THE ESTIMATION OF THE YIELDS OF CEREAL EXPERIMENTS BY SAMPLING FOR THE RATIO OF GRAIN TO TOTAL PRODUCE

The authors suggested that, if the total produce of each plot were weighed on the field, the samples need be used only to determine the ratio of the weight of grain to the weight of tota

EXPERIMENTS BY SAMPLING FOR THE RATIO OF GRAIN TO TOTAL PRODUCE

BY W G COCHRAN

Statistical Department, Rothamsted Experimental

Station, Harpenden, Herts

1 INTRODUCTION THRESHING difficulties constitute a formidable barrier in planning an extensive series of small-plot cereal experiments of modern design, or in studying the residual effects of treatments on the succeeding cereals in

a series of experiments on root crops Failing the provision of a small threshing machine, sampling is at present the only practicable method for obtaining the grain yields of small plots located on commercial farms Yates & Zacopanay (1935) summarized the work carried out at Rothamsted and its associated outside centres on the estimation of the yields of cereal crop experiments by sampling In these experiments a number of small areas (e.g £ m of each of four contiguous rows) were selected at random in each plot or subplot The standing crop in each of these areas was cut close to the ground, bagged, and transported to Rothamsted for threshing, the yields of grain and straw per unit area being estimated entirely from the samples

The authors suggested that, if the total produce of each plot were weighed on the field, the samples need be used only to determine the ratio of the weight of grain to the weight of total produce In view of the high correlation which normally exists between grain and straw yields, the sampling errors of this ratio might be expected to be con-siderably smaller than those of the yield of grain itself, so that less sampling would be required to obtain results of equal precision They found from the average of nine experiments that the sampling error per metre of row length was 7-14% for the ratio of grain to total produce, compared with 23-9 % for the yield of grain Judging from these figures, only akmt one-tenth of the number of samples is required to obtain equal infor-mation if the total produce is weighed The estiinfor-mation of yields by this method has been tried in a number of experiments since 1935 The

W G COCHRAN 263

present paper reviews their results from the point of view of sampling technique

Since total produce, when weighed on the field, usually contains some moisture, the samples must be weighed on the field as well as before threshing, to enable a correction to be made in the grain and straw figures for the loss of moisture If the samples are taken from the standing crop, this should be done immediately before the crop is reaped,

in order that the samples and the total produce may be weighed on the field at the same time This may not always be convenient, but with this method the samples can alternatively be taken from the crop while it is lying in the stooks, since there is no need to know the area of the land from which a sample was taken As the crops usually lie in the stooks for some days, this gives a wider choice in the time during which the sampling must be carried out In most of the experiments discussed below, the samples were taken from the stooks

2 METHOD OF SAMPLING FROM THE STOOKS

Total produce is first weighed on each plot A spring balance may

be used, weighing the sheaves one at a time This method is rather tedious unless the plot size is small or the crop is a poor one, since

a 1/40 acre plot may contain thirty sheaves If a portable tripod is available with a platform which can hold all the sheaves in one stook, some time will be saved After weighing, the sheaves should be laid separately on the ground, to facilitate the sampling operations

The next step is to select the samples A method which gives reasonably random samples is as follows: Suppose that there are eighteen sheaves on a plot and that each sample is to be approximately 1 % of the produce of the plot A sheaf is first selected at random The binding tape is cut and the sheaf is divided into six portions of about equal weight One of these is selected at random and constitutes the sample The division of the sheaf is usually most quickly done by successive subdivision into halves, selecting one-half at random at each stage for further subdivision, until a sample of about the required size is reached This method also has the advantage that it reduces to a minimum the number of small bundles which are scattered about the plot For selecting the halves at random, a piece of paper bearing a selection of odd and even numbers drawn from a book of random numbers may be used; alternatively, a set of disks containing an equal number of two different colours may be carried in the pocket

When the samples have been selected, labelled and bagged, they are weighed For the calculation of the grain yields on each plot, it is

necessary to know only the total weight of the samples from the plot,

but if a full investigation of sampling errors is required, each sample must be weighed individually As the samples may weigh less than one pound each, a fairly accurate balance is required, and the weighings should if possible be done indoors whenever there is any appreciable wind The average weight of a bag, with its label and string, must also

be recorded

This completes the experimental operations on the field The sheaves should be restooked unless they are being carted off immediately The taking of random samples from the stooks is a lengthy process Following a suggestion made by Yates & Zacopanay, samples were also taken by picking a few shoots from each of several sheaves until a sample

of about the agreed size had been amassed These samples, which will be called grab samples, can be taken in about one-third of the time required for random samples, since the sheaves need not even be opened unless

they are very tightly bound It is, however, not clear a priori whether

grab samples give unbiased estimates of the grain/total produce ratios

or how they compare in accuracy with random samples In grabbing,

no attempt was made to select representative shoots, as this method is known to be likely to introduce bias One might, however, expect a tendency to miss the shorter and less vigorous shoots, and possibly also

to free the shoots of weeds in pulling them from the sheaves Both factors would tend to increase the apparent grain/total produce ratio

