The Reverse Cumulative Distribution Plot: A Graphic Method for Exploratory Analysis of Antibody Data

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REFERENCES

1. Ad Hoc Group for the Study of Pertussis Vaccines. Placebo-controlled

trial of two acellular pertussis vaccines in Sweden-protective efficacy and adverse events. Lancet. 1988;1:955-960

2. Edwards KM, Meade BD, Decker MD, et al. Comparison of 13 acellular

pertussis vaccines: overview and serologic response. Pediatrics. 1995; 96(suppl):548-557

3. Blumberg DA., Mink CM, Cherry JD, et al. Comparison of acellular and

whole-cell pertussis-component diphthena-tetanus-pertussis vaccines

in infants. JPediatr. 1991;119:194-204

4. Granstr#{246}m M, Ferngren H, Linde A, GranstrOm G. lgG subclass responses

to Bordetella pertussis filamentous haemagglutinin and pertussis toxin in

whooping cough. Serodiagn Immunother Infect Dis. 19893:403-412

5. Wong KH, Skelton 5K. New, practical approach to detecting antibody to

pertussis toxin for public health and clinical laboratories. ICliti Micro-biol. 1988;26:1316-1320

6. Mink CM, O’Brien CH, Wassilak 5, Deforest A, Meade BD. Isotype and

antigen specificity of pertussis agglutinins following whole-cell pertus-sis vaccination and infection with Bordetella pertussis. Infect Immun. 1994;62:1I 18-1120

7. Medical Research Council. Vaccination against whooping cough:

rela-tion between protection in children and results of laboratory tests. Br Med J.1956;2:454-462

8. Sako W. Studies on pertussis immunization. IPediatr. 1947;30:29-40

9. Miller ii Jr. Silverberg RJ, Saito TM, Humber JB. An agglutinative

reaction for Heinophilus pertussis. II. Its relation to clinical immunity. Pediatr. 1943;22:644-651

10. Cowell JL, Zhang JM, Urisu A, et al. Purification and characterization of

serotype 6 fimbriae from Bordetella pertussis and comparison of their

properties with serotype 2 fimbriae. Infect Immun. 1987;55:916-922

11. Li ZM, Brennan MJ, David JL, Carter PH, Cowell JL, Manclark CR.

Comparison of type 2 and type 6 fimbriae of Bordetella pertussis by using agglutinating monoclonal antibodies. Infect Immun. 1988;56:3184-3188

12. Ashworth LAE, Irons LI, Dowsett AB. Antigenic relationship between

serotype-specific agglutinogen and fimbriae of Bordetella pertussis. Infect

Immun. 1982;37:1278-1281

13. Zhang JM, Cowell jL, Steven AC, Carter PH, McGrath PP. Manclark CR. Purification and characterization of fimbriae isolated from Bordetella pertussis. Infect Immun. 1985;48:422-427

14. Mooi FR. Bordetella pertussis fimbriae. In: Klemm P. ed. Fimliriae: Adhesion,

Genetics, Biogenesis and Vaccines. Boca Raton, FL: CRC Press; 1994:115-126 15. Brennan MJ, Li ZM, Cowell JL, et al. Identification of a 69-kilodalton

nonfimbrial protein as an agglutinogen of Bordetella pertussis. Infect

Immun. 1988;56:3189-3195

16. Li ZM, Cowell JL, Brennan MJ, Burns DL, Manclark CR. Agglutinating monoclonal antibodies that specifically recognize lipooligosaccharide A of Bordetella pertussis. Infect Immun. 1988;56:699 -702

17. Edwards KM. Bradley RB, Decker MD, et al. Evaluation of a new highly

purified pertussis vaccine in infants and children. IInfect Dis. 1989;160:

832-837

18. Pichichero ME, Francis AB, Blatter MM, et al. Acellular pertussis vac-cination of 2-month-old infants in the United States. Pediatrics. 1992;89:

882-887

19. Englund JA, Decker MD, Edwards KM. Pichichero ME, Steinhoff MC,

Anderson EL. Acellular and whole-cell pertussis vaccines as booster

doses: a multicenter study. Pediatrics. 1994;93:37-43

20. Blumberg DA, Pineda E, Cherry JD, Caruso A, Scott JV. The agglutinin response to whole-cell and acellular pertussis vaccines is Bordetella

pertussis-strain dependent. Am JDis Child. 1992;146:1148-1150

21. Demina AA, Devyatkina NP, Voloshina LZ, et al. Serological types of

whooping cough bacteria and their connection with immunity in

vac-cinated children. JHyg Epidemiol Microbiol Immunol. 1973;17:304-315 22. Manclark CR, Meade BD, Burstyn DG. Serological response to Bordetella

pertussis. In: Rose NR, Friedman H, Fahey JL, eds. Manual of Clinical

Laboratory Immunology. 3rd ed. Washington, DC: American Society for

Microbiology; 1986:388-394

23. Meade BD, Deforest A, Edwards KM. et al. Description and evaluation

of serologic assays used in a multicenter trial of acellular pertussis vaccines. Pediatrics. 1995;96(suppl):570-575

24. Gillenius P. Jaatmaa E, Askelof P. Granstr#{228}m M, Tiru M. The

standard-ization of an assay for pertussis toxin and antitoxin in microplate

culture of Chinese hamster ovary cells. I Biol Stand. 1985;13:61-66

25. Podda A, Nencioni L, DeMagistris MT. et a!. Metabolic, humoral, and

cellular responses in adult volunteers immunized with the genetically

inactivated pertussis toxin mutant PT-9K/129G. I Exp Med. 1990;172:

861-868

26. Edwards KM, Decker MD, Bradley RB, Taylor JC, Hager CC. Booster

response to acellular pertussis vaccine in children primed with acellular or whole-cell vaccines. Pediatr Infect Dis J.1991;10:315-318

The

Reverse

Cumulative

Distribution

Plot:

A Graphic

Method

for

Exploratory

Analysis

of

Antibody

Data

George

F. Reed,

PhD*;

Bruce

D.

Meade,

PhD;

and

Mark

C.

Stemhoff,

MD

ABSTRACT. Serologic data often have a wide range

and commonly do not approximate a normal distribution. Means, medians, SDs, or other conventional numerical summaries of antibody data may not adequately or fully describe these complex data. The reverse cumulative

dis-tribution plot is a graphic tool that completely displays

all the data, allows a rapid visual assessment of impor-tant details of the distribution, and simplifies compari-son of distributions. Pediatrics 1995;96:600-603;

anti-body, cumulative distribution curve, plot, vaccine.

From the Division of Microbiology and Infectious Diseases, National

In-stitute of Allergy and Infectious Diseases, Bethesda MD; Division of

Bac-terial Products, Center for Biologics Evaluation and Research, Food and

Drug Administration, Rockville MD; and the §Departments of International

Health and Pediatrics, Johns Hopkins University, Baltimore, MD.

Reprint requests to (G.F.R.) Division of Microbiology and Infectious

Dis-eases, National Institute of Allergy and Infectious Diseases, 6003 Executive

Blvd, MSC 7630, Bethesda, MD 20892-7630.

Acad-emy of Pediatrics.

ABBREVIATIONS. GMI, geometric mean titer or concentration;

RCD, reverse cumulative distribution; MLD, minimum level of

detection; PT, pertussis toxin.

Serum antibody levels after immunization may range in value

from virtually zero to levels that are orders of magnitude higher

than those in the middle of the range. Antibody data are

conven-tionally summarized by the geometric mean titer or concentration

(GMI) and a confidence interval. These conventional descriptors

are not always capable of accurately describing the wide

variabil-ity and skewness commonly observed in serum antibody

distributions.

The usual approach for data summary is based on a logarithmic

transformation of the data, which diminishes the range of the

values and reduces the influence of the few very large values that

may skew the distribution to the right. If the transformation is

successful in producing a symmetric and near-normal

distribu-tion, then the mean and SD are sufficient numeric descriptors of

the transformed data. They can be used to construct confidence

limits on the mean, and the antilogarithms of the mean and its

confidence limits are the GMT and its confidence limits.’

How-ever, the transformed data frequently are not well fitted to a

normal distribution, so that further descriptive measures are

needed.

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Alternative or supplemental descriptors of antibody

distribu-lions include percentiles, such as the median for estimating central

tendency, or the 90th percentile to indicate an upper limit on the

majority of values; these particular descriptors are not widely

used in current publications. Many authors report the percentage

of observed values that exceed a specified antibody level, such as

a known protective level. Siber and Ransil’ have advocated the

display of preimmunization and postimmunization data points

simultaneously, to allow inspection of all the data.

