Abstract
Background
In testing for differential gene expression involving multiple serial analysis of gene expression (SAGE) libraries, it is critical to account for both between and within library variation. Several methods have been proposed, including the t test, t_{w }test, and an overdispersed logistic regression approach. The merits of these tests, however, have not been fully evaluated. Questions still remain on whether further improvements can be made.
Results
In this article, we introduce an overdispersed loglinear model approach to analyzing SAGE; we evaluate and compare its performance with three other tests: the twosample t test, t_{w }test and another based on overdispersed logistic linear regression. Analysis of simulated and real datasets show that both the loglinear and logistic overdispersion methods generally perform better than the t and t_{w }tests; the loglinear method is further found to have better performance than the logistic method, showing equal or higher statistical power over a range of parameter values and with different data distributions.
Conclusion
Overdispersed loglinear models provide an attractive and reliable framework for analyzing SAGE experiments involving multiple libraries. For convenience, the implementation of this method is available through a userfriendly webinterface available at http://www.cbcb.duke.edu/sage webcite.
Background
Serial analysis of gene expression (SAGE) is used to measure relative abundances of messenger RNAs (mRNAs) for a large number of genes [1,2]. Briefly, mRNAs are extracted from biological samples and reversetranscribed to cDNAs. The doublestranded cDNAs are then digested by a 4cutter restriction enzyme (anchoring enzymes, usually NlaIII). After digestion, another restriction enzyme (tagging enzymes) is used to release the downstream DNA sequences at 3' of most of the anchoring enzyme restriction sites. The released sequences, usually 10–11 base pairs (bp) long, are called SAGE tags. The tags derived from many different species of mRNAs can be concatenated, cloned and sequenced. In a typical SAGE experiment, a large number of tags (often ranging from 30,000 to 100,000) are collected from each sample, with each tag representing, ideally, one gene; the tag count indicates the transcription level of the gene represented by that specific tag. A natural question of interest is whether a given tag is differentially expressed. Over the past few years, SAGE has been extensively used for expression analysis of cancer samples for identifying diagnostic or therapeutic targets [3,4].
Most SAGE studies focus on comparing expression levels between two samples. For such twolibrary comparisons, several statistical methods have been proposed, such as the simulation method of Zhang et al. [2], the Bayesian approaches [57], and the normal approximation based ztest [8] (which is equivalent to the chisquare test [9]). A comparative review by Ruijter et al. [10] has shown that all these methods perform equally well.
The comparison between two SAGE libraries can identify biologically interesting tags (or genes). However, in many cases it is essential to conduct experiments with replicates in order to account for normal background biological variation. For experiments involving multiple SAGE libraries, betweenlibrary variation beyond the binomial sampling variation is introduced. Such betweenlibrary variation can be due to additional known factors involved in the experimental design, as well as to unknown genetic or environmental variation between observations. Indeed, major differences in gene expression exist among SAGE libraries prepared from the same tissues of different individuals [11]. Statistical methods are needed for analyzing SAGE experiments involving multiple libraries. In the case of twogroup comparisons (e.g. comparisons between a normal group and a cancer group), methods such as pooling the libraries in each group and transforming to twolibrary comparisons (for example, using the chisquare test), or the twosample ttest on proportions have been proposed and discussed [1214]. The pooling approach is often problematic since it ignores gene expression variation among libraries within the same treatment group, which leads to biased estimates for the variance. The twosample ttest on proportions, however, can be problematic as well; proportions estimated from libraries with smaller sizes are known to be more variable than those from larger libraries.
