library(lme4); library(lmerTest); library(emmeans)
library(dplyr); library(performance); library(desplot)
library(broom.mixed)8 Strip Plot Design
8.1 Background
In strip plot design each block or replication is divided into number of vertical and horizontal strips depending on the levels of the respective factors. 1. Vertical strip plot for the first factor – vertical factor 2. Horizontal strip plot for the second factor – horizontal factor
Divide the experimental area into ‘A’ horizontal strips and ‘B’ vertical strips. Each level of factor A is assigned to all the plots in one row, and each level of factor B is assigned to all the plots in one column.
The statistical model:
\[y_{ijk} = \mu + \alpha_j + \beta_k + \alpha_j\beta_k + b_i + r_{ij} + c_{ik} + \epsilon_{ijk}\] Where:
\(\mu\)= overall experimental mean, \(\alpha\) and \(\beta\) are the main effects applied in a horizontal and vertical direction, and \(\alpha\)\(\beta\) represents the interaction between main factors. The random effects in above equation are \(b_i\), the random rep effect, \(r_{ij}\), the row within rep random effect, \(c_{ik}\), the column within rep random effect.
\[ b_i \sim N(0, \sigma_1^2)\]
\[ r_{ij} \sim N(0, \sigma_2^2)\]
\[ c_{ik} \sim N(0, \sigma_3^2)\]
\[ \epsilon_{ijk} \sim N(0, \sigma^2)\]
8.2 Example Analysis
We will start by loading the required libraries for this analysis for lme and lmer models, respectively.
library(nlme); library(performance); library(emmeans)
library(dplyr); library(desplot); library(broom.mixed)For this example, we will use Rice strip-plot experiment data from theagridat package. A strip-plot experiment with three reps, variety as the horizontal strip and nitrogen fertilizer as the vertical strip.
data1 <- agridat::gomez.stripplot| rep | replication unit |
| nitro | nitrogen fertilizer in kg/ha |
| gen | rice variety |
| row | row (represents gen) |
| col | column (represents nitro) |
| yield | grain yield in kg/ha |
For the sake of analysis, ‘row’ and ‘col’ variables are used to represent ‘nitrogen’ and ‘Gen’ factors. The plot below shows the application of treatments in horizontal and vertical direction in a strip plot design.
8.2.1 Data integrity checks
First thing we need to verify is the data types of the variables in data1. The ‘rep’, ‘nitro’, and ‘gen’ needs to be a factor/character variables and ‘yield’ should be numeric.
str(data1)'data.frame': 54 obs. of 6 variables:
$ yield: int 2373 4076 7254 4007 5630 7053 2620 4676 7666 2726 ...
$ rep : Factor w/ 3 levels "R1","R2","R3": 1 1 1 1 1 1 1 1 1 1 ...
$ nitro: int 0 60 120 0 60 120 0 60 120 0 ...
$ gen : Factor w/ 6 levels "G1","G2","G3",..: 1 1 1 2 2 2 3 3 3 4 ...
$ col : int 1 3 2 1 3 2 1 3 2 1 ...
$ row : int 1 1 1 3 3 3 4 4 4 2 ...Let’s convert ‘nitro’ from numeric to factor.
data1$nitro <- as.factor(data1$nitro)Let’s have a look at the balance of treatment factors by running a a cross tabulation, and it looks well balanced.
table(data1$gen, data1$nitro)
0 60 120
G1 3 3 3
G2 3 3 3
G3 3 3 3
G4 3 3 3
G5 3 3 3
G6 3 3 3apply(data1, 2, function(x) sum(is.na(x)))yield rep nitro gen col row
0 0 0 0 0 0 We don’t have any missing values in this data set.
Lastly, let’s check the dependent variable by plotting.
hist(data1$yield, main = "", xlab = "yield")
No extreme values or skewness is present in the yield values.
