statistical methods for efficiency adjusted real-time pcr

1 Introduction

Real-time PCR is a powerful, economical, rapid,and high-throughput technique for assaying geneexpression with broad applications in clinical stud-ies, diagnostics, forensics, food technology, patho-gen detection, and functional genomics [1–4]. Thetechnique has been adopted quite widely because

of the apparent advantages over other gene ex-pression methodologies such as Northern blotanalysis: a much larger dynamic range, higher sen-sitivity, smaller sample amount requirements, andless labor. Even though real-time PCR has becomeone of the most important enabling technologies inthe genomic era, the real or perceived unreliabilityof real-time PCR data has engendered serious con-cerns [1, 4]. Both proper experimental design andreliable statistical analysis are the bases for the ac-curacy and reproducibility of results. Furthermore,without these, data interpretation and conclusionscould be faulty. Proper experimental design in-

Technical Report

Statistical methods for efficiency adjusted real-time PCR quantification

Joshua S. Yuan1, 2*, Donglin Wang3 and C. Neal Stewart Jr.2

1UTIA Genomics Hub, The University of Tennessee, Knoxville, TN, USA2Department of Plant Sciences, The University of Tennessee, Knoxville, TN, USA3Department of Statistics, The University of Tennessee, Knoxville, TN, USA

The statistical treatment for hypothesis testing using real-time PCR data is a challenge for quan-tification of gene expression. One has to consider two key factors in precise statistical analysis ofreal-time PCR data: a well-defined statistical model and the integration of amplification efficiency(AE) into the model. Previous publications in real-time PCR data analysis often fall short in inte-grating the AE into the model. Novel, user-friendly, and universal AE-integrated statistical meth-ods were developed for real-time PCR data analysis with four goals. First, we addressed the defi-nition of AE, introduced the concept of efficiency-adjusted ΔΔCt, and developed a general mathe-matical method for its calculation. Second, we developed several linear combination approachesfor the estimation of efficiency adjusted ΔΔCt and statistical significance for hypothesis testingbased on different mathematical formulae and experimental designs. Statistical methods werealso adopted to estimate the AE and its equivalence among the samples. A weighted ΔΔCt methodwas introduced to analyze the data with multiple internal controls. Third, we implemented the lin-ear models with SAS programs and analyzed a set of data for each model. In order to allow otherresearchers to use and compare different approaches, SAS programs are included in the Sup-porting Information. Fourth, the results from analysis of different statistical models were com-pared and discussed. Our results underline the differences between the efficiency adjusted ΔΔCtmethods and previously published methods, thereby better identifying and controlling the sourceof errors introduced by real-time PCR data analysis.

Keywords: Amplification efficiency · Efficiency adjusted ΔΔCt · Real-time PCR · Statistical analysis

Correspondence: Professor C. Neal Stewart, Jr., Department of PlantSciences, The University of Tennessee, Knoxville, TN 37996, USAE-mail: [email protected]: +1 865 846 1989

Abbreviations: AE, amplification efficiency; PAE, percentile amplificationefficiency *E-mail: [email protected]

Received 1 August 2007Revised 29 October 2007Accepted 13 November 2007

Supporting information available online

BiotechnologyJournal DOI 10.1002/biot.200700169 Biotechnol. J. 2008, 3, 112–123

112 © 2008 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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volves the inclusion of suitable internal and exter-nal controls, sufficient replication, and appropriateand efficient cDNA synthesis methods [4, 5]. Theoptimal statistical treatment for real-time RT-PCRrequires precise data modeling, integration of am-plification efficiency (AE), and rational outlier ex-clusions. While real-time PCR experiments can beperformed for both absolute and relative quantifi-cation, the focus of this paper will be on the relativequantification, which is the more prevalent appli-cation in biology [4].

Previous publications laid down the founda-tions for mathematical methods for real-time PCRdata analysis [6, 7].The standard curve method andΔΔCt method are described in these articles. Bothmethods involve calculation based on Ct, the cyclenumber at which logarithm-transformed fluores-cence crosses a threshold. The standard curvemethod calculates the AE and expression ratiobased on Ct numbers from a serial dilution of tem-plates [6].The ΔΔCt method assumes the percentileamplification efficiency (PAE) is 100% (doubling ofPCR products per cycle) and calculates the ratiobased on 2–ΔΔCt [7]. Even though these papers havebeen significant contributions for analysis of real-time PCR data, robust statistical treatments arelargely lacking, especially in terms of confidenceinterval estimation and P value [8]. Statistical treat-ments are crucial to obtain meaningful and accu-rate interpretation of real-time PCR data, sincelarge variation may be observed with differentmathematical methods without appropriate statis-tical analysis [9].A recent influx of publications forstatistical modeling of real-time PCR data is reflec-tive of bioinformaticians’ desire to address theseissues. However, the problem is still acute for“messy” data sets.

