A bootstrap approach to estimate reference intervals of biochemical variables in sheep using reduced sample sizes

Small Ruminant Research 83 (2009) 34–41

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A bootstrap approach to estimate reference intervals of biochemical variables in sheep using reduced sample sizes Corrado Dimauro a,∗ , Nicolò P.P. Macciotta a , Salvatore P.G. Rassu a , Cristiana Patta b , Giuseppe Pulina a,c a b c

Dipartimento di Scienze Zootecniche, Università di Sassari, Italy Istituto Zooprofilattico della Sardegna “G. Pegreffi”, Sassari, Italy AGRIS Sardegna, Olmedo (SS), Italy

a r t i c l e

i n f o

Article history: Received 8 December 2008 Received in revised form 6 March 2009 Accepted 9 March 2009 Available online 10 April 2009 Keywords: Bootstrap Biochemical variable Reference interval Sample size Sheep

a b s t r a c t Laboratory analyses represent a key element in veterinary medicine diagnosis providing objective information about the health status of a patient. Analytic data are interpreted by comparing them with a specific reference intervals previously determined on a reference population of healthy animals. The International Federation of Clinical Chemistry recommends the use of nonparametric methods and, as a consequence, a reference sample of at least 120 healthy subjects, to obtain reliable reference intervals. Such order of magnitude for the reference sample is not always feasible especially if the laboratory variable under study is affected by several sources of variation, e.g., environmental conditions, physiological status of the animal, age, or gender. A convenient method to estimate reference intervals should be able to avoid assumptions on the probability distribution of the considered variable and produce robust results even with a limited sample size. This study presents a new statistical approach, based on data bootstrap, to estimate reference intervals for 12 blood biochemical variables in Sarda dairy sheep. The method was applied to real and simulated data from 120 to 40 animals. The reference intervals calculated with the new method remained quite constant as sample size decreased from 120 down to 60 animals, and became wider with fewer individuals. So, a minimum threshold of 60 animals could be considered a good limit to obtain reliable reference intervals for blood biochemical variables in Sarda dairy sheep. Moreover, comparisons between results from real and simulated data suggested that the method could be also applied to other laboratory variables. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Laboratory analyses are a key element in medicine diagnosis, because their results provide objective information about the health status of a patient. This is fundamental in veterinary medicine, where it is particularly difficult to assess the healthy status of an animal, since nobody knows

∗ Corresponding author at: Dipartimento di Scienze Zootecniche, facoltà di Agraria, Università di Sassari, Via De Nicola 9, 07100 Sassari, Italy. Tel.: +39 079 229304; fax: +39 079 229302. E-mail address: [email protected] (C. Dimauro). 0921-4488/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.smallrumres.2009.03.004

how an animal really feels. In both cases, analytical data are interpreted by comparing them with specific reference intervals (RIs) previously estimated on a reference sample (RS) extracted from a reference population of healthy subjects. The first drawback in estimating RI of a given variable is the choice of the most suitable statistical method for estimating it (Grossi et al., 2005). There are two possible ways: the parametric and the nonparametric approach. From the theoretical point of view, the parametric approach would be better than the nonparametric one, because it is a well assessed procedure and does not require a large RS (Solberg, 2004). However, it can be applied only if the

