ABSTRACT: Grazing systems can be studied with statistical validity if all practical means of increasing efficiency are considered. Heterogeneity of rangelands and range herds is high; therefore, grazing studies require all possible procedures for gaining statistical precision. These should include use of covariance and response groups, adequate replications and sample numbers, and perhaps increasing treatments and years.

INTRODUCTION

At present, numerous grazing systems have been studied and no general conclusion has been reached as to which of these is most desirable under any given set of conditions, even under rather local situations. Variability on rangelands, even within relatively small pastures, is sufficiently high to require considerable refinement of acceptable statistical reliability. Studies of grazing systems on rangelands require rather large areas as well as several years to witness normal variability in climatic fluctuation and the resulting dynamics of vegetation and animal responses. Cost and the lack of homogeneous land areas frequently limit the size and number of replications or treatments. Thus, it is difficult to design an experiment that will detect treatment responses with acceptable precision except for gross differences.

Different kinds and classes of livestock require different numbers for a given accuracy, depending upon the parameters to be measured. Small pastures reduce variation within replications, but herd size numbers in the pastures may be required for natural behavior. However, restriction of plant and animal population also restricts the applications of the research findings. Unless acceptable precision is obtained, little can be said about differences among treatments. Before planning a study comparing grazing systems, it is desirable to have information on the magnitude of the components of variance of sampling and experimental errors to ensure adequate numbers of animals and replications.

Cumulative treatment effects of grazing systems automatically require long-term studies. These effects may be measured for plants, soils, and animals. The measure of cumulative effects in animals, such as production longevity, is non-existent for the study of grazing systems on rangelands. This requires a rather complicated long-term plan since measures of body condition, general health, and reproduction would need to be evaluated. Proficiency in reproduction requires rather large numbers and longevity or productive life requires a long period of time. Longevity is related to health, reproductive failure, and quantity of offspring over the entire life span of the female. A minimum of three generations has been suggested to measure cumulative effects on animal longevity. The importance of longevity as a selected criterion for comparisons of grazing systems may appear to be of limited relevance, but it is indeed a key economic consideration.

Productive plant responses that indicate profound and useful differences resulting from grazing systems generally exhibit themselves in a relatively short period of time compared to livestock responses other than, perhaps weight gains or losses. The design for cumulative plant responses should continue through a complete weather cycle of both favorable and unfavorable years. This will present several annual production cycles and their interaction with favorable and unfavorable growing conditions that will allow for range condition and trend analysis as a result of grazing treatments.

UNEQUAL NUMBERS

In many grazing trials it is difficult to have or maintain equal numbers of animals per treatment or replication. In most cases it is, perhaps, easiest to handle this by least squares analyses for uneven subclass numbers. In some studies least squares analysis is used to estimate the components of variance due to years, fields, and random error. These estimates are then used to obtain adjusted treatment means.

MODELS

Many types of models can be used to study range beef production systems. These models vary from deterministic models for simulating range beef production with many subroutines that include range and herd dynamics that would characterize only a single phrase of the overall production system. Simulation models use input information that typifies the procedures used or recommended for a herd and range dynamics program. The information inputs would include climate, quantity of forage, nutrient content of forage, daily intake, rate of stocking, weight of animals, condition, genotype potential, class of animal, and others.

The complexity of only one subroutine, such as growth and development of the offspring, would include age, genotype potential, weight change, feed and milk intake, size, condition, lean and fat deposition, and rate of gain, all adjusted at monthly intervals with respect to plant and climatic changes.

Simulation models for range livestock production must describe as accurately as possible the biological processes involved in the plant-animal interface relationship. This interchange of dependence and responses is rather sensitive; and if left totally up to natural relief or balance, the time lapse is far too great for the range livestock enterprise. As a result, drastic management decisions must be made rather rapidly and sometimes rather drastically. Certainly an operating model could lend information for decision making when, otherwise, a literature search would be time consuming and perhaps no more accurate than an up-to-date model.

It is generally recognized that when assumptions are made because information is limited, the results may be faulty. Modification of any model will be required to simulate production programs using different grazing systems for different localities where phenology of plants and resulting nutrient content differ by seasons and climatic conditions. Modification of the model will be required to include a broad range of different genetic potentials within animals with respect to degree of finish and lean and fat carcass characteristics as maturity advances. Models to explore the efficiency of different grazing schemes, that use dynamic herd and range interface functions, presents a most interesting challenge to the practical and scientific communities.

