Bivariate Analysis Notes

Bivariate Analysis Overview

• Introduction to Bivariate Analysis
  • Focuses on relationships between pairs of variables rather than individual variables.
  • Data representation of pairs of variables (X,Y) recorded on multiple units.

Pairs of Variables

• Data can be represented as pairs
  • Each pair consists of values of X and Y represented as (xi, yi).
  • Dataset formula: (x1, y1), (x2, y2), …, (xn, yn).
• Distinct categories of X and Y:
  • X categories: c1, c2, …, ck (k distinct categories)
  • Y categories: d1, d2, …, dh (h distinct categories)

Joint Frequency Distribution

• Absolute Joint Frequency
  • Notation: njl = Absolute joint frequency of (cj, dl)
  • Total count: $ext{Total} = \frac{kj=1 ext{sum (j=1 to k)}}{h ext{sum (l=1 to h)}} njl = n$
• Relative Joint Frequency
  • Relative joint frequency is calculated as:
  • $fjl = \frac{njl}{n}$
  • Ensures all relative frequencies sum to 1: $ext{Total} = \frac{1}{n} \frac{kj=1 ext{sum (j=1 to k)}}{h ext{sum (l=1 to h)}} njl = 1$

Marginal Distribution of X

• Marginal Absolute Frequency:
  • $nj+ = \frac{h ext{sum (l=1 to h)}}{njl}$
  • Represents total number of pairs for each category of X.
• Marginal Relative Frequency:
  • $fj+ = \frac{nj+}{n}$ where $fj+ = \frac{h ext{sum}}{j=1 ext{to h}}$

Example 1: Gender and Education

• Dataset: 20 employees (n=20), X = Gender (M/F), Y = Education (E/HSG/CG)

• Contingency table absolute frequencies:
  | | E | HSG | CG | Tot |
  |---|---|---|---|---|
  | F | 2 | 5 | 3 | 10 |
  | M | 1 | 5 | 4 | 10 |
  | Tot | 3 | 10 | 7 | 20 |

• Conditional Relative Frequency:
  • Probability of X given Y = dl : $\frac{njl}{n+l}$
  • Probability of Y given X = cj : $\frac{njl}{nj+}$
• Properties:
  • All sums equal to 1 for conditional distributions.

Statistical Independence

• Definition: X independent of Y if all conditional distributions are equal.
  • $n<em>{jl} = n</em>{+l}/n_{+j}$
• Consequences:
  1. Marginal distribution of X equals conditional distributions of X.
  2. Independence is symmetric: if X independent of Y, then Y is independent of X.

Covariance and Correlation

• Covariance Formula:
  • $sxy = \frac{1}{n} ext{sum} ((xi - x̄)(yi - ȳ))$
• Interpretation:
  • Positive covariance: increasing relationship.
  • Negative covariance: decreasing relationship.

Pearson’s Correlation Coefficient

• Formula:
  • $rxy = \frac{sxy}{sx imes sy}$
• Properties:
  1. Sign indicates direction of the relationship.
  2. Equals 0 if no linear relationship.
  3. Invariant under linear transformations.
  4. Absolute value <= 1.

Conclusion

• Bivariate analysis is an essential tool for understanding relationships between two variables, with clear methodologies for calculating distributions, covariances, and correlations necessary for accurate economic and social science studies.