^{
S.L. Taylor, M.E. Payton and W.R. Raun*
Contribution from the Okla. Agric. Exp. Sta.,
Department of Plant and Soil Sciences, 044 North Agricultural Hall, and
Department of Statistics, 301 Math Sciences, Oklahoma State University,
Stillwater, OK 74078, *Corresponding author.
ABSTRACT
Large coefficients of variation (>30%) are often
associated with increased experimental variability. The objective of this
study was to examine the relationships between mean square errors, yield
means, coefficients of variation (CV) and plot size using statistical
information compiled from past wheat field research projects. Three
hundred and sixty two wheat field research projects were selected from
over 2000 published wheat experiments that included soil fertility, weed,
tillage, and variety evaluation. Little or no relationship between mean
square error (MSE) and mean yield or plot size was found. However, mean
yields and CVs demonstrated a significant negative correlation. This work
proposes decreased variability among experimental units as defined by the
CV can be accomplished simply by increasing the mean yield, suggesting
that the use of the CV may be improper when comparing variability of
trials from similar experiments. Attempts to compare variation from
similar experiments should be done using the MSE since unit differences
would not be a problem. The CV should only be used when comparing
variation from experiments using different metrics. Plot size in plant
breeding variety trials (average of 3.59m2) was much smaller than that in
fertilizer/weed/tillage trials (average of 37.2m2). The smaller plot size
employed in the variety trials reduced the variability encountered in the
estimation of the mean and is consistent with the resolution where
detectable differences in soil test parameters exist and that should be
treated independently.
INTRODUCTION
Experimental error is defined as a measure of the
variation which exists among observations on experimental units treated
alike (Steel et al. 1997, p. 129). Steel et al., (1997) further noted that
variation comes from two main sources; 1) inherent variability that exists
in the experimental material to which treatments are applied and 2)
variation which results from any lack in uniformity in the physical
conduct of the experiment.
Coefficients of variation were first employed as
relative measures of variation. The CV is defined as the standard
deviation expressed as a percentage of the mean (Tippett, 1952; Senders,
1958; Steel et al. 1997; Lewis, 1963). Mills (1924) indicated that the CV
is affected by the value of the mean, as well as by the size of the
standard deviation. Since the CV is a ratio, Zar (1984, p. 32) claimed
that the CV should only be used for ratiotype data. The CV should not be
used for strictly interval or nominal data since ratios have no explicit
meaning with these data types.
Conceptually, the CV is not defined for means equal to
zero, and the CV is unreliable for means that are close to zero relative
to the standard deviation. This often poses problems for researchers who
wish to use the CV. However, if the data is truly ratiotype data, this
scenario will not occur since small valued means (relative to the standard
deviation) cannot occur.
Moore (1958) found that if it is desired to discover
whether one distribution is relatively more variable than another, it
follows that it is necessary to find some method to eliminate the basic
units. This is achieved by using the CV. Moore (1958) also showed that the
CV does not depend on the units of measurement since both the mean and
standard deviation are linear functions of the units involved.
Ostle (1954) found that the CV is an ideal device for
comparing the variation in two series of data which are measured in two
different units (e.g., a comparison of variation in height with variation
in weight). Lewis (1963) noted that the CV may be used to compare the
dispersion of series measured in different units and that of series with
the same units but running at different levels of magnitude. Similarly,
CVs have been used to evaluate results from different experiments
involving the same units of measure, possibly conducted by different
persons (Steel et al., 1997).
Little and Hills (1978, p.18), stated that the
variability among experimental units of experiments involving different
units of measurements and/or plot sizes can be compared by CVs. Their
extrapolation suggested that a lima bean experiment (s = 5.8
seedlings/plot, yield mean of 82.7 seedlings/plot, CV = 7.0%) had 1.8
times more variability among the plots within a treatment than a sugar
beet root yield experiment (s = 1.18 t/ac, yield mean of 30.5 t/ac, CV =
3.9%). Snedecor and Cochran (1980, p.37) indicated that the CV is often
used to describe the amount of variation in a population.
Gomez and Gomez (1976) stated that the CV is an
indication of the degree of precision to which the treatments are compared
and is a good index of the reliability of the experiment. Gomez and Gomez
(1976, p. 17) further indicated that the higher the CV value, the lower is
the reliability of the experiment.
