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So let's say that I want to collect some data on a sample of people who has the same type of disease.

One group will take a drug A

Another one will take drug B

Third type will take both of them.

I want to perform a statistical test to see whether taking both of the drugs simultaneously results in getting back to health quicker.

Can anyone tell me what test would I use and what hypothesis would I state? Assumptions that I would make?

I need to plan a research project (Without actually doing it) and I thought about doing something like this but I am not sure how to start.

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    $\begingroup$ Why don't you look up an existing study and see what test(s) they used? $\endgroup$
    – Χpẘ
    Commented Mar 21, 2017 at 19:01

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First, you need some value to measure for each patient that expresses his/her progress toward recovery. Lower level of a liver enzyme, low count of bacterium or virus in blood, days for temperature return to normal, etc. Call such a measured value $Y.$

Model. The standard design that best matches the description of your study is a one-factor analysis of variance (ANOVA) with three levels of the factor, A, B, AB. One hopes it would be ethical from a medical point of view to assign, say $N = 60$ patients at random, so that there are $n = 20$ subjects in each of $g = 3$ groups (with $N = ng$).

The model for such an ANOVA design is $$Y_{ij} = \mu + \alpha_i + e_{ij},$$

where $i = 1,2,3;\; j - 1, \dots, 10$ and $e_{ij} \stackrel{iid}{\sim} \mathsf{Norm}(0, \sigma).$

Assumptions. Assumptions of the model are that the subjects are randomly selected, data are normal, and the population SD $\sigma$ is the same in each group.

Hypotheses. The null hypothesis is $H_0: \alpha_1 = \alpha_2 = \alpha_3$ and the alternative is $H_a: \alpha_i$ not all equal.

You can look up the ANOVA table, formulas, and F-test in a statistic text or online. There are also help pages online for doing ANOVA computations in various kinds of statistical software.

Power. In planning such an experiment, you might try to do a preliminary power computation. That requires guessing the numerical value of $\sigma,$ perhaps based on prior experience with recovery from this disease using drugs A and B. Also, you would need to have an idea how much difference in $Y$-values would be of interest. Then you could find out whether $n = 20$ subjects per group gives a realistic chance of detecting a difference of the desired size if it is really present.

Contrast. Also you might want to look at a 'contrast' of the form $\frac{\alpha_A + \alpha_B}{2} - \alpha_{AB}.$ That focuses on the specific goal to see whether the combination of A and B together gives better results than one drug by itself. [A 'contrast' is a linear combination of 'effects' $\alpha_i$ of the form $c_1\alpha_1 + c_2\alpha_2 + c_3\alpha_3,$ such that $\sum_i c_i = 0.$ In my example, $c_1 = c_2 = .5$ and $c_3 = -1.$]

A text on basic experimental design will include information about power computations and testing (pre-planned) contrasts. You can also find such information online, but stick to pages from authoritative sources--NIST, class notes at reputable universities, and Wikipedia pages that seem to be in final form. (Any idiot can pose as an expert on experimental design, and several do.)

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