Multivariable Linear Models Outperform 2-ΔΔCT for qPCR Data Analysis

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Stanton, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6014969/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Here we present a method based on multivariable linear models for qPCR data analysis as an alternative to the most commonly used method, 2 −ΔΔCT . It has long been understood that amplification efficiency during qPCR may be less than two, that is, the amount of DNA present may not double in each cycle, and it is also known that amplification efficiency may differ between target and reference genes. Therefore, it has long been recommended that qPCR experiments include direct assessment of amplification efficiency, and that efficiency values be included in the calculation of differential gene expression. Nonetheless, current reports that include qPCR data continue to use 2 −ΔΔCT , even though 2 −ΔΔCT assumes an efficiency of two in both reference and target genes. Using multivariable linear models for qPCR data analysis does not require direct measurement of amplification efficiency but provides correct significance estimates for differential expression even when amplification is less than two or differs between target and reference genes. We introduce the logic behind using multivariable linear models in the context of qPCR data analysis, the mathematics behind using them, and provide simulations demonstrating that multivariable linear models outperform 2 −ΔΔCT for qPCR data analysis. Molecular Biology qPCR delta-delta CT amplification efficiency Figures Figure 1 Figure 2 Introduction Throughout this report, we will make use of data from one of our recently published studies to illustrate the utility of multivariable linear models for the analysis of qPCR cycle threshold (CT) data. In that study, we exposed primary human airway epithelial cells from five cystic fibrosis (CF) donors to ETI ( e lexacaftor, t ezacaftor and Ivacaftor) or vehicle (DMSO) for 48 hours 1 . ETI is well tolerated and vastly improves lung function in people with cystic fibrosis 2 , 3 . Cycle thresholds were calculated for several target genes ( DEFB1, MMP10, MMP12, IL1B, TNF ) and reference genes ( HSP90AB, GAPDH, HPRT1, GUSB and UBC ). In that paper, we used ANCOVA, a kind of multivariable linear model (MLM) to analyze our qPCR results because ANCOVA was more consistent with the statistical approaches we used to analyze RNA-seq and proteomic results than a classic 2 −ΔΔCT approach 4 would have been. 2 −ΔΔCT remains highly popular 5 despite well documented technical limitations 6 . As will be described in greater detail, the main difference between 2 ΔΔCT and more robust qPCR data analysis approaches is that the latter incorporate amplification efficiency estimates of target and references genes into their calculations whereas 2 −ΔΔCT assumes that amplification efficiency of both genes is 2. Our simulations confirm that using analysis of covariance (ANCOVA) for qPCR data analysis represents a sensible compromise between the added effort of measuring amplification efficiency, e.g., by running a sample dilution curve, and legitimate concerns that failure to address amplification efficiency might compromise experimental results. qPCR remains popular but following data analysis best practices is not Papers mentioning qPCR in their methods section have grown steadily throughout the 21st century (Fig. 1 A) as have citations of the original 2 −ΔΔCT paper 4 but citations of the highly recommended Pfaffl method that incorporates amplification efficiency into its calculations 6 have not (Fig. 1 B). Our survey of 20 recent publications in PubMed Central containing “qPCR” as a methods keyword (Table 1 ) replicates findings reported by Bustin et al. 5 suggesting that about 75% of published qPCR results use the 2 −ΔΔCT method and fewer than 5% explicitly take amplification efficiency into account. Table 1 Survey of 20 recent papers mentioning qPCR in the text of Methods section, including PubMed Central ID, Method Ref (whether the paper cited a method for qPCR data analysis), Norm Type (type of normalization used, if stated), Reference Approach (reference genes used, if stated) and in the last column, whether amplification efficiency was used 5 , 8 . ID Method Ref Norm Type Reference Approach Efficiency Reported PMC11645144 No 2 −ΔΔCT No PMC11567002 No 2 −ΔΔCT GAPDH, ACTB No PMC11567001 No GAPDH , ACTB No PMC11566997 No No PMC11559625 No Standard Curve Mean of 2 genes No PMC11554307 No ACTB No PMC11554301 No Standard Curve ACTB No PMC11546848 No rp49 No PMC11542633 No 2 −ΔΔCT S7 rRNA No PMC11540302 No No PMC11540031 No ACT7 No PMC11566318 Yes Pfaffl ACTB No PMC11527450 No 2 −ΔΔCT ACTB No PMC11527446 No 2 −ΔΔCT B2M No PMC11537643 No 2 −ΔΔCT GAPDH No PMC11537631 No 2 −ΔΔCT GAPDH No PMC11529709 Yes QBase 7 No PMC11567019 No GAPDH No PMC11529476 No GAPDH No PMC11554380 Yes 2 −ΔΔCT GAPDH No The disconnect between what the qPCR community has been advised to do and what the community actually does is significant, but we think the simplicity and statistical efficiency of using multivariable linear models (MLM) for qPCR make it an appealing alternative to 2 −ΔΔCT . ANCOVA models of qPCR compared to 2 −ΔΔCT In this section, we compare the commonly used 2 −ΔΔCT approach to a multivariable linear model like ANCOVA for qPCR data analysis. For our data set both approaches yielded comparable results but the mathematical properties of ANCOVA may make it superior in general. To understand why, we will explore how each approach works. The mathematical basis for using ANCOVA in this context of qPCR can be found in the Appendix. 2 −ΔΔCT uses two levels of control, a control for the treatment, e.g. treatment vs control, and a control for the quality of the sample, namely the reference gene. The practice of raising 2 to the difference of cycle thresholds (CT) reflects that during each round of amplification in qPCR, the amount of material should roughly double. Raw CT values are therefore naturally on a log base 2 scale and suitably distributed for tests that assume normal distribution. 2 −ΔΔCT calculates the average difference between the target gene CT and the reference gene CT in the experimental groups and control groups. Next, the difference in these averages is typically calculated and compared using a t-test or paired t-test. Alternatively, the distribution of the target and reference gene differences can be compared between the exposed and control samples using a rank-based approach. Users may choose to “back-transform” the average difference of the differences by raising 2 to the difference in CT. In that case, because CT is inversely related to the amount of starting material, an intuitively meaningful back-transformation incorporates a minus sign in the exponent: 2 −ΔΔCT . However, remaining on the log scale throughout can avoid possible statistical problems involved with back transformation 9 . The reasoning underlying 2 −ΔΔCT is difference in differences; a ubiquitous study design in economics, social sciences and epidemiology 10 . For example, in economics it is common to evaluate the effect of a policy, such as introduction of a new therapy into Medicare or Medicaid, by comparing results between states that had the policy change and states that did not have the policy change, before and after the introduction of the policy. An example of difference-in-differences in clinical research is the randomization of patients into two groups, a treatment group and control group, with measurements of the dependent variable at baseline and post-baseline. In most difference-in-difference designs the dependent variable is the same, for instance, one might use mortality rate as the dependent variable in different states, before and after a policy change. 2 −ΔΔCT does not use the same dependent variable: CT for the target gene and reference gene are independent. Consequently, these two measures may have little or no correlation, and may have vastly different variation. Mathematically, 2 −ΔΔCT assumes that sample quality affects the value of the target gene and the reference gene by the same amount. However, it is possible, and even likely, that sample quality and other factors such as primer design and cycling conditions affect reference and target genes in different amounts. For instance, it may be that if sample quality and/or primer design and cycling conditions are impacting the reference gene by x, they impact the target gene by the amount k*x, where k is some number. 