How to interpret interaction in the absence of main effect? Multilevel Modeling

Question

What is the substantive interpretation of an interaction in the absence of a main effect? Statistically I understand that the interaction modifies the main effect, but how to interpret this from a substantive perspective?

Example output (cropped from nlme model in R):

                                                    Value   Std.Error    DF    t-value   p-value
socialinteraction_cwp                           0.1702094  0.03855630 26843   4.414567    0.5037
relstat_Single                                  0.2796805  0.23039908   485   1.213896    0.2254
socialinteraction_cwp:relstat_Single           -0.0455506  0.02069088 26843  -2.201483    0.0277

Where socialinteraction_cwp is within-person centered amount of social interaction and relstat_Single is a dummy code for relationship status single (1) compared to married (0).

Answer 1

Approximate data from an agricultural experiment investigating the effect on crop yields of levels of irrigation and of applied fertilizer. Average yields are shown. Main effects are not significant, but interaction is.

                  Irrigation
Fertilizer    Lo      Med      Hi       AVG
-----------------------------------=--------
    Lo         4       5        0        3
    Med        5       5        5        5
    Hi         0       5        5       3.3  
--------------------------------------------
    AVG        3       5       3.3      3.4  

Rationale:

• Lo fertilizer gets washed away with Hi irrigation.
• Hi fertilizer with Lo irrigation burns plants.

Both main effects obscured by variation--neither is significant. But interaction effect is strong enough to be significant. Considering costs of fertilizer and irrigation, medium levels of both may be best.