(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 205546, 8754]*) (*NotebookOutlinePosition[ 209380, 8857]*) (* CellTagsIndexPosition[ 208542, 8834]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["SALVADOR 2.0 Manual", "Title", FontSize->16], Cell["\<\ Qi Zheng Department of Epidemiology and Biostatistics School of Rural Public Health, Bryan, Texas 77802 qzheng@srph.tamhsc.edu February, 2005\ \>", "Text"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "SALVADOR is a package intended as a tool for exploring various \ quantitative aspects of fluctuation experiments. For reasons of computational \ efficiency, the package was written in a mixture of the ", StyleBox["Mathematica", FontSlant->"Italic"], " and the C language. Therefore, before using SALVADOR, two programs must \ be install from within ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Install["\"]\)], "Input"], Cell[BoxData[ \(LinkObject["salvadorV2", 2, 2]\)], "Output"] }, Open ]], Cell[BoxData[ \(<< salvadorV2.m\)], "Input"], Cell["\<\ (To avoid being distracted by warning messages against use of similar \ variable names, e.g., var1 and var2, we suppress such warning messages.)\ \>", "Text"], Cell["Off[General::spell1]", "Input"], Cell["\<\ For notation, nomenclature and other background information, the user can \ consult Zheng(1999,2002,2005) and numerous additional references contained \ therein. The sole purpose of this manual is to demonstrate the various \ functionalities of SALVADOR 2.0.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Simulation", "Section", FontSize->16], Cell[TextData[{ "Five basic parameters must be specified before you simulate the number of \ mutants in a culture. These five parameters are: 1) mutation rate (per cell \ per unit time) \[Mu]; 2) cellular birth rate for nonmutants ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_1\)]], "; 3) cellular birth rate for mutants ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\_2\)]], "; 4) initial population size ", Cell[BoxData[ \(TraditionalForm\`N\_0\)]], "; and 5) final population size ", Cell[BoxData[ \(TraditionalForm\`N\_T\)]], ". It is strongly recommended that, before running a simulation, you check \ the parameters of interest by computing the mean and variance of the number \ of mutants, for unrealistic parameters can lead to very time-consuming \ computational tasks and useless results. We now use the following five \ parameters to simulate a fluctuation experiment ." }], "Text"], Cell[BoxData[ \(mu = 10^\(-8\); b1 = 1.2; b2 = 1.1; N0 = 100; Nt = 2.4*10^8;\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MeanOfMutants\)\)], "Input"], Cell[BoxData[ \("MeanOfMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2,N\_0,N\_T\)] computes \ the mean number of mutants on each sister culture."\)], "Print", CellTags->"Info3317216010-4785616"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MeanOfMutants[mu, b1, b2, N0, Nt]\)], "Input"], Cell[BoxData[ \(16.94451339681247`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?VarOfMutants\)\)], "Input"], Cell[BoxData[ \("VarOfMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2,N\_0,N\_T\)] computes the \ variance of the number of mutants on each sister culture."\)], "Print", CellTags->"Info3317216011-1867243"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VarOfMutants[mu, b1, b2, N0, Nt]\)], "Input"], Cell[BoxData[ \(995576.079641768`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?simuMutants\)\)], "Input"], Cell[BoxData[ \("simuMutants[\!\(\[Mu],\[Beta]\_1,\[Beta]\_2,\ N\_0,N\_T\)] simulates \ the number of mutants in a sister culture. Note \!