Normal Distribution Theory Research Articles

Power and sample size in cost-effectiveness analysis.

For resource allocation under a constrained budget, optimal decision rules for mutually exclusive programs require that the treatment with the highest incremental cost-effectiveness ratio (ICER) below a willingness-to-pay (WTP) criterion be funded. This is equivalent to determining the treatment with the smallest net health cost. The designer of a cost-effectiveness study needs to select a sample size so that the power to reject the null hypothesis, the equality of the net health costs of two treatments, is high. A recently published formula derived under normal distribution theory overstates sample-size requirements. Using net health costs, the authors present simple methods for power analysis based on conventional normal and on nonparametric statistical theory.

G-cell density in the antral mucosa: a feasibility study.

In this work a new method for gastrin-secreting cell (G-cell) density measurement was employed in order to test the hypothesis that G-cell density of the antral mucosa can be estimated with an acceptable degree of error. The zone of antral mucosa where most G-cells are located was demarcated, its area was measured and the G-cells within this area were counted. The variation in the e error of the estimate according to normal distribution theory, and the size of the sampling expected to yield G-cell density values with a given error were assessed. Our data indicate that a very large surface of the G-cell zone should be explored in order to estimate G-cell density with an acceptable degree of error. G-cell density measurement appears not to be feasible on biopsy specimens or, for routine diagnosis, on gastrectomy specimens.

Preliminary report on an approach to testing for normality

The report is the introductory step to our investigation on joint probability distributions of crustal movements and geological parameters (basement depth, heat flow, seismicity and gravity anomalies) which will be carried out in the near future. Because of the remarkable role of the normal distribution, the working out of procedures testing for normality has been a continuing research subject of statisticians. Recently, Shapiro and Wilk (1965) and Shapiro et al. (1968) proposed a new test statistic and carried out a Monte Carlo comparative study with the known test statistics, and the result indicates superiority of their procedure from the test power point of view. The present report considers an approach to testing for normality which is related to the Shapiro-Wilk procedure. Also, the closed form of the test statistic is given. A brief historical outline of both the normal distribution theory and testing procedures is presented, making it of more methodological value, since the normal distribution is often involved in geodesy.

Efficacy of robust analysis of variance for the interpretation of data from collaborative trials

The Analytical Methods Committee (AMC) robust analysis of variance method has been compared with the Harmonized Protocol for the interpretation of data from collaborative trials. Many data sets, representing the extremes of concentration and precision, have been selected from published studies and subjected to the two methods for calculating estimates of the mean and the variances of repeatability and reproducibility. In addition the variabilities of the statistics have been investigated by the bootstrap method. The conclusion is that the AMC robust procedure is considerably more reliable than the Harmonized Protocol. Although the two methods give quite close results generally, the variability of the estimates produced by the Harmonized Protocol is considerably greater because of the masking of outliers. Even the robust method gives estimates that are more variable than expectations based on normal distribution theory.

A probability nomogram to predict rectal temperature in children.

The relationship between rectal and peripheral-site temperature was investigated to achieve two objectives: 1) to evaluate a prevailing practice of intersite adjustment by use of an invariant temperature difference; and 2) to develop a statistical method for intersite temperature predictions in the individual child, especially for fever as defined by rectal measurement. Rectal, oral, axillary, left abdomen skin, and forehead skin temperatures (degrees F) were measured with an electronic thermometer in 257 children. Objective 1 was not achieved because a simple temperature difference between a peripheral site and the rectal site could not be used to predict rectal temperature reliably. For objective 2, intersite differences met three statistical criteria so that normal distribution theory could be used to derive the probabilities for occurrence of each difference. Accordingly, cumulative probability nomograms were constructed to estimate rectal-site fever from measurements at peripheral sites. This nomogram method produces a clinically reliable prediction of rectal-site fever from temperature measurement at peripheral sites, especially the oral and axillary sites. These predictions offer useful assessments of febrile illness severity when rectal temperature is not available.