2 −ΔΔCT assumes that k = 1. A value of zero would indicate the target gene has no ability to inform on the quality of the sample whereas a statistically significant negative value would be hard to interpret biologically and would suggest an error. One advantage of multivariable linear models (e.g., ANCOVA) is that they control for variation due to sample quality and cycling conditions to the extent that the reference gene reflects that variability. If the reference gene is not capturing sample quality and cycling conditions, a multivariable linear model will essentially ignore it. The ability of the reference gene to capture variation in sample quality and cycling conditions can also be assessed using the related method of Pearson or Spearman (rank based) correlations applied to the reference and target genes as presented in Fig. 2 B and 2 D. If there is no correlation between the target and reference genes, the 2 −ΔΔCT method is dubious, and subtracting the reference gene CT from the target gene CT actually reduces the power of the study. Coefficients different from k = 1 can also account for amplification efficiency differences between target and reference genes, making ANCOVA significantly more robust than 2 −ΔΔCT when amplification efficiency differences are an issue. In summary, ANCOVA models of qPCR use a reference to control for differences in sample quality. However, instead of simply subtracting reference values from the gene of interest values, ANCOVA uses regression to establish the level of correction to apply. Applying ANCOVA to qPCR data requires fewer steps than using 2 −ΔΔCT to normalize results and perform a statistical test: ANCOVA uses a reference to account for sample quality variability and assesses the significance in one step. For example, here are GAPDH and MMP10 CT values from our recent study 1 (Table 2 ). Table 2 Sample annotations (Donor, Treatment, Group) and CT values for GAPDH and MMP10 from Fig. 2 . Donor Treatment Group GAPDH MMP10 KK22F DMSO KK22F DMSO 20.17 24.50 KK22F ETI KK22F ETI 20.55 25.96 KK32G DMSO KK32G DMSO 19.86 23.26 KK32G ETI KK32G ETI 19.66 23.92 KK27H DMSO KK27H DMSO 19.62 21.82 KK27H ETI KK27H ETI 19.72 22.44 KK29H DMSO KK29H DMSO 21.29 27.19 KK29H ETI KK29H ETI 21.01 28.37 KK18G DMSO KK18G DMSO 20.06 23.87 KK18G ETI KK18G ETI 20.49 24.82 An ANCOVA analysis of the data in Table 2 requires just two lines of R. The first line says: “Use the data in Table 2 .data to estimate the impact of Donor, Treatment and GAPDH on MMP10 and store the result in a variable called fit.” The second line displays the result. fit <- lm(MMP10 ~ Donor + Treatment + GAPDH, data = Table 2 .data) summary(fit) Key estimates and p-values generated by this ANCOVA model are shown in Table 3 . The second to last row includes the term “Treatment-ETI.” The name of the term includes a factor name “Treatment” and the name of the level “ETI.” The estimate for Treatment ETI is an increase of about 1 CT (0.944) compared to Treatment DMSO. This squares with Fig. 2 A: exposure to ETI raised CT by roughly 1 unit in each donor, and it was significant (p = 0.013). Table 3 ANCOVA analysis of the data in Table 2 . MMP10 CT were predicted as a function of Donor, Treatment and GAPDH . ETI treatment significantly raises CT by 0.944 (p = 0.013), as shown in the second to last row. Term Estimate p_value Donor KK22F 0.859 0.054 Donor KK27H -1.998 0.021 Donor KK29H 3.127 0.013 Donor KK32G -0.574 0.256 Treatment-ETI 0.944 0.013 GAPDH 0.352 0.595 The last row in Table 3 shows that GAPDH is not explaining a statistically significant amount of variation in the target gene (CT = 0.352, p = 0.595), but the estimate has the same (positive) sign as the treatment effect, which is what we expect. It is easy to apply ANCOVA analysis to data structured like Table 2 . For example, the following R command estimates the impact and significance of Donor, Treatment and HSP90AB on DEFFB1 CT values using data from a table called All.CT: lm(DEFB1 ~ Donor + Treatment + HSP90AB, data = All.CT) This command produced the estimates and significance shown in Table 3 : Table 3 ANCOVA analysis of the data in ALL.CT. DEFB1 CT were predicted as a function of Donor, Treatment and HSP90AB . ETI Treatment significantly raises CT by -0.803 (p = 0.031), as shown in the second to last row. Term Estimate p_value DonorKK22F -1.150 0.063 DonorKK27H -1.973 0.024 DonorKK29H -2.709 0.008 DonorKK32G -2.225 0.017 Treatment-ETI -0.803 0.031 HSP90AB 0.011 0.984 Table 3 shows that ETI significantly reduces DEFB1 CT consistent with Fig. 2 C, and that HSP90AB does not itself respond to ETI (estimate = 0.011 CT, p = 0.984). Figure 2 D shows that HSP90AB and DEFB1 are uncorrelated. This is an example of a situation where simply subtracting CT values of the reference gene from the gene of interest, which is what 2 −ΔΔCT would do, would yield inferior results compared to ANCOVA. ANCOVA handles reaction efficiency differences between target and reference gene The ANCOVA multivariable linear model offers another advantage over the 2 −ΔΔCT approach. Unlike 2 −ΔΔCT it is invariant to any difference in reaction efficiency between the target and reference genes. Let \(\:{e}_{T}\) (a number between 0 or 1, i.e. 0–100%) be the reaction efficiency of the target gene. In general, we should be using the logarithm with base \(\:1+{e}_{T}\) . Using the change of base property of logarithms, \(\:{\text{log}}_{b}c={\text{log}}_{a}c\text{*}\:{\text{log}}_{b}a,\) the cycle thresholds calculation is off by a factor \(\:{\text{log}}_{2}1+{e}_{T}\) . Similarly, if \(\:{e}_{R}\) is the reaction efficiency of the reference gene its cycle thresholds will be off by the factor \(\:{\text{log}}_{2}1+{e}_{R}\) . Fortunately, it is a property of multivariable linear models that the p-values are invariant to scale changes. That is, any difference in reaction efficiency (i.e., cycling conditions) between the target and reference gene will not change the p-values from the multivariable linear model we propose. Simulations validate multivariable models like ANCOVA for qPCR Although our published paper successfully applied a MLM to qPCR data analysis 1 , we ran simulations to establish that MLM would give unbiased results for typical qPCR applications with small sample sizes whose distributions may differ somewhat from the normal distributions linear models assume. As detailed below, simulations suggest that ANCOVA works well for qPCR and may be more powerful in practice than 2 −ΔΔCT . Simulations were also used to assess the impact of correlation between target and reference genes (Table 4 ) and what happens when the reference gene responds to treatment effects (Table 5 ). Rules of thumb for the sample size necessary to conduct a linear regression, e.g. ANCOVA, range from at least two 11 to up to eight observations 12 for every paramter estimated (e.g. variable) in your model. If this ratio of sample size to parameters estimated is too small then the p-values reported are less reliable. This is becuse the p-values from linear regression, like the p-values from two sample t-tests, assume that the underlying distribution is Gaussian (e.g. bell shaped). The p-values reported may be biased toward accepting or rejecting the null hypothesis when the underlying distribution is not Gaussian. Ideally, one should reject the null hypothesis about 5% of the time when the null hypothesis is true, e.g., when there is no true difference between experimental groups. Rejecting the null hypothesis less than 5% of the time when there is no true difference suggests that a test may too conservative. Tests that are too conservative, reduce power and inflate Type II errors. The flip side, rejecting the null hypothesis more than 5% of the time when there is no true difference suggests that a test is too liberal and inflates Type I errors. In our application there are three degrees of freedom, one for the treatment, one for the reference gene and one for the intercept, as well as one for the random intercept we used. This is four degrees of freedom (parameters to estimate) whereas our sample size is ten: just enough, because ten is greater than two times four. We assessed: (i) any bias in the rejection rate of the multivariable linear model (MLM) in comparison to 2 ΔΔCT , as well as ΔCT (which does not use a reference gene), and (ii) the power of MLM in comparison to 2 ΔΔCT and ΔCT. We evaluated the performance of the MLM when the underlying distribution is right skewed and left skewed. We consider the scenario when there are a total of 10 observations (e.g. 10 donors) in which 5 are treated and 5 are controls. To compare how MLM and 2 −ΔΔCT perform in terms of the ability of the reference gene to capture sample quality, we evaluated performance with respect to the correlation of the target and reference gene; specifically, we considered a very strong Pearson correlation of 0.9, a moderate correlation of 0.5 and zero correlation. The code used in the simulation is included in GitHub ( https://github.com/DartCF/MLM-for-qPCR ). As shown in Table 4 , the empirical type I error rate of the MLM approach is close to the nominal value (5%) when the distribution is not Gaussian (right skewed, left skewed) for each of the three correlation parameters (0, 0.5 and 0.9). This indicates that the multivariable approach is a statistically valid approach (e.g. little or no bias in Type I error). MLM showed slight bias in the conservative direction, that is, empirical Type I error rate was a bit less than 5%. 