Work by McClave and Benson (1988) indicated that it is
common for the standard deviation of a random variable to increase
proportionally as the mean increases. Snedecor and Cochran (1980, p. 37)
indicated that the CV is often used to describe the amount of variation in
a population. For data from different populations or sources, the mean and
standard deviation often tend to change together so that the CV is
relatively stable or constant (Snedecor and Cochran, 1980, p. 37). Steel
et al. (1997) stated that the CV is a relative measure of variation, in
contrast to the standard deviation, which is in the same units as the
observation.
Ostle (1954) indicated that experimental error
essentially reflects in each particular instance all the extraneous
sources of variation which, by their occurrence, help to disguise the true
effect of the "treatments" under examination.
The objective of this study was to examine the
relationships between mean square errors, yield means, coefficients of
variation and plot size using statistical information compiled from past
wheat field research projects.
MATERIALS AND METHODS
Data from 362 wheat field experiments were targeted for
additional statistical analysis. From this population of experiments, 220
were fertilizer, weed management and tillage trials and 142 were variety
trials. All experiments were conducted by field researchers, included more
than two replications from past M.S. and Ph.D. thesis and published wheat
field research projects (Agronomy Journal, Crop Science, and Soil Science
Society of America Journal). From each experiment the following
information was obtained: number of replications, number of treatments,
plot size, CV, degrees of freedom in the error term, mean square error
(variance), mean yield, and the standard error of the difference between
two treatment means. If all of the above information was not reported,
back calculation of the missing term was accomplished when possible (e.g.,
yield mean determined from the reported CV and MSE). Data for the
variables mentioned was only collected for wheat grain yield. The type of
experiment was recorded into four separate groups: soil fertility, weed
management, tillage and variety trials. From these groups, soil fertility,
weed management and tillage trials were combined into a separate group
apart from the variety trials due to distinct differences in plot size.
Average plot sizes for the fertility/weed/tillage and variety trials were
37.2±24.3 and 3.59±3.13 m2, respectively.
Where necessary, all experimental results were
converted into metric units. Correlation matrices were established between
all variables collected and simple linear regression equations were
determined for specific relationships.
RESULTS AND DISCUSSION
This work assumes that residual mean square error (MSE)
from analysis of variance is the best estimate of experimental error or
the variability present in a given field experiment. Since all of these
data are from similar experiments, MSEs can be compared in order to
ascertain the relative variability from experiment to experiment.
In general, when the sums of squares for all
independent effects included in an experiment are accounted for in the
model, residual error and experimental error are considered to be
synonymous. Very few of the experiments reported CVs that exceeded 30%;
therefore, grouping experiments with CVs less than 30% was not attempted.
Linear Relationships
No distinct relationship was found between MSE and plot
size for either group (Figures 1 and 2). However, there was a tendency for
MSE to decrease when plot sizes were between 30 and 100 m2 and 6 and 20m2
for the fertility/weed/tillage and variety data, respectively. Previous
work by Barreto and Raun, (1990), which evaluated corn field experiments
conducted in Mexico, demonstrated that increasing plot size decreased mean
square errors.
Mean yield and CV were negatively correlated for both
groups (Figures 1 and 2). Because site mean yields are used as the divisor
in calculating CVs, increasing mean yields were expected to produce
smaller coefficients of variation.
No highly significant linear relationship could be
established between MSE and CV for the fertilizer/weed/tillage trials
(Figure 1). Alternatively, MSE and CV were positively correlated for the
variety trials. Assuming that the mean square error from analysis of
variance is the best measure of variability for field experimentation, and
because CVs are considered to be a relative measure of variation (Steel et
al., 1997), MSE and CV were expected to be highly correlated. Because this
was not the case for the fertilizer/weed/tillage trials, this work
demonstrates that CVs are not measuring what some researchers are
expecting in field trials. However, there was one critical difference
between the fertilizer/weed/tillage trials and the variety trials, and
that was plot size which averaged 37.2 and 3.59 m2, respectively.
