stripPlot¶

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

Synopsis¶

stripPlot(nLocations, nStripA, nStripB, block, stripA, stripB, 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 stripPlot. See optional argument locations.
int nStripA (Input): Number of levels associated with the strip factor A. nStripA must be greater than one.
int nStripB (Input): Number of levels associated with the strip factor B. nStripB 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 factor A strip-plot identifiers for each observation in y. Each level of this factor must be assigned a different integer. This function verifies that the number of unique factor A strip-plot identifiers is equal to nStripA.
int stripB[] (Input): An array of length n containing the factor B strip-plot identifiers for each observation in y. Each level of this factor must be assigned a different integer. This function verifies that the number of unique factor B strip-plot identifiers is equal to nStripB.
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, and strip-plot B for each observation in y are identified by the corresponding values in the arguments locations, stripA, and stripB.

Return Value¶

A two dimensional, 12 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, \(\texttt{anovaTable}_{i,0} = \texttt{anovaTable}[\texttt{i}] \times 6\), 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`	\(\texttt{anovaTable}_{i,j} = \texttt{anovaTable}[\texttt{i}*6+\texttt{j}]\)
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	`BLOCK WITHIN LOCATION`
-3	`STRIP-PLOT A`
-4	`LOCATION × STRIP-PLOT A†`
-5	`STRIP-PLOT A ERROR`
-6	`STRIP-PLOT B`
-7	`LOCATION × STRIP-PLOT B†`
-8	`STRIP-PLOT B ERROR`
-9	`STRIP-PLOT A × STRIP-PLOT B`
-10	`LOCATION × STRIP-PLOT A × STRIP-PLOT B †`
-11	`STRIP-PLOT A × STRIP-PLOT B ERROR`
-12	`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 whole-plot, split-plot and sub-plot coefficients of variation. cv[0] contains the whole-plot C.V., cv[1] contains the split-plot C.V., and cv[2] contains the sub-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 factor B strip-plot means.
treatmentMeans (Output): An array of length (nStripA×nStripB) containing the treatment means. For \(i>0\) and \(j>0\), \(\text{treatmentMeans}_{i,j}\) = treatmentMeans [(i‑1)×nStripA +(j‑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 and the j‑th level of the factor B strip-plot.
stdErrors (Output): An array of length 10 containing five standard errors and their associated degrees of freedom. The standard errors are in the first five elements and their associated degrees of freedom are reported in stdErrors[5] through stdErrors[9].

Element	Standard Error for Comparisons Between Two	Degrees of Freedom
`stdErrors[0]`	Factor A Strip-Plot Means	`stdErrors[5]`
`stdErrors[1]`	Factor B Strip-Plot Means	`stdErrors[6]`
`stdErrors[2]`	Factor A Strip-Plot Means at the same level of Factor B	`stdErrors[7]`
`stdErrors[3]`	Factor B Strip-Plot Means at the same level of Factor A	`stdErrors[8]`
`stdErrors[4]`	Treatment Means (same strip-plot A and strip-plot B)	`stdErrors[9]`

nBlocks (Output): An array of length nLocations containing the number of blocks, or replicates, at each location.
locationAnovaTable (Output): A 3-dimensional array of size nLocations by 12 by 6 containing the Anova tables associated with each location. For each location, the 12 by 6 dimensional array corresponds to the Anova table for that location. For example, locationAnovaTable[(i-1)×72+(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 stripPlot is capable of analyzing a wide variety of strip-plot experiments.

The essential distinction between strip-plot and split-plot experiments is the application of factor B. In a split-plot experiment, levels of Factor B are nested within Factor A, see Table 4.28 below. In strip-plot experiments, Factors A and B are completely crossed, see Table 4.27 below. This occurs, for example, when an agricultural field is used as a block and the levels of factor A are applied in vertical strips across the entire field. Levels of factor B are assigned to horizontal strips across the same block.

Table 4.27 — Strip-Plot Experiments – Strip-Plots Completely Crossed¶
		Strip Plot Factor A
		A2	A1	A4	A3
Strip Plot Factor B	B3	A2B3	A1B3	A4B3	A3B3
	B1	A2B1	A1B1	A4B1	A3B1
	B2	A2B2	A1B2	A4B2	A3B2

Table 4.28 — Split-Plot Experiments – Split-Plot B Nested within Strip-Plot A¶
Whole Factor Plot
A2	A1	A4	A3
A2B1	A1B3	A4B1	A3B2
A2B3	A1B1	A4B3	A3B1
A2B2	A1B2	A4B2	A3B2

In some studies, a strip-plot experiment is replicated at several locations. stripPlot can analyze strip-plot experiments replicated at multiple locations, even when the number of blocks or replicates at each location are different. If only a single replicate or block is used at each location, then location should be treated as a blocking factor, with nLocations set equal to one. If nLocations=1, it is assumed that the experiment was conducted at a single location with more than one block or replicate at that location. In this case, the four 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 factor A and B 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.

