Control Group vs Treatment

The control group is what the treatment group gets compared against, and its job is to be handled exactly like the treatment group in every respect except the one factor under test. What it actually receives depends on which comparison answers the question: nothing at all, an inactive placebo, or the treatment already in use. Units land in it through the same random mechanism that fills every other group, so it is not the pile of units left over.

A design with a control group and no placebo anywhere in it: 60 identical laptops, 30 randomly assigned new battery firmware and 30 keeping the shipping firmware, every machine running the same script at the same screen brightness in the same room. Mean runtime comes out 9.4 hours for the new firmware against 8.6 for the control, so the estimated effect is $9.4 - 8.6 = 0.8$ hours. No placebo is possible here and none is wanted, because a laptop has no expectations to manage. The control group is doing its entire job by supplying the 8.6.

Two sentences to stop writing. The first is "the control group is the group that gets nothing." Often it gets the current standard treatment instead, and where an effective treatment already exists, withholding it is both unethical and the wrong comparison, since the question is whether the new treatment beats the old one. The second is "the control group is what lets you claim cause and effect." Comparison alone does not do that. Random assignment is what makes the groups alike beforehand, and a control group compared against a self-selected treatment group establishes nothing.

Not every experiment has one. Comparing three doses of the same drug, or two teaching methods already in use, satisfies the comparison principle without a control group. An observational study is sometimes described as having one too, loosely: there the comparison group was never assigned, so the two groups can differ at the start in any number of ways.

Experimental design is topic 1.13 in Unit 1.

A factor is an explanatory variable the experimenter sets, and its levels are the settings that variable can take. A treatment is one complete combination: one level of every factor, imposed on an experimental unit. With a single factor the treatments are simply its levels. With two factors the number of treatments is the product of the two level counts.

A bakery tests two oven temperatures, 190 and 210 degrees Celsius, crossed with three rising times of 30, 60 and 90 minutes. That is $2 \times 3 = 6$ treatments, running from 190 degrees for 30 minutes through to 210 degrees for 90 minutes. Six, not the two factors and not the $2 + 3 = 5$ levels. With 48 loaves available, $48/6 = 8$ loaves receive each treatment, and the response is loaf height in centimeters after baking.

The sentence that costs points: "the treatment group got the new gel and the control group got nothing, so there was one treatment." Whatever the control group receives, a placebo, the current standard product, or no gel at all, is one of the conditions being compared, which is why a well-designed experiment compares at least two treatment groups. A treatment is also a condition rather than a set of units: the 8 loaves baked at 210 degrees for 90 minutes are a treatment group, and the treatment is the baking recipe they got.

Treatments exist only where somebody imposes them. In an observational study the groups compared are formed by what the subjects were already doing, so those are levels of a variable that got recorded, not treatments, and that is exactly why such a study cannot carry a causal claim on its own.

Factors, levels and treatments are the vocabulary of topic 1.13 Experimental Design, and a design description that never names the treatments precisely has not described the experiment.