C. Dimauro et al. / Small Ruminant Research 83 (2009) 34–41

variable under investigation has a normal distribution and this widely limits its use. Actually, testing a variable for normality is often very difficult because it may have a normal distribution in the reference population, but not in the RS, due to incorrect sampling procedures or to stochastic drift. When the variable is not normally distributed, mathematical transformations of the data can be used to normalize it, but, as demonstrated by Linnet (1987), some types of data distribution cannot be normalized, often due to bi-modal or multi-modal distribution of the variable. As a consequence, the International Federation of Clinical Chemistry (IFCC) recommends the use of nonparametric methods to estimate RIs, since they do not make a priori assumptions on the probability distribution of the considered variable. However, the choice of this approach leads to other problems. First of all, even if several nonparametric methodologies are available (Horn and Pesce, 2003), none of them has been considered the most suitable one by the scientific community. Secondly, these nonparametric methods require a large RS of healthy individuals (Horn and Pesce, 2003; Solberg, 2004) and become robust when the sample size is as high as 200 units (Horn et al., 1998). For these reasons, the IFCC recommends the extraction of a RS of at least 120 healthy individuals from the reference population that represents a specific typology of patient (Solberg, 2004). However, if the variable under study is affected by several factors, e.g. environmental conditions, physiological status of the subject, age, or gender, it can be costly and difficult to achieve group sizes of that order of magnitude (Horn and Pesce, 2003). Therefore, new methods to evaluate RIs using small samples are needed. A convenient method for estimating a RI should be able to avoid assumptions on the probability distribution of the considered variable and produce robust results even with a limited sample size. With this aim, in this work we propose the use of the bootstrap technique, implemented with the parametric approach, to obtain an easy and single method to estimate RIs for blood biochemical variables in Sarda dairy sheep. Bootstrap is a computer intensive resampling technique that bases its rationale on the central limit theorem, thus allowing to make inferences on a given statistic (mean, median, etc.) using the parametric method, without considering the probability distribution of the variable. With the improvement of the computer performances, the bootstrap technique has been increasingly utilized in several fields of applied biology (e.g. human medicine, molecular biology and genetics), as well as in other sciences (e.g. astronomy, economics and engineering) (Henderson, 2005). The aims of this work were: (i) to develop a single method to estimate RIs for 12 blood biochemical variables in Sarda dairy sheep using the bootstrap technique implemented with the parametric approach and (ii) to evaluate the effects of a reduction of the required sample size of 120 healthy animals on RI estimates. 2. Materials and methods 2.1. Classic determination of reference intervals In this study the term “classic RIs” indicates RIs estimated using the parametric or nonparametric approach according to the suggestions of the

35

IFCC. The choice between the two statistical techniques depends on the distribution shape of each parameter, which can be tested by using the normal probability plot and the Anderson–Darling normality test (Linnet, 1988). For normally distributed variables, RIs and their 95% confidence intervals (CI95% ) for the lower and upper bounds are estimated using the well-known parametric procedure (Solberg, 2004). If m and s are the mean and the standard deviation of the RS, the parametric RI and the CI95% for its limits are calculated as:  RI = m ± 2s

CI95% = (m ± 2s) ± 1.96

s2

1 n

+

2 n−1



,where n is the

size of the RS and m and s are independent with variances s2 /n and s2 /2(n − 1), respectively. If the probability distribution of the variable is not normal, the RI is estimated using the nonparametric approach. In this study, the chosen nonparametric method was suggested by the IFCC (Solberg, 2004), which calculates the RI using the simple rank-based procedure described by Reed et al. (1971). In brief, values of a variable are ordered according to increasing value and assigned rank numbers. The rank number of the ˛ percentile is calculated as ˛(n + 1)/100, where n is the sample size, whereas the CI95% for the ˛ percentile can be found by using the binomial distribution method (Conover, 1999; Garcia-Perez, 2005). 2.2. The bootstrap method In this section we examine a new method to estimate RIs that could be used in alternative to the classic methods. This method combines the parametric approach with the bootstrap resampling technique (the bootstrap method). The sampling distribution of a statistics, such as the mean, is based on many samples being randomly extracted from the population. In the bootstrap approach, different samples are generated by repeatedly sampling with replacement from the same sample. Sampling with replacement means that after an observation is randomly drawn from the RS, it is put back before the next observation is drawn. As a result, any number can be drawn more than once, or not at all, and every number has the same probability of being chosen. In this way, each resample is as large as the original RS and, in practice, hundreds of thousands of resamples can be drawn. Let x1 , x2 , . . ., xn be a set of independent observations measured on n subjects and x¯ the sample mean. The estimated standard error (SE) of the mean is s (1) SE = √ n

 n   (xi −¯x)2  i=1

where the standard deviation is calculated as s = . n−1 Let y¯ j , for j = 1, 2, . . ., m, be the mean of each m-bootstrap sample and y¯ boot the mean of the bootstrap distribution. Then the bootstrap standard error (SEboot ), i.e. the standard deviation of the bootstrap distribution, is

  m  2  (y¯ j − y¯ boot )  j=1

SEboot =

m−1

.