8.2.2 Model Building
The impact of nitro, gen, and their interaction was evaluated on rice yield. The rep, gen nested in rep, and nitro nested in rep were random effects in the model.
https://stackoverflow.com/questions/36643713/how-to-specify-correlated-crossed-random-effects-in-nlme
model_lmer <- lmer(yield ~ nitro*gen + (1|rep) +
(1|rep:gen) + (1|rep:nitro),
data = data1)
tidy(model_lmer)# A tibble: 22 × 8
effect group term estimate std.error statistic df p.value
<chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 fixed <NA> (Intercept) 3572. 572. 6.24 17.8 0.00000732
2 fixed <NA> nitro60 1560. 558. 2.80 22.4 0.0104
3 fixed <NA> nitro120 3976. 558. 7.13 22.4 0.000000341
4 fixed <NA> genG2 1363. 717. 1.90 20.9 0.0714
5 fixed <NA> genG3 678. 717. 0.945 20.9 0.355
6 fixed <NA> genG4 487. 717. 0.679 20.9 0.504
7 fixed <NA> genG5 530. 717. 0.739 20.9 0.468
8 fixed <NA> genG6 -364. 717. -0.508 20.9 0.617
9 fixed <NA> nitro60:genG2 219. 741. 0.296 20.0 0.771
10 fixed <NA> nitro120:genG2 -1699. 741. -2.29 20.0 0.0328
# ℹ 12 more rowsmodel_lme <-lme(yield ~ nitro*gen,
random = list(one = pdBlocked(list(
pdIdent(~ 0 + rep),
pdIdent(~ 0 + rep:gen),
pdIdent(~ 0 + rep:nitro)))),
data = data1 %>% mutate(one = factor(1)))
summary(model_lme)Linear mixed-effects model fit by REML
Data: data1 %>% mutate(one = factor(1))
AIC BIC logLik
651.4204 686.2578 -303.7102
Random effects:
Composite Structure: Blocked
Block 1: repR1, repR2, repR3
Formula: ~0 + rep | one
Structure: Multiple of an Identity
repR1 repR2 repR3
StdDev: 393.4278 393.4278 393.4278
Block 2: repR1:genG1, repR2:genG1, repR3:genG1, repR1:genG2, repR2:genG2, repR3:genG2, repR1:genG3, repR2:genG3, repR3:genG3, repR1:genG4, repR2:genG4, repR3:genG4, repR1:genG5, repR2:genG5, repR3:genG5, repR1:genG6, repR2:genG6, repR3:genG6
Formula: ~0 + rep:gen | one
Structure: Multiple of an Identity
repR1:genG1 repR2:genG1 repR3:genG1 repR1:genG2 repR2:genG2 repR3:genG2
StdDev: 600.1711 600.1711 600.1711 600.1711 600.1711 600.1711
repR1:genG3 repR2:genG3 repR3:genG3 repR1:genG4 repR2:genG4 repR3:genG4
StdDev: 600.1711 600.1711 600.1711 600.1711 600.1711 600.1711
repR1:genG5 repR2:genG5 repR3:genG5 repR1:genG6 repR2:genG6 repR3:genG6
StdDev: 600.1711 600.1711 600.1711 600.1711 600.1711 600.1711
Block 3: repR1:nitro0, repR2:nitro0, repR3:nitro0, repR1:nitro60, repR2:nitro60, repR3:nitro60, repR1:nitro120, repR2:nitro120, repR3:nitro120
Formula: ~0 + rep:nitro | one
Structure: Multiple of an Identity
repR1:nitro0 repR2:nitro0 repR3:nitro0 repR1:nitro60 repR2:nitro60
StdDev: 235.2591 235.2591 235.2591 235.2591 235.2591
repR3:nitro60 repR1:nitro120 repR2:nitro120 repR3:nitro120 Residual
StdDev: 235.2591 235.2591 235.2591 235.2591 641.5963
Fixed effects: yield ~ nitro * gen
Value Std.Error DF t-value p-value
(Intercept) 3571.667 572.1257 36 6.242800 0.0000
