diff --git a/6_dea/dea.Rmd b/6_dea/dea.Rmd
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--- a/6_dea/dea.Rmd
+++ b/6_dea/dea.Rmd
@@ -32,51 +32,507 @@ classoption: aspectratio=169
## Hypothesis testing
-\begin{itemize}
- \item $H_0$: the gene pression is the same between the $2$ groups
- \item $H_1$: the gene pression is not the same between the $2$ groups
-\end{itemize}
+### Rejection of a null hypothesis $H_0$
+ Given the null model of our data how likely are we to observe a value? \\
+ We can compute this likelihood from the probability distribution of the null hypothesis.\\
+ We reject the hypothesis at risk $\alpha$, the probability that the null hypothesis was true for the observed value.
+
+### $p$-value
+ The $p$-value is the probability to observe a value as or more extreme under the null hypothesis model.
## Hypothesis testing
+\begin{center}
+\only<1>{\includegraphics[width=12cm]{img/dnorm}}
+\only<2>{\includegraphics[width=12cm]{img/dnorm_alpha}}
+\only<3>{\includegraphics[width=12cm]{img/dnorm_alpha_accept}}
+\only<4>{\includegraphics[width=12cm]{img/dnorm_alpha_reject}}
+\end{center}
\begin{itemize}
- \item $H_0$: the gene pression is the same between the $2$ groups
- \item $H_1$: the gene pression is not the same between the $2$ groups
+ \item distribution under the null hypothesis $H_0$
+ \pause
+ \item rejection zone at a risk $\alpha$
\end{itemize}
+## $p$-value construction
+\begin{center}
+ \only<1>{\includegraphics[width=10cm]{img/pval_1_0.05}}
+ \only<2>{\includegraphics[width=10cm]{img/pval_1_0.15}}
+ \only<3>{\includegraphics[width=10cm]{img/pval_1_0.25}}
+ \only<4>{\includegraphics[width=10cm]{img/pval_1_0.35}}
+ \only<5>{\includegraphics[width=10cm]{img/pval_1_0.45}}
+ \only<6>{\includegraphics[width=10cm]{img/pval_1_0.55}}
+ \only<7>{\includegraphics[width=10cm]{img/pval_1_0.65}}
+ \only<8>{\includegraphics[width=10cm]{img/pval_1_0.75}}
+ \only<9>{\includegraphics[width=10cm]{img/pval_1_0.85}}
+ \only<10>{\includegraphics[width=10cm]{img/pval_1_0.95}}
+\end{center}
+ \begin{center}
+ probability to observe a value as or more extreme under the null hypothesis model
+ \end{center}
+
+## $p$-value construction
\begin{center}
- {\bf We reject $H_0$ with a given risk $\alpha$ and a power $\beta$}
+ \only<1>{\includegraphics[width=10cm]{img/pval_2_0.05}}
+ \only<2>{\includegraphics[width=10cm]{img/pval_2_0.15}}
+ \only<3>{\includegraphics[width=10cm]{img/pval_2_0.25}}
+ \only<4>{\includegraphics[width=10cm]{img/pval_2_0.35}}
+ \only<5>{\includegraphics[width=10cm]{img/pval_2_0.45}}
+ \only<6>{\includegraphics[width=10cm]{img/pval_2_0.55}}
+ \only<7>{\includegraphics[width=10cm]{img/pval_2_0.65}}
+ \only<8>{\includegraphics[width=10cm]{img/pval_2_0.75}}
+ \only<9>{\includegraphics[width=10cm]{img/pval_2_0.85}}
+ \only<10>{\includegraphics[width=10cm]{img/pval_2_0.95}}
\end{center}
## Hypothesis testing
+\begin{center}
+ \href{https://en.wikipedia.org/}{
+ \includegraphics[width=0.6\textwidth]{img/alpha_beta.png}
+ }
+\end{center}
+\vspace{-2.5em}
+We specify $H_0$ not $H_1$
+
+## Hypothesis testing
+
+\begin{center}
+\begin{columns}
+\column{0.6\textwidth}
\begin{itemize}
\item $\beta$ probability of a Type II error, known as a {\it false negative}
\item $1 - \beta$ probability of a {\it true positive}, i.e., correctly rejecting the null hypothesis. $1 - \beta$ is also known as the power of the test.
