Multiple linear regression analysis Subject

Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regression analysis can provide an equation for the standardization. 

The ultimate development in the field of sample preparation is to eliminate it completely, that is, to make a chemical measurement directly without any sample pretreatment. This has been achieved with the application of chemometric near-infrared methods to direct analysis of pharmaceutical tablets and other pharmaceutical solids (74-77). Chemometrics is the use of mathematical and statistical correlation techniques to process instrumental data. Using these techniques, relatively raw analytical data can be converted to specific quantitative information. These methods have been most often used to treat near-infrared (NIR) data, but they can be applied to any instrumental measurement. Multiple linear regression or principal-component analysis is applied to direct absorbance spectra or to the mathematical derivatives of the spectra to define a calibration curve. These methods are considered secondary methods and must be calibrated using data from a primary method such as HPLC, and the calibration material must be manufactured using an equivalent process to the subject test material. However, once the calibration is done, it does not need to be repeated before each analysis. 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

If the data are collected at fixed time intervals then one trick to generate imputed values that would account for within-subject correlations is to transform the data into a columnar format with one row of data per subject. So if the data were collected at Visits 2, 3, and 4, then three new variables would be generated. Variable 1 would correspond to Visit 2, Variable 2 to Visit 3, etc. In this manner then each row of data would correspond to a single individual. Now any of the imputation techniques introduced in the chapter on Linear Regression and Modeling could be used to impute the missing data based on the new variables. Once the data are imputed, the data set can be reformatted to multiple rows per subject and the analysis proceeds with the imputed data. This approach assumes that all samples are collected at the same time interval for all subjects, i.e., assumes that all samples were assumed at Visits 1-4 in this case. 

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