nitro60 1560.333 557.9682 36 2.796456 0.0082
nitro120 3976.333 557.9682 36 7.126452 0.0000
genG2 1362.667 717.3336 36 1.899628 0.0655
genG3 678.000 717.3336 36 0.945167 0.3509
genG4 487.333 717.3336 36 0.679368 0.5012
genG5 530.000 717.3336 36 0.738847 0.4648
genG6 -364.333 717.3336 36 -0.507899 0.6146
nitro60:genG2 219.000 740.8516 36 0.295606 0.7692
nitro120:genG2 -1699.333 740.8516 36 -2.293757 0.0277
nitro60:genG3 312.333 740.8516 36 0.421587 0.6758
nitro120:genG3 -357.667 740.8516 36 -0.482778 0.6322
nitro60:genG4 -65.667 740.8516 36 -0.088637 0.9299
nitro120:genG4 -941.000 740.8516 36 -1.270160 0.2122
nitro60:genG5 -28.667 740.8516 36 -0.038694 0.9693
nitro120:genG5 -2066.000 740.8516 36 -2.788682 0.0084
nitro60:genG6 -1053.333 740.8516 36 -1.421787 0.1637
nitro120:genG6 -4691.667 740.8516 36 -6.332802 0.0000
Correlation:
(Intr) nitr60 ntr120 genG2 genG3 genG4 genG5 genG6 n60:G2
nitro60 -0.488
nitro120 -0.488 0.500
genG2 -0.627 0.343 0.343
genG3 -0.627 0.343 0.343 0.500
genG4 -0.627 0.343 0.343 0.500 0.500
genG5 -0.627 0.343 0.343 0.500 0.500 0.500
genG6 -0.627 0.343 0.343 0.500 0.500 0.500 0.500
nitro60:genG2 0.324 -0.664 -0.332 -0.516 -0.258 -0.258 -0.258 -0.258
nitro120:genG2 0.324 -0.332 -0.664 -0.516 -0.258 -0.258 -0.258 -0.258 0.500
nitro60:genG3 0.324 -0.664 -0.332 -0.258 -0.516 -0.258 -0.258 -0.258 0.500
nitro120:genG3 0.324 -0.332 -0.664 -0.258 -0.516 -0.258 -0.258 -0.258 0.250
nitro60:genG4 0.324 -0.664 -0.332 -0.258 -0.258 -0.516 -0.258 -0.258 0.500
nitro120:genG4 0.324 -0.332 -0.664 -0.258 -0.258 -0.516 -0.258 -0.258 0.250
nitro60:genG5 0.324 -0.664 -0.332 -0.258 -0.258 -0.258 -0.516 -0.258 0.500
nitro120:genG5 0.324 -0.332 -0.664 -0.258 -0.258 -0.258 -0.516 -0.258 0.250
nitro60:genG6 0.324 -0.664 -0.332 -0.258 -0.258 -0.258 -0.258 -0.516 0.500
nitro120:genG6 0.324 -0.332 -0.664 -0.258 -0.258 -0.258 -0.258 -0.516 0.250
n120:G2 n60:G3 n120:G3 n60:G4 n120:G4 n60:G5 n120:G5 n60:G6
nitro60
nitro120
genG2
genG3
genG4
genG5
genG6
nitro60:genG2
nitro120:genG2
nitro60:genG3 0.250
nitro120:genG3 0.500 0.500
nitro60:genG4 0.250 0.500 0.250
nitro120:genG4 0.500 0.250 0.500 0.500
nitro60:genG5 0.250 0.500 0.250 0.500 0.250
nitro120:genG5 0.500 0.250 0.500 0.250 0.500 0.500
nitro60:genG6 0.250 0.500 0.250 0.500 0.250 0.500 0.250
nitro120:genG6 0.500 0.250 0.500 0.250 0.500 0.250 0.500 0.500
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-1.52993309 -0.52842524 0.05394367 0.51465584 1.46902934
Number of Observations: 54
Number of Groups: 1 #tidy(model_lme)8.2.3 Check Model Assumptions
check_model(model_lmer, check = c('normality', 'linearity'))
plot(model_lme, resid(., scaled=TRUE) ~ fitted(.),
xlab = "fitted values", ylab = "studentized residuals")
qqnorm(residuals(model_lme))
qqline(residuals(model_lme))
The residuals fit the assumptions of the model well.
