[EMAIL PROTECTED] wrote:
> Gus,
>
> Have you tried predicting x1 from y and checking the residual (errors)? Be
> sure to use uniform x1 and x2 when generating
> y=x1+x2. Then, generate the residual by using y as the predictor of x1. The
> residual will equal x2 (when the predictor is the effect). Then look at the
> relationship between the extremity of the predictor (y) and the absolute
> value of the residual (error). I think you will find the error actually
> DECREASE as we move from the mean towards the extremes of the predictor (y).
> Thus flipping the regression prediction variables around will lead to very
> different error patterns.
That last statement is correct. In the regression y = b0 + b1 x1 + error,
the error has essentially the same distribution as x2: uniform (on [-1,1]
in my test), while the error in the regression x1 = b0 + b1 y + error
behaves very differently, essentially filling out a diamond with vertices
at (-2,0), (0,-1), (2,0), (0,1) if plotted against y.
> The reduction in errors when predicting x1 from y is the basis of
> corresponding correlations. Predicting from the effect decreases errors in
> the predictors extremes. Predicting from the cause increases errors in the
> extremes of the predictor.
> The asymmetry allows us to detect which is the cause, which is the effect.
But this is plain wrong. First off, the effect diappears when you use normal
distributions in place of the uniform. Moreover, you can give the appearance
of any "cause" you care to show, because you bias the results in a certain
direction. I performed the following experiment:
I generated a large sample (1,000,000) of uniform x1 and x2 and computed
y = x1 + x2. The residuals and correlations behave as you predicted, leading
you to the conclusion that y is caused by x1 (and x2).
Then I selected a subsample from this in such a way that (x2,y) are uniformly
distributed. (This takes some doing, but it is possible.) With this data set
you'd
come to the conclusion that the cause is y!
.
.
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