docs(learning-path): statistical part 4

content/docs/2021-04-07-LearningPath-Statistical-Modeling-4/index.mdx

@@ -200,20 +200,20 @@ To use ANOVA for linear regression, the data must conform to the following assum
 
 You may have heard phrases like one-way ANOVA or two-way ANOVA and the difference between them is the number of independent variables we are considering - as also suggested by their name. Of course, you can have three-way ANOVA, four-way ANOVA, etc..
 
-So how do we do ANOVA for linear regression models? Here we are using a dataset `mtcar` from [R dataset](https://www.rdocumentation.org/packages/datasets/versions/3.6.2/topics/mtcars) as an example which contains eleven different variables for thirty-two cars. Suppose we have the following two models for finding out variables which affects fuel consumption (in `mpg`) of cars, where the second linear model has a additional variable `gear` (number of gears that particular car have) we would like to include:
+So how do we do ANOVA for linear regression models? Here we are using a dataset `mtcar` from [R dataset](https://www.rdocumentation.org/packages/datasets/versions/3.6.2/topics/mtcars) as an example which contains eleven different variables for thirty-two cars. Suppose we have the following two models for finding out variables which affects fuel consumption (in `mpg`) of cars, where the second linear model has a additional variable `wt` (weight of cars) we would like to include:
 
 ```r
 lm1 <- lm(mpg ~ hp , mtcars)
-lm2 <- lm(mpg ~ hp + gear, mtcars)
+lm2 <- lm(mpg ~ hp + wt, mtcars)
 
 summary(lm1)
 summary(lm2)
 ```
 
-![Summaries of two linear models](4lms.png)  
+![Summaries of two linear models](4lm.png)  
 _Summaries of two linear models._
 
-From the summaries on above we can see the variable `gear` with a positive coefficient is a good addition to the model (somewhat suggesting cars with more gears have better fuel economy). Now let's use the `anova` function to see which model provides the best fit of the data.
+From the summaries on above we can see the variable `wt` (weight in 1000 lbs) with a positive coefficient is a good addition to the model (somewhat suggesting cars with less weight have better fuel economy). Now let's use the `anova` function to see which model provides the best fit of the data.
 
 ```r
 anova(lm1, lm2)
@@ -226,6 +226,20 @@ From the _Analysis of Variance Table_ we can see that the sum of squares has red
 
 ### Dummy Variables for normal ANOVA with Linear Regression
 
+Sometimes we may want to include categorical variables in the models but these models cannot be compared simply using ANOVA because of the [simpson paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox). Categorical variables usually have two or more categories and we generally refer each variable as _factor_ and each category as _level_.
+Dummy variables take values of zero or one, and we need to create $n-1$ dummy variables for a categorical variable that has $n$ categories. E.g. For a variable indicating whether there is rain on a certain date, we can create a dummy variable where `1` represents a rainy day and `0` represents a sunny day. In the `mtcar` dataset cars have gears ranging from 3 to 5, therefore we can create two variables indicating whether the car has four gears or five gears. But why do we not need a third variable for three gears? This is because we can deduce whether a car has three gears from the previous two dummy variables (i.e. when both dummy variables are equal to zero); we would have collinearity if we introduced a variable for cars with three gears. Note that this method of creating dummy variables is also suitable for conducting T-test.
+
+The following figure shows the effect of the `gear` variable on the second linear models that we have created earlier. The model now becomes
+
+$$
+mpg ~ hp + mt + 4gears + 5gears
+$$
+
+![Effect of dummy variables](6dummy.png)  
+_Effect of dummy variables_
+
+The p-values for dummy variables are noticeable larger than 0.05 which suggesting the `gear` variable is not suitable for predicting the fuel economy of a particular car. At the bottom of the figure we can see the ANOVA also rejecting the new variable. Therefore, we can conclude the `gear` variable has no special meaning to `mpg`.
+
 ## What's next?
 
 In this part of the learning path we have scratched the surface of statistical testing and seen two common hypothesis testing methods and their application in linear models. For more comprehensive materials on this topics, please refer to materials in the **Recommended reading** section. In <Link to="/docs/12/04/2021/LearningPath-Statistical-Modeling-5">part 5</Link> we will introduced you to the **Central Limit Theorem** which is a very important theorem in statistics, and is also the reason why we can make assumptions about the underlying distribution of samples in parametric tests.

gatsby-config.js

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     },
     /*********** END RSS Feed ************* */
     `gatsby-plugin-sass`,
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-        develop: true,
-        tailwind: true,
-        purgeOnly: [
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-          "src/pages/",
-          "src/templates/",
-          "content/"
-        ]
-      }
-    },
     {
       resolve: "gatsby-plugin-webpack-bundle-analyser-v2",
       options: {

package.json

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src/utils/hooks/backgroundMovement.jsx

@@ -42,11 +42,6 @@ export function backgroundMovement(
 
       // update mouse location
       setLocation({ x: event.clientX, y: event.clientY });
-<<<<<<< HEAD
-=======
-      // xLocation.current = event.clientX;
-      // yLocation.current = event.clientY;
->>>>>>> e863593d94a2e16add0f5533ab99d3ad00751985
 
       // move background according to difference
       let translateValues = getTranslateValues(background);