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c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163

jo ur n al hom ep age : www.int l .e lsev ierhea l th .com/ journa ls /cmpb

onte Carlo simulation of parameter confidence intervalsor non-linear regression analysis of biological data using

icrosoft Excel

onald J.W. Lamberta,∗, Ioannis Mytilinaiosa, Luke Maitlanda, Angus M. Brownb

Applied Microbiology Group, Cranfield Health, Cranfield University, Cranfield MK43 0AL, UKSchool of Biomedical Sciences, Queens Medical Centre, University of Nottingham, Nottingham NG7 2UH, UK

r t i c l e i n f o

rticle history:

eceived 3 September 2010

eceived in revised form

6 May 2011

ccepted 27 May 2011

eywords:

onte Carlo simulation

OLVER

a b s t r a c t

This study describes a method to obtain parameter confidence intervals from the fitting of

non-linear functions to experimental data, using the SOLVER and Analysis ToolPaK Add-In

of the Microsoft Excel spreadsheet. Previously we have shown that Excel can fit complex

multiple functions to biological data, obtaining values equivalent to those returned by more

specialized statistical or mathematical software. However, a disadvantage of using the Excel

method was the inability to return confidence intervals for the computed parameters or the

correlations between them. Using a simple Monte-Carlo procedure within the Excel spread-

sheet (without recourse to programming), SOLVER can provide parameter estimates (up to

200 at a time) for multiple ‘virtual’ data sets, from which the required confidence intervals

xcel and correlation coefficients can be obtained. The general utility of the method is exemplified

by applying it to the analysis of the growth of Listeria monocytogenes, the growth inhibition of

Pseudomonas aeruginosa by chlorhexidine and the further analysis of the electrophysiological

data from the compound action potential of the rodent optic nerve.

calculation and use would invalidate the aim of making NLR

. Introduction

e have previously described the use of the Microsoft Excelpreadsheet to conduct non-linear regression (NLR) analysisf biological data [1]. Direct fitting of the dose–response curve,or example, through the use of NLR techniques has beenidely advocated, but access to, and comprehension of, com-ercial software are often at odds with the direct needs of the

esearcher. We recognised the ease of access and understand-ng that most researchers have of Microsoft Excel [2], whichould allow even those with an elementary understanding of

he spreadsheet to conduct relatively sophisticated data anal-ses, without the expense of purchasing and learning a newtatistical or advanced mathematical package.

∗ Corresponding author. Tel.: +44 1234 758314.E-mail address: rjwlambert@cranfield.ac.uk (R.J.W. Lambert).

169-2607/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights resoi:10.1016/j.cmpb.2011.05.009

© 2011 Elsevier Ireland Ltd. All rights reserved.

The SOLVER Add-In package of Excel allows the user toconduct investigations of non-linear (NL) functions using theminimisation of the sum of squares of the errors between theobserved and modelled values [1]. We further described theuse of the technique for the modelling of multiple Gaussianfunctions, which described the observed electrophysiologicaldata from the compound action potential of the rodent opticnerve [3]. One particular failing of the SOLVER package wasthe inability to return parameter confidence intervals. It wasnoted that this requires the use of the Hessian matrix, whose

open to anyone [1]. Hence the error analysis of the modelledand observed data was terminated at the calculation of thestandard error of the fit.

erved.

s i n

156 c o m p u t e r m e t h o d s a n d p r o g r a m

Confidence intervals can be calculated from knowledge ofthe Hessian but can also be estimated using either the Boot-strap technique of re-sampling errors between the modelledfit and the observed, or from Monte Carlo (MC) simulation [6].The MC technique uses the standard error of the fit of thenon-linear model to the observed data to produce sets of ‘vir-tual’ data. These data are modelled using the same non-linearmodel and a new group of parameters obtained for each vir-tual set. From the statistical distribution of these parameters,confidence intervals as well as correlation coefficients can beobtained.

We describe here the use of NLR within the Excel environ-ment and augment our original method with a simple MCanalysis, and show its general utility by applying it to anal-yse the growth of Listeria monocytogenes, the growth inhibitionof Pseudomonas aeruginosa by chlorhexidine and the furtheranalysis of the electrophysiological data from the compoundaction potential of the rodent optic nerve.

