When One Answer Is Not Enough
Some questions ask a respondent to choose exactly one answer: for example, “Which one of these plans do you prefer?” Other questions invite every applicable answer: “Which of these services have you used?” A respondent might select several services. Recording that response in one ordinary categorical variable can lose information, because a single category value cannot hold all the selected options unless the values themselves are defined as combinations.
The key is to represent what was actually selected. With a fixed set of options, each option can be recorded as its own yes-or-no variable. Alternatively, a record can store the respondent’s selections as a list. These are two ways to preserve the same response, provided the choices and coding rules are clear.
Separate Yes-or-No Variables
Suppose a form asks which of four activities a person did last weekend: reading, gardening, cooking, and exercising. Define one variable for each activity. For person \(i\) and option \(j\), let \(x_{ij}=1\) if person \(i\) selected option \(j\), and let \(x_{ij}=0\) if the person did not select it. Each variable is binary: its two values indicate selection and non-selection.
The row still represents one individual, and the columns record different features of that individual’s response. A person who selected reading and cooking has a 1 in those two columns and a 0 in the other two. This is not one response split across four people; it is four related variables recorded for the same person.
| Person | Reading | Gardening | Cooking | Exercising |
|---|---|---|---|---|
| A | 1 | 0 | 1 | 0 |
| B | 0 | 1 | 1 | 1 |
| C | 1 | 0 | 0 | 0 |
Person A selected two options, person B selected three, and person C selected one. The zeros do not mean that these individuals did none of the activities; they mean only that the corresponding option was not selected. This interpretation relies on a completed response and a clear coding rule.
Worked Example: Coding a Multiple-Selection Response
In a hypothetical class survey, each student may select any of three study resources used during the previous week: lecture notes, practice problems, and video lessons. Four completed responses are: A selected notes and practice problems; B selected video lessons only; C selected all three; D selected practice problems and video lessons. Construct the binary variables and find how many resources each student selected.
Step 1: Define the columns. Let \(N_i\), \(P_i\), and \(V_i\) equal 1 when student \(i\) selected notes, practice problems, and video lessons, respectively, and 0 otherwise.
Step 2: Encode the responses. A’s selections give \((N_A,P_A,V_A)=(1,1,0)\). B’s response gives \((0,0,1)\). C’s response gives \((1,1,1)\). D’s response gives \((0,1,1)\).
Step 3: Count the selections for each student. Add the three indicators in each row. The counts are A: \(1+1+0=2\), B: \(0+0+1=1\), C: \(1+1+1=3\), and D: \(0+1+1=2\). Thus the number selected is a summary of the response, while the three columns retain which resources were selected.
Storing the Response as a List
A list representation stores the selected option names together in one field. For the survey above, the entries might be “notes; practice problems,” “video lessons,” “notes; practice problems; video lessons,” and “practice problems; video lessons.” This can be a natural way to preserve the original response, especially when the set of available options is long or changes over time.
A list is not the same as a single ordinary category such as “notes.” It contains multiple labels for one individual. If the option set is fixed and each selection is recorded once, a list can be converted to binary columns by checking, option by option, whether that label appears in the list. Conversely, the selected labels can be collected from the columns whose values equal 1.
For reliable conversion, the options need stable labels and the recording rule must be explicit. For example, “video lesson” and “video lessons” should not accidentally be treated as different options. If respondents may enter their own wording, those entries cannot be assumed to match a fixed option without a rule for handling them.
This equivalence can be justified directly. Let \(S_i\) be the set of options selected by individual \(i\), and let \(J\) be the fixed set of possible options. For every option \(j\in J\), define \(x_{ij}=1\) when \(j\in S_i\), and \(x_{ij}=0\) otherwise. Then the entries with value 1 identify exactly the members of \(S_i\). In the other direction, the list is recovered by collecting exactly those \(j\) for which \(x_{ij}=1\). Therefore neither representation loses selections under the stated conditions.
Worked Example: Converting a List to Indicators
A hypothetical questionnaire offers four choices for preferred forms of exercise: walking, swimming, cycling, and dancing. One completed response is stored as the list “walking; cycling.” Convert it to indicators, then recover the list from the indicators.
