a higher ratio of fat as catabolic substrate than normally fed or satiated ones. Feeding strategies of poikilo-therm predators are discussed. This work was partially supported by the Nippon Life Insurance Foundation Research Fund and Grant-in-Aid (No. 56480039) from the Ministry of Education, Science and Culture, Japan.