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Unformatted text preview: is. ighaptem 0 Evaluation of Analytical Data 53 /________________._————————— ;4.1 INTRODUCTION The field of food analysis, or any type of analysis, :involves a Considerable amount of time learning prin- iciples, methods, and instrument operations and per- , fecfing Various techniques. Although these areas are extremely important, much of our effort would be for tinaught if there were not some way for us to evaluate the ram obtained from the various analytical assays. Sev- eral mathematical treatments are available that provide an‘idea of how well a particular assay was performed gibrhow well we can reproduce an experiment. Fortu- nately, the statistics are not too involved and apply to ' most analytical determinations. , The focus in this chapter is primarily on how to evaluate replicate analyses of the same sample for accu- ‘acy and precision. In addition, considerable attention 3. to the determination of best line fits for stan- dard curve data. Keep in mind as you read and work : through this chapter that there is a vast array of com- . _ tel-software to perform most types of data evaluation and calculations / plots. VA ; Proper sampling and sample size are not covered _ in this chapter. Readers should refer to Chapter 5 (espe- ~ section 5.3.4.5) for sampling in general and statis- cal approaches to determine the appropriate sample . ,r'an‘d to Chapter 19, section 19.2.2 for mycotoxin } isanipling. MEASURES OF CENTRAL TENDENCY ‘ 3T9 mercase accuracy and precision, as well as to eval- * hate these parameters, the analysis of a sample is W 7 uallyiperformed (repeated) several times. At least assays are typically performed, though often the umber can be much higher. Because we are not sure hich value is closest to the true value, we determine "theemean (or average) using all the values obtained ‘and report the results of the mean. The mean is des- _ by the symbol 3': and calculated according to {,theequation below. x1+x2+x3+---+Xn 2x1 it = = [1] . . n 11 Where: 1‘ 3 5c 2 mean _ xi. 3C2, etc. 2 individually measured values (xi) 5 ' __ ,7 n = number of measurements ‘ _ For example, suppose we measured a sample . 0f uncooked hamburger for percent moisture content four times and obtained the following results: 64.53%, __64.45%, 65.10%, and 64.78%. - 64.53 + 64.45 65.10 64.78 x = + = ’ Thus, the result would be reported as 64.72% mois- ture. When we report the mean value, we are indicating that this is the best experimental estimate of the value. We are not saying anything about how accurate or true this value is. Some of the individual values may be closer to the true value, but there is no way to make that determination, so we report only the mean. Another determination that can be used is the median, which is the midpoint or middle number within a group of numbers. Basically, half of the experi- mental values will be less than the median and half will be greater. The median is not used often, because the mean is such a superior experimental estimator. 4.3 RELIABILITY OF ANALYSIS ___—.——————-—_ Returning to our previous example, recall that we obtained a mean value for moisture. However, we did not have any indication of how repeatable the tests were or how close our results were to the true value. The next several sections will deal with these questions and some of the relatively simple ways to calculate the answers. 4.3.1 Accuracy and Precision One of the most confusing aspects of data analysis for students is grasping the concepts of accuracy and preci- sion. These terms are commonly used interchangeably in society, which only adds to this confusion. If we consider the purpose of the analysis, then these terms become much clearer. If we look at our experiments, we know that the first data obtained are the individ- ual results and a mean value (it). The next questions should be: “How close were our individual measure— ments?” and “How close were they to the true value?” Both questions involve accuracy and precision. Now, let us turn our attention to these terms. Accuracy refers to how close a particular measure is to the true or correct value. In the moisture analy— sis for hamburger, recall that we obtained a mean of 64.72%. Let us say the true moisture value was actually 65.05%. By comparing these two numbers, you could probably make a guess that your results were fairly accurate because they were close to the correct value. (The calculations of accuracy will be discussed later.) The problem in determining accuracy is that most of the time we are not sure what the true value is. For certain types of materials we can purchase known samples from, for example, the National Institute of Standards and Technology, and check our assays . against these samples. Only then can we have an indica- tion of the accuracy of the testing procedures. Another approach is to compare our results with those of other labs to determine how well they agree, assuming the other labs are accurate. 54 Partl 0 General Information a 4-1 m A term that is much easier to deal with and deter— mine is precision. This parameter is a measure of how reproducible or how close replicate measurements become. If repetitive testing yields similar results, then we would say the precision of that test was good. From a true statistical view, the precision often is called error, when we are actually looking at experimental varia- tion. 50, the concepts of precision, error, and variation are closely related. The difference between precision and accuracy can be illustrated best with Fig. 4—1. Imagine shooting a rifle at a target that represents experimental values. The bull’ s eye would be the true value, and where the bullets hit would represent the individual experimental val- ues. As you can see in Fig. 4—1a, the values can be tightly spaced (good precision) and close to the bull’s eye (good accuracy), or, in some cases, there can be situations with good precision but poor accuracy (Fig. 4—1b). The worst situation, as illustrated in Fig. 4-1d, is when both the accuracy and precision are poor. In this case, because of errors or variation in the determination, interpre— tation of the results becomes very difficult. Later, the practical aspects of the various types of error will be discussed. When evaluating data, several tests are used com— monly to give some appreciation of how much the experimental values would vary if we were to repeat the test (indicators of precision). An easy way