\(\[Mu]\) is the mutation \ rate per cell per unit time; \!\(\[Beta]\_1\) is growth rate for nonmutants; \ \!\(\[Beta]\_2\) is growth rate of mutants; \!\(N\_0\) is the initial \ population size; \!\(N\_T\) is the population size prior to plating."\)], \ "Print", CellTags->"Info3317216011-7491825"] }, Open ]], Cell["\<\ Now we simulate the numbers of mutants in 20 sister cultures.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[simuMutants[mu, b1, b2, N0, Nt], {20}]\)], "Input"], Cell[BoxData[ \({9, 6, 12, 2, 89, 20, 9, 2, 5, 6, 1, 3, 3, 7, 6, 3, 65, 5, 5, 1}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Important Distributions", "Section", FontSize->16], Cell[TextData[{ "SALVADOR can compute the probability mass function (p.m.f.) of four most \ important distributions in fluctuation analysis: the Poisson distribution, \ the LD or Luria-Delbruck distribution, the M or Mandelbrot distribution, and \ the convolution of a Poisson and an LD distribution. The last distribution \ was popularized in fluctuation analysis by Cairns et al. in 1988. Using \ pmf2cdf, you can transform a list of p.m.f. to a list of c.d.f. (cumulative \ distribution function). As a demonstration we first compute ", Cell[BoxData[ \(TraditionalForm\`p\_1\)]], ", ..., ", Cell[BoxData[ \(TraditionalForm\`p\_10\)]], " of a Poisson distribution having mean 10.5." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfPoisson\)\)], "Input"], Cell[BoxData[ \("pmfPoisson[\!\(\[Lambda]\),k] returns a list of probabilities \ \!\({p\_0,p\_1,...,p\_k}\) according to a Poisson(\!\(\[Lambda]\)) \ distribution."\)], "Print", CellTags->"Info3317216011-4729632"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfPoisson[10.5, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00028913271817234516`, 0.001517946770404812`, 0.005312813696416842`, 0.013946135953094211`, 0.029286885501497845`, 0.051252049627621234`, 0.07687807444143184`, 0.1009024727043793`, 0.11771955148844249`, 0.12360552906286462`}\)], "Output"] }, Open ]], Cell["\<\ Now we plot the c.d.f. of a Luria-Delbruck distribution, so its upper tail \ behavior can be more easily examined.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfLD\)\)], "Input"], Cell[BoxData[ \("pmfLD[\!\(m,\[Phi],k\)] returns a list of probabilities \ \!\({p\_0,p\_1,...,\ p\_k}\) according to a Luria-Delbr\!\(\[UDoubleDot]\)uck \ distribution, LD(m,\!\(\[Phi]\)). Note pmfLD[m,k] is the same as \ pmfLD[m,1,k]."\)], "Print", CellTags->"Info3317216011-4887619"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LDprob = pmfLD[5.7, 150]; \ LDprob[\([Range[10]]\)]\)], "Input"], Cell[BoxData[ \({0.003345965457471272`, 0.009536001553793125`, 0.016767469398752913`, 0.02355789717184977`, 0.02910035040826871`, 0.03312787122784504`, 0.03569563178469149`, 0.03701109059727024`, 0.037328993778887536`, 0.036895826091020184`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LDcdf = pmf2cdf[LDprob]; \ LDcdf[\([Range[10]]\)]\)], "Input"], Cell[BoxData[ \({0.003345965457471272`, 0.012881967011264397`, 0.02964943641001731`, 0.05320733358186708`, 0.08230768399013579`, 0.11543555521798082`, 0.1511311870026723`, 0.18814227759994254`, 0.22547127137883008`, 0.26236709746985026`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ListPlot[Transpose[{Range[0, 150], LDcdf}], PlotJoined \[Rule] True, Frame \[Rule] True]\)], "Input"], 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TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Now let's look at both an approximate form and the exact form of an M \ distribution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfM\)\)], "Input"], Cell[BoxData[ \("pmfM[m,r,k] returns a list of probabilities \!