Measures of Nonlinearity and Diagnostics |
A "close-to-linear" nonlinear regression model, in the sense of Ratkowsky (1983, 1990), is a model in which parameter estimators have properties similar to those in a linear regression model. That is, the least squares estimators of the parameters are close to being unbiased and normally distributed, and they have minimum variance.
A nonlinear regression model sometimes fails to be close to linear due to the properties of one or several parameters. When this occurs, bias in the parameter estimates can render inferences that use the reported standard errors and confidence limits invalid.
PROC NLIN provides various measures of nonlinearity. To assess the nonlinearity of a model-data combination, you can use both of the following complementary sets of measures:
Box’s bias (Box 1971) and Hougaard’s skewness (Hougaard 1982, 1985) of the least squares parameter estimates
curvature measures of nonlinearity (Bates and Watts 1980).
Furthermore, PROC NLIN provides residual, leverage, and local-influence diagnostics (St. Laurent and Cook 1993).
In the following several sections, these nonlinearity measures and diagnostics are discussed. For this material, several basic definitions are required. Let be the Jacobian matrix for the model, , and let and be the components of the QR decomposition of of , where is an orthogonal matrix. Finally, let be the inverse of the matrix constructed from the first rows of the dimensional matrix (that is, ). Next define
where , and the acceleration array are three-dimensional matrices. The first faces of the acceleration array constitute a parameteter-effects array and the last faces constitute the intrinsic curvature array (Bates and Watts 1980). The previous and subsequent quantities are computed at the least squares parameter estimators.
The degree to which parameter estimators exhibit close-to-linear behavior can be assessed with Box’s bias (Box 1971) and Hougaard’s measure of skewness (Hougaard 1982, 1985). The bias and percentage bias measures are available through the BIAS option in the PROC NLIN statement. Box’s bias measure is defined as
where if the SIGSQ option is not set. Otherwise, is the value you set with the SIGSQ option. is the diagonal weight matrix specified with the _WEIGHT_ variable (or the identity matrix if _WEIGHT_ is not defined) and is the Hessian matrix at the th observation. In the case of unweighted least squares, the bias formula can be expressed in terms of the acceleration array ,
As the preceeding formulas illustrate, the bias depends solely on the parameter-effects array, thereby permitting its reduction through reparameterization. Example 62.4 shows how changing the parameterization of a four-parameter logistic model can reduce the bias. Ratkowsky (1983, p. 21) recommends that you consider reparameterization if the percentage bias exceeds .
In addition to Box’s bias, Hougaard’s measure of skewness, (Hougaard 1982, 1985), is also provided in PROC NLIN to assess the close-to-linear behavior of parameter estimators. This measure is available through the HOUGAARD option in the PROC NLIN statement. Hougaard’s skewness measure for the th parameter is based on the third central moment, defined as
where the sum is a triple sum over the number of parameters and
The term denotes the value in row , column of the matrix . (Hougaard (1985) uses superscript notation to denote elements in this inverse.) The matrix is a three-dimensional array
The third central moment is then normalized using the standard error as
The previous expressions depend on the unknown values of the parameters and on the residual variance . In order to evaluate the Hougaard measure in a particular data set, the NLIN procedure computes
Following Ratkowsky (1990, p. 28), the parameter is considered to be very close to linear, reasonably close, skewed, or quite nonlinear according to the absolute value of the Hougaard measure being less than 0.1, between 0.1 and 0.25, between 0.25 and 1, or greater than 1, respectively.
Bates and Watts (1980) formulated the maximum parameter-effects and maximum intrinsic curvature measures of nonlinearity to assess the close-to-linear behavior of nonlinear models. Ratkowsky (1990) notes that of the two curvature components in a nonlinear model, the parameter-effects curvature is typically larger. It is this component that you can affect by changing the parameterization of a model. PROC NLIN provides these two measures of curvature both through the STATS plot-option and through the NLINMEASURES option in the PROC NLIN statement.
The maximum parameter-effects and intrinsic curvatures are defined, in a compact form, as
where and denote the maximum parameter-effects and intrinsic curvatures, while and stand for the parameter-effects and intrinsic curvature arrays. The maximization is carried out over a unit-vector of the parameter values (Bates and Watts 1980). In line with Bates and Watts (1980), PROC NLIN takes as the convergence tolerance for the maximum intrinsic and parameter-effects curvatures. Note that the preceeding matrix products involve contraction of the faces of the three-dimensional acceleration arrays with the normalized parameter vector, . The corresponding expressions for the RMS (root mean square) parameter-effects and intrinsic curvatures can be found in Bates and Watts (1980).
