Factor vs Treatment

In experimental design a factor is an explanatory variable whose values the researcher sets rather than records. It has nothing to do with factors in arithmetic. The values chosen are the factor's levels, and the conditions actually applied to units are the treatments. With a single factor the levels are the treatments. With more than one, each treatment is one combination of levels, so the treatment count multiplies rather than adds.

A coffee study varies grind size at 2 levels, coarse and fine, and brew time at 4 levels: 2, 3, 4, and 5 minutes. That is 2 factors and $2 \times 4 = 8$ treatments, running from coarse at 2 minutes through to fine at 5 minutes. Add a third factor with 3 levels, water temperature, and the design carries $2 \times 4 \times 3 = 24$ treatments, each of which still needs several experimental units. That multiplication is why factors are expensive to add.

"The treatments are grind size and brew time" is the standard wrong answer. Those are the factors. A treatment is a complete specification of what one unit receives, so "fine grind, 4 minutes" is a treatment and "brew time" is not. The related slip is reporting the number of levels as the number of treatments: this design lists 6 levels across its two factors and applies 8 treatments to units.

Two things standing nearby are not factors. In an observational study the same variable is called an explanatory variable, because its value is recorded rather than imposed, and there are no treatments at all. A blocking variable is not a factor either: units already belong to their block by age or sex or location, nothing about it is assigned, and its categories are not treatments.

Nor is the response variable. Factors are inputs you set before the study runs; the response is the outcome measured on each unit afterward. Naming the factor, its levels, the treatments, and the response in that order is most of what a design description on the exam is asking for.

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.