Regression

Question 1

What information does b tell us about the nature of the relationship between X & Y?

Answer 1

It tells us about the slope of the regression line

Question 2

If given raw scores, which formula would we use to find b?;

What formula would we use if given r & standard deviations (Sx & Sy)?

Answer 2

b = SPxy / SSx;
b = r (Sy / Sx)

Question 3

How do we find a?

Answer 3

Get the mean of Y & subtract b multiplied by the mean of X (a = Y bar - bX bar)

Question 4

What does the regression equation become when it is standardised?

Answer 4

Z hat y = Beta (standardised correlation coefficient) times Zx; or Z hat y = r xy times Zx

Question 5

How do we find a z-score?;
If the mean of z-scores = 0 & SD = 1, what will the value of b become when standardising?;
What would a become?

Answer 5

Subtract the mean from X & divide by SD;
r (as the SD’s cancel each other out);
Zero (there is no intercept)

Question 6

If a person is at the mean on X (Zx=0), where would we predict their Y to fall?;
If there’s no correlation between X & Y (r=0), where would we predict a person’s Y to fall?;
What if there’s a perfect correlation between X & Y (r=1)?

Answer 6

At the mean (Z hat y=0);
At the mean (Z hat y=0), regardless of score on X;
We’d predict a person is the same number of SDs from the mean on Y as they are on X (Z hat y = Zx)

Question 7

If Y hat = 8, SD = 2 & r = .50, what is the predicted score for someone who is 2 SDs above the mean?

Answer 7

.50 times 2 = 1; 1 x 2 SDs = 2 + 8 = 10

Question 8

If X is not known or r = 0, what’s our best prediction of Y?;
If we use this as a prediction, what’s the average amount of error associated with this prediction?;
How do we find this?

Answer 8

Mean of Y;
Sy (the SD of y - maximum amount of error possible);
Sy = square root of SSy / df (N-1)

Question 9

If X is known & r doesn’t = 0, what’s our best prediction of Y?;
What’s the average amount of error associated with this called?;
How do we find this if given raw scores?;
What other method do we use?

Answer 9

Y hat;
Standard error of the estimate (Sy.x);
Sy.x = square root of SS error / df (remaining variability after using X to predict Y); df = N-2;
Sy.x = Sy times square root of 1 - r squared (slightly underestimates the amount of error)

Question 10

How do we partition variance in regression?;
How can SS predicted be interpreted?;
SS residual?

Answer 10

SS y (total variability in y) = SS y hat (predicted variability) + SS error (what we can’t account for);
bX(i) + a;
e(i)

Question 11

What is the standard error of the estimate?

Answer 11

Gives us an idea of the variability of the real scores around the regression line (aka standard deviation of errors of prediction or SD of the residuals)

Question 12

Explain “regression towards the mean”;
If r = 1, then…;
If r < 1, then…;
the weaker the r the more the mean becomes what?

Answer 12

It’s a phenomenon of related measurements;
Z hat y = Zx: no regression towards the mean;
Z hat y < Zx: regression towards mean: we expect the 2nd measurement to be closer to the mean than the 1st;
The best predictor of Y

Question 13

What does not influence the magnitude of the correlation coefficient?;
What does influence it?

Answer 13

Measurement scales of the variables;

Sample size, restriction of range & extreme scores or outliers

Question 14

If I know that the relationship between two variables is significantly different for men & women but I run a regression across genders anyway, why is my r value inaccurate?

Answer 14

Presence of heterogeneous subsamples

Question 15

How are residual scores represented?

Answer 15

e(i) = Y(i) - Y hat(i)

Question 16

What is r squared?;

How do we calculate Error Variance?;

Answer 16

Predictable variance; gives the proportion of variance in Y accounted for or predicted by X;
1 - r squared; error variability (SS error) / total variability (SSy)