Comparing Averaged-Out Utilities of Probability Trees Having Random Parameters

This paper develops a procedure to approximate the marginal distributions of each of the following performance measures defined on a forest of probability trees with random parameters: (a) the averaged-out utility of each individual tree in the forest, and (b) the difference between the averaged-out utilities of each pair of trees in the forest. The parameters of a probability tree are its branching probabilities and node utilities, and they may be specified as mutually independent random variables with given distributions; moreover, any of these parameters may be duplicated at several places in the forest to represent parameter dependencies within and between trees. The approximation procedure incorporates an efficient,numerically robust algorithm for computing up to the first four moments of the selected performance measure. The desired distribution is approximated by an Edgeworth expansion based on the computed moments. With respect to accuracy and computational complexity, the approximation procedure is compared to estimation techniques based on simulation and normal distribution theory. To illustrate how the analysis of probability trees can directly incorporate uncertainty or random variation in the values of tree parameters, the approximation procedure is applied to the comparison of alternative medical protocols for managing patients following myocardial infarction (heart attack). In addition to its applications in medical decision making, the procedure developed in this paper should also have applications in risk analysis, reliability, project planning, marketing, and other fields in which probability trees are used to model various processes of interest.

A New Concept of the Central Point of Intra-urban Population Distribution

This paper presents a new measure of the central point of population using two-dimensional normal distribution theory. When a previously presented measure, for example, the center of population (equation (1)) is calculated for one city, its value does not necessarily indicate a singular point for that city. This is because previous measures depend on how an observer delimits the city area. The center of concentric contour circles of population density, however, should indicate the singular point for the city, if its population forms a normal distribution. The position of this center can be estimated from the two-dimensional normal distribution type of model (equation (3)), since a pair of variables (xc, yc) in the model are identified as the coordinates of the center. Thus, this paper defines a new measure as a pair of the least square estimates (xc, yc) and calls this measure the singular point of urban population. The estimates xc and yc are part of (dc, a, xc, yc) which minimizes the error sum of squares S in equation (5). The variable di in (5) is an observed value of the population density at place (xi, yi).Although (xc, yc) in (3) is equivalent to the mean value in a two-dimensional normal distribution, Figure 1 shows that (xc, yc) in (5) is different from the mean value in the sample distribution. Accordingly, the singular point of urban population is not a concept included in “ordinary” statistics, but a concept peculiar to spatial science.The iterative method is necessary for estimating the point, since (3) is a nonlinear regression equation. This method, however, is more complicated than the algebraic method. Thus, this paper introduces the following two methods to obtain an approximate algebraic solution. One is to derive equation (7) from (3) using logarithmic linearization, and then to obtain an approximate value from (7) using linear regression analysis. This value is given by equations (12) and (13), where f=logd; z=x2+y2; Sfx=Σ(fi-f)(xi-x); and Cij is a cofactor of the square matrix in equation (8). The other is to obtain a more approximate value from (7) using weighted regression analysis. This value is given by equations (15) and (16), where Tdfx=ΣdiΣdifixi-ΣdifiΣdixi; and C'ij is a cofactor of the square matrix in equation (14). In this paper, the above three kinds of values calculated by nonlinear, linear, and weighted regression analysis are called the nonlinear solution, the linear solution, and the weighted solution respectively.These solutions and the center of population are applied to the following areas: two hypothetical cities where population forms a concentric distribution (city A) or an elliptic distribution (city B); and three actual cities in 1970, 1975, and 1980 (Utsunomiya, Koriyama, and Yamagata). Figures 2 and 3 show the limits of these cities. Hypothetical cities A and B are delimited in two and four ways respectively, so that the city limits of A0, A1, B0, B1, B2, and B3 are established; the limits of the actual cities are established within each circle centered at the central station. In the hypothetical cities, the true value of the singular point of urban population is determined at the origin (0.0, 0.0); nevertheless in the actual cities

Exposure-response relationship for a dichotomized response when the continuous underlying variable is not measured.

Radiological assessment of pneumoconiosis is an example of a dichotomized variable, namely one that is analysed as a binary response but in fact has an underlying continuum, which in this case is not measurable. Estimates of exposure-response relationships vary greatly for different observers of a dichotomized response variable because of random error of measurement and differences in the threshold implicitly chosen by each observer for categorizing cases. We present a method of using the biserial correlation coefficient and normal distribution theory to estimate exposure-response relationships at any required threshold for each observer. Exposure-response relationships can also be corrected for random observational error using the reliability coefficient, calculated as the tetrachoric correlation between repeat observations by readers.

