Statistical Relationships

Andrew Martin

PS 372

University of Kentucky

Statistical Relationships

Generally speaking, a statistical relationship between two variables exists if the values of the observations for one variable are associated with the observations for the other variable.

Statistical Relationships

Knowing two variables are related allows political scientists to make predictions.

Some Questions

How strong is the relationship?

What is the direction or shape of the relationship?Is it a causal one?

Does it change or disappear if other variables are considered?

Can we conclude the relationship holds for the population?

Some basics

The level of measurement.

The form of the relationship.

The strength of the relationship.

Numerical summaries of relationships.

Conditional relationships.

Levels of Measurement

Types of Relationships

General association. Exists when the values of one variable, X, tend to be associated with specific values of the other variable.

Monotonic Relationships

Positive monotonic correlation. High values of one variable (X) are associated with high values of another (Y), and conversely, low values (X) are associated with low values (Y).

Negative monotonic correlation. High values of X are associated with low values of Y; low values of X are associated with high values of Y.

Monotonic Relationships

In a positive monotonic relationship, the data curve never goes down once on its way up.

In a negative monotonic relationship, the data curve never goes up once on its way down.

Linear Relationships

Positive linear correlation. This type of correlation is a particular type of monotonic relationship in which plotted X-Y values fall on (or at least close to) a straight line. The line slopes upward from left to right.

Negative linear correlation. In this type of correlation, the plotted values of X and Y fall on a straight line that slopes downward from left to right.

Linear Relationships

Perhaps you remember the following from high school algebra:

Y=mX+bY = Y valueX= X valuem = slope b = intercept

Note: The textbook uses b for the slope coefficient instead of m, and a instead of b for the intercept.

Linear Relationships

Relationships may have other forms, as when values of X and Y increase together until some threshold is met when they decline.

These are known as curvilinear relationships, and will not be addressed in this class.

Strength of Relationship

Strength of relationship is an indication of how consistently the values of a dependent variable are associated with the values of an independent variable.

Strength of Relationship

12-2a. The values of X and Y are tied together tightly. You could imagine a straight line passing through or very near most points.

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Strength of Relationship

12-2b The values tend to be associated – as X increases, so does Y – but the connection is rather weak.

Measures of Association

Measures of association are statistics that summarize the relationships between two variables.

These measures are typically used to support theoretical or policy claims.

Measures of Association

However, a note of caution:

These coefficients

(1) assume a particular level of measurement – nominal, ordinal, interval and ratio

(2) rest on a specific conception of association

To interpret its numerical value one has to grasp the kind of association being measured.

Important Properties of Coefficients

(1) Null value: Zero typically indicates no association, but there are exceptions.

(Ex: Difference of the means)

(2) Maximum values: Some coefficients have a maximum value. Many are bounded, with the typical

lower bound being 0 and the upper bound being 1. (Ex: Correlation)

Important Properties of Coefficients

(3) Strength of the relationship. Subject to lower and upper boundaries, a coefficient's absolute numerical value increases with the strength of the association. (Ex: Regression coefficient)

(4) Level of measurement. Nominal, ordinal and quantitative variables require their own type of coefficient. (Ex: Stats for quantitative data)

Important Properties of Coefficients

(5) Symmetry. A symmetric measure keeps the same value no matter which variable is treated as dependent or independent.

With an asymmetric measure, the coefficient calculated with Y as dependent variable may be different from the same indicator using X as the dependent variable.

(Ex: Correlations)

Important Properties of Coefficients

Standardized vs. Unstandardized

The measurement scale affects the numerical value of most coefficients of association.

Sometimes statisticians transform variables into standardized coefficients so that they all have variances of 1.

(Ex: Standard errors)