2. Computational methods and theory

For a given data set and a particular model (yfit), the sum ofsquares of the errors is given by

SSE =n∑

i=1

(yi − yfit)2

In a regression analysis the value of SSE is minimised bychanging the parameter values of the model yfit, resulting inthe best estimates of these parameters. In linear regressionthis is solved analytically, but if using non-linear regressionthis is carried out numerically, based on the input of ini-tial parameter estimates. The square root of the mean ofthe square of the error (RMSE) is the standard error of thefit. For a given set of conditions the model will return theexpected value of yi, E(yi) = yi. With linear regression, if all theprerequisite conditions are met, then the reported 95% con-fidence intervals will contain the true value of the regressionparameters 95% of the time. With non-linear regression con-fidence intervals are found using linear approximations andthe labelled 95% confidence intervals may not contain the trueinterval as often.

If the conditions required for regression are met, e.g. con-stant variance of error (homoscedasticity), normal distributionof errors, then the RMSE is an unbiased estimator of the stan-dard deviation of the fit. A virtual data set can be calculatedby adding random error to the expected value of y;

Y′i = yi + N(0, RMSE)

This virtual data set can be analysed by NLR to give anotherset of parameters (the best fit estimates for this virtual dataset).

SSE =n∑

(Y′i − yfit)

2

i=1

Another set of virtual data can be generated and the NLR fittingrepeated. In the procedure outlined here the sum of squares of

b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163

the errors from multiple data sets are summed and the fittingof m-sets of data are conducted simultaneously

SSEtotal =m∑

j=1

n∑i=1

(Y′i − yfit)

2

From the m-sets of parameters obtained, frequency analysesof the parameter values are performed and the 95% confidenceintervals obtained from the normal quantiles; covariancebetween parameter pairs can be found by calculating theparameters’ correlation coefficient.

2.1. Fitting the modified Gompertz equation tomicrobial growth data

The modified Gompertz equation is a standard empiricalmodel for the fitting of microbial growth data [5].

log N(t) = A + C exp{−exp(B(M − t))} (1)

where A is the asymptotic number (log cfu ml−1) as t tendsto negative infinity, A + C = maximum population density as ttends to positive infinity, B is a measure of the slope and M isthe time of maximum slope. From these fitted parameters thegrowth rate and lag are calculated, respectively, as

CBexp(1)

(2)

M − 1B

(3)

Data for the growth of L. monocytogenes at 30 ◦C in growthmedia (Tryptone Soya broth) containing 9% salt in terms oflog cfu ml−1 over a period of 100 h were obtained.

2.2. Fitting the Lambert–Pearson model to microbialgrowth inhibition data

The time taken for a microbial culture to reach a specific opti-cal density (also known as the time to detection, TTD) in thepresence of an inhibitor is dependent on the concentrationand dose–response of that inhibitor. The Lambert–Pearsonmodel (LPM, Eq. (4)) [4] describes the visual growth of a cultureas an exponential decay function of the concentration of theapplied inhibitor. A plot of the log concentration against therelative rate to detection (RRTD, the ratio of the time to detec-tion of the uninhibited culture, or positive control, to the timeto detection of the test culture) gives a characteristic sinusoid,with inflexion at RRTD = 1/exp(1). A linear extrapolation from

this point to the log concentration axis allows the estimationof the minimum inhibitory concentration (MIC, Eq. (5)), and alinear extrapolation to the RRTD = 1 axis allows the estimationof the non-inhibitory concentration (NIC, Eq. (6)), the concen-

i n b i

tt

R

woaad

far

M

N

Dis

2t

Tturlo(

C

wtmafa1

C

c o m p u t e r m e t h o d s a n d p r o g r a m s

ration below which normal visual growth is observed even inhe presence of the inhibitor.

RTD =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

if[x] = 0, 1

else if[x] < [P1]

then

exp(

−(

[x]P1

)P2)else if

1e

(1 − P2(ln[x] − ln P1)) < 0, 0

else1e

(1 − P2(ln[x] − ln P1))

(4)

here RRTD = relative rate to detection, [x] is the concentrationf the given inhibitor, P1 is the concentration of inhibitor giving

relative inhibition of 1/e, where e is the exponential of 1,nd P2 is a slope parameter which has been defined as theose–response due its similarity with the Hill model.

