Recommendations for use of retransformation methods in regression models used to estimate sediment loads ("The bias correction problem")

In Reply Refer To:                               December 31, 1992
Mail Stop 415


Subject:  Recommendations for use of retransformation methods 
          in regression models used to estimate sediment 
          loads ("The bias correction problem")

Estimates of suspended-sediment loads are often derived from 
periodic data using regression models.  Many of the regression 
models involve transformation into logarithmic space but final 
results are often required to be in the original engineering 
units; therefore, retransformation of load data is needed.  This 
retransformation involves a "bias correction problem" that has 
received much attention. 

Systems Analysis Technical Report 91.01, "Estimating Loads from 
Periodic Records," by T. A. Cohn and E. J. Gilroy, identifies the 
problem of bias in computing sediment loads from transport curves 
and describes methods of handling the bias correction.  The 
authors recommended the Minimum Variance Unbiased Estimator (MVUE) 
for use when the errors can be assumed to be normally distributed 
and they recommend the Smearing Estimator (SM) when a non-normal 
error distribution is identified.  The Office of Surface Water 
endorses these recommendations and notes that they give 
considerable flexibility to the analyst to determine the most 
appropriate method for a given situation.

Although the focus of this memorandum is on an appropriate bias 
correction factor, it is well worth emphasizing that mis-
specification of the appropriate regression model in a particular 
situation can yield sizable errors and render any care taken in 
correcting for bias as a useless exercise.

Attached to this memorandum is a short example calculation for 
loads showing the application of three methods of bias correction:

1.  The Quasi-Maximum Likelihood Estimator (QMLE) 
      [Ferguson method], 

2.  The Minimum Variance Unbiased Estimator (MVUE), and 

3.  The Smearing Estimator (SM).

The Ferguson method is not recommended for use but is presented in 
the example because it has been used extensively in the past. The 
example also contains a FORTRAN program which is needed to 
implement the MVUE method.

The report by T. A. Cohn and E. J. Gilroy also addresses issues 
such as transforming the response variable, form of the model to 
be used, and other covariates to consider (e.g., time trends, 
seasonality, flow dependence, and non-linear terms).  Although not 
covered in depth in this report, attention is given to sampling 
questions such as; are data representative of target population, 
what are the dominant physical processes operating at the sites in 
question, and are there outliers in the data set?  Some background 
in statistical analysis and terminology is needed to fully 
comprehend the report.  In particular, the reader should be 
familiar with statistical notation and have an understanding of 
standard statistical analysis and simple linear regression.

                                 Charles W. Boning, Chief
                                 Office of Surface Water