QK Box 11.4

Al-Janabi et al (2016) examined the effects of ocean-acidification and warming (OAW) and nutrient levels on growth of different sibling groups of the early life stage of a seaweed, Fucus vesiculosus. There were two fixed and crossed between-plots factors: OAW (ambient pH and temperature vs future 2010 pH and temperature) and nutrient levels (ambient nutrients vs doubled nutrients). The plots were “benthocosms”, benthic mesocosm tanks in an outdoor facility in Germany; there were three mesocosms nested within each combination of OAW and nutrients. The sub-plots were 2 cm² sandstone cubes each containing germlings from one of 16 sibling groups (populations) of F. vesiculosus previously produced by mating male and female plants collected from the wild. One cube for each sibling group was placed within each mesocosm. In contrast to Al-Janabi et al who considered sibling group a random effect, we will illustrate the analysis with sibling group as fixed. The response variable was the mean relative growth rate (% change in area) of a subset of individual germlings on each cube over 8 weeks.

Fucus vesiculosus. Alex Lomas, CC BY 2.0

The paper is here

Al-Janabi, B., Kruse, I., Graiff, A., Winde, V., Lenz, M. & Wahl, M. (2016). Buffering and amplifying interactions among OAW (Ocean Acidification & Warming) and nutrient enrichment on early life-stage Fucus vesiculosus L. (Phaeophyceae) and their carry over effects to hypoxia impact. PLoS One, 11, e0152948.

Preliminaries

First, load the required packages (afex, car, lattice, lme4, lmerTest, nlme, VCA, ez, emmeans, Rmisc, MuMIn)

Import aljanabi data file (aljanabi.csv)

aljanabi <- read.csv("../data/aljanabi.csv")
aljanabi

NA

set contrasts from afex

set_sum_contrasts()

setting contr.sum globally: options(contrasts=c('contr.sum', 'contr.poly'))

Model fitting

Don’t use OLS ANOVA because of missing cell

Fit mixed effects models with lmer

aljanabi.lmer <- lmer(meanrate~oaw+nutrient+oaw*nutrient+sib+oaw*sib+nutrient*sib+oaw*nutrient*sib+(1|mesocosm), REML=TRUE, na.action=na.omit, aljanabi)

Check residuals from lmer plot

plot(aljanabi.lmer)

Looks fine

summary(aljanabi.lmer, ddf="Kenward-Roger")

Linear mixed model fit by REML. t-tests use Kenward-Roger's method ['lmerModLmerTest']
Formula: meanrate ~ oaw + nutrient + oaw * nutrient + sib + oaw * sib +      nutrient * sib + oaw * nutrient * sib + (1 | mesocosm)
   Data: aljanabi

REML criterion at convergence: 420.3

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.78235 -0.53967 -0.06754  0.55534  2.27385 

Random effects:
 Groups   Name        Variance Std.Dev.
 mesocosm (Intercept) 0.06653  0.2579  
 Residual             0.35377  0.5948  
Number of obs: 191, groups:  mesocosm, 12

