statistics III

Trend

A directional relationship between two continuous variables.

Linear regression

A statistical method used to test whether a linear trend exists between two continuous variables.

Regression equation

y=mx+c where m is slope and c is intercept.

Slope (m)

The amount by which Y changes for each unit increase in X.

Intercept (c)

The value of Y when X = 0.

Null hypothesis for trends

There is no trend; slope = 0 ( y=C ).

Alternative hypothesis for trends

There is a trend; slope ≠ 0 ( y=mx+C ).

F‑statistic

Test statistic used in linear regression to determine whether the slope is significantly different from zero.

Significance F

The p‑value for the regression model; shows whether the trend is statistically significant.

R² (coefficient of determination)

Proportion of variation in Y explained by X.

High R²

Strong relationship; X explains much of the variation in Y.

Low R²

Weak relationship; X explains little of the variation in Y.

Residuals

Differences between observed and predicted Y values.

Best‑fit line

The regression line that minimises the sum of squared residuals.

Prediction using regression

Substituting X into the regression equation to estimate Y.

Standard error (SE)

Measure of uncertainty in an estimate (e.g., mean or slope).

Standard error of the mean (SEM)

SEM=sn where s is sample standard deviation.

Confidence interval (CI)

Range of values likely to contain the true population parameter.

95% confidence interval

Mean ± 1.96 × SEM.

Effect of sample size on SEM

Larger sample size → smaller SEM → more confidence.

Normal distribution

Bell‑shaped distribution describing many biological variables.

68–95–99.7 rule

68% of values within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

Error bars

Graphical representation of variability (often SEM or CI).

Purpose of regression

To test whether X predicts Y and quantify the strength of the relationship.

Example trend (lecture)

Higher GDP per capita is associated with lower child mortality.

Interpreting slope sign

Negative slope → Y decreases as X increases; positive slope → Y increases as X increases.

Interpreting p‑value in regression

p < 0.05 → reject H₀ → significant trend.