Statistical analysis of real-time PCR data in-volves data modeling for ΔΔCt calculation, AE cal-culation, quality control, and outlier exclusion.Most statistical modeling methods include dataquality control functions based on linear regres-sion models [8]. KOD and Grubbs’ test have beenproposed to be outlier exclusion methods and canbe integrated for data analysis [9]. Statistical meth-ods have been developed to derive ΔΔCt based onpair-wise tests, ANOVA, and GEE models [10]. In2006, Yuan et al. proposed fitting real-time PCRdata into ANCOVA and multiple regression modelsfor ΔΔCt calculation, data quality control, and anequal AE test. Here, the regression models are es-pecially powerful tools for real-time PCR dataanalysis. However, the current models often in-clude inappropriate assumptions in relation to AE,which limits the models application to appropriatedata sets. Some models assume that all PCR reac-

tions should have PAE of 100%, and others assumethe equal PCR amplification efficiencies amongsamples or genes [7, 8]. Either assumption is some-times violated in actual experiments, since amplifi-cation efficiencies in real-time PCR experimentscan typically vary between 70 and 100%.When datafrom lower AE or unequal amplification efficien-cies are analyzed with the current available mod-els, the inferred result could be much differentfrom actual gene expression, which is a majorsource of unreliability of real-time PCR technolo-gy.

AE is probably the most important concept inreal-time PCR quantification, since the develop-ment of the technique is based on the assumptionthat PCR products double each cycle, but it can bea confusing issue to practitioners. When the per-centile PCR AE is not 100%, the quantification mustbe adjusted by the AE. PCR AE has been presentedas variably from 1 to 2, or 0 to 1 (0–100%) [11, 12]. Inorder to avoid the confusion, we use AE to repre-sent the multiplication of PCR product increaseduring each cycle, which is between 1 and 2, andPAE to represent the percentage of full AE capaci-ty, which is between 0 and 1 or 0 and 100%. Themathematical conversion between AE and PAE hasbeen controversial as well. Some researchers de-fine AE as 1 added to PAE as shown in Eq. (1) [11],while others define AE as 2 powered by PAE asshown in Eq. (2) [4]. In this article, we adopted thesecond definition to have AE equal to 2 powered byPAE as shown by Eq. (2), because of the advantagesin statistical modeling and estimation. AE and PAEare normally calculated through standard curve forCt or fluorescent signal strength during the ampli-fication [8, 11]. The concept and calculation of AEand PAE will be discussed in detail in the articlesince they are important in calculating AE adjustedΔΔCt and relative gene expression abundance.Even though several papers have been publishedin calculating AE, few have thoroughly discussed aprecise universal statistical model with AE inte-grated in ΔΔCt and ratio calculation. Proper AE-in-tegrated statistical models for ratio calculation andsignificance estimation are therefore the key issuefor accurate real-time PCR quantification nowa-days.

AE = 1+PAE (1)

AE = 2PAE (2)

where AE is the amplification efficiency (1–2) andPAE is the percentile amplification efficiency (0–1).

Besides the efficiency adjustment, recent ad-vances of real-time PCR quality control involving


multiple controls require a proper statistical mod-el for ΔΔCt calculation integrating more than onecontrol [13, 14]. A weighted ΔΔCt method is alsopresented, and integrating efficiency adjusted ΔΔCtshould provide optimal accuracy.

Overall, in this article, we present efficiencyadjusted and weighted ΔΔCt methods for gene ex-pression analysis with real-time PCR as well asthe statistical models implementing the method.Several linear model based methods and SASprograms will provide new approaches for effi-ciency-integrated real-time PCR data analysis.The concept of ΔΔCtadjusted and an efficiency-ad-justed quantification method are first introduced.The definition of AE and its calculation based onsimple linear regression models are discussed.Statistical models are then developed for efficien-cy adjusted ΔΔCt calculation for single concentra-tion design, standard curve design, and a mixeddesign. The weighted ΔΔCt models are includedfor experiments with more than one internal con-trol. Two-way ANOVA and multiple regressioncombined with an unbalanced linear combinationare the main statistical approaches employed.SAS programs are developed for each model anda test sample set is analyzed with different statis-tical methods. Comparison of efficiency adjustedquantification approach with previous modelshas indicated that analysis of low-quality real-time PCR data may miscalculate the target geneexpression ratio if AE effects are not explicitlyconsidered. Moreover, it highlights the impor-tance of proper experimental design and condi-tion optimization for real-time PCR.The pros andcons for each proposed models and designs arealso discussed. The new methodology also pro-vides a more precise and high-throughput alter-native for statistical analysis of low quality real-time PCR data.

2 Materials and methods

2.1 Real-time PCR experiments

Arabidopsis thaliana (Col1) plant growth, RNA ex-traction, and real-time PCR experiments were car-ried out as described [15]. The real-time PCR ex-periments were conducted using a standard proto-col recommended by the manufacturer. Basically,approximately one microgram of total RNA wassynthesized into cDNA using iScript cDNA synthe-sis kit (BioRad Laboratories). The cDNA was thendiluted into one to four and one to sixteen serial di-lutions. Real-time PCR experiments were carriedout with duplication for each concentration with an

ABI 7000 Sequence Detection System (AppliedBiosystems). After the experiment, the Ct numberwas extracted for both reference gene and targetgene with auto baseline and manual threshold of0.4613. In addition to the Ct number, the fluores-cence measurements were also downloaded. Be-sides the dataset with one internal control, we in-cluded a dataset with two internal controls (ubiqui-tin and tubulin) to illustrate weighted ΔΔCt estima-tion.

