Showing papers on "Linear model published in 1979"

The robust beauty of improper linear models in decision making.

Abstract: Proper linear models are those in which predictor variables are given weights in such a way that the resulting linear composite optimally predicts some criterion of interest; examples of proper linear models are standard regression analysis, discriminant function analysis, and ridge regression analysis. Research summarized in Paul Meehl's book on clinical versus statistical prediction—and a plethora of research stimulated in part by that book—all indicates that when a numerical criterion variable (e.g., graduate grade point average) is to be predicted from numerical predictor variables, proper linear models outperform clinical intuition. Improper linear models are those in which the weights of the predictor variables are obtained by some nonoptimal method; for example, they may be obtained on the basis of intuition, derived from simulating a clinical judge's predictions, or set to be equal. This article presents evidence that even such improper linear models are superior to clinical intuition when predicting a numerical criterion from numerical predictors. In fact, unit (i.e., equal) weighting is quite robust for making such predictions. The article discusses, in some detail, the application of unit weights to decide what bullet the Denver Police Department should use. Finally, the article considers commonly raised technical, psychological, and ethical resistances to using linear models to make important social decisions and presents arguments that could weaken these resistances. Paul MeehPs (1954) book Clinical Versus Statistical Prediction: A Theoretical Analysis and a Review of the Evidence appeared 25 years ago. It reviewed studies indicating that the prediction of numerical criterion variables of psychological interest (e.g., faculty ratings of graduate students who had just obtained a PhD) from numerical predictor variables (e.g., scores on the Graduate Record Examination, grade point averages, ratings of letters of recommendation) is better done by a proper linear model than by the clinical intuition of people presumably skilled in such prediction. The point of this article is to review evidence that even improper linear models may be superior to clinical predictions. Vol. 34, No. 7,571-582 A proper linear model is one in which the weights given to the predictor variables are chosen in such a way as to optimize the relationship between the prediction and the criterion. Simple regression analysis is the most common example of a proper linear model; the predictor variables are weighted in such a way as to maximize the correlation between the subsequent weighted composite and the actual criterion. Discriminant function analysis is another example of a proper linear model; weights are given to the predictor variables in such a way that the resulting linear composites maximize the discrepancy between two or more groups. Ridge regression analysis, another example (Darlington, 1978; Marquardt & Snee, 1975), attempts to assign weights in such a way that the linear composites correlate maximally with the criterion of interest in a new set of data. Thus, there are many types of proper linear models and they have been used in a variety of contexts. One example (Dawes, 1971) was presented in this Journal; it involved the prediction of faculty ratings of graduate students. All graduWork on this article was started at the University of Oregon and Decision Research, Inc., Eugene, Oregon; it was completed while I was a James McKeen Cattell Sabbatical Fellow at the Psychology Department at the University of Michigan and at the Research Center for Group Dynamics at the Institute for Social Research there, I thank all these institutions for their assistance, and I especially thank my friends at them who helped. This article is based in part on invited talks given at the American Psychological Association (August 1977), the University of Washington (February 1978), the Aachen Technological Institute (June 1978), the University of Groeningen (June 1978), the University of Amsterdam (June 1978), the Institute for Social Research at the University of Michigan (September 1978), Miami University, Oxford, Ohio (November 1978), and the University of Chicago School of Business (January 1979). I received valuable feedback from most of the audiences. Requests for reprints should be sent to Robyn M. Dawes, Department of Psychology, University of Oregon, Eugene, Oregon 97403. AMERICAN PSYCHOLOGIST • JULY 1979 • 571 Copyright 1979 by the American Psychological Association, Inc. 0003-066X/79/3407-0571$00.75 ate students at the University of Oregon's Psychology Department who had been admitted between the fall of 1964 and the fall of 1967—and who had not dropped out of the program for nonacademic reasons (e.g., psychosis or marriage)— were rated by the faculty in the spring of 1969; faculty members rated only students whom they felt comfortable rating. The following rating scale was used: S, outstanding; 4, above average; 3, average; 2, below average; 1, dropped out of the program in academic difficulty. Such overall ratings constitute a psychologically interesting criterion because the subjective impressions of faculty members are the main determinants of the job (if any) a student obtains after leaving graduate school. A total of 111 students were in the sample; the number of faculty members rating each of these students ranged from 1 to 20, with the mean number being 5.67 and the median being 5. The ratings were reliable. (To determine the reliability, the ratings were subjected to a oneway analysis of variance in which each student being rated was regarded as a treatment. The resulting between-treatments variance ratio (»j) was .67, and it was significant beyond the .001 level.) These faculty ratings were predicted from a proper linear model based on the student's Graduate Record Examination (GRE) score, the student's undergraduate grade point average (GPA), and a measure of the selectivity of the student's undergraduate institution. The cross-validated multiple correlation between the faculty ratings and predictor variables was .38. Congruent with Meehl's results, the correlation of these latter faculty ratings with the average rating of the people on the admissions committee who selected the students was .19; 2 that is, it accounted for one fourth as much variance. This example is typical of those found in psychological research in this area in that (a) the correlation with the model's predictions is higher than the correlation with clinical prediction, but (b) both correlations are low. These characteristics often lead psychologists to interpret the findings as meaning that while the low correlation of the model indicates that linear modeling is deficient as a method, the even lower correlation of the judges indicates only that the wrong judges were used. An improper linear model is one in which the weights are chosen by some nonoptimal method. They may be chosen to be equal, they may be chosen on the basis of the intuition of the person making the prediction, or they may be chosen at random. Nevertheless, improper models may have great utility. When, for example, the standardized GREs, GPAs, and selectivity indices in the previous example were weighted equally, the resulting linear composite correlated .48 with later faculty rating. Not only is the correlation of this linear composite higher than that with the clinical judgment of the admissions committee (.19), it is also higher than that obtained upon cross-validating the weights obtained from half the sample. An example of an improper model that might be of somewhat more interest—at least to the general public—was motivated by a physician who was on a panel with me concerning predictive systems. Afterward, at the bar with his' wife and me, he said that my paper might be of some interest to my colleagues, but success in graduate school in psychology was not of much general interest: "Could you, for example, use one of your improper linear models to predict how well my wife and I get along together?" he asked. I realized that I could—or might. At that time, the Psychology Department at the University of Oregon was engaged in sex research, most of which was behavioristically oriented. So the subjects of this research monitored when they made love, when they had fights, when they had social engagements (e.g., with in-laws), and so on. These subjects also made subjective ratings about how happy they were in their marital or coupled situation. I immediately thought of an improper linear model to predict self-ratings of marital happiness: rate of lovemaking minus rate of fighting. My colleague John Howard had collected just such data on couples when he was an undergraduate at the University of Missouri—Kansas City, where he worked with Alexander (1971). After establishing the intercouple reliability of judgments of lovemaking and fighting, Alexander had one partner from each of 42 couples monitor these events. She allowed us to analyze her data, with the following results: "In the thirty happily married ^This index was based on Cass and Birnbaum's (1968) rating of selectivity given at the end of their book Comparative Guide to American Colleges. The verbal categories of selectivity were given numerical values according to the following rale: most selective, 6; highly selective, 5; very selective (+), 4; very selective, 3; selective, 2 ; not mentioned, 1. Unfortunately, only 23 of the 111 students could be used in this comparison because the rating scale the admissions committee used changed slightly from year to year. 572 • JULY 1979 • AMERICAN PSYCHOLOGIST