2 −ΔΔCT was more conservative in every scenario, in other words, less powerful than MLM. Table 4 illustrates the ability of the MLM to control for the reference gene to the extent that it is relevant. When there is very little shared variation in sample quality or cycling conditions between the target and reference genes, their correlation is close zero, reducing the power of 2 -ΔΔCT in comparison to the best approach in that case, ΔCT (not using a reference gene at all). However, MLM has almost the power of ΔCT, even when the distribution is Gaussian or right skewed or left skewed. When there is a high correlation (e.g., 0.9), 2 -ΔΔCT outperforms ΔCT (100% vs 78.8% for Gaussian) but MLM is just as good (100%). When there is moderate correlation of the target and reference gene, MLM outperforms both 2 -ΔΔCT and ΔCT. Table 4 (Left) Empirical Type I Error Rate for each model (MLM, 2 -ΔΔCT , ΔCT). Correlation between target and reference gene, distribution type (Gaussian, Right Skewed, Left Skewed). (Right) Power for each model shown at left. Empirical Type I Error Rate Power Correlation Distribution MLM 2 −ΔΔCT ΔCT MLM 2 −ΔΔCT ΔCT 0 Gaussian 5.0% 4.5% 4.2% 71.5% 47.6% 76.4% 0.5 Gaussian 4.8% 4.1% 4.2% 92.7% 85.5% 85.7% 0.9 Gaussian 5.0% 4.7% 4.8% 100.0% 100.0% 78.8% 0 Right Skewed 4.4% 3.7% 2.8% 76.1% 52.9% 78.3% 0.5 Right Skewed 4.2% 3.4% 4.0% 91.9% 84.0% 84.9% 0.9 Right Skewed 4.7% 3.9% 4.4% 99.9% 99.9% 79.4% 0 Left Skewed 4.2% 3.5% 2.8% 75.3% 52.8% 78.1% 0.5 Left Skewed 4.6% 3.7% 4.2% 91.8% 84.2% 85.5% 0.9 Left Skewed 4.3% 3.5% 4.2% 99.9% 99.9% 78.8% Table 5 shows the results of simulations when the treatment has an effect on the reference gene equal to one standard deviation. This is a big problem. In this setting, only ΔCT (ignoring the reference gene) is a valid approach. 2 -ΔΔCT is invalid when the exposure affects the reference gene, because it will identify an effect on the target gene in the opposite direction. For instance, if the correlation of the target and reference gene is high, the 2 -ΔΔCT falsely rejects the null hypothesis with a frequency of 93%. MLM is also invalid in this setting unless there is zero correlation between target and reference gene. Table 5 (Left) Empirical Type I Error Rate for each model (MLM, 2 -ΔΔCT , ΔCT). Correlation between target and reference gene, distribution type (Gaussian, Right Skewed, Left Skewed) (Right) Power for each model shown at left. Empirical Type 1 Error Rate Power Correlation Distribution MLM 2 −ΔΔCT ΔCT MLM 2 −ΔΔCT ΔCT 0 Gaussian 5.0% 14.9% 4.8% 63.2% 15.6% 77.1% 0.5 Gaussian 12.7% 32.9% 4.1% 63.6% 32.9% 85.6% 0.9 Gaussian 75.5% 93.2% 4.4% 88.2% 93.0% 78.4% 0 Right Skewed 4.5% 18.2% 2.8% 65.1% 18.1% 78.5% 0.5 Right Skewed 17.6% 37.9% 4.2% 65.0% 37.7% 85.9% 0.9 Right Skewed 77.0% 91.0% 4.3% 88.2% 89.6% 78.9% 0 Left Skewed 4.9% 17.8% 2.7% 65.9% 17.7% 79.0% 0.5 Left Skewed 17.2% 37.6% 4.2% 65.2% 37.9% 85.8% 0.9 Left Skewed 77.1% 90.7% 4.6% 88.7% 90.8% 78.7% Limitations Both ANCOVA and 2 −ΔΔCT assume reference gene stability. Although ANCOVA models offer advantages over 2 −ΔΔCT , including offering greater statistical power and handling variable amplification efficiency better than 2 −ΔΔCT , effect estimates provided by ANCOVA models are in CT units, not fold change. CT units are only equal to 1 log2 unit of change in the special case where amplification efficiency is 100%. Discussion qPCR remains a foundational technique in molecular biology, with about 5,000 new citations every month (Fig. 1 A). Most practitioners control for possible variability using a single, well-known reference gene such as GAPDH without explaining how this gene was chosen or providing evidence of its stability 5 . This practice could obscure true treatment effects 13 , degrading reproducibility. Most researchers assess qPCR differential gene expression using the 2 −ΔΔCT method, and therefore tacitly assume perfect amplification efficiency, potentially distorting findings 6 , but it is impossible to know because researchers rarely share raw data 14 . Solutions to many reproducibility problems have been proposed and reviewed 15 but largely ignored 5 perhaps because validating reference genes, performing dilution curves to measure amplification efficiency and adhering to data standards is viewed as a low priority by study authors and peer reviewers. If the precise magnitude of fold change were important to the research question, running a sample dilution curve could establish efficiency, and those estimates could be used to create efficiency-weighted CT values suitable for downstream statistical analysis 16 . Often, what matters most is whether a treatment significantly affects gene expression in the statistical sense (p < .05) and our simulations show that multivariable linear models like ANCOVA are superior to 2 −ΔΔCT . Using ANCOVA for qPCR data analysis requires very little coding and provides p-values that are invariant to amplification efficiency differences between the reference and gene of interest. Moreover, because ANCOVA can be more sensitive than 2 ΔΔCT , ANCOVA will sometimes detect experimental differences with fewer samples, saving time and resources. Multivariable linear models naturally accommodate the more complex experimental designs that are increasingly common in science. Based on simulations, ANCOVA offers a lower false negative rate than 2 ΔΔCT , especially when the reference gene does not provide useful information about the gene of interest. Declarations Code Availability The code used in the simulation is included in GitHub (https://github.com/DartCF/MLM-for-qPCR). Acknowledgements The research of KKF, LT, TH and BAS is supported by the National Institutes of Health (R01 HL151385 and P30 DK117469), the Cystic Fibrosis Foundation (STANTO19R0, STANTO23GO), and the Flatley Foundation. The research of TM is supported by National Institutes of Health P30 DK117469. Author Contributions LT, KKF, TH and TM conceived of the project and contributed to the writing of the manuscript. BAS provided advice, support and contributed to the writing of the manuscript. All authors have read and agreed to the published version of the manuscript. Declaration of Interests The authors have no conflicts of interest. 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Multivar Behav Res 26:499–510. https://doi.org/10.1207/s15327906mbr2603_7 Bustin S, Nolan T (2017) Talking the talk, but not walking the walk: RT-qPCR as a paradigm for the lack of reproducibility in molecular research. Eur J Clin Invest 47:756–774. https://doi.org/10.1111/eci.12801 Untergasser A, Hellemans J, Pfaffl MW, Ruijter JM, van den Hoff MJB, Dragomir MP, Adamoski D, Dias SMG, Reis RM, Ferracin M et al (2023) Disclosing quantitative RT-PCR raw data during manuscript submission: a call for action. Mol Oncol 17:713–717. https://doi.org/10.1002/1878-0261.13418 Taylor SC, Nadeau K, Abbasi M, Lachance C, Nguyen M, Fenrich J (2019) The Ultimate qPCR Experiment: Producing Publication Quality, Reproducible Data the First Time. Trends Biotechnol 37:761–774. https://doi.org/10.1016/j.tibtech.2018.12.002 Yuan JS, Wang D, Stewart CN Jr. (2008) Statistical methods for efficiency adjusted real-time PCR quantification. Biotechnol J 3:112–123. https://doi.org/10.1002/biot.200700169 Additional Declarations The authors declare no competing interests. Supplementary Files Appendix.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6014969","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Method Article","associatedPublications":[],"authors":[{"id":414755250,"identity":"65adbb5d-a054-4ac2-a682-bebadeaad552","order_by":0,"name":"Thomas H.Hampton","email":"data:image/png;base64,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","orcid":"","institution":"Geisel School of Medicine at Dartmouth","correspondingAuthor":true,"prefix":"","firstName":"Thomas","middleName":"","lastName":"H.Hampton","suffix":""},{"id":414755538,"identity":"682a40cb-9fca-46fd-b6f9-63b12e06ff75","order_by":1,"name":"Lily Taub","email":"","orcid":"","institution":"Geisel School of Medicine at Dartmouth","correspondingAuthor":false,"prefix":"","firstName":"Lily","middleName":"","lastName":"Taub","suffix":""},{"id":414755650,"identity":"d654d6ba-e604-4109-949c-2be544175684","order_by":2,"name":"Kiyoshi Ferreria-Fukutani","email":"","orcid":"","institution":"Geisel School of Medicine at Dartmouth","correspondingAuthor":false,"prefix":"","firstName":"Kiyoshi","middleName":"","lastName":"Ferreria-Fukutani","suffix":""},{"id":414756004,"identity":"ffe5dece-0786-48a5-8e3e-e4d291dc88dd","order_by":3,"name":"Bruce A. Stanton","email":"","orcid":"","institution":"Geisel School of Medicine at Dartmouth","correspondingAuthor":false,"prefix":"","firstName":"Bruce","middleName":"A.","lastName":"Stanton","suffix":""},{"id":414756383,"identity":"c921e4be-a1ab-47b0-aeb3-a4204a6d52e6","order_by":4,"name":"Todd A. MacKenzie","email":"","orcid":"","institution":"Geisel School of Medicine at Dartmouth","correspondingAuthor":false,"prefix":"","firstName":"Todd","middleName":"A.","lastName":"MacKenzie","suffix":""}],"badges":[],"createdAt":"2025-02-12 11:59:40","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-6014969/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6014969/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":76223921,"identity":"1e166230-7297-425f-9a12-8769f47b6ac8","added_by":"auto","created_at":"2025-02-13 