Averaging of TNTC Counts

Transcription

1 APPLIED AND ENVIRONMENTAL MICROBIOL-OGY. Aug p /88/082)069-04$02.00/0 Copyright O 1988, American Society for Microbiology Vol. 54. No. 8 Averaging of TNTC Counts CHARLES N. HAAS* AND BARBARA HELLER Illiniois Itistitite of Technology, Chi ago, Illitiois Received 10 March 1988/Accepted 23 May 1988 When plate count methods are used for microbial enumeration, if too-numerous-to-count results occur, they are commonly discarded. In this paper, a method for consideration of such results in computation of an average microbial density is developed, and its use is illustrated by example. In the determination of microorganisms by a technique in which individual viable units are determined, such as by plate count assay of bacteria or by plaque count assay of viruses, with insufficiently diluted samples an overgrowth or dense formation of colonies is noted which is conventionally reported as TNTC (too numerous to count). For example, q,. x Nj and plates with TNTC (TNTCi). It is convenient to maximize the log likelihood function (natural log of the joint probabilities) rather than the likelihood itself. If there are a total of q plates with scorable counts (less than T colonies) and r plates with counts greater than T, then the log likelihood function can be written: Y = In (L) = E [-AVj + Nj In (AkV1) -In (Nj!)] + In E exp [(-AVk)] (AVk)'..lin! J-I- IIIJW the membrane filter method for heterotrophic plate count of waters (1) specified that only samples with counts of 20 to 200 colonies per filter be scored. However, the omission of plates in which TNTC counts are present in the estimation of the average number of microorganisms present in a sample may lead to a misestimation of their numbers. In this paper, a statistically based method for quantitative use of TNTC information in the construction of an average-density estimate for a sample is presented. THEORETICAL DERIVATION The fundamental procedure for statistical estimation is the maximum likelihood theory (2, 4, 6). For a single plate on which a sample of volume YV has been plated, if the Poisson distribution is assumed, equation 1 gives the probability that Nj colonies will arise. P(Nj) = exp[(-avj)(avj)ni/n1j! (1) In equation 1, A is the mean concentration (number/volume) of microorganisms in the water from which the sample was taken. If T is the minimum number of colonies which is regarded as TNTC (e.g., 200 colonies in the case of the heterotrophic plate count), then the probability that a sample of water Vj will plate out with TNTC colonies is given by a summation of Poisson terms as in equation 2. The summation in equation 2 is taken over all colonies in excess of the maximum countable number (T).,~~~ P(TNTC1) = E exp[(-avj) (AIj)k]/k! (2) The maximum likelihood estimator of the unknown mean concentration is obtained as the value of Ak which maximizes the probability of having obtained the series of colony counts * Corresponding author. Given a set of experimental observations (Njs and TNTCs at various sample volumes), equation 3 can be maximized to determine the maximum likelihood estimator