LATTICE_DESIGN Function
Analyzes balanced and partially-balanced lattice experiments. In these experiments, a requirement is that the number of treatments be equal to the square of an integer, such as 9, 16, or 25 treatments. Function lattice also analyzes repetitions of lattice experiments.
Usage
result = LATTICE_DESIGN (n, n_locations, n_reps, n_blocks, n_treatments, rep, block, treatment, y)
Input Parameters
n—Number of missing and non-missing experimental observations. LATTICE_DESIGN verifies that:
n = n_locations * t * r
where:
*t = n_treatments
*r = n_reps
n_locations—Number of locations or repetitions of the lattice experiments. n_locations must be one or greater. If n_locations > 1, then the Locations keyword must be included as input to LATTICE_DESIGN.
n_reps—Number of replicates per location. Each replicate should consist of t = n_treatments organized into blocks.
n_blocks—Number of blocks per location. For every location, n_blocks must be equal to n_blocks= r * k, where r = n_reps and .
n_treatments—Number of treatments t = n_treatments must be equal to k2.
rep—Array of length n containing the replicate identifiers for each observation in y. For a balanced lattice, the number of replicate identifiers must be equal to n_reps = (k+1). For a partially balanced lattice, the number of replicate identifiers depends upon whether the design is a simple lattice, triple lattice, etc. LATTICE_DESIGN verifies that the number of unique replicate identifiers is equal to n_reps. If multiple locations or repetitions of the experiment are conducted, i.e., n_locations > 1, then the replicate and block numbers contained in rep and block must agree between repetitions.
block—Array of length n containing the block identifiers for each observation in y. LATTICE_DESIGN verifies that the number of unique block identifiers is equal to n_blocks. If multiple locations or repetitions of the experiment are conducted, i.e., n_locations > 1, then block numbers must agree between repetitions. That is, the ith block in every location or repetition must contain the same treatments.
treatment—Array of length n containing the treatment identifiers for each observation in y. Each treatment must be assigned values from 1 to n_treatments. LATTICE_DESIGN verifies that the number of unique treatment identifiers is equal to n_treatments.
y—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 replicate, block, treatment, and location number for each observation in y are identified by the corrresponding values in the input parameters rep, block, treatment, and the keyword Locations.
Returned Value
result—A two dimensional, 7 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, anova_tablei,0 = anova_table(i,0), 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 convention found in Table 5-39: ANOVA Table Values.
 
ANOVA Table Values
J
anova_tablei,j = anova_table(i,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 anova_tablei,j are the only negative values in anova_table. Assignments of identifiers to ANOVA sources use the coding shown in Table 5-40: Source Identifiers.
 
Source Identifiers
Source Identifier
ANOVA Source
–1
LOCATIONS*
–2
REPLICATES
–3
TREATMENTS(unadjusted)
–4
TREATMENTS(adjusted)
–5
BLOCKS(adjusted)
–6
INTRA-BLOCK ERROR
–7
CORRECTED TOTAL
* If N_locations = 1 rows involving location are set to missing (NaN).
Input Keywords
Double—If present and nonzero, double precision is used.
Locations—Array of length n containing the location or repetition identifiers for each observation in y. Unique integers must be assigned to each location in the study. This keyword is required when n_locations > 1.
Output Keywords
N_missing—Number of missing values, if any, found in y. Missing values are denoted with a NaN (Not a Number) value.
Cv—The coefficient of variation computed by using the location standard deviation.
Grand_mean—The overall adjusted mean averaged over every location.
Treatment_means—Array of size n_treatments containing the adjusted treatment means.
Std_errors—Array of length 4 containing the standard error and associated degrees of freedom for comparing two treatment means. Std_errors(0) contains the standard error for comparing two treatments that appear in the same block at least once. Std_errors(1) contains the standard error for comparing two treatments that never appear in the same block together. Std_errors(2) contains the standard error for comparing, on average, two treatments from the experiment averaged over cases in which the treatments do or do not appear in the same block. Finally, Std_errors(3) contains the degrees of freedom associated with each of these standard errors, i.e., Std_errors(3)= degrees of freedom for intra-block error.
Location_anova_table—Array of size n_locations by 7 by 6 containing the ANOVA tables associated with each location or repetition of the lattice experiment. For each location, the 7 by 6 dimensional array corresponds to the ANOVA table for that location. For example, Location_anova_table(i,j,k) contains the value in the kth column and jth row of the ANOVA table for the ith location.
Anova_row_labels—Labels for each of the rows of the returned ANOVA table. The label for the ith row of the ANOVA table can be printed with PRINT, Anova_row_labels(i).
Discussion
LATTICE_DESIGN analyzes both balanced and partially-balanced lattice experiments, possibly repeated at multiple locations. These designs were originally described by Yates (1936). A defining characteristic of these classes of lattice experiments is that the number of treatments is always the square of an integer, such as t=9, 16, 25, etc. where t is equal to the number of treatments.
Another characteristic of lattice experiments is that blocks are organized into replicates, where each replicate contains one observation for each treatment. This requires the number of blocks in each replicate to be equal to the number of observations per block. That is, the number of blocks per replicate and the number of observations per block are both equal to .
For balanced lattice experiments the number of replicates is always k+1. For partially-balanced lattice experiments, the number of replicates is less than k+1. Tables of balanced-lattice experiments are tabulated in Cochran & Cox (1950) for t=9, 16, 25, 49, 64, and 81.
The analysis of balanced and partially-balanced experiments is detailed in Cochran & Cox (1950) and Kuehl (2000).
Consider, for example, a 3×3 balanced-lattice, i.e., k=3 and t=9. Notice that the number of replicates is 4 and the number of blocks per replicate is equal to 3. The total number of blocks is equal to:
 
