Anal. Chem. 1998, 70, 58-63
Increasing the Precision and Accuracy of Top-Loading Balances: Application of Experimental Design Thomas J. Bzik, Philip B. Henderson,* and J. Peter Hobbs
Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501
The traditional method of estimating the weight of multiple objects is to obtain the weight of each object individually. We demonstrate that the precision and accuracy of these estimates can be improved by using a weighing scheme in which multiple objects are simultaneously on the balance. The resulting system of linear equations is solved to yield the weight estimates for the objects. Precision and accuracy improvements can be made by using a weighing scheme without requiring any more weighings than the number of objects when a total of at least six objects are to be weighed. It is also necessary that multiple objects can be weighed with about the same precision as that obtained with a single object, and the scale bias remains relatively constant over the set of weighings. Simulated and empirical examples are given for a system of eight objects in which up to five objects can be weighed simultaneously. A modified PlackettBurman weighing scheme yields a 25% improvement in precision over the traditional method and implicitly removes the scale bias from seven of the eight objects. Applications of this novel use of experimental design techniques are shown to have potential commercial importance for quality control methods that rely on the mass change rate of an object. Experimental design techniques (EDT) are powerful tools for the exploration of multivariate processes/systems.1 Typically, EDT are used to understand the effect of several variables on a system by a well-defined mathematical model. Good experimental design strategies optimize the amount of information obtained from a collection of experiments. Varying one parameter at a time is, in general, the easiest to analyze (no system of equations to solve) but can be one of the least efficient. In many commercial, industrial, and research settings, it is necessary to weigh multiple objects. This operation may be used to determine the quality of production lots, monitor the progress of a process, or compare the performances of systems. Regression analysis and hence EDT can be applied to the problem of weighing multiple objects by treating the total mass on the scale as the dependent (measured) variable and the presence of each object on the scale as a (0,1) dummy variable where each variable’s coefficient is the estimated weight of that object. Weighing objects (1) Montgomery, D. C. Design and Analysis of Experiments, 3rd ed.; John Wiley and Sons: New York, 1991; pp 1-3.
58 Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
individually is equivalent to varying one parameter at a time. The choice of the optimum weighing scheme for a given number of objects, balance capacity, and limit on the number of weighings is not obvious but can be determined through the use of EDT. In this article, we will show how it is possible to obtain higher accuracy and precision on a top-loading balance with a wellstructured multivariate weighing strategy without requiring a greater number of weighings than the number of objects. We will present verification of the mathematics behind the improvements through simulated and empirical examples. We will also illustrate the impact of applying a multivariate weighing strategy versus a univariate one. Accuracy is a measure of scale bias and is defined, in this paper, as the average deviation of the determined object weight from the object’s true weight after many weighings. Scale precision, often expressed as a standard deviation, is a measure of how variable the observed weights of an object are about their average value after many weighings. Our research of polymer surface barrier treatments requires us to measure the permeation of solvents from sealed plastic containers. The weight loss from these bottles must be monitored over several weeks. Let us arbitrarily call the eight bottles in our study objects A, B, ..., and H. Each object has a true mass associated with it. Call these masses a, b, ..., h, respectively. Weighing each of the eight objects alone on the balance is an example of a univariate or “one-at-a-time” design strategy. This requires eight weighings with each observed weight recorded as ao, bo, ..., ho. An optimized strategy in terms of accuracy and precision improvement for weighing eight objects is shown in Table 1, where “yes” means the object is on the balance and “no” means the object is off the scale. A similar table for the one-at-a-time design scheme would contain yes’s only on the diagonal. The optimized scheme in Table 1 is a modified eight-experiment Plackett-Burman2 (P-B) design or Taguchi L8 array.3 The total weight of each measurement, w1, w2, ..., and w8 is recorded and a multiple linear regression analysis will yield eight estimated weights, ae, be, ..., and he. The regressions may be performed using a PC and any software package that supports multiple linear regression, such as Excel, Lotus 123, or SAS. The changes in (2) Placket, R. L.; Burman, J. P. Biometrika 1946, 33, 305-325. (3) Taguchi, G.; Konishi, S. Orthogonal Arrays and Linear Graphs; American Supplier Institute: Dearborn, MI, 1987; pp 1. S0003-2700(97)00348-X CCC: $14.00
© 1997 American Chemical Society Published on Web 01/01/1998
Table 1. Eight Experiment P-B Strategy for Determining the Weight of Eight Objects object measurement no.