2.2 ΔΔCtadjusted and efficiency adjusted quantificationmethod

As discussed in Section 1, we adopted the PAE def-inition in Eq. (2). According to the equation, PCRproduct amount during the reaction can be definedby Eq. (3), where PCR product equals AE raised tothe power of cycle number n and then multiplied byoriginal template amount. Since Ct is also a cyclenumber by definition, it can replace the n in Eq. (3)with Ct. From Eq. (3), we can derive Eq. (4).The goalof real-time PCR quantification is to calculate theabsolute or relative abundance of original templateamount (P0) as shown in Eq. (4).

(3)

where P is the PCR product amount for a givenPCR cycle, P0 the original template amount, n is thenumber of PCR cycles.

(4)

The purpose of relative quantification is to derivethe ratio of target gene expression between a treat-ment sample and a control sample after normalizedby the internal reference gene. Equation (5) pro-vides the universal solution for relative quantifica-tion, where the P0 for each sample will be first nor-malized against the internal reference gene andthen compared between the samples. Since Ct isderived from the cycle number for a given thresh-old, the PCR product amount (P) is equal among allthe samples and genes and therefore can be can-celled out. The relative abundance of original tem-plate for the target gene between samples can thusbe presented as follows in Eq. (5).

(5)

where Con is the abbreviation for control sample,Tgt the abbreviation for target gene, Trt the abbre-

Ratio Tgt_Trt 0_Ref_Trt

Tgt_Con 0_Re=

P PP P

0

0

_

_

// ff_Con

Ct_Ref_Trt P A E Ct_Tgt_Trt P A E

Ct_R=× ×2 2

2/

eef_Con P A E Ct_Tgt_Con P A E× ×/ 2

P P0 2

= ×Ct P A E

P P P Pn n n= = = × ×0 0 02 2( ) ( )A E P A E P A E

BiotechnologyJournal Biotechnol. J. 2008, 3, 112–123





viation for treatment sample, and Ref is the abbre-viation for reference gene.

In the traditional 2–ΔΔCt method without efficien-cy adjustment, the ratio can be calculated directlyfrom the Ct numbers as shown in the Eq. (6).

(6)

The differences between Eqs. (5) and (6) are thatthe Cts in Eq. (5) have been adjusted by PAE. Thegeneralized efficiency adjusted ΔΔCt (ΔΔCtadjusted)can therefore be defined as the differences for theCt numbers that are adjusted by PAE. From Eq. (5),the ΔΔCtadjusted can be calculated and defined as inEq. (7). Equation (7) is the basis of statistical mod-els in Sections 2.4 and 2.5 of the article

(7)

2.3 The estimation of PAE

A key component for the ΔΔCtadjusted calculation isthe PAE of PCR. We hereby present the linear re-gression models for PAE estimation in both stan-dard curve design and single concentration design.The PAE can be estimated for a group of reactionsor a single reaction by simple linear regressionmodel. In 2006,Yuan et al. proposed a simple linearregression model to calculate and perform qualitycontrol on the PAE based on the standard curve de-sign as shown in Eq. (8).The PAE can be defined as–βlcon [8, 12, 16].

Ct = β0 + βlconXlcon + ε (8)

where Xlcon is the logarithm 2 based transformationof serial diluted concentration.

Even though the Eq. (8) was developed for stan-dard curve design, an alternative simple linear re-gression model can be used for single concentra-tion design by estimating PAE from fluorescencemeasurements for each amplification cycle [11, 12].Linear models for AE estimation has been dis-cussed before, however, previous research eithermodels data with Eq. (1) or uses logarithm 10-based transformation, both of which lead to no di-rect estimation and test of PAE in the model.A bet-ter way to calculate PAE in fluorescence can be de-rived from logarithm 2-based transformation of Eq.(3) as shown in Eq. (9).

Plog = P0log + n × PAE (9)

ΔΔCt Ct PAECt

adjusted Tgt_Trt Tgt_Trt

Ref_Tr

= ×−(

tt Ref_Trt

Tgt_Con Tgt_Con

Ref_

PAECt PAECt

×− ×−

)(

CCon Ref_ConPAE× )

Ratio CtCt_Ref_Trt Ct_Tgt_Trt

Ct_Ref= =−2 2 22

ΔΔ /__Con Ct_Tgt_Con/ 2

where Plog is the logarithm 2 transformed PCRproduct amount (or fluorescence), P0log the loga-rithm 2 transformed original template amount, andn is the number of amplification cycle.

According to Eq. (9), a simple linear regressionmodel can be developed as shown in Eq. (10).

Plog = β0 + βxXc + ε (10)

where Plog is the logarithm 2 transformed PCRproduct amount (or fluorescence reading) asshown in fluorescence data, and Xc is the cyclenumber n.