\item $\alpha$ probability of a Type I error, known as a {\it false positive}
\item $1 - \alpha$ probability of a {\it true negative}, i.e., correctly not rejecting the null hypothesis
\end{itemize}
+\column{0.4\textwidth}
-## Hypothesis testing
+\begin{center}
+ {\bf ROC curve}
+ \href{https://en.wikipedia.org/}{
+ \includegraphics[width=\textwidth]{img/roc_curve.png}
+ }
+\end{center}
+receiver operating characteristic
+\end{columns}
+\end{center}
+
+## Differential expression analysis
+
+### Finding difference in gene expression
+\vspace{1em}
+
+\begin{center}
+\begin{columns}
+\column{0.5\textwidth}
+\begin{center}
+\begin{tikzpicture}
+ \fill
+ (0.5,3.5) node {\bf $\text{gene}_1$}
+ -- (0.5,2.5) node {\bf $\text{gene}_2$}
+ -- (0.5,1.5) node {\bf $\vdots$}
+ -- (0.5,0.5) node {\bf $\text{gene}_n$};
+ \fill
+ (1.5,4.5) node {\bf{$\text{cell}_1$}}
+ -- (1.5,3.5) node {mRNA}
+ -- (1.5,2.5) node {mRNA}
+ -- (1.5,1.5) node {$\vdots$}
+ -- (1.5,0.5) node {mRNA};
+ \fill
+ (2.5,4.5) node {\bf{$\text{cell}_2$}}
+ -- (2.5,3.5) node {mRNA}
+ -- (2.5,2.5) node {mRNA}
+ -- (2.5,1.5) node {$\vdots$}
+ -- (2.5,0.5) node {mRNA};
+ \fill
+ (3.5,4.5) node {\bf{$\cdots$}}
+ -- (3.5,3.5) node {$\cdots$}
+ -- (3.5,2.5) node {$\cdots$}
+ -- (3.5,1.5) node {$\ddots$}
+ -- (3.5,0.5) node {$\cdots$};
+ \fill
+ (4.5,4.5) node {\bf{$\text{cell}_c$}}
+ -- (4.5,3.5) node {mRNA}
+ -- (4.5,2.5) node {mRNA}
+ -- (4.5,1.5) node {$\vdots$}
+ -- (4.5,0.5) node {mRNA};
+ \draw (1,0) grid (5,4);
+\end{tikzpicture}
+\end{center}
+\column{0.5\textwidth}
+For a given gene $x_i$ we can test:
+\vspace{1em}
+\begin{itemize}
+ \item $H_0$: $E\left(x_i\right) = E\left(x_{i'}\right)$
+ \item $H_0$: $E\left(x_i\right) = E\left(x_{i'}\right) = \dots = E\left(x_{i''}\right)$
+ \item $H_0$: $E\left(x_i \times y\right) = E\left(x_i \times 1\right)$
+\end{itemize}
+\vspace{2em}
+or any combination of the above cases
+\end{columns}
+\end{center}
+
+## Differential expression analysis
+\begin{center}
+ \href{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8701051/}{
+ \includegraphics[width=0.9\textwidth]{img/dea_tree.png}
+ }
+\end{center}
+
+## Counts distributions
+
+$P(X = x)$ for $\mathcal{NB}(\lambda, \alpha = 10)$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/NB_sigma_10.png}
+\end{center}
+
+
+## Counts distributions
+
+$P(X = x)$ for $\mathcal{NB}(\lambda, \alpha = 2)$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/NB_sigma_2.png}
+\end{center}
+
+## Counts distributions
+
+$P(X = x)$ for $\mathcal{NB}(\lambda, \alpha = 1)$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/NB_sigma_1.png}
+\end{center}
+
+## Non-parametric approaches
+
+### We don't try to model the data distribution
+
+Instead we work with:
+
+- ranks of the values
+- the sign of the difference between two groups (Wilcoxon)
+- the distribution of differances
+
+If we know the distribution the parametric approach is often more powerfull
+
