Analysis of an alpha lattice (resolvable incomplete block) design.
Gives the intra-block ANOVA, genotype means adjusted for incomplete
blocks with exact standard errors and letter groupings, and the
efficiency of the design relative to a randomized complete block design.
By default replications and blocks are treated as random (REML), so
that inter-block information is recovered (Patterson and Williams,
1976); analysis = "intra" gives the classical intra-block
estimates.
Usage
anova_alpha_lattice(
data,
response,
genotype,
replication,
block,
verbose = TRUE,
analysis = c("reml", "intra"),
posthoc = "lsd",
alpha = 0.05
)Arguments
- data
Data frame containing the data.
- response
Name of the response variable.
- genotype
Name of the genotype column.
- replication
Name of the replication column.
- block
Name of the incomplete block column (nested within replication; labels may repeat across replications).
- verbose
Logical. If TRUE (default), prints the report.
- analysis
"reml"(default; replications and incomplete blocks random, fitted with nlme, so that inter-block information is recovered) or"intra"(blocks fixed; the classical intra-block analysis). If the REML fit fails, the intra-block analysis is used with a message.- posthoc
Method for the letters (default
"lsd").- alpha
Significance level. Default 0.05.
Value
An object of class aridagri_alpha with
anova_table, genotype_means (adjusted means with SE and
letters, highest first), avg_sed, cd (from the average
SEd), efficiency (percent, relative to RCBD), cv,
grand_mean, posthoc, variance_components (REML)
and model.
Details
The intra-block ANOVA is the sequential analysis of the model replication + block(replication) + genotype: genotypes are tested after blocks. Blocks within replications are also shown adjusted for genotypes. Adjusted means are least-squares means. The efficiency relative to an RCBD is the average variance of a difference under the RCBD analysis (\(2 MSE_{RCBD} / r\)) divided by the average variance of a difference of adjusted means. With REML, letters and CD use the intra-block error degrees of freedom, an approximation.
References
Patterson, H.D. and Williams, E.R. (1976). A new class of resolvable incomplete block designs. Biometrika, 63(1), 83-92. doi:10.1093/biomet/63.1.83
Examples
data(cowpea_alpha)
res <- anova_alpha_lattice(cowpea_alpha, response = "grain_yield",
genotype = "genotype", replication = "rep",
block = "block")
#>
#> ======================================================================
#> ALPHA LATTICE DESIGN
#> ======================================================================
#> Response : grain_yield
#> Analysis : REML (blocks random)
#> Genotypes : 24
#> Replications : 3
#> Incomplete blocks : 18
#>
#> Intra-block analysis of variance
#> Source Df Sum Sq Mean Sq F value Pr(>F)
#> rep 2 110849.69 55424.85
#> Blocks within rep (unadjusted) 15 359316.25 23954.42
#> Blocks within rep (adjusted) 15 170639.87 11375.99 4.407 0.0002 ***
#> genotype (adjusted) 23 247396.06 10756.35 4.167 0.0001 ***
#> Intra-block error 31 80030.44 2581.63
#> Signif. codes: *** p < 0.001, ** p < 0.01, * p < 0.05, NS not significant
#>
#> Grand mean = 1058.278 CV = 4.80% Average SEd = 46.672 CD (5%) = 95.189
#> Efficiency relative to RCBD = 166.8%
#>
#> REML variance components
#> Component Variance
#> rep 1679
#> Block within rep 3142
#> Residual 2560
#>
#> Adjusted genotype means (top 15, Fisher's LSD)
#> Genotype Mean Adjusted_mean SE Group
#> CP-03 1229.67 1210.69 42.576 a
#> CP-17 1216.67 1163.07 42.576 abc
#> CP-16 1179.67 1162.40 42.595 ab
#> CP-12 1098.67 1108.60 42.595 bcd
#> CP-14 1097.33 1103.80 42.576 bcde
#> CP-09 1063.00 1094.97 42.595 bcdef
#> CP-15 1111.67 1093.96 42.595 bcdefg
#> CP-19 1113.00 1093.20 42.576 bcdefg
#> CP-13 1094.33 1079.56 42.595 bcdefgh
#> CP-08 1013.67 1069.33 42.595 cdefgh
#> CP-20 1069.00 1063.63 42.595 defgh
#> CP-11 1069.00 1062.62 42.595 defghi
#> CP-22 1033.00 1058.60 42.576 defghi
#> CP-02 1071.67 1057.56 42.576 defghi
#> CP-24 1026.33 1037.79 42.576 defghi
head(res$genotype_means)
#> Genotype Mean Adjusted_mean SE Group
#> 1 CP-03 1229.667 1210.688 42.57639 a
#> 2 CP-17 1216.667 1163.069 42.57639 abc
#> 3 CP-16 1179.667 1162.403 42.59500 ab
#> 4 CP-12 1098.667 1108.596 42.59500 bcd
#> 5 CP-14 1097.333 1103.796 42.57639 bcde
#> 6 CP-09 1063.000 1094.972 42.59500 bcdef