A Pascal program for weighted least squares regression on a microcomputer

Weighted least-squares regression has been programmed in Pascal for a microcomputer. A double precision Pascal compiler and the Motorola 6809 assembler produce a fast machine-code program occupying 22,000 bytes of memory when appended to the Pascal run-time module. Large data sets fit in the remaining memory. A regression with 72 observations and 24 parameters runs in 7 min, excluding optional print out of large matrices. The maximum dimensions of the design matrix, X, can be altered by modifying two Pascal constants. Minor changes to the Pascal source program will make it compatible with other Pascal compilers. The program optionally orthogonalises the X matrix to detect linearly-dependent columns in X, and/or generate orthogonal parameter estimates. After orthogonalizing X and fitting the model, the parameter estimates for the original X can be retrieved by the program. Regressions on a repeatedly reduced model are performed through elimination of columns in X until the minimum adequate model is obtained.     

A program for least squares analysis of reassociation and hybridization data.

A computer program is described for the rapid calculation of least squares solutions for data fitted to different functions normally used in reassociation and hybridization kinetic measurements. The equations for the fraction not reacted as a function of Cot follow: First order, exp(-kCot); second order, (1+kCot)-1; variable order, (1+kCot)-n; approximate fraction of DNA sequence remaining single stranded, (1+kCot)-.44; and a function describing the pairing of tracer when the rate constant for the tracer (k) is distinct from the driver rate constant (kd): (formula: see text). Several components may be used for most of these functional forms. The standard deviations of the individual parameters at the solutions are calculated.     

Ordinary least squares regression is indicated for studies of allometry

When it comes to fitting simple allometric slopes through measurement data, evolutionary biologists have been torn between regression methods. On the one hand, there is the ordinary least squares (OLS) regression, which is commonly used across many disciplines of biology to fit lines through data, but which has a reputation for underestimating slopes when measurement error is present. On the other hand, there is the reduced major axis (RMA) regression, which is often recommended as a substitute for OLS regression in studies of allometry, but which has several weaknesses of its own. Here, we review statistical theory as it applies to evolutionary biology and studies of allometry. We point out that the concerns that arise from measurement error for OLS regression are small and straightforward to deal with, whereas RMA has several key properties that make it unfit for use in the field of allometry. The recommended approach for researchers interested in allometry is to use OLS regression on measurements taken with low (but realistically achievable) measurement error. If measurement error is unavoidable and relatively large, it is preferable to correct for slope attenuation rather than to turn to RMA regression, or to take the expected amount of attenuation into account when interpreting the data.     

Cue-signal-response analysis of TNF-induced apoptosis by partial least squares regression of dynamic multivariate data.

Biological signaling networks process extracellular cues to control important cell decisions such as death-survival, growth-quiescence, and proliferation-differentiation. After receptor activation, intracellular signaling proteins change in abundance, modification state, and enzymatic activity. Many of the proteins in signaling networks have been identified, but it is not known how signaling molecules work together to control cell decisions. To begin to address this issue, we report the use of partial least squares regression as an analytical method to glean signal-response relationships from heterogeneous multivariate signaling data collected from HT-29 human colon carcinoma cells stimulated to undergo programmed cell death. By partial least squares modeling, we relate dynamic and quantitative measurements of 20-30 intracellular signals to cell survival after treatment with tumor necrosis factor alpha (a death factor) and insulin (a survival factor). We find that partial least squares models can distinguish highly informative signals from redundant uninformative signals to generate a reduced model that retains key signaling features and signal-response relationships. In these models, measurements of biochemical characteristics, based on very different techniques (Western blots, kinase assays, etc.), are grouped together as covariates, showing that heterogenous data have been effectively fused. Importantly, informative protein predictors of cell responses are always multivariate, demonstrating the multicomponent nature of the decision process.     