A two sample ordered alternative test for means and variances

A two sample test of likelihood ratio type is proposed, assuming normal distribution theory, for testing the hypothesis that two samples come from identical normal populations versus the alternative that the populations are normal but vary in mean value and variance with one population having a smaller mean and smaller variance than the other. The small sample and large sample distribution of the proposed statistic are derived assuming normality. Some computations are presented which show the speed of convergence of small sample critical values to their asymptotic counterparts. Comparisons of local power of the proposed test are made with several potential competing tests. Asymptotics for the test statistic are derived when underlying distributions are not necessarily normal.

Symmetry, 0-1 Matrices and Jacobians: A Review

In this paper we bring together those properties of the Kronecker product, the vec operator, and 0-1 matrices which in our view are of interest to researchers and students in econometrics and statistics. The treatment of Kronecker products and the vec operator is fairly exhaustive; the treatment of 0–1 matrices is selective. In particular we study the “commutation” matrix K (defined implicitly by K vec A = vec A′ for any matrix A of the appropriate order), the idempotent matrix N = ½ (I + K), which plays a central role in normal distribution theory, and the “duplication” matrix D, which arises in the context of symmetry. We present an easy and elegant way (via differentials) to evaluate Jacobian matrices (first derivatives), Hessian matrices (second derivatives), and Jacobian determinants, even if symmetric matrix arguments are involved. Finally we deal with the computation of information matrices in situations where positive definite matrices are arguments of the likelihood function.

Discussion of a Comparison of Event Study Methodologies Using Daily Stock Returns: A Simulation Approach

The authors chose a simulation approach using actual returns data to address issues commonly encountered in studies. The question was raised about whether this approach was any more or less appropriate than an analytical approach for addressing the issues of interest. It was suggested that many of the conclusions reached in the paper could be derived analytically. Such an approach could begin with an analysis based on normal distribution theory and be extended to incorporate some of the nonnormal properties of actual returns data. An analytic approach might reinforce and extend the present analysis. For example, the issue of event-date uncertainty could be assessed using various alternatives to the uniform probability distribution for specifying event occurrence. A concern was raised that some of the conclusions of the paper may require further qualification or clarification. The authors refer to portfolio size considerations, but those issues should be referred to as size considerations because the authors do not compare different levels of aggregation for a given sample size. The issue that was addressed in the paper was the increase in power resulting from increased sample size, not how best to aggregate. Conclusions regarding the insensitivity of adjustments for the nonsynchronous trading problem are subject to a sample bias which tended to exclude thinly traded securities. Concerns were also raised regarding the appropriateness of the proxies for trading frequency (i.e., trading volume and exchange listing). The paper stated (but did not investigate) that the results were equally applicable for either good news or bad news events. This statement

Discussion of An Application of the Bootstrap Method to the Analysis of Squared, Standardized Market Model Prediction Errors

This is a synthesis of the major points raised in the discussion of An Application of the Bootstrap Method to the Analysis of Squared, Standardized Market Model Prediction Errors, referred to below as Marais [1984]. The discussion at the conference can be organized into three general areas: (1) motivation for the bootstrap approach and its relative advantages (and disadvantages) compared to other statistical approaches; (2) alternative approaches to the problem of bias toward rejection of the null hypothesis which arises for normal distribution theory tests using daily and weekly security returns; (3) comments about this application of the bootstrap. Each of these areas is summarized below. Since the published version of Marais [1984] has been substantially revised in the light of comments made at the conference, it is important to bear in mind that many of the comments applied more directly to the version presented at the conference. Unless otherwise indicated, all equation and section numbers refer to Marais [1984].

An Application of the Bootstrap Method to the Analysis of Squared, Standardized Market Model Prediction Errors

This paper examines significance tests on squared, standardized market model prediction errors as performed in some information content studies. It makes two principal points. First, the normal distribution theory for these statistics is biased against the null hypothesis of no information effect when applied to leptokurtic market data. Second, the bias can be corrected using the bootstrap method,' although not in the most obvious way. Patell's [1976] earnings forecast data set is used as an illustration. In section 2, I consider the effect of departures from normality on the distribution theory for standardized market model prediction errors. I show that for large estimation samples and leptokurtic disturbances the normal theory formula for the variance of squared prediction errors underestimates the true variance. This result is the focus of the paper, because it identifies the specific type of departure from normality that affects inferences based on squared prediction errors. Section 3 describes a simulation experiment which demonstrates the empirical relevance of the bias in the normal theory formula for sample

Introduction to Multivariate Analysis

Part One. Multivariate distributions. Preliminary data analysis. Part Two: Finding new underlying variables. Principal component analysis. Factor analysis. Part Three: Procedures based on the multivariate normal distribution. The multivariate normal distribution. Procedures based on normal distribution theory. The multivariate analysis of variance. The multivariate analysis of covariance and related topics. Part Four: Multi-dimensional scaling and cluster analysis. Multi-dimensional scaling. Cluster analysis.