A
B
C
D
E
F
G
H
1 2 3 4 5 6 7 8
yes yes yes no yes no no no
yes yes no yes no no yes no
yes no yes no no yes yes no
no yes no no yes yes yes no
yes no no yes yes yes no no
no no yes yes yes no yes no
no yes yes yes no yes no no
yes yes yes yes yes yes yes yes
the scale loading that occur with each measurement taken are highlighted in italic type in the table. The estimated weights from the P-B weighing scheme will have an expected standard deviation which is 25% less than that from the one-at-a-time design. To obtain a similar improvement in precision using the one-at-a-time design, one would need to independently perform every measurement in the design twice and average the pairs of observed weights for each object for a 29% improvement in precision. In addition to the precision improvement, the weighing scheme in Table 1 will yield an accuracy improvement over the one-at-atime design scheme. The P-B design strategy mathematically removes an unknown fixed scale bias in the process of estimating the weights of objects A-G. The eighth object, H, will have the same unknown fixed scale bias as does any other individually obtained weight. Besides the enhanced computational requirement, the multivariate approach also requires more handling of each object during weighing. In the P-B design, the first weighing has five objects on the balance, the second through seventh weighings take two objects off the balance and replace them with two other objects, and the final weighing takes four objects off the scale. Our experience has shown that once an operator becomes familiar with the weighing scheme in Table 1, this procedure takes no more time to complete than the individual weighings because most of the weighing time is spent waiting for the balance reading to stabilize. During the stabilization time, the operator prepares for the next weighing by setting aside the next two objects to be placed on the balance and noting which two objects will be removed. SIMULATED EXAMPLE For a simulated example, assume that the true weights, a-h, of objects A-H are all 100 g. The enhanced weighing scheme does not require that the weights of the objects be identical, but this a priori assumption makes the example more readable. Suppose that the balance used for this weighing has a fixed scale bias of +0.4 g and a precision of 0.1 g, measured as one standard deviation. Using the RANNOR function in SAS, a set of eight measurements was generated with these error characteristics. For purposes of illustration only, the identical hypothetical error sequence (bias plus precision error) was applied to both sets of eight measurements. The one-at-a-time strategy yielded the following weights (in g): 100.414, 100.491, 100.462, 100.490, 100.208, 100.390, 100.343, and 100.393. The average accuracy error is +0.399 g, with a standard deviation of 0.093 g.
The P-B design strategy (Table 1) yielded the following weights (in g): 500.414, 500.491, 500.462, 500.490, 500.208, 500.390, 500.343, and 100.393. A multiple linear regression (with the intercept forced through the origin) on the P-B data set yielded the following estimated weights (in g): 99.990, 100.071, 100.007, 99.918, 99.953, 99.954, 100.119, and 100.393. The average accuracy error of the first seven estimated weights is +0.002 g, with a standard deviation of 0.071 g, and the estimated weight of the eighth object retains the 0.4 g bias. As discussed previously, use of this design implicitly removes the scale bias from ae through ge but not from he. Estimating the scale bias from the P-B design data alone is not possible because the design is “saturated”, that is, all of the degrees of freedom in the P-B design are used to estimate the weight of the objects. At least one additional weighing is required to estimate the bias. For example, if eo (100.208 g) is combined with the P-B data, and a multiple linear regression with a nonzero intercept is performed, the weight estimates for the first seven objects would be identical to the P-B estimate values above. The bias would be explicitly estimated from the regression intercept as +0.255 g, and he would be 100.138 g. Object E having the worst bias estimate implicit to its weighing gives the least improvement to he. Incorporation of other object weights or additional object weights in the regression would improve the estimate of both the scale bias and h. EMPIRICAL EXAMPLE At the start of a typical bottle permeation study there is a brief induction period followed by a continuous increase in the weight loss rate, or permeation