Fixed effects:
                      Estimate Std. Error        df t value Pr(>|t|)    
(Intercept)            2.43430    0.08603   7.99963  28.295 2.63e-09 ***
oaw1                   0.48744    0.08603   7.99963   5.666 0.000473 ***
nutrient1             -0.31572    0.08603   7.99963  -3.670 0.006312 ** 
sib1                  -0.01593    0.16629 119.00237  -0.096 0.923860    
sib2                  -0.17518    0.16629 119.00237  -1.053 0.294281    
sib3                  -0.33308    0.16629 119.00237  -2.003 0.047455 *  
sib4                  -0.47234    0.16629 119.00237  -2.840 0.005301 ** 
sib5                   0.36989    0.17625 119.42278   2.099 0.037956 *  
sib6                  -0.57029    0.16629 119.00237  -3.429 0.000832 ***
sib7                  -0.64904    0.16629 119.00237  -3.903 0.000158 ***
sib8                  -0.11039    0.16629 119.00237  -0.664 0.508071    
sib9                   0.42231    0.16629 119.00237   2.540 0.012387 *  
sib10                 -0.14582    0.16629 119.00237  -0.877 0.382320    
sib11                  0.34078    0.16629 119.00237   2.049 0.042633 *  
sib12                 -0.06305    0.16629 119.00237  -0.379 0.705244    
sib13                  0.03526    0.16629 119.00237   0.212 0.832419    
sib14                  0.74411    0.16629 119.00237   4.475 1.76e-05 ***
sib15                  0.18688    0.16629 119.00237   1.124 0.263356    
oaw1:nutrient1         0.25402    0.08603   7.99963   2.953 0.018354 *  
oaw1:sib1              0.02257    0.16629 119.00237   0.136 0.892266    
oaw1:sib2             -0.06216    0.16629 119.00237  -0.374 0.709207    
oaw1:sib3              0.04990    0.16629 119.00237   0.300 0.764627    
oaw1:sib4              0.28582    0.16629 119.00237   1.719 0.088255 .  
oaw1:sib5             -0.21133    0.17625 119.42278  -1.199 0.232893    
oaw1:sib6             -0.02245    0.16629 119.00237  -0.135 0.892856    
oaw1:sib7              0.31027    0.16629 119.00237   1.866 0.064531 .  
oaw1:sib8             -0.06491    0.16629 119.00237  -0.390 0.696974    
oaw1:sib9              0.09421    0.16629 119.00237   0.567 0.572091    
oaw1:sib10            -0.10809    0.16629 119.00237  -0.650 0.516955    
oaw1:sib11            -0.03109    0.16629 119.00237  -0.187 0.852021    
oaw1:sib12             0.06055    0.16629 119.00237   0.364 0.716437    
oaw1:sib13            -0.15265    0.16629 119.00237  -0.918 0.360483    
oaw1:sib14            -0.08850    0.16629 119.00237  -0.532 0.595600    
oaw1:sib15             0.11386    0.16629 119.00237   0.685 0.494859    
nutrient1:sib1         0.41234    0.16629 119.00237   2.480 0.014552 *  
nutrient1:sib2         0.14611    0.16629 119.00237   0.879 0.381370    
nutrient1:sib3         0.09513    0.16629 119.00237   0.572 0.568369    
nutrient1:sib4         0.04376    0.16629 119.00237   0.263 0.792887    
nutrient1:sib5        -0.15465    0.17625 119.42278  -0.877 0.382010    
nutrient1:sib6        -0.21061    0.16629 119.00237  -1.266 0.207816    
nutrient1:sib7        -0.12472    0.16629 119.00237  -0.750 0.454724    
nutrient1:sib8        -0.09684    0.16629 119.00237  -0.582 0.561438    
nutrient1:sib9         0.26490    0.16629 119.00237   1.593 0.113816    
nutrient1:sib10       -0.16471    0.16629 119.00237  -0.991 0.323936    
nutrient1:sib11        0.03637    0.16629 119.00237   0.219 0.827255    
nutrient1:sib12        0.18524    0.16629 119.00237   1.114 0.267562    
nutrient1:sib13       -0.24403    0.16629 119.00237  -1.467 0.144884    
nutrient1:sib14       -0.15702    0.16629 119.00237  -0.944 0.346953    
nutrient1:sib15       -0.04661    0.16629 119.00237  -0.280 0.779723    
oaw1:nutrient1:sib1    0.17350    0.16629 119.00237   1.043 0.298895    
oaw1:nutrient1:sib2   -0.43361    0.16629 119.00237  -2.608 0.010287 *  
oaw1:nutrient1:sib3    0.21914    0.16629 119.00237   1.318 0.190109    
oaw1:nutrient1:sib4   -0.08261    0.16629 119.00237  -0.497 0.620253    
oaw1:nutrient1:sib5   -0.12648    0.17625 119.42278  -0.718 0.474396    
oaw1:nutrient1:sib6   -0.07806    0.16629 119.00237  -0.469 0.639622    
oaw1:nutrient1:sib7    0.39245    0.16629 119.00237   2.360 0.019901 *  
oaw1:nutrient1:sib8    0.09197    0.16629 119.00237   0.553 0.581257    
oaw1:nutrient1:sib9   -0.14914    0.16629 119.00237  -0.897 0.371616    
oaw1:nutrient1:sib10  -0.13999    0.16629 119.00237  -0.842 0.401580    
oaw1:nutrient1:sib11  -0.26137    0.16629 119.00237  -1.572 0.118663    
oaw1:nutrient1:sib12   0.20103    0.16629 119.00237   1.209 0.229104    
oaw1:nutrient1:sib13   0.04545    0.16629 119.00237   0.273 0.785107    
oaw1:nutrient1:sib14  -0.07526    0.16629 119.00237  -0.453 0.651667    
oaw1:nutrient1:sib15   0.15503    0.16629 119.00237   0.932 0.353088    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation matrix not shown by default, as p = 64 > 12.
Use print(x, correlation=TRUE)  or
    vcov(x)        if you need it

anova(aljanabi.lmer, ddf="Kenward-Roger")

Type III Analysis of Variance Table with Kenward-Roger's method
                  Sum Sq Mean Sq NumDF  DenDF F value    Pr(>F)    
oaw              11.3560 11.3560     1   8.00 32.1001 0.0004728 ***
nutrient          4.7642  4.7642     1   8.00 13.4670 0.0063124 ** 
sib              28.1276  1.8752    15 119.03  5.3006 5.024e-08 ***
oaw:nutrient      3.0841  3.0841     1   8.00  8.7178 0.0183536 *  
oaw:sib           4.0399  0.2693    15 119.03  0.7613 0.7172454    
nutrient:sib      6.1474  0.4098    15 119.03  1.1585 0.3137711    
oaw:nutrient:sib  7.7113  0.5141    15 119.03  1.4532 0.1342442    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Get variance components with CIs

aljanabi.ci <- confint.merMod(aljanabi.lmer,oldNames=FALSE)

