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## Material Information- Title:
- A guide to extensive testing on farms
- Added title page title:
- Extensive testing on farms
- Creator:
- Hopp, Henry, 1911-
United States -- Foreign Agricultural Service - Place of Publication:
- Washington
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- Foreign Agricultural Service, U.S. Dept. of Agriculture
- Publication Date:
- 1954
- Language:
- English
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- 4 pts. in 1 v. : illus., tables. ; 26 cm.
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- Field experiments ( lcsh )
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- federal government publication ( marcgt )
non-fiction ( marcgt )
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- Electronic resources created as part of a prototype UF Institutional Repository and Faculty Papers project by the University of Florida.
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extensive testing on farms in four parts PART If. Foreign Agricultural Service April 1954 U.S.DEPARTMENT OF AGRICULTURE This publication has been prepared for use in the technical cooperation program of the Foreign Operations Administration. a guide to EXTENSIVE TESTING ON FARMS by Henry Hopp in 4 parts Part IV: Using Data to Design Extensive Tests on Farms Foreign Agricultural Service UNITED STATES DEPARTMENT OF AGRICULTURE Washington, D. C. April 1954 CONTENTS Page Preface . . . . ... . . 1 Improving the estimate of number of farms required . 2 Plot variability ................... .. 3 Location variability ................... 15 Treatment variability . . . . 23 Modifying the estimate for large plots . . ... 28 Determining the best size of plot . . .. 30 PREFACE Parts II and III of this Guide describe procedures for laying out the two kinds of extensive tests--result tests and experiments on farms. One of the important items in the cost of such projects is their size, i.e., the number of farms they involve and the area of the plots. Of course you want to keep the undertaking as small as possible, and yet you want to meet the requirements for adequate design. In the previous parts, some rather crude methods were given for deciding on the number of farms to include in an extensive test: Part II, on result tests, gave a rule-of- thumb method; Part III, on farm experiments, gave a somewhat better method- making "informed guesses" and applying to them certain simple statistical procedures. Neither of these methods is likely to be very precise. They might miss the mark by a great deal. The one source of really accurate information would be data from surveys or experiments. Fortunately such data are often available: surveys already may have been made in the area; experiments already may have been conducted at one or several research stations. If such data pertinent to an extensive test are available, they can be used in estimating the number of farms as well as the size of plots. The methods involve conventional statistical pro- cedures and therefore require some knowledge of statistical analysis. For tests that are critical, costly, or time-consuming, the added inconvenience that statistical methods entail may be minor compared with the savings they accomplish. This part of the Guide (1) describes the procedure for using data in estimating the number of farms required for the extensive test, (2) tells how to modify this procedure when you want plots larger than the research plots, and (5) gives a method for determining the best size for extensive- test plots. This part, then, serves as a supplement to the three preceding parts. You will need it only when you wish to improve the design that you have already arrived at through the previous parts; and the directions that you find here can be used only in connection with that design. IMPROVING THE ESTIMATE OF NUMBER OF FARMS REQUIRED In Part III of this Guide you learned how to decide on the number of farms to have in an extensive test. You estimated three kinds of varia- bility--plot, location, and treatment--and you learned to do it by a method we might dignify as "informed guessing." With this rather crude method, your decision as to the number of farms will not be completely reliable, but you will have to be satisfied with it if you lack actual yield data from surveys or experiments. If, however, you have or can get some pertinent yield data and think it worthwhile to determine the number of farms more reliably, you can substitute more accurate methods for estimating each of the three kinds of variability. The method to use will depend on, first, the kind of variability you are estimating; second, the source of your data, i.e., whether they are from surveys or from experiments; and third, the competence and facilities available for making statistical computations, i.e., whether you have a trained technician and a calculating machine for him to work with. Three methods are presented here. Use Method 1 for plot or location variability if you have, or can obtain, survey data on yields and if you do not have a technician and machine available. Use Method 2 for plot or location variability if you have, or can obtain, survey data and if you have a technician and machine. Use Method 3 for plot, location, or treatment variability if you have data from experiments in the area and if you have a technician and machine. Although each of these methods will give you a more precise estimate of the optimum number of farms than the guessing method, the third one will give you the most precise estimate of all. This method, however, can be used only if some experiments already have been performed in your area on the practice you are testing. For Methods 1 and 2, usually, data are more easily obtained. You do not have to use the same method for estimating all three kinds of variability. Thus, you might be