Ch 3

The Scatter Diagram

• A bivariate distribution shows an individual’s score on two variables at the same time.

• A scatter diagram is a visual representation of the relationship between two variables.

Correlation

• A correlation is designed to examine a linear relationship between two variables.

• Definition: The correlation coefficient is a statistic that depicts the direction and magnitude of the relationship between two variables.

Regression

• Definition: Regression is a statistical method used to predict scores on one variable based on knowing the scores of another variable.

• Obtained through a regression line, which is the best-fitting straight line through the points on a scatter diagram.

• The regression line is calculated using the formula:
  $<br>Y' = a + bX <br>$
  where:

  • $Y'$ is the predicted score.

  • $a$ is the y-intercept.

  • $b$ is the slope of the regression line.

  • $X$ is the independent variable.

• The formula for calculating the slope (b) of the regression line is given by:
  $b = \frac{(\sum XY) - \frac{(\sum X)(\sum Y)}{N}}{(\sum X^2)-(\frac{(\sum X)^2}{N})}$
  where:

  • $N$ is the number of pairs of scores.

The Best-Fitting Line

• The best-fitting line through a series of data points involves both predicted and actual scores, which are rarely identical.

• Definition: The difference between them is called a residual, expressed as (Y – Y’).

Testing the Statistical Significance of a Correlation Coefficient

• Statistical significance determines whether a finding is likely due to chance.

• The process begins with a null hypothesis: $H_0: r = 0$ (no relationship).

• If a relationship is found, the null hypothesis is rejected: $H_1: r \neq 0$.

• Significance of the correlation coefficient is assessed using a t-distribution, with consideration to degrees of freedom:
  $t = \frac{r\sqrt{N - 2}}{\sqrt{1 - r^2}}$
  where $N$ is the sample size.

• A critical value table is used to determine if $t$ exceeds a critical value; if it does, reject the null hypothesis.

Interpretation of Regression Plots

• Regression plots visually represent the relationship between variables.

• Criterion Validity Evidence: Correlation is used to establish the relationship between a test score and a well-defined criterion.

• Normative data (representative groups) may be employed for comparison.

• Examples:

  • Association between a test of job aptitude and actual performance.

  • Association between an IQ test and academic performance.

Other Correlation Coefficients

• Spearman’s rho: Measures association between two sets of ranks.

• Biserial correlation: Measures the relationship between one continuous and one artificial dichotomous variable.

• Point biserial correlation: Measures the relationship between one continuous and one real dichotomous variable.

• Phi coefficient: Measures the relationship between two dichotomous variables, with one being a true dichotomous variable.

Terms and Issues in the Use of Correlation

Residual and Standard Error of Estimate

• A regression calls for both estimated values and actual values.

• Definition: The difference between them is called the residual (Y-Y’). The sum of the residuals always equals 0.

• The standard deviation of the residuals is known as the standard error of estimate:
  $SE = \sqrt{\frac{\sum(Y - Y')^2}{N - 2}}$

• A lower standard error indicates a more accurate prediction.

Coefficients of Determination and Alienation

• Coefficient of determination (r²): The square of the correlation coefficient, indicating the proportion of variation in one variable (X) explained by another variable (Y).

• Coefficient of alienation: A measure of nonassociation, represented as $\sqrt{1 - r^2}$, indicating the inverse of the coefficient of determination.

Shrinkage

• This refers to the decrease in the predictive accuracy of a regression equation when applied to a different population from which it was derived.

• Amount of shrinkage is calculable based on variance, covariance, and sample size of the original regression equation.

Cross Validation

• Definition: Cross-validation involves applying a regression equation to a group other than the one for which it was created, allowing for an estimate of the standard error of estimate for the relationship between predicted and actual values.

The Correlation-Causation Problem

• It is crucial to remember that a relationship between two variables does not imply causation.

• Further research is necessary to establish any cause-and-effect relationship.

Third Variable Explanation

• Explanation highlights that two variables might be correlated, but this correlation can also be influenced by a third variable.

Restricted Range

• This refers to when a sample does not adequately represent the entire range of a variable, possibly distorting the perceived correlation between variables.

Knowledge Check (Example Questions)

1. Which correlation coefficient expresses the relationship between a continuous variable and an artificial dichotomous variable?

   • Options:

     • Pearson r

     • biserial r

     • Spearman’s rho

     • phi coefficient

2. Professor Lopez's study: a significant correlation was found between the number of hours children played violent video games and their level of aggression. However, the colleague indicates that this does not imply causation.

   • This example demonstrates the correlation-causation problem.

Note: All content referenced is adapted from Robert M. Kaplan and Dennis P. Saccuzzo's "Psychological Testing: Principles, Application, and Issues, 9th Edition, 2023 Cengage."