stripSplitPlot

Analyzes data from strip-split-plot experiments. Function stripSplitPlot also analyzes strip-split-plot experiments replicated at several locations.

Synopsis

stripSplitPlot (nLocations, nStripA, nStripB, nSplit, block, stripA, stripB, split, y)

Required Arguments

int nLocations (Input)

Number of locations. nLocations must be one or greater. If nLocations>1 then the optional array locations[] must be included as input to stripSplitPlot.

int nStripA (Input)

Number of levels associated with the strip-plot A factor. nStripA must be greater than one.

int nStripB (Input)

Number of levels associated with the strip-plots B factor. nStripB must be greater than one.

int nSplit (Input)

Number of levels associated with the split factor. nSplit must be greater than one.

int block[] (Input)

An array of length n containing the block identifiers for each observation in y. Locations can have different numbers of blocks. Each block at a single location must be assigned a different identifier, but different locations can have the same assignments.

int stripA[] (Input)

An array of length n containing the strip-plot A level identifiers for each observation in y. Each level of this factor must be assigned a different integer. stripSplitPlot verifies that the number of unique strip-plot identifiers is equal to nStripA.

int stripB[] (Input)

An array of length n containing the strip-plot B identifiers for each observation in y. Each level of this factor must be assigned a different integer. stripSplitPlot verifies that the number of unique strip-plot identifiers is equal to nStripB.

int split[] (Input)

An array of length n containing the split-plot level identifiers for each observation in y. Each level of this factor must be assigned a different integer. stripSplitPlot verifies that the number of unique split-plot identifiers is equal to nSplit.

float y[] (Input)

An array of length n containing the experimental observations and any missing values. Missing values cannot be omitted. They are indicated by placing a NaN (not a number) in y. The NaN value can be set using the function machine(6). The location, strip-plot A, strip-plot B and split-plot for each observation in y are identified by the corresponding values in the argument’s locations, stripA, stripB, and split.

Return Value

A two dimensional, 22 by 6 array containing the ANOVA table. Each row in this array contains values for one of the effects in the ANOVA table. The first value in each row, \(\text{anovaTable}_{i,0}\) = anovaTable[i×6], identifies the source for the effect associated with values in that row. The remaining values in a row contain the ANOVA table values using the following convention:

j	\(\text{anovaTable}_{i,j}\) = anovaTable[i\*6+j]
0	Source Identifier (values described below)
1	Degrees of freedom
2	Sum of squares
3	Mean squares
4	F-statistic
5	p-value for this F-statistic

The Source Identifiers in the first column of \(\text{anovaTable}_{i,j}\) are the only negative values in anovaTable[]. Note that the p-value for the F-statistic is returned as 0.0 when the value is so small that all significant digits have been lost. Assignments of identifiers to ANOVA sources use the following coding:

Source Identifier	ANOVA Source
-1	LOCATION†
-2	BLOCKs WITHIN LOCATION
-3	STRIP-PLOT A
-4	LOCATION × STRIP-PLOT A †
-5	STRIP-PLOT A ERROR
-6	SPLIT-PLOT
-7	SPLIT-PLOT × STRIP-PLOT A
-8	LOCATION × SPLIT-PLOT †
-9	SPLIT-PLOT ERROR
-10	LOCATION × SPLIT-PLOT × STRIP-PLOT A †
-11	STRIP-PLOT B
-12	LOCATION × STRIP-PLOT B †
-13	STRIP_PLOT B ERROR
-14	STRIP-PLOT A × STRIP-PLOT B
-15	LOCATION × STRIP-PLOT A × STRIP-PLOT B
-16	STRIP-PLOT A × STRIP-PLOT B ERROR
-17	SPLIT-PLOT × STRIP-PLOT B
-18	STRIP-PLOT A × STRIP-PLOT B × SPLIT-PLOT
-19	LOCATION × SPLIT-PLOT × STRIP-PLOT B †
-20	LOCATION × STRIP-PLOT A × STRIP-PLOT B × SPLIT-PLOT †
-21	STRIP-PLOT A × STRIP-PLOT B × SPLIT-PLOT ERROR
-22	CORRECTED TOTAL

Notes: † If nLocations=1 sources involving location are set to missing (NaN).

