Statistics - Designs - Differences between Orthogonality and Confounding

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Miner

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This is a tricky concept. An easy way to test whether your design is orthogonal is to run a correlation of each factors levels (using -1, 1 level coding) against every other factors levels. If all correlations are 0, the design is orthogonal.

Confounding (a.k.a. aliasing) can occur in two ways.
• First, if there is some correlation between the levels of different factors (non-orthogonal design) you can have partial confounding.
• Second, in an orthogonal fractional factorial you may have partial confounding between main effects and interactions as in a Plackett-Burmann design to complete confounding as in a main effect with a 2-way interaction in a Resolution III fractional factorial or between interactions in a Resolution IV design.
The two are actually separate concepts.

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Steve Prevette

Deming Disciple
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There are two good examples of confounding in the wikipedia writeup for confounding at http://en.wikipedia.org/wiki/Confounding

As an example, suppose that there is a statistical relationship between ice-cream consumption and number of drowning deaths for a given period. These two variables have a positive correlation with each other. An evaluator might attempt to explain this correlation by inferring a causal relationship between the two variables (either that ice-cream causes drowning, or that drowning causes ice-cream consumption). However, a more likely explanation is that the relationship between ice-cream consumption and drowning is spurious and that a third, confounding, variable (the season) influences both variables: during the summer, warmer temperatures lead to increased ice-cream consumption as well as more people swimming and thus more drowning deaths.

In another concrete example, say one is studying the relation between birth order (1st child, 2nd child, etc.) and the presence of Down's Syndrome in the child. In this scenario, maternal age would be a confounding variable:
1.Higher maternal age is directly associated with Down's Syndrome in the child
2.Higher maternal age is directly associated with Down's Syndrome, regardless of birth order (a mother having her 1st vs 3rd child at age 50 confers the same risk)
3.Maternal age is directly associated with birth order (the 2nd child, except in the case of twins, is born when the mother is older than she was for the birth of the 1st child)
4.Maternal age is not a consequence of birth order (having a 2nd child does not change the mother's age)

v9991

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will it be right to say that, balanced design need not be orthogonal, but orthogonal design necessary be balanced!.

and further, does the power of design change for orthogonal and non orthogonal design! (although, statistically the technique/formula might be different, but my understanding is that, it should not matter, i.e., because the power is function of signal-noise-runs)

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Miner

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will it be right to say that, balanced design need not be orthogonal, but orthogonal design necessary be balanced!.
No. It is possible to have an orthogonal design that was originally balanced (i.e. 5 repeats per experiment), but due to insufficient materials, one experiment only has 3 repeats. This creates an unbalanced orthogonal design.
and further, does the power of design change for orthogonal and non orthogonal design! (although, statistically the technique/formula might be different, but my understanding is that, it should not matter, i.e., because the power is function of signal-noise-runs)
Good question! I'll have to research that one. The biggest impact is in the increased complexity of the analysis and in the potential for confounding(aliasing).

Miner

Forum Moderator
Staff member
I did research your question regarding power of orthogonal versus non-orthogonal designs.

If all things are equal (i.e., number of runs, number of replicates, number of factors, etc.) the two designs will have the same power.

However, orthogonal designs have a special property called the Projective Property. As terms are removed from the model, this property increases the resolution of the design until it becomes a full factorial. If additional terms are removed it becomes a replicated full factorial. This is called hidden replication.

A non-orthogonal design does not have these properties. Therefore, even though both types have the same power initially, as terms are removed an orthogonal design may increase in power while the non-orthogonal design will not.

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