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regression

Course: PHARM 508, Fall 2009
School: Washington
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Word Count: 636

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Model Regression Definition: A statistical model is a mathematical abstraction that is a representation of events in the real world. Many models are written as an equation that defines a value you want to predict (y) from one or more variables (x) that you know. Purposes of Regression Looking for a trend Adjusting for a confounding variable Curve fitting Prediction Standard curve estimation (in assay...

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Model Regression Definition: A statistical model is a mathematical abstraction that is a representation of events in the real world. Many models are written as an equation that defines a value you want to predict (y) from one or more variables (x) that you know. Purposes of Regression Looking for a trend Adjusting for a confounding variable Curve fitting Prediction Standard curve estimation (in assay analysis) Types of Regression Simple linear regression Multiple linear regression Logistic regression Proportional hazards regression Nonlinear regression Poisson regression Scatter Plot for Example 17.1 500 r = 0.74 100 18 200 Insulin Sensitivity Index (mg/m2/min) 300 400 19 20 21 %C20-22 Fatty Acids 22 23 24 Example 17.1 Scatter Plot 500 100 18 200 Insulin Sensitivity Index (mg/m2/min) 300 400 19 20 21 %C20-22 Fatty Acids 22 23 24 Simple Linear Regression The regression line can be used to estimate the dependent variable from the independent one; the difference between the actual value and the estimated value is a residual. Among all lines, the regression line makes the smallest mean squared error in estimating the dependent variable from the independent one. The slope of the regression line is the average rate at which y changes with x. It is the average change in y, per unit change in x. The equation for the regression line can be used to make all regression estimates, by substituting given values of the independent variable for x. Simple Linear Regression Model There is a linear relationship between %C20-22 fatty acid content and the insulin sensitivity index x = %C20-22 fatty acid content y = insulin sensitivity index y = + x Simple Linear Regression Model Given N observations on the explanatory variable x and the response variable y, (x1, y1), (x2, y2), . . . . . (xN, yN) the statistical model for simple linear regression states that the observed response yi when the explanatory variable takes the value xi is yi = + xi + i Here + xi is the mean response when x = xi. The deviations i are assumed to be independent and normally distributed with mean 0 and standard deviation . The parameters of the model are , , and . Calculating Linear Regression and Slope Intercept Slope = b = ( x - x )( y - y ) (x - x) i i 2 i Y - intercept = a = y - bx Calculating Linear Regression SE of slope: sb = where (x - x) i i se 2 se = = (y - predicted yi ) 2 N -2 N -2 (1 - r 2 ) ( yi - y ) 2 Calculating Linear Regression Example 17.1: Summary statistics: ( x - x )( y - y ) = 2480.154 ( x - x ) = 66.65692 ( y - y ) i i 2 i 2 i = 155642.3 x = 20.71538 , y = 284.2308 se = ( y - predicted y ) i i 2 13 - 2 = 75.895 Calculating Linear Regression Example 17.1: results from computer Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) -486.5420 193.7160 -2.5116 0.0289 fatty.acids 37.2077 9.2959 4.0026 0.0021 Residual standard error: 75.9 on 11 degrees of freedom Multiple R-squared: 0.5929 Calculating Linear Regression: Prediction Least squares line: y = -486.5 + 37.2 x Predict y when x = 20 y = -486.5 + 37.2(20) = 257.5 Predict y when x = 10 (extrapolation) y = -486.5 + 37.2(10) = -114.5 Calculating Linear Regression 95% CI of the slope: slope t*sb to slope + t*sb Example 17.1: 37.208 2.201 or (16.74, 57.67) 75.895 66.65692 Calculating Linear Regression Obtaining a P-value for the slope H0 : = 0 t= observed - expected under H 0 b - 0 = with N - 2 df yardstick sb t= 37.2077 = 4.0026 9.2959 Example 17.1: P-value = 0.0021 ...

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