Optional Arguments

locations, int[] (Input)

An array of length n containing the location identifiers for each observation in y. Unique integers must be assigned to each location in the study. This argument is required when nLocations>1.

nMissing (Output)

Number of missing values, if any, found in y. Missing values are denoted with a NaN (Not a Number) value.

cv (Output)

An array of length 3 containing the strip-plots and split-plot coefficients of variation. cv[0] contains the strip-plot A C.V., cv[1] contains the strip-plot B C.V., and cv[2] contains the split-plot C.V.

grandMean (Output)

Mean of all the data across every location.

stripPlotAMeans (Output)

An array of length nStripA containing the factor A strip-plot means.

stripPlotBMeans (Output)

An array of length nStripB containing the strip-plot B means.

splitPlotMeans (Output)

An array of length nSplit containing the strip-plot B means.

stripPlotASplitPlotMeans (Output)

A 2-dimensional array of size nStripA by nSplit containing the means for all combinations of the factor A strip-plot and split-plots.

stripPlotBSplitPlotMeans (Output)

A 2-dimensional array of size nStripB by nSplit containing the means for all combinations of strip-plot B and split-plots.

stripPlotAbMeans (Output)

A 2-dimensional array of size nStripA by nStripB containing the means for all combinations of strip-plots.

treatmentMeans (Output)

An array of size (nStripA\*nStripB\*nSplit) containing the treatment means. For i >0, j >0, and k >0, \(\text{treatmentMeans}_{i,j,k}\) = treatmentMeans (i‑1)*nStripB × nSplit + (j‑1) × nSplit + (k‑1)] contains the mean of the observations, averaged over all locations, blocks and replicates, for the i-th level of the factor A strip-plot, the j‑th level of the factor B strip-plot, and the k‑th level of the split-plot.

stdErrors (Output)

An array of length 20 containing ten standard errors and their associated degrees of freedom. The standard errors are in the first 10 elements and their associated degrees of freedom are reported in stdErrors[10] through stdErrors[19].

Element	Standard Error for Comparisons Between Two	Degrees of Freedom
stdErrors[0]	Strip-Plot A Means	stdErrors[10]
stdErrors[1]	Strip-Plot B Means	stdErrors[11]
stdErrors[2]	Split-Plot Means	stdErrors[12]
stdErrors[3]	Strip-Plot A Means at the same level of split-plots	stdErrors[13]
stdErrors[4]	Strip-Plot A Means at the same level of strip-plot B	stdErrors[14]
stdErrors[5]	Strip-Plot B Means at the same level of split-plots	stdErrors[15]
stdErrors[6]	Strip-Plot B Means at the same level of strip-plot A	stdErrors[16]
stdErrors[7]	Split-Plot Means at the same level of strip-plot A	stdErrors[17]
stdErrors[8]	Split-Plot Means at the same level of strip-plot B	stdErrors[18]
stdErrors[9]	Treatment Means (same strip-plot A, strip-plot B and split-plot)	stdErrors[19]

nBlocks (Output)

An array of length nLocations containing the number of blocks, or replicates, at each location. This value must be greater than one, nBlocks > 1.

locationAnovaTable (Output)

A 3-dimensional array of size nLocations by 22 by 6 containing the anova tables associated with each location. For each location, the 22 by 6 dimensional array corresponds to the anova table for that location. For example, locationAnovaTable[(i‑1)×132+(j-1)×6+(k-1)] contains the value in the k‑th column and j‑th row of the returned anova table for the i‑th location.

anovaRowLabels (Output)

