Patent Application: US-36372703-A

Abstract:
a method is disclosed for improving the reliability of physical measurements obtained from array hybridization studies performed on an array having a large number of genomic samples including a replicate subset containing a small number of replicates insufficient for making precise and valid statistical inferences . an error in measurement of a sample is estimated by combining estimates obtained with individual samples in the replicate subset , and utilizing the estimated sample error as a standard for accepting or rejecting the measurement of a sample under test .

Description:
we assume throughout that we observe data y ij , with i = 1 , . . . , n and j = 1 , . . . , in where : and the ε ij are independent and identically distributed . our interest is in estimating the residual distribution , the distribution of the ε ij . let ƒ , ƒ * and f denote the density , characteristic function and cumulative distribution function of the ε ij . a tacit assumption is that n is large and m is small , for instance 2 or 3 . assumptions such as these arise naturally in measurement error models . while our interest in estimating the residual distribution arose in the analysis of gene expression data , we expect the methodology to be of broader applicability . with m moderate to large , the usual estimate of the residual distribution is a discrete distribution that gives equal mass to each of the estimated residuals : f ^ e  ( e ) = 1 nm  ∑ i = 1 n  ∑ j = 1 m  i  { y ij - y _ i ≥ ɛ } ( 2 ) this estimator is biased with the bias dependent up n the residual distribution . for instance , for a n ( 0 , 1 ) residual distribution the expectation of { circumflex over ( f )} e is the n ( 0 ,( m − 1 )/ m ) distribution . for a cauchy residual distribution , the expectation is the distribution of a cauchy random variable multiplied by 2 − 2 / m . when the residual distribution has finite mean , the bias decreases with increasing m . with n large and m small however , the bias dominates the variance . in contrast the methods presented here give consistent ( large n ) estimates of the residual distribution . the basic idea uses the differences in observations , y ij1 − y ij2 which have distributions that depend , in a known way , upon the residual distribution alone . this differs from the usual way of calculating residuals . an example best illustrates this difference . consider the three replicate values of 1 , 2 , and 3 . the usual way of calculating residuals is to subtract the mean of the three values from each value in turn ( 1 - 2 ; 2 - 2 ; 3 - 2 ) yielding three residuals (− 1 , 0 , 1 ). in the preferred form of the present process , the residuals are calculated instead by subtracting each replicate value from each of the other replicate values in all possible permutations . in the present example , this would be ( replicate 1 - replicate 2 ; replicate 2 - replicate 1 ; replicate 1 - replicate 3 ; replicate 3 - replicate 1 ; replicate 2 - replicate 3 ; replicate 3 - replicate 2 ), that is , ( 1 - 2 ; 2 - 1 ; 1 - 3 ; 3 - 1 ; 2 - 3 ; 3 - 2 ) to yield six residuals (- 1 , 1 , - 2 , 2 ,- 1 , 1 ). this approach has the advantage of not including the potentially biasing effect of including the mean in the calculations . alternatively , all possible combinations ( rather than permutations ) might be used . two methodologies are proposed : inversion of an estimate of the characteristic function of the residuals and an e - m algorithm approach that seeks a residual distribution that maximizes a pseudo - likelihood for the differenced data . a key reference for the characteristic function methodology is zhang ( 1990 ). background material for the e - m algorithm is available in dempster , laird and rubin ( 1977 ) and mclachlan and krishnan ( 1997 ). the estimation of residual distributions became of interest to us in the analysis of array based gene expression intensity data . regardless of the technology used ( macroarrays , microarrays , or biochips ) or the labeling method ( radio - is topic , fluorescent , or multi - fluorescent ), the observed values reflect the total amount of hybridization ( joining ) of two complementary strands of dna to form a double - stranded molecule . the log - transformed observations ( radio - isotopic , fluorescent , fluorescent ratios ) can be labeled y gij where g denotes the experimental condition that the observed values correspond to ( for instance , drug versus control , different