rate, with time. After several days, the change in the permeation rate slows as the bottle gradually approaches a steady-state permeation rate. Both the PlackettBurman strategy and the individual weighing strategy were used to measure toluene permeation from eight half-gallon bottles in order to estimate the steady-state permeation rate for the bottles. Each bottle was filled with 1640 g of toluene, sealed, and placed in an air circulating oven at 50 °C. Periodically over the next 36 days the bottles were removed from the oven and weighed on a top-loading Mettler PC 16 balance (16 kg capacity measured to the nearest 0.1 g). During each weighing episode, weight data was collected using the one-at-a-time weighing scheme followed by the P-B scheme of Table 1. In total, 22 sets of each type of data were collected. As described in the simulated example, estimating the scale bias and precision for the two weighing strategies separately is not possible. However, we can use the combined data from both Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
59
Table 2. Days 14-35 Estimated Cumulative Weight Loss and Uncertainty (Standard Deviation) for the Two Weighing Schemes and the Combined Data (in g) bottle
one-at-a-time
Plackett-Burman
combined dataa
A B C D E F G H
1.3 ( 0.16 1.2 ( 0.16 0.7 ( 0.16 0.9 ( 0.16 1.1 ( 0.16 0.3 ( 0.16 1.5 ( 0.16 0.7 ( 0.16
1.275 ( 0.06 1.375 ( 0.06 0.775 ( 0.06 1.075 ( 0.06 1.175 ( 0.06 0.575 ( 0.06 1.575 ( 0.06 0.7 ( 0.16
1.323 ( 0.05 1.356 ( 0.05 0.789 ( 0.05 1.056 ( 0.05 1.189 ( 0.05 0.523 ( 0.05 1.589 ( 0.05 0.818 ( 0.05
a
One-at-a-time and P-B.
weighing schemes to estimate each of these quantities. The combined data weighing scheme has 16 equations and 9 unknowns: eight bottle weights and the unknown scale bias. In a multiple linear regression on this combined data (not forced through the origin), the intercept is an estimate of the scale bias. The bias estimation was performed by this method for each of the 22 weighing episodes. Over the 22 episodes, the scale bias was found to have an average value of -0.016 g. The standard deviation of this scale bias (sb) was 0.101 g. This sb represents a long-term variability in the scale’s measurement. The average of the absolute value of the estimated scale bias values was 0.081 g. In addition to obtaining an estimate of scale bias, it is also possible to use the combined data to estimate the scale precision during each weighing episode. During each episode, container H was weighed by itself twice, once for each of the two weighing schemes. Each pair of bottle H weighings provides an estimate of the scale precision which can be expressed as a standard deviation. The precision estimates from the 22 weighing episodes were pooled to give a value of 0.056 g, which is an estimate of the short-term variability, or the standard deviation of the scale precision, sp. The average bias, bias variability, and short-term variability are consistent with the maximum resolution of a scale that reports to 0.1 ( 0.05 g. By combining the values sp and sb it is possible to obtain an estimate of the standard deviation for any individual weighing (sind), whether w1-w8 or ao-ho:
sind ) xs2p + s2b ) x0.0562 + 0.1012 ) 0.116 g
(1)
For our scale, contrary to our initial expectation, the long-term variability of the bias dominated over the short-term variability. For the bottles in this permeation study, the weight loss rate started to level off at day 14. The cumulative weight loss of each bottle observed from day 14 through day 35 is shown in Table 2. The table includes the data obtained from the one-at-a-time and P-B weighing schemes as well as the results of combining the two schemes. The average estimated weight loss rate from each bottle is less than 0.1 g/day. Any weight loss calculation requires a subtraction be performed on two weight determinations separated by a time period. If the scale bias were the same at the beginning and end of the time period, determinations of weight loss would remove scale bias implicitly. The previous calculation of the variability of our 60
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
scale bias shows this was not the case for our data, and this variability must be included in any error estimates of the weight loss values in Table 2. For the one-at-a-time weighing method and bottle H of the P-B method, the uncertainty in the estimated weight loss expressed as a standard deviation, sdiff, would be the statistical propagation of errors of the standard deviation of two individual weighings, sind.