Prediction and Improved Estimation in Linear Models

Abstract: (1979). Prediction and Improved Estimation in Linear Models. Journal of the Operational Research Society: Vol. 30, No. 1, pp. 88-89.

On the generalized kinematic routing method

Abstract: Field measurements suggest that the relationship between velocity and discharge in steep rough upland channels may be nonlinear at low-flow stages but approximately linear at high-flow stages. This paper describes a generalized kinematic channel network routing model which can allow a flow relationship that describes both high- and low-flow characteristics. The model is applied to the channel network of the Institute of Hydrology's River Severn experimental catchment and, for this specific application, proves to be equivalent to a linear model based on a constant kinematic flow velocity. A strategy for choosing and applying a flow routing model for use in small catchments is suggested.

Robustness of Design Against Autocorrelation in Time I: Asymptotic Theory, Optimality for Location and Linear Regression

Abstract: A new asymptotic theory for studying the effect of dependence of the observations in experimental design for the linear model is developed. The uniform design is shown to be asymptotically optimal in a strong sense for estimating location and in a weaker sense for estimating the slope of a straight line regression. Numerical results supporting the asymptotics appear in a companion paper.

A Simultaneous Test Procedure for Contingency Table Models

Abstract: SUMMARY A simultaneous test procedure is proposed for the choice of a parsimonious model for complex contingency tables, based on a similar procedure for the ANOVA of unbalanced cross-classifications. The procedure requires a fully hierarchical partitioning of the maximized log-likelihood, or deviance, and may be applied to log-linear models for counts, and to logistic or linear models for proportions. Several examples are given.

The use of linear models and matrix least squares in clinical chemistry.