of the mean concentration from which the samples were taken. The value of A that maximizes equation 3 can be determined in two basic ways, either by searching for the maximum directly or by differentiating, equating the derivative in A to zero, and solving the resulting nonlinear equation for A. In this work, the first approach of direct search is used with the algorithm of Nelder and Mead (5) for nonlinear optimization. The calculation is performed using a BASIC program for a personal computer. A listing suitable for use with standard interpreted BASICs is given in Fig. 1. To use this program, the ordinary user need only modify the statements from lines 1070 on. The first DATA statement contains the number of countable plates, the number of TNTC plates, and the maximum number of countable colonies on a plate (T). The following DATA statements are one line for each countable plate with the volume plated and the number of colonies on that plate, followed by one line for each TNTC plate with the volume plated on each TNTC plate. Note that the number of colonies on a countable plate must not exceed the TNTC threshold. If more than 20 plates are counted, the dimension statements in lines 20 and 30 for the arrays VOLSCORE, VOLTNTC, and YOBS must be increased accordingly. Compilation of the program for execution on a machine with a math coprocessor will dramatically improve execution times. The tolerance criterion of in line 70 has been found to be generally acceptable; however, the user can lower this for a greater degree of precision. COMPUTATIONAL EXAMPLE The use of equation 3 for the particular case of two plates at volumes of 1 and 10 is illustrated. It is assumed that a countable number of colonies (including 0) occurs on the low-volume plate but that TNTC occurs on the higher- (3) Downloaded from on April 20, 2018 by guest 2069

2 2070 HAAS AND HELLER APPL. ENVIRON. MICROBIOL. - - "; 10 REM PROGRAM TO COMPUTE MAXIMUM LIKELIHOOD ESTIMATORS DIM SET(20,20),CENTROID(20),EXPAND(20),CONTR(20),X(20),FVAL(20),REFLECT(20) 30 DIM VOLSCORE(20),VOLTNTC(20),LNFACTOR(300),BEST(20),YOBS(20) 40 RESTORE:READ NSCORABLE,NTNTC,TNTCCOUNT:LNFACTOR(0)-0 50 FOR I-1 TO TNTCCOUNT:LNFACTOR(I)-LNFACTOR(I-1)+LOG(I):NEXT I 60 NVAR-1 : REM - number of parameters 70 TOL REM - convergence criterion 80 MAXITER - 50 REM - maximum iterations 90 A -.1 : REM - polygon side for initial guess 100 AM-1:BB-.5:GG-2: REM - parameters for simplex (do not change) 110 REM -g----get data FOR I-1 TO NSCORABLE:READ VOLSCORE(I),YOBS(I):NEXT I 130 FOR J-1 TO NTNTC:READ VOLTNTC(J):NEXT J 140 REM Compute Initial Guess By Neglecting TNTC's SUMCOL-0:SUMVOLO0 160 FOR I1 TO NSCORABLE:SUMCOL-SUMCOL+YOBS(I):SUMVOL-SUMVOL+VOLSCORE(I):NEXT I 170 ITER-0:PRINT:PRINT 'Working 180 SET(l,l)-(SUMCOL+.5)/SUMVOL : REM Initial Guess 190 Q-A 200 FOR I-2 TO NVAR FOR J-1 TO NVAR 220 IF I-1<>J THEN SET(I,J)-SET(1,J)*(1-Q) 230 IF I-1-J THEN SET(I,J)-SET(1,J)*(l+A) 240 CENTROID(J)-CENTROID(J)+SET(I,J)/(NVAR+l) 250 NEXT J 260 NEXT I 270 REM get values at polygon vertices FOR I-1 TO NVAR FOR J-1 TO NVAR:X(J)-SET(I,J):NEXT J 300 GOSUB FVAL(I)-F 320 NEXT I 