#StackBounty: #r #mixed-model #lme4-nlme #standard-error #post-hoc Can I add a random effect accounting for huge differences within one…

Bounty: 50

The Experiment and Data

The experiment I am working on has the following design:

A B C D E F
B A D E F C
A B E F C D
B A F C D E

  • Each Letter represents a different level of the single factor called “system” analyzed in this experiment. The dataset contains eight years and the dependent variable we are analyzing is yield. A and B can be grouped together, as well as C to F according to their system type.
  • I am aware of the missing randomization between groups AB and CDEF, which was necessary due to regulations, as well of the missing randomization within these two Groups, which has simply not been made, sadly. However the rows can be seen as complete blocks.
  • I am investigating if there are significant differences in yield between the systems (A-F)

My data looks like this:

> str(data)
'data.frame':   192 obs. of  6 variables:
 $ year  : Factor w/ 8 levels "2012","2013",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ type  : Factor w/ 2 levels "org","pest": 1 1 1 1 1 1 1 1 1 1 ...
 $ system: Factor w/ 6 levels "dgst_org","cc_pest",..: 3 3 3 3 5 5 5 5 6 6 ...
 $ row   : Factor w/ 4 levels "row_1","row_2",..: 1 2 3 4 2 3 4 1 3 4 ...
 $ column: Factor w/ 6 levels "column_1","column_2",..: 6 5 4 3 6 5 4 3 6 5 ...
 $ yield : num  26.2 41.4 43.4 45 40.8 52.3 47.1 47.2 40.1 42.4 ...

> summary(data)
      year      type             system      row          column       yield       
 2012   :24   org :128   dgst_org   :32   row_1:48   column_1:32   Min.   : 26.20  
 2013   :24   pest: 64   cc_pest    :32   row_2:48   column_2:32   1st Qu.: 52.30  
 2014   :24              cc_org     :32   row_3:48   column_3:32   Median : 62.95  
 2015   :24              manure_pest:32   row_4:48   column_4:32   Mean   : 73.79  
 2016   :24              manure_org :32              column_5:32   3rd Qu.:103.83  
 2017   :24              fmyd_org   :32              column_6:32   Max.   :127.10  

> head(data)
    year type     system   row   column yield
377 2012  org     cc_org row_1 column_6  26.2
378 2012  org     cc_org row_2 column_5  41.4
379 2012  org     cc_org row_3 column_4  43.4
380 2012  org     cc_org row_4 column_3  45.0
417 2012  org manure_org row_2 column_6  40.8
418 2012  org manure_org row_3 column_5  52.3
419 2012  org manure_org row_4 column_4  47.1
420 2012  org manure_org row_1 column_3  47.2
461 2012  org   fmyd_org row_3 column_6  40.1
462 2012  org   fmyd_org row_4 column_5  42.4

My previous attempts

  1. My first model was created according to a tutorial from Piepho and Edmondson (2018):
    m1 <- lmer(yield ~ system + (1|year/row) + (1|year:system)
    They suggest for repeated mesurements to include year as a random effect, with nested effects for replicates (row) and interactions with the main effect system
  2. I also looked at a model where year is a fixed effect since I am also interrested in the differences of years and the differences of systems within each year:
    m2 <- lmer(yield ~ system * year + (1|row), data = data)
  3. I compared both models, checked their summaries and performed post hoc tests with the emmeans() function and came to deviating results.
    • m1 had much higher std.err. than m2 and thus found less significant differences in the pairwise compairison of systems
    • m1 had a higher AIC but lower BIC as m2
    • residual plots and QQ-Plots of both models looked fine
  4. I assumed that the high increase in std. err. after adding the (1|year:system) random interaction, compared with the baseline model m0, had something to do with the huge yield differences between the two system types so I tried to account for that by adding a variable for type.
    I added it as a random interaction with year since I wanted it to be a random effect but with only two levels I wasn’t possible to add it as a single random effect:
    m3 <- lmer(yield ~ system + (1|year/row) + (1|year:system) + (1|year:type))
  5. Comparing the models now again, as well as their post hoc tests I noticed that:
    • The std. err. of m3 differnciated within system types and thus got simular results in the pairwise compairison of systems like m1
    • m1 still had the lowest AIC and m3 had a lower one than m2

