【科学実験リテラシー】Day4 相関係数

Scientific concepts / phenomena covered

Correlation coefficients (quantitative relationship)

• Correlation: a quantitative description of how two variables relate (e.g., (x) and (y)).
• Scatter plot: used to visualize the relationship between two variables ((x), (y)).

Pearson’s product-moment correlation coefficient ((r))

• The main focus is Pearson’s correlation coefficient (r).
• The value (r) is interpreted using variance and covariance.
• Key interpretation range:
  • (r \in [-1, 1])
  • (r = 1): strong positive linear relationship (points lie on a line with positive slope)
  • (r = -1): strong negative linear relationship
  • (r \approx 0): little/no linear correlation
• Absolute value reflects strength (sign gives direction):
  • roughly (0)–(0.2): almost no correlation
  • (0.2)–(0.4): weak
  • (0.4)–(0.7): strong
• Methodological note: interpret (r) together with the scatter plot, not alone.

Variance and covariance (mathematical basis)

• Variance: used to standardize/normalize the spread of each variable.
• Covariance:
  • computed from deviations from the mean of (x) and deviations from the mean of (y)
  • used in the numerator of correlation
• The lesson emphasizes that correlation uses normalized covariance, producing a value bounded between -1 and 1, which makes comparisons easier (and also relates to convenience vs. sample-size effects).

Regression vs correlation

• Correlation analysis: treats (x) and (y) symmetrically; asks whether two variables vary together.
• Regression analysis (contrasted for next week): emphasizes a directional / one-way influence, such as predicting/estimating (y) from (x).

Causation vs correlation

• A correlation coefficient indicates association, not cause-and-effect.
• Examples:
  • Height vs weight: taller people tend to weigh more, but correlation alone cannot determine which causes which.
  • Math vs science grades: correlated, but causal direction is not implied.
  • Spurious correlation (apparent correlation):
    • restaurants vs financial institution branches in Tokyo appear strongly correlated (reported (r \approx 0.9)),
    • but the lesson attributes both to a third variable: daytime population (more people flowing into business districts increases both restaurants and bank branches).
• Partial correlation:
  • introduced as a correction approach:
    • after “removing” the third variable’s effect, the correlation decreases (example: reduced from ~0.9 to 0.66).

Simpson’s paradox / grouping trap (hidden patterns)

• Aggregated data can hide or understate relationships.
• Example described: unemployment rate vs election vote share:
  • overall correlation appears weak,
  • but within subgroups/regions (e.g., England vs Scotland, Wales in the example), correlations can be strong or differ.
• Takeaway: how you group data matters, and it requires domain knowledge.

Other correlation coefficients (beyond Pearson)

• Rank correlation:
  • Spearman’s rank correlation coefficient
  • Kendall’s rank correlation coefficient
• Use case: when data are ordinal/rank-based (e.g., students ranked by math and English).
• They assess monotonic/rank-based association rather than raw numeric spacing.
• Spearman/Kendall are presented as alternatives derived from ranking relationships.

Autocorrelation for time series (correlation with time lag)

• Autocorrelation: correlation of a variable with itself at different time shifts (lag (h)).
• Time-series phenomena discussed:
  • trends (persistence or reversal depending on the sign),
  • seasonality / periodicity (repeating patterns),
  • used for predicting upward/downward trends and identifying regular variations.
• Lesson explanation:
  • compute correlation between (x_i) and (x_{i+h}),
  • as lag (h) increases, correlation typically approaches 0 if no periodicity exists.
• Example: department store sales (monthly) over multiple years:
  • periodicity detected by peaks in autocorrelation at specific lag values,
  • periodicity appears at a lag like 6, suggesting a repeating cycle roughly every 6 months.

Scientific literacy / galaxy-phenomenon correlation mention

• Mentions an example dataset from scientific literacy tests:
  • countries with higher scientific literacy scores tend to have higher reading comprehension (positive correlation).
• Mentions a galaxy/black-hole dataset:
  • axes related to galaxy/black hole masses and a “loss” measure,
  • the concept is correlation between astrophysical quantities (subtitle details are garbled, but correlation between properties is implied).

---

Methodology / workflow shown (Excel-based calculation)

How to compute correlation in Excel (as taught)

1) Create a scatter plot

• Select two columns of data ((x) and (y)).
• Insert → scatter plot with points.
• Add/format:
  • chart title
  • axis labels
  • adjust axis limits and font sizes

2) Compute the correlation coefficient

• Option 1: Excel “Data Analysis” tool
  • Data → Data Analysis
  • choose Correlation
  • set input range (including first row labels)
  • set output destination
  • read the resulting correlation matrix (diagonal should be 1)
• Option 2: Excel function for Pearson correlation
  • use a built-in correlation function (Pearson product-moment correlation)
  • calculate correlation directly from two ranges
• Option 3 (conceptual): compute via variance/covariance
  • compute standard deviations and covariance using Excel/statistical functions
  • plug into the Pearson (r) definition
  • the lesson highlights careful handling of population vs sample versions (noted as potential pitfalls with .p vs .s variants).

---

Researchers / sources featured (names at end)

• Karl Pearson (Pearson’s product-moment correlation coefficient)
• Spearman (Spearman’s rank correlation coefficient)
• Kendall (Kendall’s rank correlation coefficient)
• Murakami (teacher referenced as “Mr. Murakami” introducing parts of the lesson)