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Modeling gene expression measurement error: a quasilikelihood approach
BMC Bioinformatics volume 4, Article number: 10 (2003)
Abstract
Background
Using suitable error models for gene expression measurements is essential in the statistical analysis of microarray data. However, the true probabilistic model underlying gene expression intensity readings is generally not known. Instead, in currently used approaches some simple parametric model is assumed (usually a transformed normal distribution) or the empirical distribution is estimated. However, both these strategies may not be optimal for gene expression data, as the nonparametric approach ignores known structural information whereas the fully parametric models run the risk of misspecification. A further related problem is the choice of a suitable scale for the model (e.g. observed vs. logscale).
Results
Here a simple semiparametric model for gene expression measurement error is presented. In this approach inference is based an approximate likelihood function (the extended quasilikelihood). Only partial knowledge about the unknown true distribution is required to construct this function. In case of gene expression this information is available in the form of the postulated (e.g. quadratic) variance structure of the data.
As the quasilikelihood behaves (almost) like a proper likelihood, it allows for the estimation of calibration and variance parameters, and it is also straightforward to obtain corresponding approximate confidence intervals. Unlike most other frameworks, it also allows analysis on any preferred scale, i.e. both on the original linear scale as well as on a transformed scale. It can also be employed in regression approaches to model systematic (e.g. array or dye) effects.
Conclusions
The quasilikelihood framework provides a simple and versatile approach to analyze gene expression data that does not make any strong distributional assumptions about the underlying error model. For several simulated as well as real data sets it provides a better fit to the data than competing models. In an example it also improved the power of tests to identify differential expression.
Background
An analysis of gene expression data typically includes the application of some multivariate statistical techniques such as clustering, classification, PCA etc. These highlevel procedures all require the assumption of a lowlevel error model for the data. In practice, this model is often only specified implicitly rather than explicitely. Nevertheless, its choice has a great impact on subsequent statistical considerations.
In the case of array data, the error model characterizes the variability of gene expression intensity measurements [1]. It is essential, e.g., to get a measure of precision of an estimated expression level, to statistically evaluate the difference between treatment and control samples, to calibrate and normalize data sets using regression techniques, to increase prediction accuracy in classification, and to assess confidence in highlevel analysis (e.g. clustering). Some of these applications may be robust against a misspecified lowlevel model but most will not. Thus, a careful choice of a suitable error model is warranted.
The observed intensity I_{ P }at a microarray probe may be decomposed into
I_{ P }= I_{ T }+ I_{ S }+ error, (1)
where I_{ T }denotes the true signal (foreground, due to the transcript) and I_{ S }is the stray signal (background, not due to the transcript) [2]. On this original scale I_{ T }is approximately linearly proportional to the true transcript concentration [2–4]. Distributional assumptions for the error term in Equation 1 used in the literature include

a normal foreground and background (e.g. [5]),

a Gamma foreground with unspecified background [6],

a lognormal foreground and normal background, [1]

a lognormal foreground and background [7], and
At present, as there currently is no generally accepted mechanistic or empirical model for gene expression measurements, there is no agreement which of these suggested error models comes closest to the truth (except perhaps that a normal model on the observed scale can be ruled out). Moreover, all of the models are fairly difficult to distinguish statistically for the small sample size present in microarray data.
Because of this difficulty to choose among simple parametric error models for the observed probe intensities two other alternative ways to describe the error term in Equation 1 have been explored in the literature. First, it sometimes is possible to obtain a fully nonparametric estimate of the underlying error distribution (e.g. [10]). However, the drawback of completely empirical error models is that they generally require quite a lot of data, i.e. many measurements per gene and condition, and more than are usually available. In addition, nonparametric approaches are prone to overfitting and ignore known prior structural information on the data.
A second widely pursued alternative is to try to find a transformation of the data to a different scale where the error term follows a normal distribution and where the variance is constant and intensity independent. For gene expression data, this can often be achieved, at least approximately, using the logtransform or some other related function [8, 9, 11–13]. Thus, in this perspective the problem of finding a suitable error model is equivalent to the problem of choosing an appropriate transformation. Note, however, that an ideal scale combines ease of interpretation, constancy of variance variance, normal errors, and additivity of systematic effects. Unfortunately, these properties cannot in general be achieved simultaneously [14]. In particular, if a nonlinear transformation such as the logfunction is applied to the data, the expectation of the transformed intensity is not anymore a linear measure of the transcript concentration.
