# Manual

# Overview

rtpcr is a tool for analysis of RT-qPCR gene expression data using delta Ct (dCt) and delta delta Ct (ddCt) methods, including t-tests and ANOVA, repeated-measures models, and publication-ready visualizations. The package implements a general calculation method adopted from Ganger et al. (2017) and Taylor et al. (2019), covering both the Livak and Pfaffl methods. See the calculation method for details.

Figure 1: rtpcr is now available as shiny_rtpcr, a web application developed using R/Shiny for interactive analysis of qPCR data at https://mirzaghaderi.shinyapps.io/rtpcr/

## Running the shiny version of the rtpcr

If you have problem with connecting to https://mirzaghaderi.shinyapps.io/rtpcr/, you can follow the following steps in Rstudio to run the shiny version of the rtpcr.

    # install.packages("rtpcr")
    # install.packages("shiny")
    library(shiny)
    library(rtpcr)
    # Run the following code in Rstudio
    runApp(system.file("shinyapp/app.R", package = "rtpcr"))

# Functions

In the rtpcr package, functions with _DDCt at the end of their name (`ANOVA_DDCt()`, `TTEST_DDCt()`, `WILCOX_DDCt()`) perform expression analysis based on the delta delta Ct (ddCt) method, while `ANOVA_DCt()` function analyze gene expression using the delta Ct (dCt) method. The ANOVA prefix indicates that the function uses analysis of variance (using a default full factorial model or a user defined model) for statistical analysis, and mean comparisons. Mean comparisons is actually performed by the `emmeans()` function using the resulting model from the ANOVA analysis.

Function | Description
--- | ---
`ANOVA_DCt()` | dCt expression analysis for all the level combinations of factor(s).
`ANOVA_DDCt()` | ddCt expression analysis for levels of a factor (generally or per levels of another factors(s)), specified by the `specs` argument.
`TTEST_DDCt()` | ddCt method t.test analysis for paired or unpaired samples.
`WILCOX_DDCt()` | ddCt method wilcox.test analysis for paired or unpaired samples.
`plotFactor()` | Bar plot of gene expression for one-, two- or three-factor experiments
`Means_DDCt()` | Pairwise comparison of RE values for any user-specified effect
`efficiency()` | Amplification efficiency statistics and standard curves
`meanTech()` | Calculate mean of technical replicates. This is used if your data needs averaging over biological replicates.
`multiplot()` | Combine multiple ggplot objects into a single layout
`compute_wDCt()` | Cleaning data and weighted delta Ct (wDCt) calculation using the geometric mean of reference gene(s).
`long_to_wide()` | Converts a 4-column qPCR long data format to wide format

# Quick start

### Installing and loading

The `rtpcr` package can be installed by running the following code in R:

from CRAN:

    # Installing from CRAN
    install.packages("rtpcr")

    # Loading the package
    library(rtpcr)

Or from from GitHub (developing version):

    devtools::install_github("mirzaghaderi/rtpcr", build_vignettes = TRUE)

# Input data structure

For relative expression analysis (using `TTEST_DDCt()`, `WILCOX_DDCt()`, `ANOVA_DCt()`, and `ANOVA_DDCt()` functions), the amplification efficiency (`E`) and `Ct` or `Cq` values (the mean of technical replicates) is used for the input table. If the `E` values are not available you should use ‘2’ instead representing the complete primer amplification efficiency. The input data table should include the following columns from left to wright:

1. Experimental condition columns (and one block if available NOTE 1)
2. Replicates information (biological replicates or subjects; see NOTE 2, and NOTE 3)
3. Target genes efficiency and Ct values (a pair column for each gene).
4. Reference genes efficiency and Ct values (a pair column for each gene) NOTE 4.

The package supports one or more target or reference gene(s), supplied as efficiency–Ct column pairs. Reference gene columns must always appear last. Two sample input data sets are presented below.

Figure 2: A sample input data with one experimetal factor, replicate column and E/Ct information of target and reference genes

If there is no blocking factor, the block column should be omitted. However, a column for biological replicates (which may be named “Rep”, “id” or similar) is always required.

