Stat-Ease Blog

Ideally all variables other than those included in an experiment are held constant or blocked out in a controlled fashion. However, sometimes a variable that one knows will create an important effect, such as ambient temperature or humidity, cannot be controlled. In such cases it pays to collect measurements run by run. Then the results can be analyzed with and without this ‘covariate.’

Douglas Montgomery provides a great example of analysis of covariance in section 15.3 of his textbook Design and Analysis of Experiments. It details a simple comparative experiment aimed at assessing the breaking strength in pounds of monofilament-fiber produced by three machines. The process engineer collected five samples at random from each machine, measuring the diameter of each (knowing this could affect the outcome) and testing them out. The results by machine are shown below with the diameters, measured in mils (thousandths of an inch), provided in the parentheses:

1. 36 (20), 41 (25), 39 (24), 42 (25), 49 (32)
2. 40 (22), 48 (28), 39 (22), 45 (30), 44 (28)
3. 35 (21), 37 (23), 42 (26), 34 (21), 32 (15)

The data on diameter can be easily captured via a second response column alongside the strength measures. Montgomery reports that “there is no reason to believe that machines produce fibers of different diameters.” Therefore, creating a new factor column, copying in the diameters and regressing out its impact on strength leads to a clearer view of the differences attributed to the machines.

I will now show you the procedure for handling a covariate with Stat-Ease software. However, before doing so, analyze the experiment as planned and save this work so you can do a before and after comparison.

Figure 1 illustrates how to insert a new factor. As seen in the screenshot, I recommend this be done before the first controlled factor.

The Edit Info dialog box then appears. Type in the name and units of measure for the covariate and the actual range from low to high.

Press “Yes” to confirm the change in actual values when the warning pops up.

After the new factor column appears, the rows will be crossed out. However, when you copy over the covariate data, the software stops being so ‘cross’ (pun intended).

Press ahead to the analysis. Include only the main effect of the covariate in your model. The remainder of the terms involving controlled factors may go beyond linear if estimable. As a start, select the same terms as done before adding the covariate.

In this case, the model must be linear due to there being only one factor (machine) and it being categorical. The p-value on the effect increases from 0.0442 (significant at p<0.05) with only the machine modeled—not the diameter—to 0.1181 (not significant!) with diameter included as a covariate. The story becomes even more interesting by viewing the effects plots.

You can see that the least significant difference (LSD) bars decrease considerably from Figure 4 to Figure 5 without and with the covariate; respectively. That is a good sign—the fitting becomes far more precise by taking diameter (the covariate) into account. However, as Montgomery says, the process engineer reaches “exactly the opposite conclusion”—Machine 3 looking very weak (literally!) without considering the monofilament diameter, but when doing the covariate analysis, it becomes more closely aligned with the other two machines.

In conclusion, this case illustrates the value of recording external variables run-by-run throughout your experiment whenever possible. They then can be studied via covariate analysis for a more precise model of your factors and their effects.

This case is a bit tricky due to the question of whether fiber strength by machine differs due to them producing differing diameters, in which case this should be modeled as the primary response. A far less problematic example would be an experiment investigating the drying time of different types of paint in an uncontrolled environment. Obviously, the type of paint does not affect the temperature or humidity. By recording ambient conditions, the coating researcher could then see if they varied greatly during the experiment and, if so, include the data on these uncontrolled variables in the model via covariate analysis. That would be very wise!

PS: Joe Carriere, a fellow consultant at Stat-Ease, suggested I discuss this topic—very appealing to me as a chemical process engineer. He found the monofilament machine example, which I found very helpful (also good by seeing agreement in statistical results between our software and the one used by Montgomery).

PPS: For more advice on covariates, see this topic Help.

Here's the latest Publication Roundup! In these monthly posts, we'll feature recent papers that cited Design-Expert® or Stat-Ease® 360 software. Please submit your paper to us if you haven't seen it featured yet!