\({p\_0,p\_1,...,p\_k}\) \ according to an \!\(M(m,r,1)\) distribution; pmfM[m,\!\(\[Rho],\[Phi]\),n] \ returns the same list according to an \!\(M(m,r,\[Phi])\) distribution. Note \ m is the mean number of mutations, \!\(r=\[Beta]\_1/\[Beta]\_2\) is the ratio \ of mutant growth rate to nonmutant growth rate, and \ \!\(\[Phi]=1-N\_0/N\_T\)."\)], "Print", CellTags->"Info3317216011-5180669"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfM[10.5, 0.8, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00012850343029882007`, 0.00034573541961349213`, 0.0007047470220843672`, 0.001209956176818187`, 0.0018478624868760109`, 0.002593033991680133`, 0.0034141644273363568`, 0.0042789138833278706`, 0.005157198619975631`, 0.006023079521618202`}\)], "Output"] }, Open ]], Cell["\<\ If we assume \[Phi]=1, we can also use an algorithm devised by Koch (based \ on an algorithm of Lea and Coulson) to compute the p.m.f. of an M \ distribution.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfMKoch\)\)], "Input"], Cell[BoxData[ \("pmfMKoch[m,r,k] computes \!\({p\_0,p\_1,...,p\_k}\) for an \ \!\(M(m,r,1)\) distribution, using Koch's algorithm. Note that r is the \ reciprocal of Koch's original parameter b (called \!\(\[Rho]\) in SALVADOR \ v.1). (This function is not recommended for large k.)"\)], "Print", CellTags->"Info3317216011-5924977"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfMKoch[10.5, 0.8, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00012850343029882007`, 0.000345735419613492`, 0.0007047470220843671`, 0.0012099561768181867`, 0.0018478624868760109`, 0.0025930339916801328`, 0.0034141644273363563`, 0.00427891388332787`, 0.00515719861997563`, 0.006023079521618202`}\)], "Output"] }, Open ]], Cell["\<\ To use the exact form of the M distribution, the parameter \[Phi]<1 must be \ specified.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pmfM[10.5, 0.8, 0.9999, 10]\)], "Input"], Cell[BoxData[ \({0.000027536449349747158`, 0.00012851628179849647`, 0.000345799986016298`, 0.0007049322303118314`, 0.0012103555467003163`, 0.0018485831774866113`, 0.0025941846925863032`, 0.003415844983222769`, 0.004281207952509993`, 0.005160169530586166`, 0.006026768867509342`}\)], "Output"] }, Open ]], Cell["\<\ Finally let us take a look at the convolution of a Poisson distribution and \ an LD distribution .\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?pmfCairns\)\)], "Input"], Cell[BoxData[ \("pmfCairns[\!\(m,\[Phi],\[Lambda],k\)] returns a list of probabilities \ \!\({p\_0,...,p\_k}\) according to the convolution of an LD(\!\(m,\[Phi]\)) \ distribution and a Poisson(\!\(\[Lambda]\)) distribution."\)], "Print", CellTags->"Info3317216011-5580427"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pmfCairns[2.5, 1, 8.5, 10]\)], "Input"], Cell[BoxData[ \({0.00001670170079024566`, 0.00016284158270489518`, 0.0008008117576822997`, 0.0026513515064390232`, 0.006657036789692021`, 0.013540891094019243`, 0.023283755824621593`, 0.03488343821483244`, 0.04659431114081542`, 0.05652302198442524`, 0.06324871361500847`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Estimation of Mutation Rates", "Section", FontSize->16], Cell[TextData[{ "SALVADOR concentrates on estimation of the mean number of mutations \ denoted by m, because an estimate of the mutation rate ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_\[Beta]\)]], " can be easily obtained from that of m. The recommended estimation method \ is the maximum likelihood method. We use the experimental data of Demerec \ (1945) to illustrate the capabilities of SALVADOR 2.0. The following are \ the known experimental parameters of Demerec's experiment." }], "Text"], Cell[BoxData[ \(N0 = 90; Nt = 1.9*10^8; phi = 1 - N0/Nt;\)], "Input"], Cell["\<\ The experimental data of Demerec's experiment are given below.