My Question

  • I am now unsure which model to pick, I fear that the interaction term (1|year:system) of m2 disguise the differences inbetween systems of the same type, especially the type org ones (ABCD in the experiment design).
  • m1 seems to be a good model but has year as a fixed effect, whereas m3 meets all requirements and detects the differences between systems of the same type well (because of the different std. err. of the system types in the post hoc test)
  • but is it legitimate to add this random effect (1|year:type) to take account for the huge yield differences of the two system types

Added Summaries

#The models
> m0 <- lmer(yield ~ system + (1|row), data = data)
> m1 <- lmer(yield ~ system + (1|year) + (1|year:system) + (1|year:row), data = data)
> m2 <- lmer(yield ~ system * year + (1|row), data = data)
> m3 <- lmer(yield ~ system + (1|year) + (1|year:system) + (1|year:type) + (1|year:row), data = data)

#Model Compairison
> anova(m0,m1,m2,m3)
refitting model(s) with ML (instead of REML)
Data: data
Models:
m0: yield ~ system + (1 | row)
m1: yield ~ system + (1 | year) + (1 | year:system) + (1 | year:row)
m3: yield ~ system + (1 | year) + (1 | year:system) + (1 | year:type) + (1 | year:row)
m2: yield ~ system * year + (1 | row)
   npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
m0    8 1414.6 1440.7 -699.30   1398.6                         
m1   10 1305.3 1337.9 -642.67   1285.3 113.26  2  < 2.2e-16 ***
m3   11 1283.7 1319.6 -630.86   1261.7  23.61  1  1.180e-06 ***
m2   50 1215.6 1378.5 -557.80   1115.6 146.13 39  2.681e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Model Summaries
> summary(m0)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: yield ~ system + (1 | row)
   Data: data

REML criterion at convergence: 1380.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.9797 -0.7010  0.0885  0.6564  3.1912 

Random effects:
 Groups   Name        Variance Std.Dev.
 row      (Intercept)  2.503   1.582   
 Residual             86.550   9.303   
Number of obs: 192, groups:  row, 4

Fixed effects:
                  Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)        53.2375     1.8250  26.7856  29.172  < 2e-16 ***
systemcc_pest      56.3094     2.3258 183.0000  24.211  < 2e-16 ***
systemdgst_org      9.7438     2.3258 183.0000   4.189 4.35e-05 ***
systemfmyd_org     -0.9781     2.3258 183.0000  -0.421    0.675    
systemmanure_org    1.3750     2.3258 183.0000   0.591    0.555    
systemmanure_pest  56.8906     2.3258 183.0000  24.461  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) systmc_ systmd_ systmf_ systmmnr_r
systmcc_pst -0.637                                   
systmdgst_r -0.637  0.500                            
systmfmyd_r -0.637  0.500   0.500                    
systmmnr_rg -0.637  0.500   0.500   0.500            
systmmnr_ps -0.637  0.500   0.500   0.500   0.500    

> summary(m1)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: yield ~ system + (1 | year) + (1 | year:system) + (1 | year:row)
   Data: data

REML criterion at convergence: 1262.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.2609 -0.4988  0.0592  0.5590  2.3885 

Random effects:
 Groups      Name        Variance Std.Dev.
 year:system (Intercept) 43.868   6.623   
 year:row    (Intercept)  2.276   1.509   
 year        (Intercept) 22.305   4.723   
 Residual                26.442   5.142   
Number of obs: 192, groups:  year:system, 48; year:row, 32; year, 8

