Application of Most-Probable-Number Statistics to Direct Enumeration

Vol. 53, No. 6

APPLIED AND ENVIRONMENTAL MICROBIOLOGY, June 1987, p. 1327-1332

0099-2240/87/061327-06$02.00/0

Copyright © 1987, American Society for Microbiology

Application of Most-Probable-Number Statistics to Direct Enumeration of Microorganisms D. J. ROSER, H. J. BAVOR,*

AND

S. A. McKERSIE

Water Research Laboratory, Faculty of Food and Environmental Sciences, Hawkesbury Agricultural College, Richmond, New South Wales, Australia 2753 Received 15 October 1986/Accepted 10 March 1987

A novel method for rapid determination of total microbial cell numbers was investigated. The method involves the application of most-probable-number estimation statistics to direct microscopic counting of microbial cells by using a particle sizing graticule. Its accuracy and reliability were tested with computer simulations of bacterial cell distributions encountered in ecological studies. Good estimates of cell numbers were obtained when the cell density varied from 3 to 6,000 cells per field, i.e., over 3 orders of magnitude. Low levels of contagion did not markedly influence cell estimates, although high levels, corresponding to discrete scattered microcolonies, did. However, these could be recognized visually. Estimates of cell numbers in Breed smears confirmed its speed and good correlation with the standard quadrat counting technique under real experimental conditions.

The introduction of epifluorescence microscopy and membrane filtration techniques has greatly assisted the task of estimating microbial cell numbers in natural environments. The commonly used quadrat-counting approach applied in the acridine orange direct count method (7, 8, 11), however, is handicapped by the tedium involved in physically counting the quadrats and cells (16). Combining video systems and computerized image analysis may be one possible solution to this problem (5, 16, 18). Alternatively, improved counting methods could also speed up the accumulation and interpretation of data, especially when resources are limited. Such methods could be valuable not only for ecology studies but also for industrial uses such as Breed smear direct counting with conventional stains commonly used in the dairy industry. In a previous investigation of plotless methods for estimating bacterial cell numbers (17), it was found that indirect enumeration methods could give satisfactory estimates of microbial cell numbers. These methods involved the use of cell-to-fixed-point distance measurements to estimate cell numbers (10) and could be applied to aggregated cells in some circumstances. Like conventional counting methods, however, they were less reliable at high and low cell densities. The best results were obtained with fields having a density of 10 to 1,000 cells per 10,000 p.m2. Tippett (20) suggested that enumeration could also be speeded up by determining the frequency with which cells are observed in a large number of graticules rather than by counting the number of graticules containing 400 particles (8). His method, however, was also restricted to a limited particle density range. In this paper, we propose a second indirect approach to visual cell enumeration similar to the latter method of Tippett (20). Our method involves the application of mostprobable-number (MPN) statistics to microscopy. Unlike distance measurements or Tippett's method (20), it should give acceptable estimates of cell numbers over a wide range of cell densities. Some accuracy would be lost compared with the standard counting method, but the work required *

for examining multiple fields of view would be markedly reduced. In its more familiar form, the MPN method involves inoculating animals or tubes of media with serial dilutions of a viral or cellular agent, recording the number of positive and negative responses, and obtaining a cell density estimate with the aid of statistical tables. It is usually assumed that the response frequency follows a Poisson distribution (12). In our proposed adaptation, microbial cells would be scored as present in or absent from graticule quadrats rather than test tubes. The quadrats would enclose a series of discrete areas, decreasing in size by multiples of 2, and effectively simulate a twofold serial dilution. To assess the accuracy of this method, as well as its limitations under defined conditions, we used computer simulations of a graticule and cells distributed randomly or in aggregates. The simulated cell arrangements were comparable to those which would be encountered on membrane filters, Breed smears, and other surfaces commonly examined by microscopy. MATERIALS AND METHODS Simulation of bacterial cells in a microscope field of view. Microscope fields and bacterial size cells were simulated by using a BASIC program written for a Hewlett-Packard 86B microcomputer equipped with 128K of random access memory. Each field consisted of a two-dimensional array of 10,000 (100 by 100) memory elements mimicking a field of 10,000 p.m2. This system was used to test the accuracy of the method on groups of fields having a known "cell" density. In each case examined, a predetermined number of cells was distributed among 5, 10, or 20 such fields by changing the value of selected memory elements. Initially, we distributed cells randomly within each group of fields and then within each field by assigning a cell to a coordinate set with machine-generated pseudorandom numbers. Although these numbers were generated by formulae, they could be considered random for our purposes. Each coordinate had an equal probability of having each cell assigned to it. In these circumstances, different fields in a group did not necessarily have equal numbers of cells

Corresponding author. 1327

1328

APPL. ENVIRON. MICROBIOL.

