Robust regression methods for real-time polymerase chain reaction

Current real-time polymerase chain reaction (PCR) data analysis methods implement linear least squares regression methods for primer efficiency estimation based on standard curve dilution series. This method is sensitive to outliers that distort the outcome and are often ignored or removed by the end user. Here, robust regression methods are shown to provide a reliable alternative because they are less affected by outliers and often result in more precise primer efficiency estimators than the linear least squares method.     

Robust two-stage estimation in hierarchical nonlinear models

Hierarchical models encompass two sources of variation, namely within and among individuals in the population; thus, it is important to identify outliers that may arise at each sampling level. A two-stage approach to analyzing nonlinear repeated measurements naturally allows parametric modeling of the respective variance structure for the intraindividual random errors and interindividual random effects. We propose a robust two-stage procedure based on Huber's (1981, Robust Statistics) theory of M-estimation to accommodate separately aberrant responses within an experimental unit and subjects deviating from the study population when the usual assumptions of normality are violated. A toxicology study of chronic ozone exposure in rats illustrates the impact of outliers on the population inference and hence the advantage of adopting the robust methodology. The robust weights generated by the two-stage M-estimation process also serve as diagnostics for gauging the relative influence of outliers at each level of the hierarchical model. A practical appeal of our proposal is the computational simplicity since the estimation algorithm may be implemented using standard statistical software with a nonlinear least squares routine and iterative capability.     

IDSS: deformation invariant signatures for molecular shape comparison

Background  

The conventional superposition methods use an ordinary least squares (LS) fit for structural comparison of two different conformations of the same protein. The main problem of the LS fit that it is sensitive to outliers, i.e. large displacements of the original structures superimposed.     

A robust multiple-locus method for quantitative trait locus analysis of non-normally distributed multiple traits

Linear regression-based quantitative trait loci/association mapping methods such as least squares commonly assume normality of residuals. In genetics studies of plants or animals, some quantitative traits may not follow normal distribution because the data include outlying observations or data that are collected from multiple sources, and in such cases the normal regression methods may lose some statistical power to detect quantitative trait loci. In this work, we propose a robust multiple-locus regression approach for analyzing multiple quantitative traits without normality assumption. In our method, the objective function is least absolute deviation (LAD), which corresponds to the assumption of multivariate Laplace distributed residual errors. This distribution has heavier tails than the normal distribution. In addition, we adopt a group LASSO penalty to produce shrinkage estimation of the marker effects and to describe the genetic correlation among phenotypes. Our LAD-LASSO approach is less sensitive to the outliers and is more appropriate for the analysis of data with skewedly distributed phenotypes. Another application of our robust approach is on missing phenotype problem in multiple-trait analysis, where the missing phenotype items can simply be filled with some extreme values, and be treated as outliers. The efficiency of the LAD-LASSO approach is illustrated on both simulated and real data sets.     

Residual plots for log odds ratio regression models.

This article describes graphical diagnostic methods for log odds ratio regression models. To study the effects of an additional covariate on log odds ratio regression analysis, three types of residual plots based on weighted least squares (WLS) are discussed: (i) added variable plot (partial regression plot), (ii) partial residual plot, and (iii) augmented partial residual plot. These plots provide diagnostic procedures for identifying heterogeneity of error variances, outliers, or nonlinearity of the model. They are especially useful for clarifying whether including a covariate as a linear term is appropriate, or whether quadratic or other nonlinear transformations are preferable. A well-known data set for case-control studies is analyzed to illustrate the residual plots.     

Using least median of squares for structural superposition of flexible proteins

Background

The conventional superposition methods use an ordinary least squares (LS) fit for structural comparison of two different conformations of the same protein. The main problem of the LS fit that it is sensitive to outliers, i.e. large displacements of the original structures superimposed.

Results

To overcome this problem, we present a new algorithm to overlap two protein conformations by their atomic coordinates using a robust statistics technique: least median of squares (LMS). In order to effectively approximate the LMS optimization, the forward search technique is utilized. Our algorithm can automatically detect and superimpose the rigid core regions of two conformations with small or large displacements. In contrast, most existing superposition techniques strongly depend on the initial LS estimating for the entire atom sets of proteins. They may fail on structural superposition of two conformations with large displacements. The presented LMS fit can be considered as an alternative and complementary tool for structural superposition.

Conclusion

The proposed algorithm is robust and does not require any prior knowledge of the flexible regions. Furthermore, we show that the LMS fit can be extended to multiple level superposition between two conformations with several rigid domains. Our fit tool has produced successful superpositions when applied to proteins for which two conformations are known. The binary executable program for Windows platform, tested examples, and database are available from https://engineering.purdue.edu/PRECISE/LMSfit.     

