回归分析第四章

Assumptions about the Predictors
(3) The predictor variables X1, X2,…, Xp are assumed to be linearly independent of each other. If the assumption does not hold: The collinearity problem
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n 2 ( e e ) i i 1 n 2 e i i 1
ˆ
n p 1

n p 1

SSE n p 1
Checking Normality Assumptions
Normal probability plot of the standardized residuals If the residuals are normally distribution, the ordered residuals should be approximately the same as the ordered normal scores, which are what we would expect to obtain if we take a sample of size n from a standard normal distribution.
Normal probability plot of the standardized residuals
This plot should resemble a (nearly) straight line with an intercept of zero (i.e., the mean) and a slope of one (i.e., the standard deviation)
Assumptions about the Errors
The errors ε1, ε2,…, εn are assumed to be independently and identically distributed (iid) normal random variables each with mean zero and a common variance σ2. This implies four assumptions. (3) Constant variance (or homogeneity or homoscedasticity) assumption: The errors ε1, ε2,…, εn have the same (but unknown) variance σ2.
Regression Diagnostics
Outline
• • • • Standard Regression Assumptions Standardized Residuals Checking Linearity Assumptions Checking Normality Assumptions
Assumptions about the observations
All observations are 1) equally reliable and 2) have an approximately equal role in determining the regression results and in influencing conclusions.
A pattern indicating nonlinearity
A pattern indicating heterogeneity
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Standardized Residuals
The standardized residual is
ei e ei 0 ei ei ˆ ˆ ˆ
Assumptions about the Form of the Model
Linearity assumption:
Y 0 1 X 1 2 X 2 ... p X p yi 0 1 xi1 2 xi 2 ... p xip i , i 1, 2,..., n
This plot should resemble a (nearly) straight line with an intercept of zero (i.e., the mean) and a slope of one (i.e., the standard deviation)
Checking Normality Assumptions
Checking Linearity Assumptions
Residual plot: scatter plots of the standardized residuals against each of the predictor variables Under the standard assumptions, the standardized residuals are uncorrelated with each of the predictor variables. This plot should be a random scatter of points.
Checking the assumption: For simple regression For multiple regression If the assumption does not hold: Scatter plot of Y versus X Residual Plot Data transformation
Assumptions about the Errors
The errors ε1, ε2,…, εn are assumed to be independently and identically distributed (iid) normal random variables each with mean zero and a common variance σ2. This implies four assumptions. (1) Normality assumption: The error εi, i = 1, 2,…, n, has a normal distribution. Checking the assumption: Normal probability plot (2) The errors ε1, ε2,…, εn have mean zero.
If the assumption does not hold: The heterogeneity or heteroscedasticity problem
Assumptions about the Errors
The errors ε1, ε2,…, εn are assumed to be independently and identically distributed (iid) normal random variables each with mean zero and a common variance σ2. This implies four assumptions. (4) Independent-errors assumption: The errors ε1, ε2,…, εn are independent of each other (their pair-wise covariances are zero). If the assumption does not hold: The auto-correlation problem
Assumptions about the Predictors
(1) The predictor variables X1, X2,…, Xp are nonrandom, that is, the values x1j, x2j, …, xnj; j = 1, 2, …, p are assumed fixed or selected in advance. Cannot be validated (2) The values x1j, x2j, …, xnj; j = 1, 2, …, p are measured without error. Cannot be validated