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A simple nonparametric method for evaluating the results of half-leaf local-lesion assays
\‘IROLO(:Y
4, 130-131
(l%T)
A Simple Nonparametric Results of Half-Leaf
Method Local-Lesion
for
Evaluating Assays’
the
ROGER CT. HAKT~ ANU GLADYS I<. PERE~-MEN~EZ
The statistical analysis of half-leaf local-lesion assays of plant viruses is hampered by the fact that the lesions per half leaf are not normally distributed and vary greatly from leaf to leaf on the same plant, from plant to plant, and from day to day. Kleczkowski (1949) has been able to transform the lesion number per half leaf into another variable which, by the proper choice of one arbitrary parameter, can be adjusted so as to be normally distributed within a given assay. The t’ransformed lesion numbers can then be treated by ordinary statistical methods. Where there are not sufficient, data to establish the validity of such a t,ransformation, and in the particular cast where just two inocula are compared on opposite halves of several leaves, a simple nonparametric method used in this laboratory has been found to provide a satisfactory analysis of the result,s. Our rnethod will be illustrated here with a typical set’ of assay data. Then, for a larger set of data, the results obtained hy our method will be compared wit’h those obtained by applying an ordinary small-sample method to the same lesion numbers after transformation into normally distributed variables by Kleczkomski’s method. An assay was rnade t,o det’ermine what fraction of the originally infectious particles in a sample of tobacco mosaic virus had survived an X-ray treatmentS. The irradiated sample, I, was inoculated at three times the concentration of C, the unirradiated cont,rol. They were applied to opposite halves of twelve leaves of a variety of Nicotiana tubacalm L. having the .V. ylutinosa L. gene for local lesions (Holmes, 19):B). The lesion counts for the individual half leaves are shown in Table 1, L Aided 2 United Institute.
in part, by a grant) from the United States Public Healt,h Service. States Public: Health Service Research Fellow of the Natjion:tl Cancer
EVALUATING
HALF-LEAF
LOCAL-LESION
TABLE LESION
NUMBERS Leaf:
I Lesions C Lesions 1:C lesion ratio Ratio rank
AND 1
4 4 1.00 9
LESION
2
3
15 5 11 4 1.361.25 4
RATIOS
7
4
1 IN
5
72 54 1.33
5 7 0.71
6
11
131
ASSAYS
A HALF-LEAF 6
7
41 13 18 44 2.28 0.30 1
12
LOCAL-LESION 8
9 ______
21 14 20 16 1.05 0.875 8
10
ASSAY
10
11
24 20 15 10 1.60 2.00 3
2
12
15 11 1.36 5
together with the ratios of lesions produced by I t’o those produred by C on each of the leaves. The twelve ratios may be regarded as a random sample chosen from the population of ratios which would be obtained if a large number of similar such leaves were similarly inoculated. With no a priori knowledge or assumptions as to how this ratio would be distributed over the leaf population, its median value can be estimated from the given sample (Table 1) in the following way: From the definition of a median, it follows that any leaf chosen at random is equally likely to have an I: C lesion ratio above or below the median value. The probabiliby of obtaining twelve values all of which lie above the population median is (l/2)lz = l/4096. The hypothesis that the population median is below the entire range of sample values (below 0.30 for the data of Table 1) can therefore be rejected with 99.975 % confidence. By similar reasoning (Mood, 1950)) the probability that nine or more of the twelve sample values lie above the median, or that such a majority lie below it,, can be calculated by the binomial distribution :
The population median ran therefore be said to lie between the fourth lowest (1 .OO) and fourth highest (1.X) sample values with 85 % confidence. The I: C lesion ratio is then estimated to be 1.18 & 0.18. If allowance is made for the different concentrations of the inocula, and if the infectivity is assumed to be proportional to the lesion number, the fraction of virus particles surviving irradiation may be given as 0.39 f 0.06.3 3 The interpretation of this assay has been simplified by the avoidance of SAVera1 important experimental considerations. First, there is not, in general, a proportionality between the concentration of act,ive rxlrtirlps in an inocnlum and
132
HART
AND
PEREZ-MENDEZ
TABLE
2
ESTIMATDS MADE BY A NONPARAMETRIC AND A PARAMETRIC METHOD FOR THE RATIO OF THE LESIONS PRODUCED BY Two IDENTICAL INOCULA ON OPPOSITE HALF LEAVES IN 28 Assays EACH EMPLOYING 12 LEAVES. 85% CONFIDENCE LIMITS ARE GIVEN FOR EACH ESTIMATE Assay