SEboot represents the best estimation of the SE of the mean (or of any other statistics) (DiCiccio and Efron, 1996). Therefore, by replacing SE with SEboot in the formula (1), the bootstrap sampling standard deviation sboot can be estimated as: √ sboot = SEboot n The bootstrap accuracy is generally evaluated using the bias, i.e. the difference between the sample mean and the bootstrap estimate of the mean (bias = x¯ − y¯ boot ). The root mean square error of the mean (RMSE) is calculated as: RMSE =



bias2 + SE2boot

As a rule of thumb, Efron and Tibshirani (1993) suggested that if the absolute value of the bias is smaller than 0.25SEboot , the bias can be ignored and therefore RMSE ≈ SEboot . Finally, the bootstrap reference interval (RIboot ) of a given variable and the CI95% of its limits can be calculated as:

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Table 1 Sample mean, Anderson–Darling normality test, bootstrap mean, bootstrap standard error (SEboot ) and bias for 12 biochemical variables measured on 120 Sarda dairy sheep. Biochemical variables

Sample mean

Normality (A–D test)a

Bootstrap mean

SEboot

Bias

␥-GT (U/L) Total proteins (g/dL) Glucose (g/L) Cholesterol (mg/dL) P (mg/dL) Ca (mg/dL) Creatinine (mg/dL) Urea (mg/dL) ALT (U/L) AST (U/L) CK (U/L) LDH (U/L)

86.750 7.680 38.550 58.973 5.294 10.054 0.557 43.433 23.542 138.092 146.083 496.008

>0.25 0.06 0.09 0.11 0.05 <0.005 <0.005 <0.005 <0.005 <0.005 <0.005 <0.005

86.741 7.677 38.536 58.971 5.293 10.054 0.552 43.432 23.537 138.077 146.047 495.980

1.212 0.087 0.953 0.945 0.104 0.061 0.012 0.877 0.649 3.426 4.835 8.095

0.009 0.003 0.014 0.002 0.001 0.000 0.005 0.001 0.005 0.015 0.036 0.028

␥-GT: gamma-glutamyltransferase; P: phosphorus; Ca: calcium; ALT: alanine aminotransferase; AST: aspartate aminotransferase; CK: creatinine kinase; LDH: lactic dehydrogenase. a A–D test values ≥0.05 indicate that the variable is normally distributed. RIboot = y¯ boot ± 2sboot ,



CI95% = (y¯ boot ± 2sboot ) ± 1.96

 2 sboot

1 2 + n n−1



where n is the size of the RS (named d-size). y¯ boot and SEboot (required to estimate sboot ) were evaluated by using 10,000 resamples in the bootstrapping process (Henderson, 2005; Carpenter and Bithell, 2000). All the computations were developed using the SAS software. See Appendix for the SAS code.

the results was tested using an index (Id ) which quantifies the relationship between the accuracy of RIboot estimation and the d-size. First, a number (p) of random samples was extracted from the original RS of 120 subjects at a given d-size (<120), then the bootstrap standard error for each p-sample replicate, SEd i (i = 1, . . ., p), was calculated. The index Id was defined as boot

the mean of the ratio between SEd

booti

Id = mean

SEdboot i SE120 boot

and the SE120 boot :

 .

2.3. Estimation of reference intervals using a decreasing sample size The bootstrap method was applied to RSs with d-sizes smaller than 120 (the lower limit for d-size suggested by the IFCC) and the robustness of

This index takes into account the unavoidable sampling effects when a data set is extracted from another.