An array containing the labels for each of the nAnova rows of the returned ANOVA table. The label for the i‑th row of the ANOVA table can be printed with print anovaRowLabels[i]

Description

Function stripSplitPlot is capable of analyzing a wide variety of strip-split plot experiments, also referred to as strip-strip plot experiments. By default, stripSplitPlot assumes that both strip-plot factors, and split-plots are fixed effects, and the location effects, if any, are random effects. The nature of randomization used in an experiment determines analysis of the data. Two popular forms of randomization in strip-plot and split-plot experiments are illustrated in the following two figures. In both experiments, the strip-plot factor, factor A, has 4 levels that are randomly assigned to a block or field in four strips.

Table 4.29 — Strip-Plot Experiment - Strip-Plots Completely Crossed

		Factor A Strip-Plots
		A2	A1	A4	A3
Factor B Strip Plots	B3	A2B3	A1B3	A4B3	A3B3
	B1	A2B1	A1B1	A4B1	A3B1
	B2	A2B2	A1B2	A4B2	A3B2

In the strip-plot experiment, factor B, has 3 levels that are randomly assigned as strips across each of the four levels of factor A. In this case, factors A and B are completely crossed. The randomization applied to factor B is independent of the application of the strip-plots, factor A.

Contrast this to the randomization depicted in Table 4.30. In this split-plot experiment, the levels of factor B are nested within each level of factor A whole-plots. Factor B is randomized independently within each level of factor A. Unlike the strip-plot experiment, in the split-plot experiment different levels of factor B appear in the same row.

Table 4.30 — Split-Plot Experiment – Factor B Split-Plots Nested within Factor A Whole-Plots

Whole-Plot Factor
A2	A1	A4	A3
A2B1	A1B3	A4B1	A3B2
A2B3	A1B1	A4B3	A3B1
A2B2	A1B2	A4B2	A3B2

A strip-split plot experiment is a strip-plot experiment with a third factor randomized within each level of strip-plot factor A, see Table 4.31. The third factor, referred to as the split-plot factor, is randomly assigned to experimental units within each level of strip-plot factor A. stripSplitPlot analyzes strip-split plot experiments consisting of two strip-plot factors and one split-plot factor nested within strip-plot factors A and B.

Table 4.31 — Strip-Split Plot Experiment - Split-Plots Nested Within Strip-Plot Factors A

		Factor A Strip Plots
		A2	A1	A4	A3
Factor B Strip Plots	B3	A2B3C2 A2B3C1	A1B3C1 A1B3C2	A4B3C2 A4B3C1	A3B3C2 A3B3C1
	B1	A2B1C1 A2B1C2	A1B1C1 A1B1C2	A4B1C2 A4B1C1	A3B1C2 A3B1C1
	B2	A2B2C2 A2B2C1	A1B2C1 A1B2C2	A4B2C1 A4B2C2	A3B2C2 A3B2C1

Strip-split plot experiments are closely related to split-split plot experiments, see Table 4.32. The main difference between the two is that in strip-split plot experiments, the order of the levels for factor B are not applied randomly across factor A. Each level of factor B is constant across any row. In this example, the entire first row is assigned to the third level of factor B. In the equivalent split-split plot experiment, the levels of factor B are not constant across any row. The levels are randomized within each level of factor A.

Table 4.32 — Split-Split Plot Experiment – Sub-Plot Factor C Nested Within Split-Plot Factor B

Whole Plot Factor A
A2	A1	A4	A3
A2B3C2 A2B3C1	A1B2C1 A1B2C2	A4B1C2 A4B1C1	A3B3C2 A3B3C1
A2B1C1 A2B1C2	A1B1C1 A1B1C2	A4B3C2 A4B3C1	A3B2C2 A3B2C1
A2B2C2 A2B2C1	A1B3C1 A1B3C2	A4B2C1 A4B2C2	A3B1C2 A3B1C1

In some studies, a strip-split-plot experiment is replicated at several locations. Function stripSplitPlot can analyze strip-split plot experiments replicated at multiple locations, even when the number of blocks or replicates at each location might be different. If only a single replicate or block is used at each location, then location should be treated as a blocking factor, with nLocations=1. If nLocations=1, it is assumed that either the experiment was conducted at multiple locations, each with a single block, or at a single location with more than one block or replicate at that location. When nLocations=1, all entries associated with location in the anova table will contain missing values.

