COVARIANCE ANALYSIS. Rajender Parsad and V.K. Gupta I.A.S.R.I., Library Avenue, New Delhi

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1 COVARIANCE ANALYSIS Rajender Parsad and V.K. Gupta I.A.S.R.I., Library Avenue, New Delhi Introduction It is well known that in designed experiments the ability to detect existing differences among treatments increases as the size of the experimental error decreases, a good experiment attempts to incorporate all possible means of minimizing the experimental error. Besides proper experimentation, a proper data analysis also helps in controlling experimental error. In situations where blocking alone may not be able to achieve adequate control of experimental error, proper choice of data analysis may help a great deal. By measuring one or more covariates - the characters whose functional relationships to the character of primary interest are known - the analysis of covariance can reduce the variability among experimental units by adjusting their values to a common value of the covariates. For example, in an animal feeding trial, the initial body weight of the animals usually differs. Using this initial body weight as a covariate, the final weights recorded after the animals have been subjected to various physiological feeds (treatments) can be adjusted to the values that would have been obtained had there been no variation in the initial body weights of the animals at the start of the experiment. As another example, in a field experiment where rodents have (partially) damaged some of the plots, covariance analysis with rodent damage as a covariate could be useful in adjusting plot yields to the levels that they should have been had there been no rodent damage in any plot. Analysis of covariance requires measurement of the character of primary interest plus the measurement of one or more variables known as covariates. It also requires that the functional relationship of the covariates with the character of primary interest is known beforehand. Generally a linear relationship is assumed, though other type of relationships could also be assumed. Consider the case of a variety trial in which weed incidence is used as a covariate. With a known functional relationship between weed incidence and grain yield, the character of primary interest, the covariance analysis can adjust grain yield in each plot to a common level of weed incidence. With this adjustment, the variation in yield due to weed incidence is quantified and effectively separated from that due to varietal difference. Analysis of covariance can be applied to any number of covariates and to any type of functional relationship between variables viz. quadratic, inverse polynomial, etc. Here we illustrate the use of covariance analysis with the help of a single covariate that is linearly related with the character of primary interest. It is expected that this simplification shall not unduly reduce the applicability of the technique, as a single covariate that is linearly related with the primary variable is adequate for most of the experimental situations in agricultural research.

2 2. Uses of Covariance Analysis in Agricultural Research There are several important uses of covariance analysis in agricultural research. Some of the most important ones are: 1. To control experimental error and to adjust treatment means. 2. To aid in the interpretation of experimental results. 3. To estimate missing data. 2.1 Error Control and Adjustment of Treatment Means It is now well realized that the size of experimental error is closely related to the variability between experimental units. It is also known that proper blocking can reduce experimental error by maximizing the differences between the blocks and thus minimizing differences within blocks. Blocking, however, can not cope with certain types of variability such as spotty soil heterogeneity and unpredictable insect incidence. In both instances, heterogeneity between experimental plots does not follow a definite pattern, which causes difficulty in getting maximum differences between blocks. Indeed, blocking is ineffective in the case of nonuniform insect incidences because blocking must be done before the incidence occurs. Furthermore, even though it is true that a researcher may have some information on the probable path or direction of insect movement, unless the direction of insect movement coincides with the soil fertility gradient, the choice of whether soil heterogeneity or insect incidence should be the criterion for blocking is difficult. The choice is especially difficult if both sources of variation have about the same importance. Use of covariance analysis should be considered in experiments in which blocking couldn't adequately reduce the experimental error. By measuring an additional variable (e.g., covariate X) that is known to be linearly related to the primary variable Y, the source of variation associated with the covariate can be deducted from experimental error. This adjusts the primary variable Y linearly upward or downward, depending on the relative size of its respective covariate. The adjustment accomplishes two important improvements: 1. The treatment mean is adjusted to a value that it would have had; had there been no differences in the values of the covariate. 2. The experimental error is reduced and the precision for comparing treatment means is increased. Although blocking and covariance techniques are both used to reduce experimental error, the differences between the two techniques are such that they are usually not interchangeable. The analysis of covariance can be used only when the covariate representing the heterogeneity among the experimental units can be measured quantitatively. However, that is not a necessary condition for blocking. In addition, because blocking is done before the start of the experiment, it can be used only to cope with sources of variation that are known or predictable. Analysis of covariance, on the other hand, can take care of unexpected sources of variation that occur during the experiment. Thus, covariance analysis is useful, as a supplementary procedure to take care of sources of variation that cannot be accounted for by blocking. 600

3 When covariance analysis is used for error control and adjustment of treatment means, the covariate must not be affected by the treatments being tested. Otherwise, the adjustment removes both the variation due to experimental error and that due to treatment effects. A good example of covariates that are free of treatment effects are those that are measured before the treatments are applied, such as soil analysis and residual effects of treatments applied in the past experiments. In other cases, care must be exercised to ensure that the covariates defined are not affected by the treatments being tested. This technique can be illustrated through the following example: Example 1: A trial was designed to evaluate 15 rice varieties grown in soil with a toxic level of iron. The experiment was in a RCB design with three replications. Guard rows of a susceptible check variety were planted on two sides of each experimental plot. Scores for tolerance for iron toxicity were collected from each experimental plot as well as from guard rows. For each experimental plot, the score of susceptible check (averaged over two guard rows) constitutes the value of the covariate for that plot. Data on the tolerance score of each variety (Y variable) and on the score of the corresponding susceptible check (X variable) are shown below: Scores for tolerance for iron toxicity (Y) of 15 rice varieties and those the corresponding guard rows of a susceptible check variety (X) in a RCB trial Variety Replication-I Replication-II Replication-III Number X Y X Y X Y The usual analysis of variance without using the covariate (X variable) is as follows: Source DF SS Mean Square F Value Pr > F Replication Treatment Error Total R-Square C.V. Root MSE Y - Mean