How could plot size affect the relationship between MSE
and CV? Work by Solie et al. (1996) and Raun et al. (1998) attempted to
establish the fundamental field element size (area to which an independent
rate of a nutrient should be applied). The fundamental field element size
is essentially the resolution or scale where detectable differences in
soil test parameters exist and that should ultimately be treated
differently. Solie et al. (1996) reported that in order to optimize
nutrient inputs, areas of 1.96m2 should be treated independently, largely
because of significant microvariability in soil test parameters found in
soils (Raun et al., 1998). CVs and MSEs were positively correlated in
variety trials because the plot size was smaller than those in the
fertilizer/weed/tillage trials and nearer the size recommended by Solie et
al. (1996). We believe this positive relationship occurs for two reasons.
First, the small plot size allows for a better estimate of MSE and
minimizes the intraplot variability. Secondly, smaller plot sizes will
reduce the variability encountered in the estimation of the mean, thus
reducing the problems experienced by changes in the mean affecting the CV.
Alternatively, the fertilizer/weed/tillage trials had average plot sizes
10 times greater than the variety trials, which may have led to the
apparent independence of MSE and CV.
Although MSE and CV were positively correlated in the
variety trials, the presence of an equally significant negative
correlation between CV and mean yield (for both groups, Figures 1 and 2)
suggests that CVs were influenced by the environment (mean yield being an
indicator of the environment or environment mean). This trait (increasing
mean yield and decreasing CV), is not desirable when using the CV as a
measure of variability. It is important to remember that the CV is a
measure of relative variability. Even though an increase in mean yield
will usually result in a corresponding increase in MSE, no linear
relationship was found between mean yield and MSE for either of the groups
evaluated (Figures 1 and 2). The lack of a significant linear relationship
between mean yield and MSE clearly shows why weak relationships between CV
and MSE were found since the mean yield entered into the calculation of
CV. These results would incorrectly suggest that experimental variability
could be reduced simply by increasing the mean yield. For similar types of
trials, comparing MSE's (residual error or variance) would be more
appropriate in terms of assessing experimental variability since CVs were
greatly influenced by the value of the mean yield. The CV would be an
appropriate measure for comparing the variation differences in experiments
that have variables measured in different units.
CONCLUSIONS
The CV is useful when comparing the experimental
variation differences in experiments that have variables measured in
different units. A researcher must remember that the CV is measuring
relative variability and that it has an inverse relationship with the
sample mean. If one wishes to compare the experimental variation of trials
containing variables with common units, the MSE would be the appropriate
measure to use. The lack of a strong relationship between MSE and CV
should cause concern for researchers using the CV as a measure of the
'reliability of the experiment' or to compare results from different
experiments involving the same units of measure. The smaller plot size
employed in plant breeding variety trials (average of 3.59m2) when
compared to the fertilizer/weed/tillage trials (average of 37.2m2) reduced
the variability encountered in the estimation of the mean. The smaller
plot size employed in the variety trials is considered to be advantageous
in field plot work since it is consistent with the resolution where
detectable differences in soil test parameters exist and that should be
treated independently.
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FIGURE 1. Relationship between mean yield, mean square error (MSE),
coefficient of variation (CV) and plot size from 220 fertilizer, weed
management and tillage trials.
FIGURE 2. Relationship between mean yield, mean square error (MSE),
coefficient of variation (CV) and plot size from 142 variety trials.
}
TABLE 1. Mean, minimum,
maximum and standard deviation for selected components from wheat field
experiments that included fertilizer, weed and/or tillage variables.
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Mean
Min Max Std. Dev.
No.
of replications 3.83
2.00 6.00 0.77
No.
of treatments 11.1
4.00 49.0 8.10
Mean yield (kg/ha) 2802
379 5915 1222
MSE
(kg^{2} /ha^{2}) 113782
8256 480822 97743
CV,
% 13.3
3.09 61.8 7.51
Plot size (m^{2})
37.2 8.4 96.0 24.3
Standard error (kg/ha) 231
64.2 558 98.7
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TABLE 2. Mean, minimum, maximum and standard deviation for selected
components from wheat variety field experiments.
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Mean
Min Max Std. Dev.
No.
of replications 3.34
2.00 6.00 0.92
No.
of treatments 14.8
4.00 60.0 11.7
Mean yield (kg/ha) 2841
967 5196 924
MSE
(kg^{2}/ha^{2}) 140179
40576 342290 78607
CV,
% 13.7
5.17 30.9 5.29
Plot size (m^{2}) 3.49
0.31 19.1 3.13
Standard error (kg/ha) 290
138 532 107
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