Unit 10: Test of Association

I. Orientation: Association in Research

Association describes the extent to which two variables vary together. Tests and measures of association help researchers determine whether an observed relationship in sample data is sufficiently systematic to justify inference about a population. Association does not, by itself, establish causation: a relationship may arise from confounding variables, reverse causation, measurement procedures, or chance.

• Purpose: Measures of association describe the direction, strength, or statistical significance of a relationship between variables.
• Variable type: The appropriate method depends primarily on the level of measurement.
  • Interval or ratio data: Karl Pearson’s correlation is commonly used for quantitative variables.
  • Ordinal data: Spearman rank correlation is appropriate when observations can be ranked.
  • Nominal data: Contingency tables and chi-square tests assess association between categories.
• Direction: A relationship may be positive, negative, or absent; direction applies naturally to ordered or quantitative variables, but not generally to unordered nominal categories.
• Strength: Correlation coefficients usually range from -1 to +1, with larger absolute values indicating stronger relationships.
• Statistical significance: A significance test evaluates whether the sample evidence is inconsistent with a null hypothesis of no association.
• Assumptions: Independence, suitable measurement levels, representative sampling, and correct model conditions are necessary for valid interpretation.
• Causal restraint: Even a strong and statistically significant association is not proof that one variable causes another.

II. Correlation Coefficient — Direction and Strength of Covariation

A. Correlation coefficient

A correlation coefficient is a standardized numerical measure of the direction and strength of association between two variables.

• General notation: Population correlation is usually represented by ρ (rho), while a sample correlation is represented by r.
• Range: A conventional correlation coefficient satisfies:
  
  TEXT

  -1 ≤ r ≤ +1

  • r = +1 indicates a perfect positive relationship.
  • r = -1 indicates a perfect negative relationship.
  • r = 0 indicates no relationship of the kind measured by that coefficient.
• Positive association: When higher values of X tend to occur with higher values of Y, the coefficient is positive.
• Negative association: When higher values of X tend to occur with lower values of Y, the coefficient is negative.
• Magnitude: The absolute value |r| expresses strength; for example, -0.80 is stronger than +0.30, although its direction is negative.
• Standardization: Because correlation has no measurement unit, variables measured in different units, such as income in rupees and education in years, can be compared.
• Coefficient of determination: For Pearson correlation, r² gives the proportion of variance shared through the fitted linear relationship. If r = 0.70, then:
  
  TEXT

  r² = (0.70)² = 0.49

  
  Thus, approximately 49% of the variance is accounted for by the linear association.
• Interpretive caution: A coefficient near zero may conceal a strong non-linear relationship, and a single outlier may substantially alter some coefficients.

B. Applications and limitations

Correlation coefficients summarize relationships efficiently, but their meaning depends on the data, design, and selected coefficient.

• Applications: Correlation is used in reliability studies, scale construction, prediction, validation, and investigation of relationships such as education and income.
• Restriction of range: Studying only a narrow range of scores can reduce the observed coefficient.
• Heterogeneous groups: Combining distinct subgroups may create, conceal, or reverse an association.
• Sample size: A weak coefficient can be statistically significant in a large sample, while a substantial coefficient may be non-significant in a small sample.
• No causation: A correlation between stress and poor sleep does not identify which variable causes the other or exclude a third factor.

III. Spearman Rank Correlation — Association Between Ordered Observations

A. Spearman rank correlation

Spearman rank correlation, denoted by rₛ or ρₛ, measures the direction and strength of a monotonic relationship using the ranks of observations.

• Appropriate data: It is suitable for ordinal variables, ranked judgments, or quantitative data that do not satisfy Pearson correlation assumptions.
• Monotonic relationship: As one variable increases, the other generally moves in one consistent direction, although the rate of change need not be constant.
• Principle: Raw values are replaced by ranks, and the correspondence between the two sets of ranks is measured.
• Formula without tied ranks:
  
  TEXT

  rₛ = 1 - [6Σdᵢ² / n(n² - 1)]

  • rₛ = Spearman rank correlation coefficient.
  • dᵢ = difference between the two ranks for observation i.
  • Σdᵢ² = sum of squared rank differences.
  • n = number of paired observations.
• Ranking convention: Rank both variables in the same direction, such as assigning rank 1 to the smallest value in each set.
• Tied observations: Equal values receive the average of the ranks they would occupy. With ties, Spearman’s coefficient is most directly calculated as Pearson’s correlation between the rank values.
• Worked example: Suppose four individuals have ranks X = 1, 2, 3, 4 and Y = 1, 3, 2, 4. Then d = 0, -1, 1, 0, so Σd² = 2:
  
  TEXT

  rₛ = 1 - [6(2) / 4(4² - 1)]
     = 1 - 12/60
     = 0.80

  
  The rankings therefore show a strong positive association.

B. Applications and limitations

Spearman correlation is useful when order is meaningful but exact numerical distances are unavailable or unreliable.

• Applications: It can compare rankings by two judges, relate class position to performance ratings, or analyse skewed scores.
• Advantages: It is less affected by extreme raw values because calculations use ranks rather than original magnitudes.
• Information loss: Ranking discards the size of differences; scores of 10 and 11 may be treated like scores of 10 and 100 if their ranks are adjacent.
• Condition: Observations must be paired and independent across cases.
• Limitation: It measures monotonic association and may fail to describe a relationship that changes direction.

