section 4 : measurement and analysis of agroforestry experiments

The analysis of data collected over time

S. Langton

Statistics Department
Rothamsted Experimental Station
Harpenden, Herts AL52JQ, England, UK

Abstract

Experiments are frequently carried out on the same experimental plots over a number of years. This poses a number of problems which will be described, together with the proposed solutions. In particular, the use of response curves for describing changes in yield over time will be considered.

Introduction

One difference between experiments involving trees and those involving field crops only is that the former generally continue for a period of years, whereas the latter often only last for one year or season. Thus in agroforestry experiments the data frequently consist of sets of successive measurements of the variables of interest, one set for each year or season, with the same treatments being maintained at each plot through the experiment (this situation should not be confused with a cross-over type of experiment, where different treatments are applied successively to the same experimental units). Such data are referred to in the statistical literature as 'repeated measurements', and the best way of analysing them has been a subject of considerable controversy.

Methods of analysis

The simplest method of dealing with repeated measurements of this sort is to sum or average them over time, and then carry out an analysis of variance on these sums or means. This analysis is perfectly correct from a statistical point of view, but provides no information regarding changes in the data with time.

One procedure that has been used to produce such information is simply to produce a separate analysis of variance for each time or season. Each of these analyses, with its associated tests of significance, is completely valid but, because results in successive seasons will almost certainly be correlated, the analyses are not independent of each other. In practice this means that, for example, significant treatment effects in five successive years in the experiment do not provide such good evidence for real differences between treatments as would significant treatment effects in five totally separate experiments. This type of analysis is not as attractive as it appears at first sight.

One common, but erroneous, method of analysis is to treat time as a factor, usually a sub-plot factor in a split-plot analysis. The first objection to such an analysis is that time can never be properly randomized; each treatment is in its second year of growth in the same calendar year, for example. Randomization is, of course, essential for a valid analysis of variance. A further problem with the split-plot analysis is that the subplot error is unlikely to be homogeneous, because observations closer in time are likely to be more closely correlated than those further apart (Rowell and Walters 1976). Similar objections apply to the 'split-block' or 'criss- cross' type of analysis which is sometimes recommended.

Multivariate methods have sometimes been used for the analysis of repeated measurements (Cole and Grizzle 1966) and this approach is unquestionably valid. Unfortunately these methods are conceptually more difficult and the results are not always easy to interpret. Hence there is a strong case for avoiding a multivariate approach where there is an alternative that is statistically sound.

The fifth method provides such an alternative; it was first used by Wishart (1938) and was described in detail by Rowell and Walters (1976). It involves the analysis of contrasts over time, and is discussed below.

The analysis of contrasts over time

The hypothetical example that I will use to describe this method consists of an experiment arranged as three randomized blocks of three treatments each (Table 1). The treatments are actually three different agroforestry systems and the objective of the experiment is to compare the changes in soil moisture content between the systems. Measurements were taken every 30 days over a period of 90 days following rain; during this time there was no rain or irrigation so the soil moisture content shows a continuous decline.

Table 1 Hypothetical data for analysis

NB: Variables measured on a percentage scale should normally be transformed before analysis.

The first stage in the analysis is to plot the mean soil moisture content for each system against time (Figure 1). This graph clearly shows that the moisture content is decreasing with time and, as is the case in many of these analyses, orthogonal polynomials can be used to break the change down into linear, quadratic and cubic components.

Figure 1 Graph showing the changes in soil moisture content for the three agroforestry systems
(———system 1;-------system 2; — — system 3)

The formal analysis is best described starting with the erroneous split-plot type analysis of variance (Table 2) described in the previous section.

Table 2 Analysis of variance with split plots

As was mentioned earlier, the problem with this is that the within-plot error (error b) is heterogeneous; this problem is solved by splitting it into separate components for the linear, quadratic and cubic contrasts, in the same way that the time effects can be split. To carry out this split it is also necessary to divide error b into the two interactions that make it up (blocks x days and blocks x systems x days). The resulting anova table is shown in Table 3. Such a table can easily be produced using a computer package such as Genstat or can be calculated by hand by the methods described in any good textbook on the analysis of designed experiments (e.g., Cochran and Cox 1957).

Table 3 Analysis of variance with interactions

The contrasts can now be compared with the appropriate component of the error variance; for example quadratic contrasts will be compared with a mean square calculated from block.quadratic and block.system.quadratic. The tabular form given in Table 4 (after Rowell and Walters 1976) is a clear way of displaying this. The days and system x days mean squares are taken directly from Table 3, but the error terms are obtained by adding together the appropriate component of the block x days and block x system x days sums of squares and dividing by the degrees of freedom (e.g., for the linear contrast:

(1.64 + 7.82)/6 = 1.58 ).

Table 4 Analysis of variance of linear, quadratic and cubic components

Now by comparing the days and error rows of Table 4 it can be concluded that the overall trend is mainly linear, but with smaller quadratic and cubic components. The system x days interaction gives information about the way these trends vary between systems; it can be seen that the differences lie in the linear and quadratic components, especially the latter.

Orthogonal polynomials are not always the most appropriate form of contrast to use, and the method will work with any set of contrasts. Yates (1982) gives some alternative suggestions for a medical example where the curves are sigmoid. Rowell and Walters consider an example concerning yields of apples over a 14 year period. The graph of yield against year shows that while there is a gradual increase in yield, good and bad years alternate. Such biennial yields are best dealt with by considering sums and differences of successive years. The seven two-year totals showed the overall increase, while the seven differences demonstrated the extent of the biennial effect. Orthogonal polynomials were still used within the two analyses to reveal the changes in these effects with time.

A brief mention should be made of the number of times at which measurements should be made. Sometimes this will be dictated by other factors, but in many cases the experimenter will be free to choose. In experiments such as the soil moisture content example, where there is a regular pattern of increase or decrease, there is seldom much point in measuring at more than four or five times. Five times would permit fitting terms no high as quartic, and it is unlikely that we would be able to interpret terms higher than this, even if they were significant. Of course if there was the possibility of some more rapid change of soil moisture during the experiment (e.g., due to rain or irrigation) we might wish to make a more regular check to study this.

Other uses of this technique

Although all the literature on this method refers to measurements repeated in time, there is no reason why it should not be used for measurements repeated in space in situations where the level of correlation decreases with increasing distance apart. Examples might be where measurements of yield or microclimate are being made at different distances from a tree or where vertical profiles of CO₂ concentration or wind-speed, for example, are being compared between systems.

Acknowledgement

This contribution was prepared while the author was funded by the U.K. Overseas Development Administration.

References

Cochran, W.G. and G.M. Cox. 1957. Experimental designs. 2nd Ed. New York:

Wiley. Cole, J.W.L. and J.E. Grizzle. 1966. Applications of multivariate analysis of variance to repeated measurements experiments. Biometrics 22: 810-828.

Rowell, J.G. and D.E. Walters. 1976. Analysing data with repeated observations on each experimental unit.J. Agric. Sci. 87: 423- 432.

Wishart, J. 1938. Growth-rate determinations in nutrition studies with the bacon pig, and their analysis. Biometrika 30:16-28.

Yates, F. 1982. Regression models for repeated measurements: reader reaction. Biometrics 38: 850-853.