# How to compare three different replacements for eggs in baking?

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).

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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.

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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.

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Kyle: thank you for your response. I will be doing laboratory test: texture profile, moisture, etc. and also a sensory evaluation. –  Wendy Alfaro Jan 18 '13 at 0:31
@WendyAlfaro: Ok, so the outcomes your looking at are the subjects' evaluation of those characteristics and then a sensory evaluation? There's a lot of interesting ways you could set this up...I'm just trying to get a better sense of what you're planning. My main question is whether every subject will evaluate all three preparations--e.g., Subject1 has a value for preparation1, preparation2, and preparation3. Is this correct? –  Kyle. Jan 18 '13 at 1:01
Dear Kyle: Thank you for kindness and willingness to help. First 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, etc. Yes, every subject will need to try the samples and asign a value (Likert scale). –  Wendy Alfaro Jan 18 '13 at 1:47
No problem! This sounds like a repeated measure ANOVA, or possibly MANOVA, to me! There are several design consideration you should think about before you set this up. An important one making sure the order of exposure is properly stratified across your subjects. From your description, it sounds as though it might be a MANOVA if your outcome variable is the simultaneous acceptance/texture/etc. ratings. Does that help? –  Kyle. Jan 18 '13 at 5:42
@Wendy, you probably should add that information to the question, and kyle should add his responses to his amswer –  naught101 May 18 '13 at 6:22

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.