ROSER ET AL.

Yr

>

SIMULATED QUADRATS

OQUADRAT

M1t 024

AREA

TOTAL AREA 10000 UNITS

FIG. 1. Visual representation of a computer-simulated microscope field of 10,000 units, showing the relative position and size of the simulated particle-sizing graticule.

assigned to them, and more than one cell could be assigned to the same coordinate. Uneven distribution among fields was mimicked by altering the probability of assignment of cells to a given field. Uneven distribution within fields was simulated, either by assigning cells in blocks of 16 and 256 as well as individually, or by increasing the likelihood that a subarea within a field would have cells, compared with the remainder of that field. To simplify programming and decrease the run times required, the aggregates of simulated cells occupied rectangular areas. Simulation of a graticule. A commercially available graticule, which has the required series of quadrats of serially decreasing sizes, is the particle-sizing graticule British Standard G10 (Graticules Ltd., Tonbridge, England). Twelve different-sized areas can be distinguished on the graticule. In this study, 11 simulated quadrats were arranged in a fashion comparable to that of quadrats in the particlesizing graticule. The simulated quadrats ranged in size from 1,024 to 1 memory unit (Fig. 1). This simulated grid was superimposed on each field, and the presence or absence of cells in each quadrat was recorded. A typical result, for which 10 fields were examined, is illustrated in Table 1. Application of MPN statistics. Since suitable MPN tables were not available for all the situations of interest, cell density was estimated by using the formulae and tables of Fisher and Yates (9). This method removed the need for conventional MPN tables and involved little loss of accuTABLE 1. Frequency of occurrence of cells in simulated graticules of various sizes when 10 fields are examined (example) Quadrat no.

Quadrat size (units)

No. of quadrats with cells

1 2 3 4 5 6

1,024 512

10 10

256 128 64 32

10 10 10

7 8 9

16 8 4 2 1

10 11

7 5 2

i 0

0

TABLE 2. K' values for the calculation of log10 X, where n 2 11 x/y K(x) K(y) 1.613 -0.197 0.05 -0.108 1.324 0.1 1.045 -0.007 0.2 0.982 0.050 0.3 0.101 0.789 0.4 0.167 0.657 0.6 0.212 0.575 0.8 0.521 0.245 1.0 0.271 0.483 1.2 0.292 0.456 1.4 0.309 0.438 1.6 0.323 0.425 1.8 0.334 0.426 2.0 0.356 0.405 2.5 0.370 0.402 3.0 0.379 0.401 3.5 0.386 0.401 4.0 0.390 0.401 4.5 0.394 0.401 5.0 0.397 0.401 6.0 0.399 0.401 7.0 0.401 0.401 >7.0 aAdapted from Table VIII2 of Fisher and Yates (9). The K value is selected from the K(x) or K(y) column depending on which of x or y is the smaller.

racy. Most importantly, it could easily be incorporated into the overall BASIC program. Our adaptation of the Fisher method was as follows. The number of cells, X, in the largest quadrat (in our example, 1,024 units), is related to the total numbet of empty quadrats Y (45 in the example in Table 1),by the following formula: z

Y

n

=

s

Y1

e- a(z I)

(1)

z=

where n is the number of fields examined, s is the total number of different quadrat sizes, and a is the graticule size reduction (dilution) factor. The value of A can be obtained once x (the mean fertile level) and y (the mean sterile level are determined by using the formulae