16:08:18","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":31347,"visible":true,"origin":"","legend":"\u003cp\u003eA) Number of PubMed Central publications mentioning “qPCR” in the Methods section since 2000. B) PubMed Central citations since 2000 for the original 2\u003csup\u003e-ΔΔCT\u003c/sup\u003e paper by Livak and Schmittgen\u003csup\u003e4\u003c/sup\u003e and the seminal Pfaffl paper advocating for correcting CT values for amplification efficiency\u003csup\u003e6\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6014969/v1/633498666cba7acae19c20b9.png"},{"id":76223933,"identity":"974654e9-7c81-4aa2-8c47-cd42fbfdf836","added_by":"auto","created_at":"2025-02-13 16:08:18","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":76383,"visible":true,"origin":"","legend":"\u003cp\u003eA) Mean CT values for triplicate measurements \u003cem\u003eGAPDH \u003c/em\u003e(circles) and \u003cem\u003eMMP10\u003c/em\u003e (triangles) in samples from 5 donors (colors) exposed to DMSO or ETI. Colored lines connect samples from the same donor. Black symbols and dashed lines represent donor averages for each gene and treatment. B) \u003cem\u003eMMP10\u003c/em\u003e CT values as a function of \u003cem\u003eGAPDH\u003c/em\u003e CT value for each of the 10 observations in Figure 2A. C) Mean CT values for triplicate measurements \u003cem\u003eHSP90AB (\u003c/em\u003ecircles) and \u003cem\u003eDEFB1\u003c/em\u003e (triangles) in samples from 5 donors (colors) exposed to DMSO or ETI. Colored lines connect samples from the same donor. Black symbols and dashed lines represent donor averages for each gene and treatment. D) \u003cem\u003eDEFB1\u003c/em\u003e CT values as a function of \u003cem\u003eHSP90AB\u003c/em\u003eCT value for each of the 10 observations in Figure 2C.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6014969/v1/9d785694a45b31688bcafe35.png"},{"id":76224271,"identity":"1c49bf1f-0782-48f0-a22d-d332c7ed096c","added_by":"auto","created_at":"2025-02-13 16:08:46","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1049278,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6014969/v1/be6f62a4-a974-489d-9c63-39bf39fd0ef2.pdf"},{"id":76223919,"identity":"68d4e923-2160-43c6-8158-59e3421781fe","added_by":"auto","created_at":"2025-02-13 16:08:18","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":16430,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-6014969/v1/2ff9363b931cabb57c40d069.docx"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eMultivariable Linear Models Outperform 2\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e-ΔΔCT\u003c/strong\u003e\u003c/sup\u003e\u003cstrong\u003e for qPCR Data Analysis\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThroughout this report, we will make use of data from one of our recently published studies to illustrate the utility of multivariable linear models for the analysis of qPCR cycle threshold (CT) data. In that study, we exposed primary human airway epithelial cells from five cystic fibrosis (CF) donors to ETI (\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003ee\u003c/span\u003elexacaftor, \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003et\u003c/span\u003eezacaftor and Ivacaftor) or vehicle (DMSO) for 48 hours\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e. ETI is well tolerated and vastly improves lung function in people with cystic fibrosis\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e,\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e. Cycle thresholds were calculated for several target genes (\u003cem\u003eDEFB1, MMP10, MMP12, IL1B, TNF\u003c/em\u003e) and reference genes (\u003cem\u003eHSP90AB, GAPDH, HPRT1, GUSB\u003c/em\u003e and \u003cem\u003eUBC\u003c/em\u003e). In that paper, we used ANCOVA, a kind of multivariable linear model (MLM) to analyze our qPCR results because ANCOVA was more consistent with the statistical approaches we used to analyze RNA-seq and proteomic results than a classic 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e approach\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e would have been. 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e remains highly popular\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e despite well documented technical limitations\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. As will be described in greater detail, the main difference between 2\u003csup\u003eΔΔCT\u003c/sup\u003e and more robust qPCR data analysis approaches is that the latter incorporate amplification efficiency estimates of target and references genes into their calculations whereas 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e assumes that amplification efficiency of both genes is 2. Our simulations confirm that using analysis of covariance (ANCOVA) for qPCR data analysis represents a sensible compromise between the added effort of measuring amplification efficiency, e.g., by running a sample dilution curve, and legitimate concerns that failure to address amplification efficiency might compromise experimental results.\u003c/p\u003e\n\u003ch3\u003eqPCR remains popular but following data analysis best practices is not\u003c/h3\u003e\n\u003cp\u003ePapers mentioning qPCR in their methods section have grown steadily throughout the 21st century (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA) as have citations of the original 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e paper\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e but citations of the highly recommended Pfaffl method that incorporates amplification efficiency into its calculations\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e have not (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB). Our survey of 20 recent publications in PubMed Central containing \u0026ldquo;qPCR\u0026rdquo; as a methods keyword (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) replicates findings reported by Bustin et al.\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e suggesting that about 75% of published qPCR results use the 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e method and fewer than 5% explicitly take amplification efficiency into account.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSurvey of 20 recent papers mentioning qPCR in the text of Methods section, including PubMed Central ID, Method Ref (whether the paper cited a method for qPCR data analysis), Norm Type (type of normalization used, if stated), Reference Approach (reference genes used, if stated) and in the last column, whether amplification efficiency was used\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e,\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e ID\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMethod Ref\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNorm Type\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eReference Approach\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eEfficiency Reported\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11645144\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11567002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH, ACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11567001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e, \u003cem\u003eACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11566997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11559625\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eStandard Curve\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMean of 2 genes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11554307\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11554301\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eStandard Curve\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11546848\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003erp49\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11542633\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eS7 rRNA\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11540302\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11540031\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eACT7\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11566318\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ePfaffl\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11527450\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eACTB\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11527446\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eB2M\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11537643\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11537631\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11529709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eQBase\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11567019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11529476\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePMC11554380\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eYes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNo\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe disconnect between what the qPCR community has been advised to do and what the community actually does is significant, but we think the simplicity and statistical efficiency of using multivariable linear models (MLM) for qPCR make it an appealing alternative to 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eANCOVA models of qPCR compared to 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e\u003c/h2\u003e \u003cp\u003eIn this section, we compare the commonly used 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e approach to a multivariable linear model like ANCOVA for qPCR data analysis. For our data set both approaches yielded comparable results but the mathematical properties of ANCOVA may make it superior in general. To understand why, we will explore how each approach works. The mathematical basis for using ANCOVA in this context of qPCR can be found in the Appendix.