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How to compute significant interaction estimates when main effect is not significant?

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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marked as duplicate by whuber Dec 20 '12 at 20:11

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1 Answer 1

Welcome to the site. This has been discussed many times here, e.g. this question

Briefly, it is very rarely a good idea to include an interaction term in a model without the main effects, and the main effects do not have a simple interpretation when there is an interaction.

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Hi, thanks for the pointer! I am sorry if this is replicating a common query. I guess the reason I felt that this is more complicated than other information I have come across is that the interaction is between the person ability and the item type (i.e. the length of the word tested by the item). This effectively means that in this part of the explanation I have two different sample populations: lower ability learners and higher ability learners. I wasn't sure if this makes a difference. –  KJB Dec 20 '12 at 18:57
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Interactions are certainly tricky. You are talking about an interaction between a continuous and a categorical variable. The essence is the same, but the interpretation is now slightly simpler. The main effect of ability is now the effect of ability in whichever group was coded with a 0. To get the effect of ability in the group coded with a 1, you have to add the interaction parameter. (This assumes the groups are coded 0-1, which is usually a good thing) –  Peter Flom Dec 20 '12 at 19:15
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