Regression equation and parameter estimates for Nonlinear Regression

Find definitions and interpretation guidance for the regression equation and every statistic in the Parameter Estimates table.

Equation

Use the regression equation to describe the relationship between the response and the terms in the model. The regression equation is an algebraic representation of the regression line. Enter the value of each predictor into the equation to calculate the mean response value. Unlike linear regression, a nonlinear regression equation can take many forms.

For nonlinear equations, determining the effect that each predictor has on the response can be less intuitive than it is for linear equations. Unlike the parameter estimates in linear models, there is no consistent interpretation for the parameter estimates in nonlinear models. The correct interpretation for each parameter depends on the expectation function and the parameter's place in it. If your nonlinear model contains only one predictor, assess the fitted line plot to see the relationship between the predictor and response.

Convergence on a solution does not necessarily guarantee that the model fit is optimal or that the sum of squared errors (SSE) are minimized. Convergence on incorrect parameter values can occur due to a local SSE minimum or an incorrect expectation function. Therefore, it is crucial to examine the parameter values, fitted line plot, and residual plots, to determine if the model fit and parameter values are reasonable.

Interpretation

In these results, there is one predictor and seven parameter estimates. The response variable is Expansion and the predictor variable is temperature on the Kelvin scale. The lengthy equation describes the relationship between the response and the predictors. The effect that a 1 degree Kelvin increase has on copper expansion highly depends on the starting temperature. The effect of changing temperatures on copper expansion cannot be easily summarized. Assess the fitted line plot to see the relationship between the predictor and response.

If you enter a value for temperature in Kelvin into the equation, the result is the fitted value for copper expansion.

Equation Expansion = (1.07764 - 0.122693 * Kelvin + 0.00408638 * Kelvin ** 2 - 1.42627e-006 * Kelvin ** 3) / (1 - 0.00576099 * Kelvin + 0.000240537 * Kelvin ** 2 - 1.23144e-007 * Kelvin ** 3)

Estimate

If the algorithm converged on the parameter values correctly, the set of parameter estimates minimize the sum of squared errors (SSE).

Convergence on a solution does not necessarily guarantee that the model fit is optimal or that the sum of squared errors (SSE) are minimized. Convergence on incorrect parameter values can occur due to a local SSE minimum or an incorrect expectation function. Therefore, it is crucial to examine the parameter values, fitted line plot, and residual plots, to determine if the model fit and parameter values are reasonable.

Interpretation

For nonlinear equations, determining the effect that each predictor has on the response can be less intuitive than it is for linear equations. Unlike the parameter estimates in linear models, there is no consistent interpretation for the parameter estimates in nonlinear models. The correct interpretation for each parameter depends on the expectation function and the parameter's place in it. If your nonlinear model contains only one predictor, assess the fitted line plot to see the relationship between the predictor and response.

In these results, there is one predictor and seven parameter estimates. The response variable is Expansion and the predictor variable is temperature on the Kelvin scale. The lengthy equation describes the relationship between the response and the predictors. The effect that a 1 degree Kelvin increase has on copper expansion highly depends on the starting temperature. The effect of changing temperatures on copper expansion cannot be easily summarized. Assess the fitted line plot to see the relationship between the predictor and response.

Parameter Estimates Parameter Estimate SE Estimate 95% CI b1 1.07764 0.170702 ( 0.744913, 1.42486) b2 -0.12269 0.012000 (-0.147378, -0.09951) b3 0.00409 0.000225 ( 0.003655, 0.00455) b4 -0.00000 0.000000 (-0.000002, -0.00000) b5 -0.00576 0.000247 (-0.006246, -0.00527) b6 0.00024 0.000010 ( 0.000221, 0.00026) b7 -0.00000 0.000000 (-0.000000, -0.00000) Expansion = (b1 + b2 * Kelvin + b3 * Kelvin ** 2 + b4 * Kelvin ** 3) / (1 + b5 * Kelvin + b6 * Kelvin ** 2 + b7 * Kelvin ** 3)

SE Estimate

The standard error of the estimate (SE Estimate) estimates the variability between parameter estimates that you would obtain if you took samples from the same population again and again.

Interpretation

Use the standard error of the estimate to measure the precision of the parameter estimate. The smaller the standard error, the more precise the estimate.

95% CI

These confidence intervals (CI) are ranges of values that are likely to contain the true value of each parameter in the model.

Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. However, if you take many random samples, a certain percentage of the resulting confidence intervals contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.

The confidence interval is composed of the following two parts:
Point estimate
This single value estimates a population parameter by using your sample data. The confidence interval is centered around the point estimate.
Margin of error
The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the margin of error is added to the point estimate. To calculate the lower limit of the confidence interval, the margin of error is subtracted from the point estimate.

Interpretation

Use the confidence intervals to assess the estimate of each parameter estimate.

For example, with a 95% confidence level, you can be 95% confident that the confidence interval contains the value of the parameter for the population. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size.

If you need to determine whether a parameter estimate is statistically significant, use the confidence intervals for the parameters. The parameter is statistically significant if the range excludes the null hypothesis value. Minitab cannot calculate p-values for parameters in nonlinear regression. For linear regression, the null hypothesis value for every parameter is zero, for no effect, and the p-value is based on this value. However, in nonlinear regression, the correct null hypothesis value for each parameter depends on the expectation function and the parameter's place in it.

For some data sets, expectation functions, and confidence levels, it is possible that one or both confidence bounds may not exist. Minitab indicates missing results with an asterisk in the Session window. If the confidence interval has a missing bound, a lower confidence level might produce a two-sided interval.

Correlation matrix for parameter estimates

The matrix displays the correlation between parameter estimates. If parameter estimates are highly correlated, consider reducing the number of parameters to simplify the model.

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