\u003c/p\u003e \u003cp\u003e2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e uses two levels of control, a control for the treatment, e.g. treatment vs control, and a control for the quality of the sample, namely the reference gene. The practice of raising 2 to the difference of cycle thresholds (CT) reflects that during each round of amplification in qPCR, the amount of material should roughly double. Raw CT values are therefore naturally on a log base 2 scale and suitably distributed for tests that assume normal distribution. 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e calculates the average difference between the target gene CT and the reference gene CT in the experimental groups and control groups. Next, the difference in these averages is typically calculated and compared using a t-test or paired t-test. Alternatively, the distribution of the target and reference gene differences can be compared between the exposed and control samples using a rank-based approach. Users may choose to \u0026ldquo;back-transform\u0026rdquo; the average difference of the differences by raising 2 to the difference in CT. In that case, because CT is inversely related to the amount of starting material, an intuitively meaningful back-transformation incorporates a minus sign in the exponent: 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e. However, remaining on the log scale throughout can avoid possible statistical problems involved with back transformation\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe reasoning underlying 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e is \u003cem\u003edifference in differences;\u003c/em\u003e a ubiquitous study design in economics, social sciences and epidemiology\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. For example, in economics it is common to evaluate the effect of a policy, such as introduction of a new therapy into Medicare or Medicaid, by comparing results between states that had the policy change and states that did not have the policy change, before and after the introduction of the policy. An example of difference-in-differences in clinical research is the randomization of patients into two groups, a treatment group and control group, with measurements of the dependent variable at baseline and post-baseline.\u003c/p\u003e \u003cp\u003eIn most difference-in-difference designs the dependent variable is the same, for instance, one might use mortality rate as the dependent variable in different states, before and after a policy change. 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e does not use the same dependent variable: CT for the target gene and reference gene are independent. Consequently, these two measures may have little or no correlation, and may have vastly different variation. Mathematically, 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e assumes that sample quality affects the value of the target gene and the reference gene by the same amount. However, it is possible, and even likely, that sample quality and other factors such as primer design and cycling conditions affect reference and target genes in different amounts. For instance, it may be that if sample quality and/or primer design and cycling conditions are impacting the reference gene by x, they impact the target gene by the amount k*x, where k is some number. 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e assumes that k\u0026thinsp;=\u0026thinsp;1. A value of zero would indicate the target gene has no ability to inform on the quality of the sample whereas a statistically significant negative value would be hard to interpret biologically and would suggest an error. One advantage of multivariable linear models (e.g., ANCOVA) is that they control for variation due to sample quality and cycling conditions to the extent that the reference gene reflects that variability. If the reference gene is not capturing sample quality and cycling conditions, a multivariable linear model will essentially ignore it. The ability of the reference gene to capture variation in sample quality and cycling conditions can also be assessed using the related method of Pearson or Spearman (rank based) correlations applied to the reference and target genes as presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eB and \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eD. If there is no correlation between the target and reference genes, the 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e method is dubious, and subtracting the reference gene CT from the target gene CT actually reduces the power of the study. Coefficients different from k\u0026thinsp;=\u0026thinsp;1 can also account for amplification efficiency differences between target and reference genes, making ANCOVA significantly more robust than 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e when amplification efficiency differences are an issue.\u003c/p\u003e \u003cp\u003eIn summary, ANCOVA models of qPCR use a reference to control for differences in sample quality. However, instead of simply subtracting reference values from the gene of interest values, ANCOVA uses regression to establish the level of correction to apply. Applying ANCOVA to qPCR data requires fewer steps than using 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e to normalize results and perform a statistical test: ANCOVA uses a reference to account for sample quality variability and assesses the significance in one step.\u003c/p\u003e \u003cp\u003eFor example, here are \u003cem\u003eGAPDH\u003c/em\u003e and \u003cem\u003eMMP10\u003c/em\u003e CT values from our recent study\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSample annotations (Donor, Treatment, Group) and CT values for \u003cem\u003eGAPDH\u003c/em\u003e and \u003cem\u003eMMP10\u003c/em\u003e from Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonor\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTreatment\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGroup\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eMMP10\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK22F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK22F DMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e24.50\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK22F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK22F ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e25.96\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK32G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK32G DMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e23.26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK32G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK32G ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e23.92\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK27H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK27H DMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e21.82\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK27H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK27H ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22.44\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK29H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK29H DMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27.19\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK29H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK29H ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e28.37\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK18G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK18G DMSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e23.87\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKK18G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKK18G ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e24.82\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAn ANCOVA analysis of the data in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e requires just two lines of R. The first line says: \u0026ldquo;Use the data in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.data to estimate the impact of Donor, Treatment and \u003cem\u003eGAPDH\u003c/em\u003e on \u003cem\u003eMMP10\u003c/em\u003e and store the result in a variable called fit.