Two biologically important parameters can be obtainedrom the LPM; the minimum inhibitory concentration (MIC)nd the non-inhibitory concentration (NIC) and are defined,espectively, as

IC = P1 exp(

1P2

)(5)

IC = P1 exp(

1 − e

P2

)(6)

ata from the growth inhibition of P. aeruginosa (ATCC 15442)n the presence of chlorhexidine at 37 ◦C were obtained usingtandard, published, methods (Lambert and Pearson [4]).

.3. Further analysis of compound action potential ofhe rodent optic nerve

he compound action potential from the rodent optic nerveypically has three peaks (indicating the presence of three pop-lations of axons with different conduction velocities) with aapidly decaying transient or artefact from the initial stimu-us. Originally this phenomenon was modelled using the sumf four Gaussian functions, one for each feature of the CAP

Brown [3]).

AP =4∑

i=1

Ai

wi

√�/2

exp

{−2

(t − ci

wi

)2}

(7)

here Ai is the area under the curve, wi the width at halfhe maximum amplitude and ci is the latency to the maxi-

um amplitude of peak i. This equation models the artefacts a Gaussian; a secondary model which has some desirableeatures as described in this report, models the artefact as

simple decay, modelling from the initial recording time of.04 ms.

AP = D1 exp(−D2(t − 1.04))

2∑i=1

Ai

wi

√�/2

exp

{−2

(t − ci

wi

)2}

(8)

o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163 157

where D1 and D2 are parameters describing the decay of theinitial stimulus.

3. Program description and sample runs

The method can be split into 3-stages: Stage 1 fits the givenNL model to the observed data using SOLVER. This generatesthe initial best fit parameters and the RMSE value of the fit.Stage 2 uses the RMSE to generate a set of random numbersbased on N(0, RMSE), which are added to the predicted datafrom Stage 1. The NL model is then applied to multiple virtualdata sets simultaneously (using SOLVER) to generate multiplevalues of best-fit parameters. In Stage 3 these values are statis-tically analysed to provide the mean of the best-fit parameters,their standard errors, 95% confidence intervals and parametercorrelations. The data can also be used to provide confidenceintervals for parameters calculated from the regressed param-eters, which are often dependent on the magnitude of thecorrelation between those parameters.

3.1. NLR: Excel analysis of the growth of L.monocytogenes at 30 ◦C in high salt media

3.1.1. Monte-Carlo: Excel analysisStage 1. The initial part of the procedure generally follows thatgiven previously [3] except in this case the modified Gompertzmodel (1) was used. The data used and the initial NLR areshown in Fig. 1. The sum of the squares of the errors (SSE, CellE39) was calculated using the inbuilt “SUMXMY2 (data range1, data range 2)”, where the first data range was the modelledvalues and the second data range was the observed values. CellE41 divides this by the degrees of freedom (Cell E40) and takesthe square root to give the root mean square error (RMSE). Thisis the standard error of the curve fit. The SOLVER package wasused to minimise this value, by changing the values of the fourparameters A, C, B and M. Fig. 2 shows a plot of the observeddata and the modelled function.

Stage 2. Generation of random numbers with a distribution ofN(0, RMSE): (It is assumed the user has installed the AnalysisToolPak Add-In.) On a separate worksheet, the Random Num-ber Generator was used to generate an array of 50 columnsof 33 random numbers using the RMSE of the fitted model asthe standard deviation (Fig. 3). The number of columns used isset by the maximum number of values that SOLVER can han-dle (200) divided by the number of parameters in the model. Arandom seed number of 2 was used for illustrative purposes asthe use of this seed number will allow any reader to recreatethe exact procedure carried out here.

Generation of virtual data: The random data were added tothe modelled log cfu ml−1 data to produce a set of 50 virtualobserved data. On the spreadsheet these were convenientlylocated below the random number array.