planning an extensive test on corn ferti- lizers, and, though you may not have data on the subject, you may have some that were obtained in experiments with corn varieties. You can use these data for determining plot variability, following Method 3. But, if the research had been done at only two or three locations, the data may be insufficient for getting location variability. So, for location varia- bility, you make a little survey and obtain an estimate by Method 2. Finally, lacking specific information on corn fertilizer experiments, you may have to fall back on the guess procedure in Part III of this Guide to estimate treatment variability. LOCATION VARIABILITY Method 1 If survey data--the kind used for Method 1--do not already exist for your area, you can obtain some by making a quick survey of farms. An example of data collected in such a survey is shown in Table 6: 15 farms were selected at random, and yield per acre of the crop in question was obtained from the farms' total yield of that crop, not from small sample plots. If, however, you get yields from small plots, follow instead the procedure in Method 2 (page 18). The first step in determining location variability from the yield data in table 6 is to add all the yields and divide by the number of lo- cations, thus arriving at the mean yield--in this example, 32. Now find the difference between the mean yield and each location yield, and enter these differences, or deviations, in the third column. Rank the locations in increasing order of the deviations: Farm 2 has the smallest deviation and ranks first; Farm 4 has the largest deviation and ranks fifteenth. Now list the 15 farms in the order of this rank (table 7). The midpoint, or median, rank is 8; and the deviation corresponding to this rank is 9. Location variability is now obtained as follows: Location variability = Deviation for midpoint rank 0.7 9 0.7 = 15 To express the variability in percent, use this equation: Location variability in percent = Location variability x 100 Mean yield 13 x 100 32 =41% Use this estimate in the calculations page 15). of extensive-test error (Part III, 4/ Snedecor, op. cit., sec. 2.16. Table 6. Survey data on average yield of entire farms, for 15 randomly selected farms. Farm Yield per Deviation acre from mean 1 27 5 4 2 33 1 1 3 23 9 7 4 71 39 15 5 31 1 2 6 24 8 7 21 11 10 8 21 11 11 9 57 25 14 10 15 17 13 11 41 9 8 12 31 1 3 13 27 5 5 14 42 10 9 15 19 13 12 Total yield! 483 Mean yield 32 Table 7. Relisting of farms order of rank. Rank Farm of table 6 in Deviation Method 2 With Method 2, which calls for a calculating machine, you can make a more accurate estimate of location variability from the survey data. We will illustrate the procedure for two types of data: one in which yields of entire farms are available, and the other in which yields of sample plots are available. The data in table 6 can be used as an example of yields for entire farms. Start with the yields in the second column of the table and add their squares:2/ 272 + 332 + + 192 18,777 From this value, the uncorrected sum of squares, subtract a "correction factor" obtained by squaring the sum of the yields and dividing by the number of farms: Correction factor = 482 15 = 15,553 Subtracting 15,553 from 18,777, you get 3,224. This value is called the corrected sum of squares. Now divide by the number of farms less 1, that is, by 15 1, to get the mean square: Mean square = Sum of suares Number of farms 1 = 230 The square root of 230, or it in percent as follows: 15, is the location variability. Express Location variability in percent SPlot variability x 100 Mean yield S15 x 100 32 - 47% 5/ Snedecor, op. cit., sec. 2.8. This 47 percent is a more exact computation of the location variability than the 41 percent obtained by Method 1. This becomes, then, the estimate to use in calculating extensive-test error (Part III, page 15). Now let us illustrate the procedure when the yields are for small plots on farms rather than for entire farms.6 You must measure 2 plots side by side on each farm. Hence, you can use here the data collected for plot variability in table 1. However, summarize the data as shown in table 8. The first step is to square each observation, or the value for each plot, and add the squares to get the uncorrected sum: 312 + 212 + + 21 = 37,769 Then compute the correction factor: Correction factor = Total yield2 Number of plots 9652 50 = 31,041 The difference between the uncorrected sum of squares and this correction factor, 37,769 31,041, or 6,728, is called the total sum of squares. Enter it in the indicated place in table 9. In the next column, enter the total degrees of freedom--the total number of ob- servations less 1, or 30 1. The next value to compute is the sum of squares for farms. Add the squares of the farm totals (in last column of table 8) and divide by the number of plots per farm: 662+ + 42 + + 82 74,889 = 7,444 2 2 and then subtract the correction factor, already computed as 31,041, to get 6,403. Eater this value in table 9 as the sum of squares for farms. In the next column enter the degrees of freedom--the number of farms less 1, or 15 1. Now, for plots, obtain the sum of squares and degrees of freedom by subtracting the values for farms from the values for total. 6/ Snedecor, op. cit., sec. 10.6. Table 8. Survey data on yields of 2 adjoining plots on each of 15 farms. (Each plot is of the size to be used in the extensive test) Farm Plot 1 Plot 2 Sum 1 31 35 66 2 21 33 54 3 25 20 45 4 74 68 142 5 32 31 63 6 21 28 49 7 24 18 42 8 20 22 42 9 52 61 113 10 13 17 30 11 46 36 82 12 33 28 61 13 30 24 54 14 38 46 84 15 17 21 58 II I Total yield . . .. . 965 Number of plots . . . 50 Mean yield . . .... .. 