Featured Article

Implementation of the QbD Approach to the Analytical Method Development and Validation for the Estimation of the Treprostinil Injection Dosage Form by RP-HPLC
ACS Omega, 2025
Authors: Narasimha Raju Alluri, Mallikharjuna Rao Bandlamudi, Sujatha Kuppusamy, Shabna Roupal Morais

Mark's comments: This one hits the spot for me by deploying a solid central composite design, being succinct in presenting only the most relevant statistics on significance and model fits, showing compelling 3D pictures, and providing the data via a link to supplementary material. It is great to see how response surface methods done with Stat-Ease software led to a “novel, precise, sensitive, stable, and cost-effective” analytical method. Well done!

Be sure to check out this important study, and the other research listed below!

More new publications from April

1. Optimisation of process parameters for lignocellulosic biomass degradation by Pseudomonas sp. using response surface methodology
   International Journal of Biological Macromolecules, Volume 309, Part 1, May 2025, 142792
   Authors: Sunder, Sangita Yadav, Jitender Pal
   
2. Quality by Design Based Chromatography Technique Development and Validation for the Medicine Venetoclax (for Chronic Leukemia), in the Context of Impurities Including Degradation Products
   Biomedical Chromatography, 39: e70072, 02 April 2025
   Authors: Rajeshwari Dandabattina, Karuna Sree Merugu, Lova Gani Raju Bandaru, Haridasyam Sharathbabu, Rambabu Gundla, Naresh Kumar Katari
   
3. Optimization of the Drying Parameters for Plantain Chips using a Locally Made Tray Dryer: A Study on Drying Efficiency and Drying Rate Modeling using RSM
   Journal of Food Technology & Nutrition Sciences, 7(2):1-10, April 2025
   Authors: Arinzechukwu Hipolite Madukasi, Ifeanyichukwu Ugochukwu Onyenanu
   
4. Optimization of low-temperature nitrogen plasma in reducing fungi and aflatoxin human exposure through maize
   Scientific Reports volume 15, Article number: 11707 (2025)
   Authors: Hannah Mugure Kamano, Michael Wandayi Okoth, Wambui Kogi-Makau, Patrick Wafula Kuloba, Joshua Ombaka Owade & Patrick Murigu Kamau Njage
   
5. Smart nanocomposite of carbon quantum dots in double hydrogel (carboxymethyl cellulose/chitosan) for effectively adsorb and remove diquat herbicide: Characterization, thermodynamics, isotherms, kinetics, and optimizing through Box-Behnken Design
   International Journal of Biological Macromolecules Volume 309, Part 1, May 2025, 142806
   Authors: Wesam Abd El-Fattah, Ahlem Guesmi, Naoufel Ben Hamadi, Mohamed G. El-Desouky, Ashraf A. El-Bindary
   
6. Green Analytical Stability Indicating UHPLC Method for the Quantification of Related Impurities in Vonoprazan Formulation Applying Analytical Quality by Design
   Separation Science Plus, Volume 8 Issue 4, April 2025, e70032
   Authors: Ashwinkumar Matta, Raja Sundararajan
   
7. Developing a Model for Reducing Carbon Emissions of Construction Heavy Machinery Through ECO-Hauling and Collaboration
   Proceedings of the International Conference on Smart and Sustainable Built Environment (SASBE 2024), pp 618–628, 20 April 2025
   Authors: Milad Hosseinzadeh Moghaddam, Ehsan Asnaashari, Amrit Sagoo
   
8. Enhanced Antibacterial Effect of pH/Gelatinase-Responsive Florfenicol Nanogels Against Staphylococcus aureus
   International Journal of Nanomedicine, 2025;20:5193-5208
   Authors: Nannan Leng, Jinhuan Liu, Yongtao Jiang, Ning Du, Ali Sobhy Dawood, Samah Attia Algharib, Wanhe Luo
   
9. Amphiphilic mPEG-PLGA copolymer nanoparticles co-delivering colistin and niclosamide to treat colistin-resistant Gram-negative bacteria infections
   Communications Biology volume 8, Article number: 673 (2025)
   Authors: Kaifang Yi, Xilong Wang, Pengliang Li, Yanling Gao, Dandan He, Yushan Pan, Xiaoyuan Ma, Gongzheng Hu, Yajun Zhai

Here's the latest Publication Roundup! In these monthly posts, we'll feature recent papers that cited Design-Expert® or Stat-Ease® 360 software. Please submit your paper to us if you haven't seen it featured yet!