tissues , etc .). the index i indicates the genetic sequence tag used in the experiment and j indicates that the observation was the jth repeated measurement within the genetic sequence tag / condition . the model for the y gij is : where the ε gij are assumed independent and identically distributed . here the ε gij are measurement errors ; μ gi is the true intensity value for the gth condition and ith tag . primary interest is in μ 1i − μ 2i the difference in the intensity values , for a given genetic sequence tag , between two different conditions . a gene &# 39 ; s expression intensity reflects its activity at specific moments or circumstances according to the design of the study . a gene &# 39 ; s activity is of interest in its own right and also because it usually reflects the production of protein , which has corollaries for the function and regulation of cells , tissues , and organs in the body . differences in gene expression are of interest to the extent that they reflect differences across conditions of these biological processes . gene expression data have been characterized by large measurement error variation , large numbers of comparisons ( sequence tags ) and small numbers of measurements for each sequence tag . the number of comparisons can range between a few hundred and hundreds of thousands . the numbers of measurements for a given sequence tag and condition are often 2 or 3 . because the measurement error is non - negligible it is usually the case that confidence intervals for the differences μ 1i − μ 2i are desired . one approach is to make the common assumption that the residuals are normally distributed , in which case ( 1 − α )× 100 % confidence intervals would be provided by { overscore ( y )} 1i −{ overscore ( y )} 2i ± z α / 2 { square root }{ square root over (( σ 1 2 / m + σ 2 2 / m )} here σ g 2 is the measurement error variance for the gth condition . with known non - normal residual distributions different forms of confidence intervals would usually be considered but it would still be reasonable to consider intervals with center { overscore ( y )} 1i −{ overscore ( y )} 2i and half - width a constant multiple τ of { square root }{ square root over ( σ 1 2 / m + σ 2 2 / m )}. what value of τ to use depends upon the particular form of the residual distribution . for the normal distribution τ is z α / 2 , for the double exponential exponential distribution it would be − log ( α ). thus , for instance , to obtain a 95 % confidence interval τ = 1 . 96 would be used for a normal residual distribution and τ = 3 would be used for the double exponential . these very different values of τ indicate that the residual distribution for a given condition is important t the inferences of interest in the analysis of expression data . because of the similarities in the measurement process across comparisons and the large number of comparisons , it should be possible to obtain estimates of the residual distribution with low variability . because of the small number of measurements for each comparison , care has to be taken to avoid bias in estimation . one approach to estimation of the residual distribution is through the characteristic function for the y ij1 − y ij2 . since y ij1 − y ij2 = ε ij1 − ε ij2 this characteristic function is ƒ *( t ) ƒ *(− t ). the form of the characteristic function for the difference indicates several identifiability problems . if the residual distribution is not a symmetric distribution then the distribution of − ε ij is not the same as the distribution of ε ij . however , since the characteristic function of − ε ij is ƒ *(− t ), the characteristic function for the difference ε ij − ε ij2 is ƒ *( t ) ƒ *(− t ) whether the residual distribution is that of − ε ij or ε ij . thus skewness in the residual distribution will not be recoverable from the distribution of the difference of two errors . a common assumption for measurement error models is that the residual distribution is symmetric . recognizing that we cannot detect skewness we will make this assumption here . in this case the characteristic function of the difference becomes ƒ *( t ) 2 this creates an additional difficulty in that one cannot discern the sign of the residual characteristic function from the characteristic function of the difference . to adjust for this we make the additional assumption that ƒ *( t ) is everywhere non - negative . examples of residual distributions that satisfy the assumptions include the normal , double exponential and cauchy distributions . a direct estimate of the characteristic function for the differences is available as , for instance , f ^ c *  ( t ) = 1 nm  ( m - 1 )  ∑ i = 1 n  ∑ j 1 ≠ j 2  exp  [ it  ( y ij1 - y ij2 ) ] = 2 nm  ( m - 1 )  ∑ i = 1 n  ∑ j 1 & lt ; j 2  cos  [ t  ( y ij1 - y ij2 ) ] ( 3 ) the estimate { circumflex over ( ƒ )}* e ( t ) is unbiased but highly variable . following zhang ( 1990 ) it is valuable to consider a smoothed version of the characteristic function : { circumflex over ( ƒ )}* s ( t ; c )={ circumflex over ( ƒ )}* e ( t ) h *( t / c ) ( 4 ) where h * is a characteristic function in correspondence with density h . since h *( t )& lt ; 1 , { circumflex over ( ƒ )}* s ( t ; c ) is biased downwards . small values of c tend to give smoother characteristic function estimates . on the other hand as c →∞, { circumflex over ( ƒ )}* s ( t ; c )→{ circumflex over ( ƒ )}* e ( t ). since the characteristic function is assumed non - negative another reasonable estimate of ƒ *( t ) is f ^ z * = { f ^ e *  ( t ) if   t & lt ; z 0 otherwise ( 5 ) where z is the smallest t & gt ; 0 such that { circumflex over ( ƒ )}* e ( t )= 0 . given a characteristic function estimate { circumflex over ( ƒ )}* d ( t ) for the difference ε ij1 − ε ij2 , an estimate of the residual characteristic function is { square root }{ square root over ([{ circumflex over ( ƒ )})}* e ( t )] + . a density estimate is obtained by the inversion formula f ^  ( x ) = 1 π  ∫ 0 ∞  [ f ^ z * ( t ) ] +  cos  ( - tx )   t ( 6 ) the cumulative distribution function estimate can be obtained by integration of the density estimate . the integration cannot be performed explicitly and must be done numerically . we will refer to a density or cumulative distribution function estimate based on { square root }{ square root over ([{ circumflex over ( ƒ )})}* d ( t )] + as a icf ( inverse characteristic function ) density or cumulative distribution function estimate . the estimates vary depending upon which estimate for the characteristic function of the differences is used . we refer to the estimate based on ( 5 ) as the unsmoothed icf estimate and an estimate based on ( 4 ) as a smoothed icf estimate . the use of characteristic functions for the estimation of a density of a random variable y when y + x is observed where x is a random variable with known density has been considered by carroll and hall ( 1988 ) and zhang ( 1990 ). here we wish to estimate the density of y − x where y and x both have the same but unknown density . the problems are similar and we will use a modification of the results of zhang ( 1990 ) to obtain upper bounds for the rate of convergence of the smoothed density estimates . theorem 1 let { circumflex over ( ƒ )}( x ) be the estimator of ƒ ( x ) given by ( 6 ) with { circumflex over ( ƒ )}* d ( t ) given by { circumflex over ( ƒ )}*( t ; c n ). h ( x )= h (− x ), ∫ x 2 h ( x ) dx & lt ;∞, ∫| xh ′( x )| dx & lt ;∞, h *( t )= 0 ,∀| t |& gt ; 1 ( 7 ) 1 / n ≤ c 0   f ^ *  ( c n )  2 / c n 3 , c 0 & lt ; ∞ ,  f *  ( c n )  = min  l  ≤ c n   f *  ( t )    then ( 8 ) lim n → ∞   e   f ^ - f  2 = 0 , ∀ f ∋  f  & lt ; ∞   and ( 9 ) sup f :  tf * ( i )  ≤ m 1   e   f ^ - f  2 ≤  c 0  c 3 + ( m 1  c 1 ) 2  / 2  π , ∀ n ≥ 1 . ( 10 ) an example of a characteristic function satisfying ( 7 ) is the function h *( t ) that is proportional to the density of the average of four uniform random variables but rescaled so that h *( 0 )= 1 . we used this characteristic function for the simulations and examples in later sections . zhang ( 1990 ) shows that the normal , cauchy , and double exponential distributions satisfy the assumptions of the theorem . the resulting rates of convergence are as follows : 1 . normal : ƒ ( x )= exp (− x 2 / 2 )/{ square root }{ square root over ( 2π )}. with c n ={ square root }{ square root over ( αlog ( n ))}, αε ( 0 , 1 ), e ∥{ circumflex over ( ƒ )}− ƒ ∥ 2 = o ([ log ( n )] − 1 ). 3 . double exponential : ƒ ( x )= exp (−| x |)/ 2 . with c n = αn 1 / 7 , α & gt ; 0 e ∥{ circumflex over ( ƒ )}− ƒ ∥ 2 = o ( n − 2 / 7 ). as an alternative to the estimation using characteristic functions we consider estimation based upon maximization of a pseudo - loglikelihood e   f ^ - f  2 = o  ( [ log  ( n ) ] - 2 ) .   