Figure 3: A sample input data with two experimetal factors, blocking factor, replicate column and E/Ct information of target and reference genes

#### NOTE 1

When a qPCR experiment is done in multiple qPCR plates, variation resulting from the plates may interfere with the actual amount of gene expression. One solution is to conduct each plate as a randomized block so that at least one replicate of each treatment and control is present on a plate. Block effect is usually considered as random and its interaction with any main effect is not considered.

#### NOTE 2

For `TTEST_DDCt()` and `WILCOX_DDCt()` (independent groups), `ANOVA_DCt()`, and `ANOVA_DDCt()` each row is from a separate and unique biological replicate. For example, a data frame with 12 rows has come from an experiment with 12 individuals. The repeated measure models are intended for experiments with repeated observations (e.g. time-course data). In repeated measure experiments the replicate column contains identifiers for each individual (id or subject). For example, all rows with a `r1` at Rep column correspond to a single individual, all rows with a `r2` correspond to another individual, and so on, which have been sampled at specific time points.

#### NOTE 3

Your data table may also include a column of technical replicates. For example, using one target and one reference genes, if you want to have 4 biological and 3 technical replicates under Control and Treatment conditions, there would be a table of 24 rows containing both biological replicates and technical replicate columns in the data. In this case, the `meanTech()` function should be applied first to calculate the mean of the technical replicates. The resulting collapsed table is then used as the input for expression analysis. To use the `meanTech()` function correctly, the technical replicate column must appear immediately after the biological replicate column (see Mean of technical replicates for an example).

Figure 4: An experimental design with four biological replicates for both Control and Treatment conditions, assuming a single sampling time point, where cDNA samples were analyzed by qPCR. The diagram details the initial dataset containing technical replicates. Three technical replicates shown for biological replicate 1 under Control, with example amplification efficiencies (E) and cycle threshold (Ct) values for both target and reference genes. Technical replicates is averaged to get a condensed dataset, comprising eight rows (one per biological replicate). This final data structure is used for the downstream relative expression analysis. Green and yellow circles are control samples.

#### NOTE 4

Complete amplification efficiency (E) in the input data is denoted by 2. This means that 2 indicates 100%, and 1.85 and 1.70 indicate 0.85% and 0.70% amplification efficiencies.

# Handling missing data

Missing Ct values for target genes is Handled using the `set_missing_target_Ct_to_40` argument. If `TRUE`, missing target gene Ct values become 40; if `FALSE` (default), they become NA. missing Ct values of reference genes are always converted to NA. If there are more than one reference gene, NA in the place of the E or the Ct value of cause skipping that gene and remaining references are geometrically averaged in that replicate.

# Data Analysis

### Amplification Efficiency

The `efficiency()` function calculates the amplification efficiency (E), slope, and R² statistics for genes, and performs pairwise comparisons of slopes. It takes a data frame in which the first column contains the dilution ratios, followed by the Ct value columns for each gene.

    # Applying the efficiency function
    data <- read.csv(system.file("extdata", "data_efficiency.csv", package = "rtpcr"))
    data
    # dilutions Gene1   Gene2   Gene3
    # 1.00  25.58   24.25   22.61
    # 1.00  25.54   24.13   22.68
    # 1.00  25.50   24.04   22.63
    # 0.50  26.71   25.56   23.67
    # 0.50  26.73   25.43   23.65
    # 0.50  26.87   26.01   23.70
    # 0.20  28.17   27.37   25.11
    # 0.20  28.07   26.94   25.12
    # 0.20  28.11   27.14   25.11
    # 0.10  29.20   28.05   26.17
    # 0.10  29.49   28.89   26.15
    # 0.10  29.07   28.32   26.15
    # 0.05  30.17   29.50   27.12
    # 0.05  30.14   29.93   27.14
    # 0.05  30.12   29.71   27.16
    # 0.02  31.35   30.69   28.52
    # 0.02  31.35   30.54   28.57
    # 0.02  31.35   30.04   28.53
    # 0.01  32.55   31.12   29.49
    # 0.01  32.45   31.29   29.48
    # 0.01  32.28   31.15   29.26

