Thank you for your reply, Rick. I want to check how many years before death the decline in SBP became faster. I guess the derivative test you mentioned could be helpful. I would perform the test. Thank you again for your patience and help. ... View more

Hi Rick,   There are some different situations.  For example, the below graph. Each time point is with a smaller slope compared with the previous one.   Thanks for your help in advance.        Obs back_timescale Predicted Lower Upper slope SlopeChanged 1 -15 135.06 130.82 139.31 . . 2 -14 135.52 132.57 138.47 0.46 . 3 -13 135.94 133.72 138.16 0.42 -0.03 4 -12 136.30 134.38 138.23 0.37 -0.06 5 -11 136.59 134.78 138.41 0.29 -0.08 6 -10 136.79 135.06 138.51 0.19 -0.10 7 -9 136.87 135.27 138.46 0.08 -0.12 8 -8 136.81 135.36 138.26 -0.06 -0.14 9 -7 136.59 135.25 137.93 -0.21 -0.16 10 -6 136.20 134.90 137.51 -0.39 -0.18 11 -5 135.61 134.27 136.95 -0.59 -0.20 12 -4 134.81 133.42 136.19 -0.81 -0.22 13 -3 133.76 132.37 135.16 -1.04 -0.24 14 -2 132.46 131.09 133.83 -1.30 -0.26 15 -1 130.88 129.40 132.36 -1.58 -0.28 16 0 129.01 126.97 131.04 -1.88 -0.30 ... View more

Hi Rick,   The method you mention could deal with the graph with the changing sign of slopes, eg. from positive to negative. However, some graphs in my example are without the changing direction of slopes. I am wondering how to detect the changing point in this situation.   Below is my dataset. X= back_timescale, y = predicted.   Obs back_timescale Predicted Lower Upper slope SlopeChanged 1 -15 29.40 27.60 31.19 . . 2 -14 28.86 27.49 30.22 -0.54 . 3 -13 28.44 27.37 29.52 -0.42 0.12 4 -12 28.13 27.25 29.02 -0.31 0.11 5 -11 27.91 27.13 28.69 -0.22 0.09 6 -10 27.76 27.06 28.47 -0.15 0.07 7 -9 27.67 27.02 28.31 -0.10 0.05 8 -8 27.60 27.01 28.20 -0.06 0.04 9 -7 27.56 27.01 28.11 -0.05 0.02 10 -6 27.51 27.00 28.03 -0.05 0.00 11 -5 27.45 26.95 27.95 -0.06 -0.02 12 -4 27.35 26.85 27.85 -0.10 -0.04 13 -3 27.20 26.70 27.70 -0.15 -0.05 14 -2 26.97 26.47 27.47 -0.22 -0.07 15 -1 26.66 26.15 27.16 -0.31 -0.09 16 0 26.24 25.69 26.78 -0.42 -0.11     This is the graph between x and y.     Since the back_timescale include limited point (-15 to 0 by 1 ), could I use different point as the spline point and then compare their fit statistics (AIC, BIC) to decide the best spline point?           ... View more

Thank you for your kind reply. The paper is very interesting. ... View more

Hi,   I did not find a good definition for the "changing point" or break point. In the paper (https://support.sas.com/resources/papers/proceedings20/4739-2020.pdf), they define it as "the time when development switches from one phase to another ".   They used PROC NLMIXED to find the point. I am thinking they may detect the changing point based on the fitness (AIC, BIC) of the model. For example, using different knots to spline the curve and find the best one with best fitness.     ... View more

Hi, I know for a bell-shaped curve or U-shaped curve, it is easy to find the changing point. However, for the blue line in the below picture, the slope for each time point seems to increase over time. There is no such extreme value.  Do you know how to detect the changing point for those types of curves?   Thanks for your help in advance.           ... View more

Thank you for your quick reply.   Now, I did not try the ods output and ods select. I knew in R, the package "nlme" "segmented" can help to get the changing point. But if it is to choose the extreme slope rate, I know we could output the mixed effect model and calculate the Derivative. Is this what you mean?   ... View more

Hi, sorry but the link from Wiki is not available now. I just want to find the change point (or knot, joint, turning point), which was defined as “the time when development switches from one phase to another”. They are the points that are shown in the below graph.   I think the extreme value may be one solution.       ... View more