A comparison of the results with random and grab samples will be given later in this paper

3 MATERIAL

A list of the experiments discussed is given in Table I below The plots were not subdivided for sampling, so that the plot area given is in all cases the area to which the sampling errors apply

The random samples were taken from the stooks in five experiments The grab samples were taken from the stooks in all cases except in exp 5, where they were taken from the crop as it lay on the ground immediately after scything

W G COCHRAN 265

Table I List of experiments

Jo.

1

2

3

4

5

6

7

Year

1935

1935

1936

1936

1937

1937

1938

Place

Rothamsted

Woburn

Woburn

Rothamsted

Wye

Tunstall

Rothamsted

Type Wheat

6 x 6 L.S.*

6 x 6 L.S.

6 x 6 L.S.

Barley

4, 16 R.B.f

6 x 6 L.S.

3, 9 R.B.

Oats

4, 18 R.B.

Size of plot acres

1/40 1/100 1/100

1/40 1/120 1/40

1/60

Random samples taken from

Stooks Stooks Standing crop

Stooks Standing crop Stooks

Stooks

No of samples

per

i •

plot

* « Random Grab

2 2 2 2 2 2

3

1 1 1 1 2 2 2

Latin square.

I.e four randomized blocks of sixteen plots each.

4 SAMPLING ERRORS PER CENT PER METRE

The sampling errors per cent per metre of row length for the ratio r

of grain to total produce are shown in Table II Where the samples were taken from the stooks, the average number of metres sampled was estimated from the ratio of the weight of the sample to the weight of the whole crop on the plot The sampling errors in all cases refer to the

ratios of grain to dry total produce, as these were the figures with which

Yates & Zacopanay dealt

Table II Sampling errors of the ratio of grain to total produce

Exp.

1

3

4

5

6

7

Method of

sampling

R.

R.

R.

R.

JR.

1G.

G.

IR.

G.

Mean yield

of grain cwt per acre 32-3 29-9 20-8 251 14-7 151 5-6 33-5 33-6

Area of plot acres 1/40 1/100 1/100 1/40 1/120 1/40 1/40 Mean

Size of sampling unit metres 5-4 1-9

2 0 [5-9]

14-0

13-0

5-6 (2-6 131 3-5

Sampling error

% per metre 16-8 100 150 [29-1] (13-8 U2-5 13-7 [7-0

•3 12-6

In the first four experiments sampling errors are obtainable only for the random samples, since only one grab sample was taken per plot

In exp 6 the random samples were unfortunately bulked for threshing The sampling errors per cent per metre are considerably higher than Yates & Zacopanay's figure of 7-14 % Exp 4 may perhaps be omitted

in reaching an average figure, since 12 % of the samples were reported as damaged by mice during storage These samples were excluded from the statistical analysis, but five other samples also showed an anomalously low ratio of dry total produce to wet total produce, as well as an anomalously low grain/total produce ratio These samples might perhaps

be regarded as affected by damage which was not reported The experi-ment was, however, one in which different leys were growing under barley and the samples in question all came from plots growing a clover-ryegrass mixture, so that they may have contained a substantial amount

of the undergrowth In any case it is clear that if there is a vigorous and variable undergrowth of ley or weeds, this method is likely to give high sampling errors

Excluding exp 4, the average value for the sampling error is 12-6% per metre There are several reasons which might account, in part at least, for the higher value obtained

Size of plot

The criterion used, sampling error per cent per metre, is likely to increase as the size of the plot increases While no correlation is evident

in Table II between sampling error and size of plot, the average plot size in these experiments was considerably larger than in Yates & Zacopanay's experiments, in which, most of the sampling subplots were only 1/200 acre Since, however, Yates & Zacopanay used only a fraction

of their data for this particular calculation, some additional information

on the effect of plot size was obtained by calculating the sampling error

of r for six of their experiments in which the plots were 1/80 acre The

results are shown in Table III

Table I I I Sampling errors of the ratio of grain to total

produce (from 1/80 acre plots)

Size of Sampling error % per metre sampling unit , * , metres rf Grain

5 5-98 23-7

1 906 30-3

1 8-89 28-8

1 10-57 32-5

1 9-42 22-8

1 6-76 33-5 Mean 8-45 28-6

* In Yates & Zacopanay's notation,

•f The method by which these figures were obtained is discussed in the Appendix.

