Getting Started

The NLIN procedure performs univariate nonlinear regression using the least squares method. Nonlinear regression analysis is indicated when you have information specifying that the functional relationship between the predictor and response variables is nonlinear in the parameters. Such information might come from direct knowledge of the true model, theoretical developments, or previous studies. Nonlinear , in this sense, means that the mathematical relationship between the variables and parameters is not required to have a linear form. For example, consider the following two models:

where a and b are parameters and X and Y are random variables. The first model is linear in the parameters; the second model is nonlinear.

Estimating the Nonlinear Model

As an example of a nonlinear regression analysis, consider the following theoretical model of enzyme kinetics. The model relates the initial velocity of an enzymatic reaction to the substrate concentration.

where x _i represents the amount of substrate for n trials and f ( x, ) is the velocity of the reaction. The vector contains the rate parameters.

Suppose that you want to study the relationship between concentration and velocity for a particular enzyme/substrate pair. You record the reaction rate (velocity) observed at different substrate concentrations. Your data set is as follows :

  data Enzyme;   input Concentration Velocity @@;   datalines;   0.26 124.7   0.30 126.9   0.48 135.9   0.50 137.6   0.54 139.6   0.68 141.1   0.82 142.8   1.14 147.6   1.28 149.8   1.38 149.4   1.80 153.9   2.30 152.5   2.44 154.5   2.48 154.7   ;  

The SAS data set Enzyme contains the two variables Concentration (substrate concentration) and Velocity (reaction rate). The double trailing at sign (@@) in the INPUT statement specifies that observations are input from each line until all of the values are read.

The following statements request a nonlinear regression analysis:

  proc nlin data=Enzyme method=marquardt hougaard;   parms theta1=155   theta2=0 to 0.07 by 0.01;   model Velocity = theta1*Concentration / (theta2 + Concentration);   run;  

The DATA= option specifies that the SAS data set Enzyme be used in the analysis. The METHOD= option directs PROC NLIN to use the MARQUARDT iterative method. The HOUGAARD option requests that a skewness measure be calculated for the parameters.

The MODEL statement specifies the enzymatic reaction model

where V represents the velocity or reaction rate and C represents the substrate concentration.

The PARMS statement declares the parameters and specifies their initial values. In this example, the initial estimates in the PARMS statement are obtained as follows. Since the model is a monotonic increasing function in C , and

take the largest observed value of the variable Velocity (154.7) as the initial value for the parameter Theta1 . Thus, the PARMS statement specifies 155 as the initial value for Theta1 , which is approximately equal to the maximum observed velocity.

To obtain an initial value for the parameter theta ₂ , first rearrange the model equation to solve for ₂ :

By substituting the initial value of Theta1 for ₁ and taking each pair of observed values of Concentration and Velocity for C and V , respectively, you obtain a set of possible starting values for Theta2 ranging from about 0.01 to 0.07.

You can choose any value within this range as a starting value for Theta2 , or you can direct PROC NLIN to perform a preliminary search for the best initial Theta2 value within that range of values. The PARMS statement specifies a range of values for Theta2 , which results in a search over the grid points from 0 to 0.07 in increments of 0.01. The output from this PROC NLIN invocation are displayed in the following figures.

PROC NLIN evaluates the model at each point on the specified grid for the Theta2 parameter. Figure 50.1 displays the calculations resulting from the grid search.

  The NLIN Procedure   Dependent Variable Velocity   Grid Search   Sum of   theta1      theta2     Squares   155.0           0      3075.4   155.0      0.0100      2074.1   155.0      0.0200      1310.3   155.0      0.0300       752.0   155.0      0.0400       371.9   155.0      0.0500       147.2   155.0      0.0600     58.1130   155.0      0.0700     87.9662   The NLIN Procedure   Dependent Variable Velocity   Method: Marquardt   Iterative Phase   Sum of   Iter      theta1      theta2     Squares   0       155.0      0.0600     58.1130   1       158.0      0.0736     19.7017   2       158.1      0.0741     19.6606   3       158.1      0.0741     19.6606   NOTE: Convergence criterion met.  

Figure 50.1: Nonlinear Least Squares Grid Search from the NLIN Procedure

The parameter Theta1 is held constant at its specified initial value of 155, the grid is traversed, and the residual sums of squares are computed at each point. The best starting value is the point that corresponds to the smallest value of the residual sum of squares. Figure 50.1 shows that the best starting value for Theta2 is 0.06. PROC NLIN uses this point as the initial value for Theta2 in the following iterative phase.

PROC NLIN determines convergence using the relative offset measure of Bates and Watts (1981). When this measure is less than 10 ^{ˆ’ 5} , convergence is declared. Figure 50.1 displays the iteration history.

Figure 50.2 displays a summary of the estimation including several convergence measures R, PPC, RPC, and Object.

  The NLIN Procedure   Estimation Summary   Method                  Marquardt   Iterations                      3   R                        5.861E-6   PPC(theta2)              8.569E-7   RPC(theta2)              0.000078   Object                   2.902E-7   Objective                19.66059   Observations Read              14   Observations Used              14   Observations Missing            0  

Figure 50.2: Estimation Summary from the NLIN Procedure

The R measure is the relative offset convergence measure of Bates and Watts. A PPC value of 8.569E-7 indicates that the parameter Theta2 (which has the largest PPC value of all the parameters) would change by that relative amount were PROC NLIN to take an additional iteration step. The RPC value indicates that Theta2 changed by 0.000078, relative to its value in the last iteration. These changes are measured before steplength adjustments are made. The Object measure indicates that the objective function value changed 2.902E-7 in relative value from the last iteration.

Figure 50.3 displays the least squares summary statistics for the model. The degrees of freedom, sums of squares, and mean squares are listed.

  The NLIN Procedure   NOTE: An intercept was not specified for this model.   Sum of        Mean               Approx   Source                    DF     Squares      Square    F Value    Pr > F   Model                      2      290116      145058    88537.2    <.0001   Error                     12     19.6606      1.6384   Uncorrected Total         14      290135  

Figure 50.3: Nonlinear Least Squares Summary from the NLIN Procedure

Figure 50.4 displays the estimates for each parameter, the associated asymptotic standard error, and the upper and lower values for the asymptotic 95% confidence interval. PROC NLIN also displays the asymptotic correlations between the estimated parameters (not shown).

  The NLIN Procedure   Approx       Approximate 95%   Parameter      Estimate    Std Error      Confidence Limits     Skewness   theta1            158.1       0.6737       156.6       159.6      0.0152   theta2           0.0741      0.00313      0.0673      0.0809      0.0362  

Figure 50.4: Parameter Estimates from the NLIN Procedure

The skewness measures of 0.0152 and 0.0362 indicate that the parameters are nearly linear and that their standard errors and confidence intervals can be safely used for inferences.

Thus, the estimated nonlinear model relating reaction velocity and substrate concentration can be written as

where V represents the velocity or rate of the reaction, and C represents the substrate concentration.