However, if nLocations>1, it is assumed the experiment was repeated at multiple locations, with blocking occurring at each location. Although the number of blocks at each location can be different, the number of levels for the strip-plot and split-plot factors strip-plots must be the same at each location. The locations associated with each of the observations in y are specified in the argument locations[], which is a required input argument when nLocations>1.

By default, locations are assumed to be random effects. Tests involving strip-plots use the interaction between strip-plots and locations as the error term for testing whether there are statistically significant differences among strip-plots. However, this assumes that the interaction of strip-plots and locations is not statistically significant. A test of this assumption is included in the anova table. If any interactions between locations and strip-plot or split-plot factors are statistically significant, then the nature of these interactions should be explored since this impacts the interpretation of the significance of the treatment factors.

Similarly, when locations are assumed to be random effects, tests involving split-plots do not use the split-plot errors pooled across locations. Instead, the error term for split-plots is the interaction between locations and split-plots.

Suppose, for example, that a researcher wanted to conduct an agricultural experiment comparing the effectiveness of 4 fertilizers with 3 seed lots and 3 rates of application. One replicate of the experiment is conducted at each of the 3 farms. That is, only a single field at each location is assigned to this experiment.

Each field is divided into 4 vertical strips and 3 horizontal strips. The vertical strips are randomly assigned to fertilizers and the rows are randomly assigned to application rates. Fertilizers and application rates represent strip-plot factors A and B respectively. Seed lots are randomly assigned to three sub-divisions within each combination of strip-plots.

Table 4.33 — Strip-Split Plot Experiment – Fertilizer Strip-Plots, Application Rate Strip-Plots

		Fertilizer Strip Plots
		F2	F1	F4	F3
Application Rate Strip Plot	R3	F2R3S1 F2R3S2 F2R3S3	F1R3S3 F1R3S2 F1R3S1	F4R3S3 F4R3S2 F4R3S1	F3R3S2 F3R3S1 F3R3S3
	R2	F2R1S3 F2R1S1 F2R1S2	F1R1S2 F1R1S3 F1R1S1	F4R1S3 F4R1S1 F4R1S2	F3R1S1 F3R1S2 F3R1S3
	R1	F2R2S1 F2R2S2 F2R2S3	F1R2S1 F1R2S3 F1R2S2	F4R2S2 F4R2S3 F4R2S1	F3R2S3 F3R2S1 F3R2S2

In this case, each farm is a blocking factor, fertilizers are factor A strip-plots, fertilizer application rates are factor B strip-plots, and seed lots are split-plots. The input array rep would contain integers from 1 to the number of farms.

Example

The experiment was conducted using a 2 x 2 strip_split plot arrangement with each of the four plots divided into 2 sub-divisions that were randomly assigned one of two split-plot levels. This was replicated 3 times producing an experiment with \(n=2x2x2x3=24\) observations.

from __future__ import print_function
import sys
from numpy import *
from l_print_lsd import *
from pyimsl.stat.multipleComparisons import multipleComparisons
from pyimsl.stat.page import page, SET_PAGE_WIDTH
from pyimsl.stat.stripSplitPlot import stripSplitPlot
from pyimsl.stat.writeMatrix import writeMatrix