32 Table 9. Analysis of variance of data in table 8, for arriv- ing at location and plot variabilities. Source Sum Degrees of of of Mean Units Variance variation squares freedom square component Farms 6,403 14 457 Plots 325 15 22 1 22 Total 6,728 29 Difference due to farms 455 2 218 Obtain the mean squares for farms and plots by dividing the degrees of freedom into the sum of squares. Then, to find out what variation was due to the farms, subtract the mean square for plots from the mean square for farms. You have to do this to eliminate the plot-to-plot variation from the farm differences. The next column of table 9, "units," gives the number of individual plots that were represented in the numbers you squared. For farms, you squared a value that represented 2 plots; hence, enter 2 in the bottom line of that column. For plots, you squared the values for individual plot yields; hence, enter 1 for plots. Divide the mean squares by the respective number of units to get the values called variance components. The square roots of these variance components are the variability values we seek. For farms (locations)the variability is the square root of 218, or 14.8. Expressed in percent of the mean, it is: Location variability x 100 Location variability in percent =Loc n vy x Mean yield 14.8 x 100 32 = 46% Use this variability in calculating extensive-test error (Part III, page 15). If the plots of the survey are approximately the same size as those you will use in the extensive test, the variance component for plots can be used to determine plot variability. Follow the same procedure as with location variability: Plot variability = V = 4.7 You express this variability in percent as follows: Plot variability in percent = Plot variability x 100 Mean yield 4.7 x 100 32 = 15% Note that you have the same answer here as you obtained by Method 2 for plot variability, page 6. Use this estimate in your calculation of extensive-test error (Part III, page 15). PLOT VARIABILITY Method 1 Method 1 uses survey data and requires no calculating machine. Perhaps you already have data from surveys previously made; but, if you do not, you can get some by collecting simple crop-cutting samples from 2 adjoin- ing plots on each of 10 or more farms, having each plot of the same size as plots you later will use in the extensive test. From such data you can make a fair approximation of plot variability, even without a calcu- lating machine. Let us consider an example. You have collected data, let us say, on yields from 2 adjoining plots on each of 15 farms. Now, for each farm enter the yield for each plot in columns 2 and 3 of table 1; in column 4 enter the difference between them. When you have done this for all farms, use column 5 to number the differences consecutively in increas- ing order of magnitude. Thus, Farm 5, which has the smallest difference between plots, get the rank of 1, and Farm 2, which has the largest difference, gets the rank of 15. Now list the 15 farms in the order of this rank (table 2). The midpoint, or median, rank is 8; and the dif- ference corresponding to this rank is 6. Now use this difference to obtain the figure for plot variability, thus: Plot variability Difference at midpoint rank Plot variability = 1.4 6 4.3 To express this plot variability in percent you must first calculate the mean yield of the plots: divide the sum of all the yields in table 1 (965) by the total number of plots (30). Then use the mean yield (32) in this equation: Plot variability in percent Plot variability x 100 Mean yield 4.3 x 100 32 = 13% 1/ George W. Snedecor, Statistical Methods, 4th ed., 1946, sec. 2.16. Table 1. Survey data on yields of 2 adjoining plots on each of 15 farms. (Each plot is of the size to be used in the extensive test.) 35 33 20 68 31 28 18 22 61 17 36 28 24 46 21 Total yield 965 Number of plots 30 Mean yield 32 Table 2. Relisting of rank. farms of table 1 in order of Rank Farm Difference Substitute this value for the one you got by the "guessing" procedure in Part III and use it in calculating the extensive-test error (Part III, p. 15)- Method 2 Method 2 calls for survey data and requires a calculating machine; use of the machine will help you obtain a more accurate measure of varia- bility. Start with data like those in table 1, but omit the last column; instead, square each difference in the fourth column and add the squares : 42 + 122 + + 42 = 649 To find plot variability, divide this sum of squares by the number of plots, ant extract the square root of the quotient: Plot variability = / Lu of squsrs of differences V Number of plots V 30 4.7 Plot variability in percent is obtained in theeameway as for Method 1: Plot variability in percent = 4.7 x 100 32 = 15% Bear in mind that use of a calculating machine does not of itself assure a reliable estimate of plot variability. The estimate is good only so far as the data are representative of all the farms in the area. You should therefore evaluate both the source of the survey data and the amount that is available. You must have measures of plot variability from a sufficient number of farms, preferably selected at random, before you can place much confidence in your mathematical estimate. You may even modify this estimate by judging the representativeness of the data, and so arrive at a better estimate. For instance, your data may have come from soils that are more uniform than other soils in the region and may, in your judgment, underestimate plot variability for the region as 2/ Ibid., sec. 4.2. a whole. If so, you might increase your calculated value by a small amount; thus, in our example, you might increase it from 15 percent to 20. The estimate you finally arrive at is the one to use in calculating the extensive-test error (Part III, p. 15). Method 3 Method 3, which uses data from experiments and requires a calculating machine, gives the most exact estimate of variability. It is the method that you are likely to choose if you are working in an area that already has an agricultural research organization, for such organizations usually accumulate data