330 REM find worst point and best point 340 BEST-1:WORST FOR I-2 TO NVAR IF FVAL(I)<FVAL(BEST) THEN BEST-I 370 IF FVAL(I)>FVAL(WORST) THEN WORST-I 380 NEXT I 390 FOR I-1 TO NVAR:BEST(I)-SET(BEST,I);NEXT I 400 IF ITER>MAXITER THEN DELTA-ABS(FVAL(BEST)-FVAL(WORST)):PRINT "+ ";:IF DELTA>TOL THEN PRINT:PRINT " PRINT :PRINT:PRINT "The Estimated Mean Density - ";SET(BEST,1) ;"colonies/unit volume" 440 PRINT:PRINT:PRINT "- ln(likelihood) -";FVAL(BEST) 450 END: 460 REM ****************** SIMPLEX ALGORITHM *************************** 470 REM **************************************************************** 480 ITER-ITER FOR I1 TO NVAR:CENTROID(I)-0:NEXT I 500 FOR I-1 TO NVAR FOR J-1 TO NVAR 520 IF I-WORST THEN CENTROID(J)-CENTROID(J)+SET(I,J)/(NVAR) 540 NEXT J 550 NEXT I 560 FOR I-1 TO NVAR:REFLECT(I)-CENTROID(I)+M*(CENTROID(I)-SET(WORST,I) ):NEXT I 570 FOR I-1 TO NVAR:X(I)-REFLECT(I):NEXT I 580 GOSUB 870 :FREFLECT-F

3 VOL. 54, 1988 AVERAGING OF TNTC COUNTS IF FREFLECT<FVAL(BEST) THEN 750 : REM reflected point is best 600 IF FREFLECT>FVAL(WORST) THEN IF FREFLECT<-FVAL(WORST) THEN FOR I11 TO NVAR:CONTR(I)-CENTROID(I)-BB*(CENTROID(I)-SET(WORST,I)):NEXT I 630 GOTO FOR I-1 TO NVAR:CONTR(I)-CENTROID(I)-BB*(CENTROID(I)-REFLECT(I)):NEXT I 650 FOR I11 TO NVAR:X(I)-CONTR(I):NEXT I 660 GOSUB CONTR-F 680 IF CONTR>FVAL(WORST) THEN FOR I-1 TO NVAR : REM contracted point is an improvement 700 SET(WORST,I)-CONTR(I) 710 NEXT I 720 FVAL(W0RST)-CONTR:GOTO FOR J-1 TO NVAR:FOR I'1 TO NVAR+1:SET(I,J)-(SET(I,J)+BEST(J))/2:NEXT I:NEXT J 740 GOTO FOR I-1 TO NVAR:EXPAND(I)-CENTROID(I)+GG*(REFLECT(I)-CENTROID(I)):NEXT I 760 FOR I-1 TO NVAR:X(I)-EXPAND(I):NEXT I 770 GOSUB 870:FEXPANDF:IF FEXPAND > FREFLECT THEN FOR I-1 TO NVAR:SET(WORST,I)-EXPAND(I):NEXT I 790 FVAL(WORST)-FEXPAND 800 GOTO FOR I-1 TO NVAR:SET(WORST,I)-REFLECT(I):NEXT I 820 FVAL(WORST)-FREFLECT 830 GOTO REM******************************************************************** 850 REM * FUNCTION COMPUTATION ***************************** 860 REM **********SUM LIKELIHOOD FUNCTION FOR ALL POINTS******************** 870 IF X(1)<-0 TEEN F E+24:RETURN :REM No Negative Densities 880 F# FOR JJJ-1 TO NSCORABLE 900 TEMPf--X(1)*VOLSCORE(JJJ)+YOBS(JJJ)*LOG(X(1)*VOLSCORE(JJJ) )-LNFACTOR(YOBS(JJJ)) 910 F#-F#-TEMP# 920 NEXT JJJ 930 FOR JJJ-1 TO NTNTC 940 LOGTTERMD--(X(1)*VOLTNTC(JJJ) )+TNTCCOUNT*LOG(X(1)*VOLTNTC( JJJ) ) -LNFACTOR(TNTCCOUNT) 950 SUM?'0:IF LOGTTERMI-60 TEEN SUM#.EXP(LOGTTERM#) 960 KKK-TNTCCOUNT MULT#-LOG(X(1)*VOLTNTC(JJJ) /KIXK) :LOGTTERM#ILOGTTERM#+MULT# 980 IF LOGTTERMI>-60 THEN SUM-SUMI+EXP(LOGTTERMI) 990 KKK-KKK+1:IF KKK>500 TEEN PRINT "KKK TOO LARGE": STOP 1000 IF SUM#0 THEN GOTO LOGRATIOPLOGTTERk*-LOG(SUM#):IF MULT#>0 OR LOGRATIO#>LOG(TOL) TEEN GOTO IF SUMI>1 THEN SUtM F#-F#-LOG(SUM ) :F-F# 1040 NEXT JJJ 1050 RETURN 1060 REM ************************* DATA STATEMENTS *************************** 1070 DATA 5,3,80 :REM # Scorables, # TNTC Observations, TNTC Cutoff 1080 DATA 1,10 :REM scorable data -- volume, colonies 1090 DATA 1, DATA 1, DATA 5, DATA 5, DATA DATA 10 :REM TNTC data -- volume of sample with TNTC 1150 DATA 10 FIG. 1. BASIC program listing.

