regression analysis regression analysis is a statistical technique that is very useful for exploring...
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Regression Analysis
Regression analysis is a statistical technique that is very useful for exploring the relationships between two or more
variables (one independent variable and the one dependent variable)
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Simple Linear Regression Model
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Probabilistic Linear Regression Model
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The Least Square Method
LSM is based on the concept of minimizing L
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The Least Square Method
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Example 11.1
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See the Excel Solution
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Estimation of Variance
Where SSE = Error sum of squares
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Solution 11.1
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Problem 11.11
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Problem 11.11
Solve using Excel
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Standard Error of the Estimates
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HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
• Objective:– Assessing the adequacy of a linear regression
model by testing statistical hypotheses about the model parameters and constructing certain confidence intervals.
• Assumption:– the errors are normally and independently
distributed with mean zero and variance σ2
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HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
• Suppose we wish to test the hypothesis that the slope equals a constant
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HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
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A very important special case of the hypotheses about the slope:
Rejecting H0: Either that the straight-line model is adequate or that, although there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x
Either x is of little value in explaining the variation in Y and that the best estimator of Y for any x is Y or that the true relationship between x and Y is not linear
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Example11.2
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Analysis of Variance Approach to Test Significance of Regression
The error sum of squares
The total corrected sum of squares
The regression sum of squares
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Analysis of Variance Approach to Test Significance of Regression
The above test statistic:
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Example 11.3
See the Excel solution
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Confidence Intervals on the Slope and Intercept
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Confidence Intervals on the Slope and Intercept
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Confidence Interval on the Mean Response
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Example 11.5
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Example 11.5
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Residual Analysis• Analysis of the residuals is frequently helpful in
checking the assumption that the errors are approximately normally distributed with constant variance
• As an approximate check of normality, the experimenter can construct a frequency histogram of the residuals or a normal probability plot of residuals.
• The analysis can also be done by ploting the residuals against the independent variable x.
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Residual Analysis
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Coefficient of Determination(R2)
• Coefficient of determination is used to judge the adequacy of a regression model.
• R2 is the square of the correlation coefficient between X and Y.