Overview
Real-time polymerase chain reaction (real-time PCR), is widely used in biological research. Various analysis methods are employed on the real-time PCR data to measure the mRNA levels under different experimental conditions. I developed ‘rtpcr’ package for amplification efficiency calculation and statistical analysis of real-time PCR data in R. By accounting for up to two reference genes and amplification efficiency values, a general calculation method by Ganger et al. 2017 covering both the Livak and Pfaffl methods was used. Based on the experimental conditions, the functions of the ‘rtpcr’ package use a t-test (for experiments with a two-level factor) or analysis of variance (for cases where more than two levels or factors exist) to calculate the fold change (FC) or relative expression (RE). The functions further provide standard deviations and confidence limits for means, apply statistical mean comparisons and present letter mean grouping. To facilitate function application, different data sets were used as examples and the outputs were explained. An outstanding feature of ‘rtpcr’ package is providing publication-ready bar plots with various controlling arguments for experiments with up to three different factors. The ‘rtpcr’ package is user-friendly and easy to work with and provides an applicable resource for analyzing real-time PCR data in R. ‘rtpcr’ is a CRAN package although the development version of the package can be obtained through Github.
Calculation methods
Among the various approaches developed for data analysis in real-time PCR, the Livak method, also known as the \(2^{-\Delta\Delta C_t}\) method, stands out for its simplicity and widespread use.
$$\begin{align*} \text{Fold change} & = 2^{-\Delta\Delta C_t} \\ & = \frac{2^{-(C_{t_{\text{target}}}-C_{t_{\text{ref}}})_{Tr}}} {2^{-(C_{t_{\text{target}}}-C_{t_{\text{ref}}})_{Co}}} \\ & =2^{-[(C_{t_{\text{target}}}-C_{t_{\text{ref}}})_{\text{Tr}}- {(C_{t_{\text{target}}}-C_{t_{\text{ref}}})}_{\text{Co}}]} \\ & = 2^{-[{(\Delta C_t)_{Tr} - (\Delta C_t)_{Co}}]} \end{align*}$$
where Tr is Treatment and Co is Control conditions, respectively. This method assumes that both the target and reference genes are amplified with efficiencies close to 100%, allowing for the relative quantification of gene expression levels¹.
On the other hand, the Pfaffl method offers a more flexible approach by accounting for differences in amplification efficiencies between the target and reference genes. This method adjusts the calculated expression ratio by incorporating the specific amplification efficiencies, thus providing a more accurate representation of the relative gene expression levels².
$$\text{Fold change} = \frac{E^{-(C_{t_{\text{Tr}}}-C_{t_{\text{Co}}})_{target}}} {E^{-(C_{t_{\text{Tr}}}-C_{t_{\text{Co}}})_{ref}}}$$
A generalized calculation method
the ‘rtpcr’ packager calculates wDCt values for each experimental condition based on the efficiency and Ct values as described by Ganger et al. (2017).
$$w\Delta Ct =\log_{10}(E_{target}).Ct_{target}-\log_{10}(E_{ref}).Ct_{ref}$$ if wDCt values are calculated using the efficiencies (E) values, calculations match the formula of Pfaffl. if 2 (100% amplification efficiency) is used, calculations match the formula of Livak. Analysis of variance (ANOVA) and t-test (paired or unpaired) is done on the wDCt values. Subsequently, the mean of wCt over each condition along with the resulting statistics (e.g. confidence limits for the means) is converted by a power of ten-conversion for presentation. For example, relative expression is calculated as: $$\text{Relative Expression} = 10^{-\overline{w\Delta Ct}}$$ When there are only two conditions (Tr and Co), fold change can also be calculated:
$$\text{Fold Change}=10^{-(\overline{w\Delta Ct}_{\text{Tr}}-{\overline{w\Delta Ct}_{\text{Co}}})}$$
Installing and loading
rtpcr is a CRAN package and can be installed through install.packages().
install.packages("rtpcr")
library(rtpcr)
The development version of the rtpcr package can be obtained through:
# install `rtpcr` from github (under development)
devtools::install_github("mirzaghaderi/rtpcr", build_vignettes = TRUE)
Data structure and column arrangement
To use the functions, input data should be prepared in the right format with appropriate column arrangement. The correct column arrangement is shown in Table 1.
Table 1. Data structure and column arrangement required for ‘rtpcr’ package.
| Experiment type | Column arrangement of the input data | Example in the package |
|---|---|---|
| Amplification efficiency | Dilutions targetCt refCt | data_efficiency |
| t-test (accepts multiple genes) | condition efficiency gene Ct | data_ttest |
| Factorial (Up to three factors) | factor1 rep targetE targetCt refE refCt | data_1factor |
| factor1 factor2 rep targetE targetCt refE refCt | data_2factor | |
| factor1 factor2 factor3 rep targetE targetCt refE refCt | data_3factor_b | |
| Factorial with blocking | factor1 block rep targetE targetCt refE refCt | . |
| factor1 factor2 block rep targetE targetCt refE refCt | data_2factorBlock | |
| factor1 factor2 factor3 block rep targetE targetCt refE refCt | . | |
| Two reference genes | . . . . . . rep targetE targetCt ref1E ref1Ct ref2E ref2Ct | . |
| calculating biological replicated | . . . . . . biologicalRep techcicalRep Etarget targetCt Eref refCt | data_withTechRep |
| . . . . . . biologicalRep techcicalRep Etarget targetCt ref1E ref1Ct ref2E ref2Ct | . | |
functions built in ‘rtpcr’ package
To simplify ‘rtpcr’ usage, examples for using the functions are presented below.