The average value, 8-45, is somewhat larger than the previous figure

of 7-14 for smaller plots, but is still considerably below 12-6 The average

Exp.*

4

7

10

10

11

11

Crop Barley Barley Wheat Barley Wheat Barley

W G COCHRAN 267

sampling error for the yields of grain in the same experiments was 28-6%, so that the relative efficiency of the two methods works out at almost the same figure as Yates & Zacopanay obtained It does not appear as if the difference in the size of the plots can account for more than a small part of the increase from 7-1 to 12-6 %

Size and type of sampling unit The sampling error of r will also depend to some extent on the size

and shape of the sampling unit As a rule, it is to be expected that for the same total percentage sampled, a few large sampling units will be less efficient than a larger number of small sampling units In the present experiments the average size of the sampling unit was 3-5 m as against 2-0 m in Yates & Zacopanay's experiments, and this difference might partly account for the higher sampling error In this connexion

it would have been instructive to compare the variation in r between samples taken from the same sheaf with that between samples taken from different sheaves, but this is not possible from the way in which the samples were selected It is also possible that the reaper or scythe gives a less even cut than is obtained when small samples are cut by hand from the standing crop

Presence of weeds or undergrowth

This point has already been mentioned in discussing exp 4 in Table II, but it applies, to a less extent, to all experiments In Yates & Zacopanay's experiments, the samples were cleared of weeds before determining the weights of grain and straw, whereas in sampling for the grain/total produce ratio it is essential that the sample should not be cleaned of weeds Thus the presence of weeds, from which few experiments are entirely free, adds to the variability of r, particularly so as weeds compete with the crop and are more likely to abound in poorer patches, where the value of r is already low

5 THE COEEECTION FOE LOSS OF MOISTUEE

No discussion has so far been given for the correction which must be made for the amount of moisture in the total produce as weighed on the field Since this correction is made from the samples, it will involve some loss of information, so that the sampling errors given in the preceding

section for the ratio of grain to dry total produce do not represent the

whole of the sampling error involved in this method

The yield of dry grain of any plot is most simply obtained by multiplying the yield of wet total produce by the ratio, in samples from that plot, of the total yield of dry grain to the total yield of wet total produce The percentage sampling variance per plot of the yield of dry grain will be given (with all necessary accuracy) by the percentage sampling variance of the ratio of dry grain to wet total produce, divided by the number of samples taken per plot This can be calculated if'the samples were weighed individually on the field and threshed individually

Since the sampling errors of the ratio of dry grain to dry total produce have already been discussed, it will be more convenient to discuss here the sampling errors of the ratio of dry total produce to wet total produce, assuming these ratios to be independent In general, however, the more direct approach is preferable, since the assumption

of independence is not likely always to hold

Unfortunately, little evidence on the dry/wet ratio is obtainable from these experiments The samples were weighed individually on the field

in only three experiments, nos 3, 4 and 7, mainly because the accuracy

of the spring balance and the external conditions did not appear to justify weighing each sample Of these experiments, no 4 has already been noted as exceptionally variable, while in no 7 there appears to have been a zero error in the spring balance, since almost all the dry weights of total produce were slightly higher than the wet weights

In exp 3, the sampling error per cent per plot for the ratio of dry to wet total produce was 7-03, as compared with 7-50 for the ratio r of

dry grain to dry total produce The corresponding figures in exp 4, omitting the plots undersown with the clover-ryegrass mixture, were 7-45 and 8-50 These figures suggest that almost as much information is being lost in estimating the correction for drying as in estimating the ratio of dry grain to dry total produce If this is true, the accuracy of the method is only half that indicated by the figures in the last section There is, however, reason to believe that these results are not repre-sentative, since rain fell during the sampling of exp 3, some samples being wet when weighed, and in both experiments there was an unusual amount of drying-out, the mean values of the ratio of dry to wet total produce being 0-628 and 0-673 respectively

For the remaining experiments, the experimental error between plots for the ratio of dry to wet total produce of the samples may be used as

an upper limit to the corresponding sampling error within plots It may

be mentioned that in exp 3, the sampling variance of the dry/wet ratio

W G COCHRAN 269

was practically equal to the experimental variance, though there were significant differences between rows, columns and treatments, while in exp 4 the sampling variance was less than half the experimental

variance The results per plot for the other experiments are shown in

Table IV

Table IV

Exp.