\ \>", "Text"], Cell[BoxData[ \(\(demerec = {33, 18, \ 839, 47, 13, 126, 48, 80, 9, 71, 196, 66, 28, 17, 27, 37, 126, 33, 12, 44, 28, 67, 730, 168, 44, 50, 583, 23, 17, 24};\)\)], "Input"], Cell["\<\ We can fit an LD(m,1) distribution to the data by calling newtonLD, which \ uses the Newton-Raphson methods to compute the maximum likelihood estimate \ (m.l.e.) of m.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?newtonLD\)\)], "Input"], Cell[BoxData[ \("newtonLD[data,opts] uses the Newton-Raphson method to compute mle of m \ for the LD model."\)], "Print", CellTags->"Info3317216011-4936587"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(newtonLD[demerec]\)], "Input"], Cell[BoxData[ \(10.843826994529076`\)], "Output"] }, Open ]], Cell["\<\ Sometimes it is necessary to choose an initial guess, to set convergence \ critiria or to specify the maximum number of iterations. Thus it is helpful \ to familiarize yourselve with the options of newtonLD.\ \>", "Text"], Cell[CellGroupData[{ Cell["Options[newtonLD]", "Input"], Cell[BoxData[ \({ShowIterations \[Rule] False, Phi \[Rule] 1, InitialM \[Rule] \(-1\), MaxIterations \[Rule] 50, Tolerance \[Rule] 1\/100000000}\)], "Output"] }, Open ]], Cell["\<\ For example, we can choose two different starting points and see how the two \ converge to the same m.l.e. Note that if InitialM is set to a negative \ number, then SALVADOR 2.0 uses its own rules to deterimine an initial guess, \ which was documented in Zheng (2004).\ \>", "Text"], Cell[CellGroupData[{ Cell["newtonLD[demerec,InitialM->3,ShowIterations->True]", "Input"], Cell[BoxData[ \(3\)], "Print"], Cell[BoxData[ \(5.9410741308071096`\)], "Print"], Cell[BoxData[ \(9.34988232782199`\)], "Print"], Cell[BoxData[ \(10.733970006389027`\)], "Print"], Cell[BoxData[ \(10.843274363697045`\)], "Print"], Cell[BoxData[ \(10.843826980617358`\)], "Print"], Cell[BoxData[ \(10.843826994529072`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["newtonLD[demerec,InitialM->15,ShowIterations->True]", "Input"], Cell[BoxData[ \(15\)], "Print"], Cell[BoxData[ \(10.171664609935714`\)], "Print"], Cell[BoxData[ \(10.822551997688214`\)], "Print"], Cell[BoxData[ \(10.843806356058554`\)], "Print"], Cell[BoxData[ \(10.843826994509673`\)], "Print"], Cell[BoxData[ \(10.843826994529072`\)], "Output"] }, Open ]], Cell["\<\ If you believe that mutants grow at a rate different from that at which \ nonmutants grow, then you can try to fit a M(m,r) model to your data. Note \ that r in SALVADOR 2.0 is the reciporacol of \[Rho] in SALVADOR 1.0.\ \>", "Text"], Cell[CellGroupData[{ Cell["?newtonM", "Input"], Cell[BoxData[ \("newtonM[data,opts] uses the Newton-Raphson method to compute MLEs of m \ and r for the M(m,r) model."\)], "Print", CellTags->"Info3317216012-4476652"] }, Open ]], Cell[CellGroupData[{ Cell["newtonM[demerec]", "Input"], Cell[BoxData[ \({9.852618286896071`, 0.8938070840172462`}\)], "Output"] }, Open ]], Cell["Options of newtonM are similar to those of newtonLD.", "Text"], Cell[CellGroupData[{ Cell["Options[newtonM]", "Input"], Cell[BoxData[ \({ShowIterations \[Rule] False, MaxIterations \[Rule] 50, Tolerance \[Rule] 1\/100000000, InitialM \[Rule] \(-1\), InitialR \[Rule] \(-1\)}\)], "Output"] }, Open ]], Cell["\<\ you can also use the slower melM to accomplish the same task.