\u0026rdquo; The second line displays the result.\u003c/p\u003e \u003cp\u003efit \u0026lt;- lm(MMP10\u0026thinsp;~\u0026thinsp;Donor\u0026thinsp;+\u0026thinsp;Treatment\u0026thinsp;+\u0026thinsp;GAPDH, data\u0026thinsp;=\u0026thinsp;Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.data)\u003c/p\u003e \u003cp\u003esummary(fit)\u003c/p\u003e \u003cp\u003eKey estimates and p-values generated by this ANCOVA model are shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The second to last row includes the term \u0026ldquo;Treatment-ETI.\u0026rdquo; The name of the term includes a factor name \u0026ldquo;Treatment\u0026rdquo; and the name of the level \u0026ldquo;ETI.\u0026rdquo; The estimate for Treatment ETI is an increase of about 1 CT (0.944) compared to Treatment DMSO. This squares with Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eA: exposure to ETI raised CT by roughly 1 unit in each donor, and it was significant (p\u0026thinsp;=\u0026thinsp;0.013).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eANCOVA analysis of the data in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. \u003cem\u003eMMP10\u003c/em\u003e CT were predicted as a function of Donor, Treatment and \u003cem\u003eGAPDH\u003c/em\u003e. ETI treatment significantly raises CT by 0.944 (p\u0026thinsp;=\u0026thinsp;0.013), as shown in the second to last row.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTerm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEstimate\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ep_value\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonor KK22F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.859\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.054\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonor KK27H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-1.998\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.021\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonor KK29H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.127\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonor KK32G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.574\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.256\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTreatment-ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.944\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eGAPDH\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.352\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.595\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe last row in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows that \u003cem\u003eGAPDH\u003c/em\u003e is not explaining a statistically significant amount of variation in the target gene (CT\u0026thinsp;=\u0026thinsp;0.352, p\u0026thinsp;=\u0026thinsp;0.595), but the estimate has the same (positive) sign as the treatment effect, which is what we expect.\u003c/p\u003e \u003cp\u003eIt is easy to apply ANCOVA analysis to data structured like Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. For example, the following R command estimates the impact and significance of Donor, Treatment and \u003cem\u003eHSP90AB\u003c/em\u003e on \u003cem\u003eDEFFB1\u003c/em\u003e CT values using data from a table called All.CT:\u003c/p\u003e \u003cp\u003elm(DEFB1\u0026thinsp;~\u0026thinsp;Donor\u0026thinsp;+\u0026thinsp;Treatment\u0026thinsp;+\u0026thinsp;HSP90AB, data\u0026thinsp;=\u0026thinsp;All.CT)\u003c/p\u003e \u003cp\u003eThis command produced the estimates and significance shown in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e3\u003c/span\u003e:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eANCOVA analysis of the data in ALL.CT. \u003cem\u003eDEFB1\u003c/em\u003e CT were predicted as a function of Donor, Treatment and \u003cem\u003eHSP90AB\u003c/em\u003e. ETI Treatment significantly raises CT by -0.803 (p\u0026thinsp;=\u0026thinsp;0.031), as shown in the second to last row.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTerm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEstimate\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ep_value\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonorKK22F\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-1.150\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.063\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonorKK27H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-1.973\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.024\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonorKK29H\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-2.709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.008\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDonorKK32G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-2.225\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.017\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTreatment-ETI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.803\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.031\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHSP90AB\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.011\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.984\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows that ETI significantly reduces \u003cem\u003eDEFB1\u003c/em\u003e CT consistent with Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eC, and that \u003cem\u003eHSP90AB\u003c/em\u003e does not itself respond to ETI (estimate\u0026thinsp;=\u0026thinsp;0.011 CT, p\u0026thinsp;=\u0026thinsp;0.984). Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eD shows that \u003cem\u003eHSP90AB\u003c/em\u003e and \u003cem\u003eDEFB1\u003c/em\u003e are uncorrelated. This is an example of a situation where simply subtracting CT values of the reference gene from the gene of interest, which is what 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e would do, would yield inferior results compared to ANCOVA.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eANCOVA handles reaction efficiency differences between target and reference gene\u003c/h3\u003e\n\u003cp\u003eThe ANCOVA multivariable linear model offers another advantage over the 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e approach. Unlike 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e it is invariant to any difference in reaction efficiency between the target and reference genes. Let \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}_{T}\\)\u003c/span\u003e\u003c/span\u003e (a number between 0 or 1, i.e. 0\u0026ndash;100%) be the reaction efficiency of the target gene. In general, we should be using the logarithm with base \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1+{e}_{T}\\)\u003c/span\u003e\u003c/span\u003e. Using the change of base property of logarithms, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{log}}_{b}c={\\text{log}}_{a}c\\text{*}\\:{\\text{log}}_{b}a,\\)\u003c/span\u003e\u003c/span\u003e the cycle thresholds calculation is off by a factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{log}}_{2}1+{e}_{T}\\)\u003c/span\u003e\u003c/span\u003e. Similarly, if \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}_{R}\\)\u003c/span\u003e\u003c/span\u003e is the reaction efficiency of the reference gene its cycle thresholds will be off by the factor \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{log}}_{2}1+{e}_{R}\\)\u003c/span\u003e\u003c/span\u003e. Fortunately, it is a property of multivariable linear models that the p-values are invariant to scale changes. That is, any difference in reaction efficiency (i.e., cycling conditions) between the target and reference gene will not change the p-values from the multivariable linear model we propose.\u003c/p\u003e\n\u003ch3\u003eSimulations validate multivariable models like ANCOVA for qPCR\u003c/h3\u003e\n\u003cp\u003eAlthough our published paper successfully applied a MLM to qPCR data analysis\u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e, we ran simulations to establish that MLM would give unbiased results for typical qPCR applications with small sample sizes whose distributions may differ somewhat from the normal distributions linear models assume. As detailed below, simulations suggest that ANCOVA works well for qPCR and may be more powerful in practice than 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e. Simulations were also used to assess the impact of correlation between target and reference genes (Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e4\u003c/span\u003e) and what happens when the reference gene responds to treatment effects (Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eRules of thumb for the sample size necessary to conduct a linear regression, e.g. ANCOVA, range from at least two\u003csup\u003e\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u003c/sup\u003e to up to eight observations\u003csup\u003e\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u003c/sup\u003e for every paramter estimated (e.g. variable) in your model. If this ratio of sample size to parameters estimated is too small then the p-values reported are less reliable. This is becuse the p-values from linear regression, like the p-values from two sample t-tests, assume that the underlying distribution is Gaussian (e.g. bell shaped). The p-values reported may be biased toward accepting or rejecting the null hypothesis when the underlying distribution is not Gaussian. Ideally, one should reject the null hypothesis about 5% of the time when the null hypothesis is true, e.g., when there is no true difference between experimental groups. Rejecting the null hypothesis less than 5% of the time when there is no true difference suggests that a test may too conservative. Tests that are too conservative, reduce power and inflate Type II errors. The flip side, rejecting the null hypothesis more than 5% of the time when there is no true difference suggests that a test is too liberal and inflates Type I errors. In our application there are three degrees of freedom, one for the treatment, one for the reference gene and one for the intercept, as well as one for the random intercept we used. This is four degrees of freedom (parameters to estimate) whereas our sample size is ten: just enough, because ten is greater than two times four.