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I would like to know what is the effect of dismissing data points in a regression analysis. For example, say originally a sample of 100 data points is collected (say they are all useful, equally reliable observations, no outliers) but one decides to fit a model to only 50 or 60 points from this sample.

Let us say also, that the dismissed points are not chosen randomly. I am looking at a case where the parameters of a normal distribution $\Phi$, $\sigma$ and $\mu$, are estimated using simple linear regression:

$$ X=\sigma \Phi^{-1}(p(X)) + \mu $$

The points that are systematically dismissed in this example are the points which have probabilities $p(X)$ of 0 or 1, because the standard normal distribution does not converge there.

From reading Green (1991), I understand that dismissing data points reduces the power of the analysis.

1/ Firstly, in the context of regression analysis I am not 100% sure what this means (increases the probability of type II erros? On what?).

2/ Any other consequences of dismissing data points apart from this?

Ref.

Green, S.B. (1991). How many subjects does it take to do a regression analysis? Multivariate Behavioral Research, 26(3), 499-510.

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    $\begingroup$ What you're discussing would normally be called 'deleting' or 'removing' rather than 'dismissing'. The effect depends on how the points are chosen. Choosing at random would lower power but might otherwise do little harm. Choosing in other manners can have far more serious effects. $\endgroup$ – Glen_b Jan 30 '14 at 8:14
  • $\begingroup$ Thanks Glen, the deletion is not random. I have edited my post $\endgroup$ – Neodyme Jan 31 '14 at 8:32
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Power is defined as the probability of rejecting the null hypothesis, assuming it is false. In other words: power $= P(\text{reject } H_0| H_0\text{ is false})$ (although it's not really a conditional probability since frequentists don't like to assign probabilities to events).

The answer to your second question is harder to answer, as it depends on how the points are removed (i.e. at random or not).

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  • $\begingroup$ Thank you... but what is H0 in the context of regression analysis? Say for example simple linear regression. $\endgroup$ – Neodyme Jan 30 '14 at 8:25
  • $\begingroup$ as for my second question actually the points that are removed are points with values greater than a certain threshold (so all of the "high" values) - not random. $\endgroup$ – Neodyme Jan 30 '14 at 8:26
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    $\begingroup$ The null-hypothesis is the `boring' hypothesis. Suppose I have a very simple regression model $y = ax + b$, then the null-hypothesis concerning $a$ is that it is zero (whereas the alternative hypothesis is that it's greeater, less than or just not-equal to zero). For a primer on null-hypothesis-significance-testing, see: stattrek.com/hypothesis-test/hypothesis-testing.aspx $\endgroup$ – Stijn Jan 30 '14 at 8:29
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    $\begingroup$ @IngridCharvet If as your comment above says the points removed were those with values greater than a threshold, this is a key point. I suggest that you edit your question to make this clear and, if possible, indicate why it was decided to remove those points. This may help you to obtain answers addressing the particular case you are interested in. $\endgroup$ – Adam Bailey Jan 30 '14 at 9:11
  • $\begingroup$ Thank you Adam - I have edited my original post to help answer the second point. $\endgroup$ – Neodyme Jan 31 '14 at 8:30

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