Statistics on Marginal Distributions

Univariate vs Bivariate Data

• Univariate data involves the measurement of a single variable for each individual.
• Bivariate data entails measuring two variables for each individual.

Associations Between Variables

• Positive association indicates that as one variable increases, the other variable also increases.
• Negative association describes a scenario where an increase in one variable leads to a decrease in the other variable.

Correlation and Causation

• Correlation does not imply causation; a properly designed experiment is necessary to establish a causal relationship.
• Observational study data cannot conclusively indicate causality.

Scatter Diagrams and Linear Relationships

• A scatter diagram represents the relationship between two quantitative variables.
• Data points that do not align in a straight line do not exhibit a linear relationship.
• Linear relations can be classified as positive or negative based on the direction of the trend in the scatter diagram.

Linear Correlation Coefficient

• The linear correlation coefficient (r) ranges from -1 to +1, where:
  • r = +1 indicates a perfect positive linear relationship.
  • r = -1 indicates a perfect negative linear relationship.
  • r close to 0 suggests little or no linear relation.
• Examples of correlation coefficients:
  • r = 0.787 suggests a strong positive relationship.
  • r = -0.946 suggests a strong negative relationship.
  • r = 0.049 indicates almost no relationship.

Residuals

• A residual is the difference between the observed value of a response variable and its predicted value.
• Positive residuals imply that observed values exceed predicted values.

Least-Squares Regression Line

• Points on the least-squares regression line represent predicted y-values corresponding to x-values based on the data.

Correlation vs Regression Analysis

• The slope of the regression line corresponds with the correlation:
  • A negative correlation results in a negative slope.

Practical Examples of Correlation

• Example of negative correlation: outside temperature and the number of people wearing coats.
• Example of positive correlation: the number of doctors and administrators at a hospital.
• Example of no correlation: size of the TV in a living room and the heating bill.

Lurking Variables

• A lurking variable can influence the relationship between two variables, such as the economy affecting both cell phone prevalence and crime rates.### Scatter Diagrams and Regression Analysis
• The x-axis of the scatter diagram ranges from 0 to 6, and the y-axis is also from 0 to 6.
• Six points plotted: (1,5), (2,5.8), (4,4.8), (5,3.4), (5,3), (6,2.8).
• The plotted points suggest a negative association, indicating that as x increases, y tends to decrease.
• The least-squares regression line formula is given by (y = b_1x + b_0), where:
  • (b_1) is the slope calculated as (b_1 = r \cdot \frac{sy}{sx}).
  • (b_0) is the y-intercept given by (b_0 = \bar{y} - b_1\bar{x}).

Calculation of Regression Line

• Slope (b_1 = -0.571).
• Using (b_1) to calculate the intercept (b_0):
  • (b_0 = 4.0500 - (-0.571)(3.8333) = 6.239).
• The least-squares regression line is (y = -0.571x + 6.239).
• The line indicates a downward trajectory, represented graphically alongside the scatter diagram.

Pediatric Study on Height and Head Circumference

• A study investigates the relationship between height (x) and head circumference (y) of 11 children.
• The slope of the regression line is (0.081), implying:
  • For every 1 inch increase in height, head circumference increases by approximately 0.081 inches.
• The y-intercept, at (15.1), does not carry practical significance since height cannot be zero.

Predictions and Residuals

• For a child with a height of 24 inches, predicted head circumference (= 17.04) inches.
• If the observed head circumference of the child is 17.4 inches, the residual is calculated as:
  • (residual = observed - predicted = 17.4 - 17.04 = 0.36).
• This positive residual indicates that the observed value is above the predicted value from the regression model.

Scatter and Residual Plots

• A scatter plot with the regression line and labeled residuals enhances understanding of data discrepancies.
• Variations in head circumference among children of the same height can arise from biological differences, measurement errors, or other factors.

Coefficient of Determination

• Defined as (R^2), measures the proportion of total variation in the response variable explained by the regression line.
• Total deviation can be expressed as:
  • (Total deviation = unexplained deviation + explained deviation).

Residual Plots

• A residual plot visually assesses the adequacy of the linear model, displaying residuals on the vertical axis against the explanatory variable on the horizontal axis.
• A well-fitted linear model will show a random distribution of residuals, while patterns indicate potential issues in linearity or variance constancy.

Summary of Analysis Techniques

• Techniques used in statistical analysis involve plotting data to visualize relationships, calculating regression lines, and interpreting slopes and intercepts.
• Errors such as residuals provide insight into the accuracy of predictions made by the regression model, aiding in refining models and understanding underlying relationships.### Residual Plots and Linear Models
• A U-shaped pattern in a residual plot indicates a violation of linear model assumptions.
• Constant error variance is when the spread of residuals remains steady as the explanatory variable increases.
• Outliers are extreme observations that deviate from the data's overall pattern; they can be identified in a residual plot where residuals lie far from others.
• Absence of outliers in a residual plot suggests a potential linear relationship between explanatory and response variables.

Analyzing Scatter Diagrams

• Coefficient of determination (R²) quantifies the variation in the response variable explained by the regression line.
• Values indicate strength of correlation:
  • R² close to 1 indicates strong association.
  • R² = 0.58, 0.94, and 0.01 represent weak to strong correlations in different scatter diagrams.
• A scatter diagram with all points on a line indicates perfect correlation (R² = 1).

Influence of Data Points

• Influential points can significantly impact the least-squares regression line and correlate with a notable change in the slope or intercept.
• Analysis of whether a point is influential can be guided by comparing regression lines with and without the outlier.

Geyser Eruptions Case Study

• Data indicates a positive linear association between time between geyser eruptions and eruption length.
• The residual plot validates the linear relationship, as no patterns discernible imply good model appropriateness.
• The coefficient of determination (88.4%) suggests that the least squares regression line accounts for a substantial portion of variation in eruption length.

Planetary Sidereal Year and Distance

• A scatter diagram illustrates the relationship between distance from a star and the sidereal year of a planet.
• The linear correlation coefficient (r = 0.987) shows a strong positive relationship, indicating a significant linear correlation.
• A least-squares regression line is computed to further analyze the relationship: y = 0.0624x - 12.2.
• A residual plot is generated to verify the quality of the regression model with respect to the explanatory variable.

Summary of Key Concepts

• Residual analysis is crucial for checking linear model validity.
• Coefficient of determination (R²) is essential for quantifying explained variation.
• Influential data points require careful examination in regression analysis.
• Strong correlation coefficients indicate a high degree of linear dependence between variables.