I think confounding and OVB at least can be the same, if they are not always the same.

The first example I learned about a fractional factorial experimental design, the guy teaching it (PhD in Stats from a good program along with decades of consulting), he explained a fractional factorial as making the grid of all possible combinations of treatments to assign participants to, but then leaving out certain treatment groups for some reason (maybe cost or lack of interest in a specific combination). He said the downside is you may end up with confounded treatment effects due to omitting those portions of the factorial.

Example (what I recall of the top of my head, gist of it, he used more factor-level combos):

Treatments A, B, and C

I could give: A, B , C, AB, AC, BC, ABC.

Let's say I leave out all of the 2-way interactions (AB, AC, BC) for cost and only assign and measure A, B ,C, and ABC. I can get an estimate for the 3-way interaction, however, the 2 way interactions (such as the impact of A on Y depending on the level of B, independent of C) are

*confounded* within the 3-way interaction (the effect of A on Y depends on the specific B-C combination, for example, could be thought of as the effect of A on Y depends on the effect of B on Y for a particular setting of C). The ABC estimate may appear one way in this design, but if we included/measured AB, BC, and AC, we would then have the estimate of ABC after adjusting for the confounding that occurs by omitting AB, AC, and BC.

I think looking at the definition of confounding (obscuring [informal term] the relationship of one variable with the dependent variable

*because of unmeasured/improperly-measured variables*), this fits well with omitted variable bias. I would suspect though, that to be more rigorous, one should show that there is a calculable bias in the estimators (i.e. beta coefficient(s)) to show that omitting the (confounding) variable causes the expected value of the estimator to differ from the true value of the parameter. Pretty sure OVB and confounding are the same (at least in many cases). I vote with

@hlsmith. [And, I suppose, the omitted variable is correlated with an included variable and obviously is related to the dependent variable. This would show up doing the calculation to determine unbiasedness.]