Fig. 1. Sample probability distribution (a) and related normal probability plot (b) for ALT measured on 120 Sarda dairy sheep. Bootstrap probability distribution (c) and related normal probability plot (d) for ALT.

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Table 2 Classic and bootstrap reference intervals with 95% confidence interval for lower and upper bounds (shown in bracket), for 12 biochemical variables measured on 120 Sarda dairy sheep. Biochemical variables

Classic reference intervals

␥-GT (U/L) Total proteins (g/dL) Glucose (g/L) Cholesterol (mg/dL) P (mg/dL) Ca (mg/dL) Creatinine (mg/dL) Urea (mg/dL) ALT (U/L) AST (U/L) CK (U/L) LDH (U/L)

Bootstrap reference intervals

Lower limits

Upper limits

Lower limits

Upper limits

60 (56–64) 5.8 (5.5–6.1) 17.5 (14–21) 36 (32–40) 3.0 (2.6–3.3) 8.5 (7.0–9.1) 0.2 (0.1–0.3) 29 (26–30) 13 (10–14) 87 (63–91) 79 (74–87) 365 (328–388)

114 (109–118) 9.6 (9.3–9.9) 59.6 (56–63) 81 (78–85) 7.6 (7.2–8.0) 11.2 (11.1–12.1) 0.8 (0.7–0.9) 62 (60–68) 38 (34–46) 229 (208–250) 286 (262–340) 746 (721–816)

60 (56–64) 5.8 (5.5–6.1) 17.7 (14–21) 37 (33–40) 3.0 (2.7–3.4) 8.7 (8.5–8.9) 0.3 (0.2–0.3) 26 (23–29) 9 (7–12) 63 (52–75) 41 (24–57) 319 (291–347)

113 (109–117) 9.6 (9.3–9.9) 59.3 (56–62) 77 (74–80) 7.6 (7.2–7.9) 11.4 (11.2–11.6) 0.82 (0.8–0.9) 61 (59–64) 38 (35–40) 213 (201–224) 252 (235–268) 673(645–700)

␥-GT: gamma-glutamyltransferase; P: phosphorus; Ca: calcium; ALT: alanine aminotransferase; AST: aspartate aminotransferase; CK: creatinine kinase; LDH: lactic dehydrogenase. 2.4. Applications of the bootstrap method In this work, the proposed bootstrap method was applied to a data set of 12 biochemical variables measured on a RS of 120 healthy adult Sarda dairy sheep. The chosen variables were: gamma-glutamyltransferase

(␥-GT), total proteins, glucose, cholesterol, phosphorus (P), calcium (Ca), creatinine, urea, alanine aminotransferase (ALT), aspartate aminotransferase (AST), creatinine kinase (CK), and lactic dehydrogenase (LDH). Analyses of biochemical variables were performed by a routine clinical chemistry system using spectrophotometry (Dimension RXL, Dade

Table 3 Mean bootstrap reference intervals with their 95% confidence intervals and Id values at decreasing sample size (d-size) for 12 biochemical variables. In bold are also reported bootstrap reference intervals for the complete reference sample of 120 Sarda dairy sheep. Biochemical variables

d-size

␥-GT (U/L)

