# Understanding signficant interaction with non-significant main effects [duplicate]

I am an applied linguist and I am modelling responses to a vocabulary test taken by second language learners of English; the aim is to test theoretical hypotheses regarding the relationship between the nature of the word and the likelihood of the learners’ knowing that word. The models I am using to explore the role of explanatory variables are the random item Item Response Theory model LLTM+e (see de Boeck et al. 2011; de Boeck 2008). These are created using the lmer function in the LME4 package in R and treat item and person responses as random. The estimates shown below are effectively like those from a binary logistic regression model, indicating the log-odds of a correct response on a word, given certain properties. Covariates relate to both item and person characteristics.

The nature of my concern is actually a basic regression issue. I have found a significant interaction between ability grouping of the test taker (GRP; with two levels High and Low) and the length of the word in letters (LEN_L). As far as I can see the estimate of the fixed effects for the first model shows that (a) the lower level learners have an overall lower probability of giving a correct answer (b) that LEN_L does not provide a significant explanation across the pattern of responses for the whole test-taker population, and (c) a significant interaction between GRP and LEN_L indicates that the lower ability learners are less likely to give a correct answer for a longer word. This is in keeping with theory.

However, when I model the data without including the main effect for LEN_L, I am not seeing a significant effect for either high or low groups as shown in Model 2. LEN_L does not show as significant if modelled without interaction with grp low (not shown). I feel that I am missing something obvious, but I cannot quite grasp what is happening. And my references have run dry on this particular issue and I am thinking myself around in circles about it.

(NB: in my full model I have many other significant covariates, but this pattern holds true.) My query basically regards whether I should use Model 1, including the non-significant main effect, or whether my findings from Model 2 indicate that the finding is a little unstable. Any advice would be much appreciated! (I can post more details if necessary.) Karen

Model 1 Fixed effects:

              Estimate Std. Error z value Pr(>|z|)
(Intercept)    0.25244    0.83194   0.303  0.76156
grp low       -2.09126    0.26406  -7.920 2.38e-15 ***
LEN_L          0.02093    0.13062   0.160  0.87272
grp low:LEN_L -0.09185    0.03146  -2.919  0.00351 **


Model 2 Fixed effects:

               Estimate Std. Error z value Pr(>|z|)
(Intercept)     0.25243    0.83194   0.303    0.762
grp low        -2.09126    0.26406  -7.920 2.38e-15 ***
grp high:LEN_L  0.02093    0.13062   0.160    0.873
grp low:LEN_L  -0.07092    0.13202  -0.537    0.591


De Boeck, P., Bakker, M., Zwitser, R., Nivard, M., Hofman, A., Tuerlinckx, F. and Partchev, I. (2011) The Estimation of Item Response Models with the 'lmer' Function from the lme4 Package in R. Journal of Statistical Software (39:12) pp 1-28

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