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Interval Estimation of Mutation Rates", "Section"], Cell["\<\ The most important new feature of SALVADOR 2.0 is its capability of computing \ (profile) likelihood ratio-based confidence intervals. For the equal growth \ model, the function for constructing a CI for m is LRInervalLD.\ \>", "Text"], Cell[CellGroupData[{ Cell["?LRIntervalLD", "Input"], Cell[BoxData[ \("LRIntervalLD[X,opts] uses data given in X to construct a likelihood \ ratio (LR) based confidence interval for m that has asymptotic confidence \ coefficient \!\(1-\[Alpha]\); \!\(\[Alpha]\) is specified by Alpha->\!\(\ \[Alpha]\). Data in X is assumed to be generated by an LD(m,\!\(\[Phi]\)) \ distribution; \!\(\[Phi]\) is specified by Phi->\!\(\[Phi]\)."\)], "Print", CellTags->"Info3317216013-2836232"] }, Open ]], Cell["\<\ A 95% asymptotic confidence interval for m is obtained by the command\ \>", "Text"], Cell[CellGroupData[{ Cell["LRIntervalLD[demerec]", "Input"], Cell[BoxData[ \({8.650538086656935`, 13.194764931218957`}\)], "Output"] }, Open ]], Cell["\<\ And a 90% confidence interval would be constructed by using the Alpha option. \ Note that if a (1-\[Alpha])100% CI is desired, then the options Alpah should \ be set to \[Alpha], not \[Alpha]/2, because critical values are determined by \ 2 chi-squared distribution.\ \>", "Text"], Cell[CellGroupData[{ Cell["LRIntervalLD[demerec,Alpha->0.1]", "Input"], Cell[BoxData[ \({8.991839033899844`, 12.806736879987557`}\)], "Output"] }, Open ]], Cell["In addition to Alpha, There are a few other options.", "Text"], Cell[CellGroupData[{ Cell["Options[LRIntervalLD]", "Input"], Cell[BoxData[ \({Alpha \[Rule] 0.05`, Tolerance \[Rule] 1\/100000000, ShowIterations \[Rule] False, MaxIterations \[Rule] 50, Phi \[Rule] 1.`, InitialM \[Rule] \(-1\), InitialLowerM \[Rule] \(-1\), InitialUpperM \[Rule] \(-1\)}\)], "Output"] }, Open ]], Cell["\<\ InitialM sets an initial guess for computing the maximum likelihood estimate \ of m, which is needed for computing a CI for m. The two options, \ InitialLowerM and intialUpperM, are statrting points for computing the lower \ and upper boundary points of the desired CI for m. When these options are \ set to a negative number, SALVADOR adopts its algorithms described in Zheng \ (2004) to determine reasosnalbe starting values. For example, if we choose 3 \ as an initial guess for the lower boundary point, and 16 as an initial guess \ for the upper boudary point, the iteration process will proceed as follows.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ LRIntervalLD[demerec,InitialLowerM->3,InitialUpperM->16,ShowIterations->True]\ \ \>", "Input"], Cell[BoxData[ \("Iterating for lower limit ..."\)], "Print"], Cell[BoxData[ \(3\)], "Print"], Cell[BoxData[ \(5.699422221344831`\)], "Print"], Cell[BoxData[ \(7.603623983988979`\)], "Print"], Cell[BoxData[ \(8.454611596733105`\)], "Print"], Cell[BoxData[ \(8.64144232084982`\)], "Print"], Cell[BoxData[ \(8.65051691861597`\)], "Print"], Cell[BoxData[ \("Iterating for upper limit ..."\)], "Print"], Cell[BoxData[ \(16\)], "Print"], Cell[BoxData[ \(13.871854805032044`\)], "Print"], Cell[BoxData[ \(13.263291837867623`\)], "Print"], Cell[BoxData[ \(13.19564885862982`\)], "Print"], Cell[BoxData[ \(13.194765082626434`\)], "Print"], Cell[BoxData[ \(13.194764931218952`\)], "Print"], Cell[BoxData[ \({8.650538086541864`, 13.194764931218952`}\)], "Output"] }, Open ]], Cell["\<\ You can also construct a Wald type CI for m by first computing expected \ Fisher information.\ \>", "Text"], Cell[CellGroupData[{ Cell["?FisherInfoLD", "Input"], Cell[BoxData[ \("FisherInfoLD[m,\!\(\[Phi]\),k] computes the Fisher information for m \ according to an LD(m,\!