+### Often limited to the 2 groups setting
+
+
+## Wilcoxon rank sum test
+
+### $H_0$: the median are equal
+
+\begin{center}
+ \href{https://www.nature.com/articles/s41467-021-27464-5}{
+ \includegraphics[width=\textwidth]{img/wilcoxon_example.png}
+ }
+\end{center}
+
+## WaddR
+
+### Base on 2-Wasserstein distance
+\begin{center}
+ \href{https://pubmed.ncbi.nlm.nih.gov/33792651/}{
+ \includegraphics[width=\textwidth]{img/waddR.png}
+ }
+\end{center}
+
+## Model based approaches
+
+\begin{center}
+\begin{columns}
+\column{0.5\textwidth}
+
+\vspace{1em}
+
+{\bf Gaussian models}
+
+\[
+log(X + 1) \sim \mathcal{N}(\mu, \sigma)
+\]
+
+{\bf Poisson models}
+
+\[
+X \sim \mathcal{P}(\lambda)
+\]
+
+{\bf NB models}
+
+\[
+X \sim \mathcal{NB}(\lambda, \alpha)
+\]
+
+{\bf ZINB models}
+
+\[
+X \sim \pi \delta_0 + \left(1 - \pi\right) \mathcal{NB}(\lambda, \alpha)
+\]
+\column{0.5\textwidth}
\begin{center}
\href{https://en.wikipedia.org/}{
- \includegraphics[width=0.6\textwidth]{img/alpha_beta.png}
+ \includegraphics[width=\textwidth]{img/dirac.png}
+ }
+\end{center}
+\end{columns}
+\end{center}
+
+
+## Model based approaches
+
+### NB distributed counts with excess of zeros
+\begin{center}
+ \includegraphics[width=0.8\textwidth]{img/ziNB_1}
+\end{center}
+
+## Model based approaches
+
+### Mixture of two NB distributions
+
+\begin{center}
+ \includegraphics[width=0.8\textwidth]{img/ziNB_2}
+\end{center}
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/lm_2_groups_b0_3_b1_05.png}
+\end{center}
+
+$\beta_0 = 3$, $\beta_1 = 0.5$
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x$
+
+### Wald test:
+\[H_0: \beta_1 = 0\]
+
+### Likelihood ratio test (LTR)
+\[H_0: L\left(y = \beta_0\right) = L\left(y = \beta_0 + \beta_1 x\right)\]
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/lm_b0_3_b1_05.png}
+\end{center}
+
+$\beta_0 = 3$, $\beta_1 = 0.5$
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x$
+
+\begin{center}
+ \href{https://cole-trapnell-lab.github.io/monocle3/}{
+ \includegraphics[width=0.7\textwidth]{img/deg_pseudotime.png}
+ }
+\end{center}
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/lm_2_groups_b0_b0_3_b1_05.png}
+\end{center}
+
+$\beta_0 = 3$, $\beta_1 = 0.5$ $\beta_2 = 5$
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/lm_2_groups_b0_b0_3_b1_05_interaction.png}
+\end{center}
+
+$\beta_0 = 3$, $\beta_1 = 0.5$ $\beta_2 = 5$
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$
+
+\begin{center}
+\includegraphics[width=0.7\textwidth]{img/lm_2_groups_2_factors_b0_b0_3_b1_05_interaction.png}
+\end{center}
+
+$\beta_0 = 3$, $\beta_1 = 0.5$ $\beta_2 = 5$
+
+## Model based approaches
+
+### $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$
+
+\begin{center}
+ href{doi: 10.1093/nar/gky675}{
+ \includegraphics[width=0.6\textwidth]{img/deg_time_group.png}
}
\end{center}
-# Parametric versus non-parametric testing
-# Differential expression analysis between groups
+# Multiple hypotheses testing