For a balanced-lattice:
 
A 3x3 Balanced-Lattice for 9 Treatments in Four Replicates
Replicate I
Replicate II
Block 1 (T1, T2, T3)
Block 4 (T1, T4, T7)
Block 2 (T4, T5, T6)
Block 5 (T2, T5, T8)
Block 3 (T7, T8, T9)
Block 6 (T3, T6, T9)
Replicate III
Replicate IV
Block 7 (T1, T5, T9)
Block 10 (T1, T6, T8)
Block 8 (T2, T6, T7)
Block 11 (T2, T4, T9)
Block 9 (T3, T4, T8)
Block 12 (T3, T5, T7)
The analysis of variance for data from a balanced-lattice experiment, takes the form familiar to other balanced incomplete block experiments. In these experiments, the error term is divided into two components: the Inter-Block Error and the Intra-Block Error. For single and multiple locations, the general format of the ANOVA tables is illustrated in the Table 5-42: ANOVA Table for a Lattice Experiment at one Location and Table 5-43: ANOVA Table for a Lattice Experiment at Multiple Locations.
 
ANOVA Table for a Lattice Experiment at one Location
SOURCE
DF
Sum of Squares
Mean Squares
REPLICATES
r – 1
SSR
MSR
TREATMENTS(unadj)
t – 1
SST
MST
TREATMENTS(adj)
t – 1
SSTa
MSTa
BLOCKS(adj)
SSBa
MSBa
INTRA-BLOCK ERROR
SSI
MSI
TOTAL
SSTot
 
ANOVA Table for a Lattice Experiment at Multiple Locations
SOURCE
DF
Sum of Squares
Mean Squares
LOCATIONS
p – 1
SSL
MSL
REPLICATES WITHIN LOCATIONS
p(r – 1)
SSR
MSR
TREATMENTS(unadj)
t – 1
SST
MST
TREATMENTS(adj)
t – 1
SSTa
MSTa
BLOCKS(adj)
SSB
MSB
INTRA-BLOCK ERROR
SSI
MSI
TOTAL
SSTot
Example 1
This example is a lattice design for 16 treatments conducted at one location. A lattice design with t = k2 = 16 treatments is a balanced lattice design with r = k+1 = 5 replicates and r·k = 5(4) = 20 blocks.
; Total number of observations
n = 80
; Number of locations
n_locations = 1
; Number of treatments
n_treatments = 16
; Number of replicates
n_reps = 5
; Total number of blocks
n_blocks = 20
 
rep = [ $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, $
      4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, $
      5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ]
 
block = [ $
      1,  1,  1,  1,  2,  2,  2,  2,  3,  3,  3,  3, $
      4,  4,  4,  4,  5,  5,  5,  5,  6,  6,  6,  6, $
      7,  7,  7,  7,  8,  8,  8,  8,  9,  9,  9,  9, $
     10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, $
     13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, $
     16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, $
     19, 19, 19, 19, 20, 20, 20, 20 ]
 