col_labels = [" ", "\nID", "\nDF", "\nSSQ",
              "Mean\nsquares", "\nF", "\np-value"]
page_width = 132
n = 24               # Total number of observations
n_locations = 1      # Number of locations
n_strip_a = 2        # Number of factor A strip-plots
n_strip_b = 2        # Number of factor B strip-plots
n_split = 2          # Number of split-plots within factor A split-plot
block = [1, 1, 1, 1, 1, 1, 1, 1,
         2, 2, 2, 2, 2, 2, 2, 2,
         3, 3, 3, 3, 3, 3, 3, 3]
strip_a = [1, 1, 1, 1, 2, 2, 2, 2,
           1, 1, 1, 1, 2, 2, 2, 2,
           1, 1, 1, 1, 2, 2, 2, 2]
strip_b = [1, 1, 2, 2, 1, 1, 2, 2,
           1, 1, 2, 2, 1, 1, 2, 2,
           1, 1, 2, 2, 1, 1, 2, 2]
split = [1, 2, 1, 2, 1, 2, 1, 2,
         1, 2, 1, 2, 1, 2, 1, 2,
         1, 2, 1, 2, 1, 2, 1, 2]
y = [30.0, 40.0, 38.9, 38.2,
     41.8, 52.2, 54.8, 58.2,
     20.5, 26.9, 21.4, 25.1,
     26.4, 36.7, 28.9, 35.9,
     21.0, 25.4, 24.0, 23.3,
     34.4, 41.0, 33.0, 34.9]
alpha = 0.05
grand_mean = []
anova_row_labels = []
cv = []
treatment_means = []
strip_plot_a_means = []
strip_plot_b_means = []
split_plot_means = []
strip_a_split_plot_means = []
strip_b_split_plot_means = []
strip_plot_ab_means = []
std_err = []
equal_means = []

aov = stripSplitPlot(n_locations, n_strip_a, n_strip_b,
                     n_split, block, strip_a, strip_b, split, y,
                     grandMean=grand_mean, cv=cv,
                     treatmentMeans=treatment_means,
                     stripPlotAMeans=strip_plot_a_means,
                     stripPlotBMeans=strip_plot_b_means,
                     splitPlotMeans=split_plot_means,
                     stripPlotASplitPlotMeans=strip_a_split_plot_means,
                     stripPlotBSplitPlotMeans=strip_b_split_plot_means,
                     stripPlotAbMeans=strip_plot_ab_means,
                     stdErrors=std_err,
                     anovaRowLabels=anova_row_labels)

# Output results
page(SET_PAGE_WIDTH, page_width)

# Print ANOVA table
writeMatrix("   *** ANALYSIS OF VARIANCE TABLE ***",
            aov, writeFormat="%3.0f%3.0f%8.2f%7.2f%7.2f%7.3f",
            rowLabels=anova_row_labels,
            colLabels=col_labels)

# Print the various means
print("\nGrand mean: ", grand_mean[0])
print("\nCoefficient of Variation: ")
print("  Strip-Plot A:      ", cv[0])
print("  Strip-Plot B:      ", cv[1])
print("  Split-Plot:        ", cv[2])

# Print treatment means
print("\nTreatment Means")
l = 0
for i in range(0, n_strip_a):
    for j in range(0, n_strip_b):
        for k in range(0, n_split):
            tma = array(treatment_means, dtype='float')
            print("treatment[%d][%d][%d]   %9.4f" %
                  (i + 1, j + 1, k + 1, tma.flat[l]))
            l += 1
print("\nStandard Error for Comparing Two Treatment Means: %f \n(df=%f)" %
      (std_err[9], std_err[19]))
tma = array(treatment_means)
equal_means = multipleComparisons(tma.flat,
                                  int(std_err[19]), std_err[9] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_printLSD3Table(n_strip_a, n_strip_b, n_split,
                 equal_means, treatment_means)

# Print the Strip-plot A Means
print("\n\n*************************************************************")
writeMatrix("Strip-plot A Means", strip_plot_a_means, column=True)
print("\nStandard Error for Comparing Two Strip-Plot A Means: %f \n(df=%f)" %
      (std_err[0], std_err[10]))
equal_means = multipleComparisons(strip_plot_a_means,
                                  int(std_err[10]), std_err[0] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_print_lsd(n_strip_a, equal_means, strip_plot_a_means)

