week 7 Regression

Question 1

The Regression Equation

Answer 1

Y^{^}=bX + a

Where Y^{^}=predicted or estimated Y

b=the slope

a=the intercept

It is still very important to remember that the regression equation has been derived by fitting the data to an equation, and is not 100% accurate. Nor is there a cause and effect, but only still a correlation.

Other situations may be better served with more complex equations such as when the data appears more curvilinear.

The question is not whether a straight line can be drawn on the data, but how accurate it is and how representational it is to use as a predictor.

Question 2

b (the slope)

Answer 2

The slope of the regression line tells us the degree of difference in the predicted score(Y), which is associated with a one unit difference in the predictor (X) variable.

b=COV_xy/S²_x

Question 3

a (the intercept)

Answer 3

The intercept tells us the value of Y, when X=0. Sometimes there are situations however where to have X=0 is nonsensical.However, the intercept must still be calculated in order to obtain the regression equation.

Question 4

Centre-ing

Answer 4

The data is artifically centred around the mean, by subtracting X^- from every X value. The slope remains the same.

Question 5

Smoothing

Answer 5

Smoothing techniques are used to average the Y values closer to the target value of the predictor.

Question 6

Standard error of the Estimate

Answer 6

If we recall that we previously determined with 2 variables, how much variation in one could be explained by the other variable

(r²), (and therefore that some variation occurred due to other factors), it is logical that for any Y^{^} there is some degree of error.

Thus, SE=S_y[square root of (1-r²)]

where SE =standard error of the estimate

and S_y=standard deviation of Y.

The Standard Error tells us on average, how much our prediction is “off” by.

Question 7

Confidence limits on the prediction

Answer 7

CI_y=Y^{^} +/- (t_a/df)(SE)

Where CI_y= the confidence limits around the predicted Y value. (usually confidence limts are set at 95%)

t(_a/df) is the t_critical with a =alpha (usually 0.050) and df=degrees of freedom (N-2)

and SE is the standard error of the estimate.

Thus, given X, we can predict Y^{^} , and Y^{^} +/- (t_critical)(SE)

gives us the 95% confidence range with which we believe the Y score will be, given X.