For twogroup comparisons, Baggerly et al. introduced a test statistic, t_{w}, based on a hierarchical betabinomial model to account for both betweenlibrary and withinlibrary variation [13]. The t_{w }test statistic is assumed to have an approximate tdistribution and like the ttest, the t_{w}test is only good for twogroup comparisons. For SAGE experiments with a more general design (e.g. involving 2 or more factors), an approach based on overdispersed logistic regression has been described [15]. Overdispersed models aim to allow for the possibility of overdispersion in the tag counts, i.e., cases where the variance in tag counts exceeds what is expected for binomial or Poisson sampling alone. Besides its flexibility in modeling multiple factors and/or continuous covariates, logistic regression compares group proportions on a logit scale (log of odds) rather than a raw scale as in the t and t_{w }tests. Comparing groups in logistic regression (and any generalized linear model) is equivalent to testing the hypotheses of whether the coefficients β = 0. Baggerly et al. [15] showed evidence suggesting that "the logit scale may be more appropriate" than the original proportion scale. One drawback with overdispersed logistic regression, however, is that it can break down for cases where all the tag counts in any of groups are very small. In such cases, the deviance test rather than the ttest (on the hypothesis that the coefficient β is zero) has been proposed [15]. Besides that a systematic evaluation of the deviance test is needed, a potential drawback with the deviance test is that it may require multiple rounds of model fitting if a model contains multiple factors or covariates. Furthermore, questions still remain on exactly when the deviance test should be used in preference to the ttest.
In this report we introduce an overdispersed loglinear model approach to analyzing SAGE which is closely related to overdispersed logistic regression but has a different meanvariance relationship assumption. We compare its performance in identifying differential expression with that of three other methods, including the ttest, t_{w }test and overdispersed logistic regression. Analysis of simulated and real datasets show that both the loglinear and logistic overdispersion methods generally perform better than the t and t_{w }tests. Based on simulated data, the loglinear method is found to have better performance than the logistic method, showing equal or higher statistical power over a range of parameter values and with different data distributions. The overdispersed loglinear method also appears to have better performance on the real SAGE data which we analyze; a number of cases are seen where a tag is identified by the loglinear approach and appears to be clearly differentially expressed, but which would not have been identified as significant using the logistic regression method. Overdispersed loglinear models also offer the same flexibility as logistic regression, allowing for modeling multiple factors and/or covariates. We conclude that the overdispersed loglinear models provide an attractive and reliable framework for analyzing SAGE experiments involving multiple libraries.
Results
Overdispersed loglinear models: a case study
Overdispersed loglinear models (see details in Methods) are very similar to overdispersed logistic models, but there are two major differences. First, overdispersed loglinear models work with logarithms of proportions (the log link) with logarithms of sample sizes m_{i }as offsets. In contrast, overdisersped logistic models use the log of the odds (the logit link). Second, the assumption of an overdispersed loglinear model leads to derived weights used by iteratively reweighted least squares (IRLS) that depend on the means of the tag counts (i.e. the weights depend on both library sizes and tag proportions). The weights in overdispersed logistic regression, in contrast, are a function of library sizes only (see Methods).
Baggerly et al. [15] illustrated that the overdispersed logistic model can break down in cases where all proportions in one group are 0. Here we show that such a breakdown can also occur when the proportions in one group are small. Table 1 lists the pvalues obtained from both the deviance and t tests. Note that we are testing the hypothesis that β = 0. Artificially increasing the tag counts in group 1 so that they approach the level seen in group 2 (which are held fixed), the deviance test in logistic regression and both tests (deviance and t) in the loglinear model show the expected trend of an increasing pvalue (Table 1, columns 5, 6, and 7). In contrast, the pvalues from the ttest in logistic regression actually decrease first and then increase (Table 1, column 4). This discrepancy between results from the t and deviance tests in the logistic model (a discrepancy not seen in the loglinear case) suggests that logistic regression can be problematic when the tag counts of all samples in one group are small.
Table 1. Comparisons of t and deviance tests in overdispersed logistic regression and loglinear models and a test based on a Bayesian model
Simulation study
To systematically evaluate the performance of the various tests in the case of twogroup comparisons, we performed a simulation study. The tests compared here are the t, t_{w}, logitt and logt. For t and t_{w}, the test is whether _{p}A = _{p}B, where _{p}A and _{p}B are the mean proportion in groups A and B respectively. The logitt and logt are t tests on the hypothesis of whether β = 0 in the overdispersed logistic regression and loglinear models respectively. We do not attempt to replace the ttest with the deviance test in the overdispersed logistic regression model since this requires making a possibly subjective decision on when to use one test in preference to the other.
