Test questions for Lessons 8-11 – Multiple regression
1. A researcher conducted a study to predict the amount of sushi a person consumes per
month from his/her swimming ability. He found the following regression equation: Z'y =
-0.8Zx. He also found that the regression model is significant (p < .05). From this, he can
conclude (with a 95% confidence level) that:
a) Swimming ability explains 80% of the variance in sushi consumption in the
population
b) Swimming ability explains 64% of the variance in sushi consumption in the
population
c) Swimming ability explains more than 0% of the variance in sushi
consumption in the population
d) It’s likely that the variables are not correlated in the population
e) Answers (b) and (d) are correct
Explanation:
When we reject the null hypothesis we accept the research hypothesis with 95%
confidence. The research hypothesis is that R2 is larger than zero. That is, we explained
more than 0% of the variance in DV in the population. Even though we found R 2 of 64%,
we cannot conclude that is the precise proportion of explained variance in the population.
2. A researcher wanted to examine how openness to experience (X1) and extroversion (X2)
predict the degree to which one enjoys pop music (Y). The following Venn diagram
describes the relationship between the study variables:
Pop
enjoyment
(Y)
Openness to
experience
(X1) Extroversion
(X2)
Which of the following describes the relationship between the predictors?
a) Extroversion is partially redundant
b) Extroversion is fully redundant
c) Openness to experience is fully redundant
d) There is no redundancy between the predictors
, Explanation:
The diagram describes full redundancy of X2, as the entire overlap between X2 and Y is
covered by X1.
3. A researcher wanted to test how the amount of reflective therapeutic intervention (X1)
and interpretation therapeutic intervention (X2) predict psychological well-being of
psychological patients (Y). The researcher found the following relationship between the
variables:
Reflection Interpretation
(X1) (X2)
Well-being
(Y)
Which of the following is true?
a) sr12 = pr12
b) sr12 < r12
c) sr22 = R2
d) sr22 = r22
Answer:
Reflection Interpretation
(X1) (X2)
Well-being
(Y)
a b
error
As you can see in the diagram above, sr12 = a/(a+b+error). pr12 = a/(a+error). Therefore,
sr12 < pr12. sr22 = b/(a+b+error). R2 = (a+b)/(a+b+error). Therefore, sr22 < R2. In addition,
r12 = a/(a+b+error), and r22 = b/(a+b+error). Therefore, sr12 = r12, and sr22 = r22.
, 4. If a suppressor variable X2 is entered as an additional predictor into a simple regression
model predicting y from X1, what can possibly happen to the beta of X1?
a) It would decrease in absolute value and retain its sign
b) It would increase in absolute value and retain its sign
c) It would decrease in absolute value and change its sign
d) Answers (b) and (c) are possible
e) All of the above (a, b and c are possible)
Explanation:
When not entered to the regression equation, a suppressive variable clouds the
relationship between another predictor and the predicted variable. Adding the suppressive
variable to the regression equation allows us to control for its influence, and therefore we
can observe the relationship between the predictor and the predicted variable more
accurately. Thus, the beta (β) of the other predictor (X1) would increase in its absolute
value or change its sign from positive to negative (or vice versa) after adding the
suppressive variable to the equation.
5. A researcher performed a simple regression analysis to test whether grades in a
Multivariate Linguistics course (X1) predict happiness (on a scale of 1-100). He then
added another predictor: the number of jokes told by the teaching assistant of this course
(X2). After calculating the new model, he found that β2 is significant. Which of the
following is true regarding the researcher’s new model (as compared with his original
model)?
a) SStotal would increase
b) SStotal would decrease
c) SSregression would increase
d) SSresidual would increase
Explanation:
A significant β means that this particular variable explains unique variability in the
dependent variable. Therefore, adding another predictor whose β is significant means that
the proportion of the explained variability in the model (R 2) is increased. In this case
SSregression (the explained variability) increases and SSresidual (the unexplained
variability) decreases. Note, that SStotal is the total variability in the dependent variable,
which is not changed by adding predictors to the model.