120 80 70 60 50 40

60.31–113.17 60.39–112.65 60.28–112.38 59.97–112.42 59.29–112.87 60.15–112.26

56.20–64.41 55.41–65.37 54.97–65.59 54.19–65.75 52.81–65.76 53.10–67.20

120 80 70 60 50 40

5.79–9.57 5.76–9.56 5.72–9.57 5.73–9.54 5.70–9.53 5.69–9.50

5.49–6.08 5.40–6.12 5.33–6.11 5.31–6.15 5.24–6.16 5.17–6.20

9.28–9.86 9.20–9.93 9.18–9.96 9.12–9.96 9.07–10.00 8.99–10.02

1.23 1.34 1.43 1.58 1.76

120 80 70 60 50 40

17.74–59.34 17.87–60.03 17.45–59.94 17.72–59.79 17.24–60.32 18.18–59.87

14.51–20.97 13.85–21.89 13.12–21.78 13.09–22.36 12.03–22.45 12.54–23.82

56.11–62.57 56.01–64.05 55.60–64.27 55.16–64.43 55.12–65.53 54.23–56.51

1.24 1.34 1.44 1.61 1.75

120 80 70 60 50 40

37.06–76.53 35.90–82.17 35.84–81.90 35.98–81.40 36.20–81.56 36.27–81.67

33.85–40.26 31.49–40.31 31.14–40.53 30.97–40.98 30.72–41.68 30.12–42.41

73.33–79.73 77.77–86.58 77.20–86.59 76.39–86.40 76.08–87.04 75.52–87.81

1.38 1.47 1.51 1.71 1.92

120 80 70 60 50 40

3.02–7.57 3.00–7.55 2.97–7.55 3.01–7.51 2.97–7.54 3.07–7.39

2.66–3.37 2.57–3.43 2.50–3.43 2.51–3.51 2.42–3.52 2.48–3.65

7.22–7.92 7.11–7.98 7.09–8.02 7.01–8.00 6.99–8.09 6.81–7.98

1.23 1.32 1.40 1.56 1.66

120 80 70 60 50 40

8.73–11.38 8.70–11.38 8.66–11.40 8.61–11.43 8.56–11.45 8.64–11.37

8.52–8.93 8.45–8.96 8.38–8.94 8.30–8.92 8.21–8.91 8.27–9.01

11.17–11.59 11.13–11.64 11.12–11.68 11.12–11.74 11.10–11.80 11.00–11.74

1.24 1.36 1.49 1.70 1.80

Total proteins (g/dL)

Glucose (g/L)

Cholesterol (mg/dL)

P (mg/dL)

Ca (mg/dL)

Reference interval

␥-GT: gamma-glutamyltransferase; P: phosphorus; Ca: calcium;

CI95% for lower bound

CI95% for upper bound 109.07–117.28 107.67–117.63 107.07–117.69 106.64–118.20 106.39–119.34 105.20–119.31

Id 1.21 1.29 1.41 1.58 1.72

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Behring, Newark, DE, USA) by the Istituto Zooprofilattico Sperimentale per la Sardegna laboratory. Initially, these biochemical variables were tested for normality using the Anderson–Darling normality test (Linnet, 1988) and, according to the results of this test, classical RIs were estimated. Then, RIboot s and their CI95% were calculated using: (i) the complete RS of 120 healthy animals and (ii) samples with decreasing size (d-size = 80, 70, 60, 50 and 40) randomly extracted from the 120 available animals. In particular, for each variable and each d-size, 10 random samples were extracted and, therefore, 10 RIs were calculated. These results were used to evaluate the Id . Moreover, to assess how sensitive is Id to the probability distribution of a certain variable, four samples, each of 120 units, were generated from four different probability distributions: normal, beta, gamma and chi-square. These simulated data were used to calculate Id for different d-size, ranging from 110 to 20.

3. Results and discussion Only 5 (␥-GT, total proteins, glucose, cholesterol and P) out of the 12 biochemical variables considered had a normal distribution as evidenced by the level of significance (≥0.05) of the Anderson–Darling test reported in Table 1. The ability of the bootstrap technique to achieve a mound shaped distribution is exemplified in Fig. 1 which shows the probability distribution and normal probability plot for ALT before (Fig. 1a and b) and after (Fig. 1c

and d) bootstrap resampling. As it can be seen, initially, the frequency histogram for ALT does not have a mound shape (Fig. 1a) and the normal probability plot confirms that it is not normally distributed (Fig. 1b). After bootstrap resampling, the resultant graphs (Fig. 1c and d) show that the ALT data become normally distributed. The same occurs to the other biochemical variables. These results were expected due to the effectiveness of the central limit theorem. Indeed, bootstrap means coincide with sample means for all the biochemical variables, as confirmed by the bias values which range from 0.001 to 0.036 (Table 1). In addition, the fact that the bias was smaller than 0.25SEboot for all the variables assure that the bootstrap technique applied to this study leads to accurate results. Classical and bootstrap reference intervals, evaluated on the complete dataset of 120 animals, are reported in Table 2. As theoretically expected, RIboot s overlapped classic RIs only for normally distributed variables, whereas differed for the non-normally distributed variables. To better understand the magnitude of these differences, the specific probability distribution of each variable should be considered. Based on our previous work (Dimauro et al., 2008), non-normally distributed biochemical variables can