    # Analysis
    efficiency(data)

    # $Efficiency
    #    Gene     Slope        R2        E
    # 1 Gene1 -3.388094 0.9965504 1.973110
    # 2 Gene2 -3.528125 0.9713914 1.920599
    # 3 Gene3 -3.414551 0.9990278 1.962747
    #
    # $Slope_compare
    # $contrasts
    #  contrast          estimate    SE df t.ratio p.value
    #  C2H2.26 - C2H2.01   0.1400 0.121 57   1.157  0.4837
    #  C2H2.26 - GAPDH     0.0265 0.121 57   0.219  0.9740
    #  C2H2.01 - GAPDH    -0.1136 0.121 57  -0.938  0.6186

Figure 5: Standard curve plot displaying the relationship between the logarithm of cDNA dilution factors (ranging from -2.0 to 0.0) and their corresponding qPCR cycle threshold (Ct) values for three genes: C2H2.26, C2H2.01, and GAPDH. The accompanying table provides the precise Ct measurements, which is used to determine the amplification efficiency for each gene

### Relative expression

Single factor experiment with two levels (e.g. Control and Treatment): `TTEST_DDCt()` function is used for relative expression analysis in treatment condition compared to the control group. Both paired and unpaired experimental designs are supported. if the data doesn’t follow t.test assumptions, the `WILCOX_DDCt()` function can be used instead.

Single- or multi-factor experiments: In these cases, `ANOVA_DDCt()` and `ANOVA_DCt()` functions are used for the analysis of qPCR data. By default, statistical analysis is performed based on uni- or multi-factorial Completely Randomized Design (CRD) or Randomized Complete Block Design (RCBD) design based on `numOfFactors` and the availability of `block`. However, optional custom model formula as a character string can be supplied to these functions. If provided, this overrides the default formula (full factorial CRD or RCBD design). The formula uses `wDCt` as the response variable (wDCt is automatically created by the function). For mixed models, include random effects using `lmer` syntax (e.g., `wDCt ~ Treatment + (1 | id)`). Below are a sample of most common models that can be used.

Example models may be used in `ANOVA_DCt()` or `ANOVA_DDCt()` functions | Experimental design
--- | ---
wDCt ~ Condition | Completely Randomized Design (CRD). Can also be used for t.test with two independent groups. (default)
wDCt ~ Factor1 * Factor2 * Factor3 | Factorial under Completely Randomized Design (RCBD) (default)
wDCt ~ block + Factor1 * Factor2 | Factorial under Randomized Complete Block Design (default)
wDCt ~ time + (1 | id) | Repeated measure analysis: different time points. Also can be used for t.test with two paired groups.
wDCt ~ Condition * time + (1 | id) | Repeated measure analysis: split-plot in time
wDCt ~ wDCt ~ Condition * time + (1 | block) + (1 | id) | Repeated measure analysis: split-plot in time
wDCt ~ Type + Concentration | Analysis of Covariance: Type is covariate
wDCt ~ block + Type + Concentration | Analysis of Covariance with blocking factor: block and Type are covariates

#### NOTE

By default, data are analysed according to a full factorial CRD (if there is no blocking factor) or RCBD (if there is a blocking factor) model, so you do not need to explicitly define a model. The package automatically selects an appropriate model based on the provided arguments. If no model is specified, the default used model is printed along with the output expression table.

#### NOTE

Sometime groups are paired (e.g., in repeated measure experiments). Examples: 1) Analyzing gene expression in different time points, or before and after treatment in each biological replicate; 2) Analyzing gene expression between two tissue types within the same organism. For such analysis types, if there are only two levels or time points, we can use the `TTEST_DDCt()` with the argument `paired = TRUE`; or `ANOVA_DDCt()` (if there are two or more time points) with a repeated measure model such as `wDCt ~ Treatment + ( 1 | id)` or `wDCt ~ Treatment + ( 1 | Rep)`.

