MVDA. All Workgroup Assignment Answers
Week 1: Multiple Regression Analysis (MRA)
1.1) Multiple Regression Analysis
a) Check the assumptions linearity, homoscedasticity and normality of the
residuals. Is the regression model a reasonable approximation of the data?
From the data in the scatterplot, we can see that the linearity and homoscedasticity
assumptions are met because the residuals are scattered randomly and without forming a
systematic pattern. Therefore, yes, the regression model is a reasonable approximation of
the data.
b) Is there evidence of multicollinearity in the data? (explain)
No, there isn’t any evidence of multicollinearity in the data. This is because the tolerance is
larger than 0.10 and the VIF is smaller than 10.
c) Are there outliers, influential points, or outliers on the predictors? (explain)
There are no outliers as the Std. residual < |3|.
There are no influential points as cook’s distance < 1.
There are no outliers on the predictors as Leverage < 3(2+1)/58 = 0.155.
d) What are the null and the alternative hypothesis to test the regression model?
H0: there isn’t a relation between reading achievement (Y) and language skill (X1) and
perceptual motor skill (X2): r-squared = 0
Ha: language skill (X1) and/or perceptual motor skill (X2) predict reading achievement (Y):
r-squared > 0
e) Can the null hypothesis be rejected? (report test statistic, df, and p value)
Yes, the null hypothesis can be rejected.
F(2,55) = 37,770, p < .005
f) What are the null and the alternative hypothesis to test the individual
coefficients?
H0: βj = 0
Ha: βj ≠ 0
g) Which predictor(s) is/are significant? (report test statistic, df and p values)
Both predictors are significant:
Language skill: T(54) = 4.344, p < 0.05
, Perceptual motor skill: T(54) = 2.998, p < 0.05
h) Write down both the unstandardized and standardized regression equations.
Ŷ = -1.596 + 1.049X1 + 0.464X2.
ZŶ = 0.495ZX1 + 0.342ZX2.
i) Interpret the unstandardized and standardized coefficients (see page 20).
Unstandardized coefficients:
Language skill + 1 ---> increase of the Reading Achievement by 1.049
PercMotorSkill + 1 ---> increase of the Reading Achievement by 0.464
Overall, there is a higher language and perceptual motor skill, meaning a better reading
achievement
Standardized coefficients:
The Language Skill is a better predictor for the Reading Achievement than the Perceptual
Motor Skill.
j) How much variance of Y in total is explained by X1 and X2?
R^2 = VAF = SS(Regression) / SS(Total) = 101..845 = 0.579
VAF x 100% = 57.9%
k) How much variance of Y is uniquely explained by X1? How much variance of Y
is uniquely explained by X2? What is the best predictor?
LS = (0.380)^2 x 100% = 14.4%
PMS = (0.262)^2 x 100% = 6.9%
l) Draw a Venn diagram that illustrates the decomposition of the variance of Y
Week 1: Multiple Regression Analysis (MRA)
1.1) Multiple Regression Analysis
a) Check the assumptions linearity, homoscedasticity and normality of the
residuals. Is the regression model a reasonable approximation of the data?
From the data in the scatterplot, we can see that the linearity and homoscedasticity
assumptions are met because the residuals are scattered randomly and without forming a
systematic pattern. Therefore, yes, the regression model is a reasonable approximation of
the data.
b) Is there evidence of multicollinearity in the data? (explain)
No, there isn’t any evidence of multicollinearity in the data. This is because the tolerance is
larger than 0.10 and the VIF is smaller than 10.
c) Are there outliers, influential points, or outliers on the predictors? (explain)
There are no outliers as the Std. residual < |3|.
There are no influential points as cook’s distance < 1.
There are no outliers on the predictors as Leverage < 3(2+1)/58 = 0.155.
d) What are the null and the alternative hypothesis to test the regression model?
H0: there isn’t a relation between reading achievement (Y) and language skill (X1) and
perceptual motor skill (X2): r-squared = 0
Ha: language skill (X1) and/or perceptual motor skill (X2) predict reading achievement (Y):
r-squared > 0
e) Can the null hypothesis be rejected? (report test statistic, df, and p value)
Yes, the null hypothesis can be rejected.
F(2,55) = 37,770, p < .005
f) What are the null and the alternative hypothesis to test the individual
coefficients?
H0: βj = 0
Ha: βj ≠ 0
g) Which predictor(s) is/are significant? (report test statistic, df and p values)
Both predictors are significant:
Language skill: T(54) = 4.344, p < 0.05
, Perceptual motor skill: T(54) = 2.998, p < 0.05
h) Write down both the unstandardized and standardized regression equations.
Ŷ = -1.596 + 1.049X1 + 0.464X2.
ZŶ = 0.495ZX1 + 0.342ZX2.
i) Interpret the unstandardized and standardized coefficients (see page 20).
Unstandardized coefficients:
Language skill + 1 ---> increase of the Reading Achievement by 1.049
PercMotorSkill + 1 ---> increase of the Reading Achievement by 0.464
Overall, there is a higher language and perceptual motor skill, meaning a better reading
achievement
Standardized coefficients:
The Language Skill is a better predictor for the Reading Achievement than the Perceptual
Motor Skill.
j) How much variance of Y in total is explained by X1 and X2?
R^2 = VAF = SS(Regression) / SS(Total) = 101..845 = 0.579
VAF x 100% = 57.9%
k) How much variance of Y is uniquely explained by X1? How much variance of Y
is uniquely explained by X2? What is the best predictor?
LS = (0.380)^2 x 100% = 14.4%
PMS = (0.262)^2 x 100% = 6.9%
l) Draw a Venn diagram that illustrates the decomposition of the variance of Y
