Classify Outcomes for Inference

Start With the Outcome and the Question

A study may record several variables, but a statistical question usually focuses on one particular outcome. The appropriate parameter depends on what that outcome measures and what the question asks about it. Questions about the fraction of individuals with a categorical outcome call for proportions. Questions about the average value of a quantitative outcome call for means.

This classification is an early decision, not a complete choice of procedure. It identifies the kind of population parameter relevant to the question. Other features of the question and study design determine which specific inference method is appropriate.

Proportions for Categorical Outcomes

For a categorical response, a question often focuses on the population fraction in one specified category. For example, a researcher might ask what fraction of patients experience a particular side effect. The category of interest is “experiences the side effect,” and the parameter is the population proportion in that category.

A categorical response may have more than two categories. The question can still be about the proportion in one particular category—for instance, the fraction discharged to a rehabilitation facility. If the question concerns the distribution across all categories, it involves the separate category proportions rather than an average of category labels.

The research question matters because merely knowing that an outcome is categorical does not specify which proportion is of interest. “What fraction experienced nausea?” and “What fraction experienced a headache?” refer to different categories and therefore different population proportions, even if they come from the same recorded outcome.

Means for Quantitative Outcomes

A quantitative response records a meaningful numerical amount, such as blood pressure in millimeters of mercury or waiting time in minutes. When the question asks for the population’s average outcome, the parameter is the population mean \(\mu\). The sample mean \(\bar{x}\) summarizes the observed numerical values.

The word “average” alone is not enough to make a mean appropriate. The values must represent quantitative amounts, and differences between values must be meaningful. A number used only as a group label or code does not become a quantitative measurement just because arithmetic can be performed on it.

Why Binary Codes Still Represent a Proportion

A binary outcome is sometimes recorded using 1 for “Yes” and 0 for “No.” Although the stored values are numbers, the outcome remains categorical: the numbers mark membership in one of two categories. For this special coding, the numerical sample mean equals the sample proportion of 1s. That equality is useful, but it does not change the meaning of the outcome or turn it into a general quantitative measurement.

To see why, suppose \(n\) individuals have binary responses \(x_i\), where each \(x_i\) is either 0 or 1. Adding the responses counts the number of 1s, because adding a 0 contributes nothing and each 1 contributes one. Dividing by \(n\) therefore gives the fraction of responses coded 1:

The calculation can be treated as a mean of 0s and 1s mathematically, but its substantive interpretation is the proportion in the category coded 1. In reporting or choosing an inference family, describe it as a proportion when the question concerns how common that category is.

Multi-Category Outcomes and Meaningful Targets

Suppose a response has three categories: discharged home, transferred to rehabilitation, or transferred to another hospital. These labels do not have a meaningful numerical distance or order that would justify averaging them. A question about the fraction transferred to rehabilitation is a question about one category’s proportion. A question about the overall pattern asks about the distribution of category proportions.

A Practical Classification Process

Before choosing a specific inference procedure, make the classification in a consistent order. The response variable and the wording of the question—not the appearance of the data table or the explanatory variable—determine whether the parameter is a proportion or a mean.

For example, comparing groups does not by itself settle the proportion-versus-mean choice. If the response is whether a patient experienced an event, a question about how event rates differ concerns proportions. If the response is the number of days until an event, a question about average time concerns means. In both cases, the group labels describe how the question organizes the comparison; the outcome still determines the parameter family.

This classification does not check the assumptions or conditions needed for inference. It also does not decide whether observations are paired or independent, how many samples are involved, or which particular interval or test to use. Those choices require additional information about the question and study design.

Common Mistakes

• Choosing by the explanatory variable. A study may compare two treatments, but the comparison alone does not determine the parameter type. Check whether the response is categorical or quantitative.
• Treating every number as quantitative. Codes such as 0 and 1 may label categories. Use a proportion interpretation when the question is about the fraction in a coded category.
• Averaging category labels. Coding “home,” “rehabilitation,” and “other hospital” as 1, 2, and 3 does not make their average a meaningful discharge destination. Consider a proportion for a specified category or describe the distribution across categories.
• Leaving the category or units unstated. “The parameter is \(p\)” is incomplete without saying which population fraction \(p\) represents. For a mean, name the measured quantity and its units.
• Confusing a sample summary with a population parameter. The observed \(\hat{p}\) or \(\bar{x}\) summarizes the sample. The corresponding population parameter \(p\) or \(\mu\) is the quantity the research question seeks to learn about.