Summary statistics for bivariate data

STAT1003 – Statistical Techniques

Dr. Emi Tanaka

Australian National University

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Table of contents

• Bivariate data
• Contingency table
• Stacked barplot
• Percent stacked barplot
• Side-by-side barplot
• Group summary statistics
• Graphical statistics by group
• Case study 🌾 Wheat seed morphological characteristics
• Scatterplot
• Sample covariance
• Pearson’s correlation coefficient
• Interpretation of correlation coefficient
• Wrong interpretation of correlation coefficient
• Summary statistics can be misleading
• Summary

Bivariate data

Bivariate data involves two different variables (\(x\) and \(y\)) for each observation.

• Suppose \(x_i\) and \(y_i\) are two variables measured on the same unit \(i\) for \(i = 1, 2, \ldots, n\).
• It allows us to explore relationships and associations between the two variables.
• Examples:
  • Height and weight of individuals.
  • Temperature and ice cream sales.
  • Study time and exam scores.
  • Eye color and hair color.

	Categorical	Numerical
Categorical	Contingency table Stacked barplot Percent stacked barplot Side-by-side barplot	Group summary statistics
Numerical	Group summary statistics	Scatterplot Covariance Correlation coefficient

Contingency table

A contingency table (also known as a cross-tabulation or crosstab) display the frequency distribution of two or more categorical variables.

What do you notice between these two approaches?

Stacked barplot

A stacked barplot is used to compare the composition of different groups in a dataset, especially contribution of sub-categories to the total within each main category.

Percent stacked barplot

A percent stacked barplot is ideal for comparing the relative frequencies of subgroups within categories, rather than their absolute counts.

Side-by-side barplot

A side-by-side barplot (also called a grouped barplot) is used to visually compare the values of different subgroups across categories.

Group summary statistics

For a bivariate data where one variable is numerical and the other is categorical, you can use summary statistics for univariate data for each group.

• For example, numerical statistics for each group can be computed as:

Graphical statistics by group

• Likewise, we can compute graphical statistics for each group.

Beeswarm plot is a type of scatterplot that shows the distribution of data points while avoiding overlap, making it easier to visualize the density and spread of the data.

Case study 🌾 Wheat seed morphological characteristics

A dataset was collected to investigate morphological characteristics associated with seed weight in a line of diploid wheat (Triticum monococcum).

• DSeed - identifier for each seed
• Weight - weight of seed (mg)
• Length - length of seed (mm)
• Diameter - diameter of seed (mm)
• Moisture - mositure content of seed (as a percentage)
• Hardness - endosperm hardness

Scatterplot

A scatterplot is a graphical representation that displays the relationship between two numerical variables by plotting individual data points on a two-dimensional graph.

Sample covariance

Sample covariance is a measure of how much two numerical variables change together.

\[ s_{xy}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right) \]

• Interpretation:
  • When \(s_{xy} > 0\), the variables tend to increase together.
  • When \(s_{xy} < 0\), one variable tends to increase when the other decreases.
  • When \(s_{xy} = 0\), there is no linear relationship between the variables.

Consider the following dataset with two variables, \(x\) and \(y\):

\(i\)	\(x\)	\(y\)
1	1	10
2	2	70
3	3	100

\[s_{xy} = \frac{1}{2}\left[(1-2)(10-60)+(2-2)(70-60)+(3-2)(100-60)\right]=45\]

• But the magnitude of covariance is not easy to interpret since it depends on the units of the variables.

Pearson’s correlation coefficient

• Correlation coefficient is a normalised version of covariance.
• The sample Pearson correlation coefficient, denoted as \(r\), is a measure of the strength of a linear relationship between two variables (\(x\) and \(y\)).

\[r = \frac{\sum_{i=1}^n(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n(x_i - \bar{x})^2\sum_{i=1}^n(y_i - \bar{y})^2}}\]

• The correlation coefficient ranges from -1 to 1.

Interpretation of correlation coefficient

• The sign of the correlation coefficient indicates the direction of the relationship.
• The magnitude indicates the strength of the linear relationship.

\(|r|\)	Interpretation
0.8 - 1.0	Very strong association
0.6 - 0.8	Strong association
0.4 - 0.6	Moderate association
0.2 - 0.4	Weak association
0.0 - 0.2	Very weak association

viewof nsample = Inputs.number([20, 1000], {step: 20, value: 200, label: "Number of samples"})
viewof r = Inputs.range([-1,1], {step: 0.05, value: 0.8, label: "Correlation coefficient"})

Wrong interpretation of correlation coefficient

Source: xkcd

Just because \(x\) and \(y\) are highly correlated, it does not mean that \(x\) causes \(y\) or vice versa – correlation is not causation!

• Number of ice cream sales and the rate of drowning deaths.
• It is also easy to get spurious correlation if computing many pairwise correlations.

• Correlation also only measures a linear relationship, so low correlation doesn’t mean that there is no relationship.

\[r = 0.0414779\]

Summary statistics can be misleading

• You can have a bivariate dataset with the exact same:
  • marginal mean,
  • marginal variance and
  • correlation, but the relationship between the two variables can be very different.
• Always plot your data!

Summary

	Categorical	Numerical
Categorical	Contingency table Stacked barplot Percent stacked barplot Side-by-side barplot	Group summary statistics
Numerical	Group summary statistics	Scatterplot Covariance Correlation coefficient

• Correlation \(\neq\) Causation
• Always plot the data!