Autoregressive cross-lagged models

Question

I'm working on a research project with an autoregressive cross-lagged model with two measures three time points. The paths from $t_1$ to $t_2$ were significant, but $t_2$ to $t_3$ were all not significant. I'm very unfamiliar with coding/models/etc., and I'm wondering what the best path forward is? I've read about constraining certain variables, but I don't really understand how to best describe/interpret these nonsignificant paths. I know this may be confusing, so thanks for bearing with me.

Answer 1

I will note that there are both statistical and theoretical ways to look at this, so I express the statistical part here first. Assuming your model isn't mis-specified somehow (omitted variable bias, fitting linear paths to nonlinear ones, etc), we would interpret the non-significant paths like any other significance test. The $p$-values just say that you failed to reject the null hypothesis for each slope. This could be because the standard error is simply too high (making it a noisy estimate) or the magnitude is very low (e.g. $\beta = .0001$). Remember that a non-rejection of a $p$ value only means we suspend judgement until further inquiry (such as a follow up study on the same model). Importantly, we need to move past caring about the $p$ values alone. One can still note that the paths have certain magnitudes and why they may still be important to investigate given these magnitudes. If for example we have a standardized slope coefficient for $t_2$ and $t_3$ that is around $\beta = .40$, the effect isn't actually zero, we just can't say with complete certainty that we have ruled out that these effects were not found by chance alone.

From a theoretical side, it is easy to point out cases where the effect of a variable at an initial time point ceases to matter at a later time point. A good example is radical awareness in Chinese and its influence on reading Chinese characters. Chinese characters are made up of phonetic radicals that give it sound (e.g. the 巴 part on the bottom of the word 爸 makes the sound "ba") and semantic radicals that give it meaning (e.g. the 火 part on the left of 燒 means the character is related to fire, in this case the character means to roast or cook something). When Chinese initially learn to read, the magnitude between radical awareness and reading is moderate, because those who are able to understand the components of a character can more easily read it than those who cant. But as time goes on, this skill becomes largely unimportant, as many start to master it after early primary years, thus its predictive power for reading dwindles when reading becomes more sophisticated anyway (moving from identifying characters to reading whole texts).