\u003c/p\u003e \u003cp\u003eWe assessed: (i) any bias in the rejection rate of the multivariable linear model (MLM) in comparison to 2\u003csup\u003eΔΔCT\u003c/sup\u003e, as well as ΔCT (which does not use a reference gene), and (ii) the power of MLM in comparison to 2\u003csup\u003eΔΔCT\u003c/sup\u003e and ΔCT. We evaluated the performance of the MLM when the underlying distribution is right skewed and left skewed. We consider the scenario when there are a total of 10 observations (e.g. 10 donors) in which 5 are treated and 5 are controls. To compare how MLM and 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e perform in terms of the ability of the reference gene to capture sample quality, we evaluated performance with respect to the correlation of the target and reference gene; specifically, we considered a very strong Pearson correlation of 0.9, a moderate correlation of 0.5 and zero correlation. The code used in the simulation is included in GitHub (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://github.com/DartCF/MLM-for-qPCR\u003c/span\u003e\u003cspan address=\"https://github.com/DartCF/MLM-for-qPCR\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAs shown in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the empirical type I error rate of the MLM approach is close to the nominal value (5%) when the distribution is not Gaussian (right skewed, left skewed) for each of the three correlation parameters (0, 0.5 and 0.9). This indicates that the multivariable approach is a statistically valid approach (e.g. little or no bias in Type I error). MLM showed slight bias in the conservative direction, that is, empirical Type I error rate was a bit less than 5%. 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e was more conservative in every scenario, in other words, less powerful than MLM.\u003c/p\u003e \n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv class=\"colspec\"\u003eTable 4 illustrates the ability of the MLM to control for the reference gene to the extent that it is relevant. When there is very little shared variation in sample quality or cycling conditions between the target and reference genes, their correlation is close zero, reducing the power of 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e in comparison to the best approach in that case, \u0026Delta;CT (not using a reference gene at all). However, MLM has almost the power of \u0026Delta;CT, even when the distribution is Gaussian or right skewed or left skewed. When there is a high correlation (e.g., 0.9), 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e outperforms \u0026Delta;CT (100% vs 78.8% for Gaussian) but MLM is just as good (100%). When there is moderate correlation of the target and reference gene, MLM outperforms both 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e and \u0026Delta;CT.\u0026nbsp;\u003c/div\u003e\n \u003ctable id=\"Tab5\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cstrong\u003e(Left)\u0026nbsp;\u003c/strong\u003eEmpirical Type I Error Rate for each model (MLM, 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e, \u0026Delta;CT). Correlation between target and reference gene, distribution type (Gaussian, Right Skewed, Left Skewed).\u003cstrong\u003e\u0026nbsp;(Right)\u0026nbsp;\u003c/strong\u003ePower for each model shown at left.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eEmpirical Type I Error Rate\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003ePower\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eCorrelation\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eDistribution\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eMLM\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e2\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e\u0026minus;\u0026Delta;\u0026Delta;CT\u003c/strong\u003e\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026Delta;CT\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eMLM\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e2\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e\u0026minus;\u0026Delta;\u0026Delta;CT\u003c/strong\u003e\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026Delta;CT\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e71.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e47.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e76.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e92.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e76.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e91.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e84.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e84.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e99.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e99.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e79.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e75.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e91.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e84.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e99.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e99.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003eTable 5 shows the results of simulations when the treatment has an effect on the reference gene equal to one standard deviation. This is a big problem. In this setting, only \u0026Delta;CT (ignoring the reference gene) is a valid approach. 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e is invalid when the exposure affects the reference gene, because it will identify an effect on the target gene in the opposite direction. For instance, if the correlation of the target and reference gene is high, the 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e falsely rejects the null hypothesis with a frequency of 93%. MLM is also invalid in this setting unless there is zero correlation between target and reference gene.\u003c/div\u003e\n \u003ctable id=\"Tab6\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003e\u003cstrong\u003e(Left)\u0026nbsp;\u003c/strong\u003eEmpirical Type I Error Rate for each model (MLM, 2\u003csup\u003e-\u0026Delta;\u0026Delta;CT\u003c/sup\u003e, \u0026Delta;CT). Correlation between target and reference gene, distribution type (Gaussian, Right Skewed, Left Skewed)\u003cstrong\u003e\u0026nbsp;(Right)\u0026nbsp;\u003c/strong\u003ePower for each model shown at left.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\"\u003e\u0026nbsp;\u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eEmpirical Type 1 Error Rate\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003ePower\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eCorrelation\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eDistribution\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eMLM\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e2\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e\u0026minus;\u0026Delta;\u0026Delta;CT\u003c/strong\u003e\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026Delta;CT\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eMLM\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e2\u003c/strong\u003e\u003csup\u003e\u003cstrong\u003e\u0026minus;\u0026Delta;\u0026Delta;CT\u003c/strong\u003e\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026Delta;CT\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e14.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e63.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e77.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e12.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e63.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eGaussian\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e75.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e93.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e88.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e93.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.4%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e18.