I’m fitting a growth-curve in R (lmer) for satisfaction over four waves, with time centered at the last occasion (t runs from –8 to 0). Pooled fixed effects are:
Intercept β₀ = 5.505 Linear slope β₁ = –0.062 Quadratic slope β₂ = –0.008
Plotting the combined trajectory (black parabola) gives the expected downward-curving parabola. However, plotting the quadratic-only component (red) from t=–8 to 0 shows an increasing trend, even though β₂
The quadratic term means the rate the slope changes with changing x. If it is negative the curve will be n shaped, if it is positive the curve will be u shaped.
There will always be parts of the curve where the slope is negative and parts where it will be positive. If you just have the (negative) quadratic term then the slope will be positive for negative x and negative for positive x. This term changes the slope across the graph. If there is a linear term this sets the slope for x=0. The slope will be zero where the slope of the quadratic term cancels the slope of the linear term.
Malcolm White
Thank you so much for the explanation!
One quick follow-up question: if the quadratic coefficient were positive instead of negative, would the pattern simply flip, so that for negative x values the slope would be negative, and for positive x values the slope would be positive?
Thanks again!
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