We generated tag counts under three different distributions, choosing different tag proportions and amounts of overdispersion (Table 2). Data generated from the betabinomial and negative binomial distributions meet the assumptions (i.e. have the meanvariance relationship structure) of the overdispersed logistic regression and loglinear models approaches, respectively. The negative binomial distribution is equivalent to the gammaPoisson hierarchical model and is considered a robust alternative to the Poisson distribution [16,17]. It should be noted that the t_{w}test is also derived under the assumption that the data is generated from a betabinomial distribution [13]. The range of overdispersion parameter values was chosen based on model fits from a real dataset (see section below); we used the 25, 50, and 75 percentile values of the estimated overdispersion from these fits. Note that the overdispersion parameter φ in the logistic model is not directly related to the φ in the loglinear model; φ values from the two models should not be compared. Given an overdispersion value φ and a group mean proportion p, the α and β values for the betabinomial distribution are derived as α = p(1/φ  1), and β = (1  p)(1/φ  1). The size parameter in the negative binomial distribution is easily derived as 1/φ. We used 5 samples (libraries) for each group, and determined the sizes of each of 10 libraries by randomly sampling from a uniform distribution over the interval between 30,000 and 90,000. This yielded library sizes of 66148, 67094, 53338, 80124, 64984, 70452, 74052, 60086, 52966 and 45377; these values were not changed over the course of the simulations. Results (not shown) from a separate run using a different set of library sizes were found to be in agreement with those shown here. A total of 5,000 sets of tag counts were generated for each combination of parameter values. The sensitivity and specificity of each of the tests were then evaluated and compared through receiver operating characteristic (ROC) curves [18].
Table 2. A list of parameter values used in the simulations
The ROC curves (one for each of the four tests) shown in Figure 1 were obtained using data generated from the betabinomial distribution (with overdispersion values φ shown on the top of the figure). Given the same false positive rate (xaxis), the overdispersion models (logistic and loglinear) clearly show improved statistical power (yaxis) compared to the twosample t and t_{w }tests. In contrast, when the four tests are applied to data generated from the negative binomial distribution, the overdispersed loglinear model clearly outperforms the other three tests (Figure 2). Again, the twosample t and t_{w }tests do not perform well in general. The figures generated using other parameter values are available [see Additional files 1 and 2]. These results suggest that for SAGE data, statistics methods based on raw proportions (as in the t and t_{w }tests) show less power than the logistic or loglinear model approaches. The overdispersed loglinear model not only shows the best performance in cases where the data are generated in a manner consistent with its assumptions (i.e. from the negative binomial distribution), but also has competitive performance when the data come from a different distribution (here the betabinomial). This suggests that the overdispersed loglinear model approach is more robust.
Figure 1. Comparisons based on simulated data from the betabinomial distribution. This figure shows the receiver operating characteristic curves (ROC) of the four tests applied to datasets generated from the betabinomial distribution with various magnitudes of overdispersion (φ) (shown on the top of each graph). For a specific φ, 10,000 observations (tags) are simulated; 5,000 are generated under the assumption that p_{A }= p_{B }and the remaining from p_{B }= 2 p_{A}, where p_{A }and p_{B }are the mean proportions of the two groups and p_{A }= 0.0002 (i.e. 10 out of 50,000). For figures generated under other conditions, see 1.
Figure 2. Comparisons based on simulated data from the negative binomial distribution. The ROC curves of the four tests are based on datasets generated from the negative binomial distribution with various magnitudes of overdispersion (φ). The data are simulated by the same strategy as used in Figure 1, except that p_{B }= 4p_{A}. Note that the overdispersion parameter here is not directly comparable with that in Figure 1 (the parameter φ for the negative binomial is not directly related to that for the betabinomial). For figures generated under other conditions, see 2.
Additional File 1. This gzipped tar file contains figures showing the receiver operating characteristic curves (ROC) for the four tests applied to datasets generated from the betabinomial distribution with various magnitudes of overdispersion(φ) and mean proportions. For example, the file 2_8e06_0.0002.png shows the ROC curves when p_{B }= 2p_{A}, φ = 8e06 and p_{A }= 0.0002.
Format: GZ Size: 119KB Download file
Additional File 2. Similar to the file above, this file contains figures of ROC curves but with data generated from the negative binomial distribution.