Abstract: We present a unified approach to the use of linear models and matrix least squares with the intention of providing a better understanding of the techniques themselves and of the statistics that arise from these techniques as they are used in clinical chemistry. Emphasis is placed on the importance of appropriate experimental designs and adequately precise measurement processes for efficiently obtaining the desired information.

Interrelations Among Models For The Analysis Of Moment Structures

Abstract: Factor analysis in several populations, covariance structure models, three-mode factor analysis, structural equation systems with measurement model, and analysis of covariance with measurement model are all shown to be specializations of a general moment structure model published previously in this journal. Some new structured linear models are also described; they may be considered either generalizations or special cases of existing models. Simple representations are developed for complex linear models, and some applications to behavioral data are cited.

Bayesian analysis of the linear model with spatial dependence

Abstract: This paper develops a Bayesian analysis of the linear regression model with spatial dependence in the disturbances. The existing literature in spatial econometrics has been based entirely on sampling theory approaches (e. g. Neyman-Pearson and maximum likelihood) to statistical inference, and the Bayesian perspective has not been explored for spatial estimation. In contrast, the mainstream of econometric work has been influenced by the Bayesian approach to statistical inference during the last ten years, largely through the texts and papers of Zellner and Box (Zellner, 1971; Box and Tiao, 1973) and their associates and students. Bayesian methods have been fruitfully applied to the analysis and estimation of a wide range of econometric problems, such as simultaneous equations (Chetty, 1968), production functions (Tsurumi and Tsurumi, 1976; Zellner and Richard, 1973), distributed lag models (Zellner and Geisel, 1970), the linear model with serially correlated errors (Zellner and Tiao, 1965), and the linear model with non-normal errors (Box and Tiao, 1962).

Characterization of minimax linear estimators in linear regression

Abstract: Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.

Exploratory and explanatory statistical analysis of spatial data

Abstract: 1: Introduction.- 1. General Introduction.- 2. Operational Statistical Methods for Analysing Spatial Data.- 2.1. Introduction.- 2.1 The structure of spatial data.- 2.3. Methods based on simple correlations between cross-regional data.- 2.4. Time-series analysis applied to spatial data.- 2.5 Adaptations of time-series analysis to the spatial context.- 2.6. Single equation explanatory models.- 2.7. Simultaneous equation models with spatial data.- 2.8. Some remaining topics.- 2.9. Final remarks.- References.- 2: Exploratory statistical analysis.- 3. The Analysis of Geographical Maps.- 3.1. Introduction.- 3.2. Methods of analysis.- 3.3. Models.- 3.4. Tests for randomness.- 3.5. Examples.- 3.6. Conclusions.- References.- 4 Construction of Interregional Input-Output Tables by Efficient Information Adding.- 4.1. Introduction.- 4.2. Regional and national accounts.- 4.3. Generation of survey-tired transaction tables.- 4.4. Results of the statistical estimations.- 4.5. Results of the minimum information estimations.- 4.6. Some conclusions.- References.- 5. Further Evidence on Alternative Procedures for Testing of Spatial Auto-Correlation among Regression Disturbances.- 5.1. Introduction.- 5.2. Formulation of the statistical decision problem.- 5.3. Moran's test statistic.- 5.4. Moments of the Moran statistic using OLS and LUS estimators.- 5.5. The likelihood ratio test.- 5.6. Simulation study of the Moran and likelihood ratio tests.- 5.7. Results.- 5.8. Conclusions.- References.- 3: Explanatory statistical analysis.- 6. Multivariate Models of Dependent Spatial Data.- 6.1. Introduction.- 6.2. Decomposable covariance structures.- 6.3. Linear models.- 6.4. Principal components.- 6.5. Conclusion.- References.- 7. Bayesian Analysis of the Linear Model with Spatial Dependence.- 7.1. Introduction.- 7.2. The nature of Bayesian inference.- 7.3. Linear regression model with spatially auto-correlated disturbances.- 7.4. An empirical application.- 7.5. Concluding remarks.- References.- 8. Alternative Methods of Estimating Spatial Interaction Models and Their Performance in Short-Term Forecasting.- 8.1. Introduction.- 8.2. Description of data and models.- 8.3. Parameter estimation and model calibration in terms of 1966 and 1971 data.- 8.4. On the accuracy of short-term forecasts made by spatial interaction models.- 8.5. An evaluation of tome alternatives designed to improve model performance.- 8.6. Conclusions.- References.- 9. Two Estimation Methods for Singly Constrained Spatial Distribution Models.- 9.1. Introduction.- 9.2. The calibration of a model.- 9.3. The maximum likelihood method.- 9.4. The least-squares method for the singly constrained model.- 9.5. Numerical results.- 9.6. Conclusions.- References.- 4: The introduction of stochastics in regional control.- 10. Stochastic Control of Regional Economies.- 10.1. Introduction.- 10.2. Mathematical representation of regional systems.- 10.3. Optimal control models of regional systems.- 104 Interaction of optimal control of regional economies with national governments.- 10.5. Problems in applying optimal control to regional systems.- 106. Conclusion.- References.

