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This question is specific to a text. I am reading "Statistical Methods for Survival Data Analysis" by Lee and Wang. On p. 70 in the 4th edition (and p. 66 in the 3rd) the authors show a linear interpolation for finding the median for uncensored observations using the product limit approach, or Kaplan Meier, but I cannot understand their formulation.

Data are:

t      S(t)
6      0.7
m      0.5 (median cum prob.)
8      0.4

and they use this equality to calculate $m$:

$$\frac{8 - 6}{0.4 - 0.7} = \frac{8 - m}{0.4 -0.5}$$

to solve for $m = 7.3$. I do not understand the setup. One can rearrange this to equate the ratio of the survival times (months) to the ratios of the probabilities, and so it appears that these are slope approximations, but I do not understand the reasoning behind this setup.

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You are correct that they are using the slope. Plot the points $(6,0.7)$ and $(8,0.4)$ and draw the line segment connecting them. To estimate the median using linear interpolation we must find the point on the line segment where the ordinate is $0.5$

Usually we think of slope as the change in $y$ over the change in $x.$ All they have done is reversed this in a sense, using the change in $x$ over the change in $y$ on both sides of the equation. This doesn't really change anything; the reasoning and the result is valid.

The bottom line is that (based on the linear interpolation approach) the slope of the line segment connecting the estimated median and either of the other two points must equal the slope of the entire line segment.

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