pl  ( π )  ∑ i  ∑ j1 ≠ j2  log  [ f d  ( y ij1 - y ij2 , μ , π , h ) ] ( 11 ) where ƒ d ( y , μ , τ , h ) is calculated as the density of the difference of two random variables each having density f  ( x , μ , π , h ) = ∑ j = 1 t  π j  ϕ  ( [ x - μ j ] / h ) / h ( 12 ) here ψ ( t )= e − t 2 / 2 /{ square root }{ square root over ( 2π )} and the μ j are fixed , equally spaced points symmetrically placed about 0 . let { circumflex over ( π )} be the maximizer of pl ( π ). then ƒ ( x , μ , { circumflex over ( π )}, h ) is used as the estimate of the residual density . we will refer to an estimate that maximizes ( 11 ) as a pseudo - likelihood estimator . the form of the density given in ( 12 ) is flexible enough that almost any residual density should be identifiable with large enough t . this method of estimation avoids the numerical integration involved in the characteristic function approach but increases the computational cost by requiring that { circumflex over ( π )} be calculated as the solution of an optimization problem . indeed part of the reason for the form of the pseudo - loglikelihood is to simplify the estimation . maximization of pl ( π ) can be considered as a type of missing data problem and hence the e - m algorithm ( dempster , laird and rubin , 1977 ) can be used . the data points that we observe are the d ij1j2 := ε ij1 − ε ij2 . these can be thought of as incomplete versions of here i 1 and i 2 are artificial random variables that do not have an explicit role in the algorithm . for each ( i , j 1 , j 2 ), the rth value in 1 , . . . , t is assigned to i 1 with probability π τ independently of i 2 and ε j2 . the conditional distribution of ε ij1 given i 1 is taken as n ( μ i1 , h 2 ). the generation of i 2 is defined similarly . the complete data pseudo - loglikelihood is then 1 h 2  ∑ i , j 1 , j 2  log  [ π i 1  ϕ  ( ɛ ij 1 - μ j 1 h )  π i 2  ϕ  ( ɛ ij 1 - μ j 1 h ) ] ( 13 ) the details are omitted but the e and m steps of the e - m algorithm can be shown to be : given current estimates π j k + 1  α  ∑ ij 1  j 2  [ p ( k1 )  ( j | i , j 1 , j 2 ) + p ( k2 )  ( j | ij 1 , j 2 ) ] .  here ( 14 ) p ( k1 )  ( j | i , j 1 , j 2 ) )  α   π j ( k )  ∑ r  π r ( k )  φ  ( d ij 1  j 2 - ( μ j - μ r ) h ) / h   and ( 15 ) p ( k1 )  ( j | i , j 1 , j 2 ) )  α   π j ( k )  ∑ r  π r ( k )  φ  ( d ij 1  j 2 - ( μ j - μ r ) h ) / h ( 16 ) the constants of proportionality are determined by the constraints that the sums of the p ( k1 ) ( j | i , j 1 , j 2 ) and the p k1 ) ( j | i , j 1 , j 2 ) all equal 1 . a brief example of the results of estimation when the true density is known is given in fig1 . the data in this case were simulated from model ( 1 ) with n = 500 , m = 2 and a standard normal residual density . varying the smoothing parameters h in the case of the pseudo - likelihood estimate and c for the icf estimate give significantly different estimates . small values of h allow for more modes in the density estimate and consequently produce more variable estimates than larger values . similarly small c tend to be associated with smooth density estimates and large c with density estimates with larger numbers of modes . the smoothed icf density estimates tend to underestimate the value of the density near 0 . this is due to the smoothing factor h *( t / c )& lt ; 1 in the characteristic function estimates . smaller values of c are associated with greater bias in this region of the density . the pseudo - likelihood density estimates were better for these data . generally the pseudo - likelihood estimates can be expected to perform well when the residual distribution is close to normal since the normal density is used as the kernel in ( 12 ). the density estimates in fig1 are symmetric . generally this will always be the case : the icf estimates are symmetric since both the negative and positive differences y i1 − y i2 and y i2 − y i1 are included in the construction of ( 3 ) resulting in symmetric characteristic function estimates for ( 4 ) and ( 5 ). for the pseudo - likelihood estimates it can be shown that if the μ are chosen to be symmetric about 0 and the initial weight π j ( 0 ) for μ j is the same as the initial weight μ j in ( 14 ), then the final density estimate will be symmetric . the density estimates usually vary significantly with different smoothing parameters . the procedures for the selection of smoothing parameters discussed here were used for the expression data in the following sections relating to gene expression and simulations . the multiplication of the characteristic function estimate ( 3 ) by h *( t / c ) implies that the resultant characteristic function estimate will be 0 for | t |& gt ; c . consequently