DESIGN DEPENDENCE

Precision of a study depends upon the experimental design that is chosen. Much, of course, depends upon the objectives of the study at the time of initiation. Are both animal and plant responses to be measured; if so, will each receive equal importance? In either event, the design for plant and animal responses can follow somewhat different prescriptions but must be considered in the study in light of the statistical inference that is desirable for presentation of the results.

Variables to be Measured
The number of animals and replications needed in a grazing experiment depends to a large degree upon the variables to be measured. Different data may have different inherent characteristics and yield highly different information because of the method or techniques used in the measurement. In a research project by Cook on sandhill ranges of eastern Colorado, it was found that different parameters varied widely as shown by the magnitude of the relative standard deviation (coefficient of variation ) (Table 1). The coefficients of variation in Table 1 are a means of comparing needed versatility in designing a grazing experiment to appropriately sample treatments with similar precision when several parameters are measured.

Season or Periods within the Study
Variability of average daily gain among range animals changes considerably from season to season as can be seen in Table 2. The reasons for this are weather, palatability of plants, nutrient content of forage, and the characteristics of the animals themselves will change from period to period. Animals seem to gain weight at full potential and rather uniformly during the spring when the herbage is growing. This is true of mother cows and their nursing calves and yearling cattle as well. When the forage approaches maturity in late summer or when it is fully mature and dry during the winter, rather extreme variability is demonstrated among individual response groups and among individuals (Table 2).

Short weigh periods that are related to seasons or forage changes present greater variation among animal weight responses than longer weigh periods of six or seven months. Note in Table 1 the rather small variation among average daily gain for animals from birth to weaning and from 9 to 19 months of age compared to the variation for the same animals during short seasonal periods (Table 2). Shorter weigh periods showed rather high variances; whereas, the longer growth periods appeared to average the peaks and the lows in biological changes resulting in lower variances (Table 1).

DESIGN EFFICIENCY

When comparing grazing systems on rangelands, it is desirable to consider every possible means of gaining statistical precision. Feasibility of replicated field studies is hampered by the need for rather large areas of land and livestock numbers. There are many ways of increasing efficiency in grazing trials. These include stratification through the use of outcome groups or through the use of covariance analysis when treatment responses are suspected of being associated with various characteristics of the individual samples, such as age and size of animal or cover and size of plants.

Response Groups
Using response groups is comparable to stratification to remove variation. Animal response groups can often be identified by such characteristics as sex, age, breed, weight, or frame size. Rather than pooling variation among individuals in the error term, it is desirable to identify this variation and consider it as a main effect. This removes recognizable variation among animals from the error term. As the error term becomes smaller, relative efficiency is increased, unless the loss of degrees of freedom from the error term more than offsets the gain in reduction of the magnitude of the error term.

As shown in Table 3 (footnote), there was an increase in relative efficiency of 21.76 percent for average daily gain by identifying three response groups (frame size) among experimental animals. The mean square (error) for average daily gain from weaning to 18 months of age was 0.0277 with 186 degrees of freedom. The average daily gain for the steers referred to in Table 3 was 1.30 pounds.

In like manner, the error mean square (error) for average weight of steers at 18 months of age was 6,476 when frame size was included as normal variation among animals. The increase in relative efficiency for testing differences in animal weights among treatments was 49.72 percent by removing effects of frame size from the error term (Table 3). The average weight of steers referred to in Table 3 was 941 pounds. Thus, by removing the effects of frame size, it is shown that more than twice the relative efficiency was obtained in accumulated animal weight than in average daily gain.

Covariance
It has long been suggested that relative efficiency could be gained by measuring related characteristics of the sample (plant or animal). In many cases this is true and, because of the precision gained, the experimental design can be smaller and less expensive. However, if populations chosen at the beginning of the study are relatively uniform, use of covariance to remove variation and to adjust means may be a disappointing exercise. Using a select group of samples removes variation but it also restricts the application of the results. Therefore, it may be desirable to use normal herd composition or normal plant populations and remove variability by other means.

Covariance is indeed a useful tool and should be ever present in the minds of researchers for inclusion in the study. It is common practice to remove area of plant cover by covariance as it influences biomass production or variation in yield. Likewise, in daily animal gain or accumulated weight, variation or the influence of size or initial weight upon rate of growth or accumulated weight can be removed by covariate analysis. Sometimes, hip-height or girth-length is measured along with weight of animal to remove variability by covariance. When there is rather wide variation among sizes, condition, age, or class of animals, it is indeed wise to explore the use of covariance analysis. If considerable variance is removed by the covariance analysis, it may be that the difference among treatment means is largely because of the covariate parameter which may or may not be of interest. If the covariate item is an inherent characteristic related to the response to treatment, it is generally wise to remove the variance from the error terms because of better interpretation of data as well as the gain in relative efficiency.

