title: How to compare the slope of interaction variables in mixed effect model in r
body: <p>I want to test the effects of island area and land use, and the interaction between island area and land use on species richness. For land use, I have three groups, namely forest, farmland and mix. The data is based on transects on different islands, so the island ID is set as random effect.</p>
  <p>My model looks like this:</p>
  <pre><code>#model = glmer(SR ~ Area + land_use + Area:land_use + (1|islandID))
  
  #summary(model)
  
  Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
  Formula: SR ~ Area + land_use + Area:land_use + (1 | islandID)
  Data: transect_ZS
  
  REML criterion at convergence: 184.4
  
  Scaled residuals: 
   Min       1Q   Median       3Q      Max 
  -2.66105 -0.56159 -0.00294  0.57259  1.72096 
  
  Random effects:
  Groups   Name        Variance Std.Dev.
  islandID (Intercept) 0.1524   0.3903  
  Residual             0.6805   0.8249  
  Number of obs: 70, groups:  islandID, 34
  
  Fixed effects:
                   Estimate Std. Error      df t value Pr(&gt;|t|)   
  (Intercept)         -0.9996   0.5187 57.0061  -1.927  0.05893 . 
  Area                0.9064    0.2834 40.9977   3.198  0.00267 **
  land_useforest      0.6563    0.5569 62.0889   1.179  0.24309   
  land_usemix         0.9611    0.6373 55.3032   1.508  0.13723   
  Area:land_useforest -0.8318   0.3034 63.4045  -2.742  0.00793 **
  Area:land_usemix    -0.7756   0.4748 56.3692  -1.633  0.10795   
  ---
  Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  </code></pre>
  <p>The results told me that island area and the interaction terms have significant effect on SR:</p>
  <pre><code>#    &gt; anova(model)
  #Type III Analysis of Variance Table with Satterthwaite's method
  #                Sum Sq Mean Sq NumDF  DenDF F value  Pr(&gt;F)  
  #Area                 3.0359 3.03590     1 27.448  4.4615 0.04390 *
  #land_use             1.5520 0.77601     2 57.617  1.1404 0.32679  
  #Area:land_use        5.1658 2.58288     2 60.935  3.7958 0.02795 *
  #---
  #Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  #
  </code></pre>
  <p>And then I used lsmeans function to conduct Tukeys' pairwise comparison:</p>
  <pre><code>#lsmeans(model, pairwise ~ Area:land_use, adjust=&quot;tukey&quot;)
   
  </code></pre>
  <p>The results indicate that the species richness from farmland and forest is significantly different, right? I wonder if this difference should be seen as the significant difference of intercept of the species richness-area relationship between farmland and forest in this model? That is the species richness from farmland transects is higher than that from forest transects?</p>
  <pre><code>#$contrasts
    contrast                                            estimate    SE   df t.ratio p.value
   1.19968425045037 farmland - 1.19968425045037 forest   3.4153 0.288 62.6   1.185  0.0466
   1.19968425045037 farmland - 1.19968425045037 mix     -0.0306 0.426 64.0  -0.072  0.9972
   1.19968425045037 forest - 1.19968425045037 mix       -0.3722 0.377 63.9  -0.987  0.5087
  
  Degrees-of-freedom method: kenward-roger 
  P value adjustment: tukey method for comparing a family of 3 estimates 
  </code></pre>
  <p>But how to test if the slope of the species richness-area relationship between farmland and forest in this model is significant different? That is to testify if the species richness-area relationship from farmland transects is more steeper than that from forest transect?</p>