Lecture 4_MLR, sr^2, pr,

Question 1

What are the building blocks of

multiple linear regression (MLR)?

Answer 1

Correlation coefficients (r)

Question 2

The Regression Line (2 predictors)

Answer 2

Y’ = a + b1X1 + b2X2
• Y′ is the predicted value of the DV, Y.
• a is the intercept of the regression line.
• b1 and b2 are the partial regression coefficients for the predictor variables X1 and X2, respectively

Question 3

What is Y’ in the regression equation for 2 predictors?

Answer 3

the predicted value of the DV, Y.

Question 4

What are b1 and b2 in the regression equation for 2 predictors?

Answer 4

the Partial regression coefficients for the predictor variables X1 and X2, respectively

Question 5

In a path diagram, what does the e path represent?

Answer 5

(1 - R²) the coefficient of non-determination

• error not explained by the model

Question 6

In a path diagram, what does the curved, double-headed arrow between two predictors represent?

Answer 6

the bivariate correlation between the 2 variables.
• indicates that the researcher is not going to explain the correlation between X1 & X2, but acknowledges that it is not zero.

Question 7

In a path diagram, what do squares represent?

Answer 7

measured variables

Question 8

In a path diagram, what do circles represent?

Answer 8

Factors (latent variables hypothesized by “fun” math)

Question 9

In a path diagram, what do the straight single-headed arrows represent?

Answer 9

the standardized coefficient (β) between the particular predictor variable (IV) and the outcome variable (DV)

Question 10

A path model must have …

Answer 10

a coefficient for every arrow

Question 11

What is the interpretation for the partial regression coefficient?

Answer 11

the expected change in the outcome variable (DV) when the predictor variable (IV) changes by 1 unit, controlling for all other predictor variables (holding all other IVs constant)

Question 12

Why can we not compare the partial regression coefficients (b) to determine the relative importance of each predictor variable (IV)?

Answer 12

the size of b is influenced by the scale/metric in which the IV is measured, and may be different for each IV

Question 13

What do the standardized regression coefficients (β) indicate?

Answer 13

the relative influence of the IVs in the equation.
• the expected change in the DV (in st. dev. units) when the IV changes by one st. dev. unit, holding all other IVs constant

Question 14

Which is better for comparing across groups, b or β? Why?

Answer 14

comparisons across groups should be based on bs and not βs.

• βs are population specific; they are sensitive to fluctuations in variances and covariances across populations

Question 15

What does the Squared multiple correlation (R²) represent?

Answer 15

the proportion of variance in the DV accounted for by all the IVs

Question 16

What does the Standard Error of the Estimate express?

Answer 16

the amount, on average, that the predicted value of Y will deviate from the observed value (Obtained from R² and the standard deviation of the DV)

Question 17

When will R² be equal to the sum of the squared bivariate correlations?

Answer 17

When the 2 predictors are not correlated with one another (r12 = 0)

Question 18

What does the squared semipartial (part) correlation relate?

Answer 18

• how much R² will increase when a predictor is added last to the model
• the unique contribution of a predictor to the variance explained by the model

Question 19

When predictors are correlated with each other, what will give us the unique proportion of variance in Y that is attributable to each predictor in the model?

Answer 19

Squaring the semipartial correlations

Question 20

How can we calculate an estimate of the explained variance that is ambiguous?

Answer 20

Take the difference between R² and the sum of all semipartial correlations

Question 21

Given the correlation matrix, means, and standard deviations of the variables, what can we calculate?

Answer 21

the regression coefficients (standardized and unstandardized), R² and the St. Error of the Estimate

Question 22

What is the notation for a zero-oreder (Pearson) correlation?

Answer 22

r12, rY1, rY2, etc

Question 23

What is the notation for a partial correlation?

Answer 23

prY1 = rY1.23
prY2 = rY2.1
pr12 = r12.Y34

Question 24

What is the notation for a Semipartial correlation?

Answer 24

srY1 = rY(1.23)
srY2 = rY(2.1)
sr12 = r1(2.Y34)

Question 25

What does the order (zero-order, first-order, second order, etc.) of a correlation coefficient (r, sr, pr) indicate?

Answer 25

the number of variables that have been “partialed” out
• rY1 = zero-order
• rY2.1 = first-order [also rY(2.1)]
• rY1.23 = second-order

Question 26

What is a partial correlation (pr)?

Answer 26

the correlation between two variables with the influence of one (or more) other variables removed from Both.

Question 27

What does the squared partial correlation between the DV and an IV express?

Answer 27

unique variance of the IV as a proportion of previously unexplained variance in the DV

Question 28

For a given set of variables in a MR, which will be larger (in absolute value): the partial correlation or the semipartial correlation?

Answer 28

the partial correlation (think about the difference in the denominators)
sr = a/(a + b + c + d)
pr = a/d

Question 29

What are 2 interpretations for e?

Answer 29

• the error in prediction.

* Or a transformed version of Y with the influence of the predictor(s) removed, or partialed out.

Question 30

What is an alternative method for a partial correlation using SPSS?

Answer 30

Run 2 regressions:

1. regress one variable, IV(1), onto other IVs whose effects want removed and save residuals as a new variable (new residual variable is a transformation of the original variable with the influences of the other two removed
2. repeat for 2nd variable, DV (dido)
3. correlate new IV(1) and DV residual variables

Question 31

What does the correlation between the two residual variables tell us?

Answer 31

the partial correlation between IV(1) & DV when the influences of other IV have been removed from both of them.

Question 32

What are 2 measures of how well regression model fits our data?

Answer 32

• R²

* size of the SE of the estimate

Question 33

What are the units of the SE of the estimate?

Answer 33

the units in which the DV is measured