This model will be the basis for PAE calculationfor fluorescence data in the later part of the article.The PAE can be directly estimated thorough testingthe βx. Based on this model, we can also perform thedata quality control for the PCR to test if PAE isequal to 100%.When βx equals to 1, the PAE equalsto 100%.

The above simple linear regression models canbe readily implemented with SAS 9.1 (SAS Insti-tute, Cary, NC) as shown in Supporting InformationFile 1 and 2 for fluorescence data and standardcurve data, respectively. Supporting InformationFile 3 is the input file for the single concentrationdesign fluorescence data, and Supporting Informa-tion File 4 is the input file for Ct value from stan-dard curve design. Both data formats are as shownin Tables 1 and 2. For the fluorescence data analy-sis, multiple filtering methods can be used to de-termine the data point representing the exponen-tial phase of PCR, which is not the focus of this pa-per. In order to generate the dataset for down-stream analysis, we filtered the input data by arange based on the observation of amplificationcurve.Typical real-time PCR data can be presentedas a plot of logarithm transformed fluorescenceagainst cycle number as shown in Fig. 1. A linearrange can be observed in the plot, in which the am-plification curves for different samples are parallelto one another.The linear range represents the log-arithm phase of PCR and is used for AE calculation.After the data filter, the fluorescence intensity rep-resenting the PCR product amount is then subject-ed to logarithm 2-based transformation. The loga-rithm transformed fluoresce signal (PLOG) and cy-cle number (CY) are then fitted for a simple linearregression in Eq. (10). The slope represents thePAE, and the test of slope = 1 will render the test ofPAE’s equivalence to 100%. In the same way, we cananalyze the standard curve data with Eq. (8) exceptthat the logarithm-transformed concentrationshould be in reverse proportion to Ct.

Besides the model above, several other ways canbe used for point estimation and equivalence test of


AE.Yuan et al. [8] proposed to use Type III sums ofsquares of multiple regression model to test theequivalence of AE. More generalized linear modelsas shown in Figs. 2 and 3 can be used for estimationof equivalence of PAE for different treatment andsamples with different design.

2.4 Statistical modeling of ΔΔCtadjusted for fluorescence data

The efficiency adjusted ΔΔCt in Eq. (7) can be ana-lyzed with linear models. Previous research has fit-ted real-time PCR data into ANCOVA, multiple re-gression and ANOVA models [8–10]. Fitting real-time PCR data into linear models is a significantstep toward the precise real-time PCR data analy-sis with appropriate quality controls. However,most of the current models are designed to deriveΔΔCt directly with the assumption of 100% PAE, andtherefore cannot be used in the analysis of low-quality data. Yuan et al. [8] proposed an efficiencyadjusted ANCOVA model for low quality dataanalysis, but the model still requires equal AE foreach gene between different samples, an assump-tion which might be commonly violated. In reality,

real-time PCR data could have PCR amplificationefficiencies deviating from 1 for different cDNAsamples and different genes. This is probably therule rather than the exception. The assumptionsover AE therefore compromise the application ofthese statistical models in practice. Novel modelswith universal applications based on Eq. (7) arenecessary for AE integrated statistical analysis.Moreover, previous linear models by Yuan et al. [8]were developed for analyzing Ct values from stan-dard curve design only, statistical models need to bedeveloped for efficiency adjusted analysis of fluo-rescence data. Here we present unbalanced linearcombination of group effect for a two-way ANOVAmodel and intercept estimation of multiple regres-sion model as two options for the ΔΔCtadjusted calcu-lation for fluorescence data analysis with singleconcentration design. The same models can alsobe used for standard curve design data. The prosand cons of each statistical model will be discussedlater.

2.4.1 Two-way ANOVA model and unbalanced linearcombination

The fluorescence measurements do not yield anexplicit Ct number, however, we can consider Ct as



Table 1. Input fluorescence data of single concentration design

Cycle Fluorescence Sample Group

1 0.003895 1 1···16 0.251703 1 117 0.488688 1 118 0.925395 1 1···42 7.278038 2 4

The data include PCR cycle number and fluorescence measurements (in arbi-trary units) for each PCR cycle. We have included two samples for each group,and four groups for comparison. The four groups are the reference and gene inthe control sample, as well as the reference and target gene in the treatmentsample.

Table 2. Input data for standard curve design

Treatment Gene Concentration Ct Group

Water (2 h) UBQ 16 16.8847 1Water (2 h) UBQ 16 17.205 1Water (2 h) UBQ 4 18.972 1Water (2 h) UBQ 4 19.0513 1Water (2 h) UBQ 1 21.2514 1

···Ala (2 h) MT7 1 25.367 4

The data has the relative concentration and Ct numbers as well as differenttreatments and genes.