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Parametric
Nonparametric
1.01 1.03 1.15 1.03 1.05 1.22 0.98 0.91 1.06 1.11 0.91 0.93 1.02 0.89 1.08 0.88 1.20 1.18 0.98 0.96
f f f f f f f f f f f f f f f f f f f f
0.14 0.24 0.32 0.14 0.20 0.25 0.10 0.19 0.18 0.44 0.12 0.19 0.23 0.05 0.21 0.15 0.37 0.26 0.24 0.14
0.98 1.00 1.09 1.06 1.00 1.18 0.96 0.90 0.97 1.13 0.91 0.93 1.07 0.88 1.04 0.91 1.19 1.06 1.01 0.94 MWI
20 Nonparametric 20 Parametric
estimates estimates
1.03 1.01
f f f f f f f f f f f f f f f f f f f f
0.13 0.15 0.14 0.11 0.14 0.17 0.11 0.16 0.17 0.25 0.07 0.11 0.15 0.08 0.14 0.13 0.16 0.18 0.16 0.10 Standard deviation
0.10 0.09
For a comparison, our method and a standard small-sample method have been applied to bhe lesion numbers obtained by Nleczkowski on 480 bean half leaves inoculated with a solution of tobacco mosaic virus. Each of the 20 rows of the table in which those data were presented has been treated as a separate assay in which the identical inocula A and B the number of lesions produced per half leaf, though such a proportionality can usually be assumed as an approximation over a limited favorable range of lesion numbers per half leaf. Additional factors which should be considered are the possible interference of inact,ivated particles in the assay of active particles surviving irradiation and the possibility of sublethal irradiation damage to virus particles.
EVALUATING
HALF-LEAF
LOCAL-LESION
133
ASSAYS
are compared on opposite halves of twelve leaves. Alternate have been assigned to A and B according to the scheme: A-for B-for
half leaves
even values of (row number + column number); odd values of (row number + column number).
The A:B lesion ratios computed by the method illustrated in the above example are compared in Table 2 with the corresponding estimates of these ratios made in the following way: The lesion numbers for the individual half leaves were transformed by Kleczkowski’s method (i.e., by adding the number 40 to each and taking the logarithm of each such sum to the base 10). For each of the twelve leaves in an assay, the transformed lesion number for B was subtracted from that for A. The mean value and standard deviation were then computed for the twelve differences and these were used to determine the 85 % confidence limits for the mean from Student’s 1 distribution for 11 degrees of freedom. The antilogarithms of these limits were used to estimate the corresponding confidence limits in the A:B lesion ratio by the semiempirical formula: “A” “B”
_ 1 4 1 _ 40 + “A” lesions = lesions 1 4 40 + “B”
lesions lesions
*
This expression, which reduces to a simple identity when “B” lesions = 100, has been plotted on a scatter diagram of Kleczkowski’s data (mean number of lesions per half leaf = 102.3) and found to provide a good fit for A: B lesion ratios between 0.7 and I .4. The essential difference between the parametric and nonparametric methods was reflected in the spread of the confidence limits calculated according to each. From the relative sizes of the intervals listed in Table 2, it may be estimated that nearly thirty leaves would be required in an assay evaluated by the nonparametric method in order to assure the same degree of precision obtained by the parametric method with only twelve leaves. It will be noted, however, that the group of twenty nonparametric estimates is distributed with nearly the same mean and standard deviation as is the parametric group. This finding indicates that the two methods may determine, in fact, equally accurate values for the A:B lesion ratio, although the nonparametric method has the disadvantage of providing overly conservative evaluations of the confidence intervals. The parametric method may be expected to utilize t,he dat,a more efficiently in all assays, such as these, where Kleczkowski’s
I 34
HART
AND
PEREZ-MENDEZ
transformation can be shown to apply; relation to a normal distribution is not metric method may be invalid, and any of such a method over the nonparametric
however, in assays where the established, the use of a paraapparent advantage in precision met’hod may be fallacious.
REFERESCES 0. (1938). Inheritance of resistanre to t,ohacro-mosaic disease in t obarco. Ph&pnthology 28, 553-561. KLECZKOWSKI. A. (1949). The transformat,ion of local lesion counts for st,ot.isticnl analysis. Ann. ilpp!. Rid. 36, 139-152. Moon, A. M. (1050). I~~1wdw/ion (o the 7’hrouy Nf Slafislics. McCrawHill, Xew York. HOIXES,
F.