treatments = [ $
      1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, $
     14, 15, 16,  1,  5,  9, 13, 10,  2, 14,  6,  7, 15, $
      3, 11, 16,  8, 12,  4,  1,  6, 11, 16,  5,  2, 15, $
     12,  9, 14,  3,  8, 13, 10,  7,  4,  1, 14,  7, 12, $
     13,  2, 11,  8,  5, 10,  3, 16,  9,  6, 15,  4,  1, $
     10, 15,  8,  9,  2,  7, 16, 13,  6,  3, 12,  5, 14, $
     11,  4 ]
 
y = [ $
     147, 152, 167, 150, 127, 155, 162, 172, $
     147, 100, 192, 177, 155, 195, 192, 205, $
     140, 165, 182, 152,  97, 155, 192, 142, $
     155, 182, 192, 192, 182, 207, 232, 162, $
     155, 132, 177, 152, 182, 130, 177, 165, $
     137, 185, 152, 152, 185, 122, 182, 192, $
     220, 202, 175, 205, 205, 152, 180, 187, $
     165, 150, 200, 160, 155, 177, 185, 172, $
     147, 112, 177, 147, 180, 205, 190, 167, $
     172, 212, 197, 192, 177, 220, 205, 225 ]
 
aov = LATTICE_DESIGN(n, n_locations, n_reps, n_blocks, $
                     n_treatments, rep, block, treatments, $
                     y, Grand_mean=grand_mean, Cv=cv, $
                     Treatment_means=treatment_means, $
                     Std_err=std_err)
 
labels = ['Locations        ', $
          'Replicates       ', $
          'Treatments       ', $
          '  (unadjusted)'   , $
          'Treatments       ', $
          '  (adjusted)'     , $
          'Blocks (adjusted)', $
          'Intra-Block Error', $
          'Corrected Total  ']
; Print Analysis of Variance Table
PRINT, "             *** ANALYSIS OF VARIANCE TABLE ***"
PRINT, 'ID', 'DF', 'SSQ', 'MS', 'F-test', 'P-Value', $
  Format='(A21, A7, A8, A9, A8, A8)' & $
idx = 0
FOR i=0L, (SIZE(aov))(1)-1 DO BEGIN & $
   PRINT, labels(idx), aov(i,0), aov(i,1), $
     aov(i,2), aov(i,3), aov(i,4), aov(i,5), Format= $
     '(A17, 1X, I3, 2X, F3.0, 2X, F8.2, 2X, F7.2, 2X, ' + $
     'F5.2, 2X, F7.3)' & $
   idx = idx + 1 & $
   IF idx LT N_ELEMENTS(labels)-1 THEN $
      WHILE STRPOS(labels(idx), ' ', 0) EQ 0 DO BEGIN & $
         PRINT, labels(idx) & idx = idx + 1 & $
      ENDWHILE & $
ENDFOR
PRINT, ''
 
PRINT, grand_mean, $
  Format='("Adjusted Grand Mean     :", F8.3)'
PRINT, cv, Format='("Coefficient of Variation:", F8.3)'
PRINT, ''
 
PRINT, "Adjusted Treatment Means:"
FOR i=0L, n_treatments-1 DO $
   PRINT, (i+1), treatment_means(i), Format= $
     "(2X, 'Treatment[', I2, '] Mean:', F10.4)"
PRINT, ''
 
PRINT, std_err(0), $
  Format='("Standard Error for Comparing Two Treatment ' + $
'Means: ", F9.6, I1)'
PRINT, FIX(std_err(3)), Format='("(df=", I2, ")")'
PRINT, ''
 
; Perform multiple comparison using the LSD procedure
equal_means = MULTICOMP(treatment_means, $
                std_err(3), std_err(2)/SQRT(2.0), $
                /LSD, Alpha=0.05)
PM, equal_means, $
  Title="LSD Comparison: Size of Groups of Means"
Output
              *** ANALYSIS OF VARIANCE TABLE ***
                   ID     DF     SSQ       MS  F-test P-Value
Locations          -1  NaN       NaN      NaN    NaN      NaN
Replicates         -2   4.   6524.50  1631.12    NaN      NaN
Treatments         -3  15.  27297.00  1819.80   4.12    0.000
  (unadjusted)
Treatments         -4  15.  21271.20  1418.08   4.21    0.000
  (adjusted)
Blocks (adjusted)  -5  15.  11339.28   755.95    NaN      NaN
Intra-Block Error  -6  45.  15173.23   337.18    NaN      NaN
Corrected Total    -7  79.  60334.00      NaN    NaN      NaN
 