of the kind required for the method. These accumulated experimental data, however, will be useful to you only if they have been collected from farms varied enough to be fairly representative of plot variability in the region. If they have not, you will do better to use data obtained from surveys and to analyze them by Methods 1 or 2. Above all, avoid the error of making a plot-variability estimate from experi- ments that were all conducted at one location, as at a research station. If you are working in an area where research experiments are under way, you can enlist the assistance of the technician carrying out the experi- ments: he can obtain plot variability from the analysis of variance he makes of his data. Which item in his analysis is the measure of plot variability will depend on the experimental design he has used; one ex- ample, however, will suffice here to show how variability can be measured by analysis of experimental data. Our example is based on a hypothetical experiment set up in a randomized block design, which is a rather usual type The experiment tests four treatments, covers four farms, and has two replications, or blocks, per farm. 3/ Actual experiments may differ in design from this one, but all well-designed experiments have these characteristics (1) two or more treatments, or factors, under test; (2) two or more replications of the treatments at each location; (3) repetition of the same or closely re- lated treatments at several locations. The first step in the analysis is to tabulate the data on yields from each plot by treatment and location and to add up the total yields for treatments, farms, and replications (table 3). Next, prepare a worksheet like that shown in table 4; at the head of it note the total number of plots in the experiment (32) and the total yield from these plots (1,055). Now, using a calculating machine, proceed to get the various sums of squares, entering the figures for each step in the table. 3/ Snedecor, op. cit., sec. 11135. Table 3. Experimental data on yields of 32 plots in an experiment with 4 treatments, 4 farms, and 2 replications per farm. (Each plot is of the size to be used in the extensive test) Treatment Farm 1 Farm 2 Farm 3 Farm 4 Total Rep. Rep. Total Rep. Rep. Total Rep. Rep. Total Rep. Rep. Total 1 2 1 2 1 2 1 2__ A 39 31 70 36 32 68 27 34 61 35 38 73 272 B(check) 534 35 69 27 24 51 26 21 47 32 33 65 232 C 47 39 86 37 3 9 76 30 39 69 i 42 41 83 531 D 36 25 61 26 23 49 21 36 57 21 29 50 217 Total 156 130 286 126 118 244 104 130 254 10 31 -27-1 -1035 __l iL __ Table 4. Worksheet for calculating sums of squares of data in table 3. (Number of plots, 32; sum of yields, 1,035) Treatments Replications Item Total Treatments Farms x on farms farms Uncorrected sum of squares 34,903 273,493 269,529 69,063 135,533 Divisor 1 8 8 2 4 Quotient 34,903 34,187 33,691 4,532 33,883 Correction factor 33,476 33,476 33,476 33,476 33,476 Corrected sum of squares 1,427 711 215 1,056 407 711 215 -215 192 130 The first one is designated as "total": it is the sum of the squares of each individual yield number shown in table 3. The uncorrected value is as follows: 392 + 342 + 292= 34,903 Enter this value in the first line under the column headed "Total." Immediately beneath it enter the "Divisor," which is merely the number of plots included in each number you have just squared; in this case it is only 1. Divide the uncorrected sum of squares by this divisor to get the quotient 34,903. Before you can arrive at a corrected sum of squares, you have yet to compute a correction factor: Total yield2 Correction factor =Numer of plots Number of plots S1,0352 32 = 33,476 Subtract this correction factor from the quotient, 34,903 33,476 = 1,427, and you have the corrected sum of squares to enter as the last figure in the column. The second sum of squares is calculated from the treatment totals of table 3: 2722 + 2322 + 3142 + 2172 = 273,493 Since each treatment total includes the yields of 8 plots, the divisor is 8; From this point proceed as you did in finding the corrected sum of squares for the total, and you will arrive at the corrected sum of squares for treatment--711. The third sum of squares is calculated from the farm totals in table 3: 2862 + 2442 + 2342 + 2712 = 269,529 Again the divisor is 8 since each farm total includes 8 plots. Pro- ceeding as before, you obtain the corrected sum of 215. The fourth sum of squares is calculated from the totals of each treat- ment on each farm in table 3: 702 + 692 + + 502 = 69,063 Since each squared number in this calculation includes 2 plots, the divisor is 2. Proceeding as before, you obtain the number 1,056. But this is not the corrected sum, as it would have been in the first 5 columns. From it you must first subtract the sums of squares for treatments (711) and for farms (215) to get 150, the sum of squares for treatments x farms. The fifth sum of squares is calculated from the totals of each repli- cation on each farm in table 3: 1562 + 1502 + 1262 + + 1412 = 155,555 The divisor is 4 since each replication total includes 4 plots. Now proceed as before until you obtain the value 407; this number, however, contains not only the replication differences but the farm differences as well. Hence subtract the sum of squares for farms (215), the re- mainder, 192, is the corrected sum of squares for replications on farms. Now transfer these 5 sums of squares to the analysis-of-variance sheet (table 5). In the first column list the various sources of variation, i.e., the various factors that contributed to the yield: treatments, farms, treatments x farms, and replications on farms. For each of these, and for the total, sums of squares are entered (from table 4) in the second column. For the item just before the total, "treatments x repli- cations on farms," the sum of squares is obtained by subtracting all other sums of squares from the total sum