Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

Using nonlinear least squares to assess relative expression and its uncertainty in real-time qPCR studies

Relative expression ratios are commonly estimated in real-time qPCR studies by comparing the quantification cycle for the target gene with that for a reference gene in the treatment samples, normalized to the same quantities determined for a control sample. For the “standard curve” design, where data are obtained for all four of these at several dilutions, nonlinear least squares can be used to assess the amplification efficiencies (AE) and the adjusted ΔΔC_q and its uncertainty, with automatic inclusion of the effect of uncertainty in the AEs. An algorithm is illustrated for the KaleidaGraph program.     

A study of statistical error in isothermal titration calorimetry

In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M+X right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-are assessed from the variance-covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these "exact" estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase.     

Analysis of multitemperature isothermal titration calorimetry data at very low c: Global beats van't Hoff

Isothermal titration calorimetry data for very low c (≡K[M]₀) must normally be analyzed with the stoichiometry parameter n fixed — at its known value or at any reasonable value if the system is not well characterized. In the latter case, ΔH° (and hence n) can be estimated from the T-dependence of the binding constant K, using the van't Hoff (vH) relation. An alternative is global or simultaneous fitting of data at multiple temperatures. In this Note, global analysis of low-c data at two temperatures is shown to estimate ΔH° and n with double the precision of the vH method.     

Can you trust the parametric standard errors in nonlinear least squares? Yes,with provisos

Background

Questions about the reliability of parametric standard errors (SEs) from nonlinear least squares (LS) algorithms have led to a general mistrust of these precision estimators that is often unwarranted.

Using the EM algorithm to weight data sets of unknown precision when modelling fish stocks

Stocks of commercial fish are often modelled using sampling data of various types, of unknown precision, and from various sources assumed independent. We want each set to contribute to estimates of the parameters in relation to its precision and goodness of fit with the model. Iterative re-weighting of the sets is proposed for linear models until the weight of each set is found to be proportional to (relative weighting) or equal to (absolute weighting) the set-specific residual invariances resulting from a generalised least squares fit. Formulae for the residual variances are put forward involving fractional allocation of degrees of freedom depending on the numbers of independent observations in each set, the numbers of sets contributing to the estimate of each parameter, and the number of weights estimated. To illustrate the procedure, numbers of the 1984 year-class of North Sea cod (a) landed commercially each year, and (b) caught per unit of trawling time by an annual groundfish survey are modelled as a function of age to estimate total mortality, Z, relative catching power of the two fishing methods, and relative precision of the two sets of observations as indices of stock abundance. It was found that the survey abundance indices displayed residual variance about 29 times higher than that of the annual landings.     

Confidence regions and hypothesis tests for topologies using generalized least squares

A confidence region for topologies is a data-dependent set of topologies that, with high probability, can be expected to contain the true topology. Because of the connection between confidence regions and hypothesis tests, implicitly or explicitly, the construction of confidence regions for topologies is a component of many phylogenetic studies. Existing methods for constructing confidence regions, however, often give conflicting results. The Shimodaira-Hasegawa test seems too conservative, including too many topologies, whereas the other commonly used method, the Swofford-Olsen-Waddell-Hillis test, tends to give confidence regions with too few topologies. Confidence regions are constructed here based on a generalized least squares test statistic. The methodology described is computationally inexpensive and broadly applicable to maximum likelihood distances. Assuming the model used to construct the distances is correct, the coverage probabilities are correct with large numbers of sites.     

Microsatellite Behavior with Range Constraints: Parameter Estimation and Improved Distances for Use in Phylogenetic Reconstruction

A symmetric stepwise mutation model with reflecting boundaries is employed to evaluate microsatellite evolution under range constraints. Methods of estimating range constraints and mutation rates under the assumptions of the model are developed. Least squares procedures are employed to improve molecular distance estimation for use in phylogenetic reconstruction in the case where range constraints and mutation rates vary across loci. The bias and accuracy of these methods are evaluated using computer simulations, and they are compared to previously existing methods which do not assume range constraints. Range constraints are seen to have a substantial impact on phylogenetic conclusions based on molecular distances, particularly for more divergent taxa. Results indicate that if range constraints are in effect, the methods developed here should be used in both the preliminary planning and final analysis of phylogenetic studies employing microsatellites. It is also seen that in order to make accurate phylogenetic inferences under range constraints, a larger number of loci are required than in their absence.     

