Comprehensive Groups of Experiments Analysis for Numerous Environments

Pooled analysis of a series of designed experiments repeated over locations, seasons or years, where treatment-by-environment interaction is of primary interest. One design-specific function call performs environment-wise analysis of variance, formal testing of homogeneity of error variances, an optional variance-stabilising transformation, pooled analysis of variance with the correct error stratum for every effect, and multiple comparison of means. Five classical designs are supported: completely randomised ('PooledCRD()'), randomised block ('PooledRBD()'), Latin square ('PooledLSD()'), split plot ('PooledSPD()') and strip plot ('PooledStPD()'). Homogeneity of error variances is assessed by Bartlett's test computed on the environment-wise error mean squares as in Bartlett (1937) , and by the median-centred Brown-Forsythe form of Levene's test , which is robust to non-normality. When variances are heterogeneous, Aitken's transformation (division of each observation by the square root of the error mean square of its own environment) may be applied, and treatment means are reported on both the transformed and the original scale. For split plot and strip plot designs every effect is tested against its proper error term, and mean separation uses the error mean square and degrees of freedom of the corresponding stratum. The statistical theory is given in Cochran and Cox (1957, ISBN:9780471162032), Gomez and Gomez (1984, ISBN:9780471870928), Dean et al. (2017) and Ruiz et al. (2024) .

CANE

CANE (Comprehensive Analysis of groups of experiments for Numerous Environments) performs the classical pooled analysis of variance of a balanced series of designed experiments repeated over locations, seasons or years.

A single design-specific function call runs the whole workflow for one design. Choose the function that matches the design actually conducted; there is deliberately no omnibus function covering all five.

Each call performs:

1. analysis of variance within each environment, with every effect tested against the error term of its own stratum;
2. assessment of the homogeneity of the error variances: Bartlett’s test on the environment-wise error mean squares (for every error stratum in split plot and strip plot designs), the median-centred Brown-Forsythe form of Levene’s test, environment-wise Shapiro-Wilk tests and the ratio of the largest to the smallest error mean square;
3. Aitken’s transformation, applied automatically, always or never (transform = "auto", "always" or "never");
4. pooled analysis of variance in which the denominator of every F-test is printed in an Error column; and
5. multiple comparison of means (Tukey’s HSD, Duncan’s multiple range test or Fisher’s LSD) using the error mean square and degrees of freedom of the stratum in which the compared factor belongs.

When a treatment-by-environment interaction is significant, the result carries a note pointing to the environment-wise comparisons, which are always returned.

Installation

# from CRAN
install.packages("CANE")

# development version from GitHub
# install.packages("remotes")
remotes::install_github("Vinayaka11442/CANE")

Example

Commercial cane sugar yield of 18 sugarcane genotypes in a randomised block design with two replications in six environments:

library(CANE)
data(sugarcane)

fit <- PooledRBD(data = sugarcane, Response = "ccsy", Location = "env",
                 Treatment = "gen", Replication = "rep",
                 alpha = 0.05, Mult_Comp_Test = 1)

fit$Environment_Key
#>      Level  n   Error_MS Error_df
#> 1   SGA-IP 36 10.7245624       17
#> 2  SGA-IIP 36 20.5435136       17
#> 3    SGA-R 36 10.6075908       17
#> 4  TCSM-IP 36  0.1605052       17
#> 5 TCSM-IIP 36  1.3576430       17
#> 6   TCSM-R 36  3.3007993       17
fit$Bartlett_Test
#> 
#>  Bartlett's test for homogeneity of error variances (on error mean squares)
#> 
#> data:  environment-wise error mean squares
#> Bartlett's K-squared = 81.17, df = 5, p-value = 4.77e-16
#> ratio of largest to smallest error mean square = 128
fit$Pooled_ANOVA
#>  Source          Df  Sum Sq Mean Sq F value Pr(>F)  Sig Error         
#>  Environment       5 30753. 6151.   1654.   2.5e-09 *** Rep within Env
#>  Rep within Env    6 22.312 3.719   3.719   0.0022  **  Pooled error  
#>  Treatment        17 503.72 29.63   29.63   < 1e-16 *** Pooled error  
#>  Treatment x Env  85 1233.5 14.51   14.51   < 1e-16 *** Pooled error  
#>  Pooled error    102 102.00 1.000                                     
#>  Total           215 32614.                                           
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 'Error' names the mean square used as the denominator of each F-test.
head(fit$Treatments_Comparison$groups)
#>   Treatment     Mean Groups Mean_original_scale
#> 1  Co 12009 13.93197      a            15.89367
#> 2  Co 16018 13.66819      a            15.59433
#> 3  Co 17012 12.84143     ab            15.19867
#> 4  Co 14027 12.63899     ab            15.27950
#> 5  Co 12008 11.83662     bc            12.86358
#> 6  Co 86032 11.80427    bcd            14.23675

print(fit) shows the complete analysis. The error variances are strongly heterogeneous here, so the pooled analysis is carried out on the Aitken-transformed scale; the grouping letters refer to that scale, and the arithmetic means on the original scale are reported alongside them.

Validation

Every design is checked against reference analyses specified independently of the package (stratified aov() fits with Error() strata, sums of squares accumulated from cell means, restricted maximum likelihood in nlme, design theory for the degrees of freedom, and the base-R distribution functions behind the post hoc procedures):

vignette("validation", package = "CANE")

Scope

CANE implements the fixed-effects pooled analysis of a balanced series. For other situations use:

• unbalanced series or partly overlapping treatment sets: mixed models with lme4, nlme or sommer, with statgenSTA for the single-trial stage;
• augmented designs: augmentedRCBD (archived on CRAN in April 2026 for an administrative reason; installable from its own repository);
• AMMI, GGE biplots and stability statistics: metan or ammistability.

Version 0.2.0

Version 0.2.0 corrects defects in 0.1.1 that affected numerical results, including the error strata of the split plot and strip plot designs. Users who have published analyses produced with 0.1.1 should re-run them. See the NEWS file (news(package = "CANE")) for the full list.

Citation

citation("CANE")