Restricted Procedure Research Articles

A Simplified Algorithm for the W Transformation in Variance Component Estimation

The W transformation is needed at each step in the maximum likelihood or restricted maximum likelihood procedure for estimation of the parameters of the mixed A.O.V. model. This paper develops an efficient algorithm for computing the W transformation needing only about a dozen lines of Fortran or PL/I code.

A Simplified Algorithm for the W Transformation in Variance Component Estimation

The W transformation is needed at each step in the maximum likelihood or restricted maximum likelihood procedure for estimation of the parameters of the mixed A.O.V. model. This paper develops an efficient algorithm for computing the W transformation needing only about a dozen lines of Fortran or PL/I code.

The analysis of sequential clinical trials

SUMMARY The conduct of a sequential clinical trial is considered in two distinct stages, the planning of the trial and the analysis of the results. Tests based on straight line stopping boundaries are considered. The planning stage is described in terms of existing theory and results are presented which allow significance levels to be quoted and interval estimates to be given at the analysis stage. Tables for use in the calculation of significance levels are included. The accuracy of large sample results is assessed by means of computer simulation. We consider the comparison of two treatments using sequential tests based on straight line stopping boundaries. Two special cases which have optimal properties are described in ? 2. The results of traditional sequential analysis will be used only in the planning of the trial. The analysis of the data, including the terminal sample size as part of the data, will be viewed as a separate process, involving the ordinary tools of nonsequential statistics. Results on significance levels and interval estimation are presented in ? 3. Some similar results have been given for certain restricted procedures by Armitage (1960) and for repeated significance tests by Siegmund (1978). Repeated significance tests themselves are described in the second (1975) edition of Armitage's book. In ? 4 the problems of overshoot and overrunning are discussed. A trial will be said to overrun if some responses are observed after formal termination. If the state of the trial is not updated continuously, but at regular intervals such as once a week or once a month, some overrunning is likely. It is even more likely if patient response takes some time to manifest itself, so that when termination occurs some patients have already been treated but have not yet responded. Overrunning can be ignored in the planning of a trial, but all the data, including the 'extra' observations, can be included in the analysis. Previous work on

Estimating the functional form of travel demand models

This paper shows how many particular models found in the travel demand literature, such as the linear, multiplicative, logit forms and their formulation in terms of first differences, can de derived as special cases of the specification of the fixed and stochastic parts of more general models. Estimates of the appropriate functional form of the two equations of a cross-sectional inter-city travel demand model and of the two equations of a time-series urban travel demand model are presented. It is demonstrated, using current maximum likelihood techniques extended to take multiple-order auto-correlations into account, that estimating the functional form dominates more restricted procedures on grounds both of the statistical significance and of the economic reasonableness of the results. Notably, a standard logit market share equation and log-linear total demand equations appear to be clearly inferior to less specifications of these equations.

Mathematical programming methods for deformation analysis at plastic collapse

A variety of alternative mathematical programming procedures for first-order elastoplastic analysis at the collapse-load level for discrete structures described by piecewise-linear elastic-perfectly plastic constitutive laws are formulated and then assessed with respect to their relative computational merit. An iterative Linear Programming, three Quadratic Programming and two Restricted Basis Linear Programming procedures are developed in detail. In addition, Parametric Linear Complementarity and Parametric Quadratic Programming methods are briefly discussed. The governing relations for analysis and the various formulations are initially developed for a trusslike structure, and are then shown to apply, without formal alteration, to a broad range of frameworks and continua discretized into finite elements. The unbounded nature of deformations at plastic collapse of elastic-perfectly plastic structures is considered, as is the possibility that there may be a multiplicity of (bounded) deformation responses to a single load path prior to plastic collapse, and several devices to overcome related difficulties are illustrated. Two example structures are each analyzed by six of the procedures presented.

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.

The performance of dogs in detecting ?-ionone in the vapor phase

Concentration-response functions forα-ionone were established for four German shepherds (Figs. 3 and 4). The dogs were tested in a three choice behavioral apparatus (Fig. 2), and trained, by a restricted operant procedure, to establish an approach-avoidance discrimination between odor and air. Odorant concentrations were presented from an olfactometer calibrated by gas chromatography. Minimum detectable concentrations fell within the range 4.0×104.5–4.0×106.5 molecules/cm3. Concentration-response curves for three of the dogs show a clear double reversal in slope which is statistically significant and which divides the curve into a slowly descending upper limb, best fitted by a parabolic function, and a rapidly descending lower limb, best fitted by a cubic function (Figs. 5 and 6). There is evidence that this division might reflect a dual receptor mechanism. When performance during testing stabilized for a given concentration and the concentration was then lowered by a certain magnitude, the new performance level depended markedly on that magnitude. No evidence was found to suggest that adaptation to the test odor influenced performance. As defined by a differential threshold fraction (ΔFlow/Flow) one dog's ability to discriminate between differences in flow rate fell in the range 0.12–0.08.