to look at the variation or scattering is to report the range of the experimental values. The range is simply the dif- ference between the largest and smallest observation. This measurement is not too useful and thus is seldom used in evaluating data. Probably the best and most commonly used sta- tistical evaluation of the precision of analytical data is the standard deviation. The standard deviation mea- sures the spread of the experimental values and gives a good indication of how close the values are to each other. When evaluating the standard deviation, one has to remember that we are never able to analyze the entire food product. That would be difficult, if not impossible, and very time consuming. Thus, the calculations we use are only estimates of the unknown true value. If we have many samples, then the standard devi- ation is designated by the Greek letter sigma (a). It is calculated according to Equation [3], assuming all the food product was evaluated (which would be an infinite amount of assays). ._ 2 o = [3] n where: o = standard deviation xi = individual sample values it = true mean n = total population of samples Because we do not know the value for the true mean, the equation becomes somewhat simplified so that we can use it with real data. In this case, we now call the a term the standard deviation of the sample and designate it by SD or a. It is determined according to the calculation in Equation [4], where 56 replaces the true mean term it, and n represents the number of samples. ._ “ 2 SD = ,/ ———E(x‘n x) [4] If the number of replicate determinations is small (about 30 or less), which is common with most assays, the n is replaced by the n — 1 term, and Equation [5] is used. Unless you know otherwise, Equation [5] is always used in calculating the standard deviation of a group of assays. . _ ' 2 . SD = M [5] V n — 1 Depending on which of the equations above is used, the standard deviation may be reported as SDn or on and SDn_1 or an_1. (Different brands of software Mm ‘ "wsu's‘vzmdstmmfl ":smamm‘ml fihm,rm:‘“xii'1§;3’ Mir: b c d Comparison of accuracy and precision: (a) good accuracy and good precision, (b) good precision and poor accuracy, (c) good accuracy and poor precision, and (d) poor accuracy and poor precision. _ ._.. AM“. A.“ ‘WA. m“th Chapter 4 0 Evaluation of Analytical Data 55 Determination of the Standard Deviation 4" l V of Percent Moisture in Uncooked Hamburger Deviation > from 8 Observed the Mean 3 Measurement % Moisture (x,- — R) (x,- — {02 g u. 1 64.53 —0.1 9 0.0361 2 64.45 ~O.27 0.0729 3 65.10 +0.38 0.1444 4 64.78 +0.06 0.0036 2x. = 258.86 2(xi 402 = 0.257 .aa -20 -1a True Value +10 +20 +3., (mean) )-(_£X_i_:258.86_6472 ‘ ’ ‘ — n 4 4_2 A normal distribution curve for a population or , and scientific calculators sometimes use different labels , for the keys, so one must be careful.) Table 4-1 shows nan example of the determination of standard devia- . ,. tion. The sample results would be reported to‘ average ‘ 64.72% moisture with a standard deviation of 0.293. ' ” Once we have a mean and standard deviation, we finext must determine how to interpret these numbers. I - 'v One easy way to get a feel for the standard deviation is g to calculate what is called the coefficient of variation I. _'(CV), also known as the relative standard deviation. ' i. calculation is shown below for our example of the moisture determination of uncooked hamburger. Coefficient of Variation (CV) = 5—5?— x 100% [6] a I, ,, 7 _ 0.293 {is CV _ 6472 x 100% = 0.453% [7] The CV tells us that our standard deviation is only , - 0.453% as large as the mean. For our example, that num- ‘ _ her is small, which indicates a high level of precision or improducibility of the replicates. As a rule, a CV below ,' ‘_ ~ 15:" 5% is considered acceptable, although it depends on the ‘ L a of analysis. Another way to evaluate the meaning of the stan- 9 ,dard deviation is to examine its origin in statistical _ fl'theory. Many populations (in our case, sample values 3 ‘y _, _ ‘ 01' means) that exist in nature are said to have a normal 1 u distribution. If we were to measure an infinite num- fiber of samples, we would get a distribution similar to I I that represented by Fig. 4—2. In a population with a nor- ” ‘ _ mal distribution, 68% of those values would be within ‘ " ._ ,:l:'l standard deviation from the mean; 95% would be ‘ Within :l:3 standard deviations. In other words, there is .a probability of less than 1% that a sample in a popu- _ I ,lation would fall outside :l:3 standard deviations from E - I iglhe mean value. ‘ :Within :l:2 standard deviations, and 99.7% would be ‘ a group of analyses. 4'2 Values for Z for Checking both It!!! Upper and Lower Levels Degree of Certainty Z Value (Confidence) 80% 1 .29 90% 1 .64 95% 1 .96 99% 2.58 99.9% 3.29 Another way of understanding the normal distri- bution curve is to realize that the probability of finding the true mean is within certain confidence intervals as defined by the standard deviation. For large numbers of samples, we can determine the confidence limit or interval around the mean using the statistical param— eter called the Z value. We do this calculation by first looking up the Z value from statistical tables once we have decided the desired degree of certainty. Some Z values are listed in Table 4-2. The confidence limit (or interval) for our moisture data, assuming a 95% probability, is calculated accord- ing to Equation [8]. Since this calculation is not valid for small numbers, assume we ran 25 samples instead of four. Confidence Interval (CI) = 5c i Z value standard deviation (SD) X fl [8] CI( t95‘7) 64 72 :l: 1 96 x 02927 a o — . . m = 64.72 :t 0.115% [9] Partl 0 General Information 4‘3 Values of "or Various Levels of m Probability' Degrees of Levels of Certainty Freedom (n ~ 1) 95% 99% 99.9% 1 12.7 63.7 636 2 4.30 9.93 31.60 3 3.18 5.84 12.90 4 2.78 4.60 8.61 5 2.57 4.03 6.86 6 2.45 3.71 5.96 7 2.36 3.50 5.40 8 2.31 3.56 5.04 9 2.26 3.25 4.78 10 2.23 3.17 4.59 1 More extensive t-tables can be found in statistics books. Because our example had only four values for the mois— ture levels, the confidence interval should be calculated using statistical t tables. In this case, we have to look up the t value from Table 4—3 based on the degrees of freedom, which is the sample size minus one (71 — 1), and the desired level of confidence. The calculation for our moisture example with four samples (n) and 3 degrees of freedom (11 — 1) is given below. standard deviation (SD) CI : 