Fig 1.
Amplification curves of qPCR reactions.
In the example, five molecular targets reached the threshold of 0.5 before the 32nd cycle, i.e., Cq < 32, while two reached the threshold between the 32nd and the 40th cycle. The eighth molecular target did not reach the threshold by the 40th cycle, i.e., Cq > 40. QuantStudio Real-Time PCR software version 1.3 (Applied Biosystems by Thermo Fisher Scientific) was used to create the figure.
Table 1.
Symbolic qPCR data set.
Fig 2.
The flowchart to perform CTOT with the R coin package.
Y(ijk) denotes the Cq value reported by a qPCR assay for molecular target j (j = 1, 2, …, or g) of sample i (i = 1, 2, …, or n) in group k (k = 1 or 2). ΔY(ijk) denotes normalized Cq for target j of sample i in group k.
Table 2.
Contingency table on the observations at a detection point.
Table 3.
Connection between CTOT and qPCR data features and common time-to-event parameters.
Table 4.
One-to-one correspondence of the distributions of simulated ΔCq and eΔCq data.
Table 5.
Simulation models with respective density functions and parameters.
Fig 3.
Boxplots of simulated Cq data.
The points above the solid line would be uncertainly measured by qPCR should 40 be the cutoff for data quality control or for biological, clinical, or technical concerns in practice. The points above the dash line would be uncertainly measured by qPCR should 32 be the cutoff for quality control or for biological, clinical, or technical concerns in practice.
Fig 4.
The empirical power of the CTOT, MC, and CO methods compared with that of BFD.
BFD stands for the benchmark with full data analyzed with the current standard method, which includes t-tests for two-group comparisons. CTOT stands for the cycle-to-threshold method, while CO denotes the complete-observation method and MC denotes the method that sets uncertain and incomplete observations equal to the assay-specific maximum cycle threshold C1. Uncertain qPCR data may occur in one or both groups under comparison. % denotes the percentage of uncertainty that is observed in only one group among the replicates. nrep denotes the number of the replicates with at least one uncertain observation. β0 and β1 are parameters of the underlying models. |β1| is the absolute value of effect size. Panels A, B, and C represent the empirical power of the log-normal, Weibull, and log-logistic simulation type, respectively.
Table 6.
Empirical overall power of the CTOT, MC, and CO methods with analysis on benchmark data.
BFD stands for the benchmark with full data analyzed with the current standard method, which includes t-tests for two-group comparisons. CTOT stands for the cycle-to-threshold method, while CO denotes the complete-observation method and MC denotes the method that sets uncertain observations equal to the assay-specific maximum cycle threshold C1.
Fig 5.
An Example to Illustrate the Issue of Potential False Negatives of MC and CO.
(A) The original Cq data simulated with a normal distribution (corresponding to the log-normal simulation type in Table 2, β0 = 13.35 and β1 = 2.06; the corresponding empirical power of BFD is 0.80). (B) The normalized Cq data with the BFD, CTOT, MC, or CO methods applied. The filled diamonds denote the Cq data with BFD. BFD stands for the benchmark with full data analyzed with the current standard method, which include t-tests for two-group comparisons. The filled triangles denote the Cq data with CTOT, the cycle-to-threshold method. The vertical green arrows indicate the ranges uncertain observations belong to, e.g., being greater than or equal to the assay-specific maximum cycle threshold C1. The filled squares denote the Cq data with MC, the method that sets uncertain and incomplete observations equal to C1. The maximum quality cycle threshold C1 = 40 is highlighted with a horizontal solid line. The open circles denote the Cq data with CO, the complete-observation method. The first five simulated samples belong to Group 1. The second five simulated samples belong to Group 2. The vertical dash line separates Groups 1 and 2.
Fig 6.
An example to illustrate differences of the MC, CO, and CTOT methods.
(A) The original Cq data simulated with a normal distribution (corresponding to the log-normal simulation type in Table 5, β0 = 8.47 and β1 = 4.65; the corresponding empirical power of BFD is 0.91). (B) The normalized Cq data with the BFD, CTOT, MC, or CO methods applied. The filled diamonds denote the Cq data with BFD. BFD stands for the benchmark with full data analyzed with the current standard method, which include t-tests for two-group comparisons. The filled triangles denote the Cq data with CTOT, the cycle-to-threshold method. The vertical green arrows indicate the ranges uncertain observations belong to, e.g., being greater than or equal to the assay-specific maximum cycle threshold C1. The filled squares denote the Cq data with MC, the method that sets uncertain and incomplete observations equal to the assay-specific maximum cycle threshold C1. The maximum quality cycle threshold C1 = 40 is highlighted with a horizontal solid line. The open circle denoted CO, the complete-observation method. The first five simulated samples belong to Group 1. The second five simulated samples belong to Group 2. The vertical dash line separates Groups 1 and 2.
Fig 7.
Empirical type I error rates of CTOT, BFD, CO, and MC methods.
CTOT stands for the cycle-to-threshold method. BFD stands for the benchmark with full data analyzed with the current standard method, which includes t-tests for two-group comparisons. CO denotes the complete-observation method and MC denotes the method that sets uncertain observations equal to the assay-specific maximum cycle threshold C1. In the simulation, C1 is set to be 40.ΔCq followed normal distributions and eΔCq followed log-normal distributions. Parameter Set 1: β0 = 5, σ = 1; Parameter Set 2: β0 = 10, σ = 1; +Parameter Set 3: β0 = 5, σ = 2; and Parameter Set 4: β0 = 10, σ = 2 with the parameterization listed for log-normal distribution in Table 2.
Fig 8.
Comparison of the statistical significance between t-tests with C1 = 40 and CTOT with C1 = 32.
The sensitivity analysis was performed on 17 two-group comparisons on rat serum microRNAs miR-210-3p and miR-128-3p, where there was at least one uncertain observation in either of the two groups of comparison [3]. The p-values based on t-tests and CTOT (cycle-to-threshold method) are plotted in a -log10 scale on x-axis and y-axis, respectively. C1 denotes an assay-specific maximum cycle threshold for clinical, quality, or biological relevance, e.g., the cycle number that corresponds to LLOQ (lower limit of quantification). The solid lines are set at p-value = 0.05 and the dashed lines are set at p-value = 0.005. The inset Venn diagram illustrates statistically significant differences of levels of circulating microRNAs between control and treated groups, applying the CTOT, MC, or CO method and a maximum quality cycle threshold of C1 = 32 to the data reported by Silva et al. [3].
Table 7.
Methods to analyze qPCR data with uncertain observations.