HD Statistik

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HD Statistik Session 9 - 2

• Multipel regressionsanalyse
• Bearbejdet af Steen Lund-Thomsen, IFI
• 16. marts 2001

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• The k-Variable Multiple Regression Model
• The F Test of a Multiple Regression Model
• How Good is the Regression
• Tests of the Significance of Individual
  Regression Parameters
• Testing the Validity of the Regression Model
• Using the Multiple Regression Model for Prediction

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11-2 The k-Variable Multiple Regression Model
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Simple and Multiple Least-Squares Regression
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The Estimated Regression Relationship
The estimated regression relationship where
is the predicted value of Y, the value lying on
the estimated regression surface. The terms
b0,...,k are the least-squares estimates of the
population regression parameters ?i.
The actual, observed value of Y is the predicted
value plus an error yb0 b1 x1 b2 x2. . .
bk xke
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Least-Squares Estimation The 2-Variable Normal
Equations
Minimizing the sum of squared errors with respect
to the estimated coefficients b0, b1, and b2
yields the following normal equations
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Sommerhusdata Analyse baseret på 3 forklarende
Excel Output
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Decomposition of the Total Deviation in a
Multiple Regression Model
Total Deviation Regression Deviation Error
Deviation SST SSR
SSE
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11-3 The F Test of a Multiple Regression Model
A statistical test for the existence of a linear
relationship between Y and any or all of the
independent variables X1, x2, ..., Xk H0 ?1
?2 ... ?k0 H1 Not all the ?i
(i1,2,...,k) are 0
MSR/MSE
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Using the Computer Analysis of Variance Table
(Sommerhusdata 3 forklarende variable)
The test statistic, F 28,98, is greater than
the critical point of F3,132 for any common level
of significance (p-value ?0), so the null
hypothesis is rejected, and we might conclude
that the dependent variable is related to one or
more of the independent variables.
F-fordeling med 3 og 132 frihedsgrader
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11-4 How Good is the Regression
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Decomposition of the Sum of Squares and the
Adjusted Coefficient of Determination
SST
SSE
SSR
Sommerhusdata s 610,79 R2 40
R2(adj) 38
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Measures of Performance in Multiple Regression
and the ANOVA Table
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11-5 Tests of the Significance of Individual
Regression Parameters
Hypothesis tests about individual regression
slope parameters (1) H0 b10 H1
b1?0 (2) H0 b20 H1 b2?0 .
. . (k) H0 bk0 H1 bk?0
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Regression Results for Individual Parameters



) Alle tre forklarende variable har signifikant
indflydelse på udlejningsprisen pr. uge,
idet ?10, ?20 og ?30 alle for- kastes med
stor sikkerhed, dvs. alle 3 skal indgå i modellen.
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Sommerhusdata 6 forklarende variable
X1 Antal kvadratmeter X2 Antal personer X3
Moderniseringsår X4 Antal soverum X5 Afstand
til kyst X6 Afstand til indkøbsmulighed
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Sommerhusdata 5 forklarende variable
X1 Antal kvadratmeter X2 Antal personer X3
Moderniseringsår X4 Antal soverum X6 Afstand
til indkøbsmulighed
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Sommerhusdata 4 forklarende variable
X1 Antal kvadratmeter X2 Antal personer X3
Moderniseringsår X6 Afstand til indkøbsmulighed
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11-6 Investigating the Validity of the
Regression Model Residual Plots
Der er varianshomogenitet og ingen særlige
mønstre i forløbet, så regressionsmodellen er
brugbar!
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Histogram of Standardized Residuals Sommerhusdata