5c :l: t value X ——Jfi—— [10] CI (at 95%) = 64.72 :i: 3.18 X 02927 J4 = 64.72 :I: 0.465% [11] To interpret this number, we can say that, with 95% con— fidence, the true mean for our moisture will fall within 64.72 :t: 0.465% or between 65.185 and 64.255%. The expression SD / ¢n is often reported as the stan- dard error of the mean. It then is left to the reader to calculate the confidence interval based on the desired level of certainty. Other quick tests of precision used are the relative deviation from the mean and the relative average devi- ation from the mean. The relative deviation from the mean is useful when only two replicates have been per- formed. It is calculated according to Equation [12], with values below 2% considered acceptable. xi—ic Relative deviation from the mean = x 100 [12] x The xi represents the individual sample value, and 5c is the mean. If there are several experimental values, then the relative average deviation from the mean becomes a useful indicator of precision. It is calculated similarly to the relative deviation from the mean, except the average deviation is used instead of the individual deviation. It is calculated according to Equation [13]. Relative average deviation from the mean = x 1000 n/x parts per thousand ' [13] Using the moisture values discussed in Table 4—1, the xi —5c terms for each determination are —0.19, —0.27, +0.38, +0.06. Thus, the calculation becomes: 0.19 + 0.27 -l- 0.38 + 0.06 R l. . . = ———— 1000 e avg dev 11/6472 x 0.225 _ 64.72 X 1000 = 3.47 parts per thousand [14] Up to now, our discussions of calculations have involved ways to evaluate precision. If the true value is not known, we can calculate only precision. A low degree of precision would make it difficult to predict a realistic value for the sample. However, we occasionally may have a sample for which we know the true value and can compare our results with the known value. In this case, we can cal- culate the error for our test, compare it to the known value, and determine the accuracy. One term that can be calculated is the absolute error, which is simply the difference between the experimental value and the true value. Absolute error 2 Babs = x — T [15] where: x = experimentally determined value T = true value The absolute error term can have either a positive or negative value. If the experimentally determined value is from several replicates, then the mean (0) would be substituted for the x term. This is not a good test for error, because the value is not related to the magnitude of the true value. A more useful measurement of error is relative error. E x — T Relative error = Ere] = [25 = T T [161 The results are reported as a negative or positive value, which represents a fraction of the true value. If desired, the relative error can be expressed as percent relative error by multiplying by 100%. Then the relationship becomes the following, where x can be w——— Chapter4 0 Evaluation of Analytical Data 57 either an individual determination or the mean (0) of several determinations. ' x—T abs %E,el = x 100% = x 100% [17] Using the data for the percent moisture of uncooked hamburger, suppose the true value of the sample is 65.05%. The percent relative error is calculated using our mean value of 64.72% and Equation [17]. 5c — T 64.72 — 65.05 X 100% = 65.05 x 100% %Erel = = —0.507% [18] Note that we keep the negative value, which indicates the direction of our error, that is, our results were 0.507% lower than the true value. 4.3.2 Sources of Errors As you may recall from our discussions of accuracy and precision, error (variation) can be quite important in analytical determinations. Although we strive to obtain correct results, it is unreasonable to expect an analyt— ical technique to be entirely free of error. The best we can hope for is that the variation is small and, if possi- ble, at least consistent. As long as we know about the ‘ error, the analytical method often will be satisfactory. There are several sources of error, which can be clas— sified as: systematic error (determinate), random error - (indeterminate), and gross error or blunders. Again, note that error and variation are used interchangeably in this section and essentially have the same meaning for these discussions. Systematic or determinate error produces results that consistently deviate from the expected value in one direction or the other. As illustrated in Fig. 4-1b, the results are spaced closely together, but they are consistently off the target. Identifying the source of this serious type of error can be difficult and time ., consuming, because it often involves inaccurate instru- ments or measuring devices. For example, a pipette that consistently delivers the wrong volume of reagent Will produce a high degree of precision yet inaccurate results. Sometimes impure Chemicals or the analyt- ical method itself are the cause. Generally, we can overcome systematic errors by proper calibration of instruments, running blank determinations, or using a different analytical method. Random or indeterminate errors are always \present in any analytical measurement. This type of error is due to our natural limitations in measuring a particular system. These errors fluctuate in a random ' fashion and are essentially unavoidable. For example, reading an analytical balance, judging the endpoint ‘ 7 change in a titration, and using a pipette all contribute ‘ 4—_ to random error. Background instrument noise, which is always present to some extent, is a factor in random error. Both positive and negative errors are equally pos— sible. Although this type of error is difficult to avoid, fortunately it is usually small. Blunders are easy to eliminate, since they are ’so obvious. The experimental data are usually scat- tered, and the results are not close to an expected value. This type of error is a result of using the wrong reagent or instrument or of sloppy technique. Some people have called this type of error the “Monday morning syndrome” error. Fortunately, blunders are easily identified and corrected. ‘ 4.3.3 Specificity Specificity of a particular analytical method means that it detects only the component of interest. Analytical methods can be very specific for a certain food compo- nent or, in many cases, can analyze a broad spectrum of components. Quite often, it is desirable for the method to be somewhat broad in its detection. For example, the determination of food lipid (fat) is actually the crude analysis of any compound that is soluble in an organic solvent. Some of these compounds are glycerides, phos— pholipids, carotenes, and free fatty acids. Since we are not concerned about each individual compound when considering the crude fat content of food, it is desirable that the method be broad in scope. On the other hand, determining the lactose content of ice cream would require a specific method. Because ice cream contains other types of simple sugars, without a specific method we would overestimate the amount of lactose present. There are no hard rules for what specificity is required. Each situation is different and depends on the desired results and type of assay used. However, it is something to keep in mind as the various analytical techniques are discussed. 