OLS versus MLE Example

1
OLS versus MLE Example
Here is the data
Y X
0.8 1
2.1 2
3.94 3
4.6 4
5.1 5
6.5 6
6.9 7
8.2 8
9.3 9
10.1 10
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OLS versus MLE Example
Here is the SAS code
OLS fitting using PROC GLM
proc glm datadata model y x run
Call the procedure for data set data Regression
model y b0 b1x
MLE fitting using PROC GENMOD
Call the procedure for data set data Model y
b0 b1x Assume normally distributed errors Use
an identity link (the link function
describes the relationship between y and the
linear portion of the model)
proc genmod datadata model y x / distnormal
linkidentity run
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OLS versus MLE Example
Output from PROC GLM
Output from PROC GENMOD
Source DF SS Mean
Sq Model 1 81.2051
81.2051 Error 8 1.0533
0.1316 Corrected Total 9 82.2584 R-Square
Root MSE 0.987195 0.362857 Parameter
Estimate SE Intercept 0.297333
0.2478 X 0.992121 0.0399
Criterion DF Value
Value/DF Deviance 8 1.0533
0.1317 Scaled Deviance 8 10.0000
1.2500 Log Likelihood
-2.9362 Parameter DF Estimate
SE Intercept 1 0.2973 0.2217
X 1 0.9921 0.0357
Scale 1 0.3245 0.0726
MLE s2 SS/n 0.10533
Scale f (s2)2
Note that SE estimates differ MLE variance
estimates are biased at low sample sizes
Ln(L) -n/2ln(2pes2)
-5ln(17.079470.10533) -2.9361
K 3 (intercept, x, s)
K 3 (intercept, x, scale)