feat: new article

DiogoRibeiro7 · DiogoRibeiro7 · commit 71928041d340 · 2024-11-15T23:51:21.000Z
diff --git a/_posts/-_ideas/2039-01-01-statistics.md b/_posts/-_ideas/2039-01-01-statistics.md
@@ -46,8 +46,9 @@ title: Exploring Key Topics in Statistics
 - **TODO: Bayesian Statistics: An Introduction**
    - An introductory article on Bayesian statistics, which differs from traditional (frequentist) approaches by incorporating prior beliefs and evidence into the analysis. This piece covers Bayes’ Theorem, prior and posterior distributions, and applications in decision-making.
 
-- **TODO: Chi-Square Test: Testing Categorical Data**
-   - This article explores the chi-square test, a statistical method used to examine the association between categorical variables. It covers both the chi-square test for independence and the chi-square goodness-of-fit test, with examples of how to apply these tests in practical situations.
+
+
+
 
 - **TODO: Statistical Power and Sample Size: Designing Effective Studies**
    - Learn about the concepts of statistical power and sample size, which are critical in designing experiments and studies. The article explains how to calculate the necessary sample size for a study and the importance of achieving sufficient power to detect a true effect.
diff --git a/_posts/statistics/2023-03-01-chisquare_test_testing_categorical_data.md b/_posts/statistics/2023-03-01-chisquare_test_testing_categorical_data.md
@@ -0,0 +1,141 @@
+---
+author_profile: false
+categories:
+- Statistics
+classes: wide
+date: '2023-03-01'
+excerpt: The Chi-Square Test is a powerful tool for analyzing relationships in categorical
+  data. Learn its principles and practical applications.
+header:
+  image: /assets/images/data_science_9.jpg
+  og_image: /assets/images/data_science_9.jpg
+  overlay_image: /assets/images/data_science_9.jpg
+  show_overlay_excerpt: false
+  teaser: /assets/images/data_science_9.jpg
+  twitter_image: /assets/images/data_science_9.jpg
+keywords:
+- Chi-square test
+- Categorical data
+- Goodness-of-fit
+- Independence test
+seo_description: Discover how to use the Chi-Square Test to analyze categorical data,
+  including tests for independence and goodness-of-fit.
+seo_title: Chi-Square Test for Categorical Data
+seo_type: article
+summary: An exploration of the Chi-Square Test, focusing on its use in testing the
+  association between categorical variables and examining goodness-of-fit in statistical
+  analysis.
+tags:
+- Categorical data
+- Chi-square test
+- Independence test
+- Goodness-of-fit
+title: 'Chi-Square Test: Testing Categorical Data'
+---
+
+## Chi-Square Test: Testing Categorical Data
+
+The Chi-Square test is a fundamental statistical method used to analyze categorical data. It is widely employed to test hypotheses about the association between categorical variables and to determine how well observed data align with expected distributions. This article explores the two primary types of Chi-Square tests—the test for independence and the goodness-of-fit test—and provides practical examples to illustrate their application.
+
+---
+
+### **What is the Chi-Square Test?**
+
+The Chi-Square test is based on comparing observed frequencies (counts) in categorical data to expected frequencies derived under a specific null hypothesis. The formula for the Chi-Square statistic is:
+
+$$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$
+
+Where:
+
+- $$ O_i $$: Observed frequency in category $$ i $$,
+- $$ E_i $$: Expected frequency in category $$ i $$.
+
+The test assesses whether the differences between observed and expected frequencies are due to random variation or indicative of a systematic pattern.
+
+The Chi-Square test is non-parametric, making it suitable for categorical data without requiring assumptions about underlying distributions.
+
+---
+
+### **Types of Chi-Square Tests**
+
+#### **1. Chi-Square Test for Independence**
+
+This test evaluates whether two categorical variables are independent of each other. It is commonly used in analyzing contingency tables, where data are organized into rows and columns based on two variables.
+
+**Hypotheses:**
+
+- $$ H_0 $$ (Null Hypothesis): The variables are independent.
+- $$ H_a $$ (Alternative Hypothesis): The variables are associated.
+
+**Example:**
+A health study collects data on whether individuals exercise regularly (Yes/No) and their weight category (Underweight, Normal, Overweight). A Chi-Square test for independence can determine if exercise habits are associated with weight category.
+
+#### **2. Chi-Square Goodness-of-Fit Test**
+
+The goodness-of-fit test determines whether observed categorical data conform to a specific expected distribution. This test is frequently used to validate theoretical models or assumptions about data proportions.