8.2.4 Inference
We can evaluate the model for the analysis of variance, for main and interaction effects.
car::Anova(model_lmer, type = "III", test.statistics = "F")Analysis of Deviance Table (Type III Wald chisquare tests)
Response: yield
Chisq Df Pr(>Chisq)
(Intercept) 38.9728 1 4.298e-10 ***
nitro 51.5701 2 6.334e-12 ***
gen 6.8343 5 0.2333
nitro:gen 58.0064 10 8.621e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model_lme, type = "marginal") numDF denDF F-value p-value
(Intercept) 1 36 38.97256 <.0001
nitro 2 36 25.78512 <.0001
gen 5 36 1.36687 0.2597
nitro:gen 10 36 5.80061 <.0001Analysis of variance showed a significant interaction impact of gen and nitro on rice grain yield.
Next, We estimate marginal means for nitro and gen interaction effects using emmeans package.
emm1 <- emmeans(model_lmer, ~ nitro*gen)
emm1 nitro gen emmean SE df lower.CL upper.CL
0 G1 3572 572 17.8 2368 4775
60 G1 5132 572 17.8 3929 6335
120 G1 7548 572 17.8 6345 8751
0 G2 4934 572 17.8 3731 6138
60 G2 6714 572 17.8 5510 7917
120 G2 7211 572 17.8 6008 8415
0 G3 4250 572 17.8 3046 5453
60 G3 6122 572 17.8 4919 7326
120 G3 7868 572 17.8 6665 9072
0 G4 4059 572 17.8 2856 5262
60 G4 5554 572 17.8 4350 6757
120 G4 7094 572 17.8 5891 8298
0 G5 4102 572 17.8 2898 5305
60 G5 5633 572 17.8 4430 6837
120 G5 6012 572 17.8 4809 7215
0 G6 3207 572 17.8 2004 4411
60 G6 3714 572 17.8 2511 4918
120 G6 2492 572 17.8 1289 3695
Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95 emm1 <- emmeans(model_lme, ~ nitro*gen)Warning in model.matrix.default(trms, m, contrasts.arg = contrasts): variable
'rep' is absent, its contrast will be ignored
Warning in model.matrix.default(trms, m, contrasts.arg = contrasts): variable
'rep' is absent, its contrast will be ignoredemm1Warning in qt((1 - level)/adiv, df): NaNs produced nitro gen emmean SE df lower.CL upper.CL
0 G1 3572 572 0 NaN NaN
60 G1 5132 572 0 NaN NaN
120 G1 7548 572 0 NaN NaN
0 G2 4934 572 0 NaN NaN
60 G2 6714 572 0 NaN NaN
120 G2 7211 572 0 NaN NaN
0 G3 4250 572 0 NaN NaN
60 G3 6122 572 0 NaN NaN
120 G3 7868 572 0 NaN NaN
0 G4 4059 572 0 NaN NaN
60 G4 5554 572 0 NaN NaN
120 G4 7094 572 0 NaN NaN
0 G5 4102 572 0 NaN NaN
60 G5 5633 572 0 NaN NaN
120 G5 6012 572 0 NaN NaN
0 G6 3207 572 0 NaN NaN
60 G6 3714 572 0 NaN NaN
120 G6 2492 572 0 NaN NaN
Degrees-of-freedom method: containment
Confidence level used: 0.95 Note that, confidence intervals were not estimated through emmeans from lme model.
lme vs lmer
For strip plot experiment deisgn, fitting nested and crossed random effects is more complicated through nlme. Therefore, it’s more convinent to use lmer in this case as both models yielded same results in the example shown above.