Latencies of Sexual and Asexual Responses to Double-Entendre Words as a Function of Sex-Guilt and Social Desirability in College Females

A restricted association procedure was used to study females' latencies associated with sexual and asexual responses to double-entendre sexual words. Latencies were also examined relative to two personality variables, sex-guilt and social desirability. The results indicated that sexual responses were accompanied by longer latencies than asexual responses. There was also an interaction between sex-guilt and sexual-asexual response latencies, with subjects of high sex-guilt showing longer latencies of sexual response but not of asexual response. Contrary to prediction, individual differences on the social desirability variable were unrelated to latencies of either sexual or asexual response. The study also outlined a stimulus-encoding model for associations to double-entendre sexual words.

On Exact Multicollinearity and the Estimation of the Cobb‐Douglas Production Function

AbstractThe assumptions of the economic model underlying the traditional Cobb‐Douglas production function analysis imply that exact multicollinearity should exist among the inputs. Restricted estimation procedures may lead to parameter estimates with additional economic content. Conventional methods of input aggregation and model specification may result in biased or misinterpreted estimates.

A Restricted Sequential Test

Summary A restricted sequential procedure is presented for testing the hypothesis θ = θ0, about an unknown parameter θ, against compound alternatives. It is based upon a corresponding procedure for testing Brownian motion drift, and approximations to OC and ASN functions, and confidence intervals, are obtainable from the Brownian motion case. The procedure is applicable to many common families of distributions indexed by θ.

Interference Between Gravity Wells—Steady State Flowa

ABSTRACTUnder steady state conditions of flow, the seepage toward a single gravity well is governed by the Laplace Equation which may be written in terms of either the hydraulic head, the pressure head or the velocity potential. Although this equation is linear, the principle of superposition cannot be applied to sum up the individual effects in the case of a multiple gravity well system due to the variation of the flow domain under the effect of one or several wells. A method is presented allowing the use of the superposition principle in a restricted form. The superposition of the decrements of the base pressure heads than the initial heads before pumping is valid. Also the decrements in the areas of the pressure head diagrams across specific vertical sections than the original areas can be summed up together.The limitations of Dupuit's well formula are explained. The validity of that formula has been proven on the basis of the analysis of the hydraulic forces within the flow medium, an approach which is different than that given by Hantush and Charney. Furthermore, the derived equation is written in terms of the areas of the pressure head diagrams across vertical sections and termed as the Unified Well Formula because it has been proven that the same formula is also valid for artesian wells.The analysis of the hydraulic forces leads to the development of an equation for the free surface. This equation is then solved numerically in one iterative cycle. Due to the lack of simple available solutions, only one case, previously solved by the relaxation techniques, is compared with the presented method. The maximum percent difference in the depth of saturation within 82% of the flow region does not exceed 3.2% whereas in the remainder 18% of the flow region around the well, the percent difference varies between 2.63% to 4.67%. Even these differences do not really indicate actual errors due to the approximation implied in the relaxation method itself using a coarser grid.This distribution of the hydraulic head across a vertical section is assumed parabolic. Although Polubarinova‐Kochina presented a mathematical proof which leads to the same conclusion, yet for the reasons explained in the text, the writer preferred to use this type of distribution as a valid physical assumption.The results of the analysis of each single well are applied to determine the pattern of the interference between several gravity wells. By means of the presented approach, the resultant values of the depths of saturation can be obtained on the basis of the explained restricted procedure, of superposition. The hydraulic potential distribution within the flow medium of a multiple gravity well system can also be obtained. The assumption of the parabolic hydraulic head distribution is maintained in analyzing a group of wells. It is recommended to establish a proper computer program covering a grid system that encompasses all the wells and their individual influence regions in a certain well field.In the entire analysis, Dupuit's assumptions are eliminated. However, the two main assumptions in the given analysis are: (a) the parabolic distribution of the hydraulic head across a vertical section within the flow medium, and (b) the elimination of the circumferential velocities. These two introduced assumptions are ‐i in the writer's opinion – practically valid.