+
+## Multiple hypotheses problem
+\begin{center}
+ \only<1>{\includegraphics[width=10cm]{img/dnorm_abs}\\[-2.5em]}
+ \only<1>{\includegraphics[width=10cm]{img/pval_alpha}}
+ \only<2>{\includegraphics[width=10cm]{img/pval_alpha_random_H0_1}
+ \begin{center}
+ n = 10
+ \end{center}}
+ \only<3>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_2}
+ \begin{center}
+ n = 30
+ \end{center}}
+ \only<4>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_3}
+ \begin{center}
+ n = 60
+ \end{center}}
+ \only<5>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_4}
+ \begin{center}
+ n = 100
+ \end{center}}
+ \only<6>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_5}
+ \begin{center}
+ n = 150
+ \end{center}}
+ \only<7>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_6}
+ \begin{center}
+ n = 210
+ \end{center}}
+ \only<8>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_7}
+ \begin{center}
+ n = 280
+ \end{center}}
+ \only<9>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_8}
+ \begin{center}
+ n = 360
+ \end{center}}
+ \only<10>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_9}
+ \begin{center}
+ n = 450
+ \end{center}}
+ \only<11>{\includegraphics[width=12cm]{img/pval_alpha_random_H0_10}
+ \begin{center}
+ n = 550
+ \end{center}}
+\end{center}
+
+
+## Multiple hypotheses solutions
+
+\begin{block}{Family Wise Error Rate (FWER)}
+ \begin{itemize}
+ \item Bonferoni like procedure
+ \item Control the risk of having fewer than one false positive
+ \item $\Pr\left(FP < 1\right) < \alpha_{FWER}$
+ \item $\alpha_{FWER} = \frac{\alpha}{m}$
+ \end{itemize}
+\end{block}
+\begin{example}
+ \begin{center}
+ \emph{``We reject 14 hypothesis with a FWER of 0.05''}
+ \emph{``We reject 14 hypothesis at a level of 0.05 after Bonferoni correction''}
+ \end{center}
+ Means: 14 hypotheses are not following the null distribution and we make this statement with a probability 0.05 of having fewer than one false positives in the 14 tests.
+\end{example}
+
+## Multiple hypotheses solutions
+
+\begin{block}{False Discovery Rate (FDR)}
+ \begin{itemize}
+ \item Benjamini-Hochberg like procedure
+ \item Control the risk of having less than a proportion of false positive
+ \item $\Pr\left(\mathbb{E}\left[\frac{FP}{R}\right | R > 0]\right)\Pr\left(R > 0\right) < \alpha_{FDR}$
+ \item adaptive procedure $\alpha_{FDR} \sim f_0$
+ \end{itemize}
+\end{block}
+\only<1>{
+\vspace{2em}
+\begin{tabular}{l|ccc}
+ hypothesis & Claimed non-significant & Claimed significant & Total\\
+ \hline
+ Null & TN & FP & $m_0$\\
+ Non-null & FN & TP & $m_1$\\
+ Total & S & R & $m$
+\end{tabular}
+}
+\only<2-3>{
+ \begin{example}
+ \begin{center}
+ \emph{``We reject 254 hypothesis with a FDR of 0.05''}
+ \emph{``We reject 254 hypothesis with a level of 0.05 after BH correction''}
+ \end{center}
+ Means: 254 hypotheses are not following the null distribution and we expect on average 5\% or less of false positives in the 254.
