Lessons in biostatistics Resampling methods in Microsoft Excel® for estimating reference intervals Elvar Theodorsson Department of Clinical Chemistry and Department of Clinical and Experimental Medicine, Linköping University, Linköping, Sweden Corresponding author: [email protected] Abstract Computer- intensive resampling/bootstrap methods are feasible when calculating reference intervals from non-Gaussian or small reference sam- ples. Microsoft Excel® in version 2010 or later includes natural functions, which lend themselves well to this purpose including recommended inter- polation procedures for estimating 2.5 and 97.5 percentiles. The purpose of this paper is to introduce the reader to resampling estimation techniques in general and in using Microsoft Excel® 2010 for the purpo- se of estimating reference intervals in particular. Parametric methods are preferable to resampling methods when the distributions of observations in the reference samples is Gaussian or can tran- sformed to that distribution even when the number of reference samples is less than 120. Resampling methods are appropriate when the distribu- tion of data from the reference samples is non-Gaussian and in case the number of reference individuals and corresponding samples are in the order of 40. At least 500-1000 random samples with replacement should be taken from the results of measurement of the reference samples. Key words: reference interval; resampling method; Microsoft Excel; bootstrap method; biostatistics Received: May 18, 2015 Accepted: July 23, 2015 Description of the task Reference intervals (1,2) are amongst the essential are naturally determined in a ranked set of data tools for interpreting laboratory results. Excellent values if the number of observations is large, e.g. > theoretical and practical studies have been pub- 1000. When the number of values is small, e.g. in lished since the 1960ies for establishing reference the order of 40-120 and the data are non-Gaussian, intervals as a concept in contrast to “normal inter- resampling methods are useful to estimate the ref- vals” (3), establishing principles for selecting refer- erence interval. The reference population is com- ence persons (4) and performing the appropriate monly a population of apparently healthy individ- statistical data analysis (5-8). Reference intervals uals, but may also consist of any other well-de- continue to be an active area of research (2,9) and fined population of interest for diagnostic/com- at the core of practical work in clinical laboratories parative purposes. (6,10). The reference interval includes 95% the results/ Different methods for calculating values measured in a representative sample of ref- reference intervals erence subjects and is bounded by lower and up- per reference limits marking the 2.5 and 97.5 per- Estimating reference intervals means dealing with centiles respectively (Figure 1). The reference inter- uncertainties and probabilities. All probabilistic val including the upper and lower reference limits methods are based on assumptions about a theo- http://dx .doi .org/10 .11613/BM .2015 .031 Biochemia Medica 2015;25(3):311–9 ©Copyright by Croatian Society of Medical Biochemistry and Laboratory Medicine. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc-nd/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. 311 Theodorsson E. Microsoft Excel® for estimating reference intervals ductory literature on resampling methods in gen- eral (13) and resampling methods for Microsoft Ex- Reference population cel® in particular (14). Resampling with replacement means that a large number of samples with replacement are then tak- A representative reference sample en from the original sample and the statistic of in- e e r from the reference population terest is calculated from this pseudo-population as limit limit Lowe Upper estimate of the corresponding parameter of the referenc referenc Reference interval which includes population (Figure 2). 2.5% estimated 95% of the values 2.5% in the reference population FIGURE 1. General principles when calculating reference inter- vals. 50 77 91 107 50 53 79 91 109 retical distribution which fundamentally deter- 58 80 92 112 62 80 93 114 1000 copies 62 80 94 115 mines the conclusion that can be drawn from the 62 80 96 115 53 62 80 96 115 data. The most commonly used probabilistic 62 81 97 117 64 81 97 119 1000 copies methods are the parametric methods that assume 64 82 97 120 64 82 97 120 that observations in the population are distributed 67 82 99 120 67 83 99 124 according to the Gaussian/Normal distribution. 