\(\[Phi]\)) distribution; it retains k terms in the \ series \!\(I(m)=\[Sum]\ f\^2\%j/p\_j\)."\)], "Print", CellTags->"Info3317216015-6655442"] }, Open ]], Cell[CellGroupData[{ Cell["mhat=newtonLD[demerec]", "Input"], Cell[BoxData[ \(10.843826994529076`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["info=FisherInfoLD[mhat,50000]", "Input"], Cell[BoxData[ \(0.025357094149261867`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["{mhat-1.96 Sqrt[1/(30 info)], mhat+1.96 Sqrt[1/(30 info)]}", "Input"], Cell[BoxData[ \({8.596606419965056`, 13.091047569093096`}\)], "Output"] }, Open ]], Cell["\<\ This Wald type CI for m is very simular to the likelihood ratio based CI we \ just found.\ \>", "Text"], Cell["\<\ When fitting a differential growth model to data, use the function \ profileIntervalM to construct a profile likelihood based CI for m, and use \ the function profileIntervalR to construct a CI for r.\ \>", "Text"], Cell[CellGroupData[{ Cell["?profileIntervalM", "Input"], Cell[BoxData[ \("profileIntervalM[data,opt] computes profile likelihood based \ confidence intervals for m under the M(m,r) model."\)], "Print", CellTags->"Info3317216057-4785192"] }, Open ]], Cell[CellGroupData[{ Cell["profileIntervalM[demerec]", "Input"], Cell[BoxData[ \({6.9830651233184415`, 13.007332373950865`}\)], "Output"] }, Open ]], Cell["\<\ To get a Wald type CI for m, we have to compute the expected Fisher \ information matrix approximately. Since we know little about the converging \ speed of computing this informtion matrix, it is strongly recommended that a \ Wald type CI be corroborated by a likelihood ratio based CI.\ \>", "Text"], Cell[CellGroupData[{ Cell["?FisherInfoMatrix", "Input"], Cell[BoxData[ \("FisherInfoMatrix[m,r,n] computes the expected Fisher information \ matrix for the M(m,r) model, using n terms."\)], "Print", CellTags->"Info3317216058-9081352"] }, Open ]], Cell[CellGroupData[{ Cell["{mhatm,rhatm}=newtonM[demerec]", "Input"], Cell[BoxData[ \({9.852618286896071`, 0.8938070840172462`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["infoMat=FisherInfoMatrix[mhatm,rhatm,5000]", "Input"], Cell[BoxData[ \({{0.026539225837420636`, \(-0.2594358703137823`\)}, \ {\(-0.2594358703137823`\), 5.061358718741653`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["{{var11,var12},{var21,var22}}=Inverse[infoMat]", "Input"], Cell[BoxData[ \({{75.52286098136372`, 3.871161926289656`}, {3.8711619262896573`, 0.3960039931670454`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["{mhatm-1.96 Sqrt[var11/30],mhatm+1.96 Sqrt[var11/30]}", "Input"], Cell[BoxData[ \({6.742802522010061`, 12.96243405178208`}\)], "Output"] }, Open ]], Cell["\<\ Again the above Wald type CI is similar to the likelihood ratio based CI we \ just found.\ \>", "Text"], Cell["To get a 99% CI for M, set Alpha to 0.01, not 0.005.", "Text"], Cell[CellGroupData[{ Cell["profileIntervalM[demerec,Alpha->0.01]", "Input"], Cell[BoxData[ \({6.158961230587444`, 14.043751024099189`}\)], "Output"] }, Open ]], Cell["\<\ To see if the differential growth model is necessary, we can construct a 95% \ CI for r.\ \>", "Text"], Cell[CellGroupData[{ Cell["profileIntervalR[demerec]", "Input"], Cell[BoxData[ \({0.6910073899881216`, 1.1268801153166552`}\)], "Output"] }, Open ]], Cell["\<\ Because unit is not contained in the above CI, the equal growth model may be \ sufficient. \ \>", "Text"], Cell["\<\ Sometimes you may want to use some of the options when constructing a \ likelihood based CI.