Stage 3. Fitting multiple models simultaneously using SOLVER:Below each set of virtual data the modified Gompertz modelwas entered (Fig. 4). The regression parameters were placed

below this: the initial parameters for each set used the param-eters from the initial model fit, although it is advised to checkthat the parameters do not represent a local rather thanthe global minimum. The SSE between the virtual data and

158 c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163

Fig. 1 – Non-linear regression analysis of the observedgrowth data for Listeria monocytogenes at 30 ◦C in 9% saltfrom an initial cellular density of 7.586 × 103 cfu ml−1. Thegrowth was monitored over a 90 h period. The observednumbers as their decimal log were modelled using thestandard modified Gompertz equation [5].

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

0 20 40 60 80 100

log

num

bers

(log

cfu

ml-1

)

Time (mins)

Fig. 2 – Plot of the growth data of L. monocytogenes at 30 ◦Cin 9% salt (symbols) with the fitted modified-Gompertz

such as the calculation of the growth rate (given by BC/e), thecalculation of the confidence interval becomes complex. Theconfidence intervals for these biological parameters, however,

Fig. 3 – A portion of the (50 × 33) random number arraygenerated using the random number feature of Excel’s

the modelled data was calculated per data set (Cells C115 toAZ115). The SSE from each data set was summed and this totalvalue placed in cell C117; SOLVER was used to minimise thistotal SSE by changing all the 200 parameters concurrently. Thisprocedure gave 50 sets of modelled parameters per run. A tar-get SSE of 50 times the SSE obtained from the initial fit (cellB118) was used to monitor the progress of the fitting.

After the minimisation procedure, from the fitted param-eters the mean and standard deviations were found. The95% confidence intervals were calculated from the 95% per-

centiles using the syntax ‘=PERCENTILE(array, 0.025)’, and‘=PERCENTILE(array, 0.975)’. The correlation coefficient was

model (line).

found using the “CORREL(data range 1, data range 2)” function.Tables 1a and 1b give the results of this MC analysis.

3.1.2. Calculation of biological parametersThe growth rate, maximum population density and the lagbefore the onset of growth are important biological parame-ters and have to be calculated from the parameters obtainedfrom the fitting of the modified Gompertz. However, a sin-gular problem is the calculation of the confidence interval ofthe calculated parameter. For example the MPD = A + C, but thevariance is given by

Var(A + C) = Var(A) + Var(C) + 2Cov(AC)

Knowledge of the correlation between parameters allows thecovariance to be calculated. However in more complex cases

Analysis Addin with a distribution of N(0, 0.09047), using arandom seed number of 2.

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163 159

Table 1a – Excel MC (100 iterations), JMP (non-linear regression) and Mathematica MC (10,000 iterations) analyses for thefitting of the modified Gompertz equation to Listeria monocytogenes growth data.

Method Parameter Estimate StdErr LCL UCL

ExcelMC

A 3.948 0.0278 3.906 3.999C 4.896 0.0520 4.801 4.977B 0.095 0.0033 0.090 0.101M 30.318 0.2617 29.766 30.792

JMPNLR

A 3.951 0.0279 3.892 4.008C 4.900 0.0486 4.802 5.000B 0.095 0.0033 0.089 0.102M 30.341 0.2858 29.738 30.937

MathematicaMC

A 3.95 0.0279 3.895 4.0048C 4.9003 0.0481 4.807 4.995B 0.095 0.0033 0.089 0.102M 30.340 0.2833 29.776 30.882

Units. A and C: log10 cfu ml−1; B: 1 h−1; M: h.

Table 1b – Parameter correlation table obtained using Excel MC and JMP NLR analysis (brackets).

A C B M

A 1C −0.697 (−0.675) 1

0.650.13

cswtwsrwy

4

4

TspcS

B 0.347 (0.357) −M 0.546 (0.499) −

an be estimated from the parameter data of the MC analy-is. For each parameter set the particular biological parameteras calculated, giving 101 values (including the original fit-

ing parameters). From these values the required percentilesere calculated. In this particular case a re-parameterized ver-

ion of the modified Gompertz [5] was used to show that theanges obtained by running a non-linear analysis using JMPere equivalent to those obtained directly from the MC anal-sis (Table 2).

