Project 2: Multiple Correspondence Analysis (MCA) and 3D Scatter Plot

Research Hypothesis:

Given the analysis and visualization performed in the code, we can formulate a research hypothesis related to the relationship between the categorical variable (represented by “Group”) and the 3D coordinates (X, Y, and Z) in the context of the Multiple Correspondence Analysis (MCA).

Null Hypothesis (H0):

• “There is no significant association between the categorical variable ‘Group’ and the distribution of data points in three-dimensional space (X, Y, Z) as observed through MCA.”

Alternative Hypothesis (Ha):

• “The categorical variable ‘Group’ is significantly associated with the distribution of data points in three-dimensional space (X, Y, Z) as observed through MCA.”

This hypothesis suggests that we are investigating whether there is a statistically significant relationship between the grouping variable ‘Group’ and the spatial distribution of data points in the 3D scatter plot created through MCA. Further analysis and testing would be required to determine whether we can accept or reject the null hypothesis in favor of the alternative hypothesis.

T-Test:

I have performed a t-test to determine if there is a significant difference in some measure (X, Y, or Z coordinates) between the groups defined by the “Group” variable.

Here I will show you how you can perform a t-test for each of these measures:

Our goal is to perform t-tests for the X, Y, and Z coordinates between the different groups (A, B, C, ) defined by the “Group” variable:

Step 1: Data Preparation

The “water_data_mca” dataframe has already been prepared, which includes the X, Y, and Z coordinates and the “Group” variable.

Step 2: Perform T-tests

I will utilize the “t.test()” function in R to perform t-tests for each measure separately (X, Y, Z) between the groups.

Option1: Perform Pairwise T-Tests

If you wan to compare each group against all other groups, you can perform pairwise t-tests. Here’s how you can do it using the pairwise.t.test() function:

Option 2: Reorganize Your Data

If you want to compare specific pairs of groups, you may need to reorganize your data to create a new grouping factor with exactly two levels. For example, you could create a new grouping variable that combines groups A and B into one level and groups C, D, and E into another level, and then perform the t-test.

Here’s an example of how you could create a new grouping variable:

Each of these t-tests will calculate the t-statistic and p-value to determine if there are significant differences in the X, Y, and Z coordinates between the groups defined by the “Group” variable.

Step 3: Interpret the Results

• If the p-value for a specific t-test is less than your chosen significance level (e.g., 0.05), you can conclude that there is a significant difference in the corresponding measure (X, Y, or Z) between the groups.
• If the p-value is greater than your significance level, you do not have enough evidence to conclude a significant difference.

I have repeated the t-tests for each measure (X, Y, Z) to assess if there are any statistically significant differences in these measures between the groups based on your “Group” variable.

Step 4: Data Preparation and MCA (Multiple Correspondence Analysis):

Next in this code segment, we begin by selecting and preparing the data for analysis. Specifically:

• We create a new dataframe called “water_data_cat” that contains only one categorical variable from the original dataset “Summary_Table_by_Contaminant.” This variable represents a category of interest for our analysis.
• Next, I perform Multiple Correspondence Analysis (MCA) on the “water_data_cat” dataframe. MCA is a statistical technique used for categorical data analysis, and here I set “graph = FALSE” to prevent the automatic generation of a plot during the analysis.

Step 5: Data Visualization with ggplot2 and Plotly:

Following the data preparation and MCA, I move on to data visualization. This involves creating an interactive 3D scatter plot to visualize the results of the MCA.

• I loaded the necessary R libraries, “ggplot2” for creating static plots and “plotly” for converting these plots into interactive visualizations.
• The dataframe is called “water_data_mca” with columns X, Y, Z, Group, Strip_Label, and Color. X, Y, and Z represent coordinates in 3D space. The Group indicates the grouping variable. The Strip_Label represents a custom strip label for facet customization, and Color represents the color of data points by group.

Step 6: Creating the 3D Scatter Plot:

Creating an 3D scatter plot using ggplot2:

• I used the “ggplot” function to create the plot with the data points represented as dots in three-dimensional space, with X, Y, and Z coordinates. The “color = Group” aesthetic assigns colors to data points based on the “Group” variable, making it easier to distinguish between different groups.
• Additional customizations include adjusting point size, specifying custom colors with “scale_color_manual,” adding axis labels and a plot title, setting the plot’s theme to minimal for simplicity, and customizing the appearance of titles and axis labels for better readability.
• The Facet customization is also implemented using “facet_grid.” It divides the plot into facets based on the “Strip_Label” variable. “scales = ‘free_x’” and “space = ‘free_x’” are used to allow independent x-axis scaling and spacing between facets, ensuring clarity in the visualization.

Step 7: Converting to Interactive 3D Visualization with Plotly:

The final step is to convert the static ggplot2 plot into an interactive 3D visualization using the “ggplotly” function from the “plotly” library. This conversion enables users to interact with my plot, such as zooming, panning, and accessing additional information about data points.

Step 8: Code

The core analysis technique applied here is Multiple Correspondence Analysis (MCA). MCA is specifically suited for exploring relationships within categorical data. By performing MCA on the “water_data_cat,” we unveil underlying patterns, associations, and dependencies between the categorical variables. Setting “graph = FALSE” avoids generating default plots, allowing for more customized visualizations.

Conclusion:

In conclusion, the provided code offers a well-structured and versatile workflow for exploring and visualizing relationships within categorical data in a three-dimensional space. By beginning with data preparation, applying MCA, and concluding with an interactive 3D scatter plot, it empowers data analysts to:

• Gain insights into the underlying structure of categorical data.
• Identify patterns and associations between categorical variables.
• Present findings in a visually compelling and interactive manner.
• Customize visualizations to suit specific research objectives.
• Facilitate data-driven decision-making through comprehensive exploration.

This workflow serves as a valuable tool for researchers, analysts, and data scientists who seek to understand and communicate complex relationships within categorical data. By adapting and extending this framework, one can extract deeper insights and leverage the flexibility of R for tailored data analysis and visualization.