Question

Discuss the differences in a regression model between making the random error being multiplicative and making the random error being additive regarding how you approach estimation of the model coefficient(s), how you apply linearization for estimating the model coefficient(s), and how you obtain starting values for estimation of the model coefficient(s).

Answer 1

The additive error model is defined as

Yi=b0 + b1Xi +ei

where i is the index of a datum; Xi is the reference data,

assumed error free; Yi is a measurement; b0 is the offset; b1

is a scale parameter to represent the differences in the

dynamic ranges between the reference data and the measurements; and ei is an instance of the random error which has z ero mean and variance of sigma^2

Thus, this model is defined by three parameters, namely, b0, b1, and sigma. Both b0 and b1 specify the systematic error, which is deterministic. Therefore, once b0 and b1 are determined, the uncertainty in the measurements Yi is quantified by sigma.

On the other hand, the multiplicative model is defined as

Yi=b0* Xi^b1* exp(ei)

In this model, the random error exp(ei) is a multiplicative

factor, with the mean of ei being zero and the variance sigma^2

The systematic error, defined by b0 and b1, is a nonlinear function of the reference data. Though less frequently used in

precipitation error models, it has been widely adopted in many other fields, such as biostatistics. Apparently, the values of sigma in the additive model and in the multiplicative model will be different from one another.

Now, the additive model is a simple linear regression, the parameters can be estimated easily with the ordinary least squares (OLS) as well, assuming the random errors (or “residuals” in the case of OLS) are uncorrelated with a constant variance sigma^2

Meanwhile, if we perform a natural logarithm transformation of the variables, the multiplicative model becomes

ln(Yi) = lnb0 +b1ln(Xi) +ei

which is also a simple linear regression in the transformed

domain, and the parameters can be estimated with the same

OLS procedure.

Essentially, the additive error model defines the error as

the difference between the measurement and the truth, while the

multiplicative error model defines the error as the ratio between

the two.

Estimation of coefficients by linearization :