Featured Article

Material-sparing degradation-kinetics model for thermolabile drug stability assessment during twin-screw melt granulation – insights with gabapentin
International Journal of Pharmaceutics, Volume 674, 15 April 2025, 125421
Authors: Adwait Pradhan, Fengyuan Yang, Kapish Karan, Thomas Durig, Quyen Schwing, Brian Haight, Mark Costello, Mark Anderson, Feng Zhang

Mark's comments: "It was a pleasure to help Adwait, et al apply response surface methods (RSM) for process optimization of twin-screw melt granulation to mininimze degradation of life-enhancing drugs such as gabapentin."

Be sure to check out this important study, and the other research listed below!

More new publications from March

1. RSM and AI based machine learning for quality by design development of rivaroxaban push-pull osmotic tablets and its PBPK modeling
   Scientific Reports volume 15, Article number: 7922 (2025)
   Authors: Muhammad Talha Saleem, Muhammad Harris Shoaib, Rabia Ismail Yousuf, Fahad Siddiqui
   
2. A Comprehensive QbD Study on Bioadhesive Ocular Films for Improved Conjunctivitis Management: Insights from Design Expert Software
   Indian Journal of Pharmaceutical Education and Research, 2025; 59(1): 122-133
   Authors: Repollu Maddileti, Haranath Chinthaginjala
   
3. Novel Ketoconazole-Loaded Niosomal Gel with Carbamide for Enhanced Topical Delivery and Skin Hydration in Fungal Infections
   Journal of Pharmaceutical Innovation, Volume 20, article number 46, (2025)
   Authors: Prajitha Biju, Manjunath M. Shenoy, Rouchelle Tellis, Ramesh Bhat, Ranajit Das, Ashwini Prabhu, Mohammed Gulzar Ahmed, Vivek Ghate
   
4. Development of Bread from Different Protein Isolates, Sweet potatoes (Lam Ipomea batata) and Wheat Flour Blends
   Journal of Health, Wellness and Safety Research, Vol. 7 2025
   Authors: Adelakun O. E, Aliyu F
   
5. Optimization of the process of acetylation and carboxymethylation for a polysaccharide from Gastrodia elata and antioxidant and immunomodulatory activities test
   Scientific Reports volume 15, Article number: 8460 (2025)
   Authors: Hao Guan, Wenjie Yin, Xue Zhang, Fangyun Zhao, Tanling Cai, Xi Ling
   
6. Optimization of durability characteristics of engineered cementitious composites combined with titanium dioxide as a nanomaterial applying RSM modelling
   Scientific Reports volume 15, Article number: 9428 (2025)
   Authors: Naraindas Bheel, Imran Mir Chohan, Ahmed Saleh Alraeeini, Mamdooh Alwetaishi, Sahl Abdullah Waheeb, Loai Alkhattabi, Omrane Benjeddou
   
7. Optimizing Foam Properties of Egg White Powder-Based Foam System by Response Surface Methodology
   Black Sea Journal of Agriculture, Year 2025, Volume: 8 Issue: 2, 217 - 224, 15.03.2025
   Authors: Mehmet Güldane
   
8. Leaching Parameters Optimization and Kinetic Studies for Leaching of Copper from Zarara Hill Sulphide Ore in HCl, H₂SO₄ and HNO₃ Solutions
   Journal of Science Innovation and Technology Research, 2025 7(9)
   Authors: Mustapha Mukhtar, K. I. Omoniyi, Faizuan Abdullah

Next in our 40th anniversary “Ask an Expert” blog series is John "Jay" Davies, who's an absolute rock star when it comes to teaching and implementing DOE. He's lent us his expertise before - see this talk from our 2022 Online DOE Summit - and he shared an anecdote with our statistical experts about how he approaches switching to DOE methods when working with new groups in the Army. He kindly agreed to let us publish it as part of this series.