A robust comparison of biological shapes

Localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than at least squares. The repeated median resistant fitting algorithm is introduced. The methods are tested by comparing hominid skulls with those of the least squares fit, in that the differences are more readily identified and agree more closely with the structural differences perceived on biological grounds.     

Estimation from Censored Medical Cost Data

This paper applies the inverse probability weighted least‐squares method to predict total medical cost in the presence of censored data. Since survival time and medical costs may be subject to right censoring and therefore are not always observable, the ordinary least‐squares approach cannot be used to assess the effects of explanatory variables. We demonstrate how inverse probability weighted least‐squares estimation provides consistent asymptotic normal coefficients with easily computable standard errors. In addition, to assess the effect of censoring on coefficients, we develop a test comparing ordinary least‐squares and inverse probability weighted least‐squares estimators. We demonstrate the methods developed by applying them to the estimation of cancer costs using Medicare claims data. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local adaptation of tree lizards to canyon dwelling

How and under what situations populations adapt to local conditions remains a key question in evolutionary biology. This study tests if the particular morphology of a population of Tree lizards, Urosaurus ornatus, located in a canyon on the margin of the species range represents an adaptation to canyon habitat. Morphology was compared across 40 populations showing that relative hind limb length, tail length, and mass were all outliers for this population. The function of the relatively longer hind limbs, tail, and lower mass was proposed to be for better sprinting ability on the sheer canyon walls that provide the only available habitat structure for this population. Partial least squares regression found significant effects of tail length on top speed on a broad, steep surface. Partial least squares logistic regression identified significant effects of tail length on survival as well in males but not females of this population. Another canyon population of Tree lizards with access to alternative substrates (trees) showed no evidence of selection on the same morphological features. Ancestral state reconstruction using a phylogeny inferred for 21 populations found that the unique morphology of the focal population was evolutionarily derived compared to closely related populations and so likely arose under the present environmental conditions. Population genetic structure also supported the process of adaptive divergence as there was no evidence for migration and/or a recent genetic bottleneck in the focal population. Lizards in this population appear to have responded to selection allowing them to become specialists for running on canyon walls while other canyon populations with access to a greater variety of habitat structure have not.     

Model selection and logarithmic transformation in allometric analysis

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric (or scaling) exponent, which often is interpreted in the context of underlying principles of structural and functional design. However, interpretations of this sort are based on the implicit assumption that the original data conform with a power function having an intercept of 0 on a plot with arithmetic coordinates. Whenever this assumption is not satisfied, the resulting estimate for the allometric exponent may be seriously biased and misleading. The problem of identifying an appropriate function is compounded by the logarithmic transformations, which alter the relationship between the original variables and frequently conceal the presence of outliers having an undue influence on properties of the fitted equation, including the estimate for the allometric exponent. Much of the current controversy in allometric research probably can be traced to substantive biases introduced by investigators who followed standard practice. We illustrate such biases with examples taken from the literature and outline a general methodology by which the biases can be minimized in future research.     

Non-linear heteroscedastic regression model for determination of methotrexate in human plasma by high-performance liquid chromatography

Generalized least squares regression with variance function estimation was used to derive the calibration function for measurement of methotrexate plasma concentration and its results were compared with weighted least squares regression by usual weight factors and also with that of ordinary least squares method. In the calibration curve range of 0.05 to 100 microM, both heteroscedasticity and non-linearity were present therefore ordinary least squares linear regression methods could result in large errors in the calculation of methotrexate concentration. Generalized least squares regression with variance function estimation worked better than both the weighted regression with the usual weight factors and ordinary least squares regression and gave better estimates for methotrexate concentration.     

Least squares regression methods for clustered ROC data with discrete covariates

The receiver operating characteristic (ROC) curve is a popular tool to evaluate and compare the accuracy of diagnostic tests to distinguish the diseased group from the nondiseased group when test results from tests are continuous or ordinal. A complicated data setting occurs when multiple tests are measured on abnormal and normal locations from the same subject and the measurements are clustered within the subject. Although least squares regression methods can be used for the estimation of ROC curve from correlated data, how to develop the least squares methods to estimate the ROC curve from the clustered data has not been studied. Also, the statistical properties of the least squares methods under the clustering setting are unknown. In this article, we develop the least squares ROC methods to allow the baseline and link functions to differ, and more importantly, to accommodate clustered data with discrete covariates. The methods can generate smooth ROC curves that satisfy the inherent continuous property of the true underlying curve. The least squares methods are shown to be more efficient than the existing nonparametric ROC methods under appropriate model assumptions in simulation studies. We apply the methods to a real example in the detection of glaucomatous deterioration. We also derive the asymptotic properties of the proposed methods.     

Comment on analyzing flow cytometric data for comparison of mean values of the coefficient of variation of the G1 peak.