Table 2. Functions and examples for using them.
| function | Analysis | Example (see package help for the more arguments) |
|---|---|---|
| efficiency | Efficiency, standard curves and related statistics | efficiency(data_efficiency) |
| meanTech | Calculating the mean of technical replicates | meanTech(data_withTechRep, groups = 1:4) |
| oneFACTORfcplot | Bar plot of the average fold change of one target gene with two or more levels | oneFACTORfcplot(data_1factor, levels = c(3, 2, 1), numberOfrefGenes = 1, level.names = c(“A1”, “A2”, “A3”)) |
| oneFACTORplot | Bar plot of the relative gene expression from a one-factor experiment | out <- qpcrANOVA(data_1factor, numberOfrefGenes = 1)$Result; oneFACTORplot(out) |
| qpcrANOVA | Analysis of Variance of the qpcr data | qpcrANOVA(data_3factor_a, numberOfrefGenes = 1, p.adj = “none”) |
| qpcrTTEST | Computing the average fold change and related statistics | qpcrTTEST(data_ttest, numberOfrefGenes = 1, paired = FALSE, var.equal = TRUE) |
| qpcrTTESTplot | Bar plot of the average fold change of the target genes | qpcrTTESTplot(data_ttest, numberOfrefGenes = 1, order = c(“C2H2-01”, “C2H2-12”, “C2H2-26”)) |
| threeFACTORplot | Bar plot of the relative gene expression from a three-factor experiment | res <- qpcrANOVA(data_3factor_b, numberOfrefGenes = 1)$Result; threeFACTORplot(res, arrangement = c(3, 1, 2)) |
| twoFACTORplot | Bar plot of the relative gene expression from a two-factor experiment | res <- qpcrANOVA(data_2factor, numberOfrefGenes = 1)$Result; twoFACTORplot(res, x.axis.factor = Genotype, group.factor = Drought) |
see package help for more arguments including the number of reference genes, levels arrangement, blocking, and arguments for adjusting the bar plots.
Amplification efficiency data analysis
Sample data of amplification efficiency
To calculate the amplification efficiencies of a target and a reference gene, a data frame should be prepared with 3 columns of dilutions, target gene Ct values, and reference gene Ct values, respectively, as shown below.
data_efficiency
## dilutions C2H2.26 GAPDH
## 1 1.00 25.57823 22.60794
## 2 1.00 25.53636 22.68348
## 3 1.00 25.50280 22.62602
## 4 0.50 26.70615 23.67162
## 5 0.50 26.72720 23.64855
## 6 0.50 26.86921 23.70494
## 7 0.20 28.16874 25.11064
## 8 0.20 28.06759 25.11985
## 9 0.20 28.10531 25.10976
## 10 0.10 29.19743 26.16919
## 11 0.10 29.49406 26.15119
## 12 0.10 29.07117 26.15019
## 13 0.05 30.16878 27.11533
## 14 0.05 30.14193 27.13934
## 15 0.05 30.11671 27.16338
## 16 0.02 31.34969 28.52016
## 17 0.02 31.35254 28.57228
## 18 0.02 31.34804 28.53100
## 19 0.01 32.55013 29.49048
## 20 0.01 32.45329 29.48433
## 21 0.01 32.27515 29.26234
Calculating amplification efficiency
The following efficiency function calculates the amplification efficiency of a target and a reference gene and presents the related standard curves along with the Slope, Efficiency, and R2 statistics. The function also compares the slopes of the two standard curves. For this, a regression line is fitted using the \(\Delta C_t\) values. If \(2^{-\Delta\Delta C_t}\) method is intended, the slope should not exceed 0.2!
efficiency(data_efficiency)
## $plot
Standard curve and the amplification efficiency analysis of target and reference genes.
##
## $Efficiency_Analysis_Results
## Gene Slope E R2
## 1 C2H2.26 -3.388 1.973 0.997
## 2 GAPDH -3.415 1.963 0.999
##
## $Slope_of_differences
## [1] 0.0264574
Expression data analysis
Target genes in two conditions (t-test)
Example data
when a target gene is assessed under two different conditions (for example Control and treatment), it is possible to calculate the average fold change expression i.e. \(2^{-\Delta \Delta C_t}\) of the target gene in treatment relative to control conditions. For this, the data should be prepared according to the following dataset consisting of 4 columns belonging to condition levels, E (efficiency), genes and Ct values, respectively. Each Ct value is the mean of technical replicates. Complete amplification efficiencies of 2 have been assumed here for all wells but the calculated efficiencies can be used instead.
