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Generalizing to a Population:
MODELS: IMPORTANT DETAILS continued

Linear and Non-Linear Models
A linear model is one in which the independent variable is added or multiplied together with the parameters. A non-linear model has exponents, logarithms, or other complicated functions of the independent variable and parameters.

Some non-linear models can be reduced to linear models to make it easier to do the fitting. For example, if your Y values curve upwards like a simple quadratic in relation to your X values, then it might be appropriate to fit Y = aX2. You could reduce this model to a linear one simply by introducing a new variable called S (say), which has the same values as X2. You then fit the linear model Y = aS. Some stats programs generate these new variables automatically when you fit quadratics, cubics, or other higher order polynomials. More on these shortly.

Most non-linear models can't be reduced to a simple linear model in this way. But a good stats program can fit non-linear models as complex as you like. All you do is choose the mathematical form of the model; the stats program then calculates the values of the parameters that give the best fit to your data, as explained earlier. The usual method is to minimize the sum of the squares of the residuals.

Note: Whatever model you fit, you should check visually that it really does fit the trend in the data. In other words, plot the curve and see if your points are fairly evenly scattered about it. Or get the stats program to plot residuals against predicteds from the model, then eyeball the plot to make sure you haven't got bad residuals.

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Last updated 13 March 98