+ \end{example}
+}
+\only<3>{
+ \begin{center}
+ The number of FPs increases with the number of TPs
+ \end{center}
+}
-## Differential expression analysis between $2$ groups
+## FWER versus FDR control
+\begin{center}
+\includegraphics[width=12cm]{img/pval_hist_H0}
+\end{center}
+$$\Pr\left(FP < 1\right) < \alpha_{FWER}$$
+$$\Pr\left(\mathbb{E}\left[\frac{FP}{R}\right | R > 0]\right)\Pr\left(R > 0\right) < \alpha_{FDR}$$
+When $TP \leq 1$ FWER and FDR control are identical.\\
+The difference increases with the number of $TP$s
-## Between $n$ groups
+## FDR control
-# Regression analysis
+\begin{center}
+\includegraphics[width=12cm]{img/pval_hist_H0_H1}\\[-1em]
+\pause
+When we analyse data we hope to get a mixture between:\\
+\includegraphics[width=12cm]{img/pval_hist_H0}\\[-2em]
+\pause
+\includegraphics[width=12cm]{img/pval_hist_H1}
+\end{center}
-# Multiple testing
+## FDR control: local FDR ($\ell FDR$) of Efron
+
+\begin{center}
+\only<1>{\includegraphics[width=12cm]{img/pval_hist_H0_H1}}
+\only<2-3>{\includegraphics[width=12cm]{img/pval_hist_H0_H1_mixture}}
+\only<4>{\includegraphics[width=12cm]{img/zval_hist_H0_H1_mixture}}
+\end{center}
+\only<3-4>{
+ $$\ell FDR\left(x_i\right) = \frac{{\color{blue}\Pr\left(x_i | H_i = 0\right)}}{{\color{blue}\Pr\left(x_i | H_i = 0\right)}{\color{red}\Pr\left(x_i | H_i = 1\right)}}$$
+}
+\only<4>{
+\begin{center}
+ work with $z$-values instead of $p$-values
+\end{center}
+}
# Post-selection inference
+## Post-selection inference
+
+\begin{center}
+ \href{https://en.wikipedia.org/}{
+ \includegraphics[width=\textwidth]{img/post_inference.png}
+ }
+\end{center}
+
+## SimCD
+
+\begin{center}
+ \href{https://arxiv.org/abs/2104.01512v1}{
+ \includegraphics[width=0.6\textwidth]{img/simCD.png}
+ }
+\end{center}
+
+area under ROC curves
+
# Multivariate Differential expression analysis
\ No newline at end of file
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diff --git a/6_dea/plot.R b/6_dea/plot.R
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--- /dev/null
+++ b/6_dea/plot.R
@@ -0,0 +1,227 @@
+library(tidyverse)
+library(patchwork)
+setwd("6_dea/")
+
+for( threshold in seq(from = 0.05, to = 1, by = 0.1)) {
+data <- tibble(
+ x = seq(from = -5, to = 5, length.out = 1000)
+) %>%
+ mutate(
+ y = dnorm(x),
+ q = qnorm(y),
+ p = pnorm(x)
+ )
+data %>%
+ ggplot(aes(x = x, y = y)) +
+ geom_line() +
+ geom_area(fill = "#18BABD") +
+ geom_ribbon(data = data %>% filter(x < qnorm(threshold)),
+ aes(xmin = -5, xmax = qnorm(threshold), ymin = 0, ymax = y),
+ fill = "#F67770") +
+ geom_vline(xintercept = qnorm(threshold), size = 2) +
+ theme_bw(base_size = 22) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = str_c("img/pval_1_", threshold, ".pdf"), bg='transparent', width = 12, height = 5)
+
+p1 <- data %>%
+ ggplot(aes(x = x, y = p)) +
+ geom_line() +
+ geom_area(fill = "#18BABD") +
+ geom_ribbon(data = data %>% filter(x < qnorm(threshold)),
+ aes(xmin = -5, xmax = pnorm(threshold), ymin = 0, ymax = p),
+ fill = "#F67770") +
+ geom_vline(xintercept = qnorm(threshold), size = 2) +
+ theme_bw(base_size = 22) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+
+p2 <- tibble(
+ x = pnorm(rnorm(100000))
+ ) %>%
+ ggplot(aes(x = x, fill = x > threshold)) +
+ geom_histogram( bins = 100) +
+ geom_vline(xintercept = threshold, size = 2) +
+ theme_bw(base_size = 22) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent'),
+ legend.position = "none"
+ )
+
+p <- p1 / p2
+ggsave(plot = p, filename = str_c("img/pval_2_", threshold, ".pdf"), bg='transparent', width = 12, height = 10)