data_ttest
## Condition E Gene Ct
## 1 control 2 C2H2-26 31.26
## 2 control 2 C2H2-26 31.01
## 3 control 2 C2H2-26 30.97
## 4 treatment 2 C2H2-26 32.65
## 5 treatment 2 C2H2-26 32.03
## 6 treatment 2 C2H2-26 32.40
## 7 control 2 C2H2-01 31.06
## 8 control 2 C2H2-01 30.41
## 9 control 2 C2H2-01 30.97
## 10 treatment 2 C2H2-01 28.85
## 11 treatment 2 C2H2-01 28.93
## 12 treatment 2 C2H2-01 28.90
## 13 control 2 C2H2-12 28.50
## 14 control 2 C2H2-12 28.40
## 15 control 2 C2H2-12 28.80
## 16 treatment 2 C2H2-12 27.90
## 17 treatment 2 C2H2-12 28.00
## 18 treatment 2 C2H2-12 27.90
## 19 control 2 ref 28.87
## 20 control 2 ref 28.42
## 21 control 2 ref 28.53
## 22 treatment 2 ref 28.31
## 23 treatment 2 ref 29.14
## 24 treatment 2 ref 28.63
Data analysis under two conditions
Here, the above data set was used for the Fold Change expression analysis of the target genes using the qpcrTTEST function. This function performs a t-test-based analysis of any number of genes that
have been evaluated under control and treatment conditions. The analysis can be done for unpaired or paired conditions. The output is a table of target gene names, fold changes confidence limits, and the t.test derived p-values. The qpcrTTEST function includes the var.equal argument. When set to FALSE, t.test is performed under the unequal variances hypothesis.
qpcrTTEST(data_ttest,
numberOfrefGenes = 1,
paired = F,
var.equal = T)
## $Raw_data
## Var2 wDCt
## 1 1 0.7194617
## 2 1 0.7796677
## 3 1 0.7345132
## 4 1 1.3064702
## 5 1 0.8699767
## 6 1 1.1348831
## 7 2 0.6592557
## 8 2 0.5990497
## 9 2 0.7345132
## 10 2 0.1625562
## 11 2 -0.0632163
## 12 2 0.0812781
## 13 3 -0.1113811
## 14 3 -0.0060206
## 15 3 0.0812781
## 16 3 -0.1234223
## 17 3 -0.3431742
## 18 3 -0.2197519
##
## $Result
## Gene dif FC LCL UCL pvalue
## 1 C2H2-26 0.3592 0.4373 0.1926 0.9927 0.0488
## 2 C2H2-01 -0.6041 4.0185 2.4598 6.5649 0.0014
## 3 C2H2-12 -0.2167 1.6472 0.9595 2.8279 0.0624
Generating plot
The qpcrTTESTplot function generates a bar plot of Fold Changes and confidence intervals for the target genes. the qpcrTTESTplot function accepts any gene name and any replicates. The qpcrTTESTplot function automatically puts appropriate signs of **, * on top of the plot columns based on the output p-values.
# Producing the plot
qpcrTTESTplot(data_ttest,
numberOfrefGenes = 1)
Average Fold changes of three target genes relative to the control condition computed by unpaired t-tests. Error bars represent 95% confidence interval.
# Producing the plot: specifying gene order
qpcrTTESTplot(data_ttest,
numberOfrefGenes = 1,
order = c("C2H2-01", "C2H2-12", "C2H2-26"),
paired = FALSE,
var.equal = TRUE,
width = 0.5,
fill = "skyblue",
y.axis.adjust = 0,
y.axis.by = 2,
ylab = "Average Fold Change",
xlab = "Gene")
Average Fold changes of three target genes relative to the control condition computed by unpaired t-tests. Error bars represent 95% confidence interval.
A target gene under more than two conditions (ANOVA)
Analysis of variance (ANOVA) of factorial experiments in the frame of a completely randomized design (CRD) can be done by the qpcrANOVA function. ANOVA of qPCR data is suitable when there is a factor with more than two levels, or when there is more than an experimental factor. The input data set should be prepared as shown below. Factor columns should be presented first followed by biological replicates and efficiency and Ct values of target and reference genes. The example data set below (data_3factor_a) represents amplification efficiency and Ct values for target and reference genes under three grouping factors (two different cultivars, three drought levels, and the presence or absence of bacteria). The table can contain any number of factor columns. The factor columns should be followed by five other columns assigned to biological replicates ®, the efficiency of the target gene, Ct values of the target gene, the efficiency of the reference gene, and Ct values of the reference gene, respectively. Here, the efficiency of 2 has been used for all wells, but the calculated efficiencies can be used instead.