(2)

y = s-x

logl0

K =x

log1o

a- K

(3) and Table VIII2 in Fisher and Yates (9) to obtain values of K which vary according to the value of x and y. When the number of quadrats is 11 or greaier and a = 2, values of K are provided by Table 2. K values can also be obtained by trial-and-error combination of equations 1 and 3. In the example given in Table 1, n = 10, s = 11, a = 2, x = 6.5, y = 4.5, and K = 0.390. From these values, log1o k = 6.5 logl0 2 - 0.390 and k = 36.9. Hence the cell density is 36.9 cells per 1,024 area units, or 360 cells per field of 10,000 units. The theoretical variance V (9) for most of the range of log10 A for n fields is V = (1/n) (log1o a/loglo s) (4) For the above example, then, V = 0.1 x (0.3010/1.041) = 0.0303. Fisher and Yates (9) suggest that the fiducial limits should be calculated by using the log10 K values. Detection of nonrandom distribution. The application of the above formulae assumes that cells are randomly distributed on the observed surface and that the frequency of

MICROSCOPY APPLICATION OF MPN STATISTICS

VOL. 53, 1987

occurrence of cells in a series of quadrats follows a Poisson distribution. However, this may not always be true in practice. Hence it would be desirable to be able to test for nonrandom distribution and the loss of accuracy possible under nonideal conditions. The principal statistical test examined was Moran's T test (13, 14). Other tests considered were the D statistic of Moran (15), Stevens' R statistic (19), and the J and phi statistics of Armitage (2, 3) and Armitage and Bartsch (4). These statistics were developed for analogous purposes in viral assays. Moran's test has the form T = E fm(N - fm), where fm is the number of graticules of each size containing cells and N is the total number of graticules of each size that were examined. From this a probability value, M is derived, the use of which is described by Moran (13) and Meynell and Meynell (12). Experimental confirmation. To assess the effectiveness of our technique on randomly dispersed bacteria, we used it to enumerate cells in Breed smears (1) of contaminated milk samples. The MPN value was calculated simply by counting the total number of fertile quadrats in five fields and applying equation 3 (s = 15, n = 5, a = 2) and the appropriate K values in Table 2. Specimens were stained with Newman stain and observed under transmitted light with an Olympus binocular microscope and a 10Ox oil immersion lens. A G10 graticule was further divided so that there were 15 serially sized quadrats to examine per field, ranging in size from 0.3 to 5,760 lim2. MPN counts were compared with those obtained by the quadrat-counting method used for the acridine orange direct count technique and also for Petroff-Hauser chambers (8). The central area of the graticule, equivalent to 16 80-PLm2 quadrats, was used for the standard counting technique. The total number of cells in five fields of view was also counted for comparison of the time required for estimation. RESULTS Correlation of actual versus estimated cell densities for randomly distributed units. Initially, cells were distributed randomly among 10 fields at average densities ranging from 3 to 6,000 cells per field of 10,000 units. X was estimated from equation 3 (n = 10, a = 2, s = 11). The calculation process was repeated 20 times for each density on a freshly generated pattern of cells. Regression analysis of the log10 values showed excellent 1:1 correlation with little spread over the Log 'Cells'/Field (MOosured) 10

Log 'Cells'/Field (Actual)

FIG. 2. Regression line showing the relationship of actual to measured cell densities. The analysis was performed in loglo values. Twenty Y estimates were obtained from each X value. Y = 1.004X + 0.001; R2 = 0.992; F = 999.9.

1329

TABLE 3. Effect of varying the number of fields used for determination of density estimates

Log1o (cells/field)

No. of fields/estimate

No. of cells/field

Actual

Calculateda

SE

5

10 100 1,000

1.0 2.0 3.0

0.95 2.02 3.07

0.030 0.034 0.025

10

10 100 1,000

1.0 2.0 3.0

0.96 1.99 3.02

0.022 0.015 0.021

0.016 0.011 0.016 aValues were obtained for the arithmetic mean of 10 values of logio X. 20