Approximate Error Models
In this paper, the use of approximate semiparametric error models, rather than parametric or or nonparametric models, is advocated for gene expression data. In particular, the quasilikelihood framework is considered that allows statistical analysis even when the knowledge of the underlying error distribution is incomplete. Applied to gene expression analysis, this approach allows to model the data while at the same time avoiding strong assumptions about the underlying distribution and the optimal scale.
In the next sections the utility of these approximate error models for gene expression data is explored. First, the general quasilikelihood theory is introduced. Subsequently, a suitable quasilikelihood function for gene expression data is derived. Then simulated and real data are analyzed. Finally, some conclusions concerning lowlevel models and transformations for gene expression data are drawn.
Results and Discussion
QuasiLikelihood Framework
Quasilikelihood (QL) is a framework for statistical modeling that employs an approximate likelihood function rather using than a fully specified likelihood. The advantage of this approach is that no probability structure has to be specified, as the estimating function is constructed from the first two moments only. This is a useful strategy for dealing with nonnormal multivariate data (e.g. [15], chap. 14). Note that microarray data are nonnormal and multivariate.
The original quasilikelihood idea goes back to Wedderburn [16] who employed it in a regression setting. He introduced
as the quasiloglikelihood function for independent observations y_{ i }with expectations E(y_{ i }) = μ_{ i }and variances var(y_{ i }) = V(μ_{ i }) where V is some known function. In a regression problem μ_{ i }usually depends on a linear predictor xB via a link function g by μ_{ i }= g^{1} (xB). Wedderburn showed that with respect to μ_{ i }and the regression coefficients B the function Q(μ_{ i };y_{ i }) has properties similar to those of the loglikelihood [16].
However, the original quasilikelihood as in Equation 2 cannot be used to infer free parameters θ contained in a variance function V_{θ}(u). To fix this, Nelder and Pregibon [17] suggested to use the extended quasiloglikelihood function
Mean parameters that maximize the original QL function also maximize Q^{+}(μ_{ i },θ;y_{ i }) but variance parameters θ, as well as parameters in the link function g, can now also be estimated by maximizing the extended version.
QL and extended QL estimators have desirable finite sample and asymptotic statistical properties [18–21]. They are closely related to saddlepoint approximations from exponential families [22, 23]. As a result, maximum quasilikelihood estimates coincide frequently with inferences from an exact likelihood. For example, for V(μ) = σ^{2} the extended quasiloglikelihood Q^{+}(μ_{ i },σ^{2};y_{ i }) is exactly the normal loglikelihood (see Table 1).
Gene Expression Variance Structure and QuasiLikelihood Function
A model in the quasilikelihood framework requires only knowledge of the relationship between the mean and the variance, i.e. the function V in Equation 3.
Despite the uncertainty with regard to the exact form of the error distribution for gene expression measurements there is some agreement about the variance structure V. In the twocomponent model for the measured intensity (see Equation 1) let the expectations be E(I_{ P }) = E(I_{ T }) + E(I_{ S }) or μ = E(I_{ T }) + β. If the variance of the stray signal is assumed to be constant (var(I_{ S }) = ρ^{2}) and the true signal assumed to exhibit a constant coefficient of variation σ (so that var(I_{ T }) = E(I_{ T })^{2}σ^{2}), then the total overall variance structure is
var(I_{ P }) = (μ  β)^{2}σ^{2} + ρ^{2} := V(μ; β, σ, ρ). (4)
Such a quadratic variancemean relationship is observed in a lot of microarray data (e.g. [1, 8, 9, 12, 13]) and references therein), and therefore also assumed in the following. However, any other appropriate variance function could be used equally well in the quasilikelihood framework.
From the variance function Equation 4 the extended quasiloglikelihood function can be computed using Equation 3, resulting in
This function constitutes the approximate error model used in this paper. Note again that it is derived solely from the putative variance structure of gene expression data (Equation 4) with no further assumptions. Point estimates of the mean and variance parameters (μ, β, σ, ρ) are obtained by maximizing this function.
Ideally one would like to estimate one set of these parameters for each gene and condition separately. However, this is feasable only if there are a lot of replications, otherwise a parameter reduction is advised. Typically, the estimate of the coefficient of variation σ can be shared across all genes and conditions [24]. Similar parameter reduction may be applied to the estimates of ρ and β.
The quasilikelihood approach also allows the estimation of approximate confidence intervals for the estimated parameters (for example θ). One way is to employ the profile quasilikelihood Q^{+}(θ) = Q^{+}(_{ i }, θ; y_{ i }), where _{ i }is optimized for fixed θ, to construct the interval
{θ, 2Q^{+} ()  2Q^{+} (θ) ≤ d}.
