How to compare three different replacements for eggs in baking?

Question

I am conducting a food science test to replace eggs with other alternatives in baking applications.

• Independent variable: three different formulas (substitution percentages)
• Dependent variables: texture profile, moisture, and sensory evaluation

In terms of procedure, I would like to make a separation, my study will need to incorporate some texture analysis parameters. Let's say each muffin will need to be measured using a texture analyzer. Then I will need to set up a panel to rate overall acceptance, texture, and so on. Every subject will need to try the samples and asign a value (Likert scale).

What would be the best design in this case?

Answer 1

Use Simplex-lattice, Simplex-Centroid designs, or something like that.

First, from your question I guess this is a mixture design in which you change proportions of the components of the muffin (or some of it's components) and measure several responses, i.e. texture, acceptance, etc.

I suppose that you have some knowledge in the difference between ANOVA model and Regression model, and basic DoE.

Having said that, If this is the case of a mixture model, you can not use a regular $2^k$ or $3^k$ factorial design and ANOVA, since the level of your factors are no independent. This is, you have at least one restriction: $$\sum\limits^{q}_{i=1}x_i=1$$ where $x_i$ is the proportion of every of the $q$ components of your formulation, like ingredients, additives, coadyuvants, etc.

What you need is some kind of Regression model, i.e. Scheffé model, because you have continuous explanatory variables. On the other hand, ANOVA model requires categorical explanatory variables, which is not the case.

There are several designs for this, the most common are Simplex-lattice and Simplex-centroid designs. If you add any other linear restriction, like upper limits, lower limits, or both, or some other, what you need to do is:

1) Make sure your restrictions are consistent;

2) Find the candidate mixtures (these are all possible mixtures that meet your restrictions). This is achieved using Piepel's Xvert algorithm available from some software packages. For examle JMP, or `mixexp' library in R.

3) At this point, you will have more candidates than mixtures needed. Using an optimization criteria, find the mixtures that you will use to perform your experiment. The number of mixtures requiered equals the number of parameters to be estimated. So defining a model is mandatory at this point. For example, a model with 3 components without interaction, is:

$$\eta=\beta_1x_1+\beta_2x_2+\beta_3x_3$$

requires 3 mixtures (formulations) to estimate $\beta_1$, $\beta_2$, and $\beta_3$. A full model with 3 components like

$$\eta=\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_{12}(x_1x_2)+\beta_{13}(x_1x_3)+\beta_{23}(x_2x_3)+\beta_{123}(x_1x_2x_3)$$

Requieres $2^3-1=7$ mixtures to estimate 7 parameters. There are many different designs and strategies, but the Scheffé model is the same (a regression model without intercept $\beta_0$).

Use some experimentation strategy and BLOCK, RANDOMIZE, and REPEAT.

If you need further information, I strongly recommend:

Cornell, J.A., Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, 3rd Ed.,Wiley, 2002.

Answer 2

Are you going to do your experiment within-subject? That is, will each subject try all three formulas? If so, you might consider a latin square design, as it will help control against the ordering of exposure as a confound. Here's a brief write-up on latin square designs.

Answer 3

Since you modified your question, this is not a mixture design anymore.

In fact, this is a Completely Randomized Design (CRD) and requires an ANOVA model. The statistical model of fixed effects with one factor is: $$y_{ij}=\mu+\tau_i+\epsilon_{ij}$$ with $y_{ij}$ being the response of the $j^{th}$ observation of the $i^{th}$ treatment $j=1,2,...,r_i$; $\tau_i$ being the $i^{th}$ treatment $i=1,2,3$, and $\epsilon_{ij}$ being the error of the $y_{ij}$ measure. $\epsilon_{ij}\sim N(0,\sigma^2)$. $Y$ variable con be texture, viscosity, acceptance, etc. You have to fit one model for each and every response.

CRD Design:

Number of factors: 1 (formula)

Number of levels: 3 (formula 1, formula 2, formula 3)

Number of repetitions: to be defined (at least two to estimate $\hat{\sigma}^2$)

Experimental unit: a batch of muffins,

Observation unit: a muffin.

Please note:

1. Beware of repetitions and pseudo-repetitions. Two measurements of different muffins of the same batch are pseudo-repetitions. In order to make it a CRD, you should cook different batches for every analysis. Obviously this may not have sense from the practical point of view, in this case, a Split-plot design is preferred. Whole plot is recipe, and batch as small plot. You should also block for Judge in case you use more than one judge in the sensory analysis, block on Day in case it takes you more than one day to complete this analysis, block for Cook, in case more than one person cooks batches.

2. Block, randomize, and repeat.

For further reading:

Montgomery, D.C., Chapter three, "Experiments with a single Factor: The analysis of variance", in "Design and analysis of experiments", 8th Ed., John Wiley and Sons, 2012.

The Experimental strategy is beyond your question.