10 100 1,000

1.0 2.0 3.0

0.99 1.99 2.99

entire range tested (Fig. 2). Hence under ideal conditions of complete random distribution, MPN quadrat counting should behave like conventional dilution counting performed under ideal conditions. To determine how much accuracy would be gained or lost by examining a different number of fields, the process was repeated with either 5 or 20 fields per examination. Comparison of density estimates and their overall standard error (Table 3) indicated that while accuracy increases with an increasing number of fields examined per sample, the use of only five fields can still give a good order-of-magnitude estimate of cell density. The random cell distribution expected in these cases would be equivalent to that which is usually assumed (11) to occur on membrane filters prepared for the acridine orange direct count method. Application of the MPN method to contagiously distributed cells. Three forms of uneven distribution were considered. In the first, contagion over interfield distances was simulated, in which cells were randomly distributed within fields, but the chance that they would fall in any particular field varied. In the second case (medium-range contagion), each field had an equal chance of containing cells, but within each field, a large subarea of preset dimensions had a significantly greater chance of containing cells than the remainder of the field did. The top left-hand corner of each subarea was positioned randomly within each field, subject to the criterion that the whole subarea could be contained within that field. In the third case, short-range contagion was mimicked by assigning a number of cells to adjacent coordinates. Situations comparable to these distributions would be (i) isolated cells observed in soil smears in which some fields of view contained larger numbers as a result of uneven drying; (ii) isolated cells observed in soil samples containing small pieces of organic matter with a higher cell density than that of the surroundings; and (iii) discrete bacterial microcolonies scattered over a membrane filter. Visual representations are shown in Fig. 3. Effects of long-range contagion. The effects of contagion of varying severity between fields are shown in Table 4. In case 2, in which half the fields had a threefold higher chance than the remainder of containing cells, there was little effect on estimates of log10 k or its standard error. With increasing contagion, underestimation occurred, but it could be detected by Moran's T statistic (12, 13) as long as the average density was fairly high and most cells were not concentrated in a single field. The other statistics (not shown in Table 4) performed poorly. The phi statistic of Armitage (2, 3) and Armitage and Bartsch (4) required a good estimate of cell

APPL. ENVIRON. MICROBIOL.

ROSER ET AL.

1330

8

A 0

S

U

0~~~~~~~~~

*

S

0~~~~~~~~~~~~~~~~~~~~~~~

*~0

C

300

0

~~~~~0

0

0

.0

*

1

0

*

0

0

0

S

~~~1a 1

0

:

FIG. 3. Visual representations of microbial cell distributions simulated in tests of the MPN method. (A) Randomly distributed cells; (B) short-range contagion showing simulated aggregates of 4 by 4 and 16 by 16 units; (C) medium-range contagion in a narrow band (Ca), simulating the possible effect of a fungal hypha (there is a 20-fold greater probability of finding cells in this region than in the surroundings); (D) medium-range contagion, for which half the indicated area (Da) has a fivefold-greater probability of containing cells than the adjacent area does.

density in the first place, and underestimation of this quantity gave low phi values. To be effective, the D statistic of Moran (15) (equivalent to Stevens' R statistic) required more than the 11 dilution steps used. In the extreme cases, 5, 6, and 7, Moran's test did not perform as well as in the intermediate cases. The M value obtained was only just significant. These extreme cases would be detectable visually, however, when the results were first recorded. Effects of medium-range contagion. The presence of diffuse areas of increased cell density within a field had a remarkably small effect on MPN density estimates (Table 5). When subareas had five times the chance of containing cells, the contagion was barely detectable in its influence on density estimates, even when the subarea covered a quarter of the whole field (case 3). When areas were 20 times more likely to contain cells, effects were significant but still small in terms of a logarithmic density estimate. None of the statistics were particularly effective in detecting this form of contagion. However, it could probably be recognized visually (Fig. 3). The influence of long, narrow areas of contagion, as might result from the presence of a fungal hypha in a field (cases 5 and 6), could be tolerated to some extent, irrespective of the orientation of these zones. Effects of short-range contagion. Contagion comparable to that of microcolonies was virtually impossible to detect statistically. The density was severely underestimated when cells were present as groups of 4 by 4 or 16 by 16 units. Where cells were divided equally between colonies and individual cells, the estimate obtained was essentially the number of randomly distributed cells. Such a high degree of