Frequency domain identification of structural models from earthquake records

Abstract: The usefulness of simple linear mathematical models for representing the behaviour of tall buildings during earthquake response is investigated for a variety of structures over a range of motions including the onset of structural damage. The linear models which best reproduce the measured response of the structures are determined from the recorded earthquake motions. In order to improve upon unsatisfactory results obtained by methods using transfer functions, a systematic frequency domain identification technique is developed to determine the optimal models. The periods, dampings and participation factors are estimated for the structural modes which are dominant in the measured response. The identification is performed by finding the values of the modal parameters which produce a least-squares match over a specified frequency range between the unsmoothed, complex-valued, finite Fourier transform of the acceleration response recorded in the structure and that calculated for the model. It is possible to identify a single linear model appropriate for the entire response, or to approximate the nonlinear behavior exhibited by some structures with a series of models optimal for different segments of the response. The investigation considered the earthquake records obtained in ten structures ranging in height from seven to forty-two stories. Most of the records were from the San Fernando earthquake. For two of these structures, smaller-amplitude records from more distant earthquakes were also analyzed. The maximum response amplitudes ranged from approximately 0.025 g to 0.40g. The very small amplitude responses were reproduced well by linear models with fundamental periods similar to those measured in vibration tests. Most of the San Fernando responses in which no structural damage occurred (typically 0.2g-0.3g maximum accelerations) were also matched closely by linear models. However, the effective fundamental periods in these responses were characteristically 50 percent longer than in vibration tests. The average first mode damping identified from these records was about 5 percent of critical. Only those motions which produced structural damage could not be represented satisfactorily by time-invariant linear models. Segment-by-segment analysis of these records revealed effective periods of two to three times the vibration test values with fundamental mode dampings of 15 to 20 percent. The systematic identification technique generally achieves better matches of the recorded responses than those produced by models derived by trial-and-error methods, and consequently more reliable estimates of the modal parameters. The close reproductions of the measured motions confirm the accuracy of linear models with only a few modes for representing the behaviour during earthquake response of tall buildings in which no structural damage occurs.

Specification Tests for Portfolio Regression Parameter Stationarity and the Implications for Empirical Research

Abstract: tion in these studies is that the joint distribution of the rate of return on risky securities (or portfolios) and the market portfolio proxy is bivariate normal and stationary through time.2 That is, the true parameters of the conditional distribution of the (excess) rate of return on risky securities (or portfolios) given the (excess) rate of return on the market portfolio are the same for each observation in the sample. This assumption is a necessary condition for the econometric procedures employed in studies of market efficiency and tests of the two-parameter model, although it is not required by the underlying theory in either case. However, the theory does indicate, for example, that changes in capital structure and the adoption of new projects (or acquisitions) from a different risk class than present operations will change systematic risk. Exogenous economic information may also change the market's assessment of the parameters defining this conditional distribution. Parameter nonstationarity does represent a severe violation of econometric model specification resulting in the loss of known distributional properties of the parameter estimates and confidence in any inferences made conditional on these estimates. The purpose of this study is to provide statistical significance tests of the stationarity specification by providing a more general alternative nonstationarity specification that has, as a special case, the standard linear model. The importance of testing for systematic risk stationarity was first recognized by Blume (1971). In this paper we intend to consider several new problems by addressing the following questions: (1) Are the differences in the estimates of systematic risk in subperiods just due to sampling error or are they "significantly" different from each other? That is, can the null hypothesis that the true fB's are identical in subperiods be tested? Can we test the same null hypothesis for a's?

Recursive Estimation of the General Linear Model with Dependent Errors

Abstract: SUMARY The techniques for recursive estimation of the general linear model are extended to the case of dependent errors with known second-order properties. Recurrence relations for computation and a sequence of independent recursive residuals, useful for testing the constancy of the regression relation over time, are derived and studied. Computational savings are available when the error process is stationary.

Selection of Variables in Linear Regression Using the Minimum Sum of Weighted Absolute Errors Criterion

Abstract: An efficient implicit enumeration algorithm is proposed for the problem of selecting subsets of predictor variables in a multiple linear regression model using the minimum sum of weighted absolute errors (MSWAE) criterion. The proposed algorithm is illustrated with an example. Computational experience shows that the proposed algorithm is superior to the currently available algorithm in terms of computation time and the number of iterations required to solve a problem.