one-at-a-time sdiff ) x2s2ind ) x2 × 0.1162 ) 0.164 g
(2)
The P-B data standard deviation for the estimates ae, be, ce, de, ee, fe, and ge was obtained by using the values for sp and sb in a 1000-iteration numerical simulation on the P-B design scheme of Table 1. From the simulation, the standard deviation for an individual episode’s P-B weight estimate is 0.039 g. The statistical propagation of errors of two P-B estimates gives the standard deviation of the difference (P-B sdiff) calculated to be 0.055 g, as reported in Table 2. To confirm the validity of using a numerical simulation to calculate the P-B sdiff, the error estimates for the combined data were derived by two methods: by a numerical simulation similar to what was done for the P-B data and by pooling the standard deviations estimated from the 22 multiple linear regressions. These two sets of error estimates should be similar. For the numerical simulation using sp and sb on a combined one-at-a-time and P-B weighing scheme data set, the estimated combined sdiff obtained for bottles A-G was 0.045 g and 0.051 g for bottle H. The multiple linear regressions yield seven identical standard deviations for the weight estimates for bottles A-G and a different standard deviation for the bottle H weight estimate. The statistical propagation of errors of the standard deviations for days 14-35 gave 0.049 g for bottles A-G and 0.056 g for bottle H. For both research and commercial purposes, it is often desirable to measure or estimate the steady-state permeation rate performance for a set of containers. In bottle production facilities, the primary reason for performing permeation weighings is to determine whether a given production lot is acceptable for shipping. In order to qualify a production lot for shipment, it is necessary to estimate both the average steady-state performance and the bottle-to-bottle variability so that an estimated maximum expected permeation rate from the lot can be calculated using the average permeation rate plus 3- or 5σ. Production time constraints do not allow quality control permeation testing to run until steady state is attained. Usually, a quality control method has a specification for the maximum allowable weight loss rate after 3, 5, or 7 days. Whatever qualification time period is used, the permeation rate is obtained by establishing the weight loss from the bottles and subsequently dividing by the time taken for this weight loss. The uncertainty in estimating the weight loss value, sdiff, underlies the uncertainty in being able to estimate the performance of the containers being tested. As the qualification time period increases, both the cumulative weight loss increases and the time divisor used in calculating the permeation rate become larger. Combined, these will lower the impact of sdiff on the determined permeation rate. Table 3 presents the average permeation rates between days 14 and 35, along with the estimated impact of the sdiff values on the
Table 3. Days 14-35 Average Permeation Rates and Uncertainty Derived from sdiff for the Two Weighing Schemes (in mg/day) bottle A B C D E F G H average pooled error (obsd) estimated sdiff estimated sbottle
one-at-a-time 62.5 ( 7.89 57.7 ( 7.89 33.6 ( 7.89 43.3 ( 7.89 52.9 ( 7.89 14.4 ( 7.89 72.1 ( 7.89 33.6 ( 7.89 46.3 (18.69 (7.89 (16.95
Table 4. Days 14-23 Average Permeation Rates and Uncertainty Derived from sdiff for the Two Weighing Schemes (in mg/day)
Plackett-Burman 61.3 ( 2.64 66.1 ( 2.64 37.3 ( 2.64 51.7 ( 2.64 56.5 ( 2.64 27.6 ( 2.64 75.7 ( 2.64 33.6 ( 7.89 51.2 (16.94 (3.73 (av) (16.53
certainty with which the permeation rate can be known. Table 3 is obtained by dividing the values in Table 2 by the 20.8 day time interval for the weight loss. For this time span, a 0.1 g “error” in estimating the weight loss results in a 4.8 mg/day error in the permeation rate. The observed standard deviation of the averages (sobsd) in Table 3 is a pooled error value. It is composed of two effects, one due to weighing variations, sdiff, and the other to variations in bottleto-bottle performance (sbottle): 2 sobsd ) xs2bottle + sdiff
(3)