# See a sample dataset
data_3factor_a
## Genotype Drought SA Rep EPO POCt EGAPDH GAPDHCt
## 1 R 0.00 A1 1 1.839 33.30 1.918 31.53
## 2 R 0.00 A1 2 1.839 33.39 1.918 31.57
## 3 R 0.00 A1 3 1.839 33.34 1.918 31.50
## 4 R 0.00 A2 1 1.839 34.01 1.918 31.48
## 5 R 0.00 A2 2 1.839 36.82 1.918 31.44
## 6 R 0.00 A2 3 1.839 35.44 1.918 31.46
## 7 R 0.25 A1 1 1.839 32.73 1.918 31.30
## 8 R 0.25 A1 2 1.839 32.46 1.918 32.55
## 9 R 0.25 A1 3 1.839 32.60 1.918 31.92
## 10 R 0.25 A2 1 1.839 33.37 1.918 31.19
## 11 R 0.25 A2 2 1.839 33.12 1.918 31.94
## 12 R 0.25 A2 3 1.839 33.21 1.918 31.57
## 13 R 0.50 A1 1 1.839 33.48 1.918 33.30
## 14 R 0.50 A1 2 1.839 33.27 1.918 33.37
## 15 R 0.50 A1 3 1.839 33.32 1.918 33.35
## 16 R 0.50 A2 1 1.839 32.53 1.918 33.47
## 17 R 0.50 A2 2 1.839 32.61 1.918 33.26
## 18 R 0.50 A2 3 1.839 32.56 1.918 33.36
## 19 S 0.00 A1 1 1.839 26.85 1.918 26.94
## 20 S 0.00 A1 2 1.839 28.17 1.918 27.69
## 21 S 0.00 A1 3 1.839 27.99 1.918 27.39
## 22 S 0.00 A2 1 1.839 28.71 1.918 29.45
## 23 S 0.00 A2 2 1.839 29.01 1.918 29.46
## 24 S 0.00 A2 3 1.839 28.82 1.918 29.48
## 25 S 0.25 A1 1 1.839 30.41 1.918 28.70
## 26 S 0.25 A1 2 1.839 29.49 1.918 28.66
## 27 S 0.25 A1 3 1.839 29.98 1.918 28.71
## 28 S 0.25 A2 1 1.839 28.91 1.918 28.09
## 29 S 0.25 A2 2 1.839 28.60 1.918 28.65
## 30 S 0.25 A2 3 1.839 28.59 1.918 28.37
## 31 S 0.50 A1 1 1.839 29.03 1.918 30.61
## 32 S 0.50 A1 2 1.839 28.73 1.918 30.20
## 33 S 0.50 A1 3 1.839 28.83 1.918 30.49
## 34 S 0.50 A2 1 1.839 28.29 1.918 30.84
## 35 S 0.50 A2 2 1.839 28.53 1.918 30.65
## 36 S 0.50 A2 3 1.839 28.28 1.918 30.74
The qpcrANOVA function performs ANOVA based on both factorial arrangement and completely randomized design (CRD). For the latter, a column of treatment combinations is made first as a grouping factor followed by ANOVA. You can call the input data set along with the added wCt and treatment combinations by qpcrANOVA. CRD-based analysis is especially useful when post-hoc tests and mean comparisons/grouping are desired for all treatment combinations. The final results along with the ANOVA tables can be called by qpcrANOVA.
Reverse ordering of the grouping letters
One may be interested in presenting the statistical mean comparison result in the frame of grouping letters. This is rather challenging because in the grouping output of mean comparisons (via the LSD.test function of agricolae package), means are sorted into descending order so that the largest mean, is the first in the table and “a” letter is assigned to it. If LSD.test is applied to the wCt means, the biggest wCt mean receives “a” letter as expected, but this value turns into the smallest mean after its reverse log transformation by \(10^{-(\Delta Ct)}\). to solve this issue, I used a function that assigns the grouping letters appropriately.
Output table of the analysis
The qpcrANOVA function produces the main analysis output including mean wDCt, LCL, UCL, grouping letters, and standard deviations. The standard deviation for each mean is derived from the back-transformed raw wDCt values from biological replicates for that mean.
# If the data include technical replicates, means of technical replicates
# should be calculated first using meanTech function.
# Applying ANOVA analysis
qpcrANOVA(data_3factor_a,
numberOfrefGenes = 1,
p.adj = "none")