4, 130-131
(l%T)
A Simple Nonparametric Results of Half-Leaf
Method Local-Lesion
for
Evaluating Assays’
the
ROGER CT. HAKT~ ANU GLADYS I<. PERE~-MEN~EZ
The statistical analysis of half-leaf local-lesion assays of plant viruses is hampered by the fact that the lesions per half leaf are not normally distributed and vary greatly from leaf to leaf on the same plant, from plant to plant, and from day to day. Kleczkowski (1949) has been able to transform the lesion number per half leaf into another variable which, by the proper choice of one arbitrary parameter, can be adjusted so as to be normally distributed within a given assay. The t’ransformed lesion numbers can then be treated by ordinary statistical methods. Where there are not sufficient, data to establish the validity of such a t,ransformation, and in the particular cast where just two inocula are compared on opposite halves of several leaves, a simple nonparametric method used in this laboratory has been found to provide a satisfactory analysis of the result,s. Our rnethod will be illustrated here with a typical set’ of assay data. Then, for a larger set of data, the results obtained hy our method will be compared wit’h those obtained by applying an ordinary small-sample method to the same lesion numbers after transformation into normally distributed variables by Kleczkomski’s method. An assay was rnade t,o det’ermine what fraction of the originally infectious particles in a sample of tobacco mosaic virus had survived an X-ray treatmentS. The irradiated sample, I, was inoculated at three times the concentration of C, the unirradiated cont,rol. They were applied to opposite halves of twelve leaves of a variety of Nicotiana tubacalm L. having the .V. ylutinosa L. gene for local lesions (Holmes, 19):B). The lesion counts for the individual half leaves are shown in Table 1, L Aided 2 United Institute.
in part, by a grant) from the United States Public Healt,h Service. States Public: Health Service Research Fellow of the Natjion:tl Cancer
EVALUATING
HALF-LEAF
LOCAL-LESION
TABLE LESION
NUMBERS Leaf:
I Lesions C Lesions 1:C lesion ratio Ratio rank
AND 1
4 4 1.00 9
LESION
2
3
15 5 11 4 1.361.25 4
RATIOS
7
4
1 IN
5
72 54 1.33
5 7 0.71
6
11
131
ASSAYS
A HALF-LEAF 6
7
41 13 18 44 2.28 0.30 1
12
LOCAL-LESION 8
9 ______
21 14 20 16 1.05 0.875 8
10
ASSAY
10
11
24 20 15 10 1.60 2.00 3
2
12
15 11 1.36 5
together with the ratios of lesions produced by I t’o those produred by C on each of the leaves. The twelve ratios may be regarded as a random sample chosen from the population of ratios which would be obtained if a large number of similar such leaves were similarly inoculated. With no a priori knowledge or assumptions as to how this ratio would be distributed over the leaf population, its median value can be estimated from the given sample (Table 1) in the following way: From the definition of a median, it follows that any leaf chosen at random is equally likely to have an I: C lesion ratio above or below the median value. The probabiliby of obtaining twelve values all of which lie above the population median is (l/2)lz = l/4096. The hypothesis that the population median is below the entire range of sample values (below 0.30 for the data of Table 1) can therefore be rejected with 99.975 % confidence. By similar reasoning (Mood, 1950)) the probability that nine or more of the twelve sample values lie above the median, or that such a majority lie below it,, can be calculated by the binomial distribution :
The population median ran therefore be said to lie between the fourth lowest (1 .OO) and fourth highest (1.X) sample values with 85 % confidence. The I: C lesion ratio is then estimated to be 1.18 & 0.18. If allowance is made for the different concentrations of the inocula, and if the infectivity is assumed to be proportional to the lesion number, the fraction of virus particles surviving irradiation may be given as 0.39 f 0.06.3 3 The interpretation of this assay has been simplified by the avoidance of SAVera1 important experimental considerations. First, there is not, in general, a proportionality between the concentration of act,ive rxlrtirlps in an inocnlum and
132
HART
AND
PEREZ-MENDEZ
TABLE
2
ESTIMATDS MADE BY A NONPARAMETRIC AND A PARAMETRIC METHOD FOR THE RATIO OF THE LESIONS PRODUCED BY Two IDENTICAL INOCULA ON OPPOSITE HALF LEAVES IN 28 Assays EACH EMPLOYING 12 LEAVES. 85% CONFIDENCE LIMITS ARE GIVEN FOR EACH ESTIMATE Assay