4 Using the covariate, the analysis is the following: Source DF S.S. M.S. F-Value Pr > F Replication Treatment Covariate X Error R-Square C.V. Root MSE Y Mean It is interesting to note that the use of a covariate has resulted into a considerable reduction in the error mean square and hence the CV has also reduced drastically. This has helped in catching the small differences among the treatment effects as significant. This was not possible when the covariate was not used. The covariance analysis will thus result into a more precise comparison of treatment effects. The probability of significance of pairwise comparisons among the least square estimates of the treatment effects are given below: Pr > T H0: LSMEAN(i)=LSMEAN(j) i/j Pr > T H0: LSMEAN(i)=LSMEAN(j) i/j

5 2.2 Aid in the Interpretation of Experimental Results The covariance technique can assist in the interpretation and characterization of the treatment effects on the primary character of interest Y, in much the same way that the regression and correlation analysis is used. By examining the primary character of interest Y together with other characters whose functional relationships to Y are known, the biological processes governing the treatment effects on Y can be characterized more clearly. For example, in a water management trial, with various depths of water applied at different growth stages of the rice plants, the treatments could influence both the grain yield and the weed population. In such an experiment, covariance analysis, with weed population as the covariate, can be used to distinguish between the yield difference caused directly by water management and that caused indirectly by changes in weed population which is also caused by water management. The manner in which the covariance analysis answers this question is to determine whether the yield differences between treatments, after adjusting for the regression of yield on weeds, remain significant. If the adjustment for the effect of weeds results in a significant reduction in the difference between treatments, then the effect of water management on grain yield is largely due to its effect on weeds. Another example is the case of rice variety trials in which one of the major evaluation criteria is varietal resistance to insects. With the brown plant hopper, for example, covariance analysis on grain yield, using brown plant hopper infestation as covariate, can provide information on whether yield differences between the test varieties are primarily due to the difference in their resistance to brown plant hopper. The major difference between the use of covariance analysis for error control and for assisting in the interpretation of results is in the type of covariate used. For error control, the covariate should not be influenced by the treatments being tested; but for the interpretation of experimental results, the covariate should be closely associated with the treatment effects. We emphasize that while the computational procedures for both techniques are the same, the use of covariance technique to assist in the interpretation of experimental results requires more skill and experience and, hence, should be attempted only with the help of a competent statistician. 2.3 Missing Data The only difference, between the use of covariance analysis for error control and that for analysis of missing data, is the manner in which the values of the covariate are assigned. When covariance analysis is used to control error and to adjust treatment means, the covariate is measured along with the Y variable for each experimental unit. But when covariance analysis is used to analyze the missing data, the covariate is not measured but is assigned, one each, to a missing observation. We confine our discussion to the case of only one missing observation. The rules for the application of covariance analysis to a data set with one missing observation are: 1. For the missing observation, set Y = Assign the values of the covariate as X = 1 for the experimental unit with the missing observation, and X = 0 otherwise. 603

6 3. With the complete set of data for the Y variable and the X variable as assigned in rules 1 and 2, compute the analysis of covariance following the standard procedures. However, because of the nature of the covariate used, the computational procedures for the sums of squares of the covariate and for the some of cross product can be simplified. Example 2: An experimental trial for comparing 36 treatments was laid out as a simple lattice with two replications. There were 12 blocks in all, with 6 blocks in each replication. The observations were recorded. During the experimentation two observations were lost. The lost observations correspond to treatment 5 in block 1 of replication - I and treatment 20 in block 2 of replication - II, respectively. The layout of the design and the data recorded are given in the table below: The figures in bracket are the observations. Analyze the data and draw conclusions. Block Replication - I 1. 1 (1.325) 2 (1.140) 3 (0.950) 4 (1.200) 5 (-) 6 (1.375) 2. 7 (1.000) 8 (1.280) 9 (0.800) 10 (1.450) 11 (1.525) 12 (1.040) (1.400) 14 (1.075) 15 (1.000) 16 (1.275) 17 (1.150) 18 (0.850) (0.950) 20 (0.975) 21 (0.950) 22 (0.950) 23 (1.225) 24 (0.900) (1.275) 26 (0.875) 27 (0.975) 28 (0.975) 29 (1.000) 30 (0.925) (1.325) 32 (1.225) 33 (1.250) 34 (1.400) 35 (1.325) 36 (1.650) Block Replication - II 1. 1(1.325) 7 (1.000) 13 (1.225) 19 (0.900) 25 (1.050) 31 (1.200) 2. 2 (1.140) 8(1.225) 14(0.960) 20(-) 26 (1.525) 32 (1.450) 3. 3 (0.950) 9 (0.625) 15(0.975) 21 (1.075) 27 (0.850) 33(0.940) 4. 4 (1.200) 10 (1.525) 16 (1.075) 22 (1.250) 28 (1.150) 34 (1.225) 5. 5 (0.900) 11 (1.500) 17 (1.025) 23 (1.100) 29 (1.050) 35 (1.250) 6. 6 (1.300) 12 (0.960) 18 (0.875) 24 (0.900) 30 (0.900) 36 (1.275) The data can be analyzed by the use of covariance analysis. We define two covariates, one each for each of the missing observation. The covariates are defined in the way described above. We assign a value 1 to the missing observation and a value 0 to all the other available observations. We do the same thing for both the missing observations. The missing observations are given a value 0. The analysis of covariance performed on the data yielded the following results: ANOVA Source D.F. S.S. M.S. F-Value Pr > F Replication Blocks (repli.) Treatment Covariate X Covariate X Error Total R-Square C.V. Root MSE Yield Mean Further analysis may be performed as described in example