\ \>", "Text"], Cell[CellGroupData[{ Cell["Options[profileIntervalM]", "Input"], Cell[BoxData[ \({Alpha \[Rule] 0.05`, MaxIterations \[Rule] 50, InitialM \[Rule] \(-1\), InitialR \[Rule] \(-1\), InitialLowerM \[Rule] \(-1\), InitialUpperM \[Rule] \(-1\), Tolerance \[Rule] 1\/1000000000, InitialLowerR \[Rule] \(-1\), InitialUpperR \[Rule] \(-1\), ShowIterations \[Rule] False}\)], "Output"] }, Open ]], Cell["\<\ For example, occasionally you may have to try a few starting points in \ constructing a confidence interval.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ profileIntervalR[demerec,InitialLowerR->0.5,InitialUpperR->1.5,ShowIterations-\ >True]\ \>", "Input"], Cell[BoxData[ \("Iterating for lower limit ..."\)], "Print"], Cell[BoxData[ \({8.799956037182803`, 0.5`}\)], "Print"], Cell[BoxData[ \({2.2051609933373832`, 0.39316001239037357`}\)], "Print"], Cell[BoxData[ \({4.20025480491638`, 0.6105082612451901`}\)], "Print"], Cell[BoxData[ \({5.7695296742140645`, 0.5678451456137223`}\)], "Print"], Cell[BoxData[ \({7.23224482301614`, 0.6582003543179061`}\)], "Print"], Cell[BoxData[ \({7.692066492409306`, 0.6878755543876475`}\)], "Print"], Cell[BoxData[ \({7.73234888851331`, 0.690977939427495`}\)], "Print"], Cell[BoxData[ \({7.732677433440407`, 0.6910073874171986`}\)], "Print"], Cell[BoxData[ \({7.732677459362545`, 0.6910073899881213`}\)], "Print"], Cell[BoxData[ \("Iterating for upper limit ..."\)], "Print"], Cell[BoxData[ \({10.90528053660934`, 1.5`}\)], "Print"], Cell[BoxData[ \({13.335417373602336`, 1.337867887117464`}\)], "Print"], Cell[BoxData[ \({12.310028005062616`, 1.1633629546232844`}\)], "Print"], Cell[BoxData[ \({11.946647814278675`, 1.1289512043664727`}\)], "Print"], Cell[BoxData[ \({11.928254901550776`, 1.126887348033876`}\)], "Print"], Cell[BoxData[ \({11.928185096616975`, 1.1268801154076615`}\)], "Print"], Cell[BoxData[ \({0.6910073899881143`, 1.1268801153166508`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Methods of Historical Interest", "Section", FontSize->16], Cell["\<\ The maximum likelihood method is obviously the method of choice for \ estimating mutation rates. But there are several methods of great historical \ interest; these methods are still popular.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfP0\)\)], "Input"], Cell[BoxData[ \("MethodOfP0[data] estimates m, the expected number of mutations, using \ the \!\(P\_0\) method."\)], "Print", CellTags->"Info3317216076-5808766"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MethodOfP0[demerec]\)], "Input"], Cell[BoxData[ \("The \!\(P\_0 \) method is not applicable to this case."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfMeans\)\)], "Input"], Cell[BoxData[ \("MethodOfMeans[data,\!\(w\_0:=10\)] estimates m (the mean number of \ mutations) by the method of means as proposed by Luria and Delbr\!\(\ \[UDoubleDot]\)ck. Note \!\(w\_0\) is an initial value needed for numerically \ solving a transcendental equation."\)], "Print", CellTags->"Info3317216076-9348376"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(MethodOfMeans[demerec]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(18.940937803769877`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?MethodOfMedians\)\)], "Input"], Cell[BoxData[ \("MethodOfMedians[data,\!\(w\_0:=1\)] estimates m, the expected number \ of mutations, using the method of medians proposed by Lea and Coulson. Note \ \!\(w\_0\) is an initial value needed to solve a transcendental equation by \ Newton's method. If only the median of the data is known, say med, \ MethodOfMedians[med] also returns an estimate of m."