Unequal Numbers Of Observations Research Articles

A framework for mode classification in multimodal environments using radar-based sensors

Monitoring traffic at locations where bicyclists and pedestrians are present and studying their interactions with motorized vehicles has the potential to reveal underlying crash mechanisms and, in turn, guide the implementation of suitable countermeasures. Previous research has demonstrated the capabilities of vision-based systems in traffic monitoring; however, as these systems rely on video camera data, they remain constrained in adverse lighting and weather conditions. Although radar-based sensors can be used in traffic monitoring, they have not been tested in multimodal environments. To bridge this gap a novel framework to classify trajectories recorded by radar-based sensors in multimodal traffic environments is developed. The Support Vector Machine is used as the classifier and the following aspects allow to develop a robust, flexible, and transferable classification framework that can be applied in various traffic scenes. The SVM employs (1) a speed and length or speed and acceleration feature vector, (2) trajectory normalization scheme (i.e., select multiple measurements per trajectory), (3) training sample balancing strategy, and (4) cross-validation strategy. The framework is tested and validated using data from two different multimodal intersections. The results suggest that the mode type for every trajectory can be predicted with 95% accuracy using ten speed measurements and the vehicle’s mean length while accounting for the unequal number of observations per class. While these results are subject to change, i.e., both SVM’s input (feature vector and/or balancing approach) and performance may vary across different traffic scenes, the proposed framework is transferable and flexible.

Discrete convolution statistic for hypothesis testing

The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution statistic, is proposed and its limiting normal covariance matrix determined. To challenge null hypotheses about the distribution of this sum, the generalized Wald’s method is applied to define a testing statistic whose distribution is asymptotic to a with as many degrees of freedom as the rank of such covariance matrix. Rank analysis also reveals a connection with the roots of the probability generating functions associated to the addend variables of the linear models. A simulation study is performed to compare the convolution test with Pearson’s and to provide usage guidelines.

Statistical approach for automated weighting of datasets: Application to heat capacity data

An essential step in CALPHAD is assigning relative weights to different datasets, but there is no consensus as to the best approach regarding this issue. Currently, such an assignment of weights for experimental or first-principles data is performed manually based on the knowledge and experience of the modeler. Since the existing manual treatment is subjective and time consuming, manipulation of such data is rapidly advancing toward automated procedures through statistical and data mining tools. In the present study, we propose an automated approach to determine the weight of datasets based on the K-Fold Cross-Validation method, modified under the conditions that each fold is selected non-randomly and contains an unequal number of observations. This approach can be considered for researchers as a support tool to evaluate the reliability of each dataset involved in the CALPHAD modeling and quantify the impact of weighting by statistical analysis of the corresponding model. We demonstrate the efficacy of this method through the evaluation of heat capacity data of fcc nickel, hcp magnesium, and bcc iron.

A mixed-effects least square support vector regression model for three-level count data

ABSTRACTHierarchical study design often occurs in many areas such as epidemiology, psychology, sociology, public health, engineering, and agriculture. This imposes correlation in data structure that needs to be account for in modelling process. In this study, a three-level mixed-effects least squares support vector regression (MLS-SVR) model is proposed to extend the standard least squares support vector regression (LS-SVR) model for handling cluster correlated data. The MLS-SVR model incorporates multiple random effects which allow handling unequal number of observations for each case at non-fixed time points (a very unbalanced situation) and correlation between subjects simultaneously. The methodology consists of a regression modelling step that is performed straightforwardly by solving a linear system. The proposed model is illustrated through numerical studies on simulated data sets and a real data example on human Brucellosis frequency. The generalization performance of the proposed MLS-SVR is evaluated by comparing to ordinary LS-SVR and some other parametric models.

Variance component analysis to assess protein quantification in biomarker validation: application to selected reaction monitoring-mass spectrometry

BackgroundIn the field of biomarker validation with mass spectrometry, controlling the technical variability is a critical issue. In selected reaction monitoring (SRM) measurements, this issue provides the opportunity of using variance component analysis to distinguish various sources of variability. However, in case of unbalanced data (unequal number of observations in all factor combinations), the classical methods cannot correctly estimate the various sources of variability, particularly in presence of interaction. The present paper proposes an extension of the variance component analysis to estimate the various components of the variance, including an interaction component in case of unbalanced data.ResultsWe applied an experimental design that uses a serial dilution to generate known relative protein concentrations and estimated these concentrations by two processing algorithms, a classical and a more recent one. The extended method allowed estimating the variances explained by the dilution and the technical process by each algorithm in an experiment with 9 proteins: L-FABP, 14.3.3 sigma, Calgi, Def.A6, Villin, Calmo, I-FABP, Peroxi-5, and S100A14. Whereas, the recent algorithm gave a higher dilution variance and a lower technical variance than the classical one in two proteins with three peptides (L-FABP and Villin), there were no significant difference between the two algorithms on all proteins.ConclusionsThe extension of the variance component analysis was able to estimate correctly the variance components of protein concentration measurement in case of unbalanced design.