Adjusted Grand Mean     : 171.450
Coefficient of Variation:  10.710
 
Adjusted Treatment Means:
  Treatment[ 1] Mean:  166.4533
  Treatment[ 2] Mean:  160.7527
  Treatment[ 3] Mean:  183.6289
  Treatment[ 4] Mean:  175.6298
  Treatment[ 5] Mean:  162.6807
  Treatment[ 6] Mean:  167.6716
  Treatment[ 7] Mean:  168.3822
  Treatment[ 8] Mean:  176.5731
  Treatment[ 9] Mean:  162.6928
  Treatment[10] Mean:  118.5197
  Treatment[11] Mean:  189.0615
  Treatment[12] Mean:  190.4608
  Treatment[13] Mean:  169.4514
  Treatment[14] Mean:  197.0827
  Treatment[15] Mean:  185.3560
  Treatment[16] Mean:  168.8029
 
Standard Error for Comparing Two Treatment Means: 13.221848
(df=45)
 
LSD Comparison: Size of Groups of Means
           0
          12
          12
           0
          11
           0
           0
           0
           0
           7
           0
           0
           0
           0
           0
Example 2
This example consists of a 5 × 5 partially-balanced lattice repeated twice. In this case, the number of replicates is not k+1 = 6, it is only n_reps = 2. Each lattice consists of total of 50 observations which is repeated twice. The first observation in this experiment is missing.
a = MACHINE(/Float)
NaN = a.NAN
 
; Total number of observations
n = 100
; Number of locations
n_locations = 2
; Number of treatments
n_treatments = 25
; Number of replicates
n_reps = 2
; Total number of blocks
n_blocks = 10
 
rep = [ $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
 
block = [ $
      1,  1,  1,  1,  1,  2,  2,  2,  2,  2, $
      3,  3,  3,  3,  3,  4,  4,  4,  4,  4, $
      5,  5,  5,  5,  5,  6,  6,  6,  6,  6, $
      7,  7,  7,  7,  7,  8,  8,  8,  8,  8, $
      9,  9,  9,  9,  9, 10, 10, 10, 10, 10, $
      1,  1,  1,  1,  1,  2,  2,  2,  2,  2, $
      3,  3,  3,  3,  3,  4,  4,  4,  4,  4, $
      5,  5,  5,  5,  5,  6,  6,  6,  6,  6, $
      7,  7,  7,  7,  7,  8,  8,  8,  8,  8, $
      9,  9,  9,  9,  9, 10, 10, 10, 10, 10 ]
 
treatment = [ $
      1,  2,  3,  4,  5, $
      6,  7,  8,  9, 10, $
     11, 12, 13, 14, 15, $
     16, 17, 18, 19, 20, $
     21, 22, 23, 24, 25, $
      1,  6, 11, 16, 21, $
      2,  7, 12, 17, 22, $
      3,  8, 13, 18, 23, $
      4,  9, 14, 19, 24, $
      5, 10, 15, 20, 25, $
      1,  2,  3,  4,  5, $
      6,  7,  8,  9, 10, $
     11, 12, 13, 14, 15, $
     16, 17, 18, 19, 20, $
     21, 22, 23, 24, 25, $
      1,  6, 11, 16, 21, $
      2,  7, 12, 17, 22, $
      3,  8, 13, 18, 23, $
      4,  9, 14, 19, 24, $
      5, 10, 15, 20, 25 ]
 
locations = [ $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      1, 1, 1, 1, 1, 1, 1, 1, 1, 1, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2, $
      2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
 
