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STAT1008 Study Guide - Final Guide: Dependent And Independent Variables, Test Statistic, Confounding

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Multiple Regression
Response variable: Y (quantitative)
k predictors: X1, X2, … Xk (quantitative or 0/1 categories)
Model: Y = β0 1X1 2X2 +···kXk
ε ~ N0,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 1X1 +β2X2 +···+βkXk
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
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
eoe isigifiat.
Assessing Overall Fit: R2
R2 = % of ariailit i Y hih is eplaied  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 2X2 +···+βkXk
H0: βi = β2 =…= βk = 0 (the model is ineffective)
Ha: At least oe 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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