a reasonable upper bound for the appropriate smoothing parameter c is z , the smallest t & gt ; 0 such that { circumflex over ( ƒ )} e ( t )= 0 . in our experience ( see the simulations ) we have found that even with values of c as large as z there is significant bias in the distribution function estimates for the sample sizes of primary interest ( n ≦ 00 ). for this reason we also consider the unsmoothed icf density estimate . for the pseudo - likelihood estimates we determine h using the l ∞ distance between ( i ) the unbiased estimate ( 3 ) of the distribution for the difference between two residuals and ( ii ) the cumulative distribution of the difference of two random variables resulting from the residual density estimate ( 12 ) for the h under consideration . since the variance for a random variable from ( 12 ) is at least h 2 , a reasonable upper bound h 0 2 for the smoothing parameter is the sample variance of the differences . we select a smoothing parameter ĥ as the first h in { α k h 0 : 0 & lt ; α & lt ; 1 } such that the l ∞ distance for α k + 1 h 0 is greater than the l ∞ for α k . we illustrate the estimation of the residual distribution with the estimates obtained for gene expression data from brain tissue . the data are available at http :// idefix . upr420 . vjfcnrs . fr / hgi - bin / exgenx . sh ? clnindexx . html the expression values for brain tissue for n = 7483 genetic sequence tags were obtained as described in piétu et . al ., ( 1996 ). there were m = 2 repeated measurements for each sequence tag . plots of icf densities with various smoothing parameters ( c =∞ gives the unsmoothed estimate ) are given in fig2 . the density estimates are all very similar in this case . more important for calculating confidence intervals are the cumulative distributions which are given in fig3 . the 95 \% confidence intervals for the differences μ 1i − μ 2i described previously would be obtained as { overscore ( y )} 1i · −{ overscore ( y )} 2i · ± π { square root }{ square root over ( σ 1 2 / m + σ 2 2 / m )} where τ is the 0 . 975th quantile of the residual distribution . the estimates of τ for the icf estimate of the residual distribution with c = 5 , with no smoothing and pseudo - likelihood estimate are 2 . 37 , 2 . 27 and 2 . 21 respectively . thus one would construct a 95 \% confidence interval from the unsmoothed icf estimate as { overscore ( y )} 1i · −{ overscore ( y )} 2i · ± 2 . 27 { square root }{ square root over ( σ 1 2 / m + σ 2 2 / m )} { overscore ( y )} 1i · −{ overscore ( y )} 2i · ± 1 . 96 { square root }{ square root over ( σ 1 2 / m + σ 2 2 / m )} to further evaluate the methodologies several simulations were considered . for each set of simulations , samples from ( 1 ) were generated from a given residual distribution with n = 500 and m = 2 . the residual distributions considered were the normal , double exponential and cauchy distributions . the estimators considered were ( i ) the unsmoothed icf estimate resulting from ( 5 ) ( ii ) the smoothed icf estimate resulting from ( 4 ) with c taken as z , the smallest t & gt ; 0 such that { circumflex over ( ƒ )} e ( t )= 0 , and ( iii ) the pseudo - likelihood estimate with the smoothing parameter h chosen using the l ∞ criterion , discussed previously , with α = 0 . 8 . for ( i ) and ( ii ) 10000 simulated samples were drawn . for ( iii ) the first 1000 samples were used . a summary of the results of the simulations are given in tables 1 - 2 . the estimates of the probabilities from ( ii ) are biased downwards . in contrast the estimates of the probabilities and the quantiles from ( i ) and ( iii ) are quite reasonable for these samples sizes . the methodologies discussed in this article provide a means of estimating the residual distribution for models of the form ( 1 ). such models arise in data settings , such as the analysis of gene expression data , where there are a large number of comparisons or mean estimations with a similar measurement error process . the purposes of obtaining density estimates may vary . one could use them directly to adjust confidence intervals or to check a parametric residual distribution assumption . theorem 1 indicates that the icf estimates provide for consistent residual distribution estimation . while the upper bounds on the rates of convergence given above suggest that a large number of observations are required for consistent estimation of the density function , the simulation results indicate that reasonable estimates