y = [$
    NaN,  7,  5,  8,  6, $
     16, 12, 12, 13,  8, $
     17,  7,  7,  9, 14, $
     18, 16, 13, 13, 14, $
     14, 15, 11, 14, 14, $
     24, 13, 24, 11,  8, $
     21, 11, 14, 11, 23, $
     16,  4, 12, 12, 12, $
     17, 10, 30,  9, 23, $
     15, 15, 22, 16, 19, $
     13, 26,  9, 13, 11, $
     15, 18, 22, 11, 15, $
     19, 10, 10, 10, 16, $
     21, 16, 17,  4, 17, $
     15, 12, 13, 20,  8, $
     16,  7, 20, 13, 21, $
     15, 10, 11,  7, 14, $
      7, 11, 15, 15, 16, $
     19, 14, 20,  6, 16, $
     17, 18, 20, 15, 14 ]
 
aov = LATTICE_DESIGN(n, n_locations, n_reps, n_blocks, $
                     n_treatments, rep, block, treatment, $
                     y, Locations=locations, $
                     Cv=cv, $
                     Grand_mean=grand_mean, $
                     Location_anova_table= $
                     location_anova_table, $
                     Treatment_means=treatment_means, $
                     Anova_row_labels=anova_row_labels, $
                     Std_err=std_err, $
                     N_missing=n_missing)
 
labels = ['Locations        ', $
          'Replicates within', $
          '  Locations'      , $
          'Treatments       ', $
          '  (unadjusted)'   , $
          'Treatments       ', $
          '  (adjusted)'     , $
          'Blocks (adjusted)', $
          'Intra-Block Error', $
          'Corrected Total  ']
 
; Print Analysis of Variance Table
PRINT, "             *** ANALYSIS OF VARIANCE TABLE ***"
PRINT, 'ID', 'DF', 'SSQ', 'MS', 'F-test', 'P-Value', $
   Format='(A21, A7, A8, A9, A8, A8)' & $
   idx = 0
FOR i=0L, (SIZE(aov))(1)-1 DO BEGIN & $
   PRINT, labels(idx), aov(i,0), aov(i,1), $
     aov(i,2), aov(i,3), aov(i,4), aov(i,5), Format= $
     '(A17, 1X, I3, 2X, F3.0, 2X, F8.2, 2X, F7.2, 2X, ' + $
     'F5.2, 2X, F7.3)' & $
   idx = idx + 1 & $
   IF idx LT N_ELEMENTS(labels)-1 THEN $
      WHILE STRPOS(labels(idx), ' ', 0) EQ 0 DO BEGIN & $
         PRINT, labels(idx) & idx = idx + 1 & $
      ENDWHILE & $
ENDFOR
PRINT, ''
 
; Print the location ANOVA tables
idx = 0
FOR j=0L, n_locations-1 DO BEGIN & $
   PRINT, "LOCATION", j, Format='(A33, 1X, I1)' & $
   PRINT, "*** ANALYSIS OF VARIANCE TABLE ***", $
     Format='(A47)' & $
   PRINT, 'ID', 'DF', 'SSQ', 'MS', 'F-test', 'P-Value', $
     Format='(A23, A5, A9, A8, A8, A8)' & $
   FOR i=0L, (SIZE(aov))(1)-1 DO BEGIN & $
      PRINT, labels(idx), location_anova_table(j,i,0), $
        location_anova_table(j,i,1), $
        location_anova_table(j,i,2), $
        location_anova_table(j,i,3), $
        location_anova_table(j,i,4), $
        location_anova_table(j,i,5), Format= $
        '(A17, 2X, I3, 3X, F3.0, 2X, F7.2, 2X, F6.2, ' + $
        '2X, F5.2, 2X, F7.3)' & $
      idx = idx + 1 & $
      IF idx LT N_ELEMENTS(labels)-1 THEN $
         WHILE STRPOS(labels(idx), ' ', 0) EQ 0 DO BEGIN & $
            PRINT, labels(idx) & idx = idx + 1 & $
         ENDWHILE & $
      ENDFOR & $
   idx = 0 & $
   PRINT, '' & $
ENDFOR
PRINT, ''
 
PRINT, grand_mean, $
   Format='("Adjusted Grand Mean     :", F8.3)'
PRINT, cv, Format='("Coefficient of Variation:", F8.3)'
PRINT, ''
 
PRINT, "Adjusted Treatment Means:"
FOR i=0L, n_treatments-1 DO $
   PRINT, (i+1), treatment_means(i), Format= $
     "(2X, 'Treatment[', I2, '] Mean:', F10.4)"
PRINT, ''
 
PRINT, std_err(2), $
   Format='("Standard Error for Comparing Two Treatment ' + $
   'Means: ", F9.6, I1)'
PRINT, FIX(std_err(3)), Format='("(df=", I2, ")")'
PRINT, ''
 
