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I'm analyzing the relationship between a (log-transformed) continuous independent variable and a continuous dependent variable that has a lower and upper bound. If I scale the dependent variable to values between 0 and 1, then take the logit, the relationship becomes linear, with seemingly homogenous variance.

Is it appropriate to then use ordinary least squares regression? Can I judge the model fit by R-square, or can I use a goodness-of-fit test based on deviance? Is it meaningful to look at deviance residuals to judge individual data points?

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    $\begingroup$ I would have thought beta regression would be slightly more appealing than logistic. Perhaps you need to explain a little bit more about the nature of your variables by editing into your question? $\endgroup$
    – mdewey
    Commented Nov 14, 2016 at 18:35
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    $\begingroup$ This concerns a typical bioanalytical experiment: a range of known concentrations of a chemical (independent variable) are determined by a certain analytical method, in this case colorimetric (the dependent variable: absorption of light). The objective is to derive a calibration curve, and use this to infer concentrations in measured experimental samples. The colorimetric method is bound between a background absorption as concentrations approach 0, and a maximum absorption at ever increasing concentrations. - simonderijck $\endgroup$
    – simonderijck
    Commented Nov 14, 2016 at 20:15
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    $\begingroup$ Use fractional logit. Beta regression doesn’t allow zeros or ones in the dependent variable. $\endgroup$
    – The Laconic
    Commented Aug 31, 2018 at 20:17

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I was facing similar problems with probability of loss as my dependent variable (bounded to 0% and 100%), and I was about to use logit as smoothing function (to be unbounded) to then using OLS in estimating my independent macroeconomics parameters.

First, you have to ensure that the plot of transformed dependent variable is quite linearly scattered. Second, you need to prove that the error of response is normally distributed (otherwise the OLS estimator is suboptimal). Third, if your variance of error is heteroscedastic then you need some weighting technique to keep your OLS estimator is BLUE.

You will need to use another smoothing function if the first does not held. You will need to use maximum likelihood estimator if the second and the third are not held.

If I were you, I would take R-squared as goodness indicator as I am using OLS rather than MLE. And instead of deviance residual, maybe you could try to see Cook’s D in this manner.

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    $\begingroup$ Logit is for categorical variable? Not necessarily so. What we now call logit was used to transform continuous proportions over several decades long before it was introduced as (in more recent terms) a link function for binary responses. The main thing to worry about with log$[p/(1-p)]$ is that it is not defined for $p$ of 0 or 1. $\endgroup$
    – Nick Cox
    Commented Jan 9, 2018 at 14:26
  • $\begingroup$ @NickCox in my opinion, logit transformation works better to linearise binary responses, which change rapidly around its cut off (S-shaped plot). But yes, it doesn’t mean this transformation belongs to categorical variable only. How do you think? $\endgroup$
    – Yfendra
    Commented Jan 9, 2018 at 15:22

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