## $Final_data
## Genotype Drought SA rep Etarget Cttarget Eref Ctref wDCt
## 1 R 0.00 A1 1 1.839 33.30 1.918 31.53 -0.107644864
## 2 R 0.00 A1 2 1.839 33.39 1.918 31.57 -0.095146452
## 3 R 0.00 A1 3 1.839 33.34 1.918 31.50 -0.088576136
## 4 R 0.00 A2 1 1.839 34.01 1.918 31.48 0.094350594
## 5 R 0.00 A2 2 1.839 36.82 1.918 31.44 0.849139197
## 6 R 0.00 A2 3 1.839 35.44 1.918 31.46 0.478359439
## 7 R 0.25 A1 1 1.839 32.73 1.918 31.30 -0.193401271
## 8 R 0.25 A1 2 1.839 32.46 1.918 32.55 -0.618399091
## 9 R 0.25 A1 3 1.839 32.60 1.918 31.92 -0.403163029
## 10 R 0.25 A2 1 1.839 33.37 1.918 31.19 0.007044382
## 11 R 0.25 A2 2 1.839 33.12 1.918 31.94 -0.271237502
## 12 R 0.25 A2 3 1.839 33.21 1.918 31.57 -0.142771163
## 13 R 0.50 A1 1 1.839 33.48 1.918 33.30 -0.560662180
## 14 R 0.50 A1 2 1.839 33.27 1.918 33.37 -0.636023745
## 15 R 0.50 A1 3 1.839 33.32 1.918 33.35 -0.617137686
## 16 R 0.50 A2 1 1.839 32.53 1.918 33.47 -0.860099085
## 17 R 0.50 A2 2 1.839 32.61 1.918 33.26 -0.779534340
## 18 R 0.50 A2 3 1.839 32.56 1.918 33.36 -0.821048287
## 19 S 0.00 A1 1 1.839 26.85 1.918 26.94 -0.515921930
## 20 S 0.00 A1 2 1.839 28.17 1.918 27.69 -0.378810500
## 21 S 0.00 A1 3 1.839 27.99 1.918 27.39 -0.341580630
## 22 S 0.00 A2 1 1.839 28.71 1.918 29.45 -0.733749907
## 23 S 0.00 A2 2 1.839 29.01 1.918 29.46 -0.657203874
## 24 S 0.00 A2 3 1.839 28.82 1.918 29.48 -0.713131375
## 25 S 0.25 A1 1 1.839 30.41 1.918 28.70 -0.071824515
## 26 S 0.25 A1 2 1.839 29.49 1.918 28.66 -0.303925762
## 27 S 0.25 A1 3 1.839 29.98 1.918 28.71 -0.188423145
## 28 S 0.25 A2 1 1.839 28.91 1.918 28.09 -0.296159461
## 29 S 0.25 A2 2 1.839 28.60 1.918 28.65 -0.536575015
## 30 S 0.25 A2 3 1.839 28.59 1.918 28.37 -0.460023224
## 31 S 0.50 A1 1 1.839 29.03 1.918 30.61 -0.977188133
## 32 S 0.50 A1 2 1.839 28.73 1.918 30.20 -0.940594725
## 33 S 0.50 A1 3 1.839 28.83 1.918 30.49 -0.996162646
## 34 S 0.50 A2 1 1.839 28.29 1.918 30.84 -1.238033791
## 35 S 0.50 A2 2 1.839 28.53 1.918 30.65 -1.120792942
## 36 S 0.50 A2 3 1.839 28.28 1.918 30.74 -1.212394748
##
## $lmCRD
##
## Call:
## stats::lm(formula = wDCt ~ T, data = x)
##
## Coefficients:
## (Intercept) TR:0.25:A2 TR:0.5:A1 TR:0.5:A2 TR:0:A1 TR:0:A2
## -0.404988 0.269333 -0.199620 -0.415239 0.307865 0.878938
## TS:0.25:A1 TS:0.25:A2 TS:0.5:A1 TS:0.5:A2 TS:0:A1 TS:0:A2
## 0.216930 -0.025931 -0.566327 -0.785419 -0.007117 -0.296374
##
##
## $ANOVA_factorial
## Analysis of Variance Table
##
## Response: wDCt
## Df Sum Sq Mean Sq F value Pr(>F)
## Genotype 1 1.32885 1.32885 23.9559 3.701e-05 ***
## Drought 1 3.04576 3.04576 54.9076 4.537e-08 ***
## SA 1 0.00400 0.00400 0.0720 0.790365
## Genotype:Drought 1 0.21286 0.21286 3.8373 0.060151 .
## Genotype:SA 1 0.47334 0.47334 8.5332 0.006821 **
## Drought:SA 1 0.19253 0.19253 3.4708 0.072982 .
## Genotype:Drought:SA 1 0.27533 0.27533 4.9635 0.034099 *
## Residuals 28 1.55318 0.05547
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $ANOVA_CRD
## Analysis of Variance Table
##
## Response: wDCt
## Df Sum Sq Mean Sq F value Pr(>F)
## T 11 6.5790 0.59809 28.323 4.502e-11 ***
## Residuals 24 0.5068 0.02112
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $Result
## Genotype Drought SA RE LCL UCL letters std
## R:0.25:A1 R 0.25 A1 2.5409 3.7857 1.7054 de 1.3099
## R:0.25:A2 R 0.25 A2 1.3666 2.0362 0.9173 f 0.4422
## R:0.5:A1 R 0.5 A1 4.0235 5.9947 2.7005 cd 0.3568
## R:0.5:A2 R 0.5 A2 6.6104 9.8488 4.4368 bc 0.6135
## R:0:A1 R 0 A1 1.2506 1.8633 0.8394 f 0.0280
## R:0:A2 R 0 A2 0.3358 0.5003 0.2254 g 0.3414
## S:0.25:A1 S 0.25 A1 1.5419 2.2973 1.0349 ef 0.4179
## S:0.25:A2 S 0.25 A2 2.6972 4.0186 1.8103 de 0.7382
## S:0.5:A1 S 0.5 A1 9.3608 13.9467 6.2829 ab 0.6034
## S:0.5:A2 S 0.5 A2 15.5027 23.0975 10.4052 a 2.1351
## S:0:A1 S 0 A1 2.5829 3.8482 1.7336 de 0.5779
## S:0:A2 S 0 A2 5.0276 7.4906 3.3745 c 0.4507
##
## $Post_hoc_Test
## FC pvalue signif. LCL UCL
## R:0.25:A1 - R:0.25:A2 1.8592 0.0325 * 1.0579 3.2675
## R:0.25:A1 - R:0.5:A1 0.6315 0.1055 0.3593 1.1098
## R:0.25:A1 - R:0.5:A2 0.3844 0.0018 ** 0.2187 0.6755
## R:0.25:A1 - R:0:A1 2.0317 0.0159 * 1.1561 3.5706
## R:0.25:A1 - R:0:A2 7.5672 0.0000 *** 4.3058 13.2990
## R:0.25:A1 - S:0.25:A1 1.6479 0.0800 . 