added permutation test

LaTeX rendering needs to be fixed

docs/statistical-inference/permutation.md

@@ -0,0 +1,118 @@
+---
+sidebar_position: 2
+sidebar_label: Permutation Testing
+title: Permutation Testing
+---
+
+import DataFrameComponent from '../../components/DataFrameComponent.jsx';
+
+We use a permutation test to test whether two samples were drawn from the same population.
+
+---
+
+## 1. State the question/hypothesis
+
+Our pair of hypotheses is:
+* **Null Hypothesis:** The mean weights of dogs and cats are the *same*.
+* **Alternative Hypothesis:** The mean weights of dogs and cats are *different*.
+
+Since the alternative hypothesis is of the form "A and B are different," the test statistic should measure distance and use an absolute value.
+
+$\therefore$ Use **absolute difference in group means** as the test statistic.
+
+---
+
+## 2. Query the DataFrame
+
+Since we want to compare the distributions of only cats and dogs, we need to make sure to only include the relevant pieces of data (e.g., cats and dogs weights).
+
+```python
+# Create a boolean Series that references which rows fulfill either condition.
+querying_condition = (full_pets.get('Species') == 'dog') | (full_pets.get('Species') == 'cat')
+# Query.
+cats_dogs = full_pets[querying_condition]
+# Display the first 5 rows only.
+cats_dogs.take(np.arange(5)) 
+```
+
+<DataFrameComponent data={'{"columns":["ID","Species","Color","Weight","Age","Is_Cat","Owner_Comment"],"index":[0,1,2,3,4],"data":[["dog_001","dog","black",40.0,5.0,false,"      There are no bad dogs, only bad owners."],["cat_001","cat","golden",1.5,0.2,true,"My best birthday present ever!!!"],["cat_002","cat","black",15.0,9.0,true,"****All you need is love and a cat.****"],["dog_002","dog","white",80.0,2.0,false,"Love is a wet nose and a wagging tail."],["dog_003","dog","black",25.0,0.5,false,"Be the person your dog thinks you are."]]}'} />
+
+---
+
+## 3. Create a function to calculate test statistic
+
+Since our hypotheses depend on the test statistic, create a function to be able to calculate it during every trial of our permutation test.
+
+```python
+def difference_in_means(cats_dogs):
+    """
+    Calculate the absolute difference in the mean weight of dogs and cats.
+    ---
+    Input:
+      cats_dogs: a DataFrame containing the columns 'Species' and 'Weight'.
+    ---
+    Output:
+        The absolute difference in the mean weight of dogs and cats.
+    """
+    means = cats_dogs.groupby('Species').mean()
+    return np.abs((means.get('Weight').loc['dog'] - means.get('Weight').loc['cat']))
+```
+
+*Note: Although we can simply do this in the function, we can practice good coding habits by separating our code into readable bits!*
+
+---
+
+## 4. Simulate the permutation test
+
+```python
+n = 500 # Number of simulations.
+statistics = np.array([]) # Array to keep track of the difference in means for each iteration.
+for i in np.arange(n): # Run the simulation `n` number of times
+    # 1. Shuffle the species.
+    shuffled = cats_dogs.assign(Species=np.random.permutation(cats_dogs.get('Species')))
+
+    # 2. Compute the test statistic.
+    statistic = difference_in_means(shuffled)
+
+    # 3. Save the result.
+    statistics = np.append(statistics, statistic)
+```
+
+This code will run the permutation test 500 times, but a different reasonable number can be used instead. It is **crucial** to keep track of the difference in means each time our for-loop runs so that the number of simulated values can be displayed. 
+
+---
+
+## 5. Conclusion
+
+```python
+observed = difference_in_means(cats_dogs)
+p_value = np.count_nonzero(statistics >= observed) / n
+
+print("The observed value of the test statistic is:", observed)
+print("The p-value is:", p_value)
+```
+**The observed value of the test statistic is: 30.361111111111107** <br />
+**The p-value is: 0.004**
+
+Using a significance level of 0.05...
+
+**Conclusion:**
+* Under the null hypothesis, we rarely see a difference greater than the observed value.
+* Therefore, we reject the null hypothesis: the evidence implies that the two groups do not come from the same distribution.
+* Still, we cannot conclude that species causes a different weight because there may be other factors at play.
+
+---
+
+## 6. Extra
+
+Let's see how our observed statistic compares to the overall simulated values!
+
+```python
+# Create the histogram.
+bpd.DataFrame().assign(statistics=statistics).plot(kind='hist', bins=20, density=True, ec='w')
+# Don't worry about this line - you won't need to know it for this course!
+plt.axvline(x=observed, c='black', linewidth=4, label='population difference in means')
+```
+![Distribution](/img/statistical-inference-plots/permutation-test.png)
+
+From this graph, we can tell that there is barely any data to the **right** of the black vertical line, meaning we have a very low p-value!

docusaurus.config.js

@@ -5,6 +5,8 @@
 // See: https://docusaurus.io/docs/api/docusaurus-config
 
 import {themes as prismThemes} from 'prism-react-renderer';
+const math = require('remark-math')
+const katex = require('rehype-katex')
 
 /** @type {import('@docusaurus/types').Config} */
 const config = {
@@ -20,7 +22,7 @@ const config = {
 
   // GitHub pages deployment config.
   // If you aren't using GitHub pages, you don't need these.
-  organizationName: 'dsc-courses', // Usually your GitHub org/user name.
+  organizationName: 'dsc-courses', // Usually your GitHub org/username.
   projectName: 'bpd-reference', // Usually your repo name.
 
   onBrokenLinks: 'throw',
@@ -44,6 +46,8 @@ const config = {
       ({
         docs: {
           sidebarPath: './sidebars.js',
+          remarkPlugins: [math],
+          rehypePlugins: [katex],
           // Please change this to your repo.
           // Remove this to remove the "edit this page" links.
           editUrl:
@@ -56,6 +60,16 @@ const config = {
     ],
   ],
 
+  stylesheets: [
+    {
+      href: '/katex/katex.min.css',
+      type: 'text/css',
+      integrity:
+          'sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X',
+      crossorigin: 'anonymous',
+    },
+  ],
+
   themeConfig:
     /** @type {import('@docusaurus/preset-classic').ThemeConfig} */
     ({