Some of these alternate descriptors are used to describe the

change in antibody concentration. The fold rise is the ratio of the

postimmunization to preimmunization antibody level and often is

logarithmically transformed for statistical analysis because of

skewness. Another statistic is the number of postvaccination

val-ues that exceed their corresponding prevaccination value by more

than a specified relative amount, usually twofold or fourfold.

Alone, none of the above approaches is capable of indicating

important departures from normality of antibody data such as left

or right skewness, the presence of high or low outliers, a bimodal

distribution (more than one peak), or unusual kurtosis (flatness or

peakedness). In fact, for a comprehensive understanding of a set of

antibody values, it is necessary to view all the data as a whole, not

just summary measures. Graphic methods are well suited for this

purpose;2 given the ready access afforded by modern computer

technology to the simple generation of graphics, such methods

deserve wider use.

We have found a particular graphic technique especially useful

for displaying antibody data. The technique allows inspection of

variability and central tendencies of antibody data and permits

comparison of the proportion of specimens with high values. We

independently developed this graphic technique to compare

anti-body distributions arising from a multicenter trial of 13 acellulam

and two whole-cell pertussis vaccines.3 We since have learned that

a similar technique was used by Dr Jonas Salk in 1981 to display

distributions of antipolio antibodies.4 Salk subsequently used this

technique effectively to compare variations in the dosage of oral

and inactivated polio vaccines and in the interval between doses

and to evaluate the response to booster doses compared with

infection.6 However, we are not aware of other investigators

using this technique to describe antibody data and therefore offer

the following description.

METHODS

Construction and Description of the Graph

The horizontal axis of the graph represents antibody levels in a

logarithmic scale, and the vertical axis represents the percent of

subjects having at least that level of antibody, ranging from 0% to

100%. The graph is constructed by plotting against the vertical axis

the percentage of subjects having an antibody concentration equal

to or greater than the level shown at each point along the

hori-zontal axis (Fig 1). Consecutive plotted points are then connected

by a line. The plot thus created reverses the approach of the

well-known cumulative distribution graph,7 which plots a value

against the percentage of equal or lesser values. For this reason we

refer to our curve as the reverse cumulative distribution (RCD)

curve. A similar technique is used to generate survival curves,

which substitute survival time for antibody level on the horizontal

scale.8

The first plotted point will represent the lowest antibody level

observed for any subject (and thus would align at, or close to, the

origin of the x axis). All values, by definition, are at least as large

as this value; thus, the curve begins at 100% (the top of the y axis).

From left to right the curve then descends from 100% toward 0%

in a series of descents of varying slope (Fig 2). The rightmost point

of the curve will correspond to the highest observed value (toward

the right end of the x axis), and will be close to-but never quite

reach-0% (because at least one observation has this highest

value).

RESULTS

Approximation of Points and Parameters of the

Distribution

Every individual result is represented in the graph, and it is

possible to visually assess a value’s rank among all others.

Fur-thermore, it is possible to approximate by inspection any

percen-tile of the distribution of values. For example, to approximate the

90th percentile, which is the value that 90% of responses are equal

to or less than, find 10% (ie, 100% - 90%) on the vertical axis and

trace horizontally across to the curve, then drop a perpendicular

down to the x axis to determine the corresponding value. This is

not a precise 90th percentile, because, as a corollary of the

defini-lion of the graph, 90% of observations are less than but not equal

to this value. For use in a rapid visual determination, however, a

percentile examined in this manner usually will be as satisfactory

as if it had been estimated from the cumulative distribution graph.

Specific percentile points could be marked with lines to simplify

this estimation.

The median, of course, is simply the antibody value (on the x

axis) corresponding to a y axis value of 50%. If one could rotate the

half of the curve lying above and to the left of the median point,

pivoting it clockwise about the median, and superimpose it on the

half of the curve lying to the right and below the median, then the

distribution is symmetrical about its median. If the curve shows

such symmetry, then the mean and median are equal, and thus the

median estimates the GMT (assuming that the scale of the x axis is

logarithmic).

The graph of the RCD curve also allows direct estimation of the

fraction of values that equal or exceed a specific value, such as a

known protective level or some multiple of the minimum level of

detection (MLD). The estimate is simply the height of the curve

corresponding to the specified value on the x axis.