Format: GZ Size: 142KB Download file
A pancreatic cancer dataset
We further compared the four tests (ttest, t_{w}test, logitt, and logt) using an experimental SAGE data set obtained from the publicly available SAGE Genie website [19]. To identify genes differentially expressed between the pancreatic cancer cells and normal ductal epithelium, Ryu et al. [12] compared the gene expression levels of five pancreatic cancer cell lines and two samples of normal pancreatic ductal epithelial cells. The library sizes and numbers of unique tags for the SAGE libraries are shown in Table 3. Note that the numbers in the table are slightly different from those described in the original paper due to the different SAGE tag processing procedures [20]. In this analysis, we ignore tags with total counts less than 3.
Table 3. Library information on 5 cancer and 2 normal pancreas SAGE libraries
We first compare the four tests by examining the overlap between the top ranking genes (top 50 and 100) identified by each test (Table 4). For the t and t_{w }tests, the genes are ranked by the absolute value of the t (or t_{w}) statistics instead of by pvalues (see Discussion section for details). As shown in Table 4, the results from the logitt and logt tests show the highest agreement (~80%); moderate agreement is observed between t_{w }and logitt or logt tests (~60%) and the least agreement is seen between the t and the other three tests (~40%). The top ranking genes identified by the ttest are often those with extremely small withingroup variances (data not shown). Overall, results from the ttest differ the most from the results of the other tests, while the most similar results are seen between the logitt and logt tests. This generally agrees with the trend seen in the simulations.
Table 4. Pairwise comparisons of the four tests
Of the top 100 genes (ranked by pvalue) obtained from the logitt and logt tests, 82 genes are in common leaving 18 genes from each test that are not within the top 100 identified by the other test. To further examine the discrepancy between the logitt and logt tests, we plotted pvalues obtained from both tests for these 36 remaining tags (Fig 3). It can be seen that, while tags identified by the logitt test are also given relatively small pvalues by the logt test (all less than 0.05), those identified by the logt test show a wide range of pvalues according to the logitt test. Table 5 lists tags which are ranked among the top 100 by the logt test but which have pvalues greater than 0.05 by the logitt test; 4 of these were also identified by Ryu et al. [12]. Our analysis indicates that the logt test is relatively robust in that it not only gives reasonably small pvalues to genes identified as significant by the logitt test, but also identifies genes which would never have been considered significant by the logitt test.
Figure 3. Comparing pvalues from the logitt test and those from the logt test. Of the top 100 tags (ranked according to pvalues) identified by the logitt test and by the logt test, 82 are common to both leaving 18 tags from each test that are not within the top 100 identified by the other. The pvalues from both tests for these 36 remaining tags are plotted here. The circles represent the 18 in the top 100 by the logitt test and the triangles those from the logt test. While all the tags identified by the logitt test also have reasonably low pvalues according to the logt test, the tags identified by the logt test show a much wider range of pvalues according to the logitt test.
Table 5. A set of genes identified as significantly differentially expressed (p < 0.05 and also among the list of top 100 genes) according to the logt test but not by the logitt test (p > 0.05)
Ryu et al. [12] identified 49 up and 37 downregulated genes in cancer with the twosample ttest and a set of rulebased methods. We compared their results with those from the logt test (choosing the same number of top genes). Of the total of 86 genes, only 18 are in common (with 9 in each down and upregulated gene group). The most significant gene that is upregulated in cancer on our list (but not in the original paper) is tag, "CTTCCAGCTA", which represents the annexin A2 gene. This gene has been reported to be upregulated in human pancreatic carcinoma cells and primary pancreatic cancers [21]. Another example is tag 'TTGGTGAAGG', which corresponds to the gene encoding thymosin, beta 4. This gene also has been shown to be "expressed at high levels both in tumor cell lines and in primary cultures of normal pancreas, but not in normal tissue" [22]. A list of the top 20 genes upregulated and the top 20 genes downregulated in cancer based on the logt test are listed in Table 6.