Figure 6: A) Basic structure of independent group- or paired group-experiments. Data of paired group-experiments are analysed using the `TTEST_DDCt()` function with the argument paired = TRUE; or using the `ANOVA_DDCt()` function with a repeated measure model such as wDCt ~ Treatment + ( 1 | Rep). B) The standard error (`se`) in the `ANOVA_DDCt()` function is calculated from from weighted delta Ct (wDCt) values or model residuals (default, `modelBased_se = TRUE`) for the experimental groups according to the selected `se.type` (One of `"paired.group"`, `"two.group"`, or `"single.group"`). Note that in the `"paired.group"` method,`se` is computed from differences (wddCt or equivalent values from residuals) which resembles the standard error of a paired t.test, while `"two.group"` and `"single.group"` methods use wdCt or resudual values. In the `"two.group"` method, `se` is computed for each non-reference group from that group and the reference (calibrator) group which resembles the standard error of an unpaired t.test. `"single.group"` computes `se` within each reference or non-reference group. If a model for a repeated‐measure or a paired-group design is specified, `se.type` should be set to `"paired.group"` in the `ANOVA_DDCt()` function. In method 4, the 95% confidence interval (CI) is calculated and printed in the output expression table under LCL and UCL columns. CI uses the pooled standard error (SE) and can be used as another way of error presentation. It can be used for any experimental models as it is derived from a fitted model.

### Examples

Relative expression analysis can be done using ddCt or dCt methods through different functions (i.e. `TTEST_DDCt()`, `WILCOX_DDCt()`, `ANOVA_DDCt()`, and `ANOVA_DCt()`). Below are some examples of expression analysis using ddCt method.

    data <- read.csv(system.file("extdata", "data_Yuan2006PMCBioinf.csv", package = "rtpcr"))

    # Anova analysis
    ANOVA_DDCt(
      data,
      specs = "condition",
      numOfFactors = 1,
      numberOfrefGenes = 1,
      block = NULL)

    # An example of a properly arranged dataset from a repeated-measures experiment.
    data <- read.csv(system.file("extdata", "data_repeated_measure_1.csv", package = "rtpcr"))

    # time  id  E_Target    Ct_target   E_Ref      Ct_Ref
    #    1   1      2       18.92   2   32.77
    #    1   2      2       15.82   2   32.45
    #    1   3      2       19.84   2   31.62
    #    2   1      2       19.46   2   33.03
    #    2   2      2       17.56   2   33.24
    #    2   3      2       19.74   2   32.08
    #    3   1      2       15.73   2   32.95
    #    3   2      2       17.21   2   33.64
    #    3   3      2       18.09   2   33.40

    # Repeated measure analysis
    res <- ANOVA_DDCt(
      data,
      numOfFactors = 1,
      numberOfrefGenes = 1,
      specs = "time",
      block = NULL, model = wDCt ~ time + (1 | id))

    # Paired t.test (equivalent to repeated measure analysis, but not
    # always the same results, due to different calculation methods!)
    TTEST_DDCt(
      data[1:6,],
      numberOfrefGenes = 1,
      paired = T)

    # Anova analysis
    data3 <- read.csv(system.file("extdata", "data_2factorBlock3ref.csv", package = "rtpcr"))

    res <- ANOVA_DDCt(
      x = data3,
      specs = "Type | Concentration",
      numOfFactors = 2,
      numberOfrefGenes = 3,
      block = "block",
      analyseAllTarget = TRUE)

# Output

## Data output

All the functions for relative expression analysis (including `TTEST_DDCt()`, `WILCOX_DDCt()`, `ANOVA_DDCt()`, and `ANOVA_DCt()`) return the relative expression table which include fold change and corresponding statistics. The output of `ANOVA_DDCt()`, and `ANOVA_DCt()` also include lm model, residuals, raw data and ANOVA table for each gene. These outputs can be obtained as follow:

Per_gene Output | Code
--- | ---
expression table | `res$relativeExpression`
ANOVA table | `res$perGene$gene_name$ANOVA_table`
ANOVA lm | `res$perGene$gene_name$lm`
ANOVA lm formula | `res$perGene$gene_name$lm_formula`
Residuals | `resid(res$perGene$gene_name$lm)`

