Second order discriminant analysis in chemopreventive efficacy measurement

OBJECTIVE: To describe the use of second order discriminant analysis as a classification methodology along with the underlying assumptions and sampling requirements, with special emphasis on the use of this analysis in chemopreventive efficacy studies. STUDY DESIGN: The discriminant function score distributions derived in an analysis of 2 diagnostic groups may show such overlap that a statistically significant difference in mean values cannot be shown and, more important, that a useful case-based classification cannot be attained. By using the discriminant function score distributions from each case, it is frequently possible to derive a second order discriminant function based on case-specific characteristics, rather than characteristics of nuclei, thereby attaining improved case classification. RESULTS: Second order discriminant analysis has proven very useful in the documentation of case-level efficacy in chemopreventive trials. In a study of orally administered vitamin A, a first order discriminant analysis did not achieve a statistically significant difference in the score distributions for nuclei, but a second order discriminant analysis allowed a correct recognition of intervention effects in 85% of submitted cases. In a chemopreventive study of triamcinolone, a similarly inadequate discrimination based on discriminant function scores for nuclei resulted. After a second order discriminant analysis, a reduction in solar-actinic damage could be shown in 14/15, or 93%, of treated cases. CONCLUSION: Second order discriminant analysis can be highly effective when the discriminating information offered at the nuclear level is inadequate due to high dispersion and small differences in mean values of discriminant function scores for the diagnostic groups. Second order analysis utilizes case-specific characteristics of the discriminant function score distributions to document diagnostic group separation and/or efficacy of chemopreventive intervention by a reduction in case discriminant function scores.     

Testimation in Regression Parameter Estimation

Testimation is considered in the problem of estimation of regression parameters. The first stage sample is used to test a (null) hypothesis that specifies initial (preassumed) values for some of the regression parameters. Linear combination of the preassumed values and the ordinary least square (OLS) estimates is considered as the estimate if the data agree with the hypothesis. Otherwise, a second sample is taken and parameters are estimated only by using OLS, based on the combined sample. The procedure protects against type II error and against taking larger samples when inference can be made from a smaller sample.     

Quantifying nonlinear anisotropic elastic material properties of biological tissue by use of membrane inflation

Determination of material parameters for soft tissue frequently involves regression of material parameters for nonlinear, anisotropic constitutive models against experimental data from heterogeneous tests. Here, parameter estimation based on membrane inflation is considered. A four parameter nonlinear, anisotropic hyperelastic strain energy function was used to model the material, in which the parameters are cast in terms of key response features. The experiment was simulated using finite element (FE) analysis in order to predict the experimental measurements of pressure versus profile strain. Material parameter regression was automated using inverse FE analysis; parameter values were updated by use of both local and global techniques, and the ability of these techniques to efficiently converge to a best case was examined. This approach provides a framework in which additional experimental data, including surface strain measurements or local structural information, may be incorporated in order to quantify heterogeneous nonlinear material properties.     

Choice of Smoothing Parameters for Direct Kernels in Discrimination

Direct kernels, due to LAUDER (1983), as an alternative to the indirect kernel method in discriminant analysis are considered. It is shown that direct kernels may be based on any kernel function known in discrete density estimation. The choice of smoothing parameters is based on general loss functions and a family of loss functions which are specific for the discrimination problem is introduced. Examples with distance dependent and distance independent smoothing parameters are given to illustrate the applicability.     

Estimation of random regression parameters

The problem of the best linear unbiased estimation (BLUE) of random regression parameters is considered. It is proved that increasing informations about the mean value of the parameters both extend the class of estimable linear functionals and improve on the estimation. In all investigated cases the uniqueness of BLUE is proved. In the case of known mean values the BLUE is shown to be numerically equivalent with the MMSEE almost everywhere. A numerical example shows the improvements of BLUE due to increasing informations about the mean values of the parameters.     

Improved Predictions in Linear Regression Models with Stochastic Linear Constraints

In this article, we have considered two families of predictors for the simultaneous prediction of actual and average values of study variable in a linear regression model when a set of stochastic linear constraints binding the regression coefficients is available. These families arise from the method of mixed regression estimation. Performance properties of these families are analyzed when the objective is to predict values outside the sample and within the sample.     