# Print Strip-plot B Means
print("\n\n*************************************************************")
writeMatrix("Strip-plot B Means", strip_plot_b_means, column=True)
print("\nStandard Error for Comparing Two Strip-Plot B Means: %f \n(df=%f)" %
      (std_err[1], std_err[11]))
equal_means = multipleComparisons(strip_plot_b_means,
                                  int(std_err[11]), std_err[1] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_print_lsd(n_strip_b, equal_means, strip_plot_b_means)

# Print the Split-plot means
print("\n\n*************************************************************")
writeMatrix("Split-plot Means", split_plot_means, column=True)
print("\nStandard Error for Comparing Two Split-Plot Means: %f \n(df=%f)" %
      (std_err[2], std_err[12]))
equal_means = multipleComparisons(split_plot_means,
                                  int(std_err[12]), std_err[2] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_print_lsd(n_split, equal_means, split_plot_means)

# Print the Strip-plot A by Split-plot means
print("\n\n*************************************************************")
writeMatrix("Strip-plot A by Split-plot Means",
            strip_a_split_plot_means)
print("\nStandard Error for Comparing Two Means: %f \n(df=%f)" %
      (std_err[3], std_err[13]))
saspm_array = array(strip_a_split_plot_means)
equal_means = multipleComparisons(saspm_array.flat,
                                  int(std_err[13]), std_err[3] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_printLSD2Table(n_strip_a, n_split, equal_means,
                 saspm_array.flat)

# print the Strip-plot A by Strip-plot B Means
print("\n\n*************************************************************")
writeMatrix("Strip-plot A by Strip-plot B Means",
            strip_plot_ab_means)
print("\nStandard Error for Comparing Two Means: %f \n(df=%f)" %
      (std_err[4], std_err[14]))
spabm_array = array(strip_plot_ab_means)
equal_means = multipleComparisons(spabm_array.flat,
                                  int(std_err[14]), std_err[4] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_printLSD2Table(n_strip_a, n_strip_b, equal_means,
                 spabm_array.flat)

# Print the Strip-Plot B by Split-Plot means
print("\n\n*************************************************************")
writeMatrix("Strip-plot B by Split-plot Means",
            strip_plot_ab_means)
print("\nStandard Error for Comparing Two Means: %f \n(df=%f)" %
      (std_err[5], std_err[15]))
sbspm_array = array(strip_b_split_plot_means)
equal_means = multipleComparisons(sbspm_array.flat,
                                  int(std_err[15]), std_err[5] / sqrt(2.0),
                                  lsd=True, alpha=alpha)
l_printLSD2Table(n_strip_a, n_split, equal_means,
                 sbspm_array.flat)

Output

Grand mean:  33.87083333333333

Coefficient of Variation: 
  Strip-Plot A:       13.611592209229595
  Strip-Plot B:       17.898639589608628
  Split-Plot:         5.885378367597369

Treatment Means
treatment[1][1][1]     23.8333
treatment[1][1][2]     30.7667
treatment[1][2][1]     28.1000
treatment[1][2][2]     28.8667
treatment[2][1][1]     34.2000
treatment[2][1][2]     43.3000
treatment[2][2][1]     38.9000
treatment[2][2][2]     43.0000

Standard Error for Comparing Two Treatment Means: 1.302028 
(df=4.000000)
[A][B][Split] 	  Mean 		  LSD Grouping
[1][1]  [1] 	23.833333
[1][2]  [1] 	28.100000	*
[1][2]  [2] 	28.866667	*
[1][1]  [2] 	30.766667	*	*
[2][1]  [1] 	34.200000		*
[2][2]  [1] 	38.900000		
[2][2]  [2] 	43.000000			*
[2][1]  [2] 	43.300000			*


