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I have recently started studying Designed Experiments and have have come across non-rigorous definitions of "balanced" and "orthogonal" experiments and would be interested in understanding these concepts a bit better (both in terms of some examples and a clearer definition).

According to the book I am using to study the terms are defined as follows:

Balance and orthogonality are two useful properties of designed experiments.

  • Balance: relating to strong symmetry in the combinatorial structure of the design

  • Orthogonality: special simplifications of analysis and the efficiency arising from such balance

I'm not sure that I understand either of these definitions (given that understanding Orthogonality is predicated on me first understanding Balance).

What exactly do these terms mean in more explicit terms? Do they apply to all designed experiments? If not, are there any examples of experiments with and without these properties in order for me to better understand them?

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A "Balanced" design is simply having an equal number of experimental runs at each condition or condition combinations. For example if you are measuring someone's blood sugar during mealtime. A balanced design would collect the data for the same number of breakfasts, lunches and dinners.

An "Orthogonal" design is a design where the values of the parameters are chosen in such a way that they are at right angles to each other. Generally this means each parameter will have 1 high value and 1 low value. (I know not very clear). Maybe see this answer (What is an orthogonal design?)

Yes, you can have unbalanced orthogonal designs and balanced non-orthogonal designs. The disadvantage of not having a balanced orthogonal design is a more complicated analysis and some loss of experimental power. The ability to identify the major parameters from the minor parameters becomes more difficult as the interactions between the parameters.
Due to externals factors such as cost, time, availability and design feasibility it may not be possible to meet both conditions. For example if you are performing an experiment for baking a cake, with 2 different times and temperatures. But if the cake is raw at the short time and low temperature combination, you may want to drop that point (unbalanced) or use a third temperature (short time - medium temperature) (non-orthogonal).

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  • $\begingroup$ Thank you for your answer - that helps to clarify a lot. Just to check that I’m understanding your example at the end correctly, the reason that removing the “short time and low temperature“ data point leads to imbalance is because now we have 1 short time but 2 long times (and 1 low tempuratures but 2 high temperatures). Whereas, for balance we would want 2 of each (or 1 of each if we were to remove another data point)? $\endgroup$
    – FD_bfa
    Commented Feb 8, 2023 at 1:01
  • $\begingroup$ Thanks. And the last thing is that regarding the 2nd part of the cake example, I’m not really clear on why the third temperature makes the experiment non-orthogonal. $\endgroup$
    – FD_bfa
    Commented Feb 8, 2023 at 1:14
  • $\begingroup$ It easier to understand it graphically. If graph out the combinations of low and high, you get a square. But if you replace a low with a medium then it becomes a trapezoid thus no longer orthogonal. $\endgroup$
    – Dave2e
    Commented Feb 8, 2023 at 1:31
  • $\begingroup$ Like this? imgur.com/a/GUw9cLO $\endgroup$
    – FD_bfa
    Commented Feb 8, 2023 at 1:39
  • $\begingroup$ Yes, that is a good example. $\endgroup$
    – Dave2e
    Commented Feb 8, 2023 at 1:45

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