of squares. The third column in table 5 is headed "degrees of freedom." For each source of variation the number of degrees of freedom is as follows: Treatments: Number of treatments (4) less 1. Farms: Number of farms (4) less 1. Treatments x farms; Degrees for treatments (3) multiplied by degrees for farms (3). Replications on farms: Number of replications on each farm less one (2 1) multiplied by number of farms (4). Treatments x replications on farms: Total de- grees (number of observations less 1, or 32 1) less the degrees found thus far (3 + 3 + 9 + 4). In the last column are listed the mean squares. These are obtained by dividing the degrees of freedom into the sum of squares. For ex- ample, the mean square for "treatments x replications on farms" is 179/12, or 15. Table 5. Analysis of variance for arriving at an variability. of data in table 3, estimate of plot Sum Degrees Mean Source of variation of of square Ss squares freedom 1. Treatments 711 3 237 2. Farms 215 3 72 3. Treatments x farms 130 i 9 14 4. Replications on farms 192 4 48 5. Treatments x replications on farms (plot variability) 179 12 15 6. Total 1,427 31 -- You are now ready to calculate the plot variability: Plot variability Mean square for "treatments x Plot variability V replications on farms" = .87 To express this variability in percent, use the following equation, which calls for the mean yield of untreated plots. These plots are the ones that were shown in table 3 as receiving treatment B; i.e., the check treatment. Plot variability in percent =Plot variability x 100 Mean yield of check plots = 5.87 x 100 29.0 =15 Use this estimate in the calculations of extensive-test error (Part III, page 15). The tests at the different farms may not involve exactly the same experimental design. Often the treatments are not exactly the same. As long as they are reasonably similar, you can combine them, even if the number of treatments or replications differ. In order to combine plot variability from different experiments, calculate first the plot variability in percent for each experiment separately. Then square the percentages. Add the squares, and obtain the mean of the sum by divid- ing it by the number of experiments. Finally, take the square root of the mean. As an example: Experiment Plot Variability Squares (Percent) 1 12 144 2 15 225 5 10 100 4 13 169 SUm of squares 638 Mean of squares (638/4) 160 Mean plot variability ( -\fl6 ) 12.6% A word of caution about Method 3: The calculations are rather exact, but in using the resulting information for designing an extensive test, it is generally necessary to exercise some discretion as to applica- bility. If the plots to be used in the extensive test are much larger than those used in the experiment, additional calculations are necessary (see page 28). Furthermore, in order to feel confident about using the information from experiments, you must be fairly sure that soil variability at the experi- ment stations is reasonably representative of soil variability on farms in the region. If the soil is more uniform at the experiment stations than it is likely to be on the extensive-test farms, you would be justi- fied in raising somewhat the estimate of plot variability. Method 3 Method 3, which uses experimental data for estimating variability, should be used for location variability only if the experiments have been conducted on enough farms to constitute a fair sampling of the region. A minimum of 10 farms is a good rule-of-thumb requisite. Most experiments are conducted in only one location, or in only a few, and therefore cannot serve the purpose. Table 3 shows experimental data from only 4 farms, not enough to make a reasonably precise estimate of location variability. When data are from so few locations, do not use Method 3 for calculating location variability. Instead, use one of the other methods already described. They are based on a more inclusive sample and also require much less computation. Another disadvantage of using experimental data for determining location variability is that the calculations are usually complicated. You will have to call on the technicians at the research station to do the cal- culations. This may be feasible, and we will therefore give an example of the procedure in the next section, on treatment variability. In that section we will use the same example for both treatment and lo- cation variability. TREATMENT VARIABILITY Only data from experiments can be used to calculate treatment varia- bility. Hence, Methods 1 and 2, the ones that work with survey data, are not applicable. But, in using Method 3, be sure that your data are from an experiment that meets the following requirements: (1) Experimental farms must be representative of the region as to both anticipated treatment effects and treatment variability and (2) experi- mental treatments must be similar to those planned for the extensive test. Unless the experiment meets both these requirements, you had better use the method given in Part III (page 13) of this Guide. Let us assume that the data in table 3 are from an experiment that meets these requirements and therefore can be used to illustrate the procedure of Method 3. Start with the analysis of variance that was made of the data in table 5.1/ // You now rewrite the table, using symbols as shown in table 10. The purpose of the revision is to help 7/ William G. Cochran and Gertrude M. Cox. Experimental Designs. New York, 1950, Sec. 14.1. 