4.3.4 Sensitivity and Detection Limit Although often used interchangeably, the terms sensi- tivity and detection limit should not be confused. They have different meanings, yet are closely related. Sensi- tivity relates to the magnitude of change of a measuring device (instrument) with changes in compound con— centration. It is an indicator of how little change can be made in the unknown material before we notice a difference on a needle gauge or a digital readout. We are all familiar with the process of tuning in a radio sta— tion on our stereo and know how, at some point, once the station is tuned in, we can move the dial without disturbing the reception. This is sensitivity. In many situations, we can adjust the sensitivity of an assay to fit our needs, that is, whether we desire more or less sensitivity. We even may desire a lower sensitivity so 58 Partl 0 General Information ‘\ that samples with widely varying concentration can be analyzed at the same time. ' Detection limit, in contrast to sensitivity, is the low- est possible increment that we can detect with some degree of confidence (or statistical significance). With every assay, there is a lower limit at which point we are not sure if something is present or not. Obviously, the best choice would be to concentrate the sample so we are not working close to the detection limit. However, this may not be possible, and we may need to know the detection limit so we can work away from that limit. There are several ways to measure the detection limit, depending on the apparatus that is used. If we are using something like a spectrophotometer, gas chro- matograph, or high performance liquid chromatograph (HPLC) the limit of detection often is reached when the signal to noise ratio is 3 or greater. In other words, when the sample gives a value that is three times the magnitude of the noise detection, the instrument is at the lowest limit possible. Noise is the random signal fluctuation that occurs with any instrument. A more general way to define the limit of detection is to approach the problem from a statistical viewpoint, in which the variation between samples is considered. A common mathematical definition of detection limit is given below. XLD = XBlk + 3 X SDBlk [19] where: XLD = the minimum detectable concentration Xglk = the signal of a blank SDglk = the standard deviation of the blank readings In this equation, the variation of the blank values (or noise, if we are talking about instruments) deter- mines the detection limit. High variability in the blank values decreases the limit of detection. 4.4 CURVE FITTING: REGRESSION ANALYSIS '————_— Curve fitting is a generic term used to describe the rela- tionship and evaluation between two variables. Most scientific fields use curve fitting procedures to evaluate the relationship of two variables. Thus, curve fitting or curvilinear analysis of data is a vast area as evidenced by the volumes of material describing these procedures. In analytical determinations, we are usually concerned with only a small segment of curvilinear analysis, the standard curve or regression line. A standard curve or calibration curve is used to determine unknown concentrations based on a method that gives some type of measurable response that is proportional to a known amount of standard. It typ- ically involves making a group of known standards in increasing concentration and then recording the partic- ular measured analytical parameter (e. g., absorbance, area of a chromatography peak, etc.). What results when we graph the paired x and y values is a scatterplot of points that can be joined together to form a straight line relating concentration to observed response. Once we know how the observed values change with concen— tration, it is fairly easy to estimate the concentration of an unknown by interpolation from the standard curve. As you read through the next three sections, keep in mind that not all correlations of observed values to stan- dard concentrations are linear (but most are). There are many examples of nonlinear curves, such as antibody binding, toxicity evaluations, and exponential growth and decay. Fortunately, with the vast array of computer software available today, it is relatively easy to analyze any group of data. 4.4.1 Linear Regression So how do we set up a standard curve once the data have been collected? First a decision must be made regarding onto which axis to plot the paired sets of data. Traditionally the concentration of the standards is rep- resented on the x-axis and the observed readings are on the y-axis. However, this protocol is used for rea— sons other than convention. The x-axis data are called the independent variable and assumed to be essen- tially free of error, while the y-axis data (the dependent variable) may have error associated with them. This assumption may not be true because error could be incorporated as the standards are made. With modern day instruments the error can be very small. Although arguments can be made for making the y-axis data con— centration, for all practical purposes the end result is essentially the same. Unless there are some unusual data, the concentration should be associated with the x-axis and the measured values with the y-axis. Figure 4—3 illustrates a typical standard curve used in the determination of caffeine in various foods. Caffeine is analyzed readily in foods by using HPLC coupled with an ultraviolet detector set at 272 nm. The area under the caffeine peak at 272 nm is directly proportional to the concentration. When an unknown sample (e.g., coffee) is run on the HPLC, a peak area is obtained that can be related back to the sample using the standard curve. The plot in Fig. 4—3 shows all the data points and a straight line that appears