The estimate
maximizes Q^{+}(θ) and the threshold d may be chosen as some percentage point of a χ^{2} distribution (for a oneparameter approximate 95% confidence interval d = 3.84). Alternatively, a variety of standard bootstrap procedures are applicable to construct confidence intervals for the quasilikelihood point estimates [17]. However, the bootstrap intervals tend to be more conservative (i.e. wider) than the above likelihoodbased interval.
Effect of Calibration
Prior to any analysis the raw microarray data generally need to be calibrated (or normalized). This also affects the error model. If linear locationscale transformation y' = a + by is assumed for each chip, see e.g. [25–27], then the transformed variance structure is
E(I'_{ P }) = E(a + bI_{ P }) = a + b μ = μ'
var(I'_{ P }) = (μ'  a  b β)^{2}σ^{2} + (b ρ)^{2}
Thus, for uncalibrated data the background parameter β is confounded with the shift and scale parameters a and b, while the background error ρ is confounded with scale parameter b. The coefficient of variation a is not affected by the transformation.
It may be desired to estimate calibration parameters a and b in addition to the error model itself. In this case, however, it is necessary to assume that β and ρ are shared parameters across all chips or channels. It is unclear, however, whether this is a realistic assumption for real data. Note, however, that this is the basis for estimating a and b in the transformationbased approach by Huber et al. [8].
Simulation Study
To explore the adequacy of the quasilikelihood approximate error model for gene expression data a simulation study was performed.
Data were generated according to three different schemes. First, as true error model a convolution of normal and lognormal distribution was assumed [1]. As a second model an asinhnormal (ANL) distribution was assumed [8, 9]. Note that in these two cases the approximate error model provided by quasilikelihood is misspecified as both distributions are not part of the exponential family. As third true error model a Gamma distribution was considered [6].
The simulated wholechip data consisted of 7000 genes, with the coefficient of variation set to σ = 0.25, the background parameters set to β = 25000 and ρ = 5000. The 7000 true expression levels μ_{ i } β were drawn randomly from a lognormal distribution (with logmean 8 and standard deviation 2). These values were chosen to match the molecular data analyzed in [1]. Data y from the convolution model and the Gamma model were generated directly on the observed scale. To generate data y from the asinhnormal distribution, data x were drawn from a normal distribution N(u, s^{2}) and subsequently transformed to the observed scale via y = asinh(a + bx). Note that this is possible because there exists a onetoone mapping of the parameters on the normal scale (u, s^{2}, a, b) and those of the transformed scale (μ, σ, β, ρ), see Table 2. For each variant 4, 10 and 20 replicates per gene were drawn. Figure 1 shows the observed meanvariance relationship of an example with 10 replicates per gene.
Subsequently, the extended quasilikelihood (EQL) model (Equation 5) and the ANL model were fitted to the simulated data by maximizing the extended quasilikelihood function and the likelihood function derived from the asinhnormal distribution, respectively. The results are shown in Table 3 for the convolution model as the underlying true distribution, in Table 4 for the asinhnormal distribution, and in Table 5 for the Gamma distribution as the true error model.
The colored lines in Figure 1 indicate the true variancemean relationships and the maximumlikelihood and maximum quasilikelihood estimates, respectively.
If the true error model was the asinhnormal distribution (Table 4) or the similar convolution model (Table 3) then – as expected – the fit of the correctly specified model (ANL) was always better than that of the EQL model, though in terms of the loglikelihood only by a small amount. Parameter estimation using this correct probability model was more efficient than with EQL. In both cases parameters μ_{ i }were well estimated. Variance parameters β, σ and ρ were underestimated by the approximate error model. The sample size (4, 10, 20 replicates per gene) did not greatly impact EQL estimates.
On the other hand, if the true model followed a Gamma distribution (Table 5) the EQL model fitted the data consistently better than the ANL model. However, the difference in loglikelihood between the two candidate models was again comparatively small. As the Gamma model does not contain background signal the true values for β and ρ are zero. In this case the parameter estimates based on the ANL model are highly biased upwards, whereas estimates from the EQL approach were almost unbiased.
Thus, while the EQL model was based only on the postulated variance structure with no additional information on higher distributional moments, it nevertheless provided a reasonable fit to the ANL and convolution generated data and a very good fit to the Gammagenerated data.