Figure 1. Linear region for PCR quantification.The exponential phase of PCR reaction can beused to analyze PAE, and it is a linear region in theplot of logarithm-based fluorescence against cyclenumber.




the cycle number for a given PCR product amount.When the sample effect is randomized, the sampledataset in Supporting Information File 3 can be fitinto the models in Fig. 2B. The traditional ΔΔCt canbe calculated by the ANOVA model by contrastingand estimating the linear combinations of meansfor cycle number of different groups as shown inEq. (11).The ΔΔCtadjusted can be tested and estimat-ed in the same way except that the combination ofmeans needs to be adjusted by PAE as shown in Eq.(12). The adjustment in combination often resultsin an unbalanced linear combination.

ΔΔCt = (μ1 − μ2) − (μ3 − μ4) = μ1 − μ2 − μ3 + μ4 = 0 (11)

ΔCtadjusted = μ1 × PAE1 − μ2 × PAE2 − μ3 ×PAE3 + μ4 × PAE4 = 0

(12)

where μ1–μ4 is the mean of cycle number for groups1–4, respectively, PAE1–PAE4 are the percentile am-plification efficiency for groups 1–4, respectively.

Both equations fit into the general linear com-bination model as shown in Eq. (13). For ΔΔCt esti-mation in Eq. (11), the linear combination is bal-anced with ci equals to 1 or −1, representing 100%

PAE. For ΔΔCtadjusted estimation in Eq. (12), the lin-ear combination is unbalanced with ci equals toPAE or –PAE for each group.

(13)

Equation 12 can be realized by unbalanced linearcombination of group effects as shown in Fig. 2B forfluorescent data of single concentration design.The null hypothesis for the test in Eq. (12) is thatthe ΔΔCtadjusted equals to 0 when offset by combinedmean of PAE, which indicates no changes in targetgene expression between the treatment and con-trol sample. The alternative hypothesis is that thetarget gene expression changes significantly be-tween the samples. A low P value will favor the al-ternative hypothesis. The model is implemented inSAS as shown in Supporting Information File 1.The model statement establishes the statisticalmodel for the two-way ANOVA, where CY (cyclenumber) depends on the logarithm 2-based trans-formation of PCR product amount (fluorescencereading), different groups, and their combinatorialeffects. The ΔΔCtadjusted are then derived from con-trasting and estimating the unbalanced linear com-

ΔΔCt ==∑ci ii

r

μ1

Figure 2. Statistical estimation of real-time PCR data using ANOVA model withcycle or Ct number as dependent vari-able. (A) Presents the efficiency adjustedΔΔCt estimation from unbalanced linearcombination of two-way ANOVA modelfor data from standard curve design. (B)Presents efficiency adjusted ΔΔCt estima-tion for fluorescence data from the singleconcentration design.

Figure 3. Statistical estimation of real-time PCRdata and balanced linear combination of interceptto derive efficiency adjusted ΔΔCt. (A) Presentsthe multiple regression model for standard curvedesign and the estimation efficiency adjustedΔΔCt. (B) Presents the same model for analyzingfluorescence data with single concentrationdesign.


bination of PAE adjusted means. The SAS outputprovides both point and confidence level estima-tion for ΔΔCtadjusted as shown in Table 4. Moreover,the SAS output also gives a P value to estimate thestatistical significance of gene expression differ-ences.

2.4.2 Multiple regression model and linear combination of in-tercepts

The ANOVA model provides a conceptually simpleapproach to estimate the ΔΔCtadjusted. However, themethod needs to calculate the grouped PAE foreach gene and treatment combination before theΔΔCtadjusted calculation. Moreover, since the PAE isestimated for a group and the variation of PAEwithin the group is not integrated in the model, themodel might potentially underestimate the varia-tion of ΔΔCtadjusted. A more conceptually complicat-ed, but more direct approach for ΔΔCtadjusted esti-mation is through linear combination of interceptsin multiple regression model. Considering the sim-ple linear regression model in Eq. (10), the βcXcrepresents the cycle number multiplied by PAE. Ifwe take into account the simple linear regressionfor the four groups respectively as shown in Eq.(14), a linear combination of these four groups byadding groups 1 and 4 and subtracting groups 2 3will give the Eq. (14), since the Plog for each groupwill be equal at a given Ct number and terms can becancelled out.

(14)

In Eq. (14), the right portion of the equation is theΔΔCtadjusted as shown in Eqs. (12 and 13), and the leftportion of the equation is the linear combination ofthe intercepts, which will be equal to the linearcombination of the intercept of the multiple re-gression model as shown in Fig. 3B if no interactionamong groups existed. Basically, for multiple linearregression model, ΔΔCtadjusted can be estimatedfrom linear combination the intercept of differentgroups. Conceptually, for either model in Fig. 3, ifcycle or Ct number equals to 0, the product or tem-plate concentration represents the initial amount.Therefore, the contrast and combination of inter-cepts will derive ΔΔCtadjusted as also visualized inFigs. 4A and B.