    # Relative expression table for the specified column in the input data:
    data3 <- read.csv(system.file("extdata", "data_2factorBlock3ref.csv", package = "rtpcr"))

    res <- ANOVA_DDCt(
      x = data3,
      specs = "Concentration",
      numOfFactors = 2,
      numberOfrefGenes = 3,
      block = "block",
      analyseAllTarget = TRUE)

    # Relative Expression
    # gene contrast    ddCt     RE  log2FC    LCL    UCL      se L.se.RE U.se.RE L.se.log2FC U.se.log2FC  pvalue sig
    #   PO       L1  0.0000 1.0000  0.0000 0.0000 0.0000 0.13940 0.90790 1.10144     0.00000     0.00000 1.00000
    #   PO L2 vs L1 -0.9461 1.9266  0.9461 1.2586 2.9493 0.14499 1.74245 2.13036     0.85564     1.04613 0.00116  **
    #   PO L3 vs L1 -2.1919 4.5693  2.1919 3.0806 6.7772 0.29402 3.72685 5.60221     1.78783     2.68748 0.00000 ***
    #  NLM       L1  0.0000 1.0000  0.0000 0.0000 0.0000 0.91809 0.52921 1.88962     0.00000     0.00000 1.00000
    #  NLM L2 vs L1  0.8656 0.5487 -0.8656 0.3983 0.7561 0.36616 0.42577 0.70734    -1.11579    -0.67163 0.00018 ***
    #  NLM L3 vs L1 -1.4434 2.7196  1.4434 1.9467 3.7994 0.17132 2.41511 3.06256     1.28179     1.62542 0.00000 ***
    #
    # The L1 level was used as calibrator.
    # Note: Using default model for statistical analysis: wDCt ~ block + Concentration * Type

    ANOVA_table <- res$perGene$PO$ANOVA_table
    ANOVA_table

    lm <- res$perGene$PO$lm
    lm

    lm_formula <- res$perGene$gene_name$lm_formula
    lm_formula

    residuals <- resid(res$perGene$gene_name$lm)
    residuals

Figure 7: A) All the functions for relative expression analysis (including `TTEST_DDCt()`, `WILCOX_DDCt()`, `ANOVA_DDCt()`, and `ANOVA_DCt()`) return the relative expression table which include fold change and corresponding statistics. B) The output of `ANOVA_DDCt()`, and `ANOVA_DCt()` also include lm model, residuals, raw data and ANOVA table for each gene.

## Plot output

A single function of `plotFactor()` is used to produce bar plots for one- to three-factor expression tables.

## Plot output: example 1

    data <- read.csv(system.file("extdata", "data_3factor.csv", package = "rtpcr"))
    #Perform analysis first
    res <- ANOVA_DCt(
      data,
      numOfFactors = 3,
      numberOfrefGenes = 1,
      block = NULL)

    df <- res$relativeExpression
    df
     # Generate three-factor bar plot
    plotFactor(
      df,
      x_col = "SA",
      y_col = "log2FC",
      group_col = "Type",
      facet_col = "Conc",
      Lower.se_col = "Lower.se.log2FC",
      Upper.se_col = "Upper.se.log2FC",
      letters_col = "sig",
      letters_d = 0.3,
      col_width = 0.7,
      dodge_width = 0.7,
      fill_colors = c("palegreen3", "skyblue"),
      color = "black",
      base_size = 14,
      alpha = 1,
      legend_position = c(0.1, 0.2))

Figure 8: The bar plots of the log2 fold change expression expression of a gene produced by the `plotFactor()` function.

