Exploring Relationships Between Variables

Explanatory vs Response Variables

Understanding how one variable affects another is fundamental in statistics. Variables are classified based on their roles in analysis:

• Explanatory variable: Also called the predictor, this variable is used to explain or predict changes in another variable.

• Response variable: Also called the outcome, this variable is the result or effect being studied.

Examples:

• Does the height of a roller coaster influence its speed?

• Is there a relationship between smoking and lung cancer?

• Does time spent studying affect exam grade?

Association and Correlation

Two variables are said to be associated or correlated if changes in one variable relate to changes in another. However, correlation does not automatically imply causation.

• Association: Used for categorical variables.

• Correlation: Used for quantitative variables.

Examples of correlation:

• Taller roller coasters reach higher maximum speeds.

• Smoking increases probability of lung cancer.

• Studying more will result in higher exam grades.

Correlation does not automatically imply causation.

Technical Wording

• For two categorical variables, use association (predictor and outcome variables).

• For two quantitative variables, use correlation (explanatory vs response).

Contingency Tables

Analyzing Categorical Variables

Contingency tables summarize the relationship between two categorical variables by displaying frequencies or proportions.

• Example: Food Type (Organic/Conventional) and Pesticide Status (Present/Not Present).

• Conditional proportions help determine if an association exists.

Interpretation: 23% of organic food contains pesticide residues, while 73% of conventional food does. This suggests an association between food type and pesticide presence.

Contingency Table Example: Smoking and Lung Cancer

This suggests an association between smoking and lung cancer.

If conditional proportions are approximately the same for both groups, no association is implied.

Risk and Relative Risk (Not on Final Exam)

Risk is the probability of an outcome occurring in a group.

• Risk for smokers:

• Risk for non-smokers:

Relative Risk compares the risk between two groups:

• Relative risk:

• Interpretation: Smokers are ~26 times more likely to develop lung cancer than non-smokers.

Relationship Between Two Quantitative Variables

Visualizing Relationships

• 2 categorical variables: contingency table

• 1 categorical, 1 quantitative: side-by-side box plots

• 2 quantitative variables: scatterplot

A scatterplot displays the relationship between two quantitative variables, with the explanatory variable on the x-axis and the response variable on the y-axis.

Trends in scatterplots:

• Linear positive

• Linear negative

• Non-linear positive

• Non-linear negative

• No relationship

Correlation Coefficient ()

The correlation coefficient measures the strength and direction of the linear relationship between two quantitative variables.

• ranges from -1 to 1.

• : perfect positive linear relationship

• : perfect negative linear relationship

• : no linear relationship

Simple Linear Regression (SLR)

Regression Equation

If a linear relationship exists between and , use linear regression to predict given .

• SLR equation:

• is the predicted value of for a given

• = intercept, = slope

Example: Femur length and height

• Regression line:

• When femur length = 10, predicted height =

Interpreting Slope and Intercept

• Intercept (): Predicted value of when

• Slope (): Change in for a one-unit increase in

Sometimes, the intercept may not have a meaningful interpretation if is not realistic.

Residuals

Residual is the difference between the observed value and the predicted value:

• Measures how far an observation is from the regression line

Coefficient of Determination ()

represents the proportion of variability in explained by the linear relationship with .

• ranges from 0 to 1.

• If is given, is simply squared.

• Example: means 59% of the variation in is explained by .

Linear Regression Assumptions (L.I.N.E.)

• L: Linear relationship between and

• I: Independence of observations

• N: Normal residuals (residuals follow an approximate normal distribution)

• E: Equal variance (homoscedasticity) of residuals

Quadratic Regression

Modeling Curvature

When data shows curvature, quadratic regression can be used:

• Quadratic regression equation:

• Equation for a polynomial of degree 2

Example: Predicting fish length from age:

• Use the equation to predict length for a given age

To choose between linear and quadratic models, visually inspect the plot and compare values (higher indicates a better fit).

Cautions with Associations

Extrapolation

Extrapolation involves using a prediction equation to estimate values for values outside the observed data range. This can lead to unreliable predictions.

• Only use prediction equations within the observed range of .

Influential Outliers

Outliers are data points that do not follow the pattern of the rest of the data. They can significantly affect regression lines and correlation coefficients.

• Always plot the data first to check for outliers.

• Regression and correlation are not resistant to outliers.

Correlation ≠ Causation

Correlation between two variables does not imply that one causes the other. There may be other factors involved.

• Example: Ice cream sales and drownings both increase in summer, but buying ice cream does not cause drowning.

Multiple linear regression can be used to examine more than two variables:

Lurking and Confounding Variables

• Lurking variable: Not measured in the study but influences the relationship between and (e.g., season in the ice cream/drowning example).

• Confounding variable: Measured and considered in the study, influences the relationship between and .

Identifying lurking and confounding variables is crucial for accurate interpretation of associations.