+}
+
+tibble(
+ x = c(rep(1, 500), rep(20,500))
+) %>%
+ mutate(
+ y = 3 + x * 0.5,
+ y_rand = y + rnorm(n())
+ ) %>%
+ ggplot(aes(x = x, y = y)) +
+ geom_point(aes(y = y_rand)) +
+ geom_point(size = 10, color = "blue") +
+ theme_bw(base_size = 22) +
+ ylim(c(0, 20)) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = "img/lm_2_groups_b0_3_b1_05.png", bg='transparent', width = 12, height = 6)
+
+tibble(
+ x = seq(from = 0, to = 20, length.out = 1000)
+) %>%
+ mutate(
+ y = 3 + x * 0.5,
+ y_rand = y + rnorm(n())
+ ) %>%
+ ggplot(aes(x = x, y = y)) +
+ geom_point(aes(y = y_rand)) +
+ geom_line(size = 2, color = "blue") +
+ theme_bw(base_size = 22) +
+ ylim(c(0, 20)) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = "img/lm_b0_3_b1_05.png", bg='transparent', width = 12, height = 6)
+
+
+tibble(
+ x = seq(from = 0, to = 20, length.out = 1000),
+ group = rbinom(1000, 1, 0.5)
+) %>%
+ mutate(
+ y = 3 + x * 0.5 + group * 5,
+ y_rand = y + rnorm(n())
+ ) %>%
+ ggplot(aes(x = x, y = y, group = group)) +
+ geom_point(aes(y = y_rand)) +
+ geom_line(size = 2, color = "blue") +
+ theme_bw(base_size = 22) +
+ ylim(c(0, 20)) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = "img/lm_2_groups_b0_b0_3_b1_05.png", bg='transparent', width = 12, height = 6)
+
+tibble(
+ x = seq(from = 0, to = 20, length.out = 1000),
+ group = rbinom(1000, 1, 0.5)
+) %>%
+ mutate(
+ y = 3 + x * 0.5 + group * 5 + (group + 1) * x * -0.4,
+ y_rand = y + rnorm(n())
+ ) %>%
+ ggplot(aes(x = x, y = y, group = group)) +
+ geom_point(aes(y = y_rand)) +
+ geom_line(size = 2, color = "blue") +
+ theme_bw(base_size = 22) +
+ ylim(c(0, 20)) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = "img/lm_2_groups_b0_b0_3_b1_05_interaction.png", bg='transparent', width = 12, height = 6)
+
+tibble(
+ x = seq(from = 0, to = 20, length.out = 1000),
+ group = rbinom(1000, 1, 0.5),
+ time = as.factor(c(rep(1, 250), rep(2, 250), rep(3, 250), rep(4, 250)))
+) %>%
+ mutate(
+ y = 3 + x * 0.5 + group * 5 + (group + 1) * x * -0.4,
+ y_rand = y + rnorm(n())
+ ) %>%
+ ggplot(aes(x = x, y = y, group = group, color = time)) +
+ geom_point(aes(y = y_rand)) +
+ geom_line(size = 2, color = "blue") +
+ theme_bw(base_size = 22) +
+ ylim(c(0, 20)) +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent')
+ )
+ggsave(filename = "img/lm_2_groups_2_factors_b0_b0_3_b1_05_interaction.png", bg='transparent', width = 12, height = 6)
+
+
+tibble(
+ x = rep(1:20, 2),
+) %>%
+ mutate(
+ pNB = dnbinom(x, mu = 1, size = 10) + dnbinom(x, mu = 5, size = 10)
+ ) %>%
+ ggplot(aes(x = x, y = pNB)) +
+ geom_point(size = 6) +
+ geom_col(width = 0.2, position = "dodge") +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent'),
+ legend.position = "none"
+ )
+ggsave(file = paste0("img/ziNB_1.pdf"), width = 16, height = 9)
+
+tibble(
+ x = rep(1:20, 2),
+ group = c(rep(0, 20), rep(1, 20))
+) %>%
+ mutate(
+ pNB = dnbinom(x, mu = 1 + 4 * group, size = 10)
+ ) %>%
+ ggplot(aes(x = x, y = pNB, color = as.factor(group))) +
+ geom_point(size = 6) +
+ geom_col(width = 0.2, position = "dodge") +
+ theme(
+ panel.background = element_rect(fill='transparent'),
+ plot.background = element_rect(fill='transparent', color=NA),
+ panel.grid.major = element_blank(),
+ panel.grid.minor = element_blank(),
+ legend.background = element_rect(fill='transparent'),
+ legend.box.background = element_rect(fill='transparent'),
+ legend.position = "none"
+ )
+ggsave(file = paste0("img/ziNB_2.pdf"), width = 16, height = 9)