68 84 100 127 69 84 100 127 These are the methods of choice if the data are 71 84 101 141 71 87 106 146 Gaussian or can be transformed to that distribu- 71 88 106 147 tion since the data themselves with the added 72 88 106 168 177 72 89 107 177 knowledge of the distribution of the data in the 75 90 107 1000 copies sample and population enables the user of the Reference sample values data to draw firmer conclusions than if only the data are known. When the data are non-Gaussian and cannot be transformed to the Gaussian distribution the ana- lyst is left with non-parametric or resampling methods for determining reference intervals. Data Samples 177 88 72 from 120 reference intervals are needed for relia- 80 114 50 bly determining non-parametric reference inter- 120 84 89 112 67 115 vals (7,11). Resampling/bootstrap methods (12) 82 64 93 take numerous repeated sub- samples with re- 106 84 141 placement of the available data in order to esti- 82 127 100 62 79 69 mate the distribution of the data in the popula- 80 85 81 tion, including the reference intervals. 81 107 84 Resampling methods with replacement are com- FIGURE 2. Simplified illustration of the principles of resampling monly called bootstrap methods by many be- techniques as employed by resampling methods with replace- lieved to refer to the fairytale of Baron Munchaus- ment. This figure illustrates the use of 83 raw data observations ob- en pulling himself and his horse out of a swamp by tained from Geffré et al. (24) resampled 1000 times in this case. his hair. Bootstraps are in fact the tab, loop or han- One thousand copies of each observation are created and thor- dle at the top of boots allowing the use of fingers oughly mixed giving each of them identical possibility of being pull the boots on. The concept of bootstrap is selected. A multitude of random samples are then taken from the mixture, calculating the 0.025 and 0.975 percentiles for each therefore a metaphor for clever, self-induced sal- sample in order to create an estimate of these percentiles in the vaging efforts. There is a wealth of current intro- intended population. Biochemia Medica 2015;25(3):311–9 http://dx .doi .org/10 .11613/BM .2015 .031 312 Theodorsson E. Microsoft Excel® for estimating reference intervals Percentiles Geffré et al. published “Reference Value Advisor”, a set of comprehensive freeware macros for Micro- A proper start when calculating reference intervals soft Excel® to calculate reference intervals (24). It – is to estimate the observation(s) corresponding to together with the original paper - can be down- 2.5% and 97.5% in a ranked list of reference value loaded at http://www.biostat.envt.fr/spip/spip. data. Both International Federation of Clinical php?article63 which implements the procedures Chemistry (IFCC) (5-7,15) and Clinical and Labora- recommended e.g. by EP28-A3c. Increased availa- tory Standards Institute (CLSI) (16) recommend the bility of low-cost computing machinery and resa- following formulas: lower limit has the rank num- mpling methods has made distribution independ- ber 0.025 x (n+1) and the upper limit the rank num- ent computer intensive methods accessible in rou- ber 0.975 x (n+1). This method is practicable mainly tine work (25). when the number of reference samples is 120 or more. The fundamental purpose of statistics in general and statistics in reference interval estimation is to While there is agreement on how to calculate the extrapolate from a limited number of observa- median, agreement on how to calculate quantiles tions/data to the whole population of observa- including percentiles is lacking due to the fact that tions. All statistical methods are based on assump- interpolation is frequently needed, especially tions which fundamentally decide the conclusion when the number of observations is small. Unfor- that can be made from the data. Parametric statis- tunately there is no general agreement on the tical methods which are the most widely used, as- best way of performing interpolations. In 1996 sume that the population values are distributed Hyndman and Fan published an influential paper according to a mathematically well-defined distri- on calculating quantiles (17). It evaluated the bution - the Gaussian/Normal distribution. Para- methods used by popular statistics packages with metric methods were initially described in the first the intention to find a consensus and basis for half of the twentieth century when methods for standardization. Of the nine formulas commonly performing extensive calculations were primarily used, four satisfied five of the six properties desir- manual and semi-automated at best. Using distri- able for a sample quantile. Hyndman and Fan felt bution functions to describe natural phenomena that the “Linear interpolation of the approximate including research data, substantially simplified medians for order statistics” (used by the Excel the calculations required for statistical analysis. =PERCENTILE.EXC function) was best due to the Without these known statistical distribution func- approximately median-unbiased estimates of the tions it would have been impossible to use ad- quantiles, regardless of the distribution (17).