Invariant measures and a linear model of turbulence

Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Restoration of Noisy Images Using a Two-Dimensional Linear Model

Abstract: A restoration technique of noisy images by using a two-dimensional (2-D) linear model is presented. A method of identifying the parameters in the 2-D autoregressive moving average (ARMA) model that is derived from the approximate 2-D recursive filtering algorithm in [10] is developed. The estimates for the performance of image restoration are given in terms of the stationary filter gain and the variance of prediction errors. Simulation studies are also carried out to show the applicability of the present technique.

Non-linear genotype × environment interactions arising from response thresholds

Abstract: Examples of genotype × environment interactions in Nicotiana rustica that appear to be simply linearly related to the additive environmental value (ej) and others in which this does not appear to be the case have been analysed by testing the goodness-of-fit of linear, quadratic and two intersecting-straight-line models of this relationship. In varieties 1 and 5 and their F1 cross there is no significant improvement in goodness-of-fit over the linear model from either the quadratic or the two straight-line models. In a stratified sample of 10 inbred lines derived from this cross a pair of intersecting-straight-lines was the best model for the only two inbreds to show significant non-linearity. In varieties 2 and 12, their F1 cross and the four inbred selections derived from it, the best fit is always obtained with the two intersecting-straight-lines. The reasons, however, differ for the different genotypes. In variety 2 it is because the linear rate of response is relatively higher in the better than in the poorer environments, in variety 12 it is because of a change in the reverse direction, while in the F1 it is because the rate of response more than doubles in the very best environments. Among the four inbreds it is because one of them, D10, reaches a limit to its response in the environments with above average ej values while of the remaining three inbreds two increase their linear responses in these environments while the third shows a small decrease. While, therefore, two intersecting-straight-lines appear to be a widely applicable model of the relationship between the interaction of genotype and environment and the additive environmental value, in only one case, D10, which is a low mean performance and low environmental sensitivity selection, is the use of this model necessitated by a genotypic limit to further response to environmental improvement.

Analysis of stationary spatial-temporal processes: Estimation and prediction

Abstract: SUMMARY The paper analyses a simultaneous linear model representing stationary spatial-temporal processes. For processes with unit spatial orders, a simple method to compute the likelihood function is developed. Unfortunately, this technique does not generalize to higher order processes. Alternative methods applicable to higher order processes are developed. Finally, it is shown how predictions can be made over time if the parameters are known. Whittle (1954) analysed a simultaneous linear spatial model and discussed likelihood inference for its unknown parameters. The major difficulty in analysing such a model is to evaluate a Jacobian which arises because of the multilateral nature of the process, i.e. because the random variable at a location depends on the random variables at the locations around it. Whittle developed an ingenious method for approximating this determinant. Ord (1975) treated a special case of the simultaneous model with one parameter defined on irregular lattices and showed an exact method for computing the relevant determinant. In ? 2, we generalize Whittle's two-dimensional spatial model to formulate a spatialtemporal model and derive the likelihood function from a single realization on finite rectangular lattices over time. Section 3 analyses spatial-temporal models with unit spatial orders, where the Jacobian has been put into a simple computational form. Section 4 develops alternative methods for computation of the Jacobian applicable to higher-order processes. We take up the prediction aspect of the model in ? 5.

Quadratic versus Linear Models for Land-Use Plan Design:

Abstract: Quadratic assignment models and linear-programming models have been proposed for land-use plan design. The models represent transportation and divisibility of production differently and have solution algorithms with very different properties, but the most important distinction is that quadratic assignment models can handle externalities whereas linear models without integer side conditions cannot. Therefore quadratic models are useful for plan-making in response to externalities problems, whereas linear models can be used only for plan-making in response to dynamics problems.

Generation of linear dynamic models from a digital nonlinear simulation

Abstract: The results and methodology used to derive linear models from a nonlinear simulation are presented. It is shown that averaged positive and negative perturbations in the state variables can reduce numerical errors in finite difference, partial derivative approximations and, in the control inputs, can better approximate the system response in both directions about the operating point. Both explicit and implicit formulations are addressed. Linear models are derived for the F 100 engine, and comparisons of transients are made with the nonlinear simulation. The problem of startup transients in the nonlinear simulation in making these comparisons is addressed. Also, reduction of the linear models is investigated using the modal and normal techniques. Reduced-order models of the F 100 are derived and compared with the full-state models.