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【答案】应助回帖
感谢参与,应助指数 +1 09yschen(金币+10): 2012-01-08 20:51:17
见帖
http://en.wikipedia.org/wiki/Lack-of-fit_sum_of_squares
http://muchong.com/bbs/viewthread.php?tid=4020335&pid=7&page=1#pid7
LOFRTEST Lack-of-fit test for regression model with independent replicate values.
LOFRTEST(D,alpha) is a statistical test that gives information on the form
of the model under consideration. A significant lack-of-fit suggest that there
may be some systematic variation unaccounted for in the hypothesized model
(chosen model does not well describe the data). It arises when there are exact
replicate values of the independent variable in the model that provide an estimate
of pure error. Pure error is in essence the amount of error that cannot be accounted
for by any model. Then allows a test on whether there is error present aside
from pure error. For the construction of the lack-of-fit test we need to examine
three common types of linear models:
- single mean (one parameter)
- slope and intercept or common regression model (two parameters)
- separate means for each x-value or one-way ANOVA (many parameters).
So, the pure error is the error of the separate means on ANOVA and the total error
in the residual resulting in the regression analysis: the lack-of-fit results
to be the difference between this two sources of error,
SS(LOF) = SSR(Model) - SSE(ANOVA).
Syntax: lofrtest(D,alpha)
Inputs:
D - matrix data (=[X Y]) (last column must be the Y-dependent variable).
(X-independent variable entry can be for a simple [X], multiple [X1,X2,X3,...Xp]
or polynomial [X,X^2,X^3,...,X^p] regression model).
alpha - significance level (default = 0.05).
Outputs:
A complete summary (table) of analysis of variance partitioning sources of
variation for testing lack-of-fit.
Example from the data on height and weight of 19 students in Psy 202. Assigment 3 of Psych
3030 from the Department of Psychology of the York University. Available on Internet at
the URL address http://www.psych.yorku.ca/lab/psy3030/assign/assign3.htm
We are interested to test with a significance-value = 0.05 if there is a lack-of-fit on the
regression model due to the height replicate values.
------------------- -------------------
Height Weight Height Weight
------------------- -------------------
60 90 68 140
60 100 68 135
62 110 70 160
62 116 70 145
62 120 70 148
66 140 71 143
66 170 71 135
68 130 74 195
68 117 74 164
68 155
------------------- -------------------
Data matrix must be:
D=[60 90;60 100;62 110;62 116;62 120;66 140;66 170;68 130;68 117;68 155;68 140;68 135;
70 160;70 145;70 148;71 143;71 135;74 195;74 164];
Calling on Matlab the function:
lofrtest(D)
Answer is:
Lack-of-fit test for regression model with independent replicate values.
--------------------------------------------------------------------------
SOV SS df MS F P
--------------------------------------------------------------------------
Model 7658.359 1 7658.359 31.720 0.0000
Residual 4104.378 17 241.434
--------------------------------------------------------------------------
Lack-of-fit 2142.011 5 428.402 2.620 0.0797
Pure error 1962.367 12 163.531
--------------------------------------------------------------------------
Total 11762.737 18
--------------------------------------------------------------------------
If the associated P-value for any F test is equal or larger than 0.05
The corresponding null hypothesis is met. Otherwise it is not met.
Created by A. Trujillo-Ortiz, R. Hernandez-Walls, A. Castro-Perez and
F.J. Marquez-Rocha
Facultad de Ciencias Marinas
Universidad Autonoma de Baja California
Apdo. Postal 453
Ensenada, Baja California
Mexico.
atrujo@uabc.mx
Copyright (C) March 4, 2005.
To cite this file, this would be an appropriate format:
Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez and F.J Marquez-Rocha.
(2005). lofrtest:Lack-of-fit test for regression model with independent replicate
values. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/
fileexchange/loadFile.do?objectId=7074
References:
Department of Psychology of the York University. Available on Internet at the
URL address http://www.psych.yorku.ca/lab/psy3030/assign/assign3.htm
Zar, J. H. (1999), Biostatistical Analysis (2nd ed.).
NJ: Prentice-Hall, Englewood Cliffs. p. 345-350. |
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