IV. Karl’s Pearson Correlation — Linear Association Between Quantitative Variables

A. Karl's Pearson correlation

Karl Pearson’s product-moment correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables.

• Conceptual basis: Pearson correlation standardizes the covariance of X and Y by their standard deviations.
  
  TEXT

  r = cov(X,Y) / (sₓsᵧ)

  • cov(X,Y) = sample covariance between X and Y.
  • sₓ and sᵧ = sample standard deviations.
• Computational form:
  
  TEXT

  r = Σ[(xᵢ - x̄)(yᵢ - ȳ)] /
      √{Σ(xᵢ - x̄)² Σ(yᵢ - ȳ)²}

  • xᵢ, yᵢ = paired observations.
  • x̄, ȳ = sample means.
  • Σ = summation over all paired observations.
• Linear interpretation: r = +1 occurs when all points lie exactly on an upward-sloping straight line; r = -1 occurs on a downward-sloping line.
• Assumptions for inference: Paired quantitative observations should be independent, the relationship should be approximately linear, and serious outliers should be absent. Conventional significance procedures also assume approximate bivariate normality.
• Significance test: To test H₀: ρ = 0, Pearson’s r can be transformed into:
  
  TEXT

  t = r√[(n - 2) / (1 - r²)],  df = n - 2

  
  Here, n is the number of pairs and df denotes degrees of freedom.

B. Applications and limitations

Pearson correlation is most informative when a scatterplot supports a linear model.

• Applications: It is used for relationships such as height and weight, study time and test score, or two continuous scale scores.
• Outlier sensitivity: One unusually large or small observation can inflate, weaken, or reverse r.
• Linearity requirement: A strong curved relationship can produce r ≈ 0.
• Scale invariance: Changing centimetres to metres does not alter r; reversing a scale’s direction changes its sign.
• Interpretation: Pearson correlation evaluates linear association, not agreement between measurements and not causal influence.

V. Nominal Data — Association Between Categories

A. Test of association between nominal data

Association between nominal variables is examined by comparing observed category combinations with the combinations expected if the variables were independent.

• Nominal variables: Categories such as residence type, employment status, blood group, or treatment group have labels but no inherent numerical order.
• Contingency table: Frequencies are arranged in an r × c table, where r is the number of row categories and c is the number of column categories.
• Null hypothesis:
  
  TEXT

  H₀: The two nominal variables are independent.
  H₁: The two nominal variables are associated.

• Expected frequency: Under independence, the expected count in row i and column j is:
  
  TEXT

  Eᵢⱼ = (Row totalᵢ × Column totalⱼ) / N

  • Eᵢⱼ = expected frequency in cell (i,j).
  • N = total number of observations.
• Decision principle: Large discrepancies between observed frequencies Oᵢⱼ and expected frequencies Eᵢⱼ provide evidence against independence.
• Effect-size measures: For a 2 × 2 table, the phi coefficient is useful; for larger tables, Cramér’s V provides a standardized measure from 0 to 1.
• Interpretive limit: Nominal association has strength but usually no positive or negative direction because category order is undefined.

B. Conditions and interpretation

A valid nominal association test requires count data with independent observations.

• Frequency requirement: Cells must contain counts, not percentages, means, or repeated entries for the same independent case.
• Sampling condition: Each observation should contribute to only one cell.
• Sparse tables: Very small expected frequencies make the usual chi-square approximation unreliable.
• Substantive interpretation: Statistical significance should be accompanied by examination of percentages, residuals, and effect size to identify where the association occurs.

VI. Chi-Square Test — Comparing Observed and Expected Frequencies

A. Chi-square test

The chi-square test evaluates whether differences between observed and expected categorical frequencies are too large to attribute reasonably to sampling variation.

• Test statistic:
  
  TEXT

  χ² = Σ[(Oᵢ - Eᵢ)² / Eᵢ]

  • χ² = chi-square statistic.
  • Oᵢ = observed frequency in cell i.
  • Eᵢ = expected frequency in cell i.
• Degrees of freedom for independence:
  
  TEXT

  df = (r - 1)(c - 1)

  
  Here, r is the number of rows and c is the number of columns.
• Decision rule: Reject H₀ when the calculated χ² produces a p-value below the chosen significance level α, commonly 0.05.
• Contributions: Each cell contributes (O - E)²/E; cells with large contributions account most strongly for the test result.
• Assumptions: Observations must be independent, categories mutually exclusive, and expected counts sufficiently large. A common guideline is that no expected count is below 1 and no more than 20% are below 5.
• Related forms: The test of independence examines two categorical variables, while the goodness-of-fit test compares one observed categorical distribution with a specified distribution.

B. Significance and limitations

The chi-square test identifies evidence of association but does not directly measure its magnitude or causal meaning.

• Sample-size effect: Large samples may make minor differences significant; small samples may lack power to detect meaningful association.
• Effect size: Cramér’s V supplements significance:
  
  TEXT

  V = √[χ² / {N × min(r - 1, c - 1)}]

  
  N is the total sample size, and min selects the smaller dimension.
• Residual analysis: Standardized residuals indicate which cells have counts notably above or below independence expectations.
• Small samples: Fisher’s exact test is generally preferred for a sparse 2 × 2 table.
• Reporting: A complete result states χ², degrees of freedom, sample size, p-value, and an effect-size measure.