Probabilistic reinforcement of reversal and dimensional shifts

Thirty subjects learned 120 two-dimensional, 2-valued concept-identification (CI) problems over 5 days. After each problem was learned, the solution was shifted by interchanging responses (reversal shift, RS) or by picking a new dimension (dimensional shift, DS). Two groups with complementary probabilities of the type of shift were run. The probabilities of an RS for one group on the 5 days were: 1, 0, .7, .7, .7. The shift was so introduced that the subject's first response indicated the type of shift expected. The subjects learned to predict the shift and showed a tendency to follow the probability of the shift but with a bias toward the RS prediction. Negative recency and alternation tendencies occurred, but seemed to adapt out. The subjects improved greatly in trials to solution for each CI problem. A generalization of the cue-selection model designed to handle initially correct concept selection did not adequately describe the statistic of the trial of the last error, but a restricted randomization procedure accounted for the major deviations. The error distributions were more accurately described. The parameter which measured the initial probability of selecting the correct concept on the first trial also indicated that subjects approached probability matching in their expectations of the type of shift.

Short assay periods and restricted feeding procedures in caries assays with cotton rats

Three varieties of rice hulls were assayed under equivalent conditions by either long, 98-day assays or by short, 28-day assays. In the long assays, 18-day-old weanlings were used, and caries scores were evaluated in all three molar teeth. In the short assays, 12-day-old weanlings were used, and caries was observed in first and second molars. A mixture of three rice hull varieties was tested in two 18-day assays under the same conditions as the 28-day assays. All assays readily detected anticariogenicity in the hull samples. It was concluded that for detecting rice hull activity, assays longer than 18 days and inclusion of third molar scores were unnecessary. The usefulness of assays employing only short preliminary feeding of rice hulls was also studied. An initial, 3-day feeding of hulls did not influence caries scores measured 15 days later at 30 days of age, but 18 days of hulls significantly decreased scores at 30 days and for as long as 30 days after feeding hulls was discontinued. These studies confirmed the advantages of such assays. They allow a substance to be evaluated for its ability to exert prolonged protection. If extended to 50 days of age or longer, they permit an evaluation of anticariogenic effects on unerupted teeth. The assay periods may be short if the agent being tested is sufficiently potent to protect after a few days of feeding. Sensitivity to short feedings can be increased by controlling dietary sucrose. Evidence reviewed indicated that the short assays and the restricted feeding procedures described would be useful for evaluating agents other than rice hulls and that the results of the assays would be of significance in caries studies other than those using cotton rats.

Closed sequential t-tests

Schneiderman & Armitage (1962) described a family of sequential procedures, called 'wedge plans', which could be used for testing the mean ,.t of a normal population with known variance CJ2. For tests of a null hypothesis Ho (that ,.t = 0) against two alternatives H1 (that It = ,t1) and H_1 (that It = -,tl), the boundaries favouring H1 and H_1 (i.e. for rejecting Ho) are the same as in the procedure of Sobel & Wald (1949), but the acceptance boundary (favouring Ho) is in general curved, and the boundaries are closedwith a maximum sample size N'. For one extreme member of the family (with N' = cx) the acceptance boundary coincides with that of the two-sided open sequential test of Wald (1947, Chapter 9); the other extreme is the 'restricted procedure' of Armitage (1957), where the acceptance boundary provides for termination at a fixed sample size N1 if neither rejection boundary has previously been crossed. These wedge plans, which derive their name from the shape of the acceptance boundary, thus provide a bridge between the open plans derived from Wald's theory and the restricted plans of Armitage (1957). The purpose of this note is to derive, from these results, some approximate procedures for use when o-2 is unknown. At one extreme one may use the open sequential t-test derived by Arnold & Goldberg, in the National Bureau of Standards (1951) tables, NBS-AMS-7. (These tables are unfortunately out of print; although some of the tabulated values can be derived from other tables (e.g. Resnikoff & Lieberman, 1957, and Slater, 1960), the tables in NBS-AMS-7 remain the most convenient source of reference for this test.) The other extreme, a restricted sequential t-test, has not previously been considered in any detail, and is discussed below. Between these extremes is a family of closed procedures, analogous to the family of wedge plans available when o-2 is known, and referred to above.

Restricted Sequential Procedures

Restricted Sequential Procedures