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e18.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e17.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eRight Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e77.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e91.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e88.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e89.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e17.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e17.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e79.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.5\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e17.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e85.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003e0.9\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cstrong\u003eLeft Skewed\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e77.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e90.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e88.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e90.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"Limitations","content":"\u003cp\u003eBoth ANCOVA and 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e assume reference gene stability. Although ANCOVA models offer advantages over 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e, including offering greater statistical power and handling variable amplification efficiency better than 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e, effect estimates provided by ANCOVA models are in CT units, not fold change. CT units are only equal to 1 log2 unit of change in the special case where amplification efficiency is 100%.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eqPCR remains a foundational technique in molecular biology, with about 5,000 new citations every month (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA). Most practitioners control for possible variability using a single, well-known reference gene such as \u003cem\u003eGAPDH\u003c/em\u003e without explaining how this gene was chosen or providing evidence of its stability\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e. This practice could obscure true treatment effects\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e, degrading reproducibility. Most researchers assess qPCR differential gene expression using the 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e method, and therefore tacitly assume perfect amplification efficiency, potentially distorting findings\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e, but it is impossible to know because researchers rarely share raw data\u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. Solutions to many reproducibility problems have been proposed and reviewed\u003csup\u003e\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u003c/sup\u003e but largely ignored\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e perhaps because validating reference genes, performing dilution curves to measure amplification efficiency and adhering to data standards is viewed as a low priority by study authors and peer reviewers.\u003c/p\u003e \u003cp\u003eIf the precise magnitude of fold change were important to the research question, running a sample dilution curve could establish efficiency, and those estimates could be used to create efficiency-weighted CT values suitable for downstream statistical analysis\u003csup\u003e\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u003c/sup\u003e. Often, what matters most is whether a treatment significantly affects gene expression in the statistical sense (p\u0026thinsp;\u0026lt;\u0026thinsp;.05) and our simulations show that multivariable linear models like ANCOVA are superior to 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e. Using ANCOVA for qPCR data analysis requires very little coding and provides p-values that are invariant to amplification efficiency differences between the reference and gene of interest. Moreover, because ANCOVA can be more sensitive than 2\u003csup\u003eΔΔCT\u003c/sup\u003e, ANCOVA will sometimes detect experimental differences with fewer samples, saving time and resources. Multivariable linear models naturally accommodate the more complex experimental designs that are increasingly common in science. Based on simulations, ANCOVA offers a lower false negative rate than 2\u003csup\u003eΔΔCT\u003c/sup\u003e, especially when the reference gene does not provide useful information about the gene of interest.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCode Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe code used in the simulation is included in GitHub (https://github.com/DartCF/MLM-for-qPCR).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe research of KKF, LT, TH and BAS is supported by the National Institutes of Health (R01 HL151385 and P30 DK117469), the Cystic Fibrosis Foundation (STANTO19R0, STANTO23GO), and the Flatley Foundation. The research of TM is supported by National Institutes of Health P30 DK117469.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eLT, KKF, TH and TM conceived of the project and contributed to the writing of the manuscript. \u0026nbsp;BAS provided advice, support and contributed to the writing of the manuscript. \u0026nbsp;All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no conflicts of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of generative AI and AI-assisted technologies\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNeither AI nor AI-assisted technologies were used in this report.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eHampton TH, Barnaby R, Roche C, Nymon A, Fukutani KF, MacKenzie TA, Charpentier LA, Stanton BA (2024) Gene expression responses of CF airway epithelial cells exposed to elexacaftor/tezacaftor/ivacaftor suggest benefits beyond improved CFTR channel function. Am J Physiology-Lung Cell Mol Physiol 327:L905\u0026ndash;L916. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1152/ajplung.00272.2024\u003c/span\u003e\u003cspan address=\"10.1152/ajplung.00272.2024\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBarry PJ, Mall MA, \u0026Aacute;lvarez A, Colombo C, de Winter-de Groot KM, Fajac I, McBennett KA, McKone EF, Ramsey BW, Sutharsan S et al (2021) Triple Therapy for Cystic Fibrosis Phe508del-Gating and -Residual Function Genotypes. 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Biotechnol J 3:112\u0026ndash;123. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1002/biot.200700169\u003c/span\u003e\u003cspan address=\"10.1002/biot.200700169\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"qPCR, delta-delta CT, amplification efficiency","lastPublishedDoi":"10.21203/rs.3.rs-6014969/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6014969/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHere we present a method based on multivariable linear models for qPCR data analysis as an alternative to the most commonly used method, 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e. It has long been understood that amplification efficiency during qPCR may be less than two, that is, the amount of DNA present may not double in each cycle, and it is also known that amplification efficiency may differ between target and reference genes. Therefore, it has long been recommended that qPCR experiments include direct assessment of amplification efficiency, and that efficiency values be included in the calculation of differential gene expression. Nonetheless, current reports that include qPCR data continue to use 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e, even though 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e assumes an efficiency of two in both reference and target genes. Using multivariable linear models for qPCR data analysis does not require direct measurement of amplification efficiency but provides correct significance estimates for differential expression even when amplification is less than two or differs between target and reference genes. We introduce the logic behind using multivariable linear models in the context of qPCR data analysis, the mathematics behind using them, and provide simulations demonstrating that multivariable linear models outperform 2\u003csup\u003e\u0026minus;ΔΔCT\u003c/sup\u003e for qPCR data analysis.\u003c/p\u003e","manuscriptTitle":"Multivariable Linear Models Outperform 2-ΔΔCT for qPCR Data Analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-02-13 16:08:07","doi":"10.21203/rs.3.rs-6014969/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"695f5e9e-02f0-4719-ac00-5200b4d5af05","owner":[],"postedDate":"February 13th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":44227417,"name":"Molecular Biology"}],"tags":[],"updatedAt":"2025-02-13T16:08:07+00:00","versionOfRecord":[],"versionCreatedAt":"2025-02-13 16:08:07","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6014969","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6014969","identity":"rs-6014969","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}