For both data sets in Table 3, most of the observed variation is clearly attributable to the bottle-to-bottle variability (sbottle is 16.94 mg/day using the one-at-a-time design and 16.53 mg/day using the P-B design strategy). The P-B design provides more precise results, but over this long a time interval the results from either design would be adequate in making a rational decision as to whether to ship the lot of bottles from which these eight were selected. It would also be a long enough time period for estimating the bottle-to-bottle variability for research purposes. To demonstrate the advantage of using an information-intense weighing strategy such as the P-B design of Table 1, we estimated the average permeation rate between days 14 and 23, which uses an 8.5- instead of a 20.8-day time divisor. The estimated permeation rates are given in Table 4, where the P-B sdiff value for bottles A-G is 6.47 mg/day and for bottle H is 19.30 mg/day. The average permeation rates in Table 4 are generally lower than in Table 3 because the bottles were further from steady state at day 23 than at day 35. In Table 4, a major advantage of the P-B design strategy becomes evident. With the P-B weighing design, one can solve for the bottle-to-bottle variability sbottle using eq 3. The value of 16.57 mg/day agrees well with the 16.53 mg/ day value obtained from the 20.8 day time interval. Thus, using the P-B design one can make a rational decision as to the performance of this lot of bottles. With the one-at-a-time weighing design it is not possible to determine the bottle-to-bottle variability, because it is hidden in the inaccuracies due to the weighing technique. This implicit time savings would be applicable to either a shortterm QC method or to a long-term study desiring a precise
bottle
one-at-a-time
Plackett-Burman
A B C D E F G H
70.66 47.11 23.55 35.33 47.11 11.78 58.88 23.55
55.94 55.94 26.50 50.05 50.05 20.61 73.60 23.55
average pooled error (obsd) estimated sdiff estimated sbottle
39.74 19.84 19.30 4.60
44.53 18.91 9.12 (av) 16.57
estimation of the steady-state permeation rate in the shortest possible time. To show the impact of an information-intense weighing strategy on a short-term quality control measurement, we calculated the permeation rates from the start of testing to 6.6 and 8.6 days on test for the two weighing designs. The rates are presented in Table 5 , where the P-B sdiff value for bottles A-G are 8.33 and 6.4 mg/day for days 6.6 and 8.6, respectively. The P-B design values are superior in that they clearly are more indicative of the bottle performance and variability shown in Tables 3 and 4. Due to the poor, high estimate for sdiff from the one-at-a-time design, an estimate could not be obtained from sbottle from eq 3. The largest error in the Table 5 P-B design estimations is introduced with bottle H. Excluding this bottle from the estimations gives observed average permeation of 55.49 ( 8.06 mg/day at day 6.6 and 50.08 ( 10.27 mg/day at day 8.6. From the one-at-a-time study, one has little indication of the bottle-tobottle variability and the calculated average permeation rate is both low and unstable. From a quality assurance perspective, it is necessary that the permeation of the containers be below a set value; i.e., the average steady-state permeation plus 3- or 5-σ must be less than a set threshold. If the 5 σ threshold value for these containers was less than 150 mg/day at steady state, then the uncertainty associated with the one-at-a-time design data in Table 5 would require additional time on test to reach a rational decision. From the Table 5 P-B design data, one would correctly conclude that the bottles would meet the acceptance criteria. To verify that the calculated advantages reported in Tables 2-5 were actually obtained for the empirical example, a number of 5-day time period estimates of the individual bottle permeation rates between days 14 and 35 were computed. The permeation rate of individual bottles over this 21-day period is nearly constant as they approach steady state. The observed variability of the permeation rate for a specific bottle should be primarily attributable to measurement-derived errors and be predictable by the preceding analysis. The average steady-state permeation rate in Table 3 was ∼50 mg/day and the bottle-to-bottle variability was ∼15-20 mg/day. A 5-day time period was selected to reduce the expected impact of the one-at-a-time sdiff on an individual estimate of the steadystate permeation rate to ∼33 mg/day. The impact of the P-B sdiff was estimated as ∼11 mg/day. From the nine weighing Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
61
Table 5. Permeation Rates from Day 0 to Days 6.6 and 8.6 for the Two Weighing Schemes (in mg/day) day 6.6 bottle
a
one-at-a-time
day 8.6 Plackett-Burman
one-at-a-time
Plackett-Burman
A B C D E F G H
45.25 15.08 15.08 30.17 15.08 15.08 45.25 15.08
64.11 56.57 49.03 64.11 56.57 41.48 56.57 0.00
46.35 34.77 23.18 46.35 34.77 23.18 57.94 34.77
60.84 49.25 43.46 49.25 55.05 31.87 60.84 23.18
average pooled error (obsd) estimated sdiff estimated sbottle
24.51 13.82 24.86 a
48.55 20.99 11.75 (av) 17.40
37.66 12.00 19.08 a
46.72 13.45 9.02 (av) 9.98
Due to the imprecision of the standard deviation estimates using the one-at-a-time design, it was not possible to calculate sbottle from eq 3.