to pass through most of the points. The line almost passes through the origin, which makes sense because zero concentration should produce no signal at 272 nm. However, the line is not perfectly straight (and never is) and does not quite pass through the origin. To determine the caffeine concentration in a sample that gave an area of say 4000, we could extrapolate to the Chapter4 0 Evaluation otAnaIytical Data 59 Peak Area 6 272 nm ‘0 60 Concentration (ppm) 80 A typical standard curve plot showing the data points and the generated best fit line. The data used to plot the curve are presented on the graph. line and then draw a line down to the x-axis. Following a line to the x-axis (concentration), we can estimate the solution to be at about 42—43 ppm of caffeine. We can mathematically determine the best fit of the line by using linear regression. Keep in mind the equa- tion for a straight line, which is: y = ax + b, where a = slope and b = y-intercept. To determine the slope and y-intercept, the regression equations shown below are used. We determine a and b and, thus, for any value of X y (measured), we can determine the concentration (x). 2(xi h 5C)(yi - 3?) mm—mz y-intercept b = g — aft [21] slope a = [20] The xi and yi parameters are the individual values, and 5c and g are the means of the individual values. Low- Cost calculators and computer spreadsheet software can , readily calculate regression equations so no attempt is made to go through the mathematics in the formulas. The formulas give what is known as the line of regression of y on x which assumes that the error occurs in the y direction. The regression line represents the average relationship between all the data points and thus is a balanced line. These equations also assume that the straight line fit does not have to go through the origin, which at first does not make much sense. However, there are often background interferences so that even at zero concentration a weak signal may be ‘ Observed. In most situations, calculating the origin as going through zero will yield the same results. > Using the data from Fig. 4—3, calculate the concen- tration of caffeine in the unknown and compare with the graphing method. As you recall, the unknown had an area at 272 nm of 4000. Linear regression analysis Y x x A4_4 Examples of standard curves showing the rela- tionship between the x and y variables when IE3. there is (a) a high amount of correlation and (b) a lower amount of correlation. Both lines have the same equation. of the standard curve data gave the y-intercept (b) as 84.66118 and the slope (a) as 90.07331. y = ax + b [22] or b x=y_ mm a 4000 ~ 84.66118 = —— = 4 .4 ff ' x (conc) 90.07331 3 68 ppm ca e1n[e ] 24 The agreement is fairly close when comparing the calculated value to that estimated from the graph. Using high—quality graph paper with many lines could give us a line very close to the calculated one. However, as we will see in the next section, additional informa- tion can be obtained about the nature of the line when using computer software or calculators. 4.4.2 Correlation Coefficient In observing any type of correlation, including linear ones, questions always surface concerning how to draw the line through the data points and how well the data fit to the straight line. The first thing that should be done with any group of data is to plot it to see if the points fit a straight line. By just looking at the plotted data, it is fairly easy to make a judgment on the linearity of the line. We also can pick out regions on the line where a linear relationship does not exist. The figures below illustrate differences in standard curves; Fig. 4—4a shows a good correlation of the data and Fig. 4-4b shows a poor correlation. In both cases, We can draw a straight line through the data points. Both curves yield the same straight line but the precision is poorer for the latter. There are other possibilities when working with standard curves. Figure 4—5a shows a good correlation between x and 3/ but in the negative direction, and Fig. 4—5b illustrates data that have no correlation at all. 60 Partl 0 General Information X X 4_5 , Examples of standard curves showing the rela— tionship between the x and y variables when m there is (a) a high amount of negative correlation and (b) no correlation between x and y values. The correlation coefficient defines how well the data fit to a straight line. For a standard curve, the ideal situation would be that all data points lie per- fectly on a straight line. However, this is never the case, because errors are introduced in making standards and measuring the physical values (observations). The correlation coefficient and coefficient of deter- mination are defined below. Essentially all spread- sheet and plotting software will calculate the values automatically. correlation coefficient = r = v [2(xi — 3021mm — W] [25] For our example of the caffeine standard curve from Fig. 4—3: r = 0.99943 (values are usually reported to at least 4 significant figures) For standard curves, we want the value of r as close to +1.0000 or —1.000 as possible, because this value is a perfect correlation (perfect straight line). Generally, in analytical work, the r should be 0.9970 or better. (This does not apply to biological studies.) The coefficient of determination (r2) is used quite often because it gives a better perception of the straight line even though it does not indicate the direction of the correlation. The r2 for the example presented above is 0.99886, which represents the proportiOn of the vari- ance of absorbance (y) that can be attributed to its linear regression on concentration (x). This means that about 0.114% of the straight line variation (1.0000 — 0.99886 2 0.00114 X 100% = 0.114%) does not vary with changes in x and y and, thus, is due to indeterminate variation. A small amount of variation is expected normally. 