contagion could, however, be recognized visually, as in the case of medium-scale contagion (Fig. 3). In this case, the clumps could be sized and enumerated separately. Enumeration of bacteria in Breed smears. Estimates were made of the number of bacteria in duplicate Breed smears from five different samples having cell densities ranging from 1 to 5,000 cells per field. The standard quadrat-counting method (method 1) was used to estimate cell density on the basis of the number of quadrats per 100 cells. Examination of the same samples by the MPN method and five fields per smear (method 2) gave a close 1:1 correlation (Fig. 4), as had been expected. The time taken to count the number of fertile fields was about 2 min, irrespective of cell density. By contrast, the time taken to apply method 1 or to count the total number of cells in five fields (method 3) varied greatly, depending on cell density, and often greatly exceeded the time taken for the MPN method (Fig. 5). As the standard method normally involves counting 400 cells, the actual time saved could, in practice, be much greater. Another feature of MPN counting was the ease with which scoring could be performed. This appeared to place far less strain on the operator, since it did not involve many decisions about whether a cell had been counted in any particular quadrat.

TABLE 4. Comparison of density estimates obtained for sets of fields varying in the probability with which each field contains cells Loglo in fields"

cells/field)'

Actual

Estimated

MC

SE

1

0.1 (10)

0.98 2.01 3.21

0.98 2.02 3.19

0.028 -0.9 0.028 0 0.043 -0.3

2

0.15 (5), 0.05 (5)

0.98 2.01 3.21

0.95 1.93 3.17

0.028 0.023 0.032

3

0.2 (2), 0.15 (2), 0.1 (2), 0.05 (2), 0 (2)

0.98 2.01 3.21

0.90 1.75 2.70

0.027 -0.8 2.8 0.033 6.1 0.039

4

0.2 (5), 0 (5)

0.98 2.01 3.21

0.84 1.43 2.04

0.027 -0.8 0.027 4.1 9.6 0.029

5

0.46 (2), 0.01 (8)

0.98 2.01 3.21

0.65 1.35 2.46

0.30 -1.6 2.6 0.042 -0.41 2.9

6

0.91 (1), 0.01 (9)

0.98 2.01 3.21

0.46 1.22 2.31

0.055 -2.5 1.7 0.040 0.040 1.8

7

0.5 (2), 0 (8)

0.98 2.01 3.21

0.61 0.94 1.21

0.025 -1.7 1.3 0.018 4.5 0.013

0.81 0.46 0.07

Cell distribution gives the number of fields out of 10 having a particular probability of being assigned a cell; e.g., 0.1 (10) means that 10 fields each have a probability of 0.1 of containing a computer-generated cell. b The three rows of values shown for each case indicate the extent of error and contagion occurring for different cell densities, which are expressed as logarithms. c The M probability value is derived from Moran's T value. Ideally, the M value should be 0. The M value shown is the average of 10 such values. A value of >1.65 shows significant deviation from random distribution at the 0.05 level, and a value >2.3 is significant at the 0.01 level (12).

VOL. 53, 1987

MICROSCOPY APPLICATION OF MPN STATISTICS

TABLE 5. Effects of medium-scale contagion density estimates

E2astimation

on

1331

Time (minutes)

18 Case

Areaa (by1 (X by

1

2

3

99 by 5

10 by 50

Log1o densityb

Increased

16

prbbltMC probability 5

5

Actual

Estimated

SE

0.98

0.99

0.038

-0.4

2.02 3.21

1.95 3.18

0.02 0.034

-0.1 -0.3

0.98 2.02 3.21

0.98 1.96 3.18

0.041

0.031 0.024

-0.4 0.6 -0.3

50 by 50

5

0.98 2.02 3.21

1.10 2.08 3.22

0.027 0.045 0.053

-0.1 0 -0.5

50 by 50

20

0.98 2.02 3.21

1.10 2.05 3.04

0.028 0.06 0.060

0 1.5 1.6

1412

10 8-

6

2 2.7

4

5

99 by 5

20

0.98 2.02 3.21

0.94 1.87 2.95

0.38 0.025 0.044

-0.3 0.6 -0.6

6

5 by 99

20

0.98 2.02 3.21

0.95 1.86 3.04

0.051 0.029 0.031

-0.5 -0.86 0.34

0.98 2.02 3.21

0.94 1.84 3.04

0.047 0.013 0.053

-0.7 0.75 -0.1

7

10 by 50

20

a The area X by Y may be anywhere within each field that it will fit. Its position is varied from field to field by using pseudorandom numbers. The numbers given are the lengths of the X and the Y sides, respectively. b See Table 4, footnote b. c See Table 4, footnote b.