0.9377 2.8961
## R:0.25:A1 - S:0.25:A2 0.9420 0.8288 0.5360 1.6556
## R:0.25:A1 - S:0.5:A1 0.2714 0.0001 *** 0.1545 0.4770
## R:0.25:A1 - S:0.5:A2 0.1639 0.0000 *** 0.0933 0.2880
## R:0.25:A1 - S:0:A1 0.9837 0.9527 0.5598 1.7289
## R:0.25:A1 - S:0:A2 0.5054 0.0197 * 0.2876 0.8882
## R:0.25:A2 - R:0.5:A1 0.3397 0.0006 *** 0.1933 0.5969
## R:0.25:A2 - R:0.5:A2 0.2067 0.0000 *** 0.1176 0.3633
## R:0.25:A2 - R:0:A1 1.0928 0.7482 0.6218 1.9205
## R:0.25:A2 - R:0:A2 4.0701 0.0000 *** 2.3159 7.1530
## R:0.25:A2 - S:0.25:A1 0.8863 0.6627 0.5043 1.5577
## R:0.25:A2 - S:0.25:A2 0.5067 0.0202 * 0.2883 0.8905
## R:0.25:A2 - S:0.5:A1 0.1460 0.0000 *** 0.0831 0.2566
## R:0.25:A2 - S:0.5:A2 0.0882 0.0000 *** 0.0502 0.1549
## R:0.25:A2 - S:0:A1 0.5291 0.0285 * 0.3011 0.9299
## R:0.25:A2 - S:0:A2 0.2718 0.0001 *** 0.1547 0.4777
## R:0.5:A1 - R:0.5:A2 0.6087 0.0817 . 0.3463 1.0697
## R:0.5:A1 - R:0:A1 3.2173 0.0003 *** 1.8306 5.6541
## R:0.5:A1 - R:0:A2 11.9828 0.0000 *** 6.8183 21.0591
## R:0.5:A1 - S:0.25:A1 2.6095 0.0018 ** 1.4848 4.5860
## R:0.5:A1 - S:0.25:A2 1.4917 0.1562 0.8488 2.6216
## R:0.5:A1 - S:0.5:A1 0.4298 0.0050 ** 0.2446 0.7554
## R:0.5:A1 - S:0.5:A2 0.2595 0.0000 *** 0.1477 0.4561
## R:0.5:A1 - S:0:A1 1.5578 0.1178 0.8864 2.7377
## R:0.5:A1 - S:0:A2 0.8003 0.4228 0.4554 1.4065
## R:0.5:A2 - R:0:A1 5.2857 0.0000 *** 3.0076 9.2894
## R:0.5:A2 - R:0:A2 19.6869 0.0000 *** 11.2020 34.5986
## R:0.5:A2 - S:0.25:A1 4.2872 0.0000 *** 2.4394 7.5344
## R:0.5:A2 - S:0.25:A2 2.4508 0.0032 ** 1.3945 4.3071
## R:0.5:A2 - S:0.5:A1 0.7062 0.2151 0.4018 1.2411
## R:0.5:A2 - S:0.5:A2 0.4264 0.0047 ** 0.2426 0.7494
## R:0.5:A2 - S:0:A1 2.5593 0.0021 ** 1.4563 4.4978
## R:0.5:A2 - S:0:A2 1.3148 0.3264 0.7481 2.3107
## R:0:A1 - R:0:A2 3.7245 0.0001 *** 2.1193 6.5457
## R:0:A1 - S:0.25:A1 0.8111 0.4509 0.4615 1.4254
## R:0:A1 - S:0.25:A2 0.4637 0.0096 ** 0.2638 0.8149
## R:0:A1 - S:0.5:A1 0.1336 0.0000 *** 0.0760 0.2348
## R:0:A1 - S:0.5:A2 0.0807 0.0000 *** 0.0459 0.1418
## R:0:A1 - S:0:A1 0.4842 0.0139 * 0.2755 0.8509
## R:0:A1 - S:0:A2 0.2487 0.0000 *** 0.1415 0.4372
## R:0:A2 - S:0.25:A1 0.2178 0.0000 *** 0.1239 0.3827
## R:0:A2 - S:0.25:A2 0.1245 0.0000 *** 0.0708 0.2188
## R:0:A2 - S:0.5:A1 0.0359 0.0000 *** 0.0204 0.0630
## R:0:A2 - S:0.5:A2 0.0217 0.0000 *** 0.0123 0.0381
## R:0:A2 - S:0:A1 0.1300 0.0000 *** 0.0740 0.2285
## R:0:A2 - S:0:A2 0.0668 0.0000 *** 0.0380 0.1174
## S:0.25:A1 - S:0.25:A2 0.5717 0.0518 . 0.3253 1.0047
## S:0.25:A1 - S:0.5:A1 0.1647 0.0000 *** 0.0937 0.2895
## S:0.25:A1 - S:0.5:A2 0.0995 0.0000 *** 0.0566 0.1748
## S:0.25:A1 - S:0:A1 0.5970 0.0711 . 0.3397 1.0491
## S:0.25:A1 - S:0:A2 0.3067 0.0002 *** 0.1745 0.5390
## S:0.25:A2 - S:0.5:A1 0.2881 0.0001 *** 0.1640 0.5064
## S:0.25:A2 - S:0.5:A2 0.1740 0.0000 *** 0.0990 0.3058
## S:0.25:A2 - S:0:A1 1.0443 0.8753 0.5942 1.8353
## S:0.25:A2 - S:0:A2 0.5365 0.0318 * 0.3053 0.9428
## S:0.5:A1 - S:0.5:A2 0.6038 0.0772 . 0.3436 1.0612
## S:0.5:A1 - S:0:A1 3.6242 0.0001 *** 2.0622 6.3693
## S:0.5:A1 - S:0:A2 1.8619 0.0321 * 1.0594 3.2722
## S:0.5:A2 - S:0:A1 6.0021 0.0000 *** 3.4152 10.5483
## S:0.5:A2 - S:0:A2 3.0835 0.0004 *** 1.7545 5.4191
## S:0:A1 - S:0:A2 0.5137 0.0226 * 0.2923 0.9029
Barplot with the (1-alpha)% confidence interval as error bars
# Before plotting, the result needs to be extracted as below:
res <- qpcrANOVA(data_2factor, numberOfrefGenes = 1)$Result
res