4.4.3 Errors in Regression Lines While the correlation coefficient tells us something about the error or variation in linear curve fits, it does __————-——-— not always give the complete picture. Also, neither lin- ear regression nor correlation coefficient will indicate that a particular set of data have a linear relationship. They only provide an estimate of the fit assuming the line is a linear one. As indicated before, plotting the data is critical when looking at how the data fit on the curve (actually, a line). One parameter that is used often is the y—residuals, which are simply the differences between the observed values and the calculated or computed values (from the regression line). Advanced computer graphics software can actually plot the residuals for each data point as a function of concentration. How- ever, plotting the residuals is usually not necessary because data that do not fit on the line are usually quite obvious. If the residuals are large for the entire curve, then the entire method needs to be evaluated carefully. However, the presence of one point that is obviously off the line while the rest of the points fit very well probably indicates an improperly made standard. One way to reduce the amount of error is to include more replicates of the data such as repeating the obser- vations with a new set of standards. The replicate x and 3/ values can be entered into the calculator or spreadsheet as separate points for the regression and coefficient determinations. Another, probably more desirable, option is to expand the concentrations at which the readings are taken. Collecting observations at more data points (concentrations) will produce a better standard curve. However, increasing the data beyond seven or eight points usually is not beneficial. Plotting confidence intervals, or bands or limits, on the standard curve along with the regression line is another way to gain insight into the reliability of the standard curve. Confidence bands define the statistical uncertainty of the regression line at a chosen probabil— ity (such as 95%) using the t—statistic and the calculated standard deviation of the fit. In some aspects, the con- fidence bands on the standard curve are similar to the confidence interval discussed in section 4.3.1. However, in this case we are looking at a line rather than a con- fidence interval around a mean. Figure 4—6 shows the caffeine data from the standard curve presented before, except some of the numbers have been modified to enhance the confidence bands. The confidence bands (dashed lines) consist of both an upper limit and a lower limit that define the variation of the y-axis value. The upper and lower bands are narrowest at the center of the curve and get wider as the curve moves to the higher or lower standard concentrations. Looking at Fig. 4—6 again, note that the confidence bands show what amount of variation we expect in a peak area at a particular concentration. At 60 ppm concentration, by going up from the x-axis to the bands and extrapolating to the y—axis, we see that with our data the 95% confidence interval of the observed peak area will be 4000 to 6000. In this case, the variation is P Chapter4 0 Evaluation of Analytical Data 61 ’______________________________———_————— Upper limit . E. I , sooo y s 30.386 (.x) . 244.29 \, I r = 0.98183 1 E 5000 ’ ’ , ’ \ N y .a , N - Lower limit a z , I a 7 2 “mo I 2 Peak- < I : x : WM 3 I ; 5 :no “- 2000 , ’ 5 25 2100 r g 50 4795 ' 75 6950 7565 o 20 40 so so too Concentration (ppm) 4 _ 6 A standard curve graph showing the confidence bands. The data used to plot the graph are pre- sented on the graph as are the equation of the line and the correlation coefficient. Real Outlier Measured Value Concentration 4_7 A standard curve plot showing possible devi- ations in the curve in the upper and lower limits. { large and would not be acceptable as a standard curve, ‘ and is presented here only for illustration purposes. g Error bars also can be used to show the variation of I 'y at each data point. Several types of error or variation ' statistics can be used such as standard error, stan— dard deviation, or percentage of data (i.e., 5%). Any of these methods give a visual indication of experimental variation. Even with good standard curve data, problems can arise if the standard curve is not used properly. One common mistake is to extrapolate beyond the data points used to construct the curve. Figure 4—7 illustrates Some of the possible problems that might occur when extrapolation is used. As shown in Fig. 4-7, the curve or line may not be linear outside the area where the data were collected. This can occur in the region close to the origin or especially at the higher concentration level. Usually a standard curve will go through the ori- gin, but in some situations it may actually tail off as zero concentration is approached. At the other end of the curve, at higher concentrations, it is fairly common for a plateau to be reached where the measured parameter does not change much with an increase in concentra- tion. Care must be used at the upper limit of the curve to ensure that data for unknowns are not collected out- side of the curve standards. Point Z on Fig. 4-7 should be evaluated carefully to determine if the point is an outlier or if the curve is actually tailing off. Collect- ing several sets of data at even higher concentrations should clarify this. Regardless, the unknowns should be measured only in the region of the curve that is linear. 4.5 REPORTING RESULTS In dealing with experimental results, we always are confronted with reporting data in a way that indicates the sensitivity and precision of the assay. Ideally, we do not want to overstate or understate the sensitivity of the assay, and thus strive to report a meaningful value, be it a mean, standard deviation, or some other number. The next three sections discuss how we can evaluate experimental values so as to be precise when reporting results. 4.5.1 Significant Figures The term significant figure is used rather loosely to describe some judgment of the number of reportable digits in a result. Often, the judgment is not soundly based, and meaningful digits are lost or meaningless digits are retained. Exact rules are provided below to help determine the number of significant figures to report. However, it is important to keep some flexibility when working with significant figures. Proper use of significant figures is meant to give an indication of the sensitivity and reliability of the ana- lytical method. Thus, reported values should contain only significant figures. A value is made up of signifi- cant figures when it contains all digits known to be true and one last digit that is in doubt. For example, a value reported as 64.72 contains four significant figures, of which three digits are certain (64.7) and the last digit is uncertain. Thus, the 2 is somewhat uncertain and could be either 1 or 3. As a rule, numbers that are presented in a value represent the significant figures, regardless of the position of any decimal points. This also is true ' for values containing zeros, provided they are bounded on either side by a number. For example, 64.72, 6.472, 0.6472, and 6.407 all contain four significant figures. Note that the zero to the left of the decimal point is — 62 Partl 0 General Information M used only to indicate that there are no numbers above 1. We could have reported the value as .6472, but using the zero is better, since we know that a number was not inadvertently left off our value. Special considerations are necessary for zeros that may or may not be significant. 