Anant M. Kshirsagar

Influence functions for certain parameters in multivariate analysis

The influence function introduced by Hampe1 (1968, 1973, 1974) is a tool that can be used for outlier detection. Campbell (1978) has obtained influence function for Mahalanobis’s distance between two populations which can be used for detecting outliers in discrim-inant analysis. In this paper influence functions for a variety of parametric functions in multivariate analysis are obtained. Influence functions for the generalized variance, the matrix of regression coefficients, the noncentrality matrix Σ-1 δ in multivariate analysis of variance and its eigen values, the matrix L, which is a generalization of 1-R2 , canonical correlations, principal components and parameters that correspond to Pillai’s statistic (1955), Hotelling’s (1951) generalized To2 and Wilk’s Λ (1932), which can be used for outlier detection in multivariate analysis, are obtained. Delvin, Ginanadesikan and Kettenring (1975) have obtained influence function for the population correlation co-efficient in the bivariate case. It is shown in...

Body composition, age and femoral bone mass of young adult women

Maximum bone mineral density of the femur was measured by dual-photon densitometry in 282 healthy white women, aged 20 to 40 years. Femoral sites included the neck, Wards triangle, and the trochanter. Quetelet Index was used as a measure of weight adjusted for height, and body composition was measured using four-point bioelectrical impedance and anthropometry. Maximum bone mass is believed to be an important measure if the level established which remain characteristic or predict bone mineral density during the aging process. Body weight was correlated with each measure of femoral bone density, including the femoral neck (r = .42), Wards triangle (r = .34), and the trochanter (r = .44). Weight was more highly correlated with bone mass than with other measures of body composition, including fat-free mass, percent body fat, humeral muscle area, and humeral fat area. We observed that age was negatively associated with bone mass at all three femoral sites, even in subjects within the age range of 20 to 40 years, and the relationship was significant after controlling for Quetelet Index. There was no evidence of a nonlinear relationship that would indicate when maximal femoral bone mass reaches its peak within this age range.

Distances between normal populations when covariance matrices are unequal

The definition of distance between two populations of equal covariance matrices is extended to two and more than two populations with unequal covariance matrices and Rao’s U test for testing the conditional contribution of a subset of variables to the distance is extended to this situation, even when sample sizes are not necessarily the same.

Classification procedures using principal component analysis and stepwise discriminant function

Classification procedures are examined in the case when the dimensionality exceeds the sample size. Two particular suggestions are (i) Principal components analysis and (ii) Two-step discriminant analysis. Comparisons are made in the two sample and the several sample cases. Extensions to growth curve model are investigated using the two stage discriminant analysis.

Terminating markov renewal processes

Several kinds of terminating Markov Renewal Processes are defined. Of interest in these processes are the time T until termination and the number of transitions NT until termination. For several kinds of terminating processes, the distribution and moments of T and NT are obtained along with their covariance. The distributions of associated cumulative processes are also considered. A Markov Renewal model is compared with results of Markov Chains used to model epidemics, and other examples are examined in compartmental modeling and competing risks.

Distribution of the biased hypothesis sum of squares in linear models with missing observations

In a linear model with missing observations, one can substitute algebraic quantities and then minimize the error sum of squares for the augmented model. This gives the correct error sum of squares. But this method does not produce the correct hypothesis sum of squares for testing a linear hypothesis about the parameters. The sum of squares obtained is biased but practitioners still use it. The distribution of this biased sum of squares is derived in this paper and the consequences of using this biased sum of squares on the type I and II errors is examined.

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The resu lts as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.

Uniformly better estimators with application in two-way designs

The problem of uniformly better estimators from p linear models is considered. Sufficient conditions are given for constructing an estimator from data in p samples which is uniformly (for all values of the unknown variances) better than the estimator from one of the samples. It is suggested the results be applied to recovery of inter-row and inter-column information in two-way designs.

Estimation of relative potency in multivariate parallel line bioassays

Finney (1978) has proposed using a composite response to estimate the relative potency of a test treatment with respect to a standard treatment, in the case of a multivariate parallel line bioassay. However, he assumes the coefficients in the composite response to be constants though they are estimated from the data itself. We provide two different methods of obtaining confidence intervals for this relative potency by using the exact variance. A method based on the use of delta technique is also proposed.

A note on optimum grouping and the relative discriminating power of qualitative to continuous normal variates

Ogawa (1951) considered the efficiency of estimation of the population mean from suitably chosen order statistics in large samples. Cox (1957) has considered the relative amount of information retained by grouping the normal curve. Cochran and Hopkins (1961) determined the discriminating power retained after partitioning normally distributed variates into qualitative ones in multivariate classification problems. And Connor (1972) discussed the asymptotic efficiencies of the test for the trend using m groups formed from a continuous variable. The same expression appears in all these investigations. This note throws some more light on the occurrence of the same expression in these seemingly unrelated problems.