Table 6. A list of top 40 genes differentially expressed between pancreatic cancer and normal ductal epithelium
Discussion
In this report we introduced a loglinear model with overdispersion for testing differential gene expression in SAGE. This model is closely related to the overdispersed logistic model proposed by Baggerly et al. [15] but with a different meanvariance relationship assumption. The differences between two models can be seen clearly in the weight (used by IRLS) associated with each observation: assuming library sizes are reasonably close, the overdispersed loglinear model tends to assign higher weights to observations in the group with the smaller mean proportion; in contrast, approximately equal weights are assigned to all the observations in the overdispersed logistic model. Although for real SAGE data the true meanvariance relationship is unknown, it has been observed that "for the higher counts data, the betweenlibrary variability is the dominant part of the variation" [13]; this suggests that the magnitude of the overdispersion in the group with higher counts is probably larger than that in the group with low counts so that the assumptions of the overdispersed loglinear model is probably more appropriate for SAGE data.
We also compared the model fits of the overdispersed logistic and loglinear models. Due to the introduction of the overdispersion parameter, the deviance statistic is no longer a valid basis for model fit comparison. An alternative is to use the standardized Pearson residuals, which have an asymptotic standard normal distribution [23]. Williams [24] proposed the approach of plotting the standardized Pearson residuals against the predicted proportions; a problem with a model fit is indicated by a significant decrease in the variance of the standardized residuals as estimated proportions approach zero. Figure 4 shows the residual plots from the logistic and loglinear model fits for two tags (the tag counts are listed in Table 5). In the overdipersed logistic regression case (left panels of Figure 4), the variance of the standardized Pearson's residuals is seen to be much smaller in the normal group than in the cancer group. Such a difference is not evident in the overdispersed loglinear model fits (right panels of Figure 4). Although the sample size is very small in this example (only 2 in the normal group), the residual plots give further indication that loglinear models provide a better fit to SAGE data than logistic models.
Figure 4. Plot of standardized residuals against estimated proportions. Standardized Pearson's residuals (yaxis) plotted vs. the proportion estimates (xaxis) for the two groups. The standardized Pearson's residuals are asymptotically distributed as a standard normal. The model fits of two tags (among the list of genes in Table 5) are shown here; the left is from the fit using the overdispersed logistic model and the right from the overdispersed loglinear model. A lower variance of residuals in the group (normal) with lower mean proportion is an indication of poor model fit.
From the simulation study we have shown that, besides their limitation to twogroup comparison, both the t and t_{w}tests, in general, are not as powerful as tests which allow for the possibility of overdispersion. We mention one specific problem that can arise with the t and t_{w}tests if the number of samples in the data set is small. Note that the rank orders from the ttest and the t_{w }test in Table 4 are based on test statistics instead of pvalues. The rank orders based on pvalues can be different from those based on test statistics if the residual degrees of freedom differ among tests. Both the ttest and the t_{w}test use the Satterthwaite approximation [25] for the number of degrees of freedom since the variances are assumed to be different in the two groups. An example of how this can be problematic is given by tag "AGCTGTCCCC", which has tag counts 70, 82 in the two normal samples, and 4, 1, 1, 1, 0 in the five cancer cell line samples. The differential expression is highly significant based on the logitt (pvalue 0.0003) and logt (pvalue 0.0005) tests. In contrast, if the t_{w}test with the Satterthwaite approximation to the degrees of freedom is used, this tag is barely significant at the 5% level (pvalue 0.050). The reason is that, while the magnitude of the t_{w }statistic for this tag is actually extremely high (t_{w} = 12.01), the calculated degrees of freedom is only about 1 (which leads to low significance). The small value for the degrees of freedom arises here because the estimated variance in the cancer group is very small; the approximated degrees of freedom is then about equal to the sample size of the normal group minus 1 (here, 21 = 1). Cases like this occur frequently in this data set since the number of libraries (samples) in one group is very small. It is not uncommon to have small sample numbers with SAGE data.