The model can be easily implemented in SAS asshown in Supporting Information Files 1 and 2.The

P XP X

G G xG cG

G G xG cG

log

log

1 0 1 1 1

2 0 2 2 2

= + += + +

β β εβ β εεβ β εβ β

P XP X

G G xG cG

G G xG cG

log

log

3 0 3 3 3

4 0 4 4

= + += + 44

0 1 0 2 0 3 0 4

1 1 2

+⇒ − − − += −

εβ β β β

β β( )G G G G

xG cG xG cX X GG xG cG xG cGX X2 3 3 4 4− +β β

model statement in the procedure helps to estab-lish multiple regression for logarithm transformedPCR product amount (PLOG) with group and cyclenumber (CY).The output of linear combination in-cludes the mean and confidence interval estima-tion of ΔΔCtadjusted as shown in Table 3. It should benoted that the variation for the ΔΔCtadjusted is muchlarger in this analysis as compared to the two-wayANOVA model, and the resulting test is not signif-icant.

2.5 Statistical modeling of ΔΔCtadjusted for standard curve

Even though the ΔΔCtadjusted can be calculated inthe real-time PCR experiments with fluorescencedata from single concentration design, the designmay lead to more variation than the standard curvedesign, which allows a large dynamic range of thequantification [5]. Moreover, the standard curve al-lows to observe amplification inhibition at earlierstage of PCR, while fluorescence data only reflectthe efficiency above the baseline level. The previ-ously published linear models can be modified toestimate ΔΔCtadjusted for standard curve design [8].

The principle and procedure for fluorescencedata analysis can be applied to analyze ΔΔCtadjustedfor the standard curve design as shown in Figs. 2and 3. The same unbalanced linear combination ofcycle number can be used for ΔΔCtadjusted estima-tion in the statistical model with Ct as the depend-ent variable. The linear combination of interceptsin the statistical model with logarithm transformedconcentration as the dependent variable can alsobe used for ΔΔCtadjusted estimation. The only differ-ence between the modeling of fluorescence dataand standard curve data is that logarithm trans-formed fluorescence data are in linear proportionto the cycle, while the logarithm transformed con-centration is in reverse linear proportion to the Ctnumber. Therefore, the linear combination equa-tion is reversed in modeling the two types of data.



Table 3. PAE estimation

Group Design PAE STD P value

1 Standard curve 0.9216 0.0349 0.0892 Standard curve 0.8774 0.0348 0.024*3 Standard curve 0.9067 0.0223 0.014*4 Standard curve 0.8983 0.0305 0.029*1 Single concentration 0.9571 0.0352 0.2542 Single concentration 0.9469 0.0284 0.0953 Single concentration 0.8782 0.0105 <0.0001**4 Single concentration 0.8902 0.0271 0.023*

The PAE estimation for each group is listed for both designs. Ct values wereused for standard curve design and single concentration stands for fluores-cence data from single concentration design. * indicates a p value smaller than0.05 and ** indicates a p value smaller than 0.01.




Figure 4. The visualization of data. The intercepts for the multiple regression models in single concentration design and standard curve design are shownin (A) and (B), respectively. (C) Presents the 95% confidence interval (D) and (E) are the regression data point for fluorescence data and Ct value of stan-dard curve, which shows clear differences among four groups. (F) Is the residual plot for regression model. PLOG is the abbreviation for logarithm trans-formed PCR product, lcon is the abbreviation for logarithm transformed concentration, and LPFL ist the abbreviation for logarithm transformed fluores-cence signal.


The statistical models are implemented in SAS asshown in Supporting Information file 2, and thedata output is summarized in Table 4.

2.6 Statistical modeling of ΔΔCtadjusted for mixed design

The fluorescence data from a standard curvedesign can also be analyzed: a mixed design. Athree-way ANOVA model can be used and the ΔΔCtadjusted can be estimated through the linearcombination of intercepts. The main advantage isthat the design enlarges the sample size for ΔΔCtadjusted estimation and allows estimation of AEusing two approaches. The three-way ANOVAmodel is implemented in the Supporting Informa-tion File 5, which is based on dataset in SupportingInformation File 6. The analysis results yield asmaller P value as compared to the linear combi-nation of intercepts from either fluorescence dataof single concentration design or Ct value fromstandard curve design because the increase in theeffective sample size.

2.7 Statistical modeling of ΔΔCtweighted for data with multiplecontrols

Besides the AE, another important issue of real-time PCR analysis is having the proper internal

control. Multiple genes as internal controls havebeen proposed to reduce systemic error and im-prove the accuracy of analysis [14]. Most of the cur-rent models are developed for experimental de-signs including only one control. A modified linearcombination of group effects in the ANOVA modelallows the analysis of real-time PCR data with mul-tiple controls as shown in Eq. (15), where W is theweight equaling to 1 divided by number of controlgenes. In the case inclusion of two internal genes,the weight should be 0.5. Supporting InformationFile 8 provides a data set with multiple controls,and Supporting Information File 7 is the SAS im-plementation of the Eq. (15).

ΔΔCtadjusted = μ1 × PAE1 − μ2 × PAE2 − μ3 ×PAE3 × W + μ4 × PAE4 ×W − μ5 × PAE5 × W + μ6 ×PAE6 × W

(15)

where 3, 4, 5, and 6 represent the data from two ref-erence genes for both control and treatment sam-ples. When multiple reference genes are included,W can be equal to 1 divided by total number of ref-erence genes.