Karyometry in rectal mucosa of patients with previous colorectal adenomas

OBJECTIVE: To determine whether karyometric measurements taken in biopsies from histologically normal-appearing rectal mucosa could serve as a biomarker for the risk of recurrence of polyps. MATERIALS AND METHODS: Biopsies were taken from the rectal mucosa of cases with a prior history of colonic polyps at the baseline of the study. In 57 cases recurrent polyps occurred (R cases); in 72 cases no recurrent disease was found at the end of the study (NR cases). From each biopsy 100 nuclei were recorded at high resolution. After segmentation, feature extraction and selection of a discriminating subset of features, a number of discriminant functions were derived. Also, measures of nuclear abnormality were computed. RESULTS: The differences in karyometricfeature values for nuclei from biopsies of cases with recurrent or nonrecurrent disease were very small and not notably expressed in the majority of nuclei. It was possible by focusing on nuclei showing clear deviations from normal to derive a discriminant function that exhibited a shift for the NR and R data sets. The distributions of discriminant function scores were then subjected to a second-order discriminant analysis to separate cases according to recurrence status. This function showed a statistically highly significant correlation with recurrence. At one extreme of its score distribution were 11 of 57 cases that had a recurrence, and at the other extreme were 8-10 of 72 cases that had no recurrence. The distributions of nuclear abnormality values for these subsets of cases were drastically different, with an average value of 1.72 for the group that may be at high risk for another recurrence and 1.02 for the group possibly at low risk. All cases with a prior history of colonic polyps showed a nuclear abnormality deviating from normal. CONCLUSION: Measurement of a sample of 100 nuclei from the rectal mucosa will suggest, for approximately 10% of cases, that a high risk for recurrence of adenomatous polyps exists and, for a slightly lower proportion, confirm that the nuclei deviate only slightly from those from individuals with no history of colonic polyps. For the majority of cases with a prior history of adenoma, the nuclei in the biopsy show a notable deviation from normal, but the deviation is practically the same for cases that had a recurrence and those that did not. However, a tentative discriminant function (DF I,3) derived from the characteristics of the extreme cases correctly classified approximately 64% of nonrecurrent and 83% of recurrent cases using a Bayesian decision boundary.     

A new method for dealing with measurement error in explanatory variables of regression models

We introduce a new method, moment reconstruction, of correcting for measurement error in covariates in regression models. The central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by "adjusted" values. In regression calibration the adjusted value is the expectation of the true value conditional on the measured value. In moment reconstruction the adjusted value is the variance-preserving empirical Bayes estimate of the true value conditional on the outcome variable. The adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved "true" covariate values. We show that moment reconstruction is equivalent to regression calibration in the case of linear regression, but leads to different results for logistic regression. For case-control studies with logistic regression and covariates that are normally distributed within cases and controls, we show that the resulting estimates of the regression coefficients are consistent. In simulations we demonstrate that for logistic regression, moment reconstruction carries less bias than regression calibration, and for case-control studies is superior in mean-square error to the standard regression calibration approach. Finally, we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables.     

Identification of components of fibroadenoma in cytology preparations using texture analysis: a morphometric study

S. Singh and R. Gupta Identification of components of fibroadenoma in cytology preparations using texture analysis: a morphometric study Objectives: To evaluate the utility of image analysis using textural parameters obtained from a co‐occurrence matrix in differentiating the three components of fibroadenoma of the breast, in fine needle aspirate smears. Methods: Sixty cases of histologically proven fibroadenoma were included in this study. Of these, 40 cases were used as a training set and 20 cases were taken as a test set for the discriminant analysis. Digital images were acquired from cytological preparations of all the cases and three components of fibroadenoma (namely, monolayered cell clusters, stromal fragments and background with bare nuclei) were selected for image analysis. A co‐occurrence matrix was generated and a texture parameter vector (sum mean, energy, entropy, contrast, cluster tendency and homogeneity) was calculated for each pixel. The percentage of pixels correctly classified to a component of fibroadenoma on discriminant analysis was noted. Results: The textural parameters, when considered in isolation, showed considerable overlap in their values of the three cytological components of fibroadenoma. However, the stepwise discriminant analysis revealed that all six textural parameters contributed significantly to the discriminant functions. Discriminant analysis using all the six parameters showed that the numbers of pixels correctly classified in training and tests sets were 96.7% and 93.0%, respectively. Conclusion: Textural analysis using a co‐occurrence matrix appears to be useful in differentiating the three cytological components of fibroadenoma. These results could further be utilized in developing algorithms for image segmentation and automated diagnosis, but need to be confirmed in further studies.     

