Performs the classical pooled analysis of variance for a split plot design (SPD) repeated over several environments, with every effect tested against its proper error term.
PooledSPD(
data,
Response,
Location,
Replication,
MainPlot,
SubPlot,
alpha = 0.05,
Mult_Comp_Test = 1,
transform = c("auto", "always", "never"),
env_effect = c("fixed", "random")
)An object of class "cane". The structure is as described in
PooledCRD, except that Location_wise and
Treatments_Comparison are each a list with components
MainPlot and SubPlot, computed from the main plot and
subplot error strata respectively.
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 replication factor.
Character string naming the main plot (whole plot) factor, that is the factor requiring the larger experimental units.
Character string naming the subplot 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.
A split plot design has two error strata. Main plot effects belong to the main plot stratum and must be tested against the main plot error, \(Environment \times Replication \times MainPlot\); subplot effects and all interactions involving the subplot factor belong to the subplot stratum and are tested against the subplot (residual) error. Testing a main plot effect against the subplot error, as happens if a single-stratum model is fitted and its residual mean square is used for everything, gives an F-ratio with the wrong denominator and the wrong denominator degrees of freedom.
The pooled partition implemented here is $$Environment,\; Rep(Env),\; A,\; A \times Env,\; \mathrm{main\ plot\ error},$$ $$B,\; B \times Env,\; A \times B,\; A \times B \times Env,\; \mathrm{subplot\ error},$$ where \(A\) is the main plot factor and \(B\) the subplot factor. With \(l\) environments, \(r\) replications, \(a\) main plot levels and \(b\) subplot levels the main plot error carries \(l(r-1)(a-1)\) degrees of freedom and the subplot error \(la(r-1)(b-1)\).
Mean separation is carried out in the stratum appropriate to each factor:
main plot means are compared using the main plot error mean square and its
degrees of freedom, subplot means using the subplot error. The stratum used
is recorded in the parameters component of each comparison.
Homogeneity of error variances is assessed for both strata. The main
Bartlett test, and the Aitken transformation when it is applied, use the
subplot error mean square of each environment; the main plot error mean
squares are tested separately and reported in
Bartlett_Test$other_strata, with a note in the transformation report
if they are heterogeneous.
At least two replications per environment are needed for a main plot error to exist, and three or more are strongly advised: with \(r = 2\) and \(a = 2\) the main plot error has only \(l\) degrees of freedom and the main plot test has almost no power.
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.
PooledCRD, PooledRBD,
PooledLSD, PooledStPD
data(spd_example)
out <- PooledSPD(spd_example, Response = "Yield", Location = "Location",
Replication = "Replication", MainPlot = "MainPlot",
SubPlot = "SubPlot", alpha = 0.05, Mult_Comp_Test = 1)
out
out$Pooled_ANOVA
out$Treatments_Comparison$MainPlot$parameters
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