. Samples of program runs

.1. Inhibition of P. aeruginosa by chlorhexidine

he LPM (Eq. (4)) can be written in Excel as a nested series of IF-

tatements (e.g. see Fig. 6). Initial estimates for the regressionarameters can be obtained from an analysis of a plot of thehlorhexidine concentration against the RRTD (Fig. 5). UsingOLVER estimates for parameters P1 and P2 and the RMSE were

Table 2 – Calculation of biological parameters from the modifiedfitting of the re-parameterized modified Gompertz equation for(9%).

Method Parameter Estimate

ExcelMC

Growth rate 0.172

Lag 19.806

MPD 8.845

JMPNLR

Growth rate 0.171

Lag 19.829

MPD 8.850

Units. Growth rate: log10 cfu ml−1 h−1; lag: h; MPD: log10 cfu ml−1.

8 (−0.650) 15 (−0.088) 0.0122 (0.161) 1

obtained. The fitted parameters and the RMSE were used toprime the MC analysis. The results of the Excel MC analysisare given in Table 3 and Fig. 6 shows the spreadsheet used.

A table (81 × 100) of random numbers based on N(0, RMSE)was produced (cells C4 to CX84). These random numbers wereadded to the modelled values (cells C88 to CX168), and thenon-linear fitting repeated (C172 to CX252) by regressing all200 parameters at once (Cells C255 to CX256). Fig. 6 shows aportion of the calculation performed. Cell C259 sums the SSEfor each regression performed (cells C258 to CX258). Cells B262to CX262 and C263 to CX263 calculate the MIC and NIC valuesfrom the regressed parameters, respectively.

4.2. Further analysis of compound action potential ofthe rodent optic nerve

The stimulus-evoked compound action potential from themouse optic nerve was successfully modelled using multipleGaussian functions (Brown [3]). The original work used fourGaussian functions to simulate the three peaks of the CAP and

Gompertz using Excel MC, compared to the JMP NLR the growth of Listeria monocytogenes at 30 ◦C in high salt

StdErr LCL UCL

0.005 0.162 0.1820.470 18.733 20.5890.039 8.765 8.9180.005 0.162 0.1820.499 18.765 20.8780.036 8.778 8.924

160 c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163

Fig. 4 – A portion of the (50 × 33) NLR array; column Breproduces the NLR fitting of the original data, with cellsB111 to B114 reproducing the regressed parameters. WellC77 shows the syntax used for the formula, which isreproduced over the array. Cells C115 to AZ115 calculate theSSE between the modelled data and the respective virtualdata set. Cell B117 sums all the 50 individual SSE values.This value is then minimised using the SOLVER utility.

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0. 1 1 10 10 0

RR

TD

Chlorhexidine (mg l-1 )

Fig. 5 – A plot of chlorhexidine concentration (mg l−1)against the observed relative rate to detection for thegrowth of Pseudomonas aeruginosa (ATCC 15442) in TSB at37 ◦C (symbols) and the fitted NLR model (line).

Table 3 – Excel MC (100 iterations), JMP (non-linear regression)

fitting of the LPM to the inhibition of Pseudomonas aeruginosa in

Method Parameter Estimate

ExcelMC

P1 7.265

P2 1.150

JMPNLR

P1 7.257

P2 1.149

MathematicaMC

P1 7.257

P2 1.149

ExcelMC

MIC 17.34

NIC 1.630

MathematicaMC

MIC 17.333

NIC 1.626

Units. P1, MIC and NIC (mg l−1); P2 (dimensionless).

the brief stimulus artefact (Eq. (7)). This model was set up inExcel and the 16 parameters regressed. The sum of squaresobtained was 0.09987; the parameter values for Peaks 1, 2 and 3were essentially identical to those published (nb, a typograph-ical error in the publication gave the area of Peak 1 as 7.663,whereas it should have read 0.633). The estimated parame-ter values for the artefact were, however, different from thosepublished. The peak area found in this analysis was 5.577 vs.7.180 found previously and a calculated amplitude of 30.8 vs.38.042.

