Piecewise Regression

A regression of this kind, although rather simple to perform, is time-consuming. The process is greatly facilitated by using a computer. 

The researcher can always take each x point and perform a t-test confidence interval, and this is often the course chosen. Although from a probability perspective, this is not correct from a practical perspective, it is easy, useful, and more readily understood by audiences. We discuss this issue in greater detail using indicator or dummy variables in the multiple linear regression section of this book. 

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Two approaches for interpolation function have been used. In one, polynomials, e.g., in powers of w", are fit to impedance data. Usually, a piecewise regression is required. While piece-wise polynomials are excellent for smoothing, the best example being splines, they are not very reliable for extrapolation and result in a relatively large number of peirameters. A second approach is to use interpolation... 

However, such a highly derived process would likely make the data abstract. A preferred way, and a much simpler method, is to perform a piecewise regression, using indicator or dummy variables. We employ that method in a later chapter, where we make separate functions for each linear portion of the data set. 

Polynomial regression models are useful in situations in which the curvilinear response function is too complex to linearize by means of a transformation, and an estimated response function fits the data adequately. Generally, if the modeled polynomial is not too complex to be generalized to a wide variety of similar studies, it is useful. On the other hand, if a modeled polynomial overfits the data of one experiment, then, for each experiment, a new polynomial must be built. This is generally ineffective, as the same type of experiment must use the same model if any iterative comparisons are required. Figure 7.1 presents a dataset that can be modeled by a polynomial function, or that can be set up as a piecewise regression. It is impossible to linearize this function by a simple scale transformation. 

Figure 7.2 shows another problem—that of inadequate sample points within the x,s. The large gaps between the x,s represent unknown data points. If the model were fit via a polynomial or piecewise regression with both replication and repeated measurements, the model would still be inadequate. This is because the need for sufficient data, specified in step 1, was ignored. 

The goal in piecewise regression is to model a nonlinear model by linear pieces, for example, by conducting a microbial inactivation study that is nonlinear. 

Figure 9.15 provides a residual plot e vs. Xi) of the piecewise regression residuals, one that appears far better than the previous residual plot (Figure 9.13). 

Figure 9.16 depicts schematically the piecewise regression functions. 

The extension of the piecewise regression to more complex designs is straightforward. For example, in bioequivalence studies, absorption and elimination rates are often evaluated over time, and the collected data are not linear. Figure 9.17 shows one possibility. 

FIGURE 9.15 Residual plot, piecewise regression. Example 9.2. 

The use of piecewise regression beyond two pivots is merely a continuation of the two-pivot model. The actual value y is, however, given at any one point on the entire equation range, without deconstructing it, as in Figure 9.22. 

This phenomenon can be modeled using piecewise regression analysis (Figure 9.24). 

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