Sexual dimorphism in human skulls. A comparison of sexual dimorphism in different populations

Application and comparison of sex discriminant functions in different populations led to the conclusion that a certain combination and weighting of a few sex dimorphism variables (in this study we only used craniometric variables) can give a good discrimination between male and female individuals, independent of the racial group to which this function is applied. In our study, the sex-discriminatory power of five discriminant functions which were based on different ordination and selection procedures (e.g. professional knowledge, stepwise discriminant analysis, literature) of the cranial variables is compared. These discriminant functions were applied to three different data sets, the first being skull measurements from an Amsterdam series (Europids), the second skull measurements of a Zulu series (Negrids) and the third skull measurements of a Japan series (Mongolids). Our decision as to whether a function is a good or less good sex-discriminating function is determined by the Dt values (these values give an idea about the discriminatory value of the discriminant function when applied to a new test sample), the number of variables necessary to obtain this Dt and the location of the sectioning point (i.e. comparison between the estimation of the sectioning point and the ”real” sectioning point). These discriminant functions were compared withGiles Elliot's (1962, 1963) “race-independent” sex function.     

Comparison of discriminant function and classification tree analyses for age classification of marmots

We evaluated the predictive power of two classification techniques, one parametric – discriminant function analysis (DFA) and the other non-parametric – classification and regression tree analysis (CART), in order to provide a non-subjective quantitative method of determining age class in Vancouver Island marmots ( Marmota vancouverensis ) and hoary marmots ( Marmota caligata ). For both techniques we used morphological measurements of known-age male and female marmots from two independent population studies to build and test predictive models of age class. Both techniques had high predictive power (69–86%) for both sexes and both species. Overall, the two methods performed identically with 81% correct classification. DFA was marginally better at discriminating among older more challenging age classes compared to CART. However, in our test samples, cases with missing values in any of the discriminant variables were deleted and hence unclassified by DFA, whereas CART used values from closely correlated variables to substitute for the missing values. Therefore, overall, CART performed better (CART 81% vs DFA 76%) because of its ability to classify incomplete cases. Correct classification rates were approximately 10% higher for hoary marmots than for Vancouver Island marmots, a result that could be attributed to different sets of morphological measurements. Zygomatic arch breadth measured in hoary marmots was the most important predictor of age class in both sexes using both classification techniques. We recommend that CART analysis be performed on data-sets with incomplete records and used as a variable screening tool prior to DFA on more complete data-sets.     

An association study using AFLP markers and application to a beef cattle breeding population

Using amplified fragment length polymorphism (AFLP) fingerprinting, selective genotyping was performed to determine if this method was effective for selecting superior breeding stock. Forty-eight cows with extreme genetic merit for beef marbling score (BMS) were selected from a population of Japanese Black cattle (n = 4462), including 25 with the highest for predicted breeding value (PBV) and 23 with the lowest. Sixteen AFLP fragments were selected for further analysis based on fragment frequency differences between the high and low groups. A linear discriminant analysis using these AFLP fragments was applied in order to derive a discriminant function that classified the cows into high and low groups. Seven of the 16 fragments were included in the resulting function and the discriminant scores (general genetic values, GGV) of the 48 cows were calculated using the function. These cows were clearly separated into high and low groups by GGV with a correlation ratio of 0.91 (discriminative error of 2.1%). The same function was then applied to 121 additional cows that were randomly selected from the original population. A significant regression coefficient of GGV on BMS-PBV (R2 = 0.45) was obtained, which indicates that the GGV can be used as a selection criterion for BMS in this population. These results suggest that AFLP fingerprinting can be used for animal breeding without identifying the underlying genes affecting the trait of interest.     

