# Methods of Correlation and Regression Analysis, Linear and Curvilinear

By Mordecai Ezekiel; Karl A. Fox | Go to book overview

CHAPTER 9
Three measures of correlation and regression--the meaning and use for each

So many different statistical coefficients have been introduced in the discussion of correlation that there may be some confusion as to the meaning and use of the different statistics. Particularly in the linear situation, there are three statistics which summarize nearly all that a regression analysis reveals.

First, the standard error of estimate indicates how nearly the estimated values agree with the values actually observed for the variable being estimated. This coefficient is stated in the same units as the original dependent variable, and its size can be compared directly with those values.

Second, the coefficient of determination (r2) shows what proportion of the variance in the values of the dependent variable can be explained by, or estimated from, the concomitant variation in the values of the independent variable.1 Since this coefficient is a ratio, it is a "pure number"; that is, it is an arbitrary mathematical measure, whose values fall within a certain limited range, and it can be compared only with other statistics like itself, derived from similar problems.

Third, the coefficient of regression measures the slope of the regression line; that is, it shows the average number of units increase or decrease in the dependent variable which occur with each increase of a specified unit in the independent variable. Its exact size thus depends not only on the relation between the variables but also on the units in which each is stated. It can be reduced to another form, however, by stating each of the variables in units of its own individual standard deviation. In this

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1
These statements are all subject to the error limitations set forth later, in Chapters 17 and 19.

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