PSYC3010 – Correlation & Regression Review

Question-and-answer style flashcards covering key definitions, formulas, and concepts from the PSYC3010 lecture on correlation, simple regression, and introduction to standard multiple regression.

What does covariance (COVXY) tell us about two variables X and Y?

The degree to which X and Y vary together (i.e., whether high/low scores on one are associated with high/low scores on the other).

How do you calculate covariance between X and Y?

COVXY = Σ[(X − X̄)(Y − Ȳ)] ÷ (N − 1).

What does a positive covariance indicate?

A positive relationship: higher-than-mean scores on X pair with higher-than-mean scores on Y (and vice-versa).

What does a negative covariance indicate?

A negative relationship: higher-than-mean scores on X pair with lower-than-mean scores on Y (and vice-versa).

Why is covariance considered scale-dependent?

Its magnitude changes with the measurement scales of the variables, making direct comparisons across studies impossible.

Which statistic standardises covariance and removes scale dependence?

Pearson’s correlation coefficient (r).

Between what two values can Pearson’s r range?

–1 to +1.

What does r² represent?

The proportion of variance in one variable that can be explained by the other (shared variance).

What statistical test is used to determine if an observed r significantly differs from zero?

A t-test with df = N − 2.

What is the general form of the simple regression equation?

Ŷ = bX + a.

In regression, what does the slope (b) represent?

The expected change in the predicted value of Y for each one-unit increase in X.

In regression, what does the intercept (a) represent?

The predicted value of Y when X equals zero.

How can b be calculated from covariance and variance?

b = COVXY / sX² (or equivalently b = r·sY / sX).

What is a residual (ei) in regression analysis?

The difference between an individual’s actual Y score and the score predicted by the regression line (ei = Yi − Ŷi).

What does the standard error of the estimate (sY·X) quantify?

The average magnitude of prediction error; calculated as √[Σ(Y − Ŷ)² / (N − 2)].

How does the size of r affect the standard error of the estimate?

Larger |r| → smaller sY·X, meaning more accurate predictions.

Why can’t unstandardised b alone tell us how well X predicts Y?

Its value depends on both the strength of the relationship and the measurement units of X and Y.

What is the form of the standardised regression equation?

zŶ = β zX (intercept = 0, slope = β).

When, in simple regression, is β numerically identical to r?

Always in bivariate (one-predictor) regression.

Into which two components is SSY partitioned in simple regression?

SSregression (explained variance) and SSresidual (unexplained variance).

How is total sum of squares (SSY) computed?

SSY = Σ(Y − Ȳ)².

What does SSregression represent?

The portion of SSY that can be predicted from X (Σ[Ŷ − Ȳ]²).

What is the degrees of freedom for SSregression in simple regression?

p = 1 (one predictor).

What does SSresidual represent?

Variation in Y that cannot be predicted by X (Σ[Y − Ŷ]²).

How do you obtain R² from sums of squares?

R² = SSregression / SSY.

Which test evaluates whether the overall regression model explains significant variance in Y?

An F-test: F(1, N−2) = MSregression / MSresidual.

What is the main difference between bivariate and multiple regression?

Multiple regression uses two or more predictors simultaneously to predict a single criterion variable.

In multiple regression, what does capital R denote?

The multiple correlation between Y and all predictors taken together.

Why is adjusted R² reported alongside R²?

It provides a less-biased estimate of explained variance by adjusting for sample size and number of predictors.

What is a partial correlation (pr)?

The correlation between predictor Xj and Y after removing variance due to other predictors from BOTH Xj and Y.

What is a semi-partial (part) correlation (sr)?

The correlation between predictor Xj (with shared variance removed from Xj) and the original Y; indicates Xj’s unique contribution to total variance in Y.

How do you interpret pr²?

The proportion of residual variance in Y (after other predictors are removed) uniquely explained by Xj.

How do you interpret sr²?

The proportion of TOTAL variance in Y uniquely explained by Xj.

Why might the zero-order correlation (r) be misleading in multiple regression?

Because it ignores overlap among predictors and may overstate a predictor’s unique contribution to Y.

In simple regression, dfregression equals the number of predictors. What is dfresidual?

N − p − 1.

Why can’t causality be inferred from correlational (non-experimental) designs?

Because variables are measured, not manipulated; directionality and third-variable explanations remain possible.

What criterion does the regression line satisfy to be the "line of best fit"?

It minimises the sum of squared residuals (least-squares criterion).

Which test is used to evaluate if a regression slope (b or β) differs from zero?

A t-test with df = N − p − 1 (N − 2 in simple regression).

Under what circumstance will sY·X equal zero?

When X and Y are perfectly correlated (r = ±1).

How are positive, negative, and zero correlations visually identified on a scatterplot?

Positive: points trend upward left-to-right; negative: points trend downward; zero: no discernible linear trend.

When graphing two variables, which axes display X and Y?

Predictor (X) on the X-axis; criterion (Y) on the Y-axis.

In regression terminology, what are predictor and criterion variables?

Predictors (Xs) are assumed causes/inputs; criterion (Y) is the outcome being predicted.

Write the symbolic formula for covariance.

COVXY = Σ[(X − X̄)(Y − Ȳ)] / (N − 1).

How is β obtained when both variables are z-scored?

β equals the correlation r because b = r when sX = sY = 1.

State the fundamental relationship among SSY, SSregression, and SSresidual.

SSY = SSregression + SSresidual; their degrees of freedom also add (N−1 = p + (N−p−1)).