A new regularized least squares support vector regression for gene selection

Background  

Selection of influential genes with microarray data often faces the difficulties of a large number of genes and a relatively small group of subjects. In addition to the curse of dimensionality, many gene selection methods weight the contribution from each individual subject equally. This equal-contribution assumption cannot account for the possible dependence among subjects who associate similarly to the disease, and may restrict the selection of influential genes.     

Estimation of time-varying respiratory mechanical parameters by recursive least squares.

Continuous estimation of time-varying respiratory mechanical parameters is required to fully characterize the time course of bronchoconstriction. To achieve such estimation, we developed an estimator that uses the recursive linear least-squares algorithm to fit the equation Ptr = RV + EV + K to measurements of tracheal pressure (Ptr) and flow (V). The volume (V) is obtained by numerical integration of V. The estimator has a finite memory with length into the past at each point in time that varies inversely with the difference between the current measurement of Ptr and that predicted by the model, to allow the algorithm to track rapidly varying parameters (R, E, and K). V usually exhibits significant drift and must be corrected. Of the several correction methods investigated, subtraction of the recursively weighted average of V before integration to V was found to perform best. The estimator was tested on simulated noisy data where it successfully followed a fivefold increase in R and a twofold increase in E occurring over 10 s. Three dogs and two cats were anesthetized, paralyzed, tracheostomized, and challenged with a bolus of methacholine (approximately 13 mg/kg iv). Increases of 3- to 10-fold were observed in R and 2- to 3-fold in E, beginning within 10-40 s after the bolus injection. In some animals we found that the increase in E occurred more slowly than that in R, which the V signal suggested was due to dynamic hyperinflation of the lungs. These results demonstrate that our recursive estimator is able to track rapid changes in respiratory mechanical parameters during bronchoconstrictor challenge.     

Pharmacokinetic curve fitting and parameter determination by non-linear,iterative least squares regression analysis using a programmed minicalculator

A pharmacokinetic non-linear, iterative least-squares program for the minicalculator TI-59 with adapted printer is described. The program utilizes the Gauss-Newton gradient method in an iterative, non-linear regression analysis of up to 18 data pairs. Single-dose plasma concentration data of 5-hydroxytryptophan, theophylline and prednisolone were comparatively analysed using both the described program (called NONTI-59) and Metzlers well-established digital computer program NONLIN.     

A weighted least squares method for inverse dynamic analysis

Internal forces in the human body can be estimated from measured movements and external forces using inverse dynamic analysis. Here we present a general method of analysis which makes optimal use of all available data, and allows the use of inverse dynamic analysis in cases where external force data is incomplete. The method was evaluated for the analysis of running on a partially instrumented treadmill. It was found that results correlate well with those of a conventional analysis where all external forces are known.     

Analysis of genetic marker-phenotype relationships by jack-knifed partial least squares regression (PLSR)

The utility of a relatively new multivariate method, bi-linear modelling by cross-validated partial least squares regression (PLSR), was investigated in the analysis of QTL. The distinguishing feature of PLSR is to reveal reliable covariance structures in data of different types with regard to the same set objects. Two matrices X (here: genetic markers) and Y (here: phenotypes) are interactively decomposed into latent variables (PLS components, or PCs) in a way which facilitates statistically reliable and graphically interpretable model building. Natural collinearities between input variables are utilized actively to stabilise the modelling, instead of being treated as a statistical problem. The importance of cross-validation/jack-knifing as an intuitively appealing way to avoid overfitting, is emphasized. Two datasets from chromosomal mapping studies of different complexity were chosen for illustration (QTL for tomato yield and for oat heading date). Results from PLSR analysis were compared to published results and to results using the package PLABQTL in these data sets. In all cases PLSR gave at least similar explained validation variances as the reported studies. An attractive feature is that PLSR allows the analysis of several traits/replicates in one analysis, and the direct visual identification of individuals with desirable marker genotypes. It is suggested that PLSR may be useful in structural and functional genomics and in marker assisted selection, particularly in cases with limited number of objects.