Adjusting the Scott-Knott cluster analyses for unbalanced designs

Abstract: The Scott-Knott cluster analysis is an alternative approach to mean comparisons with high power and no subset overlapping. It is well suited for the statistical challenges in agronomy associated with testing new cultivars, crop treatments, or methods. The original Scott-Knott test was developed to be used under balanced designs; therefore, the loss of a single plot can signifi-cantly increase the rate of type I error. In order to avoid type I error inflation from missing plots, we propose an adjustment that maintains power similar to the original test while adding error protection. The proposed adjustment was validated from more than 40 million simulated experiments following the Monte Carlo method. The results indicate a minimal loss of power with a satisfactory type I error control, while keeping the features of the original procedure. A user-friendly SAS macro is provided for this analysis. Key words : Type I error rate, unequal number of observations, Monte Carlo simulations, means clustering procedures, SAS macro.

Adapted Bland–Altman method was used to compare measurement methods with unequal observations per case

ObjectivesTo describe an adjustment of the Bland–Altman approach to evaluate possible patterns of discord between two measurement methods with an unequal number of observations per case. Study Design and SettingTwo methods of adaptation were compared using self-assessed general well-being scores (one individual rater) and scores given by multiple external raters for illustration. Both empirical data derived from the Older Person's Relevant Outcome of Care Score study and simulated data were used. ResultsWhen the mean of a number of assessments [X¯] is compared with a single assessment [Y], the variation [X¯ − Y] (vertical axis) will be correlated with the mean of X¯ and Y (horizontal axis) because the means of scores given by a larger group of raters tend to be less extreme than the scores given by individual raters. In contrast, in the absence of discord patterns, the variation [X¯ − Y] will not be correlated with the mean of all observations Xi and Yj made irrespective of who made them as denominator in the Bland–Altman plot. Thus, the second approach should be used for the valuation. ConclusionThe Bland–Altman approach needs correct adaptations, else artificial biases may occur.

Variance component estimation uncertainty for unbalanced data: application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous data set comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity.

On efficient estimators of two seemingly unrelated regressions

In this paper, following the results presented in Liu’s work [Liu, A.Y., 2002. Efficient estimation of two seemingly unrelated regression equations. Journal of Multivariate Analysis 82, 445–456], we first represent the Gauss–Markov estimator of the regression parameter as a matrix series, and hence we conclude that the observation vectors should appear in any efficient estimator in pairs. Second, we prove that the simpler form of the two-stage Aitken estimator is unique. Finally we generalize our results to the system of two seemingly unrelated regressions with unequal numbers of observations and briefly summarize our conclusions.

An approach for the two-group discriminant analysis: An application of DEA

This paper proposes a novel procedure based on a benchmarking model of data envelopment analysis (DEA) to solve the two-group classification problem. The idea was inspired by the fact that benchmarks dominate their own production possibility set (PPS). If observations belong to the same group, they should be in the same PPS and dominated by the same benchmarks. Therefore, the benchmarks of two groups were identified to construct a pair of nonlinear discriminant frontiers without pre-specifying the classification function form as in other parametric discriminant analysis (DA) approaches. Since the discriminant frontiers are established by the boundary of two groups, the proposed procedure differs from some existing discriminant approaches that establish the discriminant function by minimizing the total deviation of all observations from their group mean. Two applications are illustrated to support the validation of the proposed procedure even though the data sets may have outliers or an unequal number of observations.

Applications of multilevel modeling in sport injury rehabilitation research

Sport injury rehabilitation researchers are using increasingly sophisticated and complex prospective longitudinal research designs. Unfortunately, traditional statistical analysis techniques (e.g., analysis of variance [ANOVA], regression) are inadequate to thoroughly analyze the data sets these designs produce. In this article, the use of multilevel modeling (MLM) is described. MLM is shown to be an effective alternative to regression and ANOVA techniques for two main reasons: (a) MLM is particularly well suited to analyzing repeated‐measures designs when the data are clustered; and (b) MLM results are not adversely affected by random missing data and unequal numbers of observations. Results of MLM analyses highlighting the ability to examine relationships among variables pertaining to both individuals (i.e., within‐subjects analyses) and groups (i.e., between‐subjects analyses) are presented in the context of a longitudinal study of persons with knee injuries. Suggestions for applying MLM to other sport injury rehabilitation research questions are provided.