Fixed effects:
                  Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)        53.2375     3.0281 28.2596  17.581  < 2e-16 ***
systemcc_pest      56.3094     3.5524 34.9998  15.851  < 2e-16 ***
systemdgst_org      9.7438     3.5524 34.9998   2.743  0.00954 ** 
systemfmyd_org     -0.9781     3.5524 34.9998  -0.275  0.78467    
systemmanure_org    1.3750     3.5524 34.9998   0.387  0.70105    
systemmanure_pest  56.8906     3.5524 34.9998  16.015  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) systmc_ systmd_ systmf_ systmmnr_r
systmcc_pst -0.587                                   
systmdgst_r -0.587  0.500                            
systmfmyd_r -0.587  0.500   0.500                    
systmmnr_rg -0.587  0.500   0.500   0.500            
systmmnr_ps -0.587  0.500   0.500   0.500   0.500    

> summary(m2)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: yield ~ system * year + (1 | row)
   Data: data

REML criterion at convergence: 944.7

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.5152 -0.5168  0.0678  0.5333  2.5714 

Random effects:
 Groups   Name        Variance Std.Dev.
 row      (Intercept)  3.787   1.946   
 Residual             24.931   4.993   
Number of obs: 192, groups:  row, 4

Fixed effects:
                           Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)                  39.000      2.679  79.240  14.555  < 2e-16 ***
systemcc_pest                77.475      3.531 141.000  21.943  < 2e-16 ***
systemdgst_org               16.750      3.531 141.000   4.744 5.08e-06 ***
systemfmyd_org                0.425      3.531 141.000   0.120 0.904359    
systemmanure_org              7.850      3.531 141.000   2.223 0.027782 *  
systemmanure_pest            73.775      3.531 141.000  20.895  < 2e-16 ***
year2013                      9.200      3.531 141.000   2.606 0.010152 *  
year2014                     11.850      3.531 141.000   3.356 0.001015 ** 
year2015                      0.525      3.531 141.000   0.149 0.882006    
year2016                     20.250      3.531 141.000   5.735 5.70e-08 ***
year2017                     21.350      3.531 141.000   6.047 1.26e-08 ***
year2018                     37.575      3.531 141.000  10.642  < 2e-16 ***
year2019                     13.150      3.531 141.000   3.724 0.000282 ***
systemcc_pest:year2013      -14.950      4.993 141.000  -2.994 0.003252 ** 
systemdgst_org:year2013       3.350      4.993 141.000   0.671 0.503368    
systemfmyd_org:year2013       6.175      4.993 141.000   1.237 0.218255    
systemmanure_org:year2013     1.975      4.993 141.000   0.396 0.693040    
systemmanure_pest:year2013  -10.450      4.993 141.000  -2.093 0.038152 *  
systemcc_pest:year2014      -15.325      4.993 141.000  -3.069 0.002575 ** 
systemdgst_org:year2014       4.300      4.993 141.000   0.861 0.390600    
systemfmyd_org:year2014       5.400      4.993 141.000   1.081 0.281328    
systemmanure_org:year2014     0.800      4.993 141.000   0.160 0.872937    
systemmanure_pest:year2014  -13.900      4.993 141.000  -2.784 0.006110 ** 
systemcc_pest:year2015      -16.550      4.993 141.000  -3.315 0.001167 ** 
systemdgst_org:year2015      -0.725      4.993 141.000  -0.145 0.884761    
systemfmyd_org:year2015       2.650      4.993 141.000   0.531 0.596442    
systemmanure_org:year2015    -8.025      4.993 141.000  -1.607 0.110246    
systemmanure_pest:year2015  -10.925      4.993 141.000  -2.188 0.030316 *  
systemcc_pest:year2016      -22.675      4.993 141.000  -4.541 1.19e-05 ***
systemdgst_org:year2016     -13.825      4.993 141.000  -2.769 0.006383 ** 
systemfmyd_org:year2016       2.050      4.993 141.000   0.411 0.682016    
systemmanure_org:year2016   -10.625      4.993 141.000  -2.128 0.035083 *  
systemmanure_pest:year2016  -22.000      4.993 141.000  -4.406 2.07e-05 ***
systemcc_pest:year2017      -39.100      4.993 141.000  -7.831 1.05e-12 ***
systemdgst_org:year2017     -15.025      4.993 141.000  -3.009 0.003104 ** 
systemfmyd_org:year2017     -10.100      4.993 141.000  -2.023 0.044987 *  
systemmanure_org:year2017    -9.975      4.993 141.000  -1.998 0.047668 *  
systemmanure_pest:year2017  -26.750      4.993 141.000  -5.357 3.36e-07 ***
systemcc_pest:year2018      -49.825      4.993 141.000  -9.979  < 2e-16 ***
systemdgst_org:year2018     -20.625      4.993 141.000  -4.131 6.17e-05 ***
systemfmyd_org:year2018     -13.250      4.993 141.000  -2.654 0.008877 ** 
systemmanure_org:year2018   -19.025      4.993 141.000  -3.810 0.000207 ***
systemmanure_pest:year2018  -47.400      4.993 141.000  -9.493  < 2e-16 ***
systemcc_pest:year2019      -10.900      4.993 141.000  -2.183 0.030691 *  
systemdgst_org:year2019     -13.500      4.993 141.000  -2.704 0.007701 ** 
systemfmyd_org:year2019      -4.150      4.993 141.000  -0.831 0.407299    
systemmanure_org:year2019    -6.925      4.993 141.000  -1.387 0.167660    
systemmanure_pest:year2019   -3.650      4.993 141.000  -0.731 0.465990    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation matrix not shown by default, as p = 48 > 12.
Use print(x, correlation=TRUE)  or
    vcov(x)        if you need it


