update dea

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