Define the coding. Use the column order walking, swimming, cycling, dancing, with 1 meaning selected and 0 meaning not selected. Walking and cycling appear in the list, while swimming and dancing do not. The indicator row is therefore \((1,0,1,0)\).
Recover the list. Select the option labels in the columns with value 1. Those columns are walking and cycling, so the recovered list is “walking; cycling,” exactly the original response.
The conversion works because the four option labels are fixed and distinct. If “cycling” were recorded under inconsistent labels, or if one indicator column were omitted, the row might no longer preserve the original selections.
Why a Single Category Can Be Misleading
A single categorical variable assigns one category value to each individual. That format works naturally when exactly one option is allowed. For a multiple-response item, however, a respondent may select any subset of options: one choice, several choices, or possibly none, depending on the question. Coding only the first choice, or forcing everyone into one “main” choice without an explicit rule, discards the other selections.
It is possible to make a single variable whose categories are all possible combinations. With four options, for instance, combinations could include “walking only,” “walking and cycling,” and “swimming, cycling, and dancing.” But the category set grows quickly: with \(k\) options, there are \(2^k\) possible subsets, including the empty subset. This approach may be cumbersome and does not make option-specific summaries as straightforward as separate indicators. It is not equivalent to treating each original option as a mutually exclusive category.
Worked Example: Option Counts Need Not Sum to the Number of People
Six hypothetical respondents may select any combination of three community activities: gardening, sports, and music. Their completed responses are: gardening and music; sports; all three; gardening only; sports and music; and none. Find the number and percentage selecting each activity, using all six respondents as the denominator.
Count each option separately. Gardening was selected by respondents 1, 3, and 4, so its count is 3. Sports was selected by respondents 2, 3, and 5, so its count is 3. Music was selected by respondents 1, 3, and 5, so its count is also 3.
Calculate the percentages. For each option, divide its count by the six respondents. Gardening: \(3/6=0.50\), or 50%. Sports: \(3/6=0.50\), or 50%. Music: \(3/6=0.50\), or 50%.
These percentages add to 150%, not 100%, because a respondent can contribute to more than one option’s count. They answer three separate questions: what proportion selected gardening, what proportion selected sports, and what proportion selected music. They do not describe mutually exclusive shares of one whole.
A Practical Encoding Workflow
The representation should follow the response rule rather than the convenience of a spreadsheet. For a completed multiple-response item with a known option set, the following steps keep the data interpretable.
Keep one record associated with each person or other observational unit, as appropriate to the data collection.
Use distinct, consistent labels so each selection has an unambiguous place in the data.
Use one binary variable per option, a list of selected labels, or both when it is useful to retain the response and make option-specific summaries.
For each individual, confirm that the selected labels match the indicators equal to 1. A count of selected options cannot replace the option identities.
Common Mistakes
- Forcing overlapping answers into exclusive categories. A respondent who selects both “gardening” and “music” cannot be faithfully represented as only “gardening” or only “music.” Use separate indicators or preserve the list.
- Assuming option percentages must total 100%. Each percentage uses the respondent count as its denominator, but respondents may select multiple options. The percentages can sum to more than 100%.
- Replacing selections with only a total. A count such as “selected two options” does not show which two were chosen. Keep option identities if the analysis needs them.
- Confusing a list with a single choice. A list field can contain several selected labels even though it occupies one cell in a data table. The number of cells does not determine how many responses were allowed.
Check Your Understanding
Try these questions before reviewing the explanations in the tutorial.
- A respondent selects “email” and “text message” from a question allowing any number of contact methods. What are the values of the two corresponding binary variables?
- Explain why storing only the number of options selected may lose information.
- Five people answer a multiple-response question. Four select option A and three select option B. Why can the two option percentages sum to more than 100%?
- Give one condition needed to convert a list of selections into separate binary variables without losing selection information.
- Why is “walking and cycling” not automatically a single mutually exclusive category in a question allowing multiple choices?