Generalized Walker-Duncan Procedure for Obtaining Maximum Likelihood Estimates of Parameters of Logistic Discriminant Function

A recursive method of obtaining the maximum likelihood estimates of the parameters of the quadratic logistic discriminant function is presented. This method is an extension of the Walker and Duncan procedure (1967) proposed for the linear logistic discriminant function in a dichotomous case. A generalization of the method to the problem of discrimination between several populations is also given in the paper. It works for both linear and quadratic logistic discriminant function. After an estimation of the parameters of the logistic function a classification can be performed. An example of application of the method to automatic diagnosis of some respiratory diseases is presented. Comparison with the standard procedures used for the estimation is done by a short simulation study.     

Calcaneal measurement in estimation of stature of South African blacks

Stature (height) is an important factor in establishing the identity of a person in the living as well as in the skeletonized state. When stature is estimated from the bones of the limbs, regression equations, which estimate the ratios of the lengths of bones to the height of the individual, are generated. The majority of bones that were used previously were the long bones. The calcaneus was used for estimating stature only in American whites and blacks (Holland [1995] Am. J. Phys. Anthropol. 96:315-320). The regression equations that he generated were found to be useful for stature estimation in these population groups. Since the calcaneus has not been used for the same purpose in South Africa, the aim of this study was to derive regression equations that will allow this bone to be used for stature estimation in South African blacks. In total, 116 complete skeletons (60 males and 56 females) were selected from the Raymond A. Dart Collection of Human Skeletons, School of Anatomical Sciences, University of the Witwatersrand (Johannesburg, South Africa). The skeletal heights of these sets of skeletons were calculated using the anatomical method of Fully ([1956] Ann. Med. Leg. 35:266-273). Nine parameters of the calcaneus were measured and matched against skeletal heights, using univariate and multivariate regression methods. Regression equations were obtained for estimation of the stature of the South African black population from the calcaneus. The standard error of estimate that was obtained with univariate regression analysis was higher than the corresponding values using multivariate regression analysis. In both cases, the standard errors of estimate compared well with the values obtained for fragmentary long bones by previous authors.     

On classifying observations when one population is a mixture of normals

Some simple models for the mixture of distribution problems are considered. Two possible alternatives to the rather complex optimal discriminant function rule are mentioned. The performance of the BC method is never as satisfactory as the QDF method and in some cases it is far worse than the QDF method. The QDF can be used whenever sufficient numbers of observations are available to provide reasonably good estimates of means and covariances.     

Local approximation and its applications in statistics

For the discrete and for the continuous case, the problem of evaluating the derivatives of a function f(x) in a given interval of x is solved by local approximation method. Examples of application of the resulting numerical procedures are quoted relating the estimation of smooth function and its derivative for measured values (of a growth process), internal regression, trend elimination of time series, Bernstein polynomial, and kernel estimation of a density function.     

Estimation of Population Regression Coefficient in Successive Sampling

An attempt has been made to derive the theory of successive sampling for estimation of regression coefficient. The situations considered are estimation of regression coefficient for current occasion, estimation of change of regression coefficients over two occasions and estimation of average of regression coefficients over two occasions. The expressions of optimum estimators along with their variances have been worked out. On comparing their efficiencies empirically, it has been observed that similar to the estimation of mean, successive sampling can also be used with advantage for estimation of regression coefficient.     

中国汉族锁骨的性别判定

本文通过对279副中国汉族成人锁骨的测量,其中男性241副,女性38副。使用性别判别分析的方法,建立了锁骨性别判别的分析式,使中国汉族锁骨性别判别的准确率达到了90－94．7％,该文提出的性别判别方法,是锁骨性别判别的准确可靠的方法。     

Exact Confidence Bounds for Comparing Two Regression Lines with a Control Regression Line on a Fixed Interval

The problem of finding exact simultaneous confidence bounds for comparing simple linear regression lines for two treatments with a simple linear regression line for the control over a fixed interval is considered. The assumption that errors are iid normal random is considered. It is assumed that the design matrices for the two treatments are equal and the design matrix for the control has the same number of copies of each distinct row of the design matrix for the treatments. The method is based on a pivotal quantity that can be expressed as a function of four t variables. The probability point depends on the size of an angle associated with the interval. We present probability points for various sample sizes and angles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)