W1: Multiple Linear Regression

Question 1

In multiple linear regression, there are:

Answer 1

Multiple independent variables (X) and one dependent variable (Y)

Question 2

In multiple linear regression, there are:

Answer 2

Multiple independent variables (X) and one dependent variable (Y)

Question 3

The multiple R-squared value for a regression represent the proportion of the variation in the Y variable that can explained by its regression on the X variables.

True or False?

Answer 3

True

Question 4

The assumptions which we need to check when we perform a multiple linear regression are (3):

Answer 4

Normality of the errors
Common variance of the errors
Independence of the errors

Question 5

For the Kolmogorov-Smirnov and Shapiro-Wilk tests of Normality, if p < 0.05 then we conclude that the Normality assumption has been satisfied.
True or False?

Multiple linear regression

Answer 5

False

Question 6

If the p-value for a correlation coefficient was p = 0.036 then the correlation would be significant at

Answer 6

5% level

Question 7

We can use multiple linear regression to allow the use of several X-variables (predictors/IV) to predict the

Answer 7

response Y

Question 8

What is the multiple linear regression model equation?

Answer 8

Y = a + (b1 * X1) + (b2 * X2) + … + e

Question 9

What is the multiple linear regression model equation - Y?

Answer 9

Y is the response (DV)

Question 10

What is the multiple linear regression model equation? - X

Answer 10

X is predictors/IV

Question 11

What is the multiple linear regression model equation? - B1/B2

Answer 11

B1/B2 is the slope/gradient

Question 12

What is the multiple linear regression model equation? - a

Answer 12

A is constant

Question 13

What is the multiple linear regression model equation? - e

Answer 13

e is error term

Question 14

The multiple linear regression has predictor variables (X) with its own

Answer 14

coefficient (b1/b2)

Question 15

Why is their an error term ( e ) in multiple linear regression?

Answer 15

Knowing the values of X1,X2…. does not allow us to predict the value of Y exactly

Question 16

What is a residual?

Answer 16

Difference between the observed Y-value and its prediction (fitted value) based on corresponding X-values

Question 17

How to calculate residual?

Multiple linear regression

Answer 17

Residual = Observations - Fitted Valeu

Question 18

If the scatterplot of residuals are not independent + common variance (funnel effect graph)

Multiple linear regression

Answer 18

Question 19

If the scatterplot of residuals are not independent + common variance (funnel effect graph)

Graph does not have independence

Answer 19

Question 20

Test signifiance of each predictor, test null and alternate hypothesis that:

Multiple linear regression

Answer 20

H0: b = 0 vs H1 : b≠ 0 (for each particular X variable)

Question 21

Generally, an R-Squared above 0.6 (2)

Multiple linear regression

Answer 21

makes a model worth your attention
Means that most of the variability in Y var can be explained by X var/multiple linear regression model

Question 22

Step 1 (In SPSS): Writing Regression Equation (2)

Answer 22

The regression equation is:
MRI Count = 237.598 + 55.236(Gender) + 1.280 (PIQ) + 6.515 (Height)

Question 23

Step 1 (In R): Writing Regression Equation (2)

Answer 23

The regression equation is:
Costs = -3085.657 -86.774(Region) + 511.084(Sex) + 115.61(Age) -2.62(Martial) + 51.16 (Alcohol) + 138.00 (Cigs) -269.264(Exercise)

Question 24

How can you tell Y and X variables utilised in multiple linear regression model in R? (4)

Answer 24

• Costs = Y
• X = Region, Sex, Age, Marit, Alco
• Data is from ex.data
• This is all stored in variable called model

Question 25

Step 2: Writing R^2 and Interpreting it (In R) - (2) where R^2 is less than 60% (11.3%) Multiple linear regression

Answer 25

R^ = 0.113 and so 11.3% of the variation in Y var (name it) can be explained by our multiple linear regression model using X variable (e.g., using X2 and X4 var) Most of the variation remains unexplained

Question 26

Step 2: Writing R^2 and Interpreting it (In SPSS) Multiple linear regression

Answer 26

We see R^2 is 0.618 and so 61.8% of the variability in MRI count is explained by our multiple linear regression model

Question 27

Step 3 Rule: What P value to include or not? Multiple linear regression

Answer 27

p

Question 28

Step 3 Rule: How to interpret signifiance in R? (5)

Answer 28

* Anything with ' ' = significant at 100% (non-sig for mul linear reg) * Anything with . = significant at 10% (non-sig for mul linear reg) * Anything with one * = significant at 5% * Anything with two ** = significant at 1% * Anything with three *** = significant at 0.1%

Question 29

Step 3: Interpreting p-value of predictor and whether to include them (In R) - (3) Multiple linear regression

Answer 29

The coefficient for X2 is significant at 5% level ( p = 0.0397) whereas the coefficient for X4 is not significant (p = 0.123) Only X2 should be kept in model

Question 30

Step 3: Interpreting p-value of predictor and whether to include them (In SPSS) - (3)

Answer 30

The coefficients for Gender, PIQ and Height are all significant at the 5% level or greater, and so all can be kept in the model

Question 31

How to write B value?

Answer 31

Question 32

Step 4: Interpreting assumptions - histogram normally disturbed Multiple linear regression

Answer 32

Question 33

Step 4: Interpreting assumptions - histogram is not normally disturbed Multiple linear regression

Answer 33

Question 34

Step 4 - Interpreting assumptions - scatterplot random scattor Multiple linear regression

Answer 34

Question 35

Step 4 - Interpreting assumptions - scatterplot no random

Answer 35

Question 36

Step 5: Making a prediction and find residual for following squireel Time = 52.9382 + 21.6954 (Mass) -0.8899 (Length) + 2.9466 + 0.5157(Distance) - (5) Multiple linear regression

Answer 36

Input values into the equation Time = 52.9382+21.6954(1.1)+−0.8899(17)+2.9466(1.2)+0.5157(42.37) Time = 87.060969 (Fitted value) Residual = Observation - Fitted Value Residual = 78 (from table) - Fitted

Question 37

The Kolomorgorv and Smirnov test should be greater than 0.05 so

Answer 37

assumption of normality of errors are satisfied

Question 38

Written assumptions (2) Multiple linear regression that is satisfied

Answer 38

The histogram of residuals and normality tests ( p = 0.749 and p = 0.182) suggest that we have no evidence against the assumption of normal errors The scatterplot of predicted against residuals doesnt show any pattern suggesting the independence and constant variance assumptions on the errors are reasonable.

Question 39

What would your next steps in modelling confidence based on multiple regression analysis? (4) Grade and income covariates not significantly

Answer 39

Try to remove covariants from the regression In backgrounds elimination strategy we would remove the least significant covariants (income) and consider its effect on R^2 and signing ace of remaining covariates Following we could remove grade to see it’s effect on regression model Best regression model is one with high R^2 with fewest covariates