The standard error of the fit (0.02986) was used to producean array (96 × 124) of normally distributed random numbers.These values were added to the modelled data (on a sepa-

rate Excel sheet) to produce 96 virtual data sets (Cells DH5 toGY128), Fig. 7 (nb, columns and rows have been ‘hidden’ toshow the full sheet). The model was added to each cell M5 to

and Mathematica MC (10,000 iterations) analyses for the the presence of chlorhexidine.

StdErr LCL UCL

0.093 7.109 7.4550.022 1.107 1.1870.085 7.092 7.4300.019 1.114 1.1850.085 7.095 7.4270.019 1.113 1.1860.428 16.61 18.180.044 1.550 1.7090.372 16.62 18.090.037 1.554 1.698

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163 161

Fig. 6 – Spreadsheet used for the Excel MC analysis of the fitting of the LPM to the data for the growth inhibition of P.aeruginosa by chlorhexidine.

e fitt

Detpl

Fig. 7 – Spreadsheet used for the Excel MC analysis of th

D128, the parameters were placed below each set of mod-

lled data: cells M132 to DD143. The calculated SSE betweenhe modelled data and their respective virtual data set waslaced in cells M145 to DD145. The sum of these SSE was calcu-

ated in Cell L146. Due to the SOLVER limit of 200 parameters,

ing of a multiple Gaussian function (Eq. (7)) to CAP data.

the MC analysis had to be done in batches of 16. The results

of the Monte-Carlo analysis (96 runs; 6 runs of 16) are given inTable 4a. The values obtained for the parameters are very sim-ilar to those previously published, apart for the values for theartefact. The confidence interval for the area of the artefact

162 c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163

Table 4a – Excel MC (96 iterations) analysis of CAP data (Eq. (7)).

Peak Parameter Estimate Stdev LCL UCL

Pk1 A1 0.633 0.007 0.620 0.648w1 0.243 0.002 0.238 0.247c1 1.397 0.001 1.394 1.398

Pk2 A2 1.552 0.017 1.517 1.579w2 0.418 0.004 0.410 0.423c2 1.875 0.001 1.872 1.877

Pk3 A3 1.288 0.013 1.262 1.313w3 0.609 0.006 0.598 0.621c3 2.566 0.004 2.559 2.574

Artefact A4 8.214 2.675 4.002 10.697w4 0.152 0.009 0.135 0.162c4 0.883 0.021 0.863 0.920

Table 4b – Excel MC analysis (96 iterations) of CAP data (Eq. (8)).

Peak Parameter Estimate Stdev LCL UCL

Pk1 A1 0.621 0.008 0.606 0.636w1 0.238 0.002 0.234 0.243c1 1.397 0.001 1.394 1.399

Pk2 A2 1.564 0.019 1.522 1.595w2 0.421 0.005 0.411 0.427c2 1.874 0.002 1.871 1.877

Pk3 A3 1.283 0.015 1.254 1.313w3 0.607 0.007 0.596 0.621

c3 2.568

Artefact D1 4.996

D2 30.906

ranged from 4 to 10.7 and all correlations between parametersA4, w4 and c4 had magnitudes greater than 0.994, suggestingthat the model was over-parameterized.

A NLR analysis using the JMP statistical package gave anestimate for A4 of 5.5788 (95% CI of 2.595–21.835), and correla-tions greater than 0.994 between A4, w4 and c4.

In a second Excel MC study, the artefact was modelled by asimple exponential decay, replacing the Gaussian for the arte-fact by the function D1 exp(−D2(t − 1.04)). A similar MC analysiswas undertaken; the parameter estimates found for the threeprincipal peaks were relatively unchanged, but the parameterestimates for the artefact now had narrower confidence inter-vals and the correlation between D1 and D2 = 0.246 (Table 4b).

4.3. Caveats to using MC analysis within Excel

The MC analysis within Excel is initially primed using theparameter estimates from the initial SOLVER minimisation

Table 4c – JMP NLR analysis of CAP data (Eq. (8)).

