Statistical methods to compare batch flotation grade-recovery curves and rate constants

Abstract

Grade-recovery curves obtained from kinetic batch flotation testing are, like any other measurement, subject to experimental error. This leads to uncertainty in the true position of each cumulative grade-recovery point, the curve itself, and the kinetics. This uncertainty is rarely if ever taken into account when interpreting such curves, in particular when comparing curves obtained under different conditions. This paper proposes a methodology to deal with this problem.The standard formula is used to establish true confidence intervals for the grade and recovery at each replicated timed concentrate point, and the 2-sample t-test is used to compare these point values between tests conducted under different conditions. The properties of the grade-recovery curves can be compared by fitting an appropriate model to the two data sets and using a bootstrap to create distributions of differences between the model parameters and the model predictions of recovery at any chosen concentrate grade, reflecting the uncertainty in the original data. It is then easy to construct hypothesis tests on the parameter differences and on the mean difference at the chosen grade(s) between the two curves. The same approach can be used to construct confidence intervals on the fitted curves and to test differences in estimated flotation rates. An extra sum of squares test can be used to compare the fitted grade-recovery curves as a whole. Details of the methods are presented, suitable for spreadsheets.These methods are relatively easy to apply but require that all batch flotation tests be replicated. The alternative (single tests under each condition) ignores the existence of experimental error and renders the data susceptible to subjective and perhaps erroneous interpretation. Using these methods it is not unusual to find that grade-recovery curves thought to represent truly different flotation performance are not in fact statistically different, especially at the longer flotation times (high recoveries).