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a/6_dea/dea.Rmd b/6_dea/dea.Rmd index bb790f7..7658d3c 100644 --- 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 diff --git a/6_dea/img/NB_sigma_.pdf b/6_dea/img/NB_sigma_.pdf new file mode 100644 index 0000000000000000000000000000000000000000..486287438f8491587770307c6050e74236b2a560 GIT binary patch literal 8204 zcmY!laB<T$)HC5yZG6=Dxbb}B<(MKaLj?nc{G=={8yhZt=c3falFa-(m&B4(1q~M~ 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= 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) -- GitLab