BACKGROUND: The coefficient of variation (CV) is often used to characterize and summarize the flow cytometry analysis of nuclear DNA of the Go/G1 peak in a cell population within an individual organism. CV values are frequently used in subsequent statistical analysis to compare experimental groups of individuals. METHODS: We explain why the conventional analysis of variance, linear comparisons and regressions that employ the F and t-tests are not appropriate for analyzing CV data sets. The weighted least squares procedure which utilizes the chi-square test is presented as an adequate method. We further explain why this type of data needs to be analyzed by this procedure. RESULTS: To illustrate the application of the weighted least squares procedure, we analyzed a real data set that had been previously analyzed by conventional methods. We found that a non-significant result (p = 1) using the latter was significant when re-analyzed with the weighted least squares procedure (p = 0.032). CONCLUSIONS: Significant differences between treatments established by the weighted least squares often go unidentified by the conventional analysis. Use of the weighted least squares procedure is recommended for analyzing CV data sets.     

The Impact of Outliers on Net-Benefit Regression Model in Cost-Effectiveness Analysis

Ordinary least square (OLS) in regression has been widely used to analyze patient-level data in cost-effectiveness analysis (CEA). However, the estimates, inference and decision making in the economic evaluation based on OLS estimation may be biased by the presence of outliers. Instead, robust estimation can remain unaffected and provide result which is resistant to outliers. The objective of this study is to explore the impact of outliers on net-benefit regression (NBR) in CEA using OLS and to propose a potential solution by using robust estimations, i.e. Huber M-estimation, Hampel M-estimation, Tukey''s bisquare M-estimation, MM-estimation and least trimming square estimation. Simulations under different outlier-generating scenarios and an empirical example were used to obtain the regression estimates of NBR by OLS and five robust estimations. Empirical size and empirical power of both OLS and robust estimations were then compared in the context of hypothesis testing.Simulations showed that the five robust approaches compared with OLS estimation led to lower empirical sizes and achieved higher empirical powers in testing cost-effectiveness. Using real example of antiplatelet therapy, the estimated incremental net-benefit by OLS estimation was lower than those by robust approaches because of outliers in cost data. Robust estimations demonstrated higher probability of cost-effectiveness compared to OLS estimation. The presence of outliers can bias the results of NBR and its interpretations. It is recommended that the use of robust estimation in NBR can be an appropriate method to avoid such biased decision making.     

Maximum likelihood vs. minimum chi-square--a general comparison with applications to the estimation of recombination fractions in two-point linkage analysis.

Some relationships between the estimates of recombination fraction in two-point linkage analysis obtained by maximum likelihood, minimum chi-square, and general least squares are derived. These theoretical results are based on an approximation for the multinomial distribution. Applications (theoretical and experimental) with RFLP (restriction fragment length polymorphism) markers for a segregating F2 population are given. The minimum chi-square estimate is slightly larger than the maximum likelihood estimate. For applications, however, both estimates must be considered to be approximately equal. The least squares estimates are slightly different (larger or smaller) from these estimates.     

Procedures for comparing samples with multiple endpoints

Five procedures are considered for the comparison of two or more multivariate samples. These procedures include a newly proposed nonparametric rank-sum test and a generalized least squares test. Also considered are the following tests: ordinary least squares, Hotelling's T2, and a Bonferroni per-experiment error-rate approach. Applications are envisaged in which each variable represents a qualitatively different measure of response to treatment. The null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others. In all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test, and is recommended for general use. Alternatively, the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples. A convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated.     

Bias compensated least squares estimate of the center of rotation

It is shown that a recently published least squares method for the estimation of the average center of rotation is biased. Consequently, a correction term is proposed, and an iterative algorithm is derived for finding a bias compensated solution to the least squares problem.The accuracy of the proposed bias compensated least squares method is compared to the previously proposed least squares method by Monte-Carlo simulations. The tests show that the new method gives a substantial improvement in accuracy.     

Highly resistant regression and object matching

In many disciplines, it is of great importance to match objects. Procrustes analysis is a popular method for comparing labeled point configurations based on a least squares criterion. We consider alternative procedures that are highly resistant to outlier points, and we describe an application in electrophoretic gel matching. We consider procedures based on S estimators, least median of squares, and least quartile difference estimators. Practical implementation issues are discussed, including random subset selection and intelligent subset selection (where subsets with small size or near collinear subsets are ignored). The relative performances of the resistant and Procrustes methods are examined in a simulation study.     

Estimation of Individual Growth Curves by Empirically Weighted Least Squares

The simultaneous estimation of individual growth curves and a mean growth curve is accomplished by weighted least squares. A polynomial curve is fitted for each individual and the polynomial parameters are linear functions of parameters corresponding to covariates. A simple, computationally efficient variance-covariance estimator is derived. The resultant estimate is used in the weighted least squares estimation. The results are compared to empirical Bayes estimation.