North American Journal of Fisheries Management 28:847–848, 2008 Ó Copyright by the American Fisheries Society 2008 DOI: 10.1577/M07-180.1
[Comment]
Use of Piecewise Regression Models to Estimate Changing Relationships in Fisheries: Response to Comment
Brenden and Bence (Comment, this issue) reexamined the use of piecewise nonlinear regression to estimate variable size-related mortality rates as reported in Maceina (2007). We agree with them that both fitting piecewise regression models and conducting nonlinear analyses are challenging and that analysts should exercise caution when using these techniques. Brenden and Bence provide some valid insights into and useful additional information about the method as originally presented. Brenden and Bence are correct in their assertion that varying the initial starting value for the knot from 10 to 11 years slightly changes the estimated parameter coefficients and the value of the knot. Initial knot values of 5–10 in the PARMS statement in Statistical Analysis System (SAS; SAS Institute 2003) yield a knot of 10.3 years, identical to the results originally presented. Initial knot values of 11–15 yield a knot of 11.2 (as Brenden and Bence state), and initial values of 16 and greater yield even higher estimates of the knot in the relationship between age and number at age. As Brenden and Bence infer, the lowest residual sum of squares (RSS, or error) for the model was observed when the estimated knot was 10.3 years. Thus, minimization of RSS can be one criterion for optimal knot and model estimation in piecewise regression. The analyst can simply employ different initial starting values for the knot and compare the resulting RSS values. Brenden and Bence further recommend the use of grid-search methods for choosing the initial starting values and identify software that can facilitate this. Brenden and Bence bring up a good point on model selection and whether to use a simple linear catch curve or piecewise nonlinear regression in the case at hand. Maceina (2007) reported that the coefficient of determination (R2) increased from 0.87 with simply linear regression to 0.94 with piecewise nonlinear regression, which justifies selection of the nonlinear model as the better model. However, an improvement in R2 should not be the only criterion for selecting the best model. In the present example, the analyst could plot the residual values against the predicted values from the linear catch-curve regression using number at age to visually assess whether the relation is linear. For the simple linear regression of loge transformed number at age on age (data in Maceina 2007), the plot clearly
shows that this relationship is not linear—a pattern of positive, then negative, then positive residuals emerges for progressively higher predicted values for number at age (Figure 1). This strongly suggests that a nonlinear approach be explored for these two variables. Brenden and Bence state that the R2 values computed for the two piecewise models originally presented were ‘‘apparently unadjusted.’’ In nonlinear regression, the sum of the residuals may not equal 0 and there may be no ‘‘true’’ intercept; hence, the corrected sum of squares (SS) may have no meaning (Freund and Littell 1991). In addition, the RSS and regression SS may not sum to the total SS for nonlinear models and thus the R2 values and associated mean squares used in F-tests will probably be incorrect (Neter et al. 1996; Maceina and Pereira 2007). To circumvent this dilemma, Maceina and Pereira (2007) recommended regressing the predicted values generated from nonlinear regression against the observed values to compute the R2 value. This method also provides a more accurate mean square error and F-test, as the parameters in nonlinear regression are simultaneously determined and are not independent. In the present example, four parameters (including the intercept) were computed, so the SAS (SAS Institute 2003) output showed 3 model or regression degrees of freedom (2 for the slopes and 1 for the knot [as in linear regression, the intercept was not included in the determination of df) for the piecewise model, but age was the only independent variable. To improve the model selection process using piecewise nonlinear regression, Brenden and Bence recommend using information tools such as permutation tests and the Bayes information criterion (BIC). As they indicate, the software program Joinpoint 3.0 (National Cancer Institute 2005) provides both of these statistics to assist in fitting the best piecewise regression. Permutation tests compute the statistical significance of adding additional knots (or joints) and segments to the regression (the minimum and maximum number of knots are chosen in Joinpoint 3.0), providing the analyst with a ‘‘final’’ model that is statistically valid for the number of joints. This test controls the error probability of selecting the wrong model at a certain level. The BIC weighs the trade-off between reducing the RSS and overspecifying the
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FIGURE 1.—Scatterplot of the residuals versus the predicted values of loge transformed number at age from a linear catchcurve regression with data for blue catfish Ictalurus furcatus (source: Maceina 2007).
model by the addition of parameters. The PROC NLIN program in SAS (SAS Institute 2003) does not provide permutation tests or the BIC statistic, but the latter could be programmed in SAS (SAS Institute 2003) or easily computed with a hand calculator from the SAS (SAS Institute 2003) output. Joinpoint 3.0 only reports the RSS and does not provide the model or total corrected sums of squares. Thus, the coefficient of determination cannot be computed with Joinpoint 3.0. We agree with the information provided by Brenden and Bence, and Joinpoint 3.0 provides the analyst with a useful statistical tool for conducting piecewise regression. However, we feel that analysts should report the fit or coefficient of determination for any regression, and PROC NLIN in SAS (SAS Institute 2003) provides data from which this value can easily be estimated. Alternatively, the methods presented by Maceina and Pereira (2007) can be used to compute the coefficients of determination for all nonlinear regres-
sions in which predicted values of the dependent variable generated from nonlinear regression are linearly regressed on the observed values, and at times more accurately. Maceina and Pereira (2007) also recommend other computational and statistical tools that can be computed in SAS (SAS Institute 2003) to examine the validity any nonlinear regression. Thus, we recommend that both Joinpoint 3.0 and SAS (SAS Institute 2003) be used to conduct piecewise regression analysis. Finally, although a model may be the best statistical model regardless of what selection criteria or computer program are used, ultimately the analyst is responsible for selecting the model that provides the best biological explanation of the data. MICHAEL J. MACEINA* RYAN H. HUNTER Department of Fisheries, Auburn University, Auburn, Alabama 36849, USA * Corresponding author:
[email protected] Published online June 9, 2008
References Freund, R. J., and R. C. Littell. 1991. SAS system for regression. SAS Institute, Cary, North Carolina. Maceina, M. J. 2007. Use of piecewise nonlinear models to estimate variable size-related mortality rates. North American Journal of Fisheries Management 27:971–977. Maceina, M. J., and D. L. Pereira. 2007. Recruitment. Pages 121–185 in C. Guy and M. L. Brown, editors. Analysis and interpretation of freshwater fisheries data. American Fisheries Society, Bethesda, Maryland. National Cancer Institute. 2005. Joinpoint regression program, version 3.0. National Cancer Institute, Silver Spring, Maryland. Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. 1996. Applied linear statistics models, 4th edition. McGraw-Hill, Boston. SAS Institute. 2003. SAS/STAT user’s guide, version 9.1. SAS Institute, Cary, North Carolina.
