# Reciprocal Variance

With most data sets, specification of appropriate weight for each data point or weight scheme for a data set are recommended. The best weight is the reciprocal of the variance of the data point. This value may be available for each point or may be available as an a equation for a data set.

Large range of data values, variable error in the data or relatively large error in the data are all good reasons for considering applying a weight to each data point. It is essential to apply an appropriate weight to data points when different data sets are being fit simultaneously. This is especially true when the data sets have different values.

Figure 13.4.1 Drug Concentrations after IV Bolus Dose Administration

There is a considerable range of concentration values in Figure 13.4.1. Weighting these data points properly during non linear regression analysis would be an important part of the process.

Figure 13.4.2 Plasma and Urine Data after IV Bolus Dose Administration

When data values in different data set (fit simultaneously) are quite different in magnitude appropriate weighting is essential. In Figure 13.4.2 the magnitude of the urine data is much higher than the plasma data. If all the data were weighted equally the urine data would be fit well at the detriment of the fit to the plasma data. we could reverse this result by re-scaling the urine data to units of gram instead of mg. The now, smaller magnitude of the urine data would cause these data to be undervalued during any non linear regression analysis. The answer is to appropriately weight each data set, each data point.

Equation 13.4.1 Ideally the Weight for each data point will be equal or proportional to the Variance of the Data Point

If we know the variance of each data point we can simply calculate the weight for each data point. However, this is not always possible but we might know the way in which the variance varies with the value of the data. That is, the relationship or equation relating the value of the data point to the variance and this the weight. These relationships or weighting schemes may take a number of forms.