PooledRBD: Pooled Analysis of a Series of Randomised Block Designs

Description

Performs the classical pooled analysis of variance for a randomised block design (RBD) repeated over several environments. This is the workhorse analysis of multi-environment variety and agronomy trials.

Usage

PooledRBD(
  data,
  Response,
  Location,
  Treatment,
  Replication,
  alpha = 0.05,
  Mult_Comp_Test = 1,
  transform = c("auto", "always", "never"),
  env_effect = c("fixed", "random")
)

Value

An object of class "cane", a list with components

Design: Name of the design analysed.
Individual_ANOVA: List of environment-wise ANOVA tables.
Location_wise: Environment-wise multiple comparisons, each computed from the error mean square of that environment.
Bartlett_Test: Bartlett's test on the error mean squares. For split plot and strip plot designs, element other_strata gives the same test for the main plot or strip error strata.
Levene_Test: Median-centred Levene (Brown-Forsythe) test.
Normality_Test: Environment-wise Shapiro-Wilk tests.
Transformation: Whether Aitken's transformation was applied, why, the environment-wise error mean squares and the ratio of the largest to the smallest.
Pooled_ANOVA: Pooled ANOVA table, including the error term used as the denominator of each F-test.
Treatments_Comparison: Pooled treatment means, groups and all pairwise differences. When a transformation was applied, means are given on the transformed scale (on which the test was performed) and the arithmetic means on the original scale are reported alongside.
GEI_Note: Advice issued when any interaction of a treatment factor with environment is significant, naming the terms concerned.
Environment_Key: Mapping of environment levels to their error mean square and error degrees of freedom.
Settings: Arguments and software versions used.

Arguments

data: A data frame containing the experimental data. Rows with missing values in the variables used are dropped with a warning.
Response: Character string naming the numeric response variable.
Location: Character string naming the factor that identifies environments (locations, seasons or years).
Treatment: Character string naming the treatment factor.
Replication: Character string naming the replication (block) factor.
alpha: Significance level used for the variance-homogeneity test and for the multiple comparison procedure. Default 0.05.
Mult_Comp_Test: Integer selecting the multiple comparison procedure: 1 = Tukey's honestly significant difference (HSD) test, 2 = Duncan's multiple range test (DMRT), 3 = Fisher's least significant difference (LSD) test. Option 3 makes no adjustment for multiplicity and is intended for a small number of pre-planned comparisons; a warning is issued if it is requested for more than three treatments.
transform: One of "auto" (default; transform only when the variances are judged heterogeneous at level alpha), "always" or "never".
env_effect: One of "fixed" (default) or "random". With "random" each treatment main effect is tested against its interaction with environment, and in split plot and strip plot designs the interaction of the two treatment factors is tested against the three-way interaction with environment. The pooled comparison of means then uses the same mean square and degrees of freedom as the F-test of the factor compared. With few environments that interaction has few degrees of freedom; if it has fewer than two, the Tukey and Duncan comparisons for that factor are not computed and a warning is issued.

Details

The model fitted within environment $j$ is $$Y_{ijk} = \mu_j + \rho_{jk} + \tau_{ij} + \epsilon_{ijk}$$ with $\rho_{jk}$ the effect of replication $k$ in environment $j$. The pooled model is $$Y_{ijk} = \mu + \beta_j + \rho_{k(j)} + \tau_i + (\tau\beta)_{ij} + \epsilon_{ijk},$$ replications being nested within environments, which gives $l(r-1)$ degrees of freedom for replication rather than $r-1$.

Following the standard treatment of a series of block experiments, the environment main effect is tested against the replication-within-environment mean square, because replications are the units within which environments are compared. Treatment and the treatment-by-environment interaction are tested against the pooled error when env_effect = "fixed". When env_effect = "random" the treatment main effect is tested against the treatment-by-environment interaction instead.

References

Aitken, A. C. (1935). On least squares and linear combination of observations. Proceedings of the Royal Society of Edinburgh, 55, 42--48. tools:::Rd_expr_doi("10.1017/S0370164600014346")

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London A, 160, 268--282. tools:::Rd_expr_doi("10.1098/rspa.1937.0109")

Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. John Wiley and Sons, New York.

Dean, A., Voss, D. and Draguljic, D. (2017). Design and Analysis of Experiments, 2nd ed. Springer, Cham. tools:::Rd_expr_doi("10.1007/978-3-319-52250-0")

Gomez, K. A. and Gomez, A. A. (1984). Statistical Procedures for Agricultural Research, 2nd ed. John Wiley and Sons, New York.

Examples

Run this code

data(sugarcane)
out <- PooledRBD(sugarcane, Response = "ccsy", Location = "env",
                 Treatment = "gen", Replication = "rep",
                 alpha = 0.05, Mult_Comp_Test = 1)
out
out$Pooled_ANOVA
head(out$Treatments_Comparison$groups)

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