REGIONAL DETECTION OF CHANGE IN WATER QUALITY VARIABLES1

ABSTRACT: The detection of change in a hydrologic varaible, particularly water quality, is a current problem. A method is presented for testing whether there has been a shift in the mean of a hydrologic variable based on the well established bivariate normal distribution theory. In this technique, the dependent, or target, and the independent, or control, variables are formed as weighted linear combinations of the mean values at a number of locations in a selected target and control area. The weighting factors are determined based on a mathematical programming technique which minimizes the conditional coefficient of variation thereby minimizing the number of observations required to detect a change of a preselected magnitude in the mean of the target area. The result is a situation where a savings in the number of observations required to detect a change is a consequence of adding more stations: the space‐time tradeoff. Two applications of the technique are presented, the first using electrical conductivity (EC) data from two sets of river basins and the second using EC data from a set of basins as the target variable and annual discharge as the control. The results indicate that a significant savings in time can be achieved by using this method.

Ridge Analysis Following a Preliminary Test of the Shrunken Hypothesis

RIDGE ANALYSIS is of interest when some generalized Ridge regression coefficient vectors are significantly more likely to have Mean Squared Error (MSE) optimality properties than is any uniformly SHRUNKEN version of the ordinary least squares estimator. The normal distribution theory likelihood ratio statistic for the corresponding composite hypothesis is derived, and a small sample F-test is shown to be conservative. The asymptotic distribution of the likelihood ratio provides a large sample test of the RISE optimality of any restricted Ridge FAMILY of solutions. The likelihood approach for solution selection WITHIN a given family is then compared and contrasted with some snggestions of Mallows [8] and Allen [1] and with a new, non-stochastic criterion, SSCBC.

Ridge Analysis Following a Preliminary Test of the Shrunken Hypothesis

RIDGE ANALYSIS is of interest when some generalized Ridge regression coefficient vectors are significantly more likely to have Mean Squared Error (MSE) optimality properties than is any uniformly SHRUNKEN version of the ordinary least squares estimator. The normal distribution theory likelihood ratio statistic for the corresponding composite hypothesis is derived, and a small sample F-test is shown to be conservative. The asymptotic distribution of the likelihood ratio provides a large sample test of the RISE optimality of any restricted Ridge FAMILY of solutions. The likelihood approach for solution selection WITHIN a given family is then compared and contrasted with some snggestions of Mallows [8] and Allen [1] and with a new, non-stochastic criterion, SSCBC.

Quadratic logistic discrimination

SUMMARY Previous papers on logistic discrimination have only considered linear discriminant functions, although there are situations where quadratic discriminators are important and useful. The full quadratic logistic discriminant approach has too many parameters to be estimated if the dimensionality, p, is not small, say p > 4. An approximation is suggested here which gives a quadratic term in the discriminant function but with a greatly reduced number of parameters. The method is applied to the data published by Smith (1947) and compared with his results obtained using normal distribution theory. Other approximations are suggested in some special cases.

Statistical Analysis Of Density Tests

The principles of statistical analysis are explained and used to make a comparison of the relative effectiveness of different testing and compaction methods for various types of subgrade and base course materials for roadway work. The nuclear and conventional density testing methods are compared; the compaction obtained by sheepsfoot rollers, pneumatic-tired construction traffic, and hand operated vibratory-impact equipment is analyzed and compared statistically for volcanic tuff subgrades and variable soil type subgrades. In the light of the normal distribution theory, the minimum density requirement found in most specifications is seen to be unrealistic. Statistical analysis of earthwork can also be used as a tool in keeping construction control charts to achieve better product or quality control.

The Effect of Truncation on Tests of Hypotheses for Normal Populations

Loss of hypothesis testing power for truncated normal populations when complete normal distribution theory is used