The four methods compared in this study follow the frequentist approach of hypothesis testing, and can be broadly considered as examples of linear models. For twogroup comparisons, Vencio et al. [26] introduced a Bayesian approach to rank tags by the Bayes Error Rate. We compared their approach with the methods based on linear models by looking at differences in gene rankings determined using the pancreatic dataset. Considering the top 100 genes identified by the different tests, the two overdispersed models show the best agreement with the Bayesian method (~70% in common); 63 genes (of the top 100) are identified by all three tests. We also evaluated the Bayesian method using the artificial data in Table 1; as the tag counts in group 1 are increased, the evidence for differential expression decreases (i.e. the Bayes Error Rate goes up), which follows the expected trend. Furthermore, if we recognize tags with p < 0.05 or E<0.1 as being significantly differentially expressed [26], the results from the Bayesian approach are more consistent with those from the loglinear model than from the logistic models (see Table 1). Since the evidence measures used are conceptually very different, to perform a direct comparison between "Pvalue"based methods and the Bayesian approach is not straightforward. Our results, however, suggest that the Bayesian approach of Vencio is a competitive Bayesian alternative for analyzing SAGE data in the case of twogroup comparisons.
The current study has not considered the issue of multiple testing problems which is still under active research [27,28]. We note that one possible area for further improvement is to use information across genes (tags) with similar magnitude of dispersion to obtain potentially more robust and accurate overdispersion (and therefore, error) estimates. In all the methods compared here, everything is done one tag at a time, i.e., estimates of the amount of overdispersion are done for each tag individually and these can vary widely (see Figure 5). For expression data with continuous values, strategies on information sharing have been proposed [2931] and these strategies may be adapted for discrete data such as in SAGE.
Figure 5. The distribution of overdispersion estimates (). The estimates are from the overdispersed loglinear model fit to the pancreas data. Tags with the overdispersion estimate 0 are not shown in the figure.
Methods
Data
Suppose that there are a total of n SAGE libraries in an experiment. Let m_{i }be the size (total tag counts) of library i (i = 1..n) and r_{i }be the tag counts for a specific tag in that library.
Also, let x_{i }be the associated vector of explanatory variables and β the vector of coefficients. The comparison of two groups of SAGE libraries is a special case where there is only one explanatory variable associated with each observation (i.e. one factor with 2 levels).
The twosample ttest
The ttest proposed by Welch [25] was used to test whether the mean of the proportions in one group equals the mean of the other. The proportions are assumed to have unequal variances in the two groups and the degrees of freedom is calculated based on the Satterthwaite approximation as in the t_{w}test (see below).
The t_{w}test
Baggerly et al. [13] introduced a betabinomial sampling model to account for the extrabinomial variation for a simple design in which the comparison is between two groups of SAGE libraries. This is a special case of a linear model that contains one explanatory variable. Briefly, unobserved random variables P_{i }are introduced to account for the betweenlibrary variation. For a given group, P_{i }is assumed to have a beta distribution (α, β) with mean and variance E(P_{i}) = α/(α+β), and Var(P_{i}) = αβ / [(α+β)^{2 }(α+β+1)]. Notice that this is a special case of the form Var(P_{i}) = φ p_{i}(1  p_{i}) as in the overdispersed logistic model, where φ = 1/(α+β+1). Next, the group proportion is estimated by a weighted linear combination of individual proportions within the group , where = r_{i}/m_{i }and w_{i }are weights associated with each individual proportion. The unbiased variance estimator of is given as
To avoid having an estimated variance that is less than the binomial sampling variance, a lowerbound for the variance is also provided. All the parameters (i.e. α, β and w_{i}) are obtained through an iterative procedure. The same estimation procedure is applied to data from the other group. For testing whether the proportion in one group (say group A) equals the proportion in the other group (group B), a tlike statistic t_{w }is constructed, where
The t_{w }statistic is assumed to have a tdistribution with the degrees of freedom (df) calculated from the Satterthwaite approximation:
where n_{A }and n_{B }are the number of SAGE libraries in the group A and B respectively. This test is called the t_{w}test here. The implementation of both the t and t_{w}test can be found in [13].