8/ Snedecor, opo cit., sec. 11.135 you isolate the variance component for treatment variability from the rest of the information obtained in the experiment. The revision looks imposing but is not difficult if you follow directions stepby step. Copy the first column of table 5 into the first column of table 10. Filling in the next section of the table, "Calculating mean square in symbols," is accomplished in 3 steps that will be easy if you simply follow directions. By the time you have finished the third step, you will have written equations that show the various components in symbols for each mean square. Step 1 is to assign a letter as a symbol for each source of variation thus: T for treatments; F for farms; R(F) for replications, or plots on farms; and all the symbols--TFR(F) --for the total. After each symbol, write the number that applied in the experiment: table 3 will remind you that there were 4 treatments, 4 farms, and 2 replications per farm, making 32 plots in all. In Step 2, write the cofficient for each variance in step 1. This con- sists of the symbols that are missing. To clarify: all the symbols, T, F, and R(F) are found in the total; but not all are found in each variance. Whichever ones are missing in each are now to be written under Step 2, in small letters. Thus, for variance T, both F and R in are missing; write them so--fr,) When you get down to the fourth line, note that the variance, R(F) lacks both T and F; but the F is not written in Step 2 because there is a rule that when a symbol appears in parenthesis in the variance of Step 1, this symbol shall not be repeated in the coefficient. Nowyou come to Step 3, which, when finished, will give you the complete formulas. First copy each variance into this column, preceding it with its coefficient; for "treatments," for instance, write frt)T. Now add to it all other variances that also contain the identifying symbol T, not forgetting to precede each variance with its coefficient. In this case, the other variances that contain T are TF and TR(F) ; the latter, not having a coefficient to precede it, is added all by itself. Con- tinue thus to the end of the column. When you get to the last line, all you will have to enter will be TR () for there is no other vari- ance with all these symbols, neither does it have a coefficient to precede it. Having calculated the mean squares in symbols, copy in the last column the numerical mean squares from table 5; and then you have everything you need to solve the symbol equations for the numerical value of the variance components. Start with TR() the plot variability on the fifth line. You need do no more than look in the last column to find that its value is 15. Now you can solve for R ( on line 4: tR(F) + TR(F = 48 Table 10. Revision of table 5 to obtain variance components for determining variability by Method 3. Mean Source of variation Calculating mean square in symbols quare Step 1 Step 2 Step 3 in (Variance) (Coefficient) (Completed formula) numbers Treatments T (4) fr(f fr(f) T + r (f TF + TR(F 237 Farms (location variability) F (4) tr(r) tr(f) F + r (f) TF + tR(F) + TR(F) 72 Treatments x farms (treatment variability) TF r(f) r(f) TF + TR (F) 14 Replications on farms R (F) (2) t tR(F) + TR(F) 48 Treatments x repli- cations on farms (plot variability) TR (F- TR (F) 15 Total TFR(F) (32) __ Since t = 4 (from the second column) and TR(r) = 15, rewrite the equation thus: 4R(F) + 15 = 48 48 15 R(F) .25 = 8.25 Now proceed to solve the bottom: for treatment variability, on the third line from r() TF + TR(F) = 14 2TF + 15 = 14 14 15 TF = 2 = -0.5 This negative value is unusual; generally TF is positive. variance component should be taken as zero. A negative When you obtain a positive value for TF--25, for example--simply take its square root to obtain the treatment variability: Treatment variability = \ TF S'V25 = 5 To express this treatment variability in percent, use the following equation, which calls for the mean yield of check, or untreated, plots (see table 3 for yields of plots receiving treatment B): Treatment variability in percent Treatment variability x 100 Mean yield, check plots 5 x 100 29 =17% Use this estimate in your calculation of extensive-test error (Part III, page 15). While we were discussing plot variability early in this section, we pointed out that experimental data can be used for estimating location variability also, provided the experiment covers enough locations to provide an adequate sample. The experiment summarized in table 5 had only 4 locations, hardly enough for a region of any size. Nevertheless we can use this experiment to illustrate the procedure for determining location variability. Refer again to table 10, and write the equation for farms: tr(r) F + r(f) TF + tR(F) + TR(F) = 72 Substitute the coefficients and the variabilities already solved: (4 x 2F) + (2 x 0) + (4 x 8) + 15 = 72 8F + 0 + 52 + 15 = 72 F = 72 2 15 8 = 2 = 3.12 Then, since location variability is simply the square root of F-- Location variability = 'VF = 1.77 To express this location variability in percent, use this equation: Location variability in percent Location variability x 100 Mean yield of check plots = 1.77x 100 29 = 6% This is the estimate that you will use error (Part III, page 15). in calculating extensive-test MODIFYING THE ESTIMATE FOR LARGE PLOTS If you have determined plot variability from data taken from plots of the same size as those you will use in the extensive test, you will not need the information in this section. But you may wish to use larger plots in the extensive test, especially in order to enhance its demon- strational value. If so, you will need the information given here. When the extensive-test plots are to be much larger than the experi- mental plots, there is an additional adjustment to make. The procedure is quite simple. If the experimental plots were 5 square feet and the extensive-test plots will be 500 square feet, each of the latter will contain 100 experimental-plot units. Now refer to table 11. In the first column are listed a range of ratios between the sizes of the extensive-test plots and the experimental plots. The remaining columns give corresponding factors: one for crop tests, the other for animal tests. For a plot-size ratio of 100, the factor for crop tests is 3.2. Now, if the plot