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I am attempting to build a model a for which the only valid output in the range [0,100]. I was wondering if would be possible to reduce the penalty on values under 0 and over 100 as they will be constrained anyway in order to get a better fit.

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    $\begingroup$ (1) Could you please explain what you mean by "input"? What is the form of your model and what aspect(s) of it constitute "input"? (2) In ordinary least squares regression there is no penalty on (dependent) values, per se, but only on the residuals. In what sense can "values" (presumably data) be "constrained"? They are, after all, your data! Are you proposing to change them or are you really talking about constraining the fit? Or maybe constraining any predicted values? All in all, it's hard to guess what you're asking. $\endgroup$
    – whuber
    Mar 25, 2013 at 19:59
  • $\begingroup$ Input in this case is a sparse vector of ~1500 binary variables. The output is a number between 0-100. Essentially I am attempting to "learn to rank" given a human generated ranking. But, I never mentioned the inputs above. Maybe you misread? $\endgroup$ Mar 25, 2013 at 20:04
  • $\begingroup$ Yes, I misread "output" as "input" (during multiple readings--a strange trick of the mind!). But although your problem is now clearer, many of my questions remain: it's hard to guess what you're doing when you have supplied such a broad and abstract description of your problem and it would be difficult (in any responsible manner) to suggest a solution without knowing more. $\endgroup$
    – whuber
    Mar 25, 2013 at 20:12
  • $\begingroup$ It's just standard least squares linear regression, nothing fancy. $\endgroup$ Mar 25, 2013 at 20:29
  • $\begingroup$ "Standard least squares regression" is a procedure, not a problem. If you restrict yourself to a procedure that happens to be inapplicable or inferior for your data, you may get bad answers (or the good answers will try to make the same points I'm making). That seems to be the case here. By describing your problem and allowing respondents to suggest solutions, you are more likely to get useful and correct answers. $\endgroup$
    – whuber
    Mar 26, 2013 at 1:23

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If I understand you correctly, this model can be fitted using logit ideas.

Scale your response by dividing by 100. Then use an appropriate generalised linear model for proportional or fractional responses. There is an accessible miniature review at http://www.stata-journal.com/sjpdf.html?articlenum=st0147 although the ideas are older than there implied. Bartlett was taking logits of proportions back in 1937.

This is easy in Stata and no doubt in any major statistical environment.

However, I think you are wrong: this can't, or shouldn't, be standard least squares regression as any hyperplane predicts values for the response outside a bounded range, even if the prediction is bounded for your data points.

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