> summary(m3)
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: yield ~ system + (1 | year) + (1 | year:system) + (1 | year:type) +      (1 | year:row)
   Data: data

REML criterion at convergence: 1241.7

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.3528 -0.5194  0.0820  0.5278  2.5522 

Random effects:
 Groups      Name        Variance Std.Dev.
 year:system (Intercept) 12.001   3.464   
 year:row    (Intercept)  2.254   1.501   
 year:type   (Intercept) 50.459   7.103   
 year        (Intercept)  0.000   0.000   
 Residual                26.453   5.143   
Number of obs: 192, groups:  year:system, 48; year:row, 32; year:type, 16; year, 8

Fixed effects:
                  Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)        53.2375     2.9504 20.2592  18.044 5.93e-14 ***
systemcc_pest      56.3094     4.1555 19.9300  13.550 1.62e-11 ***
systemdgst_org      9.7437     2.1572 28.3095   4.517 0.000102 ***
systemfmyd_org     -0.9781     2.1572 28.3095  -0.453 0.653703    
systemmanure_org    1.3750     2.1572 28.3095   0.637 0.528989    
systemmanure_pest  56.8906     4.1555 19.9300  13.690 1.35e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) systmc_ systmd_ systmf_ systmmnr_r
systmcc_pst -0.704                                   
systmdgst_r -0.366  0.260                            
systmfmyd_r -0.366  0.260   0.500                    
systmmnr_rg -0.366  0.260   0.500   0.500            
systmmnr_ps -0.704  0.865   0.260   0.260   0.260    
convergence code: 0
boundary (singular) fit: see ?isSingular

#The Post Hoc Tests
> emmeans(m0, list(pairwise ~ system), adjust = "tukey") 
$`emmeans of system`
 system      emmean   SE   df lower.CL upper.CL
 cc_org        53.2 1.82 26.8     48.1     58.4
 cc_pest      109.5 1.82 26.8    104.4    114.7
 dgst_org      63.0 1.82 26.8     57.8     68.2
 fmyd_org      52.3 1.82 26.8     47.1     57.4
 manure_org    54.6 1.82 26.8     49.4     59.8
 manure_pest  110.1 1.82 26.8    104.9    115.3

Degrees-of-freedom method: kenward-roger 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 6 estimates 

