Russ P. answered • 11/28/14
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Patient MIT Grad For Math and Science Tutoring
Judy,
Your description of the problem may be incomplete or somewhat unclear to me, but let me take a stab at it. It may stimulate something in your thought process to get the correct answer since you have more info than you've presented here (e.g., the data points).
Usually an inflection point occurs where your regression curve changes character, from concave to convex or the other way around.
Here your regression curve (or best fit to the data) is a straight line with a very high R2 of 0.994 implying that the line explains almost all the variation in the data. It's not clear to me whether this was before or after removing the two points?
If before (with all data present that was your R2), then there is no inflection point in a straight line since it doesn't change its character. For example, imagine a regression curve y = x3 which opens upward for positive x and downward for negative x. It clearly changes character and has an inflection point at x = 0.
If after, then maybe the 2 data points you removed represented a "new domain" that had a kink and a different regression line? I still would hesitate because 2 points determine a line with no variation around it so it is scant data to base a conclusion on.
My Comment Later:
Judy, This is my third try. This WyzAnt software is crap! Twice, I tried to add my comment to my original answer. I typed it all in, save commented it and it all got lost. Now I'll try a comment to your comment, and it will probably work. The software should either not have allowed the comment option on my answer that I entered earlier to type it in, or given an error message on the attempt to save comment so I could have clip boarded my text to place it here. Now I'll copy it before I conclude.
On to your question. You're basically correct. We can use the "kink" approach of two lines crossing each other at the inflection point. I get y = -1.019 x + 0.307 for the regression line based on your first 3 data points, and the line segment y = -0.06386 x - 3.4281 between data point 4 & 5.
The crossover point (when the y's match) is x = + 3.26 and y = - 3.02.
Also, if you look at the slopes of the successive line segments connecting adjacent points, they start out at -1.167, keep getting less negative and finally approach zero by data point 4 & 5. Thus, it looks like the drug is stable for x greater than the inflection point.
Judy A.
these are the data
X= 1.5,2.45,3.46,4.24,7.17
Y= -1.1737,-2.2823,-3.1739,-3.6989,-3.886
Actually , i need straight line to find out the intercept value and also need point of reflection . In this case , can i use the value between the last point of straight and the adjacent point.
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11/28/14
Russ P.
Judy,
Given your data below, I think you were originally correct, so you can determine the crossover point of the two lines in my "kink" above which occurs between x = 3 & x=4.
Your fitted regression line for the first 3 data points is y = mx + b or approximately y = -1.019 x + 0.307, using the formulas (where I removed the subscripts for clarity) and N=3:
m = {N Σ(xy) - (Σx)(Σy)}/{N Σ(x2) - (Σx)2} and b = {Σy - m Σx}/N
And the straight line for data points 4 & 5 i y = -0.06386 x - 3.4281 from m = Δy/Δx and b = yi - mxi i = 5.
So set the Y's of these 2 lines equal, solve for x at the crossover point, and then the y from either line and you get:
x = 3.26 & y = 3.02 for the kink point. where the slope changes as line 1 is crossed over to line 2.
NOTE: Five data points is rather sparse, and the precision of the data is questionable, so realistically that point occurs somewhere in the neighborhood of the calculated kink point. Also, by looking at how the slope (of the line segment between 2 successive data points) changes (-1.167, -0.883, -0.673, -0.064), it is pretty clear that it is flattening out approaching zero so the drug becomes stable with x. Hence, our "kink" approach is not a bad simplification and approximation.
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11/28/14
Russ P.
Judy,
I must have forgotten to save the long version of my comment, so here is the short version:
The "kink" approach provides a reasonable estimate of where the slope changes dramatically from the regression line based on the first 3 data points ( y = -1.019 x + 0.307) and the second line through data points 4 & 5
(y = -0.06386 x - 3.4281). Setting the y's equal, solving for x, then y in either line, we get x = 3.26 & y = -3.02 as the approximate "kink" inflection point. So the drug is stable as the slope approaches zero beyond that point.
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11/28/14
Judy A.
11/28/14