1

2

5

6

Experimental

of dry to

Mean ratic dry/wet 0-849 0-707 0-878 0-859

errors per cent per wet total produce

Experimental

i error %

of dry/wet 2-91 4-44 2-26 2-59

plot of the ratio

Sampling error %

o f / 5-17 5-17 4-87 4-07

In exps 1, 5 and 6 the percentage sampling variance of the dry/wet ratio cannot exceed about one-third of the percentage sampling variance

of r, and may be substantially less In exp 2, in which the amount of drying was much greater, the additional loss of information was probably also greater

If the dry/wet ratios are very variable the question arises whether the use of some average correction figure will improve matters Clearly such an average can only be properly employed if the dry/wet ratios are unaffected by the treatments, for if they are so affected the use of an average will distort the treatment differences Actually four of the six

experiments considered showed clear differences between treatments,

and exp 3 also falls into this category if the clover-ryegrass plots are included The use of an average correction figure is therefore inadvisable Such distortion can of course be avoided by using a separate correction factor for each treatment, based on the average dry/wet ratio for all replicates of that treatment There is no point in following this course, however, since the results will be almost the same as if each plot is corrected separately The only effect will be to give a spuriously low estimate of experimental error

6 COMPARISON OF GRAB WITH RANDOM SAMPLES

Direct comparison of the sampling errors of r for random and grab

samples can be made in only two of the experiments in Table II, nos 5 and 7

In exp 5 grab sampling was somewhat more accurate, though not significantly so, while in exp 7 there was little to choose between the two methods

A less direct comparison may be obtained by calculating the between-plot errors of the yields of grain given by the two methods (after elimination of treatment and block effects) Some allowance must be made for the difference in the amounts which were sampled under the two methods In exp 1, for example, two random samples each of about

954 g total produce were taken, as against one grab sample of 794 g

The sampling and experimental errors per cent per plot for the random

samples were 5-15 and 8-67 respectively The estimated experimental error per cent, if only one random sample of 794 g had been taken is

and this figure is comparable with the experimental error per cent for grab sampling The adjustment for the size of the individual sample in the above formula is open to question, since a sample of twice the size,

taken from the same sheaf j would probably not be twice as accurate.

Since the grab samples were usually the larger, the adjustment possibly favours the random samples slightly

Table V Experimental errors per cent per plot of the

yields of grain

Method of sampling Exp.

1 2 3 4 5 6 7 Mean

Random 10-8 7-6 15-5 17-2 8-3 14-7 6-8 11-6

Grab 7-9 10-4 12-8 171 9-3 140 6-9 11-2 Random samples gave a smaller experimental error in two experi-ments, grab samples in three, while the remaining two experiments showed practically identical results Thus grab sampling appears to be

no less accurate than random sampling

The mean yields of grain obtained by random and grab sampling are shown in Table VI The right-hand column shows the difference between the yields from grab and random sampling as a percentage of the yield given by random sampling

Except in exp 4 the grab samples gave slightly higher yields of grain than the random samples The biases are, however, in no case large Both random and grab sampling gave a positive bias in yields as

1

2

3

4

5

6

7

Crop

Wheat

Wheat Wheat Barley Barley Barley Oats

Grain:

Full harvesting 30-59 18-57

—

—

—

cwt per acre Random sampling 32-36 29-88 20-83 25-07 14-74 5-44 33-50

Grab sampling 34-33 31-55 21-45 23-95 15-14 5-57 33-60

W G COCHRAN 271

compared with full harvesting in exps 1 and 3 In exp 3 the difference

is due almost entirely to a greater drying out of the total produce than

of the samples

Table VI Comparison of mean yields by random

and grab sampling

% bias

in grab + 6 1 + 5-6 + 3 0 -4-5 + 2-7 + 2-4 + 0-3

A more detailed examination of the treatment means in these experiments shows close agreement between results from random and grab sampling

7 DISCUSSION OF RESULTS

Owing to the uncertainty about the sampling variance of the ratio

of dry to wet total produce, the total sampling error involved in sampling for the ratio of grain to total produce cannot be fixed definitely for these experiments An average figure of 14-5 % per metre of row length for the ratio of dry grain to •wet total produce is probably not far wrong (This represents an increase in the average sampling variance in Table II

by one-third to allow for the sampling variance of the dry/wet ratio.) With this figure, a sample of 25 m per plot gives a sampling error of 2-9 % per plot With an experimental error of 7-5 % per plot, the loss

of information is about 13 %, i.e an amount which could be more than offset by adding an extra replication to an experiment with between four and seven replications This amount of sampling represents about 5 %

of the total produce in a 1/40 acre plot

This figure is subject to qualification according to the conditions of the experiment If the crop is fairly dry and free from weeds or under-growth when it is being sampled, or if the plot size is only 1/100 acre, some reduction may perhaps be allowed in the number of metres sampled, though it would be advisable to collect more experimental data on this point

The size of the samples taken in these experiments was probably too large It might be better to take not more than 2 m of row length for Journ Agric Sci xxx 18