For the past 14 years, I’ve been a Research Physicist with the U.S. Army DEVCOM Chemical Biological Center at Aberdeen Proving Grounds, MD as a member of the Decontamination Sciences Branch, which specializes in developing techniques/chemistries to neutralize chemical warfare agents. I’m dedicated to applied statistical analysis ranging from multi-laboratory precision studies to design of experiments (DOE). The Decontamination Sciences Branch has been integrating DOE methods into many of their chemical agent decontamination research programs.

I’m happy to report that the DOE methods here at the Chemical Biological Center are really catching on. I collaborated on 24 DOEs from 2014 through 2021. Then, in 2021-22, we completed 26 DOEs across 10 different programs. I’ve been doing a lot of mixture-process DOEs with the Bio Sciences groups for synthetic biology and bio manufacturing applications, and once the other groups saw the information that we were getting from just a single day’s worth of data, they too wanted to try DOE.

Lately, I’ve changed the formula that I use for the initial consultations when visiting a group that has expressed an interest in DOE but has never used DOE before. Previously we’d go right into their project, and I’d tell them how we might construct a DOE for their application. However, I’ve found that it’s too much of a culture shock if we go right into talking about what a DOE for their application might look like. Instead, especially if I’m working with a group that has no DOE experience at all, I now devote about 1 hour to discuss DOE methods in general before we even mention their actual application. In this discussion, I reveal the major differences that they are going to see with a DOE, which are:

• We’re not going to replicate every sample, we may even have zero exact replication.
• We’re not going to test every possible combination of factors.
• Sample sizes are going to be 70% to 95% smaller than what they are used to doing.
• We are not going to change “one factor at a time”, in fact we’ll be changing all factors at once.
• The designs might look chaotic, but they are strategically created to contain a hidden orthogonal structure, along with hidden replication, that is not apparent.
• You will see that many of the DOE samples contain factor combinations that don’t seem to make sense. This is because each sample is not designed as a stand-alone shot at optimization. Rather, the samples cumulatively are working in concert to tease out the influences of each factor. This will let us fit a predictive model that we will then use to predict the optimal settings of factors.

Recently, I was following this format with a group that had never used DOE before. We had a great back-and-forth dialog as I went through the bullets above and explained a bit about each point. They asked many questions and were really following along. Then, after about an hour we got into their application and I just sketched out a prototype quadratic mixture-process DOE that I thought would give them a good idea of what the initial DOE might look like, with 30 samples in total. I then went over what some simulated outputs for the DOE generated prediction model might look like. At this point one of them stopped me and with a very perplexed look on his face, said “hold on, hold on…wait a minute here. Are you telling us that if we run just those 30 samples, we would be able to predict the optimal formulation and the optimal process setting for this system?“

This scientist had been following along, asking questions and really absorbing the information in the past hour as we walked through the DOE basics, but I could see that at that moment things were just sinking in. He realized the ramifications of what we had just discussed – typically, this group might have had to run several hundreds of samples to characterize similar systems, but with DOE they would only need about 30 samples. I responded to his question saying, “Yes that’s exactly what I’m telling you. We’ve run dozens of these mixture-process DOEs, many of them much more complex than this system, and they do work.” This individual, a mid-career researcher, then responded, “How is it possible that we have not heard of this stuff before?” I told him, “I can’t give you a good answer to that one.”

And there you have it! Let us know if you want to talk about saving time & money with DOE: our statistical experts and first-in-class software make it easier than ever.

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Here's the latest Publication Roundup! In these monthly posts, we'll feature recent papers that cited Design-Expert® or Stat-Ease® 360 software. Please submit your paper to us if you haven't seen it featured yet!

Mark's comment: Make sure to check out article #10, where the authors deploy response surface methods (with lots of impressive 3D plots!) to produce an eco-friendly material for civil engineering. 

