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I am currently trying to build a model using a data set that has large gap between data points. When I look for the correlation I clearly see a negative regression line. But I am worried about the gap that exist between the poins.

I build a simple linear model though this has high R squared I don't think simple linear regression is the best model to that fits the data. This looks like it has a negative exponential behavior. I thought to post here to get some expert thoughts on what I should do when you deal with the data that has a large gap between points and does this data has a linear relationship or strong non linear relationship?

Data Set:

   density  co2
1     20.4 38.8
2     27.4 31.5
3    106.2 10.6
4     80.4 16.1
5    141.3  7.7
6    130.9  8.3
7    121.7  8.5
8    106.5 11.1
9    130.5  8.6
10   101.1 11.1
11   123.9  9.8
12   144.2  7.8
13    29.5 31.8
14    30.8 31.6
15    26.5 34.0
16    35.7 28.9
17    30.0 28.8
18   106.2 10.5
19    97.0 12.3
20    90.1 13.2
21   106.7 11.4
22    99.3 11.2
23   107.2 10.3
24   109.1 11.4

Plot: Simple Linear Regression Plot

Summary of Linear Model:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 38.12948    1.21768   31.31  < 2e-16 ***
density     -0.24247    0.01261  -19.22 3.04e-15 ***

In addition if I transfer both density and co2 as log transform variables, then I see following behavior. Since data is missing at the middle its really hard to stick to a log transformed model or the base model.

enter image description here

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  • $\begingroup$ Draw two regression lines, one for each set of data. $\endgroup$ – Peter Flom - Reinstate Monica May 12 '13 at 17:41
  • $\begingroup$ Did you mean one with the log and one without? $\endgroup$ – add-semi-colons May 12 '13 at 17:51
  • $\begingroup$ I didn't see your log graph when I posted my comment. I meant divide the data set in two where the gap is; but if taking the log of both variables makes sense, that works too. $\endgroup$ – Peter Flom - Reinstate Monica May 12 '13 at 18:00
  • $\begingroup$ Any technique to divide data into two sets? $\endgroup$ – add-semi-colons May 12 '13 at 18:07
  • $\begingroup$ Just split them at the big gap. $\endgroup$ – Peter Flom - Reinstate Monica May 12 '13 at 19:00
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Presumably co2 means "carbon dioxide" and density means what it says. Even so, it would help to have more detail on what is happening here. Is there no physics or chemistry or engineering background to help us, or you, or everyone?

Why is there a gap? Is there no hint from the background to the data?

Are these the results of an experiment in which one variable is controlled, or something else? Which variable do you want to predict and/or regard as the response or outcome (dependent variable, if you will)? You appear to be regarding co2 as the outcome. Is that prescribed by the problem?

Some rough experiments indicate that logging just one variable might make sense too. Linear is a lousy model because if you extrapolate you soon produce negative predictions for one or other variable, which is surely unphysical.

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  • $\begingroup$ It is results from an experiment but since I am not the one who did the experiment I don't have much information on that. What I am interest is statistically describing the behavior of the data based on whats given to me. Yes. co2 is the reponse and I want to model how co2 depends on density. One tiny thing. co2 is actually co2 rate per minute $\endgroup$ – add-semi-colons May 12 '13 at 19:36
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    $\begingroup$ co2 ~ density as a power function (log co2 linear in log density, as in your second graph) and as exponential decline (log co2 linear in density) both seem fairly reasonable, with a little more curvature away from the second. The choice between them should depend on the underlying science, especially limiting behaviour of the physical relationship, as well as goodness of fit. $\endgroup$ – Nick Cox May 13 '13 at 6:24

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