Performs the classical pooled analysis of variance for a completely randomised design (CRD) repeated over several environments (locations, seasons or years). The function carries out the environment-wise analysis, tests homogeneity of the error variances, optionally applies Aitken's variance-stabilising transformation, fits the pooled model including the treatment-by-environment interaction, and separates treatment means.
PooledCRD(
data,
Response,
Location,
Treatment,
alpha = 0.05,
Mult_Comp_Test = 1,
transform = c("auto", "always", "never"),
env_effect = c("fixed", "random")
)An object of class "cane", a list with components
DesignName of the design analysed.
Individual_ANOVAList of environment-wise ANOVA tables.
Location_wiseEnvironment-wise multiple comparisons, each computed from the error mean square of that environment.
Bartlett_TestBartlett'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_TestMedian-centred Levene (Brown-Forsythe) test.
Normality_TestEnvironment-wise Shapiro-Wilk tests.
TransformationWhether Aitken's transformation was applied, why, the environment-wise error mean squares and the ratio of the largest to the smallest.
Pooled_ANOVAPooled ANOVA table, including the error term used as the denominator of each F-test.
Treatments_ComparisonPooled 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_NoteAdvice issued when any interaction of a treatment factor with environment is significant, naming the terms concerned.
Environment_KeyMapping of environment levels to their error mean square and error degrees of freedom.
SettingsArguments and software versions used.
A data frame containing the experimental data. Rows with missing values in the variables used are dropped with a warning.
Character string naming the numeric response variable.
Character string naming the factor that identifies environments (locations, seasons or years).
Character string naming the treatment factor.
Significance level used for the variance-homogeneity test and
for the multiple comparison procedure. Default 0.05.
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.
One of "auto" (default; transform only when the
variances are judged heterogeneous at level alpha),
"always" or "never".
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.
The model fitted within environment \(j\) is
$$Y_{ijk} = \mu_j + \tau_{ij} + \epsilon_{ijk}$$
and the pooled model is
$$Y_{ijk} = \mu + \beta_j + \tau_i + (\tau\beta)_{ij} + \epsilon_{ijk},$$
where \(\beta_j\) is the environment effect, \(\tau_i\) the treatment
effect and \((\tau\beta)_{ij}\) their interaction. A completely randomised
design has a single error stratum, so every effect is tested against the
pooled error mean square when env_effect = "fixed". With
env_effect = "random" the treatment main effect is tested against the
treatment-by-environment interaction, which is the appropriate denominator
when the environments are regarded as a random sample from a target
population.
Homogeneity of error variances is tested by Bartlett's statistic computed
on the environment-wise error mean squares, each carrying its own error
degrees of freedom. Because Bartlett's test is sensitive to non-normality,
the median-centred (Brown-Forsythe) form of Levene's test and
environment-wise Shapiro-Wilk tests are reported alongside it. Levene's
test is computed on the environment-wise residuals; because the residuals of
environment \(j\) carry only its error degrees of freedom
\(\nu_j\) (with two replications they occur in exact plus-minus
pairs), its denominator degrees of freedom are \(\sum_j \nu_j\)
rather than \(N - l\). The transform argument lets the user
override the automatic decision.
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.
PooledRBD, PooledLSD,
PooledSPD, PooledStPD
data(crd_example)
out <- PooledCRD(crd_example, Response = "Yield", Location = "Loc",
Treatment = "Treatment", alpha = 0.05, Mult_Comp_Test = 1)
out
out$Pooled_ANOVA
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