Table 4 Mean bootstrap reference intervals with their 95% confidence intervals and Id values at decreasing sample size (d-size) for 12 biochemical variables. In bold are also reported bootstrap reference intervals for the complete reference sample of 120 Sarda dairy sheep. Biochemical variables

d-size

Reference interval

Creatinine (mg/dL)

120 80 70 60 50 40

0.28–0.82 0.30–0.82 0.29–0.82 0.29–0.82 0.28–0.83 0.28–0.84

Urea (mg/dL)

120 80 70 60 50 40

25.93–61.52 26.09–61.42 24.37–62.69 24.71–62.20 24.04–62.41 24.47–62.65

ALT (U/L)

120 80 70 60 50 40

AST (U/L)

CI95% for lower bound 0.24–0.33 0.25–0.35 0.24–0.35 0.23–0.35 0.22–0.35 0.20–0.35

CI95% for upper bound

Id

0.78–0.86 0.77–0.87 0.77–0.87 0.77–0.88 0.76–0.90 0.76–0.91

1.19 1.29 1.42 1.59 1.83

23.17–28.70 22.72–229.46 20.47–28.28 20.58–28.84 19.40–26.68 19.30–29.64

58.76–64.29 58.06–64.79 58.78–66.59 58.07–66.33 57.77–67.05 57.48–67.82

1.22 1.31 1.39 1.56 1.74

9.37–37.70 9.40–37.58 9.15–37.68 9.23–37.53 8.87–38.08 9.04–37.80

7.17–11.57 6.71–12.08 6.24–12.06 6.11–12.35 5.34–12.40 5.15–12.93

35.50–39.90 34.90–40.27 34.78–40.59 34.41–40.65 34.55–41.61 33.90–41.68

1.22 1.32 1.42 1.61 1.77

120 80 70 60 50 40

63.31–213.83 62.69–214.80 62.41–213.76 63.13–211.35 61.40–213.04 61.08–212.30

51.70–74.93 48.20–77.19 46.98–77.84 46.80–79.46 43.07–79.72 41.48–80.69

201.21–224.44 200.30–229.29 198.33–229.19 195.02–227.69 194.72–231.37 192.70–231.91

1.25 1.33 1.41 1.58 1.63

CK (U/L)

120 80 70 60 50 40

40.56–251.53 40.54–248.72 39.49–247.65 39.92–245.35 40.42–241.52 40.94–241.30

24.17–56.95 20.70–60.38 18.28–60.71 17.29–62.56 16.11–64.71 13.82–68.05

235.14–267.92 228.88–268.56 226.43–268.86 222.72–267.99 217.22–265.83 214.18–268.41

1.21 1.30 1.38 1.49 1.66

LDH (U/L)

120 80 70 60 50 40

319.33–672.57 317.55–672.92 319.10–669.39 319.33–665.05 316.97–668.34 318.95–663.99

291.89–346.78 283.69–351.41 283.39–354.8 281.23–357.42 274.51–359.43 272.26–365.65

645.23–700.01 639.06–706.78 633.69–705.10 626.96–703.14 625.88–710.80 617.29–710.68

1.23 1.30 1.39 1.55 1.71

ALT: alanine aminotransferase; AST: aspartate aminotransferase; CK: creatinine kinase; LDH: lactic dehydrogenase.