1. A researcher conducted a study to predict the amount of sushi a person consumes per
month from his/her swimming ability. He found the following regression equation: Z'y =
-0.8Zx. He also found that the regression model is significant (p < .05). From this, he can
conclude (with a 95% confidence level) that:
a) Swimming ability explains 80% of the variance in sushi consumption in the
population
b) Swimming ability explains 64% of the variance in sushi consumption in the
population
c) Swimming ability explains more than 0% of the variance in sushi
consumption in the population
d) It’s likely that the variables are not correlated in the population
e) Answers (b) and (d) are correct
Explanation:
When we reject the null hypothesis we accept the research hypothesis with 95%
confidence. The research hypothesis is that R2 is larger than zero. That is, we explained
more than 0% of the variance in DV in the population. Even though we found R 2 of 64%,
we cannot conclude that is the precise proportion of explained variance in the population.
2. A researcher wanted to examine how openness to experience (X1) and extroversion (X2)
predict the degree to which one enjoys pop music (Y). The following Venn diagram
describes the relationship between the study variables:
Pop
enjoyment
(Y)
Openness to
experience
(X1) Extroversion
(X2)
Which of the following describes the relationship between the predictors?
a) Extroversion is partially redundant
b) Extroversion is fully redundant
c) Openness to experience is fully redundant
d) There is no redundancy between the predictors
, Explanation:
The diagram describes full redundancy of X2, as the entire overlap between X2 and Y is
covered by X1.
3. A researcher wanted to test how the amount of reflective therapeutic intervention (X1)
and interpretation therapeutic intervention (X2) predict psychological well-being of
psychological patients (Y). The researcher found the following relationship between the
variables:
Reflection Interpretation
(X1) (X2)
Well-being
(Y)
Which of the following is true?
a) sr12 = pr12
b) sr12 < r12
c) sr22 = R2
d) sr22 = r22
Answer:
Reflection Interpretation
(X1) (X2)
Well-being
(Y)
a b
error
As you can see in the diagram above, sr12 = a/(a+b+error). pr12 = a/(a+error). Therefore,
sr12 < pr12. sr22 = b/(a+b+error). R2 = (a+b)/(a+b+error). Therefore, sr22 < R2. In addition,
r12 = a/(a+b+error), and r22 = b/(a+b+error). Therefore, sr12 = r12, and sr22 = r22.
, 4. If a suppressor variable X2 is entered as an additional predictor into a simple regression
model predicting y from X1, what can possibly happen to the beta of X1?
a) It would decrease in absolute value and retain its sign
b) It would increase in absolute value and retain its sign
c) It would decrease in absolute value and change its sign
d) Answers (b) and (c) are possible
e) All of the above (a, b and c are possible)
Explanation:
When not entered to the regression equation, a suppressive variable clouds the
relationship between another predictor and the predicted variable. Adding the suppressive
variable to the regression equation allows us to control for its influence, and therefore we
can observe the relationship between the predictor and the predicted variable more
accurately. Thus, the beta (β) of the other predictor (X1) would increase in its absolute
value or change its sign from positive to negative (or vice versa) after adding the
suppressive variable to the equation.
5. A researcher performed a simple regression analysis to test whether grades in a
Multivariate Linguistics course (X1) predict happiness (on a scale of 1-100). He then
added another predictor: the number of jokes told by the teaching assistant of this course
(X2). After calculating the new model, he found that β2 is significant. Which of the
following is true regarding the researcher’s new model (as compared with his original
model)?
a) SStotal would increase
b) SStotal would decrease
c) SSregression would increase
d) SSresidual would increase
Explanation:
A significant β means that this particular variable explains unique variability in the
dependent variable. Therefore, adding another predictor whose β is significant means that
the proportion of the explained variability in the model (R 2) is increased. In this case
SSregression (the explained variability) increases and SSresidual (the unexplained
variability) decreases. Note, that SStotal is the total variability in the dependent variable,
which is not changed by adding predictors to the model.