Computing profile confidence intervals ...

aljanabi.vc <- (aljanabi.ci)^2
print(aljanabi.vc)

                               2.5 %       97.5 %
sd_(Intercept)|mesocosm 1.414172e-02 1.340031e-01
sigma                   1.925128e-01 2.914894e-01
(Intercept)             5.221280e+00 6.676068e+00
oaw1                    1.141991e-01 4.054257e-01
nutrient1               2.162333e-01 2.762645e-02
sib1                    8.014782e-02 6.313439e-02
sib2                    1.956770e-01 8.466865e-03
sib3                    3.603048e-01 4.340758e-03
sib4                    5.468907e-01 4.208697e-02
sib5                    7.514840e-03 4.262480e-01
sib6                    7.013570e-01 9.187027e-02
sib7                    8.394601e-01 1.458103e-01
sib8                    1.425595e-01 2.458590e-02
sib9                    2.406772e-02 4.754202e-01
sib10                   1.705653e-01 1.473162e-02
sib11                   5.417612e-03 3.696323e-01
sib12                   1.090511e-01 4.167337e-02
sib13                   5.378329e-02 9.148038e-02
sib14                   2.274617e-01 1.022725e+00
sib15                   6.447379e-03 2.061831e-01
oaw1:nutrient1          1.092379e-02 1.626620e-01
oaw1:sib1               5.983984e-02 8.395370e-02
oaw1:sib2               1.084748e-01 4.203087e-02
oaw1:sib3               4.721429e-02 1.005404e-01
oaw1:sib4               3.470090e-04 3.058064e-01
oaw1:sib5               2.443486e-01 5.165619e-03
oaw1:sib6               8.389045e-02 5.989326e-02
oaw1:sib7               1.855729e-03 3.334458e-01
oaw1:sib8               1.102938e-01 4.091086e-02
oaw1:sib9               2.992213e-02 1.306022e-01
oaw1:sib10              1.408352e-01 2.530941e-02
oaw1:sib11              8.897105e-02 5.573817e-02
oaw1:sib12              4.270291e-02 1.074021e-01
oaw1:sib13              1.762714e-01 1.311544e-02
oaw1:sib14              1.265143e-01 3.192690e-02
oaw1:sib15              2.351014e-02 1.451910e-01
nutrient1:sib1          2.107344e-02 4.617697e-01
nutrient1:sib2          1.465707e-02 1.708194e-01
nutrient1:sib3          2.960111e-02 1.312756e-01
nutrient1:sib4          4.991489e-02 9.669181e-02
nutrient1:sib5          1.917193e-01 1.646882e-02
nutrient1:sib6          2.282767e-01 3.202021e-03
nutrient1:sib7          1.535854e-01 2.029763e-02
nutrient1:sib8          1.325079e-01 2.902017e-02
nutrient1:sib9          5.174440e-06 2.831248e-01
nutrient1:sib10         1.865302e-01 1.050176e-02
nutrient1:sib11         5.327237e-02 9.214953e-02
nutrient1:sib12         6.714394e-03 2.046912e-01
nutrient1:sib13         2.613337e-01 5.364507e-04
nutrient1:sib14         1.799454e-01 1.213737e-02
nutrient1:sib15         9.846505e-02 4.865469e-02
oaw1:nutrient1:sib1     8.777533e-03 1.941997e-01
oaw1:nutrient1:sib2     4.911279e-01 2.770083e-02
oaw1:nutrient1:sib3     2.309298e-03 2.365016e-01
oaw1:nutrient1:sib4     1.223636e-01 3.406399e-02
oaw1:nutrient1:sib5     1.676627e-01 2.456186e-02
oaw1:nutrient1:sib6     1.192011e-01 3.576423e-02
oaw1:nutrient1:sib7     1.569045e-02 4.351130e-01
oaw1:nutrient1:sib8     3.070269e-02 1.289870e-01
oaw1:nutrient1:sib9     1.733308e-01 1.393328e-02
oaw1:nutrient1:sib10    1.657957e-01 1.617712e-02
oaw1:nutrient1:sib11    2.793794e-01 3.370952e-05
oaw1:nutrient1:sib12    4.377607e-03 2.192168e-01
oaw1:nutrient1:sib13    4.917165e-02 9.773293e-02
oaw1:nutrient1:sib14    1.172755e-01 3.683111e-02
oaw1:nutrient1:sib15    1.258065e-02 1.782579e-01