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Parametric
Nonparametric
1.01 1.03 1.15 1.03 1.05 1.22 0.98 0.91 1.06 1.11 0.91 0.93 1.02 0.89 1.08 0.88 1.20 1.18 0.98 0.96
f f f f f f f f f f f f f f f f f f f f
0.14 0.24 0.32 0.14 0.20 0.25 0.10 0.19 0.18 0.44 0.12 0.19 0.23 0.05 0.21 0.15 0.37 0.26 0.24 0.14
0.98 1.00 1.09 1.06 1.00 1.18 0.96 0.90 0.97 1.13 0.91 0.93 1.07 0.88 1.04 0.91 1.19 1.06 1.01 0.94 MWI
20 Nonparametric 20 Parametric
estimates estimates
1.03 1.01
f f f f f f f f f f f f f f f f f f f f
0.13 0.15 0.14 0.11 0.14 0.17 0.11 0.16 0.17 0.25 0.07 0.11 0.15 0.08 0.14 0.13 0.16 0.18 0.16 0.10 Standard deviation
0.10 0.09
For a comparison, our method and a standard small-sample method have been applied to bhe lesion numbers obtained by Nleczkowski on 480 bean half leaves inoculated with a solution of tobacco mosaic virus. Each of the 20 rows of the table in which those data were presented has been treated as a separate assay in which the identical inocula A and B the number of lesions produced per half leaf, though such a proportionality can usually be assumed as an approximation over a limited favorable range of lesion numbers per half leaf. Additional factors which should be considered are the possible interference of inact,ivated particles in the assay of active particles surviving irradiation and the possibility of sublethal irradiation damage to virus particles.
EVALUATING
HALF-LEAF
LOCAL-LESION
133
ASSAYS
are compared on opposite halves of twelve leaves. Alternate have been assigned to A and B according to the scheme: A-for B-for
half leaves
even values of (row number + column number); odd values of (row number + column number).
The A:B lesion ratios computed by the method illustrated in the above example are compared in Table 2 with the corresponding estimates of these ratios made in the following way: The lesion numbers for the individual half leaves were transformed by Kleczkowski’s method (i.e., by adding the number 40 to each and taking the logarithm of each such sum to the base 10). For each of the twelve leaves in an assay, the transformed lesion number for B was subtracted from that for A. The mean value and standard deviation were then computed for the twelve differences and these were used to determine the 85 % confidence limits for the mean from Student’s 1 distribution for 11 degrees of freedom. The antilogarithms of these limits were used to estimate the corresponding confidence limits in the A:B lesion ratio by the semiempirical formula: “A” “B”
_ 1 4 1 _ 40 + “A” lesions = lesions 1 4 40 + “B”
lesions lesions
*
This expression, which reduces to a simple identity when “B” lesions = 100, has been plotted on a scatter diagram of Kleczkowski’s data (mean number of lesions per half leaf = 102.3) and found to provide a good fit for A: B lesion ratios between 0.7 and I .4. The essential difference between the parametric and nonparametric methods was reflected in the spread of the confidence limits calculated according to each. From the relative sizes of the intervals listed in Table 2, it may be estimated that nearly thirty leaves would be required in an assay evaluated by the nonparametric method in order to assure the same degree of precision obtained by the parametric method with only twelve leaves. It will be noted, however, that the group of twenty nonparametric estimates is distributed with nearly the same mean and standard deviation as is the parametric group. This finding indicates that the two methods may determine, in fact, equally accurate values for the A:B lesion ratio, although the nonparametric method has the disadvantage of providing overly conservative evaluations of the confidence intervals. The parametric method may be expected to utilize t,he dat,a more efficiently in all assays, such as these, where Kleczkowski’s
I 34
HART
AND
PEREZ-MENDEZ
transformation can be shown to apply; relation to a normal distribution is not metric method may be invalid, and any of such a method over the nonparametric
however, in assays where the established, the use of a paraapparent advantage in precision met’hod may be fallacious.
REFERESCES 0. (1938). Inheritance of resistanre to t,ohacro-mosaic disease in t obarco. Ph&pnthology 28, 553-561. KLECZKOWSKI. A. (1949). The transformat,ion of local lesion counts for st,ot.isticnl analysis. Ann. ilpp!. Rid. 36, 139-152. Moon, A. M. (1050). I~~1wdw/ion (o the 7’hrouy Nf Slafislics. McCrawHill, Xew York. HOIXES,
F.