DISCUSSION A major challenge presented by epifluorescence microscopy is how to gather the abundance of information which can be obtained about microbial cell size and distribution. Log10 CIals/Field

(Method

2)

Log1o C-lls/Fi1ld

(Method 1)

FIG. 4. Regression line showing the correlation between loglo estimates of bacterial numbers in Breed smears obtained either by counting the number of graticules per 100 cells (method 1) or by the MPN technique (method 2), where n = 5, s = 15, and a = 2. Two MPN estimates were performed on each duplicate smear. Y = 1.024X 0.09; R2 = 0.98; F = 977; the number of degrees of freedom = 19. From Equation 4, for each estimate, V = 0.0512 and the standard deviation is 0.226 loglo units. -

10.s1133

366

3170

CQ1lls/Field

FIG. 5. Comparison of the time taken to score the cell numbers field for fields of different density by determining the number of quadrats per 100 cells (method 1; =); by determining the mean ); and by counting and averaging fertile level (n = 5) (method 2; the total number of cells in five fields (method 3; E ). per

Refinements in computer technology now allow extensive analysis of any data which can be obtained. Hence, improved methods for cell enumeration are increasingly desirable. The standard technique, adapted from hemacytometer counting, gives high accuracy, but the value of this accuracy is sometimes questionable. A precision of +5% can be desirable in some specialized circumstances, but is counterproductive in others. Microbial biomass estimates, for example, cannot be accurately determined, despite the accuracy of the technique, because there will always be some uncertainty about average cell size and dry weight, especially if the cells are from natural environments. This has been confirmed experimentally by Bjornsen (5). The presence of any obscuring material also introduces error. The latter effect can be allowed for by introducing a correction factor of 2 (6), but this procedure itself could introduce unknown errors. Even under well-defined conditions, the value of extreme accuracy can be questioned. Unlike populations of macroscopic organisms, those of bacteria undergo changes best measured in orders of magnitude over relatively short periods. In such circumstances, estimates of the logarithm of the cell density would usually be satisfactory and even desirable when resources are limited. We are not suggesting here that it is always inappropriate to use a highly precise counting technique or that MPN techniques should replace it in all circumstances. Rather, we would argue that techniques should be applied according to the situation or aim of a study. We see our technique as filling a gap between very precise measurements and qualitative observations, for which growth is scored as + or + + +, etc. Such would be the case when large numbers of samples must be screened or large differences are expected between samples, e.g., survey work estimating microbial biomass in a range of related but undescribed habitats, and screening of sterile products such as milk for contamination. For examining small population changes in a well-described environment, however, the quadrat counting technique would be more appropriate. The method described in this paper seems to offer a potential way of estimating cell density varying over 3 or

1332

ROSER ET AL.

more orders of magnitude. In addition, Moran's T statistic provides a test of its reliability and an indication of marked cell contagion occurring between fields. Tippett (20) considered the enumeration problem as applied to small, randomly distributed particles and concluded, as we did, that valuable time could be saved by scoring the presence or absence of cells instead of by counting. His method, however, was developed for the situation in which one examines a cluster of identically sized quadrats. Consequently, it was limited to a particle density of 0.2 to 5.0 cells per quadrat. Below this limit, estimation error increases dramatically, while above the limit, there are too many cells per quadrat to make the method easy to use. Our method seeks to specifically overcome the density limitation by having quadrats of different sizes. While it suffers from the effects of extreme contagion, it may, however, be improved. Tippett (20) pointed out that probability theory predicts the number of quadrats with cell densities larger than unity. Consequently, his approach to counting might be combined with ours to address the problem posed by cell clumping and how to record it. Application of the MPN technique with dispersed bacterial cells was very successful. The results of the Breed smear comparison showed that this method correlates well with the standard method. Further, no obvious problems were found when we used it for acridine orange direct counts. When we applied it to enumeration of the populations of epilithic and planktonic bacteria in a model water treatment system (S. Hudson, unpublished report, 1985), it demonstrated that 20% were cells 0.3 ,.m in diameter. Sizing of cells was particularly simple to perform, because the different-sized areas acted as references. Scoring was easily performed, since only presence in or absence from a field had to be recorded. The number of fields examined could be increased to 15 by enclosing additional quadrat areas on the graticule. Examination of unhomogenized samples, containing aggregated bacterial cells, also yielded useful data. By treating such aggregates as separately countable entities, it was possible to gain additional information about biomass distribution. As it stands, the MPN graticule method now requires investigation with different sample types to determine the situations in which it can best be applied. It could also benefit from further investigation of statistical analysis techniques, particularly the incorporation of Tippett's ideas (20). Already, however, it is promising as a means of surveying changes in microbial distribution and abundance and screening of food material. ACKNOWLEDGMENT We thank the referees for their valuable comments, particularly in relation to clarifying our suggested applications and the drawing of our attention to the similarity of Tippett's work.

