The LCGC Blog: Would Your Process Benefit from Multivariate Optimization?

If you have a method or process that involves a number of different variables, multivariate optimization approaches can provide a faster route to optimum conditions and can lead to a more reliable outcome than using a one-factor-at-a-time approach. With a little study and practice, students and researchers can apply these optimization techniques, even if a complete understanding of the underlying statistical treatments is not immediately apparent.

Some years ago, when my first student, the now Dr. Misjudeen Raji, told me he would like to learn to use factorial design to study and optimize electrospray ionization response, I was intrigued. Through my own training, the concepts of factorial design were foreign to me. But through that work and since, I have come to have a great appreciation of the power of experimental design through multivariate optimization.

From a very basic viewpoint, I am talking about a process where, if multiple variables exist to be optimized, then a very clear framework can be easily established for experimental design to explore the responses induced by changes in that variable space. The alternative approach is to painstakingly optimize the process one factor at a time. The one-factor-at-a-time (OFAT) approach may be viable, but it can also be time consuming, and ultimately the desired maximum or minimum may not be reached if some of the variables are codependent, or interact with each other. OFAT optimization may not account for the possibility that multiple combinations of some variables can generate multiple maxima and minima, unless the process is iterated. The desired global maxima or minima may be missed.

There are many methods that can be used to perform multivariate optimization. How to begin depends on a compromise between how many samples can be practically run in a given amount of time and how comprehensive a coverage of the variable space is needed or desired. If there are a large number of variables, and each variable’s importance to the process is not clear, then it may be desirable to perform multiple stages of

Factorial design experiments are often a good place to start, because they have a lot of flexibility in terms of finding a compromise between the number of analyses needed and the variable coverage. In 2009, we published a paper on the use of full factorial design to study the main effects and interaction effects of electrospray ion source variables on ion response for two different commercial systems (1). We found a key variable for optimization was the voltage of the ion lens located just behind the desolvation line, in the entrance of the high vacuum region of the mass spectrometers. However, key to understanding the effect of this variable was that its optimal setting was codependent with other variables, such as spray capillary voltage. So, these factorial design experiments can provide knowledge about the importance of variables and about variable interactions and codependence, as well as a means to better locate global maxima and minima compared to OFAT approaches.

In 2013, associated with the development of a new ambient ionization approach, continuous flow–extractive desorption electrospray ionization (CF-EDESI) (2), a partial factorial design experiment was used to evaluate the effect of different instrumental settings (flow rates, voltages, temperatures) on ion response (3). A particular novelty of this work was its use for generating ions from solvents not amenable to electrospray ionization (ESI). Again, a plethora of variables and their interactions could be systematically investigated. A partial factorial design was used to limit the number of analytical runs; the compromise was less complete coverage of the variable space. For example, if each variable is investigated at a high and low setting, this presupposes a linear trend in the measured response. For the most important variables, it may be necessary, as a second step, to evaluate whether any maxima or minima in response exist between the high and low variable settings.

Very recently, we used a multistep multivariate process to optimize a multicomponent derivatization scheme for liquid chromatography–mass spectrometry (LC–MS) determination of benzene, toluene, ethyl benzene, and xylene (BTEX) metabolites (4). Derivatization facilitates significantly improved detection limits relative to the native metabolites. The derivatization scheme has been designed to simultaneously modify both carboxylic acid and phenolic metabolites. With this approach, a method boasting the largest coverage of BTEX metabolites has been developed. First, a factorial design was used to refine the variable space. A specific combination of the optimum amounts of reagents was desired. Next, a central composite design (CCD) experiment was performed. CCD expands the variable space investigated by traditional factorial design, especially including intentional “star” points, which are outside the typical variable range, to provide better confidence in the maxima and minima located by running the prescribed experiments.

Currently, we are using factorial design and CCD methodology to generate method development guidelines for the application of online supercritical fluid extraction–supercritical fluid chromatography– mass spectrometry. This is a perfect opportunity for multivariate optimization, given the wide applicability of supercritical fluid extraction and separations, as well as the large number of variables a user can change to develop a method. The multivariate methods allow for a more systematic assessment of the effects of different variables than can be reasonably accomplished with a OFAT approach.

A variety of affordable and commercial multivariate design software packages are available. Many of these also have limited-timeframe freeware licenses to test their capabilities. With a little study and practice, students and researchers can apply the different techniques, even if a complete understanding of the underlying statistical treatments is not immediately apparent. If you have a process that involves a number of different variables, multivariate optimization approaches can provide a faster route to optimum conditions. Further, the outcomes will be more reliable, because such strategies will also account for variable interactions, which can be often overlooked by traditional OFAT optimization.


  1. M.A. Raji, K.A. Schug, Int. J. Mass Spectrom.279, 100–106 (2009).
  2. S.H. Yang, A.B. Wijeratne, L. Li, B.L. Edwards, K.A. Schug, Anal. Chem. 83, 643–647 (2011).
  3. L. Li, S.H. Yang, K. Lemr, V. Havlicek, K.A. Schug, Anal. Chim. Acta769, 84–90 (2013).
  4. Y.Z. Baghdady, K.A. Schug, Anal. Chim. Acta1036, 195–203 (2018).

Kevin A. Schug is a Full Professor and Shimadzu Distinguished Professor of Analytical Chemistry in the Department of Chemistry & Biochemistry at The University of Texas (UT) at Arlington. He joined the faculty at UT Arlington in 2005 after completing a Ph.D. in Chemistry at Virginia Tech under the direction of Prof. Harold M. McNair and a post-doctoral fellowship at the University of Vienna under Prof. Wolfgang Lindner. Research in the Schug group spans fundamental and applied areas of separation science and mass spectrometry. Schug was named the LCGC Emerging Leader in Chromatography in 2009 and the 2012 American Chemical Society Division of Analytical Chemistry Young Investigator in Separation Science. He is a fellow of both the U.T. Arlington and U.T. System-Wide Academies of Distinguished Teachers.