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Given below the attached summary statistics of a linear regression model.

Summary statistics of a linear regression model

Noting that Adjusted R-square is only 15% but p(F-statistic) is very low (almost equal to zero):

  • Can it be said that it is a significant fit?
  • Can it be said that the relationship is linear?
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$R^2=0.15$ might not sound so great, but it might be quite acceptable for what you're doing. If that value is too low to be interesting, then you have a practically insignificant result resulting in a statistically significant p-value due to a large sample size that detects subtle differences. This means that there probably is a difference (statistical significance) but not enough of a difference for the scientist to care (practical significance).

You're not a lazy or negligent scientist to say, "Sure, $p<\alpha$, but it's not enough for me to be interested. They're basically the same."

Answering explicitly:

1) Yes, your fit is significantly better than just estimating every value as the mean of all response variables pooled together. This is what the p-value tells you.

2) No, you definitely cannot conclude that. Imagine fitting a line to the right half of a parabola. You will get some kind of fit, but the relationship between the predictor and response certainly isn't linear. Let's look at some R code.

set.seed(2020)
x <- seq(0,3,0.01)
err <- rnorm(length(x),0,1)
y <- x^2 + err
plot(x,y)
L <- lm(y~x^2)
lines(x,predict.lm(L),col='red')
summary(L)

I get $R^2 = 0.933$ and $p<2.2\times 10^{-16}\approx 0$, which sound pretty good, but the plot shows a clearly nonlinear relationship.

enter image description here

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