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I have a linear model with the following form:

$$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 $$

As $y$ is considered a "moral" behavior, I expect resistance and skepticism of the following kind:

  • the sole cause of $y$ is the permanently latent variable $x_3$.
  • $y$ does not depend on $x_1$ or $x_2$ in any way.
  • $x_1$ and $x_2$ neither cause $y$, nor do they cause $x_3$!!
  • $x_3$ does not cause either $x_1$ or $x_2$!!
  • $x_1$ and $x_2$ routinely show chance strong correlations with both $y$ and $x_3$.
  • yes, $x_3$ is the sole cause of $y$ (there is no other latent variable).

E.g., AOK = monthly number of Acts of Kindness with a specific coding process.

$$ AOK = \beta_0 + \beta_1Parents' AOK + \beta_2Best friend's AOK $$ $$ x_3 = "Individual Moral Kindness" $$

I reply:

  • The p-values of $x_1$ and $x_2$ are both <0.01, suggesting there is less than a 1 in 100 chance that the connection between them and $y$ is a product of chance.
  • If I independently replicate this study 10 times with the same results, it is more likely that your sandwich flies off the table like a bird than it is that random chance shows this relationship between $y$ and $x_1$ and $x_2$.
  • are you really claiming that $x_3$ is the sole cause of $y$?

In the role of frequentist statistician, what better reply can I give in our conceptual disagreement about the role of $x_3$?

What would be a good reply from the frequentist statistician if the skeptic tried to strengthen their case by saying: If we could measure $x_3$, you would see that even if your p-values stay the same, $x_3$ explains 99.999%, so $x_1$ and $x_2$ are meaningless even if statistically significant?

jtd
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    Are you conflating association with causation here? There are all sorts of reasons why two variables may be associated that are not causal. – Glen_b Mar 11 '15 at 00:07
  • @Glen_b: I will have to write that ("why two variables..") as another question (https://stats.stackexchange.com/questions/141253/can-two-variables-be-perfectly-correlated-but-not-share-a-single-causal-chain-an). I think $x_1$ is at least correlated with $y$, but then I want to say: you can't assign all causal power to $x_3$ because if $x_1$ not a cause, then there is some latent variable(s) $X_4$ with some causal power over $x_1$ and $y$ too. Sorry if unclear! – jtd Mar 11 '15 at 03:44

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