$`pairwise differences of system`
 contrast                 estimate   SE  df t.ratio p.value
 cc_org - cc_pest          -56.309 2.33 183 -24.211 <.0001 
 cc_org - dgst_org          -9.744 2.33 183  -4.189 0.0006 
 cc_org - fmyd_org           0.978 2.33 183   0.421 0.9983 
 cc_org - manure_org        -1.375 2.33 183  -0.591 0.9915 
 cc_org - manure_pest      -56.891 2.33 183 -24.461 <.0001 
 cc_pest - dgst_org         46.566 2.33 183  20.021 <.0001 
 cc_pest - fmyd_org         57.288 2.33 183  24.631 <.0001 
 cc_pest - manure_org       54.934 2.33 183  23.620 <.0001 
 cc_pest - manure_pest      -0.581 2.33 183  -0.250 0.9999 
 dgst_org - fmyd_org        10.722 2.33 183   4.610 0.0001 
 dgst_org - manure_org       8.369 2.33 183   3.598 0.0054 
 dgst_org - manure_pest    -47.147 2.33 183 -20.271 <.0001 
 fmyd_org - manure_org      -2.353 2.33 183  -1.012 0.9136 
 fmyd_org - manure_pest    -57.869 2.33 183 -24.881 <.0001 
 manure_org - manure_pest  -55.516 2.33 183 -23.869 <.0001 

Degrees-of-freedom method: kenward-roger 
P value adjustment: tukey method for comparing a family of 6 estimates 

> emmeans(m1, list(pairwise ~ system), adjust = "tukey") 
$`emmeans of system`
 system      emmean   SE   df lower.CL upper.CL
 cc_org        53.2 3.03 28.3     44.7     61.8
 cc_pest      109.5 3.03 28.3    101.0    118.1
 dgst_org      63.0 3.03 28.3     54.4     71.5
 fmyd_org      52.3 3.03 28.3     43.7     60.8
 manure_org    54.6 3.03 28.3     46.0     63.2
 manure_pest  110.1 3.03 28.3    101.6    118.7

Degrees-of-freedom method: kenward-roger 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 6 estimates 

$`pairwise differences of system`
 contrast                 estimate   SE df t.ratio p.value
 cc_org - cc_pest          -56.309 3.55 35 -15.851 <.0001 
 cc_org - dgst_org          -9.744 3.55 35  -2.743 0.0919 
 cc_org - fmyd_org           0.978 3.55 35   0.275 0.9998 
 cc_org - manure_org        -1.375 3.55 35  -0.387 0.9988 
 cc_org - manure_pest      -56.891 3.55 35 -16.015 <.0001 
 cc_pest - dgst_org         46.566 3.55 35  13.108 <.0001 
 cc_pest - fmyd_org         57.288 3.55 35  16.126 <.0001 
 cc_pest - manure_org       54.934 3.55 35  15.464 <.0001 
 cc_pest - manure_pest      -0.581 3.55 35  -0.164 1.0000 
 dgst_org - fmyd_org        10.722 3.55 35   3.018 0.0494 
 dgst_org - manure_org       8.369 3.55 35   2.356 0.1998 
 dgst_org - manure_pest    -47.147 3.55 35 -13.272 <.0001 
 fmyd_org - manure_org      -2.353 3.55 35  -0.662 0.9849 
 fmyd_org - manure_pest    -57.869 3.55 35 -16.290 <.0001 
 manure_org - manure_pest  -55.516 3.55 35 -15.628 <.0001 

Degrees-of-freedom method: kenward-roger 
P value adjustment: tukey method for comparing a family of 6 estimates 

> emmeans(m2, list(pairwise ~ system), adjust = "tukey") 
NOTE: Results may be misleading due to involvement in interactions
$`emmeans of system`
 system      emmean   SE   df lower.CL upper.CL
 cc_org        53.2 1.31 7.65     48.6     57.9
 cc_pest      109.5 1.31 7.65    104.9    114.2
 dgst_org      63.0 1.31 7.65     58.4     67.6
 fmyd_org      52.3 1.31 7.65     47.6     56.9
 manure_org    54.6 1.31 7.65     50.0     59.2
 manure_pest  110.1 1.31 7.65    105.5    114.7