Table 6. Estimated Steady-State Permeation Rate and Observed Standard Deviation Based on Five-Day Separations After Day 14 for the Two Weighing Schemes (in mg/day) bottle A B C D E F G H average pooled std dev (obsd) pooled std dev (calcd) average plus 3s
one-at-a-time
Plackett-Burman
83.1 ( 32.07 83.0 ( 27.58 58.6 ( 21.63 68.4 ( 23.90 73.4 ( 17.06 38.7 ( 47.51 93.0 ( 23.38 58.7 ( 27.49
62.5 ( 5.17 77.2 ( 16.35 40.4 ( 10.37 57.7 ( 9.25 65.1 ( 17.48 40.4 ( 12.98 82.1 ( 12.03 58.6 ( 27.11
69.62 28.90 32.22 156.3
60.50 12.57 15.23 (av) 98.2
episodes available after day 14, four pairs of independent weighing episodes (eight episodes in all) were selected that were separated by ∼5 days. The permeation rates calculated from these pairs may be thought of as individual estimates of the permeation rates in Tables 3 and 4 using a short-time baseline. The results are given in Table 6. The calculated pooled standard deviations for the permeation rates are also given in Table 6, where the value for the P-B design bottles A-G would be 10.81 mg/day. Not only are the calculated pooled values in reasonable agreement with the observed data, but the standard deviation for bottle H in the P-B design data is much larger than all of the others and matches well to the expected one-at-a-time design value. This confirms the advantages of using an information-intense weighing scheme like the P-B design. When it is considered that the 5-day baseline data were drawn preferentially from toward the end of the experiment, then the observed average permeation rates are also consistent with those of Tables 3 and 4. The actual steady-state permeation rate for these containers is likely to be closer to 60 mg/day than to 40 or 50 mg/day. SELECTING AN OPTIMAL DESIGN The experimental design construction strategy of D-optimality was used to optimize the precision improving properties of 62 Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
Table 7. Results of OPTEX Comparison of One-at-atime and Plackett-Burman Design Strategies design property
one-at-a-time
Plackett-Burman
D-efficiency (%) average variance of coefficients bias removal
65.80 0.222 no
100.0 0.125 yesa
a
A-G, Table 1.
multivariate weighing strategies versus their one-at-a-time counterparts. For a design of given size, D-optimality locates the weighing strategy that provides the most precise set of parameter estimates. The parameter estimates from the regression analysis are the object weights; hence, use of D-optimality provides the best possible average precision improvement for the entire set of weighings. The OPTEX procedure (SAS System for Windows, version 6.11) was used to evaluate the precision improving properties of proposed designs as well as the construction of D-optimal weighing strategies of a given size. Table 7 compares use of the one-at-a-time weighing strategy versus the use of the eight weighing P-B design. The traditional P-B design layout is modified in that a column was added for object H, the eighth object. As previously discussed, the P-B design is much more efficient than the one-at-a-time design. For precision improvement, the smaller the average variance of coefficients is (Table 7) the better, since the coefficients are the estimates of the bottle weights. An eight measurement P-B design was selected as being optimal for our purposes. It is the most efficient eight weighing design allowing a maximum of five objects on the scale. Its achievement of a 100% D-efficiency indicates that no other eight weighing strategy offers better precision improvement. This design was also selected because of its accuracy-improving properties. If the scale has an unknown fixed bias, use of the P-B design will implicitly remove this bias from the estimated weights of objects A-G, but not H (Table 1). This arises from object H being the only object weighed by itself (weighing eight) in conjunction with object H always being on the scale when multiple objects were weighed (weighings 1-7). Weighing eight provides both the weight of H as well as the scale bias and this total quantity is subtracted out from measurements 1-7. The information on
accuracy improvement was not provided by the OPTEX analysis. The efficiency of the P-B design does not simply originate with the use of multiple objects in each weighing, but rather from the structure of the entire set of weighings. Some multiweighing designs can result in a set of linear equations that do not have a unique solution; i.e., the individual weights cannot be calculated. Other designs can provide worse precision than even the one-ata-time strategy. For example, a simple five-bottle rotation weighing strategy; where objects A, B, C, D, and E are weighed followed by B, C, D, E, and F then C, D, E, F, and G and so on, is not particularly efficient, with an expected 50% degradation in precision as compared to the one-at-a-time weighing strategy. This bottle rotation design does however reduce any fixed scale bias by 80% on average since the same bias is now always associated with five bottles rather than a single bottle. The P-B design reduced scale bias by 87.5% on average (100% for objects A-G, 0% for object H). Prior to analysis of the data, our design focus was principally directed at precision improvement as it was anticipated that the weight loss determination would implicitly subtract out any longterm fixed bias. Consequently, direct bias removal