Table 4. Comparison of different methods

Type Design Test Mean Error 95% CI P Ratio Reference

ΔΔCt Single ANOVA −1.90 0.17 (−2.24, −1.55) <0.01 3.73 [9]concentration

Standard curve T-test −2.34 0.27 (−2.53, −2.15) <0.01 5.06 [8]

Wilcoxon −2.11 N/A (−2.56, −2.36) <0.01 4.32 [8]

ANCOVA −2.26 0.19 (−2.65, −1.86) <0.01 4.78 [8]

ΔΔCtadjusted Single ANOVA −1.25 0.16 (−1.57, −0.93) <0.01 2.38 This paperconcentration (PAE adjusted, UBLC)

Multiple −1.17 0.95 (−3.1, 0.76) 0.23 2.25 This paperregression (LCI)

Standard curve ANCOVA −1.47 0.17 (−1.82, −1.11) <0.01 2.77 This paper(PAE adjusted, UBLC)

Multiple regression −3.41 1.68 (−6.79, 0.15) 0.06 7.42 This paper(LCI)

Mixed design Multiple regression −1.19 0.63 (−2.43, 0.05) 0.06 2.28 This paper(LCI)

ΔΔCtweighted Standard curve ANOVA weighted −2.84 0.39 (−3.64, −2.67) <0.01 7.16 This paper

Standard curve ANOVA Weighted −2.23 0.36 (−2.98, −1.49) <0.01 4.69 This paperPAE adjusted

Statistical analysis results are presented for different models for ΔΔCt and ΔΔCtadjusted. The mean estimation, standard error, 95% confidence interval, P value, predict-ed ratio, and references are presented in the table. UBLC stands for unbalanced linear combination. LCI stands for linear combination of intercepts.




3 Results and discussion

The main advantage of the statistical models pre-sented is the ability to perform analysis without AErestrictions, which is an important improvementfrom our previous statistical treatment of real-timePCR data [8]. The concepts and modeling of ΔΔCtadjusted and ΔΔCtweighted are novel.

3.1 The estimation of PAE

The estimation of AE is presented in Table 3, wherethe statistical test of slope’s equivalence to 1 ren-ders the P value. A small P value indicates PAE issignificantly different from 1 or 100%. Even thoughthe interaction effects between group and concen-tration are not significant under the ANOVA mod-el for standard curve design, the data in Table 3 in-dicate real differences among sample AE. More-over, the PAEs are significantly different from 100%in several cases.The efficiency adjusted data analy-sis is therefore highly recommended for such data.It would, therefore, not be appropriate to apply ourpreviously reported models [8] in this case. Hence,we analyzed the data with different statistical mod-els for efficiency adjusted ΔΔCt presented above.

3.2 The comparison of statistical models for ΔΔCtadjusted andΔΔCt

The comparison of the ΔΔCt and ΔΔCtadjusted asshown in Table 4 highlights the importance forefficiency adjustment in the analysis of this dataset. The ΔΔCt estimation is different from the ΔΔCtadjusted. The ΔΔCt estimation gives a lower Pvalue and larger differences among gene expres-sion values as compared to the ΔΔCtadjusted withPAE adjustment by unbalanced linear combina-tion. The efficiency adjustment for this particulardataset is important for appropriate data interpre-tation. As shown by Table 1, most of the PCRs forthis dataset have PAE less than 1 and amplificationefficiencies also differ from one another. In thecase of low AE, the unadjusted ΔΔCt might overes-timate or underestimate the target gene expressiondifferences depending on the PAE for each groupof PCR. Since the error for PCR increases expo-nentially during the reaction, a 10% PAE differencemight result in huge data distortion if the Ct differ-ences are large enough.

If we examine Figs. 4A and B, the intercepts rep-resent the initial template amount and are used tocalculate ΔΔCtadjusted in linear combination of inter-cepts for multiple regression.The differences of in-tercepts could be quite different from those in ourobserved range, which illustrates the effects of am-

plified error. For ΔΔCt analysis, we are analyzingdata based on the observed regions. If calculationsare not adjusted by PAE, the resulting interpreta-tion could be very misleading.The efficiency prob-lem shown in this dataset is quite common from ourexperience, and it underlines the fact that it is im-possible to make assumptions of equal PCR ampli-fication efficiencies or 100% PAE unless propertests are performed. Thus, the appropriateness ofusing the quantification method is highly depend-ent on data quality. In the case where high qualitydata are available, ΔΔCt and ΔΔCtadjusted would yieldsimilar results [8], and the traditional ΔΔCt methodcould be used since the analysis would not over-estimate the errors. However, for messy data, theΔΔCtadjusted method is a much better choice, becauseotherwise assuming a 100% PAE will miscalculatethe ratio estimation significantly.