From the Help Desk: Seemingly Unrelated Regression with Unbalanced Equations

This article demonstrates how to estimate the parameters of a system of seemingly unrelated regressions when the equations are unbalanced, i.e., when the equations have an unequal number of observations. With estimators that require the data to be in wide format, such as Stata's sureg, the equations must be balanced. Any additional observations that are available for some equations, but not for all, are discarded, potentially resulting in a loss of efficiency. Reshaping and scaling the data allows us to use Stata's xtgee command to fit the model and obtain estimates utilizing all the available data. The resulting estimator is potentially more efficient when the equations are unbalanced.

A permutation procedure for testing the equality of pattern hypotheses across groups involving correlation or covariance matrices

This paper describes a permutation procedure to test for the equality of selected elements of a covariance or correlation matrix across groups. It involves either centring or standardising each variable within each group before randomly permuting observations between groups. Since the assumption of exchangeability of observations between groups does not strictly hold following such transformations, Monte Carlo simulations were used to compare expected and empirical rejection levels as a function of group size, the number of groups and distribution type (Normal, mixtures of Normals and Gamma with various values of the shape parameter). The Monte Carlo study showed that the estimated probability levels are close to those that would be obtained with an exact test except at very small sample sizes (5 or 10 observations per group). The test appears robust against non-normal data, different numbers of groups or variables per group and unequal sample sizes per group. Power was increased with increasing sample size, effect size and the number of elements in the matrix and power was decreased with increasingly unequal numbers of observations per group.

Exact Distribution of the Regression Esti Mator in Double Sampling

Asymptotic results about the distribution of the regression estimator in a double sampling scheme when using an auxiliary covariate have been established by Engel and Walstra (1991). Conniffe (1985) studied in the finite sample case the closely related model of simultaneous regression equations in econometrics with unequal numbers of observations. The aim of this paper is to provide a normal approximation of the distribution of the regression estimator. First, the exact distribution is derived in the Gaussian framework and the exact moments of the regression estimator are deduced. An Edgeworth expansion of the exact distribution is then calculated in order to propose a normal approximation. The exact distribution, the Edgeworth expansion and the normal approximation are compared throughout an example.

Macroeconomic conditions and agribusiness profitability: An analysis using pooled data

Theoretical and empirical insights into the linkages between firm profitability and macroeconomic conditions are developed for nineteen agribusinesses. The hypothesis investigated in this analysis is that firm financial performance is a function of firm specific factors and macroeconomic conditions common to all firms. Seemingly unrelated regression with an unequal number of observations is used to estimate macroeconomic linkages. Empirical results indicate that macroeconomic conditions have differing affects on firm profitability dependent on a firm's financial structure and the market segment in which it operates. Capital intensive industries and highly leveraged firms have higher business risk and are more susceptible to macroeconomic conditions.

ON A MULTIPLE THREE‐DECISION PROCEDURE FOR COMPARING SEVERAL TREATMENTS WITH A CONTROL

summaryFor the two‐sided comparisons of several treatments with a control, a common statistical problem is to decide which treatments are better than the control and which are worse than the control. This paper studies a multiple three‐decision procedure for this purpose, proposed by Bohrer (1979) and Bohrer et al. (1981), and provides tables of critical points to facilitate the application of the procedure. The paper defines a power function of the procedure, and tabulates sample sizes necessary to guarantee a given power level. It addresses the problem of optimal sampling allocation in order to maximize the power for a given total sample size, and considers generalization to the situation where the treatments might have unequal numbers of observations.

Multivariate Regression with Unequal Number of Observations—Solution

Multivariate Regression with Unequal Number of Observations—Solution

Multivariate Regression with Unequal Number of Observations

Multivariate Regression with Unequal Number of Observations

Unequal numbers of observations and partial efficiency gain

This paper investigates the efficiency gain from Aitken estimation in the seemingly unrelated regression equations with unequal numbers of observations and identical regressors. We find that there is efficiency gain for the equation with a smaller number of observations only.

A simulation study of estimators of sur models with unequal numbers of observationsand with non-normal disturbances

This paper reports the results of a Monte Carlo experiment conducted to compare the small sample properties of selected feasible estimators of the seemingly unrelated regressions model with unequal numbers of observations and with non-normal disturbances. Our results indicate that the choice of estimator depends mainly on the size of the sample, the number of extra observations and the degree of correlation among explanatory variables across equations. For instance, methods which use the extra observations in estimating all elements of the disturbance covariance matrix are relatively more efficient when data on explanatory variables are highly correlated. However, the usual estimator which is based on only the matched observations in estimating the disturbance covariance matrix performs well and competes closely with the other feasible estimators when the correlation of explanatory variables across equations is low and the sample is small or the extra number of observations is large relative to the sample size.