was assumed to be of lesser importance. Only designs that required no more weighings than the one-at-a-time approach were considered. Such designs can only have an implicit accuracy improvement since scale bias cannot be explicitly estimated when no additional weighings are made. Our research indicates that precision improvement beyond that provided by the one-at-a-time weighing strategy was not possible until at least six objects are to be weighed. For five or fewer objects, the only gain from using a multivariate strategy (where the total number of weighings equals the number of objects) will come from bias removal/reduction. The amount of bias removal from any given individual estimated weight depends strongly on the structure of the multivariate design. A four weighing P-B design (Taguchi L4 array) can remove bias from three of the four objects being weighed. In general, no multivariate design can have worse bias removal properties than the one-at-a-time design when there is fixed scale bias. Once six or more objects are to be weighed with the number of weighings equal to the number of objects, precision improvement is always possible. For exactly six objects, the improvement is slight but increases quickly as the number of objects to be weighed increases. The more objects being weighed within an optimal multivariate design, the greater the precision improvement. For example, five uses of an eight weighing P-B design will not yield 40 weights that are as precisely estimated as the 40 weights from four D-optimal 10 weighing designs. However, practical limitations enter into the designs in terms of the maximum number of objects that the scale can handle as well as the on-off logistics of execution. A combination of these issues as well as the clearly defined accuracy improving properties of the eight weighing P-B design led to its utilization herein. As a consequence of finding the importance of bias removal for our scale, it is prudent to consider the bias removal properties of any design as well. D-Optimality can be used to locate an optimal precision and accuracy improving multivariate design under certain conditions. If an additional weighing beyond the minimum required is performed, it is possible either to further
improve precision or to explicitly estimate scale bias. This can be done by including scale bias as an intercept term in the system of equations to be solved or alternatively can be done through use of a standard. For example, if the object always on the scale in the P-B design consisted of a known weight, then the remaining seven weights can be estimated as well as the scale bias. Whether or not a bias term is included in the modeling, the precision improvement that can be obtained is a function of several things: (1) restrictions on the maximum number of objects that can be placed on the scale (fewer restrictions infer better or equal precision); (2) the total number of objects to be weighed (larger designs provide more efficiency, but become operationally more difficult); (3) how many, if any, measurements beyond the minimum number required are to be collected. Neither scale precision nor scale bias can be estimated when the number of weighings equals the number of objects to be weighed. We were able to explicitly estimate both precision and bias only because we had both the P-B and one-at-a-time data available from the same set of bottles. Such a combination of the P-B and one-at-a-time designs does not provide the optimal collection of 16 weighings for the estimation of eight object weights. Better precision and accuracy would have been obtained by simply repeating the P-B design twice (still no explicit estimate of scale bias) or by construction of a 16 weighing D-optimal design for eight objects and estimation of the scale bias. A simple way of maximizing the information from two uses of an eight weighing P-B design would be to swap columns A and H in Table 1 in the second execution of the design. This 16 weighing strategy allows explicit estimation of scale bias if required. Implicit to all prior discussion are a series of assumptions about the behavior of the scale: (1) Feasibility. It must be possible to weigh more than one object at a time on the scale. (2) Precision improvement. The scale weighs multiple objects with about the same precision as it does a single object. (3) Accuracy improvement. If there is an unknown scale bias, it is assumed to be fixed over a weighing episode or the set of weighings called for in the design. When these conditions are met, an optimized weighing design strategy can provide significant improvements in the weight determinations, improvements that can be of significant commercial importance. Violations to assumptions 2 and 3 may or may not eliminate the viability of the method. If precision degrades slowly enough with increased mass, then it is still possible to obtain a precision improvement. Analogously, if scale accuracy drifts little over time or is only a weak function of mass, then improvement is still possible. Our scale exhibited the required behaviors for both precision and accuracy improvement in the collected data.
Received for review April 1, 1997. Accepted October 6, 1997.X AC970348L X
Abstract published in Advance ACS Abstracts, December 1, 1997.
Analytical Chemistry, Vol. 70, No. 1, January 1, 1998
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