The time to relapse, in months, for patients on two...

The time to relapse, in months, for patients on two treatments for lung cancer is compared using the following log normal regression model: $Y = ln(X) = 2 + 0.5Z + 2W$, where $W$ has a standard normal distribution and $Z = 1$ if treatment A and 0 if treatment B. (a) Compare the survival probabilities of the two treatments at 1, 2, and 5 years. (b) Repeat the calculations if $W$ has a standard logistic distribution. Compare your results with part (a).

Transcript

00:01 We're given that y equals the natural log of x equals 2 plus 0 .5 z plus 2 w.

00:08 So the treatment a, where z is 1, would be 2 plus 0 times 1, which is 2 .5 plus 2 .5 plus 2w.

00:21 And for treatment b, that would be 2 plus 0 plus 2 .5 times 0 plus 2w.

00:27 And this means the log of the relapse time follows the normal distribution, where for treatment a, a mean of 2 .5 and a standard deviation of 2 because the variance is 2 squared or 4.

00:47 And for treatment b, the mean is 2 instead of 2 .5.

00:53 Now the survival function for a log normal variable x is given by the function 1 minus phi of the natural log of t minus the mean over the standard deviation.

01:07 And so we want one, two, and five years, but we're going to do this in months.

01:12 So first, we're going to find for treatment a at 12 months.

01:17 So that's 1 minus 5, the natural log of 12 minus 2 .5 over 2 .2.

01:25 And that's 1 minus 5 negative 0 .0075.

01:34 And you look that up in a standard normal table or use a calculator.

01:38 And that's 0 .497.

01:40 Subtract and you get 0 .503, which is 50 .3%.

01:46 And for treatment b, that'd be 1 minus phi of the natural log of 12 minus 2 over 2, which is is 1 minus phi of 2 .25, and 1 minus, looking that up on the table, you get 0 .596, and subtracting you get 0 .404.

02:23 So that's 404 for the survival probabilities, which means a is looking better so far.

02:34 So for the two year you're going to plug in 24 for t.

02:39 So for treatment a, that would now be 1 minus 5 of the natural log of 24 minus 2 .5.

02:50 So back to treatment a.

02:54 That calculates to 0 .339.

02:59 Look that up in the table and you get 0 .633.

03:04 And so that's 0 .367 or after two years, 36 .7 % survival.

03:13 And for treatment b, after two years, that is 0 .589.

03:31 Look that up at the table, and you'll see 0 .721.

03:36 Subtraction, you get 0 .279, meaning 27 .9 % survival for treatment b.

03:43 Again, a is outperforming...

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