+
+**Hypotheses:**
+
+- $$ H_0 $$: The observed data fit the expected distribution.
+- $$ H_a $$: The observed data do not fit the expected distribution.
+
+**Example:**
+A geneticist expects a 3:1 ratio of dominant to recessive traits in offspring based on Mendelian inheritance. A goodness-of-fit test can verify whether experimental data align with this expectation.
+
+---
+
+### **Steps to Perform a Chi-Square Test**
+
+1. **Formulate Hypotheses:**
+   Define the null and alternative hypotheses for the test.
+   
+2. **Calculate Expected Frequencies:**
+   Use theoretical distributions or proportions to compute $$ E_i $$.
+
+3. **Compute the Chi-Square Statistic:**
+   Substitute $$ O_i $$ and $$ E_i $$ into the formula to calculate $$ \chi^2 $$.
+
+4. **Determine Degrees of Freedom:**
+   - For independence tests: $$ \text{df} = (r-1)(c-1) $$, where $$ r $$ and $$ c $$ are the number of rows and columns.
+   - For goodness-of-fit tests: $$ \text{df} = k-1 $$, where $$ k $$ is the number of categories.
+
+5. **Compare with the Critical Value or p-Value:**
+   Use a Chi-Square distribution table or software to determine significance.
+
+6. **Interpret Results:**
+   If $$ \chi^2 $$ exceeds the critical value or the p-value is below the threshold (e.g., 0.05), reject $$ H_0 $$.
+
+---
+
+### **Applications of the Chi-Square Test**
+
+#### **Analyzing Contingency Tables**
+
+Contingency tables provide a structured format for examining the relationship between two categorical variables. For example, in market research, a company might analyze whether purchase preferences differ by age group.
+
+#### **Evaluating Survey Data**
+
+The test is often used to analyze survey results, such as examining whether opinions on a policy differ across demographic groups.
+
+#### **Validating Theoretical Distributions**
+
+In scientific experiments, the goodness-of-fit test helps confirm whether observed data match theoretical predictions, such as phenotypic ratios in genetics.
+
+---
+
+### **Considerations for Using the Chi-Square Test**
+
+1. **Sample Size:**  
+   Small sample sizes may lead to unreliable results. The expected frequency in each category should typically be at least 5.
+
+2. **Independence of Observations:**  
+   Observations must be independent. Violations require alternative statistical methods.
+
+3. **Interpretation of Results:**  
+   A significant Chi-Square value indicates a deviation from expectations, but further analysis may be needed to understand the underlying causes.
+
+---
+
+### **Conclusion**
+
+The Chi-Square test is a versatile tool for analyzing categorical data, offering insights into relationships and patterns within datasets. By applying the test for independence and the goodness-of-fit test, researchers can evaluate hypotheses and validate theoretical distributions across diverse fields, from survey analysis to genetics. Mastery of the Chi-Square test empowers analysts to make data-driven decisions and uncover meaningful associations in categorical data.
diff --git a/_posts/statistics/2024-12-07-chisquare_test_exploring_categorical_data_goodness_fit.md b/_posts/statistics/2024-12-07-chisquare_test_exploring_categorical_data_goodness_fit.md
@@ -4,8 +4,7 @@ categories:
 - Statistics
 classes: wide
 date: '2024-12-07'
-excerpt: Dive into the Chi-Square Test, a statistical method for evaluating categorical
-  data. Understand its applications in survey analysis, contingency tables, and genetics.
+excerpt: Dive into the Chi-Square Test, a statistical method for evaluating categorical data. Understand its applications in survey analysis, contingency tables, and genetics.
 header:
   image: /assets/images/data_science_20.jpg
   og_image: /assets/images/data_science_20.jpg
@@ -18,12 +17,10 @@ keywords:
 - Goodness-of-fit test
 - Categorical data analysis
 - Independence test
-seo_description: Learn about the Chi-Square Test, its role in analyzing categorical
-  data, and its applications in testing goodness-of-fit and independence.
+seo_description: Learn about the Chi-Square Test, its role in analyzing categorical data, and its applications in testing goodness-of-fit and independence.
 seo_title: Chi-Square Test for Categorical Data Analysis
 seo_type: article
-summary: An in-depth exploration of the Chi-Square Test, focusing on its uses for
-  goodness-of-fit and independence testing in categorical data analysis.
+summary: An in-depth exploration of the Chi-Square Test, focusing on its uses for goodness-of-fit and independence testing in categorical data analysis.
 tags:
 - Chi-square test
 - Goodness-of-fit
@@ -33,6 +30,28 @@ title: 'Chi-Square Test: Exploring Categorical Data and Goodness-of-Fit'
 