Interpretation of the Shape of the Curve

Most RCD plots will tend to have a backward S shape (Fig 3),

indicating that there is a peak about which relatively many values

cluster. When values in the distribution occur with more uniform

frequency, then the plot will approximate a downward-sloping

straight line. Steepness in the plot is a reflection of the spread or

variance of values. A middle section that is steep relative to the

Increasing Antibody Level

-Fig 1. Rectangular-shaped curve, indicating that a high

propor-tion of vaccinees have antibody levels near the high end of the

included range.

Increasing Antibody Level

-*-Fig 2. Triangular-shaped curve, indicating a uniform distribution

of antibody levels within the included range.

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C

a) Si

0.

100

90

80

70

60

50

40

30

20

10

0

\

‘\ B

\

-‘

C

a)

0

a)

0.

a) 0

a)

0.

-Fig 5. Curves that intersect at the median.

a)

0.

-*-Increasing Antibody Level -k

Fig 6. Curves that intersect at the 80th percentile.

100

90

80

70

60 C

a)

0 50

a)

0.

40

30 20

10

-b-Fig 3. Parallel curves, indicating a shift in percentiles with

van-ance, skewness, and kurtosis remaining the same.

Fig 4. Curves having the same range of antibody response, but

with one (B) dominating the other (A) everywhere within the

range.

end sections means less variance, whereas a shallower middle

means more spread and larger variance. In the extreme case, an

RCD curve that terminates in a vertical line means that there is no

further variation in the data set-all subsequent values are equal.

A rectangular-shaped curve that remains high and is fairly flat

until it reaches some large value on the horizontal scale and then

descends rapidly toward the baseline (Fig 1) indicates that a large

fraction of vaccinees have high antibody levels. A more triangular

shape (Fig 2), with a plot descending in a relatively straight

diagonal line to the x axis indicates values that are more evenly

spread throughout the range; that is, there is wide variation

among the individual responses.

Figure 3 depicts two curves with the same shape, with one

shifted to the right of the other. Variability, skewness, and kurtosis

are the same for the two distributions. The difference between

them is attributable entirely to the fact that the antibody levels of

group B all are higher than those of group A. Consequently, every

percentile of B is higher than that of A, and, in particular, the

median and GMT of B exceed those of A. In Fig 4 the overall range

of values within the two groups is the same, but within that range

the individual values in group B all are higher than those in group

A. Thus, group B has a higher median response and a smaller

variance (as shown by the steeper midsection).

As shown in Fig 5, two vaccines may have similar medians, and

even have similar GMTs, but one (B) may be preferred because of

lower variability or another (A) because it elicits more high-level

responses (although it also has more low-level responses). The

vaccines shown in Fig 6 share a single percentile, as do those in Fig

5; for the vaccines in Fig 6, the shared value is the 80th percentile.

For most of the range of response, vaccine B dominates (ie, is

vertically greater) and has a smaller variance, but after the 80th

percentile, vaccine A has a greater fraction of the high values.

Although the highest individual responses occur with vaccine A,

vaccine B shows a higher median response with many fewer

subjects having low antibody responses.

Vaccines that produce essentially the same distribution of

me-sponses, except for sampling variability, will produce curves that

intersect at many places but have roughly the same shapes and

positions (Fig 7).

A Two-Vaccine Comparison

Figure 8 plots the prevaccination RCD curve for antibody to

pertussis toxin (PT) for two vaccines from the Multicenter

Ace!-lular Pentussis Vaccine Trial.3 We note that these distributions are

virtually identical, as one would expect, because prevaccination

antibody levels should differ only randomly. Because the majority

of infants have no antibody, their values are below the MLD (2

enzyme-linked immunosorbent assay units for this assay) and are

Increasing Antibody Level -*

Fig 7. Curves of vaccines with equal response distributions,

in-tersecting at several points.

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PT

in

base 10 log scale

1000

PT in base 10 log scale

SUPPLEMENT 603

a)

0.

10000

Fig 8. Reverse cumulative distribution plot of prevaccination

1ev-els of antibody to PT among infants receiving the SmithKline

Beecham two-component acellu!ar pentussis vaccine (A) or the

Lederle whole-cell vaccine (B).3

a)

IL

Fig 9. Reverse cumulative distribution plot of postvaccination

levels of antibody to PT among infants receiving the SmithKline

Beecham two-component acellular pertussis vaccine (A) or the

Lederle whole-cell vaccine (B).3

imputed a value of I.The next observed value is 2, but only about

30% of infants have values that high or higher, so the curve

descends steeply to that point. As noted before, the steepness

signals low variability, ie, that a large fraction of observations

have a single value (in this case, 1). The remaining values above

the MLD are so few that the rightmost portions of the curves are

low and relatively flat.

