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Can you see any S-curve in the below scatter plot?

If there is the S-curve is it correct to use a scatterplot plot? If the linear correlation is 0.85, does it mean the S-curve is unlikely?

I am trying to understand the relationship between the two variables.

enter image description here

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    $\begingroup$ 1) Yes I see an S-curve, on the right picture. Anyway, strange questions... $\endgroup$
    – ttnphns
    Commented Apr 9, 2013 at 14:04
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    $\begingroup$ 1) Only the green one. If I just see the left one, I'd guess linear first. 2) S-shape usually implies your Y goes up abruptly after a certain level of X, but then dies down (can be due to saturation or out of resources). Phenomenon like enzymatic reactions and adoption of new technology are often of S-shape. 3) Not sure what is asked. 4) See Anscombe's Quartet. Use graph and correlation in conjunction. 5) Get more data, start with a research question, have a reasonable mechanism behind, use lowess curve, and non-linear regression models. $\endgroup$ Commented Apr 9, 2013 at 14:24
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    $\begingroup$ What is this, a survey? $\endgroup$
    – grssnbchr
    Commented Apr 9, 2013 at 14:54
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    $\begingroup$ I see no suggestion of an S-curve in the left hand plot. If I saw that alone, my first instinct would have been 'close to linear, at least over the range we can see' (though the small sample size means it could really be almost anything). However, if the variable on the Y-axis is necessarily bounded above and below (if it's a percentage, say), we could well argue that the appearance linearity can't continue and it should start to increase and decrease more slowly as we go nearer the boundaries. $\endgroup$
    – Glen_b
    Commented Apr 9, 2013 at 23:09

2 Answers 2

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1) I see no suggestion of an S-curve in the left hand plot. If I saw that alone, my first instinct would have been 'close to linear, at least over the range we can see' (though the small sample size means it could really be almost anything).

However, if the variable on the Y-axis is necessarily bounded above and below (if it's a percentage, say), we could well argue that the appearance linearity can't continue and it should start to increase and decrease more slowly as we go nearer the boundaries.

2) You're going to have to narrow the scope of this down a little; there's all kinds of things that could be said here, depending on what you're after.

3) I don't see how having an S-shape would prevent doing a scatter plot - people plot curved relationships all the time. However, if there are natural boundaries to the y-axis variable, plotting with a transformed Y or X may be more informative.

4) The correlation isn't meaningless, but ordinary Pearson correlation measures the strength of linear relationship. If the actual relationship were monotonic instead you may prefer a measure that captured the strength of that monotonicity; perhaps a Spearman or Kendall correlation.

5) The obvious thing to do is to fit a linear relationship and examine residuals, perhaps with a superimposed loess curve on a plot of residuals vs fitted values - if there's really an "S" curve that plot should clearly look like an S tipped on its side.

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wanted to verify our analysis and saying even if there was a S-curve, linear regression cannot be incorrect

Well of course fitting a linear model to a nonlinear relationship, if there is one, is 'not correct', in a quite direct sense. However it would be very rare indeed for a relationship to be exactly anything in particular, and especially the case with linear relationships.

Over the range looked at, fitting a linear relationship - even though it's wrong - might nevertheless still be useful, if the range of values over which you want to make any conclusions/predictions remains within the range of the data.

If on the other hand, you have external knowledge about bounds on the variable, you should use it.

For data like the data shown it won't make much difference which you do in terms of fitted values. The two will look very similar. When it comes to prediction outside the range of the data, of course things will look very different.

You could always humor the other person and do what they suggest and see how things compare.

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  • $\begingroup$ If you've missed an S-curve, it should be obvious in the display I suggest in my answer to (5). Is there any argument besides the appearance of the plot for the existence of an S-curve*? *(e.g. see the second half of my answer to (1)) $\endgroup$
    – Glen_b
    Commented Apr 9, 2013 at 23:41
  • $\begingroup$ "wanted to verify our analysis and saying even if there was a S-curve, linear regression cannot be incorrect" -- uh, why didn't you put that anywhere in your five questions? I will address it above. $\endgroup$
    – Glen_b
    Commented Apr 9, 2013 at 23:57
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    $\begingroup$ @Glen_b, your struggles here are admirable, but I suspect this isn't worth any more of your time. I voted to close as not a real question. $\endgroup$ Commented Apr 10, 2013 at 2:37
  • $\begingroup$ Your words were " the given example is very well-drawn to demonstrate our concept even it is not a real scatter". Two things in that very clearly suggest the data was fake ('well-drawn' and 'to demonstrate our concept'). Apparently I misunderstood, I apologize. I assume this is just an English-not-as-a-first-language issue leading to the text conveying an incorrect impression. I will edit. $\endgroup$
    – Glen_b
    Commented Apr 10, 2013 at 9:35
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To piggy back on the comment by @Penguin_Knight, and to perhaps interpret the OP's antagonist's (for lack of a better term -- which could be "adviser") comments, the antagonist may have a theoretical reason to believe the underlying process is, for example, a growth process, or some other process that ought to produce an S. In that case, and scientifically speaking, it is reasonable to claim a linear model is incorrect from the get-go. But if you don't have any idea what it is, a scatter plot (or some other flexible visualization) is, without a doubt, the best way to start exploring. Though, even a loess line could steer you wrong with this little data and if you choose too sensitive of parameters.

I'll also say there is definitely not enough data to tell if it should be an S. The lower-right hump of the potential S is made possible by a single point in the data and that could easily be error in a linear relationship.

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