How can I spot if, say, a simple, linear regression will adequately describe some causal relationship purely from the model's output (say, from the command summary(lm( y ~x )) in R), as opposed to from looking at QQ plots or residual plots? Is it possible? Perhaps something to do with the estimated variance?
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1$\begingroup$ There is the usual $R^2$ value which indicates how good the linear fit is. This will be a function of the estimated variance, so that might give a good idea. $\endgroup$– GreenparkerCommented Mar 7, 2016 at 12:16
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5$\begingroup$ Nothing in the usual output tells you anything about causal relationships. Causality is all in the design and substantive context and interpretation. $\endgroup$– Nick CoxCommented Mar 7, 2016 at 12:23
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2$\begingroup$ You cannot tell from that output whether the model is suitable or not. See the four data sets in the Anscombe quartet which have identical output. $\endgroup$– Glen_bCommented Mar 7, 2016 at 14:03
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1 Answer
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As Nick Cox already commented, you can't. Here is a little example where I generate data according to a "causal" relationship between $y$ and $x$, and yet, because that relationship is noisy, the output neither gives you a statistically significant relationship nor an $R^2$ that differs perceptibly from zero.
n <- 100
beta <- 0.1
x <- runif(n)
u <- rnorm(n,sd=10)
y <- x*beta + u
> summary(lm(y~x))
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-23.9617 -6.8501 -0.9274 8.0018 28.7454
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.134 2.155 -0.99 0.324
x 2.383 3.608 0.66 0.511
Residual standard error: 10.83 on 98 degrees of freedom
Multiple R-squared: 0.004431, Adjusted R-squared: -0.005728
F-statistic: 0.4361 on 1 and 98 DF, p-value: 0.5105
answered Mar 7, 2016 at 12:28