Locations are assumed to be random effects, then tests involving factor A strip-plots use the interaction between factor A strip-plots and locations as the error term for testing whether there are statistically significant differences among the levels of factor A. However, this assumes that the interaction of factor A and locations is not statistically significant. A test of this assumption is included in the ANOVA table. If the interaction between factor A strip-plots and locations is statistically significant, then the nature of that interaction should be explored since it impacts the interpretation of the significance of the factor A.

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

Example¶

This example uses data from a strip-plot design with two levels for the first strip and four for the last strip.

from __future__ import print_function
import sys
from numpy import *
from pyimsl.stat.page import page, SET_PAGE_WIDTH
from pyimsl.stat.multipleComparisons import multipleComparisons
from pyimsl.stat.stripPlot import stripPlot
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 = 4        # Number of factor B strip-plots
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, 2, 3, 4, 1, 2, 3, 4,
           1, 2, 3, 4, 1, 2, 3, 4,
           1, 2, 3, 4, 1, 2, 3, 4]
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]
grand_mean = []
anova_row_labels = []
cv = []
treatment_means = []
strip_plot_a_means = []
strip_plot_b_means = []
std_err = []
n_missing = []
equal_means = []

aov = stripPlot(n_locations, n_strip_a, n_strip_b,
                block, strip_a, strip_b, y,
                grandMean=grand_mean, cv=cv,
                nMissing=n_missing,
                stripPlotAMeans=strip_plot_a_means,
                stripPlotBMeans=strip_plot_b_means,
                treatmentMeans=treatment_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("\nGrand mean: ", grand_mean[0])

# Print treatment means
writeMatrix("Treatment Means", treatment_means)
print("\n\nStandard Error for Comparing Two Treatment Means:")
print("  Same Level of Factor B          %f (df=%f)" %
      (std_err[2], std_err[7]))
print("  Same Level of Factor A          %f (df=%f)" %
      (std_err[3], std_err[8]))
print("  Different Factor A and B Levels %f (df=%f)\n" %
      (std_err[4], std_err[9]))

# Print factor A means
writeMatrix("Factor A Means", strip_plot_a_means, column=True)
print("\nStandard Error for Comparing Two Factor A Means: \n  %f (df=%f)" %
      (std_err[0], std_err[5]))
equal_means = multipleComparisons(strip_plot_a_means,
                                  int(std_err[5]), std_err[0] / sqrt(2),
                                  lsd=True, alpha=.05)

# Print multiple comparison results
writeMatrix("LSD Comparison : Size of Groups of Means",
            equal_means)

# Print factor B means
writeMatrix("Factor B Means", strip_plot_b_means, column=True)
print("\nStandard Error for Comparing Two Factor B Means: \n  %f (df=%f)" %
      (std_err[1], std_err[6]))
equal_means = multipleComparisons(strip_plot_b_means,
                                  int(std_err[6]), std_err[1] / sqrt(2),
                                  lsd=True, alpha=.05)

# Multiple comparison results
writeMatrix("LSD Comparison : Size of Groups of Means",
            equal_means)

Output¶

Grand mean:  33.87083333333333


Standard Error for Comparing Two Treatment Means:
  Same Level of Factor B          2.417643 (df=4.772558)
  Same Level of Factor A          2.639322 (df=9.140634)
  Different Factor A and B Levels 3.121075 (df=8.405351)


Standard Error for Comparing Two Factor A Means: 
  1.882171 (df=2.000000)

Standard Error for Comparing Two Factor B Means: 
  2.330465 (df=6.000000)
 
                         *** ANALYSIS OF VARIANCE TABLE ***
                                                          Mean                  
                                    ID   DF       SSQ  squares        F  p-value
Location                            -1  ...  ........  .......  .......  .......
Block Within Location               -2    2   1310.28   655.14    19.89    0.009
Strip-Plot A                        -3    1    858.01   858.01    40.37    0.024
Location x Strip-Plot A             -4  ...  ........  .......  .......  .......
Strip-Plot A Error                  -5    2     42.51    21.26     4.62    0.061
Strip-Plot B                        -6    3    227.73    75.91     4.66    0.052
Location x Strip-Plot B             -7  ...  ........  .......  .......  .......
Strip-Plot B Error                  -8    6     97.76    16.29     3.54    0.075
Strip-Plot A x Strip-Plot B         -9    3     13.40     4.47     0.97    0.466
Location x Strip-Plot A            -10  ...  ........  .......  .......  .......
   x Strip-Plot B                                                               
Strip-Plot A x Strip-Plot B Error  -11    6     27.63     4.60  .......  .......
Corrected Total                    -12   23   2577.33  .......  .......  .......
 
                   Treatment Means
             1            2            3            4
1        23.83        30.77        28.10        28.87
2        34.20        43.30        38.90        43.00
 
Factor A Means
1        27.89
2        39.85
 
LSD Comparison : Size of Groups of Means
                         0
 
Factor B Means
1        29.02
2        37.03
3        33.50
4        35.93
 
LSD Comparison : Size of Groups of Means
            1            2            3
            2            3            0