# What are appropriate statistic tests for top-down and bottom-up knockout experimental designs?

This concernes a two staged experiment on cell toxicality.

A: Aim is to find the lowest concentration of the toxic agent 'CTA', that kills the target cells with at least 99% security. Use of lowest effective concentration is to be preferred and the number of experiments shall be minimized.

M: Stage 1: Find CTA candidate from 5-step concentration cascade "Top-Down"-Testing

(It is known that CTA100% has the desired effect. In a pilot study CTA91% proved to be 71.2% effective.)

1. CTA99%
2. CTA98%
3. CTA97%
4. CTA96%
5. CTA95%

Each CTA is tested on 8 cell cultures. Result might be dichotom 1/0: effective (= all cells destroyed) vs. not effective (some cells intact)

Top-down design means that the experiment starts with the highest concentration CTA99% on 1 .. 8 cell cultures then CTA98% on 1 .. 8 successively and so on. After each single test on a cell culture the result is evaluated and the result must be positiv (=all cells destroyed) before starting the next experiment. If that one result is negative the corresponding CTA% and all CTA% lower than this CTA are referred to as "inefficient". Consequently further stage 1 experiment will be skipped and stage 2 can be started to harden the result.

possible outcome:

1. CTA99% 8/8 (8 of 8 cell cultures effectively destroyed)
2. CTA98% 8/8
3. CTA97% 8/8
4. CTA96% 5/6 (5 of 6 cell cultures effectively destroyed) BREAK
5. CTA95% -/- skipped

This would lead to stage2 with the candidates CTA99%, CTA98% and CTA97%

Stage 2: Replicate experiment with candidate CTAs from stage 1 in a "bottom-up" fashion

The effective candidate from stage 1 with the lowest CTA% is tested first using 20 cell cultures. Again each application of the CTA is evaluated before conducting the next experiment. If one application proves to be ineffective the rest of the experiments with this CTA% is skipped (CTA% disqualified) and the next higher CTA% is used. If all twenty cell cultures are destroyed by a CTA% we have found the useful CTA%.

following the outcome above it might look like this:

1. CTA97% 8/9 (8 of 9 cell cultures effectively destroyed) BREAK
2. CTA98% 20/20 (20 of 20 cell cultures effectively destroyed) <---***
3. CTA99% -/- skipped

CTA98% it is: goal accomplished - everything is great, but ..!

My (somehow desperate) questions:

1. Which statistical tests may be applied to describe this experimental design (stage 1 and stage 2 seperately?) and how to calculate the power of a results in stage 1 / 2 and in respect to the overall aim to "identify a concentration of the toxic agent that destroys the cells with 99% security"?

2. Is there "a priori" method to optimize the sample sizes for stage 1 & 2? (can you point at any way to use G*Power for this?)

3. Is there any literature on this kind of design (top-down, bottom-up) ?