Peak Parameter Estimate

Pk1 A1 0.620

w1 0.238

c1 1.396

Pk2 A2 1.566

w2 0.421

c2 1.874

Pk3 A3 1.281

w3 0.607

c3 2.568

Artefact D1 4.997

D2 30.907

0.004 2.559 2.5770.034 4.928 5.0580.194 30.447 31.455

procedure. It is possible that the estimates relate not to aglobal minimum but to a local minimum, especially if thereis a high degree of correlation between given parameters (aswas observed in the first analysis of the CAP data). One methodto overcome any such possibility is to use different initialparameter estimates for each virtual dataset. This can be doneusing Excel’s “RANDBETWEEN(a,b)” function, where the usergenerates a scaled value of a parameter between two integerextrema (a and b), and rescales to accommodate the desiredmagnitude of the initial parameter.

4.4. Comparison of the Excel NLR and MC analysiswith Mathematica

The non-linear regression capability of JMP was used to fitthe three sets of sample data; Tables 1, 2, 3 and 4c give theparameters, and their confidence intervals obtained using thissophisticated software; Table 1b also compares the param-

Approx StdErr LCL UCL

0.009 0.603 0.6370.003 0.233 0.2430.001 1.394 1.3990.019 1.529 1.6030.004 0.413 0.4300.002 1.871 1.8770.015 1.252 1.3120.007 0.593 0.6210.004 2.560 2.5760.034 4.931 5.0630.414 30.124 31.712

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 1 0 7 ( 2 0 1 2 ) 155–163 163

Table 4d – Mathematica MC (10,000 iterations) analysis of CAP data (Eq. (8)).

Peak Parameter Estimate Approx StdErr LCL UCL

Pk1 A1 0.621 0.008 0.603 0.636w1 0.238 0.002 0.233 0.242c1 1.397 0.001 1.394 1.399

Pk2 A2 1.564 0.018 1.530 1.601w2 0.421 0.004 0.413 0.430c2 1.874 0.016 1.871 1.877

Pk3 A3 1.282 0.015 1.253 1.312w3 0.607 0.007 0.594 0.621c3 2.568 0.004 2.560 2.576

eNMsptNeficwsIfdrc

tyora

5

Twue(MiU

r

[

[[

[

[

Artefact D1 4.997

D2 30.925

ter correlations obtained between the Excel MC and JMPLR analysis. A small program was written to conduct aonte-Carlo analysis within Mathematica (Version 8) with the

ubsequent analysis of 10,000 virtual data sets (approx 2–5 miner 10,000 runs) for each of the fitted models. The results ofhese MC analyses are also given in Tables 1a, 3 and 4d. TheLR analysis using JMP essentially gave the same parameterstimates as that from the NLR Excel analysis; the con-dence intervals calculated using the Excel MC techniquelosely agree with those calculated using the Hessian methodithin JMP. A comparison of the results of the MC analy-

is of Mathematica also compares well with the Excel output.t should be noted that the confidence intervals obtainedrom the MC analysis and the direct NLR analysis using JMPo not completely agree with each other and this simplyeflects the differences in the techniques used. Each is equallyorrect.

The comparison between the parameter estimates andheir confidence intervals as obtained from the Excel MC anal-sis with either the direct NLR or the MC analyses using JMPr Mathematica demonstrates the capacity of Excel to produceesults equivalent to those from more sophisticated pack-ges.

. Hardware and software specification

he method was carried out on a basic desktop computerith an AMD Phenom 9750 Quad core processor (2.4 GHz),sing Microsoft Excel 2007 under Windows 7. Non lin-ar regression comparisons were carried out using the JMP

4.0.4) Statistical Software (SAS Institute Inc, Cary, NC) and

onte-Carlo comparisons were carried out using Mathemat-ca Version 8.0.0.0 (Wolfram Research Inc, Champaign, IL,SA).

[

0.034 4.931 5.0630.413 30.141 31.752

6. Program availability

Spreadsheets with the worked examples are available from theauthor on request. The Mathematica coding used to producethe NLR fits and Monte Carlo simulations are also availablefrom the corresponding author.

Acknowledgment

We would like to acknowledge the financial support given bythe Food Standards Agency (UK) to Mr. Ioannis Mytilinaios(post graduate scholarship PG 1025).

e f e r e n c e s

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56 (1990) 1875–1881.

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