# 1 Introduction

Paul Boyd

One of the requirements in attempts to establish ‘causality’ is that there exists *concomitant variation* between the *Independent* (or ‘influencing’) variable and the *Dependent* (or ‘influenced’) variable.^{1} What that means is that we need to test statistically to determine if we can conclude that when the Independent variable changes there is also a (subsequent) change in the dependent variable. And, just as important, just because it looks to be true in our sample data, can we draw the same conclusion (via an ‘inference’) about the population at large.^{2} For example, that when the temperature goes down, do heating bills tend to go up? Or, do structured employee interviews lead to reduced long-term turnover?

The statistics that test for these types of relationships depend on what is known as the ‘level of measurement’ for each of the two variables. There are three ‘levels’ that we measure: *Categorical*, *Ordinal* or *Numeric* (UCLA Statistical Consulting, Date unknown).

*Categorical* variables are those where the values of the variables are groups.

- New England, Southeast, Midwest, etc. regions of the U.S., or
- SUV, Sedan, Coupe, Minivan, etc. automobile body styles, or
- Business, Arts & Sciences or Engineering undergraduate college degrees.

*Ordinal* variables are those where the categories increase in value along some scale, but the amount of ‘increase’ from one category to the next is undefined or inconsistent.

- very slow, somewhat slow, somewhat fast, etc. or
- <$50k/year, $50k — $100k/year, 100k — $200k/year annual household income, or
- High School, Undergraduate or Graduate degrees.

*Numeric* variables are those where the categories of the variables are specific *quantified* values. (Note that combined here into the single *Numeric* category are the traditional *Interval* and *Ratio* levels of measurement, as the statistical tests of concomitance are largely the same.)

- -10°, 68°, or 98.6° or
- 0 or 1 or 2 or 3, etc. children in the household or
- the dollar value of corporation’s debt.

See Appendix A for further examples of these variable types.