variability in the experimental plots is 15 percent (for example, page 15), then-- Plot variability for = Plot variability in experiment (% extensive test (%) Factor 13 5.2 = 4% The plot variability thus obtained is the one to use in calculating extensive-test error (Part III, page 15). A word of explanation will clarify the two sets of factors in table 11. In crop tests, the plot variability does not decrease in proportion to the size of the plots: as the size of the plots is increased, more variable ground is likely to be included. The variability between two plots lying side by side is less than the variability between two plots some distance apart. Thus, the set of factors for field crops, which is the fourth root of the ratios shown in the first column, takes into consideration this increased variability of soil with increasing plot size.2/ In animal tests, on the contrary, this circumstance does not apply; therefore, the factors used are simply the square roots of the ratios shown in the first column. The factors for animal tests can be usedalso for any other tests in which increased "plot size" does not mean a proportionate increase in the area of land per plot. 9/ An adaptation from "An Empirical Law Describing the Heterogeneity in the Yields of Agriculture Crops," by H. Fairfield Smith, Jour. Agric. Sci. 28(Part 1):1-25, January 1958. In table 11, an average heterogeneity factor of 0.50 is used. Table 11. Factors to be used in determining the plot error when the extensive-test plots are larger than the experimental plots on which estimate of variability was based. Ratio: Extensive-test plot size Factor for Factor for Experimental plot size crop tests 1/ animal tests 1.0 1.5 1.6 1.7 1.8 2.5 2.7 100 1.0 1.4 1.7 2.0 2.2 2.4 2.8 3.2 4.5 5.5 7.1 10.0 l/ This factor is the fourth root of column: ~i -2 etc. .J/ This factor column: is the square root of V\f / 2F etc. the number in the first the number in the first 1.3 DETERMINING THE BEST SIZE OF PLOT Most often you will not be concerned with the best size of the plots for an extensive test. Sometimes, however, it is necessary to limit the size of the plots as much as possible. For example, in a test of a new insecticide, you may not be able to get a sufficient quantity to permit large plots in the test. The problem of best plot size arises also when you are testing large things--trees, fruits, animals, or even people. You may want, then, to use the smallest plot that you can. In this section a procedure will be given to help you determine the best number of individuals, or units, to have in a plot. You recall from Part III of this Guide, page 15, that the error for an extensive test is the sum of several variability components. But, if we have a small number of individuals in a plot, the variability of these individuals also becomes important, and it, too, must be added to the other variabilities in our error calculations. You will now see how to do this. First, clearly designate the individuals, or units, that you are dealing with--individual plants and trees, for example, or short lengths of row in field crops, single hills with several plants per hill, individual animals in cattle experiments, small quadrats in pasture and forage experiments, students in a school, persons in a family, or farmers in a community. Next, obtain an estimate of the variability of these units. The valueyou seek is the variability among units receiving no experimental treatment. The procedure you follow is the same as the one we have discussed for determining plot variability except that now you are dealing with differences between individuals instead of with differ- ences between plots. The question you must answer is this, "If I took 2 units per plot at random at a number of farms in the region, what difference in yield would there be between them?" This question may be answered by judgment (Part III, pages 15-14) or from survey data (see the section on plot variability in this Part, Methods 1 and 2, pages 5-7). This question may be answered also by experimental data, but then a somewhat different procedure is used. To begin with, get measures of individual, or unit, yields in the experiment as well as yields of the whole plots. In table 3, for instance, yields are given for the plots as a whole, but these must now be supplemented with data for units. Such data can be gathered by harvesting either every unit (plant, hill, etc.) or just two units at random in 20 or so of the plots. You will probably do the latter since it is less work, and we will therefore show this procedure in detail. Let us start with the experiment shown in table 3 and assume that each plot contains 27 plants. Now, go into 20 of these plots and, from each, harvest 2 plants at random. List the yield data as in table 12. Get the difference between each pair of plants and write the differences in the last column of the table. Nov square each difference, add the squares, divide by 2 (the number of plants per plot actually measured) and multiply by 27 (the total number of plants per plot). This calculation gives the sum of squares for plants in plots: Sum of squares (plants in plots) = (1012 + 0.12 + ... + 0.212)27 2 = 57.7287 Next make the analysis of variance of data as it is shown in table 15. For plots, the values can be merely copied from the plot-variability line in table 5. For plants within plots, insert values as follows: Sum of squares = the sum of squares youhave Just now computed. Degrees of freedom = the number of differences listed in table 12. Mean square = the quotient resulting from dividing the sum of squares by the degrees of freedom. Square root of mean square = V 2.89 or 1.70. Units = the number of plants in each plot. Factor = the fourth root of the number of units, since this is a crop test (see table 11, second column). You have yet to enter the values of differences due to plots alone, which