4 2072 HAAS AND HELLER APPL. ENVIRON. MICROBIOL..) c a0) L ID -0 4-,- E 0 U i Counts on low volume plote FIG. 2. Estimated average densities per unit volume for a single observation at 1 and 10 volume units with TNTC at 10 units. Numbers in the graph are the assumed lower bounds for TNTC. volume plate. The threshold of TNTC is varied from 80 to 300 colonies, illustrating the effect in the range of practical interest. From application of maximum likelihood, the estimated mean density is reported as a function of the counts read on the low-volume plate and the threshold for reporting as TNTC on the high-volume plate. Figure 2 summarizes the results of these calculations. At low counts on the low-volume plate (less than those given by the TNTC threshold divided by the dilution factor, e.g., 8 colonies when 80 is the threshold), the estimated mean density is much greater than that given by the direct use of only the countable plate. In other words, the exclusion of TNTC results tends to bias the estimated density below that given from the use of the maximum likelihood equation (equation 3). As the number of colonies on the low-volume plate increases, however, there is greater agreement between the estimate that would be obtained from the exclusion of the TNTC results and that obtained from its inclusion in the maximum likelihood estimator. Execution of the program shown in Fig. 1 leads to a solution for a problem in which there are 8 plates taken and the maximum countable number is 80 colonies. On a 1-ml sample, counts for three replicate plates are 10, 12, and 23. On a 5-mi sample, counts are 40, 63, and TNTC, and on a 10-ml sample, duplicate counts are TNTC. The maximum likelihood estimated count is 13.28/ml, with a -log(likelihood) of Note that, if the TNTC results were ignored in this case and an average (total colonies per volume in countable plates) were taken, one would obtain a density of 11.9/ml, underestimating the density by about 10%. DISCUSSION Conventional practice generally discards TNTC data obtained at large volumes and retains only data in countable ranges. As shown, however, particularly when the counts in the countable ranges are low, this may lead to an underestimate of the microbial concentration from the sample. It is proposed that the procedure based on equation 3 be used for the evaluation of data. In conclusion, it is noted that the maximum likelihood approach assumes intrinsic Poisson replication errors. The authors have discussed tests of this assumption elsewhere (3). If the Poisson distribution does not characterize the distribution of microbial counts between samples, then the proposed procedure should be modified accordingly. ACKNOWLEDGMENT We thank Christine Moe of the University of North Carolina for posing this problem. LITERATURE CITED 1. American Public Health Association Standard methods for the examination of water and wastewater, 16th ed. American Public Health Association, Washington, D.C. 2. Finney, D. J Statistical method in biological assay, 2nd ed. Hafner Press, New York. 3. Haas, C. N., and B. Heller Statistics of enumerating total coliforms in water samples by membrane filter techniques. Water Res. 20: Maul, A., J. C. Block, and A. H. El-Shaarawi Statistical approach for comparison between methods of bacterial enumeration in drinking water. J. Microbiol. Methods 4: Nelder, J. A., and R. Mead A simplex method for function minimization. Computer J. 7: Rohatgi, V. K Statistical inference. John Wiley & Sons, Inc., New York.

5 ERRATUM Averaging of TNTC Counts CHARLES N. HAAS AND BARBARA HELLER Illinois Institute of Technology, Chicago, Illinois Volume 54, no. 8, p. 2069, equation 1: the equation should read "P(Nj) = exp[(-avj)](avj)nj/n1!'' Page 2069, equation 2: the equation should read "P(TNTCj) = k=r exp[(-avj)](avj)/k!" 2884