Hi iddrisw (plus Borel Fields Bobshmob and all here),
I didn’t want to be a slouch and finally decided to do a bit of work on the M&M problem. I did it to try to help the group and to entertain myself as I anxiously await for the efficacy results. I based my approach on the stochastic method you described while adding a few minor modifications based on the Geek Squad discussions. Below is a short summary of what I did. Apologies in advance if I put a few here to sleep.
1) I generated 56 random simulations of initial scores for the Leronlimab arm and another 28 random simulations for the placebo arm. I decided to make all of them normally distributed by using the “=ROUND((NORMVIN(RAND(),AVERAGE,STDEV),0) function in Excel. This function generates a uniformly distributed random number between 0 and 1, which is later transformed into a normally distributed random number, it is scaled by an assumed average and standard deviation, and it is rounded so it looks more consistent with the study’s integer scale. New simulations can be obtained instantly with the F9 key as you had suggested. I played with both the average and standard deviation until I obtained what looked like reasonable results. All the initial numbers fall between 2 and 11 at an average of about 7, so all initial conditions fall within the specified 0-12 scale. Hence, I think I generated somewhat reasonable simulated data for mild and moderate patients, but it is not difficult to make additional adjustments.
2) I applied the same “=ROUND((NORMVIN(RAND(),AVERAGE,STDEV),0)” function to generate normally distributed Delta TSS improvements and declines for each simulation. Once again, I adjusted the AVERAGE and STDEV to obtain what looked like plausible results and also specified a 1.0 average difference between the Leronlimab and placebo arms to cause a statistical difference, hopefully significant. I then calculated averages and standard deviations of the simulated DELTA TSS for each arm. The averages of the simulations vary a bit due to the stochastic nature of the simulations, but they often average about 1.0 in average TSS difference as I had set them. I used the standard deviations of both arms to calculate the standard error, while making sure that the standard deviations in the simulated data would jive with my assumptions. In addition, I recalculated the final score of each simulation (Final TSS = Initial TSS + DELTA TSS), truncated values less than 0 to equal to 0 and values greater than 12 to equal to 12, and then modified DELTA TSS values accordingly for the few outliers. Finally, I calculated the Standard Error (SE) using the equation Borel Fields shared and multiplied it by the 1.64 (1-sided) and 1.96 (2 sided) factors to calculate the corresponding minimum widths of the 95% confidence intervals. These widths represent the minimum differences between the two groups’ average DELTA TSS values to ensure statistical significance with p<0.05.
3) The F9 function allowed me to iterate many times and only select simulations with a 1.0 difference in average DELTA TSS between the two groups. A typical simulation may look as follow in the Leronimab arm: 42 patients improving, 9 patients with no change and 5 patients getting worse. The corresponding Placebo arm showed 19 patients improving, 4 patients with no change and 5 patients getting worse. This simulation showed a DELTA TSS difference between the two arms of 1.0, a SE of 0.42, and confidence intervals of 0.68 and 0.82. All these numbers change a bit when I run new simulations, but they are fairly consistent. Borel Fields had initially suggested width intervals between 1.5 and 1.8, which also appeared reasonable to me. However, I am calculating numbers that are generally about 50% smaller (1-sided = 0.6 to 0.9, 2-sided = 0.7 to 1.1). I could have easily made a mistake somewhere and plan to keep checking. However, I am wondering if the lower numbers may be the result of my normal distribution assumption to generate the data. It is my understanding that the normal distributions should have lower standard deviations than those in uniform distributions and should therefore result in a lower standard errors.
If my results are actually ok, then this would imply that a smaller difference of just under 1 between the Leronlimab and Placebo arms may be sufficient to demonstrate statistical significance. More importantly, it would increase the odds of success for some serious party time (Vegas Baby! with NP, BP, Kelly, Mulholland, Les. and all other doctors including Dr. Yo and Dr. Been who looks like a real party animal! ;-) ). Obviously, this is pure speculation because we need the actual results, but it hopefully provides additional hope that our chances of success are good.
Please let me know if this makes sense, if you notice obvious flaws or if you have any suggestions. I am also willing to try other ideas for pure entertainment and to help me keep my sanity during this waiting period.
Enjoy the rest of the weekend!
All the above is IMHO, DYOR, blah, blah, blah...
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