are as follows: Mean square = the difference between mean square for plots and the mean square for plants in plots = 14.9 2.89 = 12.0. Square root of mean square = /12 or 3.46. Units = 1 Factor = the fourth root of the number of units (see table 11, second column). Table 12. Yields of 2 plants selected at random from a total of 27 plants in each of 20 plots of the experiment shown in table 3. Plot Plant 1 Plant 2 Difference 0.67 1.37 0.94 1.51 1.66 1.25 1.39 1.27 1.28 1.20 1.19 1.28 1.00 1.76 1.11 1.58 1.30 1.04 1.21 0.99 1.68 1.23 1.26 0.79 1.04 1.21 0.84 1.39 1.10 1.16 1.07 1.19 1.43 1.03 1.50 1.13 1.24 1.55 1.90 0.78 1.01 0.14 0.32 0.72 0.62 0.04 0.55 0.12 0.18 0.04 0.12 0.09 0.43 0.73 0.39 0.45 0.06 0.51 0.69 0.21 Table 15. Analysis of variance of data for 20 plots and for 2 plants within each plot, with 27 plants per plot. Source Sum Degrees Mean Square root Number of of of square of of Factor variation squares freedom mean square units Plots 179 12 14.9 Plants within plots 57.7287 20 2.89 1.70 27 2.28 Difference due to plots 12.0 3.46 1 1.00 The variabilities are obtained by multiplying the square roots of the mean squares by the factors in the last column of table 13: Plant variability = 1.70 x 2.28 = 3.88 Plot variability = 3.46 x 1.00 = 3.46 These variability components are now to be expressed in percent. Simply multiply the variability by 100 and divide by the mean yield for the check plots. This mean yield is obtained from table 3, which indicates the check plots as those receiving treatment B; divide the total yield from these plots (232) by the number of plots (8). Then-- Plant variability x 100 Plant variability in percent = Mean yield (check plots) 5.88 x 100 29 = 13.4% Plot variability in percent =Plot variability x 100 Mean yield (check plots) 3.46 x 100 29 = 11.9% Now we are ready to determine the error of the extensive test. You do this by adding the variability components. You have just found the components for plants and plots; let us assume values of 46 percent and 10 percent respectively for the location and treatment components in order to illustrate the rest of the procedure. It is like the one shown in Part III, page 15, with the addition of the plant component and the factor (F) for the number of plants per plot, which you obtain from table 11. For Plan A use all four components (the various plans are given in Part III, in the Appendix): Plant variability = 13/F%; squared = (13/F)2 Plot variability = 12%; squared = 144 Location variability = 46%; squared = 2,116 Treatment variability = 10%; squared = 100 Total of the squares (15/F)2 + 2,560 Extensive-test error \ (13/F)2 + 2,560 For Plans B to H, omit the location variability: Plant variability = 13/F%; squared = (13/F)2 Plot variability = 12%; squared = 144 Treatment variability = 10%; squared = 100 Total of the squares (1 -~ 2 + 244 Extensive test error // (15/F)2 + 24 Now, set up table 14 to aid in determining the best number of units per plot. Record first the specifications of the extensive test. The first column in table 14 indicates the plan. The second column lists various numbers of units per plot, a range from 2 to 40. You can try whatever numbers you are interested in. In column 3, the units are transposed into factors by referring to table 11. Since this is an experiment with a crop, the factors in the second column of table 11 are used. For tests in which the plots are not units of land, the factors in column 3 of table 11 would be used. Column 4 is the plant variability (13 percent) divided by the factors of column 3. Column 5 is the square of column 4. Column 6 is column 5 plus the remainder of the error shown at the head of the table: 2,560 for plan A and 244 for plans B to H. Column 7 is the square root of column 6; these values are the errors for the plans. Column 8 is the difference to be tested, shown at the head of the table as 30 percent, divided by the plan errors shown in column 7. In column 9 record from Part III, table 2, the number of replications required for the values in column 8. Column 9 gives the answer you seek. For plan A note that the number of replications required is rather large and does not decrease beyond 5 plants per plot. Hence, 5 plants per plot are ample if you desire to keep the plots as small as possible. For plans B to H the number of replications required is much fewer. They decrease up to 10 plants per plot but not thereafter. You have little to gain, then, by having more than 10 plants in a plot. Once you have selected the minimum number of plants per plot--5 for plan A, and 10 for plans B to H--go back to Part III, table 3. Enter the corresponding number of replications in column 4 of that table. Then complete the rest of the summary in that table to determine the total requirements of the extensive test with the stipulated number of plants per plot. Table 14. SPECIFICATIONS Number of treatments: 7 (1 is a check) No. of plots on each farm: 1, 2, 3, or 4 Minimum difference: 30% Anticipated variability components: Plai nts, Plots, Locations, Treatments, 15% 12% 46% 10% Error (Plan A) = 6 (13/F)2 + 2,560 Error (Plans B to H) = '(13/F)2 + 244 (1) (2) (5) (4) (5) (6) (7) (8) (9) Units Plant Remainder Error % Difference Replications Plan per Factor variability Column 4 of error t error required plot Col. 5 squared + Col. 5 ( Col. 6) (Col. 7) (Part III, Table 2) A 2 1.2 10.8 117 2477 49.8 .60 60 5 1.5 8.7 76 2456 49.4 .61 58 10 1.8 7.2 52 2412 49.1 .61 58 15 2.0 6.5 42 2402 49.0 .61 58 20 2.1 6.2 58 2598 49.0 .61 58 30 2.5 5.7 32 '2595 48.9 .61 58 40 2.5 5.2 27 2587 48.9 .61 58 B to H 2 1.2 10.8 117 361 19.0 1.58 11 5 1.5 8.7 76 520 17.9 1.68 10 10 1.8 7.2 52 296 17.2 1.74 9 15 2.0 6.5 42 286 16.9 1.78 9 20 2.1 6.2 58 282 16.8 1.79 9 50 2.5 5.7 32 277 16.6 1.81 9 40 2.5 5.2 27 271 16.5 1.82 9 Example of a summary to aid in determining the minimum number of plants per plot for several designs. |