; Perform multiple comparison using the LSD procedure
equal_means = MULTICOMP(treatment_means, std_err(3), $
                        std_err(2)/SQRT(2.0), $
                        /LSD, Alpha=0.05)
PM, equal_means, $
   Title="LSD Comparison: Size of Groups of Means"
PRINT, ''
 
PRINT, n_missing, Format= $ 
   '("Number of missing observations: ", I2)'
Output
             *** ANALYSIS OF VARIANCE TABLE ***
                   ID     DF     SSQ       MS  F-test P-Value
Locations          -1   1.     12.19    12.19   0.25    0.622
Replicates within  -2   2.    203.99   101.99   7.44    0.001
  Locations
Treatments         -3  24.    795.46    33.14   0.02    1.000
  (unadjusted)
Treatments         -4  24.    951.20    39.63   2.89    0.006
  (adjusted)
Blocks (adjusted)  -5  16.    770.50    48.16   3.51    0.000
Intra-Block Error  -6  55.    753.82    13.71    NaN      NaN
Corrected Total    -7  98.   2535.95      NaN    NaN      NaN
 
                         LOCATION 0
             *** ANALYSIS OF VARIANCE TABLE ***
                     ID   DF      SSQ      MS  F-test P-Value
Locations           -1   NaN      NaN     NaN    NaN      NaN
Replicates within   -2    1.   203.67  203.67    NaN      NaN
  Locations
Treatments          -3   24.   567.13   23.63   0.78    0.721
  (unadjusted)
Treatments          -4   24.   661.08   27.54   2.04    0.078
  (adjusted)
Blocks (adjusted)   -5    8.   490.51   61.31    NaN      NaN
Intra-Block Error   -6   15.   202.93   13.53    NaN      NaN
Corrected Total     -7   48.  1464.24     NaN    NaN      NaN
 
                         LOCATION 1
             *** ANALYSIS OF VARIANCE TABLE ***
                     ID   DF      SSQ      MS  F-test P-Value
Locations           -1   NaN      NaN     NaN    NaN      NaN
Replicates within   -2    1.     0.32    0.32    NaN      NaN
  Locations
Treatments          -3   24.   622.52   25.94   1.43    0.196
  (unadjusted)
Treatments          -4   24.   707.51   29.48   2.83    0.018
  (adjusted)
Blocks (adjusted)   -5    8.   269.76   33.72    NaN      NaN
Intra-Block Error   -6   16.   166.92   10.43    NaN      NaN
Corrected Total     -7   49.  1059.52     NaN    NaN      NaN
 
Adjusted Grand Mean     :  14.011
Coefficient of Variation:  26.423
 
 
Adjusted Treatment Means:
  Treatment[ 1] Mean:   17.1507
  Treatment[ 2] Mean:   19.2200
  Treatment[ 3] Mean:   11.1261
  Treatment[ 4] Mean:   14.6230
  Treatment[ 5] Mean:   12.6543
  Treatment[ 6] Mean:   11.8133
  Treatment[ 7] Mean:   11.9045
  Treatment[ 8] Mean:   11.3106
  Treatment[ 9] Mean:    9.5576
  Treatment[10] Mean:   11.5889
  Treatment[11] Mean:   22.1320
  Treatment[12] Mean:   12.7232
  Treatment[13] Mean:   13.1293
  Treatment[14] Mean:   17.8763
  Treatment[15] Mean:   18.6576
  Treatment[16] Mean:   14.6568
  Treatment[17] Mean:   11.4980
  Treatment[18] Mean:   13.1540
  Treatment[19] Mean:    5.4010
  Treatment[20] Mean:   12.9323
  Treatment[21] Mean:   15.4108
  Treatment[22] Mean:   17.0020
  Treatment[23] Mean:   13.9081
  Treatment[24] Mean:   17.6550
  Treatment[25] Mean:   13.1864
 
Standard Error for Comparing Two Treatment Means:  4.617282
(df=55)
 
LSD Comparison: Size of Groups of Means
          16
          22
          22
           0
           0
           0
           0
           0
           0
           0
          15
           0
           0
           0
           0
           0
           0
           0
           0
           0
           0
           0
           0
           0
 
Number of missing observations:  1