Adding Replications or Samples
It is often believed that continually adding sub-plots or animals will compensate for the lack of replications. This is a false concept since adding numbers adds little to precision after a reasonable level has been reached. If the sampling error (variation among individuals) is small and the experimental error (replication X treatment) is large, it means comparatively little can be accomplished by adding numbers. Adding a replication should perhaps be the first consideration in increasing precision of the experimental design. It decreases the magnitude of the variance among treatment means and adds degrees of freedom to the experimental error.

The analysis of variance for a study of grazing intensities in the mountain brush type of southern Colorado is shown in Table 4. Here, it is found that when the experimental error (replications x treatments) is used to make the "F" test, the differences among means is significant at the P.<.05 level; when the sampling error (remainder) or unexplained variance is used it is highly significant (P.<.01). From the data in Table 4 for the original experiment, it can be demonstrated how the addition of a replication can increase efficiency compared to adding an equivalent number of animals as sampling units within treatments to the original design. The calculations from data in Table 4 for relative efficiency are as follows: Components of variance for experimental error include the sampling error (s²) and the variance of a mean (s_e²). From Table 4 it is found that

.376 + s² + 40s_e² where s²=0.106;

therefore, s_e²= .376/40 - .106/40 = .00676 when 5 samples (animals) per treatment are used.

The variance of a treatment mean is

V_{(mean x)} = (s²+k_ss_e²)/ (R k_s);

Thus, in the origninal experiment

V_{(mean x)} = [.106 + (40)(.00676)]/[(2)(40)] =.00471 with a total of 320 animals;

but, with one additional replication V_{(mean x)} = [.106+ (40)(.00676)]/[(3)(40)] = .00314 with a total of 480 animals.

By adding more animals (3 per treatment), for a total of 192 more animals compared to 160 more animals for one additional replication, it was found the relative efficiency was increased by (.00471-.00421/.00471) only 10.62 percent. It is assumed in these comparisons that s and s e do not change values from the original experiment. This demonstrates that when replications are small (2 to 3) the addition of a replication generally increases relative efficiency considerably more than adding an equivalent number of animals as sample numbers. The addition of replications also adds degrees of freedom to the experimental error term A which also adds precision in testing the difference among treatment means. In either case, the land needed to graze the additional numbers are about the same, but cross-fencing may be more expensive for an additional replication in comparing grazing systems. There are some rather fast methods of calculating the relative efficiency gained by adding another replication. For instance: It is frequently asked how many replications or numbers are needed to obtain a predetermined difference among treatment means at a particular probability. As a general procedure, access to data that would reveal a variance for the sampling error (s² ) and the variance of a treatment mean (s_e²) would be needed. For this purpose, data from Table 4 can again be used where s² was 0.106 and s_e² was 0.00676. The standard formula for solving this problem is

Mean X₁-Mean X₂ = t_.05((2)[s² + (k)(s_e²)]/[(R)(k)])^(1/2)

where Mean X₁-Mean X₂ represents the difference among means desired for statistical reliability such as 10 percent of the overall mean, R represents replications and k the samples or animals for each treatment within replications (Table 4). The t_.05 value is 1.96 for 158 degrees of freedom, (n₁-1) + (n₂-1) or (80-1) + (80-1). The calculations for determining the needed replications follows:

(10%)(1.60)=1.96*((2)[.106 +(40)(.00676)]/[R (40)])^(1/2)

R= .00941/.00333+2.83

In a similar fashion, number of samples k could be determined.

Adding Treatments
It is sometimes debated that it would be better to add another treatment rather than adding another replication. In some cases, adding a treatment may be more effective than adding a replication in increasing precision of the experiment. It is also argued that the added treatment adds information to the research effort; whereas, the replication adds mainly precision. For instance, if there are now two treatments and two replications, and when either a treatment or replication is added, significance is tested with about equal efficiency. Likewise, equal area and an equal number of animals are required in either case.

