F05-Parameters

F05-Parameters - PubH 7405: REGRESSION ANALYSIS SLR:...

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PubH 7405: REGRESSION ANALYSIS SLR: PARAMETER ESTIMATION
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REGRESSION MODEL Model : Y = 0 + 1 x + where 0 and 1 are two new parameters called regression coefficients , the Intercept and the Slope , respectively. The last term, , is the “error” representing the random fluctuation of y-values around their mean, 0 + 1 x , when X=x. The presence of the error term is an important characteristic of a statistical relationship; the points on a scatter diagram do not fall perfectly on the line. The scatter diagram is an useful diagnostic tool for checking out the Model ( e.g. to see if it is linear ).
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2 2 1 0 σ Var( ε and 0 E( ε to weakened be sometimes can assumption normal" " The ) σ N(0, ε ε x β β Y ) ) The Normal Error Regression Model
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REGRESSION COEFFICIENTS The error term - with variance σ 2 -would tell how spread the dots are around the regression line. The regression coefficients, 0 and 1 , determine the position of the line and are important quantities in the analysis process. In “correlation analysis”, we need to know only the coefficient of correlation r which is proportional to the slop 1 (we’ll see); but in a “regression analysis”, with new emphasis on prediction , so we need them both, 0 and 1 . As parameters , both 0 and 1 are unknown; but they can be “estimated” by statistics from data
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too! important very & role specific a has but hidden, is It : parameter third the is line) regression the (around σ Variance The 2 ) σ N(0, ε ε x β β Y 2 1 0
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THE INTERCEPT If the scope of the model include X = 0, 0 gives the Mean of Y when X = 0 ; otherwise, it does not have any particular meaning as a separate term. If the scope of the model does not include X = 0, we may choose a “ transformation such as: (New) x = x - x Under this transformation, 0 gives the Mean of Y when X = x , i.e. a “ typical subject (value = x)
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) | ( ) 0 | ( ) 0 | ( ) | ( ) 0 | ( ) | ( _ * * 0 * * * 1 * 0 * * _ * 0 1 0 x X Y E X Y E X Y E x x X Y E X X X X Y E x x X Y E  : Model d Transforme : Model Original
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THE SLOPE The Slope is a more important parameter: (i) If X is binary (=0/1) representing an exposure, 1 represents the increase in the mean of Y associated with the exposure (or a decrease if 1 is negative); (ii) If X is on a continuous scale, 1 represents the increase in the mean of Y associated with one unit increase in the value of X, X=x+1 vs. X=x , (or a decrease if 1 is negative).
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1 β 0) X | E(Y - 1) X | E(Y : X Variable t Independen Binary 1 0 0 1 0 1) X | E(Y 0) X | E(Y x) X | E(Y  x The change in the mean of Y associated with the exposure
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