Correlation
A starting point to understanding a model is the statistical concept of correlation. A correlation is a measure of relationship between two variables. A correlation measures that relationship in terms of how the values of one variable is related to another variable. The scale of this relationship is the abstract scale of standard deviation units. Scale units refers to the conceptual construct used to measure. Scales we are familiar with include weight measured in pounds and height measured in inches. A more abstract scale is calories, a measure of energy (heat) in food.The following exploration utilizes a common sense example, and an educationally related example. We start with the common sense example. A doctor tells a patient, "If you exercise, you will lose weight." This is a simple correlation statement. More exercise (one variable) will generally lead to loss of weight (another variable). Expanding on this statement the doctor continued, "When you exercise you burn more calories." "The burning of calories through exercise tends to cause one to loose weight."First, let's examine the measure of correlation. The relationship is expressed as a coefficient, a number on the standard deviation scale. A correlation coefficient varies from -1 to +1, in other words, the coefficient is bounded from -1 to +1.
- Positive coefficients indicate that the two variables tend to vary in the same direction. That is, as one variable increases so does the second one.
- Negative coefficients indicate that as one variable increases, the second variable decreases.
- Coefficients in the middle, near 0, indicate no significant relationship between the two variables.
In our common sense example, the first statement, "If you exercise, you will lose weight." is a statement that indicates a negative correlation. Increase exercise, loss of weight, one variable increases (exercise) second variable deceases (weight). The second statement, "When you exercise you burn more calories." is a statement that indicates a positive correlation. As one variable increases (exercise) the second one also increases (use of calories).Translating this to a research example, a researcher is interested in the relationship between the number of math courses and the math skills of a student. (OK, this is not a very complex question, yet.) The figure below represents an examination of this relationship.
The rectangle labeled "Math" is the sum of math courses hours taken and the "Math Skills Score" is an independent measure of math skill from a standardized test. The label of the path (arrow) between these two is .48. The direction of the arrow indicates that the model assigns "math" as an independent variable and "skills" is the dependent variable. In a simple correlation the relationship is symmetrical. However, in more complex models this assignment of order does make a difference in the coefficient's value. The .48 is the correlation coefficient of math courses to math skills. The other number that is important at this point is the .23 in the upper right corner of the dependent variable "skills" is the proportion of the variance in skills that can be explained by knowing how many math hours have been completed. This value is referred to as "explained variance."
The next step is to examine regression and more about correlation