After vaccination, both groups show an antibody response (Fig

9); 90% of group B vaccinees have measurable antibody (ie, greater

than or equal to the MLD), and 100% of group A vaccinees have at

least 20 U of antibody. In contrast to the prevaccination

equiva-lence of the two groups, the postvaccination distributions are

different at every level, except the median. The most interesting

observation is that results in group A, which has a steep

midsec-tion to its RCD curve, are fan less variable than those in group B.

As in the example of Fig 5, the judgment of which vaccine is

superior depends on the level of antibody to PT that is thought

necessary for protection. If, for instance, 20 U is sufficient, then

vaccine A is preferable to vaccine B, even though B has a larger

fraction that exceeded the median value of about I 10 U. On the

other hand, if 110 U or more of antibody is required for protection,

then B is the better vaccine. At present such a judgment is

impos-sible to make for pertussis vaccines, because protective levels have

not been established. Note that the curve for vaccine A is nearly

symmetric, in the sense defined earlier. Under this condition, the

mean and the median are equal, so that the GMT can be estimated

to be about 110 (in fact, the calculated GMT was 104).

DISCUSSION

In theory, this visual representation allows comprehension of

the complete distribution of the values, limited only by the

quality and resolution of the graph. Whereas numerical

sum-manes and statistical tests of antibody data address for the

most part the equality of the mean or median values, these

graphs display all the differences in antibody distributions.

This technique allows direct comparison of data sets, which is

not convenient with other graphic tools such as histograms or

stem-and-leaf displays.9 It also conveys the importance of each

data point, whereas the box plot9’0 displays more of a visual

summary for the majority of data points. The RCD plots meet

the criteria for techniques used for exploratory data analyses:

they need no prior assumptions about the data distribution;

they display outliers without distortion; they highlight

impor-tant aspects of the data; and they allow comparison of complete

distributions of antibody data.

The RCD curve is proposed as a useful tool for data exploration

and presentation, allowing display of the full distribution of data.

These plots have a number of advantages and should be more

widely used.

REFERENCES

I. Siber GR, Ransil BJ. Methods for the analysis of antibody responses to vaccines or other immune stimuli. Methods Enzymol. 1983;93:60-78 2. Tufte ER. The Visual Display of Quantitative Information. Cheshire, CT:

Graphics Press; 1983

3. Edwards KM, Meade BD, Decker MD, et al. Comparison of 13 acellular

pertussis vaccines: overview and serologic response. Pediatrics. 1995; 96(suppl):548-557

4. Salk J, Van Wezel AL, Stoeckel P. et al. Theoretical and practical con-siderations in the application of killed poliovirus vaccine for the control of paralytic poliomyelitis. Dcv Biol Stand. 1981;47:181-198

5. Salk J. Content of inactivated poliovirus vaccine for use in a one- or

two-dose regimen. Ann Clin Res. 1982;14:204-212

6. Salk J. One-dose immunization against paralytic poliomyelitis using a

noninfectious vaccine. Rev Infect Dis. 1984;6:S444-S450

7. Chambers JM, Cleveland WS, Kleiner B, Tukey PA. Graphical Mehthods

for Data Analysis. Belmont, CA: Wadsworth International Group; 1983:

194-195

8. Lee ET. Statistical Methods for Survival Data Analysis, 2nd ed. New York: John Wiley & Sons; 1992:8-9

9. Tukey JW. Exploratory Data Analysis. New York: Addison Wesley; 1977

10. Williamson DF, Parker RA, Kendrick JS. The box plot: a simple visual

method to interpret data. Ann Intern Med. 1989;110:916-921

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1995;96;600

Pediatrics

George F. Reed, Bruce D. Meade and Mark C. Steinhoff

Analysis of Antibody Data

The Reverse Cumulative Distribution Plot: A Graphic Method for Exploratory

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1995;96;600

Pediatrics

George F. Reed, Bruce D. Meade and Mark C. Steinhoff

Analysis of Antibody Data

The Reverse Cumulative Distribution Plot: A Graphic Method for Exploratory

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