In the present case (Table 4) it requires considerably less land area to add a treatment than is required to add a replication. One additional treatment adds one additional degree of freedom or precision to both treatment and error term to the design. The addition of one replication adds three degrees of .freedom to the error term only, but does add somewhat more efficiency than the added treatment. The gain in precision can be evaluated by using an F distribution table. In the original analysis of variance (Table 4), the experimental error for treatment has 3 degrees of freedom which shows, by the F table, the value of 9.28 is necessary for significance among treatments at the .05 level of probability. If one treatment is added, the degrees of freedom for treatment are 4 and the degrees of freedom for experimental error for treatment are 4. The F value for this comparison is 6.39 for significance at the .05 level of probability. Thus (9.28 - 6.39 = 2.89 / 9.28), there was a gain in efficiency of 31.14 percent by adding the additional treatment with all factors being equal, including the variance and differences for treatments and their interactions. As the probability changes in the F distribution table, the approximation of gain in efficiency changes for adding a treatment.

Adding Years
Generally, a study of grazing systems on rangelands requires three or more years to encounter some of the environmental variations that are certain to occur. Adding another year actually does not affect the degrees of freedom in either the experimental error or for treatments. It does, however, add degrees of freedom for the sampling error (remainder), but this is generally already large because each animal in the experiment adds a degree of freedom in this category. The "k" items added in the components of the variance by adding another year in Table 4 would be 5 for T x R error term and 10 for treatment variance in the components of variance. These higher k values (90 compared to 80) give a better mean for treatments and reduce the standard error of a treatment mean (S_T) by division of the square root of 90 instead of 80. Likewise, the standard error among treatment means (s_e) is reduced by the division of the square root of 45(k) instead of the square root of 40(k) as was the case in the original design with 8 years. In this case, the variance of a treatment mean is as follows for 9 years:

V_{(mean x)} = [.106 + (45)(.00676)]/[(2)(45)] =.00456 with a total of 360 animals.

The relative efficiency gained by adding the 9th year is (.00471 - .00456/.00471) 3.18 percent compared to the original design with only 8 years.

As was suggested in the discussion dealing with adding replications for increased precision, when numbers of treatment and/or replications are low, increased efficiency can be gained by adding replications or treatments. Likewise, when only a few years are included in the design, along with a relatively small number of treatments and few replications, considerable precision can be gained by adding another year. This assumes that adding a year will not affect variances for error terms, or for treatment means especially, by changing the latter. For instance, if there were only three years in the experimental design (Table 4) and a fourth year is added, the calculation of relative efficiency gained is:

V_{(mean x)} = [.106 + (15)(.00676)]/[(2)(15)] =.00691 for 3 years, and

V_{(mean x)} = [.106 + (20)(.00676)]/[(2)(20)] =.00603 for 4 years.

Thus, the relative efficiency gained is (.00691 - .00603/.00691) 12.74 percent by adding the fourth year.

SUMMARY AND CONCLUSIONS

In answering the question posed in the title, it can be stated that grazing systems on rangelands can be studied with statistical validity if preplanning has considered all of the practical means of increasing efficiency in the study. It may be that some desired parameters are impractical to measure with a herd system. Likewise, short duration responses (seasonal) of either plants or animals may be better studied with limited variation among individual samples by using smaller areas with fewer and more uniform animals.

Characteristically, the heterogeneity of rangelands is high and, as a result, grazing studies require all possible procedures for gaining statistical precision in design to obtain statistical reliability. The overall precision will depend upon the experimental design, and the choice of designs will depend upon the parameters to be measured, the season or period of study, class and species of animals to be used, plant response data to be collected, and the objectives of the study. Methods of maximizing efficiency include removing variation among animals by use of covariance and recognizing response groups such as size, age, sex, etc. In an example, it is shown that relative efficiency was increased as much as 50 percent by identifying response groups by means of frame size of yearling steers.

In most grazing studies it is difficult to obtain sufficient replications because of cost of fencing, area required, and need to restrict heterogeneity. Therefore, important alternatives for lack of desired replications are adding treatments or adding number of years the experiment will be carried on. An approximation of relative efficiency gained by adding a replication can be a simple calculation such as if there are now 2 and 1 more is to be added, 1/3 or 33 percent is gained, or if there are now 3 and 1 is added, 1/4 or 25 percent is gained. Adding another treatment may gain as much precision as adding a replication and sometimes with less animals and acreage. Another treatment generally adds information that would not occur from adding a replication.

An actual example shows that adding a fourth year increased relative efficiency by 13 percent, but adding a ninth year increased relative efficiency by only 3 percent. In all cases, the fewer the replications, treatments, and/or years, the greater the relative efficiency gained by increasing anyone of the three.