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We suspected that the association between an ordinal predictor and a continuous outcome had a threshold at some point: that is, that after a positive correlation (the higher the predictor, the higher the outcome), at some point the direction of the relationship would disappear (that is, the outcome would stay constant irrespective of the predictor) or change direction (the higher the predictor, the lower the outcome). Indeed, look at my data:

enter image description here

Is there a simple statistical test to prove that a regression line changes direction after a certain point? Even better, since we had no prior hypothesis as to where the direction change would occur, to check whether a regression line changes direction at some point and automatically find that point? I cannot just publish a chart; there must be a way to check the statistical significance of this relationship. I am well acquainted with regression techniques and R.

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If you had several points on both sides of the break point and if you could treat the predictor as continuous, I would suggest segmented regression (as implemented in package segmented).

Another idea, which works for the data shown in the graph would be an outlier test. Again, this treats the predictor as continuous since you'd have a perfect fit if you modeled it as an ordered factor.

DF <- data.frame(x = 0:4,
                 y = c(42, 45, 47.8, 52, 42.1))

fit <- lm(y ~ x, data = DF)

library(car)
outlierTest(fit)
#   rstudent unadjusted p-value Bonferonni p
#5 -16.38872          0.0037025     0.018512
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