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
...