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I'm having trouble understanding how to interpret the output of polynomial model when the lower order term (linear term) is significant, but the quadratic term is not.

For instance, create a random dataset:

set.seed(10)
x <- 0:51
y <- 1 - 0.7*x + 0.1*x^2 + rnorm(length(x), 7, 10)
testdf <- data.frame(x,y)
testdf$Treat <- c(0, 1)
testdf$Treat <- as.factor(testdf$Treat)
testdf$y <- ifelse(testdf$Treat == 0 & (testdf$x >= 40 & testdf$x <= 75), testdf$y + 20, testdf$y)

Interaction plot:

library(ggplot2) 
ggplot(testdf, aes(x, y, col=Treat)) + geom_point() + geom_smooth()
[![interaction plot][1]][1]

Model:

test.mod <- lm(y ~ poly(x,2)*Treat, data=testdf)
summary(test.mod)
Coefficients:
                   Estimate    Std.Error t-value Pr(>|t|)    
(Intercept)          78.913      1.851  42.643  < 2e-16 ***
poly(x, 2)1         524.521     13.382  39.197  < 2e-16 ***
poly(x, 2)2         174.842     13.374  13.073  < 2e-16 ***
Treat1               -4.405      2.617  -1.683  0.09912 .  
poly(x, 2)1:Treat1  -54.606     18.924  -2.885  0.00593 ** 
poly(x, 2)2:Treat1  -11.238     18.914  -0.594  0.55532 

Does this mean that the shapes of the curves do not differ between treatments, and so I should try to visualize (conceptually) the interaction in terms of a linear relationship between the groups? In simple terms, how can I explain this result to someone?

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In the model you formulated the coefficients at Treat interaction terms (last three coefficients in your output) tell you the differences for the three polynomial terms (of orders 0, 1 and 2) between the Treat == 0 and Treat == 1 groups.

In other words. For an observation from Treat == 0 group your model is:

y = 78.913 + 524.521*x + 174.842*x^2

and for an observation from Treat == 1 group your model is:

y = 78.913 - 4.405 + (524.521-54.606)*x + (174.842-11.238)*x^2

So in either case you have a parabolic model, just with a different shape of the parabola for both groups. The fact that the order 1 interaction term is significant and order 0 and 2 terms not does not have any particular meaning. The only meaning is that you may want to try to estimate another model without these terms and get as a result something like (of course the values of the estimates will change):

y = 78.913 + (524.521-54.606)*x + 174.842*x^2

but still it will be just two parabolas with different shapes for both groups. I don't see here any deeper meaning in the significance of particular order terms.

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