Weighted least-squares approach for comparing correlated kappa

In the medical sciences, studies are often designed to assess the agreement between different raters or different instruments. The kappa coefficient is a popular index of agreement for binary and categorical ratings. Here we focus on testing for the equality of two dependent kappa coefficients. We use the weighted least-squares (WLS) approach of Koch et al. (1977, Biometrics 33, 133-158) to take into account the correlation between the estimated kappa statistics. We demonstrate how the SAS PROC CATMOD can be used to test for the equality of dependent Cohen's kappa coefficients and dependent intraclass kappa coefficients with nominal categorical ratings. We also test for the equality of dependent Cohen's kappa and dependent weighted kappa with ordinal ratings. The major advantage of the WLS approach is that it allows the data analyst a way of testing dependent kappa with popular SAS software. The WLS approach can handle any number of categories. Analyses of three biomedical studies are used for illustration.     

Studies on Nucleoside Analogs. XXV.1) Synthesis of 5, 7-Dioxopyrimido[5,4-e]-AS-Triazine Glycosides

Abstract

The reaction of glycosyl isothiocyanates (la, b, c, d, e) with 5,6-diamino-1-3-dimethyluracil gave the respective 1-glycosyl-3-(6-amino-1, 3-dimethyl-2, 4-dioxopyrimidine-5-yl) thioureas (2a, b, c, d, e) in excellent yields. Treatment of these thioureas with NBS afforded the respective 5,7-dioxopyrimido-[5,4-e]-as-triazine glycosides (4a, b, c, d, e) in good yields.     

Least median of squares and iteratively re‐weighted least squares as robust linear regression methods for fluorimetric determination of α‐lipoic acid in capsules in ideal and non‐ideal cases of linearity

This study outlines two robust regression approaches, namely least median of squares (LMS) and iteratively re‐weighted least squares (IRLS) to investigate their application in instrument analysis of nutraceuticals (that is, fluorescence quenching of merbromin reagent upon lipoic acid addition). These robust regression methods were used to calculate calibration data from the fluorescence quenching reaction (?F and F‐ratio) under ideal or non‐ideal linearity conditions. For each condition, data were treated using three regression fittings: Ordinary Least Squares (OLS), LMS and IRLS. Assessment of linearity, limits of detection (LOD) and quantitation (LOQ), accuracy and precision were carefully studied for each condition. LMS and IRLS regression line fittings showed significant improvement in correlation coefficients and all regression parameters for both methods and both conditions. In the ideal linearity condition, the intercept and slope changed insignificantly, but a dramatic change was observed for the non‐ideal condition and linearity intercept. Under both linearity conditions, LOD and LOQ values after the robust regression line fitting of data were lower than those obtained before data treatment. The results obtained after statistical treatment indicated that the linearity ranges for drug determination could be expanded to lower limits of quantitation by enhancing the regression equation parameters after data treatment. Analysis results for lipoic acid in capsules, using both fluorimetric methods, treated by parametric OLS and after treatment by robust LMS and IRLS were compared for both linearity conditions.     

Application of multivariate curve resolution-alternating least squares (MCR-ALS) for secondary structure resolving of proteins

Fourier transform infrared (FTIR) spectroscopy is the most common spectroscopic technique used for study of protein structure. Initially, band deconvolution techniques were applied to determine the secondary structure of proteins. Recently, several multivariate regression methods have been used to predict the secondary structure of proteins as an alternative to the previous methods. Multivariate curve resolution-alternating least squares (MCR-ALS) was applied on the FTIR spectra of proteins to resolve the fraction and spectral profiles of different structural motifs. Initial estimates of spectral profiles of different protein motifs were built using orthogonal projection approach (OPA). Predicted fractions of α-helix and β-sheet obtained by MCR-ALS technique were compared with those from partial least squares (PLS) modeling which revealed superiority of the former. If we consider the possibility of pure spectra prediction in addition to the prediction of secondary structure from the data set, MCR-ALS can be proposed as a very valuable alternative for qualitative and quantitative study of protein structures.