# How to edit ouptput plots?

The plot can further be edited by adding new layers (see examples bellow):

Task | Example Code
--- | ---
Change y-axis label | `p + ylab("Relative expression ($\Delta\Delta Ct$ method)")`
Add a horizontal reference line | `p + geom_hline(yintercept = 0, linetype = "dashed")`
Change y-axis limits | `p + scale_y_continuous(expand = expansion(mult = c(0, 0.1)))`
Relabel x-axis | `p + scale_x_discrete(labels = c("A" = "Control", "B" = "Treatment"))`
Change fill colors | `p + scale_fill_brewer(palette = "Set2")`
x.axis line width | `p + theme(axis.line.x = element_line(linewidth = 0))`
y.axis line width | `p + theme(axis.line.y = element_line(linewidth = 0))`
panel line width | `theme(panel.border = element_rect(color = "black", linewidth = 1))`

### Plot output: example 2

    data <- read.csv(system.file("extdata", "data_2factorBlock.csv", package = "rtpcr"))
    res <- ANOVA_DCt(data,
          numOfFactors = 2,
          block = "block",
          numberOfrefGenes = 1)

    df <- res$relativeExpression

    plotFactor(
      data = df,
      x_col = "Concentration",
      y_col = "RE",
      group_col = "Type",
      Lower.se_col = "Lower.se.RE",
      Upper.se_col = "Upper.se.RE",
      letters_col = "sig",
      letters_d = 0.2,
      fill_colors = c("aquamarine4", "gold2"),
      color = "black",
      alpha = 1,
      col_width = 0.7,
      dodge_width = 0.7,
      base_size = 16,
      legend_position = c(0.2, 0.8))

Figure 9: The bar plots of the relative expression expression and log2 fold change of a gene produced by the `plotFactor()` function.

### Plot output: example 3

    # Using data from Heffer et al., 2020, PlosOne
    library(dplyr)

    res <- ANOVA_DDCt(
      data_Heffer2020PlosOne,
      numOfFactors = 1,
      specs = "Treatment",
      numberOfrefGenes = 1,
      block = NULL)

    data <- res$relativeExpression

    # Selecting only the first words in 'contrast' column to be used as the x-axis labels.
    data$contrast <- sub(" .*", "", data$contrast)

    plotFactor(
      data = data,
      x_col = "contrast",
      y_col = "RE",
      group_col = "contrast",
      facet_col = "gene",
      Lower.se_col = "Lower.se.RE",
      Upper.se_col = "Upper.se.RE",
      letters_col = "sig",
      letters_d = 0.2,
      alpha = 1,
      fill_colors = palette.colors(4, recycle = TRUE),
      color = "black",
      col_width = 0.5,
      dodge_width = 0.5,
      base_size = 16,
      legend_position = "none")

Figure 10: The bar plot of the fold change expression of 6 genes produced by the `plotFactor()` function.

# Post-hoc analysis

Although all the expression analysis functions perform statistical comparisons for the levels of the analysed factor, further post-hoc analysis is still possible. The `Means_DDCt()` function performs post-hoc comparisons using a fitted model object produced by `ANOVA_DCt()` and `ANOVA_DDCt()`. It applies pairwise statistical comparisons of relative expression (RE) values for user-specified effects via the `specs` argument. Supported effects include simple effects, interactions, and slicing, provided the underlying model is an ANOVA. For ANCOVA models, only simple effects are returned.

    res <- ANOVA_DDCt(
      data_3factor,
      numOfFactors = 3,
      numberOfrefGenes = 1,
      specs = "Conc",
      block = NULL)

    model <- res$perGene$E_PO$lm
    # Relative expression values for Concentration main effect
    Means_DDCt(model, specs = "Conc")

    # contrast        RE        SE df       LCL       UCL p.value sig
    # L vs H   0.1703610 0.2208988 24 0.1242014 0.2336757 <0.0001 ***
    # M vs H   0.2227247 0.2208988 24 0.1623772 0.3055004 <0.0001 ***
    # M vs L   1.3073692 0.2208988 24 0.9531359 1.7932535  0.0928 .
    #
    #Results are averaged over the levels of: Type, SA
    #Confidence level used: 0.95