## Genotype Drought RE LCL UCL letters std
## R:0 R 0 0.2852 0.4026 0.2020 d 0.0072
## R:0.25 R 0.25 0.6271 0.8853 0.4441 bc 0.3508
## R:0.5 R 0.5 0.9885 1.3956 0.7002 b 0.0979
## S:0 S 0 0.7955 1.1232 0.5635 b 0.2190
## S:0.25 S 0.25 0.4147 0.5854 0.2937 cd 0.1289
## S:0.5 S 0.5 2.9690 4.1918 2.1030 a 0.1955
# Plot of the 'res' data with 'Genotype' as grouping factor
twoFACTORplot(res,
x.axis.factor = Drought,
group.factor = Genotype,
width = 0.5,
fill = "Greens",
y.axis.adjust = 0.5,
y.axis.by = 2,
ylab = "Relative Expression",
xlab = "Drought Levels",
legend.position = c(0.09, 0.8),
show.letters = TRUE)
Average relative expression of a target gene under two different factors of genotype (with two levels) and drought (with three levels). Error bars represent standard deviations. Means (columns) lacking letters in common have significant difference at alpha = 0.05 as resulted from the `LSD.test` of agricolae package.
# Plotting the same data with 'Drought' as grouping factor
twoFACTORplot(res,
x.axis.factor = Genotype,
group.factor = Drought,
xlab = "Genotype",
fill = "Blues",
show.letters = FALSE)
Average relative expression of a target gene under two different factors of genotype (with two levels) and drought (with three levels). Error bars represent standard deviations. Means (columns) lacking letters in common have significant difference at alpha = 0.05 as resulted from the `LSD.test` of agricolae package.
A three-factorial experiment example
# Before plotting, the result needs to be extracted as below:
res <- qpcrANOVA(data_3factor_b, numberOfrefGenes = 1)$Result
res
## Type Conc SA RE LCL UCL letters std
## R:H:A1 R H A1 0.9885 1.5455 0.6323 cd 0.0979
## R:H:A2 R H A2 1.7371 2.7159 1.1110 bc 0.1747
## R:L:A1 R L A1 0.2852 0.4459 0.1824 f 0.0072
## R:L:A2 R L A2 0.0641 0.1002 0.0410 g 0.0773
## R:M:A1 R M A1 0.6271 0.9804 0.4011 de 0.3508
## R:M:A2 R M A2 0.3150 0.4925 0.2015 f 0.1105
## S:H:A1 S H A1 2.9690 4.6420 1.8990 ab 0.1955
## S:H:A2 S H A2 5.1934 8.1197 3.3217 a 0.7893
## S:L:A1 S L A1 0.7955 1.2438 0.5088 d 0.2190
## S:L:A2 S L A2 1.5333 2.3973 0.9807 c 0.1562
## S:M:A1 S M A1 0.4147 0.6483 0.2652 ef 0.1289
## S:M:A2 S M A2 0.7955 1.2438 0.5088 d 0.2368
# Arrange the first three colunms of the result table.
# This determines the columns order and shapes the plot output.
threeFACTORplot(res,
arrangement = c(3, 1, 2),
legend.position = c(0.9, 0.85),
xlab = "condition")
Average relative expression of a target gene under three different conditions with two, two and three levels. Error bars can be standard deviation or confidence interval. Means lacking letters in common have significant difference at alpha = 0.05 resulted from the `LSD.test` of agricolae package.
threeFACTORplot(res,
arrangement = c(1, 2, 3),
bar.width = 0.5,
fill = "Greys",
xlab = "Genotype",
ylab = "Relative Expression")
Average relative expression of a target gene under three different conditions with two, two and three levels. Error bars can be standard deviation or confidence interval. Means lacking letters in common have significant difference at alpha = 0.05 resulted from the `LSD.test` of agricolae package.
# releveling a factor levels first
res$Conc <- factor(res$Conc, levels = c("L","M","H"))
res$Type <- factor(res$Type, levels = c("S","R"))
# Producing the plot
threeFACTORplot(res,
arrangement = c(2, 3, 1),
bar.width = 0.5,
fill = "Reds",
xlab = "Drought",
ylab = "Relative Expression",
errorbar = "std",
legend.title = "Genotype",
legend.position = c(0.2, 0.8))
Average relative expression of a target gene under three different conditions with two, two and three levels. Error bars can be standard deviation or confidence interval. Means lacking letters in common have significant difference at alpha = 0.05 resulted from the `LSD.test` of agricolae package.