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Fig. 2. Lower (a) and upper (b) RIboot limits for creatinine kinase (CK) at decreasing sample size (d) evaluated on 10 random samples. The straight line represents the lower (a) and upper (b) limits calculated on the reference sample of 120 Sarda dairy sheep.

be divided into three groups, according to the mathematical function that can be used to normalize them: (a) no normalization function available (Ca, creatinine and urea); (b) natural logarithm function (ALT and AST); (c) reciprocal function (CK and LDH). In group (a), classic RIs and RIboot s slightly differ for both lower and upper limits. This is because these variables have a multi-modal distribution, which does not admit a normalization function but is quite symmetric around the median value (Solberg, 2004). In group (b), lower limits differ strongly between classic RIs and RIboot s. This is expected, if we think that a log-normal distributed variable has a heavy positive skewness. In group (c), differences occur for both lower and upper limits, being RIboot s shifted towards greater values than those of classic RIs. This can be explained by the fact that when the inverse function is used, a small value is enlarged. Thus, the bootstrap technique, being not sensitive to the probability

distribution of the variable, allows to calculate reference intervals using the bootstrap approach as a single method. The bootstrap method applied to reduced sample sizes leads again to reliable results. Tables 3 and 4 show mean RIboot s evaluated for each d-size on 10 replicates, at decreasing sample d-size (d-size = 80, 70, 60, 50 and 40). These intervals are quite similar to RIboot s calculated using the complete RS until d-size is as small as 60 and become wider as d-size decreases. The variation at decreasing d-size of the reference limits due to sampling effects is exemplified in Fig. 2a and b, where CK is used as an example. The straight line represents the lower (Fig. 2a) and upper limits (Fig. 2b) of the RI evaluated at a d-size = 120, whereas the other lines represent the same limits at decreasing d-size for each of the 10 random sample replicates. The variation of reference limits due to sampling effect is little until d-size gets down to 60, but becomes greater for smaller

Table 5 Id values for decreasing sample d-size calculated on simulated data with normal, chi-squared, gamma and beta probability distribution. Scale and shape parameters for gamma and beta distribution are also reported. d-size

Normal

Chi-squared

Id Values to respect probability distributions 110 1.04 1.06 100 1.09 1.11 90 1.15 1.18 80 1.19 1.23 70 1.27 1.31 60 1.38 1.43 50 1.52 1.50 40 1.66 1.69 30 1.92 1.93 20 2.36 2.32

Gamma

Beta

k = 1,  = 2

k = 5,  = 2

k = 2,  = 2

˛ = 0.5, ˇ = 0.5

˛ = 2, ˇ = 5

˛ = 5, ˇ = 2

1.04 1.09 1.15 1.25 1.35 1.46 1.56 1.72 1.98 2.48

1.04 1.09 1.14 1.17 1.26 1.33 1.50 1.69 1.96 2.29

1.05 1.11 1.18 1.27 1.37 1.49 1.65 1.84 2.08 2.29

1.06 1.11 1.17 1.24 1.32 1.39 1.56 1.72 1.97 2.34

1.05 1.10 1.15 1.21 1.29 1.36 1.53 1.71 1.94 2.24

1.04 1.10 1.17 1.22 1.32 1.42 1.55 1.71 2.00 2.38

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sample sizes. At the same time, Id s values increase as dsize decreases, but remain smaller than 1.5 until d-size is 60 (Tables 3 and 4). Id s values calculated for simulated data (Table 5) show the same trend as that of real data, thus confirming that the bootstrap results are not subjected to the probability distribution of the considered variable. Therefore, an Id = 1.5 could be considered a good threshold to obtain reliable references intervals, requiring a minimum d-size = 60.

of RIs estimation for biochemical variables, compared to the larger sample sizes used by the methods currently available. Moreover, since the same Id values were obtained for real and simulated data, we think that the bootstrap method could also be used for other laboratory variables, even if specific studies would then be required. Acknowledgment We thank Ana H.D. Francesconi for critically revising the manuscript.

