Correlation Analysis and Scatter Plots

Correlation Analysis

• Correlation analysis measures the strength of the linear relationship between two variables.
• Pearson Correlation Coefficient (r) measures the degree of linear association between two continuous variables (x and y).
• Values of the Pearson Correlation Coefficient (r) range from -1 to +1.
• Positive values indicate a positive correlation (variables move together).
• Negative values indicate a negative correlation (one variable increases as the other decreases).
• A value of 0 indicates no linear association.

Scatter Plots

• Scatter plots visually represent the relationship between two variables.
• Different patterns of scatter plots depict different degrees of correlation.
• Strong positive correlation shows the points clustered along a positive slope.
• Strong negative correlation shows the points clustered along a negative slope.
• Weak positive correlation shows scattered points with a slight positive trend.
• Weak negative correlation shows scattered points with a slight negative trend.
• No correlation shows scattered points with no discernible trend.

Correlation Effect Sizes

• Weak correlation effect sizes are ±0.1.
• Moderate correlation effect sizes are ±0.3.
• Strong correlation effect sizes are ±0.5.

Correlation Analysis Example (Pizza Hut)

• Pizza Hut wants to know if customer satisfaction is related to customers' likelihood to recommend.
• The correlation coefficient of 0.91 suggests a strong positive correlation.

Conditions for Causation

• A correlation between two variables does not imply causation.
• To establish causation, three conditions must be met:
  • Association between the independent variable(IV) and dependent variable(DV).
  • Time precedence (IV must happen before DV).
  • Elimination of extraneous variables.

Regression Analysis

• Regression analysis measures the nature and degree of association between variables.
• It can establish causation along with mathematical models and underlying knowledge.
• Regression analysis relates (or predicts) one or more independent variables (IV) to the effect or change in the dependent variable (DV).
• IVs (predictors) can be either continuous or categorical.
• DV (effect/outcome) can only be a continuous variable.

Simple Linear Regression

• Used when only one independent variable is in the model.
• The equation is: Yi = β0 + β1Xi + Ei
• β0: intercept (value of Y when X is zero)
• β1: coefficient of X (change in Y for a unit change in X)
• Ei: error term.

Multiple Linear Regression

• Used when there is more than one independent variable.
• The equation is: Yi = β0 + β1X1i + β2X2i +…+ βnXni + Ei
• βj: coefficient related to variable Xj.

Interpreting Regression Parameters

• β0 (Intercept): Represents the mean value of the dependent variable (DV) when all independent variables (IVs) are zero.
• βj (coefficients): Represents the average change in the DV for a one-unit change in an IV, holding other IVs constant.

Coding Categorical Variables

• Categorical variables with more than two levels are coded using dummy variables.
• Code n−1 dummy variables, where n is the total categories.
• One category is set as the baseline (coded as zero).
• The other categories are coded as either 1 or 0.

Regression Analysis (Example)

• Variables like location, pricing, promotion, variety, and service affect customer satisfaction. This analysis helps to uncover if these factors matter.
• The regression equation (example): satisfaction = β₀ + β₁ Location + β₂ Price + β₃ Promotion + β₄ Variety + β₅ Service + ε.
• β coefficient values indicate how a one-unit change in an IV affects the DV.

Testing Significance of Regression Parameters

• Statistical tests assesses if parameter coefficients are significantly different from zero.

• P-values < α (significance level, typically 0.05), coefficients are significant.

• ANOVA test evaluates the overall fit of the model. A low p-value suggests a significant association.

• The p-value for a significant regression coefficient represents the probability of observing the coefficient under the null hypothesis (no relationship between variables).

• The significance is dependent on a chosen significance level, typically 0.05.

• Interpret coefficients in the context of their specific variable and understand what they represent in relationship to the dependent variable (DV).