Department

Statistics

Course Code

STAT1008

Professor

Bronwyn

Study Guide

Final

Multiple Regression

• Response variable: Y (quantitative)

• k predictors: X1, X2, … Xk (quantitative or 0/1 categories)

• Model: Y = β0 +β1X1 +β2X2 +···+βkXk +ε

• ε ~ N0,σε) and independent.

What to do?

• Estimate the coefficients: b0, b1,…, k (estimate the same way as a simple model).

• Test the individual predictors: t-tests (use ŷ?.

• Assess the overall fit: R2, ANOVA.

T-tests for Individual Predictors:

• Y = β0 +β1X1 +β2X2 +···+βkXk +ε

• Testing whether X2 be included in the model or not, eg: should exam 2 be in the model

given the presence of exam 1.

• i.e. Testing the effectiveness of any predictor, say Xi, in a multiple regression model.

• H0: βi = 0 vs Ha: βi 0

• Test statistic: t = bi/Sebi

• We find a p-value using a t-distribution with n - k - 1 df, where k is the number of

predictors.

• If we reject the null we see that the predictor is an effective contributor to this model.

• Individual t-tests assess the importance of a predictor after the other predictors are

already in the model, eg: if I add another predictor into the model, exam 1 may

eoe isigifiat.

Assessing Overall Fit: R2

• R2 = % of ariailit i Y hih is eplaied the odel.

• SSTotal = (as for one predictor)

• SSE = (as for one predictor)

• SSModel = SSTotal – SSE

• R2 = SSModel/SSTotal

• Adjusted R2 – usually lower as it contains an additional value.

• R2 is interpreted as the percent of variability in the response values in the sample that

is explained by the fitted regression model.

• Eg: If R2 =0.525, this means that 52.5% of the variability in Final exam scores is

explained by the model based on exam 1 and exam 2.

Assessing Overall Fit: ANOVA

• To test for the overall effectiveness of a regression model: Y = β0 + β1X1 +β2X2 +···+βkXk

+ε

• H0: βi = β2 =…= βk = 0 (the model is ineffective)

• Ha: At least oe βi 0 (at least one predictor in the model is effective)

• Mean square = SS/df

10.2 Checking Regression Conditions

Conditions for a Regression Model:

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