F10-Goodness-of-Fit-Remedies

# F10-Goodness-of-Fit-Remedies - PubH 7405 REGRESSION...

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PubH 7405: REGRESSION ANALYSIS SLR: GOODNESS-OF-FIT & REMEDIES

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ONGOING QUESTION Does the Regression Model fit the data? Then what if the Regression Model, or certain part of the Regression Model, does not fit the data ? i.e. (1) If it does not fit , could we do something to make it fit? And (2) Does it matter if it still does not fit?
In doing statistical analyses, a “ statistical model such as the “normal error regression model”- is absolutely necessary . However, a “model” is just an assumption or a set of assumptions ; they may or may not fit the observed data . Certain part or parts of a model may be violated and, as a consequence, the results may not be valid.

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POSSIBLE DEPARTURES FROM THE NORMAL REGRESSION MODEL The regression function is not linear Variance (of error terms) is not constant Model fits all but a few “ outliers Responses (error terms) are not independent Responses terms are not normally distributed An important predictor (independent variable) including time - has been omitted.
Diagnostics could be informal using plots/graphs or could be based on formal application of statistical tests ; graphical method is more popular and, most of the times, would be sufficient .

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In general, I’m not an enthusiastic supporter of tests of goodness-of-fit ”; We need to “ accept a model but statistical tests only allow us to reject or not to reject the Null Hypothesis under investigation, the model. We can only tell when a model does not fit the data – if we have enough information; we cannot formally tell when a model fits the data.
TESTS FOR NORMALITY Goodness-of-fit tests such as the Kolmogorov- Smirnov test – can be used for examining the normality of the error terms; but they are a bit advanced for first-year students. A more simple – but also formal – test for normality can be conducted by calculating the coefficient of correlation between the residuals and their expected values under normality . High value of the coefficient of correlation is indicative of normality. This is a supplement to Q-Q plot. “Critical value” for various sample sizes are in Appendix Table B6 (page 673).

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When the distribution (of the response) is only near normal, most of the dots (on the Q-Q plot” are already very close to a straight line; the “cut-point” for rejection is quite high . Again, as mentioned, a formal statistical test may not really be needed here; but could use to supplement the Q-Q plot – more valuable when sample size n is small .
LotSize WorkHours 80 399 30 121 50 221 90 376 70 361 60 224 120 546 80 352 100 353 50 157 40 160 70 252 90 389 20 113 110 435 100 420 30 212 50 268 90 377 110 421 30 273 90 468 40 244 80 342 70 323 EXAMPLE #3: Toluca Company Data (Description on page 19 of Text) Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 62.3658586 26.17743389 2.382428 0.025851 8.21371106 116.518006 X Variable 1 3.57020202 0.346972157 10.28959 4.45E-10 2.85243543 4.28796861 Residuals 25 .

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## This note was uploaded on 11/21/2011 for the course PUBH 7405 taught by Professor Staff during the Fall '08 term at Minnesota.

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F10-Goodness-of-Fit-Remedies - PubH 7405 REGRESSION...

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