12.2 Describe the Strength of Association
Relationship between correlation and slope
- If data has the same amount of variability for each variable, with sx=sy, then r=b - the correlation r does not depend on the unit of measurement - the correlation represents the value that the slope equals if the two variables have the same standard deviation
Properties of the Correlation, r
- correlation r has the same sign as slope b (r>0 upward trend, r<0 downward trend) - correlation r always falls between -1 and +1 - the larger the absolute value of r, the stronger the linear association
What is the prediction error?
- for any given subject, it is the difference between the observed and predicted values of y - the error using regression line to make a prediction is: y-yhat-> RESIDUAL SUM OF SQUARES RSS - the error using y(mean) to make a prediction is: y-y(mean) -> TOTAL SUM OF SQUARES TSS -> when a strong linear association exists, the regression equation prediction (yhat) are better than merely using the mean y(mean): (y-yhat)^2<(y-ymean)^2
The correlation is a standardized slope
- the slope b is not able to describe the strength of the association because the slope's numerical value depends on the units of measurement. - the correlation is a standardized slope, and does not depend on the measurement R=b(sx/sy)
R^2= proportional reduction error says..
... the proportion falls between 0 to 1, and the error using yhat to predict y is % smaller than the error using ymean to predict Or „% of the variability in y is explained by x"
regression to the mean
At any particular x value, the predicted value if y is closer to its mean than x is to its mean That is, if an x value is a certain number of standard deviations from its mean, then the predicted y is r times that many standard deviations from its mean.
Minus point of the slope
It does not discribe the strengh of the association, it only indicates whether its positive or negative, and its also unresistant.
The Squared Correlation (r^2) describes predictive power
The variables are strongly associated if you can predict y much better by substituting x values into the prediction equation yhat=a+by than by merely using the sample mean and ignoring x