*************************************************************

Standard Error for Comparing Two Strip-Plot A Means: 1.882171 
(df=2.000000)
[group] 	  Mean 		LSD Grouping
  [1] 		27.891667
  [2] 		39.850000


*************************************************************

Standard Error for Comparing Two Strip-Plot B Means: 2.474972 
(df=2.000000)
[group] 	  Mean 		LSD Grouping
  [1] 		33.025000	  *
  [2] 		34.716667	  *


*************************************************************

Standard Error for Comparing Two Split-Plot Means: 0.813813 
(df=4.000000)
[group] 	  Mean 		LSD Grouping
  [1] 		31.258333
  [2] 		36.483333


*************************************************************

Standard Error for Comparing Two Means: 1.150905 
(df=4.000000)
[A][B] 	Mean 		LSD Grouping
[1][1] 	25.966667
[1][2] 	29.816667
[2][1] 	36.550000
[2][2] 	43.150000


*************************************************************

Standard Error for Comparing Two Means: 2.074280 
(df=2.000000)
[A][B] 	Mean 		LSD Grouping
[1][1] 	27.300000	*
[1][2] 	28.483333	*
[2][1] 	38.750000		*
[2][2] 	40.950000		*


*************************************************************

Standard Error for Comparing Two Means: 0.920673 
(df=4.000000)
[A][B] 	Mean 		LSD Grouping
[1][1] 	29.016667
[2][2] 	33.500000	*
[2][1] 	35.933333	*	*
[1][2] 	37.033333		*
 
                       *** ANALYSIS OF VARIANCE TABLE ***
                                                      Mean                  
                                ID   DF       SSQ  squares        F  p-value
Location ....................   -1  ...  ........  .......  .......  .......
Blocks ......................   -2    2   1310.28   655.14    14.53    0.061
Strip-Plot A ................   -3    1    858.01   858.01    40.37    0.024
Location x A ................   -4  ...  ........  .......  .......  .......
Strip-Plot A Error ..........   -5    2     42.51    21.26     1.48    0.385
Split-Plot ..................   -6    1    163.80   163.80    41.22    0.003
Split-Plot x A ..............   -7    1     11.34    11.34     2.85    0.166
Location x Split-Plot .......   -8  ...  ........  .......  .......  .......
Split-Plot Error ............   -9    4     15.90     3.97     1.56    0.338
Location x Split-Plot x A ...  -10  ...  ........  .......  .......  .......
Strip-Plot B ................  -11    1     17.17    17.17     0.47    0.565
Location x B ................  -12  ...  ........  .......  .......  .......
Strip-Plot B Error ..........  -13    2     73.51    36.75     2.85    0.260
A x B .......................  -14    1      1.55     1.55     0.12    0.762
Location x A x B ............  -15  ...  ........  .......  .......  .......
A x B Error .................  -16    2     25.82    12.91     5.08    0.080
Split-Plot x B ..............  -17    1     46.76    46.76    18.39    0.013
Split-Plot x A x B ..........  -18    1      0.51     0.51     0.20    0.677
Location x Split-Plot x B ...  -19  ...  ........  .......  .......  .......
Location x Split-Plot x A x B  -20  ...  ........  .......  .......  .......
Split-Plot x A x B Error ....  -21    4     10.17     2.54  .......  .......
Corrected Total .............  -22   23   2577.33  .......  .......  .......
 
Strip-plot A Means
  1        27.89
  2        39.85
 
Strip-plot B Means
  1        33.02
  2        34.72
 
Split-plot Means
 1        31.26
 2        36.48
 
Strip-plot A by Split-plot Means
                1            2
   1        25.97        29.82
   2        36.55        43.15
 
Strip-plot A by Strip-plot B Means
                 1            2
    1        27.30        28.48
    2        38.75        40.95
 
Strip-plot B by Split-plot Means
                1            2
   1        27.30        28.48
   2        38.75        40.95