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Suppose I have a sample with two covariates whose levels are associated with some response; e.g., the mean height of a plant at 6 months is modeled by two factors: cultivar of plant (say, $A$ versus $B$) and size of pot (e.g., "small" versus "large"). Furthermore, suppose that the truth is that cultivar $A$ in general grows taller than cultivar $B$ irrespective of the size of the pot in which it is planted. However, the pot size has no effect on height at 6 months.

It so happens, though, that through random chance or some underlying hidden relationship, twice as many cultivar $A$ plants got the large pot as cultivar $B$, whereas in the small pot, the split is 50/50. Now, if we naively do a $t$-test for $H_0 : \mu_L = \mu_S$ versus $H_a : \mu_L \ne \mu_S$, we would find significance not because the pot size has an effect, but because it happened that more $A$ cultivars were potted in the large pots. Clearly such a test is not appropriate.

However, the question I'm faced with is this: is it correct to say that cultivar and pot size have an interaction? Note that pot size doesn't influence the outcome. I know I should be using a 2x2 factorial design, but I have some difficulty interpreting the meaning of the tests: If I compute a confidence interval based on $(\bar x_{AL} - \bar x_{AS}) - (\bar x_{BL} - \bar x_{BS})$, what is the related hypothesis, and is this test accounting for the fact that $A$ gets planted into large pots more frequently than $B$?

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The statistic

$(\bar{X}_{AL} - \bar{X}_{AS})$ corresponds to an estimate of the effect of pot size when using cultivar A. Similarly

$(\bar{X}_{BL} - \bar{X}_{BS})$ corresponds to an estimate of the effect of pot size when using cultivar B.

Thus, the statistic

$(\bar{X}_{AL} - \bar{X}_{AS}) - (\bar{X}_{BL} - \bar{X}_{BS})$

corresponds to testing the hypothesis that the effect of large pots vs small pots when cultivar A is used is the same as the effect of large vs small when cultivar B is used. This test statistic would be testing for an interaction. It is looking to see if there is a different effect of large vs small depending on the type of cultivar, which is one way to interpret the definition of an interaction.

Notice that with a little algebra, the statistic can be rewritten as

$(\bar{X}_{AL} - \bar{X}_{BL}) - (\bar{X}_{AS} -\bar{X}_{BS})$, which is looking to see if the effect of cultivar A vs B depends on pot size. These are two equivalent ways of how to think about what it means to test for an interaction between two variables.

You are correct that if you test the hypothesis $\mu_L=\mu_S$ that it will look like there is an effect of pot size, but the perceived effect is completely explained, as you point out, by the imbalance in the distribution of cultivar between the large and small pots. The lurking variable, the variable that is actually driving the results, in this hypothesis test is cultivar. It is not, however, a confounding variable.

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    $\begingroup$ Okay, so since pot size does not affect plant height, there should not be an interaction; I also see why we can't call cultivar a confounder when testing for equality of means among pot sizes. Say that after testing for interaction I find none; then I go ahead with calculating the CI under the 2x2 factorial design with the point estimate $$\frac{W_1 (\bar X_{AL} - \bar X_{BL}) + W_2 (\bar X_{AS} - \bar X_{BS})}{W_1 + W_2}, \quad W_1 = \frac{n_{AL}n_{BL}}{n_{AL}+n_{BL}}, W_2 = \frac{n_{AS}n_{BS}}{n_{AS}+n_{BS}}.$$ Does this estimate control for the cultivar effect? $\endgroup$
    – heropup
    May 13, 2014 at 13:17
  • $\begingroup$ That term appears to be estimating the effect of cultivar in the two pot sizes, then takes a weighted average of the two estimates of the effect of cultivar to produce a single estimate of the effect of cultivar. $\endgroup$
    – jsk
    May 13, 2014 at 15:17
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However, the pot size has no effect on height at 6 months.

If there's an interaction, it's not correct to say 'pot size has no effect'; it's already been stated to have an interaction effect!

because it happened that more A cultivars were potted in the large pots.

If pot size has no effect what difference does it make which pots the plants were in?

Your discussion seems to be self-contradictory. What do you mean to say?

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    $\begingroup$ I think the OP is confusing confounding with interactions. $\endgroup$
    – jsk
    May 13, 2014 at 7:03
  • $\begingroup$ @jsk Maybe, but then that still raises further questions - it's a hard to have confounding with something that has no effect. $\endgroup$
    – Glen_b
    May 13, 2014 at 7:25
  • $\begingroup$ @Glen_b no it is not its called an illusion of control $\endgroup$
    – user11279
    May 13, 2014 at 7:27
  • $\begingroup$ @Glen_b Ah, the "confounding" is coming from the large and small pot groups differing in the distribution of the cultivar covariate! But I see your point that if pot size has no effect, then the relationship cannot technically be confounded! It's really just an imbalance in the covariates that needs to be controlled for upon which the effect of pot size should disappear. $\endgroup$
    – jsk
    May 13, 2014 at 7:28
  • $\begingroup$ @Glen_b Of course, but if she ignores cultivar and performs a t-test of large vs small without taking into account cultivar, then it will look like pot size has an effect, but the effect is explained by cultivar! $\endgroup$
    – jsk
    May 13, 2014 at 7:41

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