3.3 The errors for real-time PCR

The differences between ΔΔCtadjusted and ΔΔCt alsostress the importance of experimental conditionoptimization and experimental design. In this par-ticular dataset, we used a 2 × 3 design where dupli-cate biological samples and three concentrationswere tested. We found that increasing sample sizesignificantly decreases the error and makes the testmore robust. For this dataset, there should be dif-ferential target gene expression among differenttreatments, however, it was not significant for sev-eral test models using our small sample size. More-over, it is also important to optimize the cDNA syn-thesis and the primer design. If the PCRs for allgroups have PAE approximately 1, the ΔΔCt methodcan be directly applied and the test will be more ro-bust.

3.4 The comparison of different statistical methods forΔΔCtadjusted

Different methods can be used for ΔΔCtadjusted cal-culation. As discussed before, it becomes apparentthat integrating PAE in the ANOVA model with un-balanced linear combination will give a muchsmaller error than the linear combination of inter-cepts from the multiple regression model. The dis-crepancy could result from two effects, the under-estimation of error for caused by the pooling of PAEin ANOVA model, and the inflation of the error bylinear combination of intercepts.The results shownin Table 3 confirmed our concerns.All the standarderrors in PAE integrated ANOVA models weresmaller than that from the multiple regressionmodel.When sample sizes are small, PAE-adjustedANOVA may be a better choice of analysis because


error values will be less. However, it is a less con-servative approach.

How could error be inflated in the linear combi-nation of intercepts? In Fig. 4C, we can see the 95%confidence interval for the predicted value willform a curve with the smallest interval at the meanobservation. This imposes a particular problem forthe linear combination of intercepts in the multipleregression model, since the errors at the interceptmay be much inflated because of their large dis-tance from the mean of observed values.The linearcombinations of intercepts by multiple regressionmodel may overestimate the error for ΔΔCtadjusted.As we can see from Figs. 4D and E, there is a cleardifference in the target gene expression betweenthe samples, however, the error is enlarged by thelinear combination of intercepts resulting in a sta-tistically nonsignificance result in the analysis. Un-less we have a larger sample size and smaller stan-dard error for the predicted variables, the multipleregression models should be used only with greatcaution. However, multiple regression is the mostconservative model for the ΔΔCtadjusted calculationsince errors for all the effects are analyzed. In or-der to increase the sample size and dynamic range,a mixed design of analysis of fluorescence at dif-ferent concentrations might be a valid alternativefor linear combination of intercepts with multipleregression model.

Considering the limitations for both types ofΔΔCtadjusted, efficiency adjusted quantificationshould not be the default choice for real-time PCRdata analysis. If experiments are carefully con-trolled, and equal amplification efficiencies are as-sured that are required by the traditional ΔΔCtmethod, the ΔΔCt method may be a better choice fordata analysis because it is effective within the ob-served data range and would not introduce addi-tional errors. The test provided both in MethodsPart III and our previous publication [8] will help totest whether PAE is 100% and equal among allgroups.

3.5 The weighted ΔΔCt

Besides the efficiency adjustment, more internalcontrol genes in the experiment may also result inmore robust data [14]. The ΔΔCtweighted can be usedto perform linear combination of multiple controlsto derive point estimation and test for the gene ex-pression ratio. As shown in Table 4, adding anotherinternal control gene coding for tubulin resulted ina different ratio estimation. The assumption thatthe internal control gene is always expressed con-sistently can be as easily violated, which would leadto interpretive errors.Therefore, including multiple

internal control is always a recommended practice,especially for clinical studies requiring accuracyfor diagnoses, treatments, etc. Regardless of thenumber of control genes, proper efficiency adjust-ment can often improve the data quality (Table 4).

3.6 The future of real-time PCR data analysis

Since the metric of statistical analysis of real-timePCR data is ΔΔCtadjusted, not the ratio of gene ex-pression, the confidence interval and standard er-ror should therefore be presented for ΔΔCtadjustedinstead of the ratio. However, the confidence levelfor the ratio can be derived from the confidence in-terval of ΔΔCtadjusted.

Our research herein highlights the importanceof integrating AE in the statistical analysis of real-time PCR data. In addition, it helps to explain acommon phenomenon in functional genomics, inwhich the real-time PCR-derived gene expressionratio (treated vs. control) is often higher than thatof an equivalent microarray experiment [17, 18].One explanation is that real-time PCR tends tooverestimate the gene expression ratio if AE is notintegrated into the data analysis, and thus, the typ-ical interpretation of gene up-regulation appears tobe greater than it is in reality. An alternative expla-nation was that the ratio expansion for real-timePCR could be due to lower AE for the gene at a low-er expression level in the two groups being com-pared. In either scenario, the problem becomesmore severe when the gene expression ratios arehigh, since the larger Ct differences can result inamplified error. Using the methods outlined hereshould yield real-time RT-PCR estimates that aremore congruent with microarray data. With somemodification, the method can also be used to esti-mate the AE at different cycle number as previous-ly described, or to analyze the multiplex data withmore accuracy [19, 20]. The research also under-scores the importance for the scientific communityto establish standard procedures for real-time PCRdata analysis, since an improper treatment maylead to amplified error in case of real-time PCRdata analysis. In turn, this could lead to faulty bio-logical interpretations and conclusions.

The authors have declared no conflict of interest.

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