Question

are the assumptions behind any multiple regression model? (b). For a multiple regression model Y-Bo + βιΧ. + β2X2 +β3Xs + € where is the error term, to represent the relationship between Y and the four X- variables. We got the following results from the data: Source Sum of Squares degrees of freedom mean squares Regression 1009.92 Residual Total 2204.94 34 And also you are given: Variable X1 Σ.tx-xr 123.74 72.98 12.207 -Pr values -11.02 5.13 X2 X3 Y-intercept is Bo - 2.96 (and is tested to be statistcally significant) Write down the regression estimate for this multiple regression a. b. Would you modify the model (i.e. would you want to include all the X variables)? Explain why or why not Compute the p-value c.

Answer 1

1)

i)

Assumptions

Linear relationship: The model is a roughly linear one. This is slightly different from simple linear regression as we have multiple explanatory variables. This time we want the outcome variable to have a roughly linear relationship with each of the explanatory variables, taking into account the other explanatory variables in the model.

Homoscedasticity: Ahhh, homoscedasticity - that word again (just rolls off the tongue doesn't it)! As for simple linear regression, this means that the variance of the residuals should be the same at each level of the explanatory variable/s. This can be tested for each separate explanatory variable, though it is more common just to check that the variance of the residuals is constant at all levels of the predicted outcome from the full model (i.e. the model including all the explanatory variables).

Independent errors: This means that residuals should be uncorrelated.

As with simple regression, the assumptions are the most important issues to consider but there are also other potential problems you should look out for:

Outliers/influential cases: As with simple linear regression, it is important to look out for cases which may have a disproportionate influence over your regression model.

Variance in all predictors: It is important that your explanatory variables... well, vary! Explanatory variables may be continuous, ordinal or nominal but each must have at least a small range of values even if there are only two categorical possibilities.

Multicollinearity: Multicollinearity exists when two or more of the explanatory variables are highly correlated. This is a problem as it can be hard to disentangle which of them best explains any shared variance with the outcome. It also suggests that the two variables may actually represent the same underlying factor.

Normally distributed residuals: The residuals should be normally distributed.

ii)

Source	SS	df	MS	F	p-value
regression	1009.92	3	336.64	8.732774	0.00024
Residual	1195.02	31	38.54903
Total	2204.94	34

Formulas in Excel

Source	SS	df	MS	F	p-value
regression	1009.92	3	=B2/C2	=D2/D3	=F.DIST.RT(E2,3,31)
Residual	=B4-B2	=C4-C2	=B3/C3
Total	2204.94	34

MS = SS/df

F = MS regression / MS Error

a)

y^ = 2.96 -11.02 x1 + 5.13 x2 -1.15 x3