Leukemia Data
Next, the EQL and ANL model was fitted to the Leukemia data from Golub et al. [28]. After preprocessing and filtering as in [28] 3051 genes and 38 samples remained. The estimation results based on the EQL and ANL error models are shown in Table 6. The data available from the Golub et al. website were already calibrated and backgroundcorrected, hence the parameter β was set close to zero both for the approximate error model EQL as well as the ANL model. For this data set, the fit of the approximate error model EQL is better than of the parametric ANL model, i.e. the EQL models achieves a much higher (quasi) loglikelihood. The estimated EQL and ANL parameter values are similar, with EQL estimates being slightly smaller than the corresponding ANL parameter values.
The Leukemia data set contains subsets of samples from two tumor classes, AML and ALL [28]. A statistical test can then be employed to reveal which genes are differentially expressed between the two groups. One approach that explicitely takes account of the underlying error model is based on the likelihood ratio test [29]. For normal errors this approach is (asymptotically) equivalent to the standard ttest, but unlike the ttest it can also be applied for any other assumed error model. To determine pvalues the test distribution for the likelihood ratio was assumed to be χ^{2} (a more accurate distribution may be obtained, e.g., using a bootstrap approach). Figure 2 shows the number of differentially expressed genes given a nominal α value (type I error) for the individual pairwise tests, estimated using the approximate error model (open triangles) and the ANL model (filled triangles). In the data set there are a large number of differentially expressed genes. As the type I error is controlled by α the percentage of statistically significant differentially expressed genes shows the power of the test in dependence of the chosen error model. The approximate error model EQL fits the data better than the parametric model ANL, and Figure 2 shows that this leads to a subsequent improved power to select differentially expressed genes.
Conclusions
Error models play an important though often implicit part in the analysis of gene expression data. There are a lot of possibilities to model the error of intensity measurements, and these are mirrored by the wide choice of parametric models and corresponding data transformations. As nonparametric approaches are not always applicable due to lack of sufficient replicate data, in this paper the use of approximate error models based on quasilikelihood is suggested as a further alternative.
Quasilikelihood is a versatile and simple framework for semiparametric modeling that requires only a partial specification of the underlying probability structure. This is ideal for microarray data where there is agreement on the variance versus mean relationship of the measured intensities but no suitable mechanistic model available to guide the search for the true underlying error distribution. A further advantage of quasilikelihood is also that it is scaleneutral, i.e. it can be used to analyze data on any preferred scale. Thus, using an approximate error model within the quasilikelihood framework allows to analyze data on the original observed scale, where the expected intensity corresponds directly to transcript concentration, without the need for a complicated transformation. Third, quasilikelihood can be viewed as a compromise between traditional parametric and nonparametric approaches.
In this paper both for simulated and molecular data the approximate error model fitted the data as good or better than a competing parametric model derived from an transformationbased approach. Moreover, in a modelbased test for differential expression the approximate error model had more power on the same level of type I error than the parametric model. It is expected that the favorable properties of quasilikelihood also hold for other data sets.
Employing an approximate error model in a statistical analysis comprises a tradeoff between the a priori available information on the true model and the efficacy of an inference from the data. If the true underlying model is fully known, using an approximate model such as quasilikelihood inevitably entails loss of efficiency and leads to bias in parameter estimation. However, if a suitable error model is not readily available and if multiple unknown sources of error have to be taken into account, then the quasilikelihood approach is advantageous as it provides an optimal estimating equation under very general conditions, and may thus outperform other adhoc parametric models.
While in this paper quasilikelihood was used for modeling and inference purposes, it is generally applicable also in a regression setting [14]. This points towards further possible applications of the quasilikelihood framework in gene expression analysis. For instance, normalization procedures may benefit from using an approximate error model (e.g. [30]). Systematic effects in the data such as those due to different arrays, dyes, etc. can also be inferred by regression and ANOVA techniques [27, 31] and hence are amenable to analysis by quasilikelihood, too. In a related line, the affinity of probes on a chips may thus also be estimated by using quasilikelihood, rather than assuming a normal error as in [5]. Finally, highlevel analysis such as classification can incorporate quasilikelihood models.
In summary, approximate error models such as provided by the quasilikelihood framework enable the analysis of gene expression data despite our ignorance of the true underlying lowlevel processes generating the observed data.
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Acknowledgments
This work was supported by an Emmy Noether research grant (STR 624/12) from the Deutsche Forschungsgemeinschaft. I thank Anja von Heydebreck and Wolfgang Huber for discussion.
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Strimmer, K. Modeling gene expression measurement error: a quasilikelihood approach. BMC Bioinformatics 4, 10 (2003). https://doi.org/10.1186/14712105410
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DOI: https://doi.org/10.1186/14712105410