APPL. ENVIRON. MICROBIOL. LITERATURE CITED 1. American Public Health Association. 1967. Standard methods for the examination of dairy products, 12th ed. American Public Health Association, New York. 2. Armitage, P. 1959. Host variability in dilution experiments. Biometrics 15:109-133. 3. Armitage, P. 1959. An examination of some experimental cancer data in light of the one-hit theory of infectivity titrations. J. Natl. Cancer Inst. 23:1313-1330. 4. Armitage, P., and G. E. Bartsch. 1960. The detection of host variability in a dilution series with single observations. Biometrics 16:582-590. 5. Bjornsen, P. K. 1986. Automatic determination of bacterioplankton biomass by image analysis. Appl. Environ. Microbiol. 51:1199-1204. 6. Bott, T. L., and L. A. Kaplan. 1985. Bacterial biomass, metabolic state, and activity in stream sediments: relation to environmental variables and multiple assay comparisons. Appl. Environ. Microbiol. 50:508-522. 7. Daley, R. J. 1979. Direct epifluorescence enumeration of native bacteria: uses, limitations and comparative accuracy, p. 29-45. In J. W. Costerton and R. R. Colwell (ed.), Native aquatic bacteria: enumeration, activity and ecology. American Society for Testing and Materials, Philadelphia. 8. Eisenhart, C., and P. W. Wilson. 1943. Statistical methods and control in bacteriology. Bacteriol. Rev. 7:57-137. 9. Fisher, G. L., and F. Yates. 1963. Statistical tables for biological, agricultural and medical research, 6th ed. Oliver and Boyd, Ltd., London. 10. Greig-Smith, P. 1964. Quantitative plant ecology. Butterworths, London. 11. Jones, J. G. 1979. A guide to methods for estimating microbial numbers and biomass in fresh water. Freshwater Biological

Association, Ambleside, England. 12. Meynell, G. G., and E. Meynell. 1970. Theory and practice in experimental bacteriology, 2nd ed. Cambridge University Press, Cambridge. 13. Moran, P. A. P. 1954. The dilution assays of viruses. I. J. Hyg. 52:189-193. 14. Moran, P. A. P. 1954. The dilution assays of viruses. II. J. Hyg. 52:444-446. 15. Moran, P. A. P. 1958. Another test of heterogeneity of host resistance in dilution assays. J. Hyg. 56:319-322. 16. Pettipher, G. L., and U. M. Rodrigues. 1982. Semi-automated counting of bacteria and somatic cells in milk using epifluorescence microscopy and television image analysis. J. Appl. Bacteriol. 53:323-329. 17. Roser, D. J., D. B. Nedwell, and A. Gordon. 1984. A note on 'plotless' methods for estimating bacterial cell densities. J. Appl. Bacteriol. 56:343-347. 18. Sieracki, M. E., P. W. Johnson, and J. M. Sieburth. 1985. Detection, enumeration, and sizing of planktonic bacteria by image analyzed epifluorescence microscopy. Appl. Environ. Microbiol. 49:799-810. 19. Stevens, W. L. 1958. Dilution series: a statistical test of technique. J. R. Stat. Soc. 20:205-214. 20. Tippett, L. H. C. 1932. A modified method of counting particles. Proc. R. Soc. Lond. A 137:434 446.