1. Zeros after a decimal point are always signifi- cant figures. For example, 64.720 and 64.700 both contain five significant figures. 2. Zeros before a decimal point with no other pre- ceding digits are not significant. As indicated before, 0.6472 contains four significant figures. 3. Zeros after a decimal point are not significant if there are no digits before the decimal point. For example, 0.0072 has no digits before the decimal point; thus, this value contains two significant figures. In contrast, the value 1.0072 contains five significant figures. 4. Final zeros in a number are not significant unless indicated otherwise. Thus, the value 7000 contains only one significant figure. However, adding a decimal point and another zero gives the number 7000.0, which has five significant figures. A good way to measure the significance of zeros, if the above rules become confusing, is to convert the number to the exponential form. If the zeros can be omitted, then they are not significant. For example, 7000 expressed in exponential form is 7 x 103 and con- tains one significant figure. With 70000, the zeros are retained and the number becomes 7.0000 x 103. If we were to convert 0.007 to exponent form, the value is 7 x 10—3, and only one significant figure is indicated. As a rule, determining significant figures in arithmetic operations is dictated by the value having the least number of significant figures. The easiest way to avoid any confusion is to perform all the calculations and then round off the final answer to the appropriate digits. For example, 36.54 x 238 x 1.1 = 9566.172, and because 1.1 contains only two significant figures, the answer would be reported as 9600 (remember, the two zeros are not significant). This method works fine for most cal- culations, except when adding or subtracting numbers containing decimals. In those cases, the number of sig— nificant figures in the final value is determined by the numbers that follow the decimal point. Thus, adding 7.45 + 8.725 = 16.175; because 7.45 has only two num- bers after the decimal point, the sum is rounded to 16.18. Likewise, 433.8 — 32.66 gives 401.14, which rounds off to 401.1. A word of caution is warranted when using the simple rule stated above, for there is a tendency to underestimate the significant figures in the final answer. For example, take the situation in which we determined the caffeine in an unknown solution to be 43.5 ppm ________ (see Equation [24]). We had to dilute the sample 50-fold using a volumetric flask in order to fit the unknown within the range of our method. To calculate the caffeine in the original sample, we multiply our result by 50 or 43.5 jig/m1 x 50 = 2175 jig/ml in the unknown. Based on our rule above, we then would round the number to one significant figure (because 50 contains one sig- nificant figure) and report the value as 2000. However, doing this actually underestimates the sensitivity of our procedure, because we ignore the accuracy of the volu- metric flask used for the dilution. A Class-A volumetric flask has a tolerance of 0.05 ml; thus, a more reasonable way to express the dilution factor would be 50.0 instead of 50. We now have increased the significant figures in the answer by two, and the value becomes 21 80 mg / ml. As you can see, an awareness of significant figures and how they are adopted requires close inspection. The guidelines can be helpful but they do not always work, unless each individual value or number is closely inspected. 4.5.2 Rejecting Data Inevitably, during the course of working with experi- mental data we will come across a value that does not match the others. Can you reject that value, and thus not use it in calculating the final reported results? The answer is "sometimes," but only after careful consideration. If you are routinely rejecting data to help make your assay look better, then you are misrepresent- ing the results and the precision of the assay. If the bad value resulted from an identifiable mistake in that par- ticular test, then it is probably safe to drop the value. Again, caution is advised, because you may be reject- ing a value that is closer to the true value than some of the other values. Consistently poor accuracy or precision indicates that an improper technique or incorrect reagent was used or that the test was not very good. It is best to make changes in the procedure or change methods rather than try to figure out ways to eliminate undesirable values. There are several tests for rejecting an aberrant value. One of these tests, the Q-test, is commonly used. In this test, a Q-value is calculated as shown below and compared to values in a table. If the calculated value is larger than the table value, then the questionable mea- surement can be rejected at the 90% confidence level. 962—361 Q-value = [26] where: x1 = the questionable value x2 = the next closest value to x1 W=the total spread of all values; obtained by subtracting the lowest value from the highest value. w Chapter4 0 Evaluation of Analytical Data 63 4‘4 _ Q—Values for the Rejection ot Results Number of O of Rejection Observations (90% level) 3 0.94 4 0.76 5 0.64 6 0.56 7 0.51 8 0.47 9 0.44 ‘l 0 0.41 Reprinted with permission from RB. Dean and W..J. Dixon. 1951. Simplified statistics for small num- bers of observations. Analytical Chemistry 23: 686— 638. Copyright 1951, American Chemical Society. Table 4—4 provides the rejection Q-values for a 90% confidence level. The example below shows how the test is used for the moisture level of uncooked hamburger for which four replicates were performed giving values of 64.53, 64.45, 64.78, and 55.31. The 55.31 value looks as if it is too low compared to the other results. Can that value be rejected? For our example, x1 = the questionable value = 55.31, and xz is the closest neighbor to x1 (which is 64.45). The spread (W) is the high value minus the low measurement, which is 64.78 — 55.31. 64.45 — 55.31 9.14 64.78 — 55.31 ‘ E1? 