 The Chi-Square test is a cornerstone of statistical analysis for categorical data. It enables researchers to examine how well observed data align with expected distributions, assess the independence of categorical variables, and test hypotheses in a wide range of fields. This article delves into the mechanics of the Chi-Square test, its two primary applications—goodness-of-fit and independence testing—and its use in practical scenarios such as survey data, contingency tables, and genetics.
 
+---
+
+### **Understanding the Chi-Square Test**
+
+The Chi-Square test evaluates the disparity between observed and expected frequencies in categorical data. It is based on the Chi-Square statistic:
+
+$$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$
+
+Where:
+
+- $$ O_i $$ represents the observed frequency in category $$ i $$,
+- $$ E_i $$ is the expected frequency for category $$ i $$.
+
+This test assumes that:
+1. The data are in the form of counts or frequencies.
+2. Observations are independent.
+3. Expected frequencies are sufficiently large, typically $$ E_i \geq 5 $$.
+
+The Chi-Square distribution, which is positively skewed, is used to determine the significance of the calculated statistic. The degrees of freedom (df) are determined based on the test type and data structure.
+
+---
+
 ### **Types of Chi-Square Tests**
 
 #### **1. Goodness-of-Fit Test**
@@ -60,36 +79,27 @@ The hypotheses for this test are:
 - **Alternative hypothesis ($$ H_a $$)**: The variables are associated.
 
 ---
-author_profile: false
-categories:
-- Statistics
-classes: wide
-date: '2020-01-01'
-excerpt: Dive into the Chi-Square Test, a statistical method for evaluating categorical
-  data. Understand its applications in survey analysis, contingency tables, and genetics.
-header:
-  image: /assets/images/data_science_20.jpg
-  og_image: /assets/images/data_science_20.jpg
-  overlay_image: /assets/images/data_science_20.jpg
-  show_overlay_excerpt: false
-  teaser: /assets/images/data_science_20.jpg
-  twitter_image: /assets/images/data_science_20.jpg
-keywords:
-- Chi-square test
-- Goodness-of-fit test
-- Categorical data analysis
-- Independence test
-seo_description: Learn about the Chi-Square Test, its role in analyzing categorical
-  data, and its applications in testing goodness-of-fit and independence.
-seo_title: Chi-Square Test for Categorical Data Analysis
-seo_type: article
-summary: An in-depth exploration of the Chi-Square Test, focusing on its uses for
-  goodness-of-fit and independence testing in categorical data analysis.
-tags:
-- Chi-square test
-- Goodness-of-fit
-- Categorical data
-title: 'Chi-Square Test: Exploring Categorical Data and Goodness-of-Fit'
+
+### **Applications of the Chi-Square Test**
+
+#### **1. Survey Data Analysis**
+
+In surveys, the Chi-Square test is often used to analyze responses to categorical questions. For instance, a political survey may examine whether voter preference is independent of demographic factors such as age or region.
+
+#### **2. Contingency Tables**
+
+Contingency tables summarize relationships between two categorical variables. The Chi-Square test helps identify significant associations within these tables, making it a powerful tool in fields like market research and public health.
+
+**Example:**  
+A study might analyze whether vaccine acceptance rates differ by age group using a contingency table. The results can guide targeted awareness campaigns.
+
+#### **3. Genetics**
+
+The test plays a critical role in genetics for examining inheritance patterns. For instance, Mendelian inheritance can be tested by comparing observed and expected phenotypic ratios in offspring.
+
+**Example:**  
+Consider a dihybrid cross in pea plants, where expected offspring phenotypes follow a 9:3:3:1 ratio. A goodness-of-fit test can confirm whether experimental results align with this prediction.
+
 ---
 
 ### **Key Considerations and Limitations**
