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37

Weighted Least Squares

KEY WORDS calibration, confidence interval, constant variance, ion chromatography, least squares,
nitrate, regression, simulation, variance, weighted least squares, weights, weighted regression.

Weighted least squares (weighted regression) is used to fit a model y = f(β, x) when the variance of the
response y is not constant. This method is often needed to fit calibration curves where x and y cover
wide ranges of concentration and response signal. As the range of values gets larger, it becomes more
likely that there will be some proportionality between the level of the response and the variance of the
response.
Figure 37.1 shows four possible situations: (a) a straight line with random measurement errors of equal
variance at all concentrations, (b) a straight line with random measurement errors that are proportional
to concentration, (c) a curved function with random measurement errors of constant variance, and (d) a
curved function with random measurement errors that are proportional to concentration.
For curves a and c the magnitude of the error in y is the same over the full range of x. We can think
of these curves as having random measurement error in the range of ± ∆ mg/L added or subtracted from
the true response, where ∆ does not change as the concentration changes. In contrast, for curves b and
d the magnitude of the error increases as x and y increase. The error for these curves tends to be a
percentage of the response. We can think of these curves as having random measurement error in the
range of ± ∆% of the true response level added or subtracted from the true response. Then the measure-
ment error in units of milliliters per liter (mg/L) does change over the range of concentration values.
Calibration curve a is the most familiar (see Chapter 36) but there is no theoretical reason for assuming
that a straight line will be the correct calibration model. There are factors inherent in some measurement
processes that make curve b much more likely than curve a (e.g., ICP instruments). Also, there are some
instruments for which a straight-line calibration is not expected (e.g., HPLC instruments). Therefore,
the analyst must know how to recognize and how to treat calibration data for these four patterns.
The analyst must decide whether the data have constant variance and if not, what weights should be
assigned to each y value. In addition, the analyst is also trying to decide the form of the fitted function
(straight line or some curved function). These decisions will have a profound effect on confidence
intervals and prediction intervals. The confidence intervals of parameter estimates will be wrong if
weighted regression has not been used when there is nonconstant variance. Also, prediction intervals
for the model line and specimen concentrations will be wrong unless proper weighting and model form
have been used.

Case Study: Ion Chromatograph Calibration
The nitrate calibration data in Table 37.1 were made on a Dionex-120 ion chromatograph. There are
triplicate measurements at 13 concentration levels covering a range from 0 to 40 mg/L nitrate.
Suppose we assume that the calibration model is a straight line and fit it using ordinary least squares
to obtain:
y ˆ = −3246 + 8795x

© 2002 By CRC Press LLC