2 0'97 [27] Q—value = From Table 4—4, we see that the calculated Q-value must be greater than 0.76 to reject the data. Thus, we make the decision to reject the 55.31% moisture value and do not use it in calculating the mean. 4.6 SUMMARY This chapter focused on the basic mathematical treat- ment that most likely will be used in evaluating a group of data. For example, it should be almost second nature to determine a mean, standard deviation, and CV when evaluating replicate analyses of an individ- ual sample. In evaluating linear standard curves, best line fits always should be determined along with the indicators of the degree of linearity (correlation coeffi- cient or coefficient of determination). Fortunately, most computer spreadsheet and graphics software will read- ily perform the calculations for you. Guidelines are available to enable one to report analytical results in a way that tells something about the sensitivity and confidence of a particular test. These include the proper use of significant figures, rules for rounding off num- bers, and use of the Q—test to reject grossly aberrant individual values. 4.7 STUDY QUESTIONS 1. Method A to quantitate a particular food component was reported to be more specific and accurate than method B, but method A had lower precision. Explain what this means. 2. You are considering adopting a new analytical method in your lab to measure moisture content of cereal prod- ucts. How would you determine the precision of the new method and compare it to the old method? Include any equations to be used for any needed calculations. 3. A sample known to contain 20 g/ L glucose is analyzed by two methods. Ten determinations were made for each method and the following results were obtained: Method A Method B Mean 2 19.6 Std. Dev. = 0.055 Mean = 20.2 Std. Dev. 2 0.134 a. Precision and accuracy: i. Which method is more precise? Why do you say this? ii. Which method is more accurate? Why do you say this? b. In the equation to determine the standard deviation, n — 1 was used rather than just 11. Would the standard deviation have been smaller or larger for each of those values above if simply 11 had been used? c. You have determined that values obtained using Method B should not be accepted if outside the range of two standard deviations from the mean. What range of values will be acceptable? d. Do the data above tell you anything about the speci- ficity of the method? Describe what “specificity” of the method means, as you explain your answer. 4. Differentiate “standard deviation” from "coefficient of variation,” “standard error of the mean,” and "confidence interval.” 5. Differentiate the terms “absolute error” versus “relative error.” Which is more useful? Why? 6. For each of the errors described below in performing an analytical procedure, classify the error as random error, systematic error, or blunder, and describe a way to overcome the error. at. Automatic pipettor consistently delivered 0.96 ml rather than 1.00 ml. b. Substrate was not added to one tube in an enzyme assay. 7. Differentiate the terms “sensitivity” and “detection limit.” 8. The correlation coefficient for standard curve A is reported as 0.9970. The coefficient of determination for standard curve B is reported as 0.9950. In which case do the data better fit a straight line? — 64 pan] o General lnformatio 4.3 PRACTICE-PROBLEMS 1. How many significant figures are in the following num— ’ bers: 0.0025, 4.50, 5.607? 2. What is the correct answer for the following calculation expressed in the proper amount of significant figures? 2.43'x 0.01672 _ 1.83215 ‘ 3. Given the following data on dry matter (88.62, 88.74, 89.20, 82.20), determine the mean, standard deviation, and CV. Is the precision for this set of data acceptable? Can you reject the value 82.20 since it seems to be different than the others? What is the 95% confidence level you would expect your values to fall within if the test were repeated? If the true value for dry matter is 89.40, what is the percent relative error? 4. Compare the two groups of standard curve data below for sodium determination by atomic emission spectroscopy. Draw the standard curves using graph paper or a com— puter software program. Which group of data provides a better standard curve? Note that the absorbance of the emitted radiation at 589 nm increases proportionally to sodium concentration. Calculate the amount of sodium in a sample with a value of 0.555 for emission at 589 nm. Use both standard curve groups and compare the results. Sodium Concentration Emission at (pug/ml) 589 nm Group A—Sodium Standard Curve 1.00 0.050 - 3.00 0.140 5.00 0.242 10.0 0.521 20.0 0.998 Group B—Sodium Standard Curve 1.00 0.060 3.00 0.113 5.00 0.221 10.0 0.592 20.0 0.917 Answers 1. 2, 3, 4 2. 0.0222 3. mean : 87.19, SDn_1 : 3.34, CV = 3.83%; thus the prec sion is acceptable. Qcalc = 0.92, therefore the value 82.2 can be rejected. CI = 87.19 :t 5.31. %E,el = —2.47%. 4. Group A is the better standard curve (group A, r2 : 0.9991 group B, r2 = 0.9708). Sodium in the sample using grou A standard curve : 11.1 ug/ ml; with group B standar curve = 11.5 ug/ml. ‘ 4.9 RESOURCE MATERIALS 1. Garfield, F.M., Klestra, E., and Hirsch, ]. 2000. Qua ity Assurance Principles for Analytical Laboratories. AOA International, Gaithersburg, MD. This book covers most quality assurance issues but has a good chapter (chapter on the basics of statistical applications to data. 2. Meier, RC, and Richard EZ. 2000. Statistical Methods Analytical Chemistry, 2nd ed. John Wiley & Sons, New Yor This is another excellent text for beginner and advancr analytical chemists alike. It contains a fair amount of deta sufficient for most analytical statistics, yet works throug the material starting at a basic introductory level. T] authors also discuss the Q—test used for rejecting da: A big plus of this book is the realistic approach tak; throughout. 3. Skoog, D.A., West, D.M., Holler, ].F., Crouch, S. 2000. Analytical Chemistry: An Introduction, 7th e Brooks/ Cole, Pacific Grove, CA. Section 1, chapters , 7 do an excellent job of covering most of the stat: tics needed by an analytical chemist in an easy—to-rea style. 4. Willard, H.H., Merritt, L.L., In, Dean, ].A., and Sett F.A., Ir. 1988. Instrumental Methods of Analysis, 7th e Wadsworth Publishing, Belmont, CA. This gives a rig ous treatment of instrumentation and has a very use! chapter (chapter 2, Measurements, Signals and Data) r types of error generated by instruments. ...
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