Overdispersed logistic regression approach
Baggerly et al. [15] provided a thorough description on this approach and details can be found in [24]. Briefly, unobserved continuous random variables P_{i }are introduced to account for the betweenlibrary variation, where the mean and variance of P_{i }have the following forms: E(P_{i}) = p_{i }; Var(P_{i}) = φ p_{i}(1  p_{i}). Here φ is a nonnegative scale parameter. Conditional on P_{i}= p_{i}, the r_{i }have a binomial distribution (m_{i}, p_{i}). The unconditional mean and variance of r_{i }can be shown to be E(r_{i}) = m_{i }p_{i }and Var(r_{i}) = m_{i }p_{i}(1  p_{i}) [1+(m_{i}1) φ]. Notice that if φ is 0 (i.e. there is no betweenlibrary variation or overdispersion), the variance of r_{i }is the usual binomial variance m_{i }p_{i}(1  p_{i}). The estimation of coefficients β is carried out by the iteratively reweighted leastsquares (IRLS) procedure, where the weights w_{i }are 1/ [1+(m_{i } 1) φ]. Note that the weights w_{i }are equal if the library sizes m_{i }are equal.
The parameter φ is estimated by equating the goodness of fit Pearson's chisquare statistic X^{2 }to its approximate expected value, which is
where v_{i }= m_{i }p_{i}(1  p_{i}), and d_{i }is the variance of the linear predictor . An iterative procedure is introduced to estimate φ and β, where the estimates of φ (and accordingly, the weights w_{i}) and β are updated at each step. Given the estimated coefficients, the testing hypothesis is whether one (or more if there are more than two groups) of the coefficients (β) is 0. For this, the ttest rather than the ztest is recommended due to the introduction of the overdispersion parameter into the model [15,32].
The hypothesis test based on overdispersed logistic regression is called the logitt test here. The implementation including source code can be found in [15]. We consider overdispersion models (logistic or loglinear) only if the Pearson's chisquare statistic from the usual logistic regression (or loglinear) fit (i.e. without overdispersion) is greater than or equal to its expected value, np.
Overdispersed loglinear models
This model is closely related to the overdispersed logistic regression model. One way to derive it is based on the gammaPoisson hierarchical model assumption [16]. Assume that an unobserved random variables θ_{i }is distributed according to
θ_{i }~ Gamma(μ_{i}, 1/φ),
where μ_{i }= m_{i }p_{i}, φ >0, E(θ_{i}) = μ_{i }and Var(θ_{i}) = . Given p_{i}, the response variable r_{i }is assumed to be distributed as
r_{i } p_{i }~ Poisson(μ_{i}).
The unconditional mean and variance of r_{i }can be shown to be E(r_{i}) = μ_{i }= m_{i }p_{i }and Var(r_{i}) = μ_{i }(1+μ_{i}φ). Notice that as φ decreases to 0, the variance of r_{i }approaches the usual Poisson variance μ_{i }(i.e. m_{i }p_{i}). The same meanvariance relationship can also be derived if we assume r_{i }has a negative binomial distribution [16]. The mean μ_{i }of the response variable r_{i }and the covariates x_{i }are connected through the log link function,
log μ_{i }= log(m_{i }p_{i}) = x_{i}β.
As in the overdispersed logistic regression model, the estimates of the coefficients β are obtained by the iteratively reweighted leastsquares procedure, where the weights w_{i }are 1/(1+μ_{i }φ) [33]. Note that, in contrast to the overdispersed logistic regression model where the weights only depend on library sizes m_{i}, the weights in the loglinear model depend on μ_{i }(i.e. both m_{i }and p_{i}).
The hypothesis test based on overdispersed loglinear models is called the logt test here. The R [34] source code and a webinterface for implementing this approach are available [35].
Authors' contributions
JL developed the method. JL and JKT carried out the simulation and data analysis. JKT and JL set up the web interface for implementing this approach. TBK supervised the study, and assisted with the methodology. All authors contributed to the writing, read and approved the final manuscript.
Acknowledgements
The authors would like to thank anonymous reviewers for several constructive comments. We thank Gregory Riggins for introducing us to SAGE. We gratefully acknowledge the financial support of the NIH through the Duke University Center for Translational Research (5 P30 AI05144503) and through the Southeast Regional Center of Excellence in Biodefense and Emerging Infections (U54 AI05715702); and of the NSF through a grant to our collaborator David Bird (NCSU; DBI 0077503) as well as the Duke Center for Bioinformatics and Computational Biology for support through a postdoctoral fellowship to JL.
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