1. Application of Novel Biochar Derived from Experimental Sewage Sludge Gasification as an Adsorbent for Heavy Metals Removal
   Sustainability 2025, 17(3), 997
   Authors: Domagoj Nakić, Hana Posavčić, Katarina Licht, and Dražen Vouk
   
2. Azadirachta indica Fruit Mucilage Aided Mucoadhesive Microspheres of Acyclovir for Drug Entrapment and Mucoadhesive Time Assets with Design-Expert Software
   Indian Journal of Pharmaceutical Education and Research, 2025; 59(1s): s243-s255.
   Authors: Gorantla Naresh Babu, M Menaka and Hindustan Abdul Ahad
   
3. Optimizing Water Hyacinth Compost and Chicken Manure for Enhancing Spinach Growth An Eco Friendly Approach to Sustainable Agriculture
   Indian Journal of Natural Sciences, 2024, 15(87), 976-977
   Authors: Mahmuda Parveen, Sujit Ghosh
   
4. Enhancing the tensile strength and morphology of Sansevieria trifasciata Laurentii fibers using liquid smoke and microwave treatments: an RSM approach
   Scientific Reports 15, Article number: 4420 (2025)
   Authors: Muhammad Arsyad Suyuti, Djarot B. Darmadi, Winarto, Putu Hadi Setyarini
   
5. Optimization of Biodiesel Production from Tannery Industry Fleshing Wastes Using Response Surface Methodology
   SAE Technical Paper 2025-28-0115
   Authors: Kanthasamy P, Arul Mozhi Selvan, Shanmugam P
   
6. Efficiency comparison of natural coagulants (Cactus pads and Moringa seeds) for treating textile wastewater (in the case of Kombolcha textile industry)
   Heliyon Volume 11, Issue 4, 28 February 2025
   Authors: Getahun Demeke Worku, Shimeles Nigussie Abate
   
7. Physico-chemical Characterisation of Ultrasound Processed Finger Millet Malt
   Asian Journal of Dairy and Food Research 14 February 2025
   Authors: Almas Begum Adoni, Madhava Mondru, Kaliramesh Siliveru, Vishnuvardhan Sidlagatta, Sandeep Raja Donepudi, Swapna Motamarri
   
8. Preparation of Gluten Free Cookie using Chestnut and Foxnut Flour Blend: Composition Optimization Through Response Surface Methodology
   Asian Journal of Dairy and Food Research Volume 44 Issue 1 (February 2025) : 84-91
   Authors: Divya Singh Chauhan, Apoorva Behari Lal, Anto Pradeep Raja Charles, Amit Pratap Singh, Ashish Khare, Pranav Vashisht
   
9. Optimization of Biochar Production from Cassava Peels: An Application of Response Surface Methodology
   Archives of Advanced Engineering Science 1-9
   Authors: Timothy Adekanye, Abiodun Okunola, Adeolu Adediran, Afolabi Tokunbo Yemisi, Aisha Aderibigbe
   
10. Investigations on mechanical and stress strain characteristics of geopolymer concrete reinforced with glass fibers
    Scientific Reports volume 15, Article number: 6335 (2025)
    Authors: Thunuguntla Chaitanya Srikrishna, Venkatesh Noolu, B. Sudheer Kumar Reddy, George Uwadiegwu Alaneme, Ravella Durga Prasad & M. VishnuPriyan
    
11. Study on hydration mechanism and ratio optimization of slag powder modified high-water material
    Scientific Reports volume 15, Article number: 6175 (2025)
    Authors: Xiang Ma, Chenyang Liu, Liwei Zhai, Shengrong Xie, Chaowen Wu & Jian Yang
    
12. Extraction and purification of total flavonoids from Zanthoxylum planispinum Var. Dintanensis leaves and effect of altitude on total flavonoids content
    Scientific Reports volume 15, Article number: 7080 (2025)
    Authors: Jiyue Wang, Xianqi Huang, Zhenyu Chen, Nian Chen, Mingli Yang, Chenggang Liang, Yanghua Yu & Denghong Shi
    
13. Preparation and characterization of inorganic foam modified with nano-magnesium hydroxide for inhibiting coal spontaneous combustion
    Physics of Fluids 37, 022152 (2025)
    Authors: Jingxia Tang, Jiawen Cai, Shengqiang Yang, Zhaoyang Yu, Kexin Chen, Xincheng Hu
    
14. Experimental approach and assessment of Zr conversion coatings on Al alloy using response surface methodology
    Corrosion Science, Volume 249, 2025, 112824
    Authors: Ana Kraš, Davorin Kramar, Ingrid Milošev