# When using ci as error, increase y.axis.adjust to see the plot correctly!
threeFACTORplot(res,
arrangement = c(2, 3, 1),
bar.width = 0.8,
fill = "Greens",
xlab = "Drought",
ylab = "Relative Expression",
errorbar = "ci",
y.axis.adjust = 8,
y.axis.by = 2,
letter.position.adjust = 0.6,
legend.title = "Genotype",
fontsize = 12,
legend.position = c(0.2, 0.8),
show.letters = TRUE)
Average relative expression of a target gene under three different conditions with two, two and three levels. Error bars can be standard deviation or confidence interval. Means lacking letters in common have significant difference at alpha = 0.05 resulted from the `LSD.test` of agricolae package.
Checking normality of residuals
If the residuals from a t.test or a lm object are not normally distributed, the grouping letters (deduced from the LSD.test) might be violated. In such cases, one could apply another data transformation to the wDCt data for ANOVA and mean comparison purposes or use non-parametric tests such as the Mann-Whitney test (also known as the Wilcoxon rank-sum test), wilcox.test(), which is an alternative to t.test, or the kruskal.test() test which alternative to one-way analysis of variance, to test the difference between medians of the populations using independent samples. However, the t.test function (along with the qpcrTTEST function described above) includes the var.equal argument. When set to FALSE, perform t.test under the unequal variances hypothesis.
residualsCRD <- qpcrANOVA(data_3factor_b, numberOfrefGenes = 1)$lmCRD$residuals
shapiro.test(residualsCRD)
##
## Shapiro-Wilk normality test
##
## data: residualsCRD
## W = 0.88893, p-value = 0.001724
qqnorm(residualsCRD)
qqline(residualsCRD, col = "red")
QQ-plot for the normality assessment of the residuals derived from `t.test` or `lm` functions.
Mean of technical replicates
Calculating the mean of technical replicates and getting an output table appropriate for subsequent ANOVA analysis can be done using the meanTech function. For this, the input data set should follow the column arrangement of the following example data. Grouping columns must be specified under the groups argument of the meanTech function.
# See example input data frame:
data_withTechRep
## factor1 factor2 factor3 biolrep techrep Etarget targetCt Eref refCt
## 1 Line1 Heat Ctrl 1 1 2 33.346 2 31.520
## 2 Line1 Heat Ctrl 1 2 2 28.895 2 29.905
## 3 Line1 Heat Ctrl 1 3 2 28.893 2 29.454
## 4 Line1 Heat Ctrl 2 1 2 30.411 2 28.798
## 5 Line1 Heat Ctrl 2 2 2 33.390 2 31.574
## 6 Line1 Heat Ctrl 2 3 2 33.211 2 31.326
## 7 Line1 Heat Ctrl 3 1 2 33.845 2 31.759
## 8 Line1 Heat Ctrl 3 2 2 33.345 2 31.548
## 9 Line1 Heat Ctrl 3 3 2 32.500 2 31.477
## 10 Line1 Heat Treat 1 1 2 33.006 2 31.483
## 11 Line1 Heat Treat 1 2 2 32.588 2 31.902
## 12 Line1 Heat Treat 1 3 2 33.370 2 31.196
## 13 Line1 Heat Treat 2 1 2 36.820 2 31.440
## 14 Line1 Heat Treat 2 2 2 32.750 2 31.300
## 15 Line1 Heat Treat 2 3 2 32.450 2 32.597
## 16 Line1 Heat Treat 3 1 2 35.238 2 31.461
## 17 Line1 Heat Treat 3 2 2 28.532 2 30.651
## 18 Line1 Heat Treat 3 3 2 28.285 2 30.745
# Calculating mean of technical replicates
meanTech(data_withTechRep, groups = 1:4)
## # A tibble: 6 × 8
## # Groups: factor1, factor2, factor3 [2]
## factor1 factor2 factor3 biolrep Etarget targetCt Eref refCt
## <chr> <chr> <chr> <int> <dbl> <dbl> <dbl> <dbl>
## 1 Line1 Heat Ctrl 1 2 30.4 2 30.3
## 2 Line1 Heat Ctrl 2 2 32.3 2 30.6
## 3 Line1 Heat Ctrl 3 2 33.2 2 31.6
## 4 Line1 Heat Treat 1 2 33.0 2 31.5
## 5 Line1 Heat Treat 2 2 34.0 2 31.8
## 6 Line1 Heat Treat 3 2 30.7 2 31.0
Citation
citation("rtpcr")
Contact
Email: gh.mirzaghaderi@uok.ac.ir
References
Livak, Kenneth J, and Thomas D Schmittgen. 2001. Analysis of Relative Gene Expression Data Using Real-Time Quantitative PCR and the Double Delta CT Method. Methods 25 (4). doi.org/10.1006/meth.2001.1262.
Ganger, MT, Dietz GD, and Ewing SJ. 2017. A common base method for analysis of qPCR data and the application of simple blocking in qPCR experiments. BMC bioinformatics 18, 1-11. doi.org/10.1186/s12859-017-1949-5.
Yuan, Joshua S, Ann Reed, Feng Chen, and Neal Stewart. 2006. Statistical Analysis of Real-Time PCR Data. BMC Bioinformatics 7 (85). doi.org/10.1186/1471-2105-7-85.
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