The statistical significance of and and the corresponding RMS values can be assessed by comparing these values with , where is the upper quantile of an distribution with and degrees of freedom (Bates and Watts 1980).
One motivation for fitting a nonlinear model in a different parameterization is to obtain a particular interpretation and to give parameter estimators more close-to-linear behavior. Example 62.4 shows how changing the parameterization of a four-parameter logistic model can reduce the parameter-effects curvature and can yield a useful parameter interpretation at the same time. In addition, Example 62.6 shows a nonlinear model with a high intrinsic curvature and the corresponding diagnostics.
In contrast to linear regression, there are several measures of leverage in nonlinear regression. Furthermore, in nonlinear regression, the effect of a change in the th response on the th predicted value might depend on both the size of the change and the th response itself (St. Laurent and Cook 1992). As a result, some observations might show superleverage —namely, leverages in excess of one (St. Laurent and Cook 1992).
PROC NLIN provides two measures of leverages: tangential and Jacobian leverages through the PLOTS option in the PROC NLIN statement and the H= and J= options of OUTPUT statement. Tangential leverage, , is based on approximating the nonlinear model with a linear model that parameterizes the tangent plane at the least squares parameter estimators. In contrast, Jacobian leverage, , is simply defined as the instantaneous rate of change in the th predicted value with respect to the th response (St. Laurent and Cook 1992).
The mathematical formulas for tangential and Jacobian leverages are
where is the vector of residuals, is the diagonal weight matrix if you specify the special variable _WEIGHT_ and otherwise the identity matrix, and indexes the corresponding quantities for the th observation. The brackets indicate column multiplication as defined in Bates and Watts (1980). The preceeding formula for tangential leverage holds if the gradient, Marquardt, or Gauss methods are used. For the Newton method, the tangential leverage is set equal to the Jacobian leverage.
In a model with a large intrinsic curvature, the Jacobian and tangential leverages can be very different. In fact, the two leverages are identical only if the model provides an exact fit to the data () or the model is intrinsically linear (St. Laurent and Cook 1993). This is also illustrated by the leverage plot and nonlinearity measures provided in Example 62.6.
St. Laurent and Cook (1993) suggest using , the direction that yields the maximum normal curvature, to assess the local influence of an additive perturbation to the response variable on the estimation of the parameters and variance of a nonlinear model. Defining the normal curvature components
where is the Jacobian leverage matrix and , you choose the that results in the maximum of the two curvature components (St. Laurent and Cook 1993). PROC NLIN provides through the PLOTS option in the PROC NLIN statement and the LMAX= option in the OUTPUT statement. Example 62.6 shows a plot of for a model with high intrinsic curvature.
If a nonlinear model is intrinsically nonlinear, using the residuals for diagnostics can be misleading (Cook and Tsai 1985). This is due to the fact that in correctly specified intrinsically nonlinear models, the residuals have nonzero means and different variances, even when the original error terms have identical variances. Furthermore, the covariance between the residuals and the predicted values tends to be negative semidefinite, complicating the interpretation of plots based on (Cook and Tsai 1985).
Projected residuals are proposed by Cook and Tsai (1985) to overcome these shortcomings of residuals, which are henceforth called raw (ordinary) residuals to differentiate them from their projected counterparts. Projected residuals have zero means and are uncorrelated with the predicted values. In fact, projected residuals are identical to the raw residuals in intrinsically linear models.
PROC NLIN provides raw and projected residuals, along with their standardized forms. In addition, the mean or expectation of the raw residuals is available. These can be accessed with the PLOTS option in the PROC NLIN statement and the OUTPUT statement options PROJRES=, PROJSTUDENT=, RESEXPEC=, RESIDUAL= and STUDENT=.
Denote the projected residuals by and the expectation of the raw residuals by . Then
where is the th observation raw residual, is an n-dimensional identity matrix, is the projector onto the column space of , and . The preceeding formulas are general with the projectors defined accordingly to take the weighting into consideration. In unweighted least squares, reduces to
with being the last columns of the matrix in the QR decomposition of and the dimensional vector being defined in terms of the intrinsic acceleration array
Standardization of the projected residuals requires the variance of the projected residuals. This is estimated using the formula (Cook and Tsai 1985)
The standardized raw and projected residuals, denoted by and respectively, are given by
The use of raw and projected residuals for diagnostics in nonlinear regression is illustrated in Example 62.6.