4. Conclusions Results of this study confirm the experimental hypotheses: (i) the bootstrap technique, getting over the problem of the variable’s probability distribution, allows to calculate reference intervals for biochemical variables using the bootstrap approach as a single method; (ii) the bootstrap method leads to reliable results also for reduced reference samples, fixing a minimum threshold of 60 healthy subjects. This reduced sample size can increase the efficiency

Appendix A A SAS code with the macro boot (Bergstralh, 2004) was developed to provide researchers with an easily accessible tool to calculate bootstrap reference intervals. This code consists of a collection of regular SAS program statements and a macro contained within a %MACRO and a %MEND statement.

THE CODE

THE COMMAND MEANING

data bio; set‘C: \bio.sas7bdat’; run;

With this data step, the SAS data set (bio) containing the values of the biochemical parameters (the reference sample) is imported.

%let rif = alt; %let size = 120; %let sample = 10,000;

Three macro variables are created: rif = the biochemical parameter (e.g. ALT); size = the sample size; sample = number of bootstrap resamplings

%MACRO boot(data, x, n, samples, seed, outdata);

The macro begins with the command %macro and a name. In brackets, the inputs to the macro are listed. Then the input data set bio is renamed and transposed to obtain the data set tboot

data boot; set &data; keep &x; proc transpose prefix = &x out = tboot; var &x; data &outdata; set tboot; keep sample n bs&x.1-bs&x.&n; array x &x.1-&x.&n; array bsx bs&x.1-bs&x.&n;

The macro creates a new dataset (outdata) with one row and n + 1 columns (sample n, bsx1, . . ., bsxn).

do i = 1 to &samples;

The do cycle allows to iterate the computation from 1 to samples (number of resampling)

sample n = i; do j = 1 to &n; k= int(ranuni(&seed)*&n) +1; bsx(j) = x(k); end; output; end; run; quit; %MEND boot;

For a given sample i, a new do cycle is performed with j from 1 to n (sample size) The SAS function ranuni generates random numbers between 1 and n. k indicates which element in the x array (the reference sample) will be selected in correspondence with j value.

%boot(data = bio,x = &rif, n = &size, samples = &sample, seed = 57573, outdata = bstrap);

The macro boot is recalled, thus obtaining the output dataset =bstrap. This dataset has a number of rows equal to the number of resampling and a number of columns equal to N + 1

proc transpose data = bstrap out = perm; by sample n; run; data perm; set perm; rename col1 = &rif; run;

The bstrap dataset is transposed and the output dataset (perm) contains two columns: sample n, which indicates the bootstrap sample (n = 1, . . ., 10,000), and col1, which contains the bootstrap values. The column col1 is then renamed with the name of the biochemical parameter

proc means data = perm noprint; var &rif; by sample n; output out = meanboot; run; data meanboot; set meanboot; if stat ne ‘MEAN’ then delete; run;

The proc means statement calculates the means for each bootstrap sample

%mend statement closes the macro.

The following data steps allow to obtain an output dataset (meanboot) that contains one column with the means by sample bootstrap

C. Dimauro et al. / Small Ruminant Research 83 (2009) 34–41

41

Appendix A (Continued ) THE CODE

THE COMMAND MEANING

data meanboot; set meanboot; keep &rif; run; proc sort data = meanboot; by &rif; run; proc univariate data = meanboot normal; var &rif; histogram &rif; output out = boot&rif mean = meanbootstrap std = bsem; run;

Proc univariate provides univariate statistics and information about the distribution of the numeric variables. The output file (boot & rif) contains the bootstrap mean and standard error.

data boot&rif; set boot&rif; devst = bsem*(&size-1)**0.5; run; data boot&rif; set boot&rif; low = meanbootstrap-2*devst; up = meanbootstrap + 2*devst; run; data boot&rif; set boot&rif; er = devst*(1/&size + 2/(&size-1))**0.5; run; data boot&rif; set boot&rif; lower = low-1.96*er; upper = low + 1.96*er; run; data boot&rif; set boot&rif; lowerup = up-1.96*er; upperup = up + 1.96*er; run;

These data steps calculate the bootstrap standard deviation

Finally, the reference interval and the IC95% for lower and upper limits are calculated.

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