    # Relative expression values for Concentration sliced by Type
    Means_DDCt(model, specs = "Conc | Type")

    # Type = R:
    #  contrast       RE        SE df       LCL      UCL p.value sig
    #  L vs H   0.103187 0.3123981 24 0.0659984 0.161331 <0.0001 ***
    #  M vs H   0.339151 0.3123981 24 0.2169210 0.530255 <0.0001 ***
    #  M vs L   3.286761 0.3123981 24 2.1022126 5.138776 <0.0001 ***
    #
    # Type = S:
    #  contrast       RE        SE df       LCL      UCL p.value sig
    #  L vs H   0.281265 0.3123981 24 0.1798969 0.439751 <0.0001 ***
    #  M vs H   0.146266 0.3123981 24 0.0935518 0.228684 <0.0001 ***
    #  M vs L   0.520030 0.3123981 24 0.3326112 0.813055  0.0059 **
    #
    # Results are averaged over the levels of: SA
    # Confidence level used: 0.95

    # Relative expression values for Concentration sliced by Type and SA
    Means_DDCt(model, specs = "Conc | Type * SA")

# Checking normality of residuals

If the residuals from a `t.test` or an `lm` object are not normally distributed, the significance results might be violated. In such cases, non-parametric tests can be used. For example, the Mann–Whitney test - also known as the Wilcoxon rank-sum test, (implemented via `WILCOX_DDCt()` in the rtpcr package), is an alternative to t.test, and `kruskal.test()` is an alternative to one-way analysis of variance. These tests assess differences between population medians using independent groups. However, the rtpcr `TTEST_DDCt()` function includes the `var.equal` argument. When set to `FALSE`, performs `t.test` under the unequal variances hypothesis. Residuals from `ANOVA_DCt()` and `ANOVA_DDCt()` functions can be extracted from `lm`and plotted as follow:

    data <- read.csv(system.file("extdata", "data_repeated_measure_1.csv", package = "rtpcr"))
    res3 <- ANOVA_DDCt(
      data,
      numOfFactors = 1,
      numberOfrefGenes = 1,
      specs = "time",
      block = NULL,
      model = wDCt ~ time + (1 | id))

    residuals <- resid(res3$perGene$Target$lm)
    shapiro.test(residuals)
    par(mfrow = c(1,2))
    plot(residuals)
    qqnorm(residuals)
    qqline(residuals, col = "red")

# Mean of technical replicates

Calculating the mean of technical replicates and generating an output table suitable for subsequent expression analysis can be accomplished using the `meanTech()` function. The input dataset must follow the column structure illustrated in the example data below. Columns used for grouping should be explicitly specified via the `groups` argument of the `meanTech()` function.

    # Example input data frame with technical replicates
    data1 <- read.csv(system.file("extdata", "data_withTechRep.csv", package = "rtpcr"))

    # Calculate mean of technical replicates using first four columns as groups
    meanTech(data1,
             groups = 2,
             numOfFactors = 1,
             block = NULL)

Figure 11: The mean of technical replicates can be computed using the `meanTech()` function.

# Contact

Email: gh.mirzaghaderi at uok.ac.ir

# Citation

    citation("rtpcr")

    To cite the package ‘rtpcr’ in publications, please use:

      Ghader Mirzaghaderi (2025). rtpcr: a package for statistical analysis and graphical
      presentation of qPCR data in R. PeerJ 13:e20185. https://doi.org/10.7717/peerj.20185

    A BibTeX entry for LaTeX users is

      @Article{,
        title = {rtpcr: A package for statistical analysis and graphical presentation of qPCR data in R},
        author = {Ghader Mirzaghaderi},
        journal = {PeerJ},
        volume = {13},
        pages = {e20185},
        year = {2025},
        doi = {10.7717/peerj.20185},
      }

# 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, 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.

Mirzaghaderi G. 2025. rtpcr: a package for statistical analysis and graphical presentation of qPCR data in R. PeerJ 13, e20185. doi.org/10.7717/peerj.20185.

Pfaffl MW, Horgan GW, Dempfle L. 2002. Relative expression software tool (REST©) for group-wise comparison and statistical analysis of relative expression results in real-time PCR. Nucleic acids research 30, e36-e36. doi.org/10.1093/nar/30.9.e36.

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 in Biotechnology, 37(7), 761-774. doi.org/10.1016/j.tibtech.2018.12.002.

Yuan, JS, 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.