Predictor variables random measurement errors

The ultimate goal of multivariate calibration is the indirect determination of a property of interest (y) by measuring predictor variables (X) only. Therefore, an adequate description of the calibration data is not sufficient the model should be generalizable to future observations. The optimum extent to which this is possible has to be assessed carefully when the calibration model chosen is too simple (underfitting) systematic errors are introduced, when it is too complex (oveifitting) large random errors may result (c/. Section 10.3.4). 

The basic parameter estimation, or regression, problem involves fitting the parameters of a proposed model to agree with the observed behavior of a system (Figure 8.1). We assume that, in any particular measurement of the system behavior, there is some set of predictor variables jc e 91 that fiiUy determines the behavior of the system (in the absence of any random noise or error). For each experiment, we measure some set of response variables yf.r) g fp— single-response data mAilL >, multiresponse data. t NxiXe... 

Again, we perform a number N of experiments, where in the Ath experiment, we have a known set of M predictor variables, e 91, and we observe the L responses j[t] g We wish to estimate the values of P unknown parameters 0 e in a model whose predicted responses for each experiment form a vector /(x l 0) We assume that the measured responses are equal to the model predictions plus a random error vector. 

Classic univariate regression uses a single predictor, which is usually insufficient to model a property in complex samples. Multivariate regression takes into account several predictive variables simultaneously for increased accuracy. The purpose of a multivariate regression model is to extract relevant information from the available data. Observed data usually contains some noise and may also include irrelevant information. Noise can be considered as random data variation due to experimental error. It may also represent observed variation due to factors not initially included in the model. Further, the measured data may carry irrelevant information that has little or nothing to do with the attribute modeled. For instance, NIR absorbance... 

Figure 6.5 Simulated time-effect data where the intercept was normally distributed with a mean of 100 and a standard deviation of 60. The effect was linear over time within an individual with a slope of 1.0. No random error was added to the model—the only predictor in the model is time and it is an exact predictor. The top plot shows the data pooled across 50 subjects. Solid line is the predicted values under the simple linear model pooled across observations. Dashed line is the 95% confidence interval. The coefficient of determination for this model was 0.02. The 95% confidence interval for the slope was —1.0, 3.0 with a point estimate of 1.00. The bottom plot shows how the data in the top plot extended to an individual. The bottom plot shows perfect correspondence between effect and time within an individual The mixed effects coefficient of determination for this data set was 1.0, as it should be. This example was meant to illustrate how the coefficient of determination using the usual linear regression formula is invalid in the mixed effects model case because it fails to account for between-subject variability and use of such a measure results in a significant underestimation of the predictive power of a covariate.

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