\)], "Print", CellTags->"Info3317216076-1494160"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MethodOfMedians[demerec]\)], "Input"], Cell[BoxData[ \(11.851891765263119`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(?semiMLE\)\)], "Input"], Cell[BoxData[ \("semiMLE[data,\!\(m\_0:=1\)] computes an estimate of m by solving \!\(\ \[Sum]Y\_i=0\), where \!\(Y\_i\) are the Lea-Coulson transforms of the \ original data. Note that \!\(m\_0\) is a starting value used to solve the \ equation."\)], "Print", CellTags->"Info3317216076-2304299"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(semiMLE[demerec]\)], "Input"], Cell[BoxData[ \(11.553643621646893`\)], "Output"] }, Open ]], Cell["\<\ It is noteworthy that Jones has proposed a method based on the median that \ does not require an initial guess.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ JonesMedian[demerec]\ \>", "Input"], Cell[BoxData[ \(10.43365243358806`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Another Example", "Section"], Cell["\<\ Boe et al. (1994) meticulously conducted a large scale fluctuation \ experiment, producing the following 1104 observations.\ \>", "Text"], Cell["\<\ boe={0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3,4,4,4,4,\ 6,8,14,15,17,27,29,39,39,66,74,265,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,\ 1,1,1,1,2,2,2,2,3,3,4,4,4,5,5,5,7,8,12,16,18,19,30,42,132,152,513,0,0,0,0,0,0,\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,4,5,5,6,\ 7,8,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,\ 2,3,4,4,5,6,6,13,18,19,26,36,68,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,\ 1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,5,6,6,7,12,16,24,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,3,4,4,5,5,6,10,11,12,14,24,27,29,30,\ 73,482,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,\ 1,1,1,1,2,3,3,5,11,16,30,31,49,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,\ 1,1,1,1,2,2,2,2,2,2,2,2,2,4,5,5,6,6,6,7,12,12,12,16,513,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5,7,8,11,11,18,52,\ 73,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,\ 3,3,4,4,4,5,7,7,7,151,513,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,\ 1,1,1,1,1,1,2,2,3,3,3,4,4,5,6,11,13,16,40,258,320,513,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,11,11,17,107,\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,\ 3,4,4,6,6,16,18,20,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,1,1,1,1,1,1,1,2,2,3,3,5,8,8,9,21,21,26,37,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,3,3,3,4,5,5,6,6,7,7,8,10,12,14,0,0,0,0,0,0,\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,3,3,4,5,5,6,13,16,\ 19,26,41,105,116,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,\ 1,1,1,1,2,2,2,3,6,6,6,11,13,19,21,34,140,146,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,9,10,10,11,13,19,20,23,59,69,0,0,\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,\ 8,8,9,10,21,26,192,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,\ 2,2,2,2,2,3,3,4,4,6,9,9,10,14,25,26,29,57,137,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\ 0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,5,6,7,8,10,14,16,28,0,0,0,0,0,\ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,4,6,6,7,\ 7,18,32,35,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,\ 2,3,3,3,3,4,4,5,5,8,10,11,12,19,21};\ \>", "Input"], Cell["\<\ Assuming equal growth rate we have the maximum likelihood estimate of m as \ follows.\ \>", "Text"], Cell[CellGroupData[{ Cell["newtonLD[boe]", "Input"], Cell[BoxData[ \(0.7365908051572939`\)], "Output"] }, Open ]], Cell["\<\ The fitting is less than perfect, but still an LD distribution catches the 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