title: Discrepancy between summary() and anova() of linear mixed model
body: <p>I have fitted a linear mixed model with split-plot design to assess the effects of water, nitrogen and phosphorus on BWC (biomass-weighted 2c-value, achieved by summing the product of each species' 2C-value(DNA content) with its biomass fraction (species subplot biomass/total subplot biomass)：</p>
  
  <p><code>model1.1&lt;-lmer(log(BWC)~W*N*P+(1|year)+(1|W:Block),data=BWC)</code></p>
  
  <p>There are two levels for W(0,1), N(0,1) and p(0,1) I would like to use boxplot to report my results with the output of the linear mixed model. However, I'm confused with the output of the linear mixed model.</p>
  
  <ol>
  <li><p>The estimated value (slope) for W<em>N</em>P in model1.1 is negative, Does that mean WNP treatment will decrease BWC comparing to control plot? But we can see the BWC was highest in boxplot under the WNP treatment. </p></li>
  <li><p>There is a discrepancy between summary() and anova(), for example,  the significance for N and P effects. Estimate value for N is-4.0911 which means N addition decreased BWC But N effect was insignificant. How can I report the treatment effects like N?  </p></li>
  </ol>
  
  <p>Many thanks for any comments.
  Boxplot of WNP treatment on BWC:
  <a href="https://i.stack.imgur.com/xrzBI.png" rel="nofollow noreferrer">enter image description here</a></p>
  
  <p><a href="https://i.stack.imgur.com/cKOFt.png" rel="nofollow noreferrer">https://i.stack.imgur.com/cKOFt.png</a>
  (Sorry for the links,it seem I need at least 10 reputations to post images)</p>
  
  <p>The summary() and anova() output:</p>
  
  <pre><code>&gt; summary(model1) 
  
  Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
  Formula: BWC ~ W * N * P + (1 | year) + (1 | W:Block)
     Data: BWC
  REML criterion at convergence: 2969.1
  Scaled residuals: 
       Min       1Q   Median       3Q      Max 
  -2.93847 -0.71228 -0.07573  0.68191  2.92589 
  Random effects:
   Groups   Name        Variance Std.Dev.
   W:Block  (Intercept)  0.9169  0.9575  
   year     (Intercept)  0.8346  0.9136  
   Residual             18.2966  4.2774  
  Number of obs: 515, groups:  W:Block, 14; year, 10
  
  Fixed effects:
              Estimate Std. Error       df t value Pr(&gt;|t|)    
  (Intercept)  10.8498     0.6985  46.5200  15.532  &lt; 2e-16 ***
  W1            2.0844     0.8969  45.9613   2.324  0.02460 *  
  N1           -4.0911     0.7364 486.0288  -5.556 4.56e-08 ***
  P1           -2.0460     0.7600 490.1120  -2.692  0.00734 ** 
  W1:N1         4.6738     1.0394 485.9800   4.497 8.65e-06 ***
  W1:P1         0.9695     1.0687 485.9809   0.907  0.36478    
  N1:P1         5.7550     1.0687 485.9773   5.385 1.13e-07 ***
  W1:N1:P1     -3.3306     1.5100 485.9541  -2.206  0.02788 *  
  ---
  Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  
  Correlation of Fixed Effects:
           (Intr) W1     N1     P1     W1:N1  W1:P1  N1:P1 
  W1       -0.645                                          
  N1       -0.531  0.414                                   
  P1       -0.515  0.401  0.488                            
  W1:N1     0.376 -0.582 -0.708 -0.346                     
  W1:P1     0.366 -0.566 -0.347 -0.706  0.488              
  N1:P1     0.366 -0.285 -0.689 -0.706  0.488  0.502       
  W1:N1:P1 -0.259  0.400  0.488  0.499 -0.688 -0.708 -0.708
  
  &gt; anova(model1)
  Type III Analysis of Variance Table with Satterthwaite's method
        Sum Sq Mean Sq NumDF  DenDF F value    Pr(&gt;F)    
  W     750.15  750.15     1  11.90 40.9995 3.519e-05 ***
  N      10.84   10.84     1 485.95  0.5926   0.44177    
  P      29.14   29.14     1 494.92  1.5926   0.20755    
  W:N   290.51  290.51     1 485.95 15.8778 7.793e-05 ***
  W:P    15.54   15.54     1 485.96  0.8493   0.35721    
  N:P   536.85  536.85     1 485.95 29.3415 9.562e-08 ***
  W:N:P  89.01   89.01     1 485.95  4.8648   0.02788 *  
  ---
  Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
  </code></pre>
  
  <pre><code>&gt; emmeans::emmeans(model1,pairwise~N*P*W)
  $emmeans
   N P W emmean    SE   df lower.CL upper.CL
   0 0 0  10.85 0.699 46.9     9.44    12.26
   1 0 0   6.76 0.696 46.2     5.36     8.16
   0 1 0   8.80 0.721 52.1     7.36    10.25
   1 1 0  10.47 0.721 52.1     9.02    11.91
   0 0 1  12.93 0.696 46.2    11.53    14.33
   1 0 1  13.52 0.696 46.2    12.12    14.92
   0 1 1  11.86 0.721 52.1    10.41    13.30
   1 1 1  14.86 0.721 52.1    13.42    16.31
  
  Degrees-of-freedom method: kenward-roger 
  Confidence level used: 0.95 
  
  </code></pre>