Results are averaged over the levels of: year 
Degrees-of-freedom method: kenward-roger 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 6 estimates 

$`pairwise differences of system`
 contrast                 estimate   SE  df t.ratio p.value
 cc_org - cc_pest          -56.309 1.25 141 -45.109 <.0001 
 cc_org - dgst_org          -9.744 1.25 141  -7.806 <.0001 
 cc_org - fmyd_org           0.978 1.25 141   0.784 0.9699 
 cc_org - manure_org        -1.375 1.25 141  -1.102 0.8800 
 cc_org - manure_pest      -56.891 1.25 141 -45.575 <.0001 
 cc_pest - dgst_org         46.566 1.25 141  37.304 <.0001 
 cc_pest - fmyd_org         57.288 1.25 141  45.893 <.0001 
 cc_pest - manure_org       54.934 1.25 141  44.008 <.0001 
 cc_pest - manure_pest      -0.581 1.25 141  -0.466 0.9972 
 dgst_org - fmyd_org        10.722 1.25 141   8.589 <.0001 
 dgst_org - manure_org       8.369 1.25 141   6.704 <.0001 
 dgst_org - manure_pest    -47.147 1.25 141 -37.769 <.0001 
 fmyd_org - manure_org      -2.353 1.25 141  -1.885 0.4156 
 fmyd_org - manure_pest    -57.869 1.25 141 -46.359 <.0001 
 manure_org - manure_pest  -55.516 1.25 141 -44.474 <.0001 

Results are averaged over the levels of: year 
Degrees-of-freedom method: kenward-roger 
P value adjustment: tukey method for comparing a family of 6 estimates 

> emmeans(m3, list(pairwise ~ system), adjust = "tukey") 
$`emmeans of system`
 system      emmean   SE   df lower.CL upper.CL
 cc_org        53.2 2.95 19.9     44.6     61.9
 cc_pest      109.5 2.95 19.9    100.9    118.2
 dgst_org      63.0 2.95 19.9     54.4     71.6
 fmyd_org      52.3 2.95 19.9     43.6     60.9
 manure_org    54.6 2.95 19.9     46.0     63.2
 manure_pest  110.1 2.95 19.9    101.5    118.7

Degrees-of-freedom method: kenward-roger 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 6 estimates 

$`pairwise differences of system`
 contrast                 estimate   SE   df t.ratio p.value
 cc_org - cc_pest          -56.309 4.16 10.1 -13.550 <.0001 
 cc_org - dgst_org          -9.744 2.16 28.0  -4.517 0.0013 
 cc_org - fmyd_org           0.978 2.16 28.0   0.453 0.9973 
 cc_org - manure_org        -1.375 2.16 28.0  -0.637 0.9871 
 cc_org - manure_pest      -56.891 4.16 10.1 -13.690 <.0001 
 cc_pest - dgst_org         46.566 4.16 10.1  11.206 <.0001 
 cc_pest - fmyd_org         57.288 4.16 10.1  13.786 <.0001 
 cc_pest - manure_org       54.934 4.16 10.1  13.220 <.0001 
 cc_pest - manure_pest      -0.581 2.16 28.0  -0.269 0.9998 
 dgst_org - fmyd_org        10.722 2.16 28.0   4.970 0.0004 
 dgst_org - manure_org       8.369 2.16 28.0   3.879 0.0069 
 dgst_org - manure_pest    -47.147 4.16 10.1 -11.346 <.0001 
 fmyd_org - manure_org      -2.353 2.16 28.0  -1.091 0.8809 
 fmyd_org - manure_pest    -57.869 4.16 10.1 -13.926 <.0001 
 manure_org - manure_pest  -55.516 4.16 10.1 -13.359 <.0001 

Degrees-of-freedom method: kenward-roger 
P value adjustment: tukey method for comparing a family of 6 estimates 


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