of the cumulative distribution probability estimates can be obtained with n ≧ 500 , which is usually the situation for gene expression data . the simulation results further favor less smoothing than one might expect . the pseudo - likelihood density estimates give reasonable density estimates as well . in contrast to the characteristic function based estimates however , more computational power is required to obtain them . it should be appreciated that the outcome of the process of the invention can be applied to the original data set or array or it may be applied to a new one . moreover , the process may be applied in three different ways : 1 . it can be used to determine the reliability of differences across two different samples ( obtained , say , from two different tissues ), i . e . different outcomes of a physical measurement . this can be done with the original data set or array on which the process was applied . since the original data set has repeated measurements , the process would typically be applied to the mean of the repeated measurements . it can also be applied to a new data set . the new data set may have only one measurement . or in the case of repeated measurements in the new data set , the outcome of the original data set can be applied to the mean of the measurements or of course the process may be repeated with the new data set . 2 . it can also be used to determine if a measured value deviates from all of the other measured values in the distribution . this is not the same as point 1 . here the comparison is not between two measured values but rather between one measured value and all of the others in a distribution . the idea here is that the measured values &# 39 ; “ place ” in the distribution is assessed relative to a threshold established by the random error estimation process . if the measured value exceeds the threshold , it is then said to represent a different physical measurement relative to the other values in the distribution . for example , most genes in an array may not be expressed above the background noise of the system . these genes would form the major portion of the distribution . other genes may lie outside of this distribution as indicated by their values exceeding a threshold determined by the random error estimation . these genes would be judged to represent a different physical process . 3 . the process may also be used to establish “ outlier ” values . in the preceding description , they are also described as “ an extreme value in a distribution of values .” outlier data often result from uncorrectable measurement errors and are typically deleted from further statistical analysis .” point 2 , above , also refers to detecting an extreme value but in that case the extreme value is based on the intensity of the measurement . that is not an outlier as intended here . here , outlier refers to an extreme residual value . an extreme residual value often reflects an uncorrectable measurement error . although preferred forms of the invention have been disclosed for illustrative purposes , those skilled in the art will appreciate that many additions , modifications and substitutions are possible without departing from the scope and spirit of the invention as defined by the accompanying claims . [ 0080 ] table 2 the estimated mean pth quantile with n = 500 , m = 2 based on simulation . method ( i ) is the unsmoothed characteristic function based estimate ( iii ) the pseudo - likelihood estimate . estimated standard deviations are given in parentheses . p residual distribution method 0 . 75 0 . 9 0 . 95 0 . 975 normal actual 0 . 67 1 . 28 1 . 64 1 . 96 ( i ) 0 . 66 ( 0 . 07 ) 1 . 32 ( 0 . 08 ) 1 . 68 ( 0 . 10 ) 1 . 97 ( 0 . 15 ) ( iii ) 0 . 66 ( 0 . 05 ) 1 . 28 ( 0 . 05 ) 1 . 66 ( 0 . 07 ) 1 . 99 ( 0 . 09 ) double exponential actual 0 . 69 1 . 61 2 . 30 3 . 00 ( i ) 0 . 7 ( 0 . 08 ) 1 . 6 ( 0 . 12 ) 2 . 29 ( 0 . 2 ) 3 . 01 ( 0 . 32 ) ( iii ) 0 . 73 ( 0 . 08 ) 1 . 58 ( 0 . 11 ) 2 . 29 ( 0 . 18 ) 3 . 2 ( 0 . 28 ) cauchy actual 1 . 00 3 . 08 6 . 31 12 . 71 ( i ) 1 . 04 ( 0 . 11 ) 3 . 09 ( 0 . 39 ) 6 . 28 ( 0 . 83 ) 12 . 4 ( 2 . 01 ) ( iii ) 1 . 44 ( 1 . 18 ) 3 . 7 ( 2 . 02 ) 7 . 3 ( 2 . 31 ) 14 . 74 ( 3 . 7 ) carrol , r . j . and hall , p . 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( 1990 ). “ fourier methods for estimating mixing densities and distributions ”, annals of statistics }, 18 , 806 - 831 . the disclosures of the preceding references are incorporated herein in their entirty .