A Two-Visit Dropout Factorization Worked Example

Setting Up a Two-Visit Dropout Example

With two outcome visits, monotone dropout allows a participant to miss the second visit only after having responded at the first. A useful way to understand the resulting factorizations is to begin with one complete joint distribution, enumerate the response patterns, and calculate each factor directly. This separates algebraic identities from assumptions about how missingness arose.

Let \(Y=(Y_1,Y_2)\) denote the two outcomes, and let \(R=(R_1,R_2)\), where \(R_t=1\) means \(Y_t\) is observed and \(R_t=0\) means it is missing. Under monotone dropout, \(R_2\leq R_1\). Suppose, purely for illustration, that both outcomes are binary. The possible response patterns are \(11\), \(10\), and \(00\); pattern \(01\) is excluded by monotonicity.

The example below specifies a hypothetical joint distribution for the complete outcome pair and response pattern. In rows with \(R=10\) or \(R=00\), \(Y_2\) is included to describe the full joint law even though it is not observed. This is a mathematical description of the complete-data distribution, not a claim that missing outcomes were actually measured.

The entries are joint probabilities \(P(Y_1=y_1,Y_2=y_2,R=r)\). Each row sums to its pattern probability, and the three pattern probabilities sum to one. For example, the \(10\) row sums to \(0.08+0.04+0.10+0.08=0.30\). The outcome margins, found by adding down each outcome column, are \(P(Y=00)=0.32\), \(P(Y=01)=0.19\), \(P(Y=10)=0.22\), and \(P(Y=11)=0.27\); these also sum to one.

Calculating Both Full-Joint Factorizations

For each complete outcome \(y=(y_1,y_2)\), the outcome margin is \(p(y)=\sum_r p(y,r)\). The selection factorization uses this margin and the conditional pattern probabilities \(p(r\mid y)=p(y,r)/p(y)\). The pattern-mixture factorization instead uses \(p(r)=\sum_y p(y,r)\) and \(p(y\mid r)=p(y,r)/p(r)\). These divisions are defined here because all the relevant margins are positive.

For example, at \(Y=10\), the four quantities needed for the selection factorization are:

The conditional pattern probabilities sum to one, up to rounding. For the pattern-mixture factorization, the conditional probabilities of the four outcome pairs within each pattern are:

Within each line, the entries are ordered \(00,01,10,11\). Each line sums to one. These are distributions for the complete outcome pair conditional on a pattern, including outcomes that would be unobserved under that pattern.

Why the Two Products Agree

The equality can be justified for any cell with positive joint probability. By the definition of conditional probability, \(p(r\mid y)=p(y,r)/p(y)\) and \(p(y\mid r)=p(y,r)/p(r)\). Multiplying by the corresponding marginal recovers the joint cell:

This calculation is the complete justification for the conversion in this discrete example. In a model with zero-probability margins, a conditional distribution given that zero-probability event is not defined by division; the displayed identity applies to cells for which the relevant conditionals are defined. For the table here, all outcome and pattern margins are positive.

The distinction between the full-joint factorizations and the observed-data law matters. The two products above describe \(p(y,r)\), including complete outcomes that are missing under some patterns. The observed-data law describes only the quantities available from the recorded data, and is obtained by summing the full joint probabilities over the missing outcome components.

Marginalizing to the Observed-Data Law

For pattern \(11\), both outcomes are observed, so the observed-data probability for a record \((y_1,y_2,11)\) is the corresponding joint cell. For pattern \(10\), only \(Y_1\) is observed, so sum over both possible values of \(Y_2\). For pattern \(00\), neither outcome is observed, so sum over all four outcome pairs.

The notation \(p(00,10)\), for instance, denotes \(p(Y=00,R=10)\). The observed-data probabilities for \(R=10\) sum to \(0.12+0.18=0.30=P(R=10)\); adding the \(11\) probability mass \(0.40\) and the \(00\) mass \(0.30\) gives one. This checks that the observed-data distribution is normalized.

Assumptions, Checks, and Common Errors

The numerical exercise assumes that the displayed entries form a joint probability distribution: every entry is nonnegative and all entries sum to one. It also uses the monotone response-pattern restriction, the binary outcome support, and positive margins wherever a conditional probability is calculated. No missing-at-random, independence, or ignorability assumption is needed for the algebraic conversion. In an application, additional modeling assumptions might be required to interpret or estimate factors, but they are not implied by this table.

Useful checks for this calculation are direct and should be done before interpreting any factor:

• Each joint-table entry must be nonnegative, and all cells must sum to one.
• Pattern probabilities must equal their row sums, and outcome margins must equal their column sums.
• Conditional probabilities within a fixed conditioning value must sum to one, allowing for rounding.
• For a pattern with missing outcomes, the observed probability must sum the joint cells across every possible value of each missing outcome.
• Each record type in a likelihood must use its observed-data probability, rather than a probability for an unobserved complete outcome.

A common mistake is to call the observed-data law a third factorization of the full joint law. It is not: the selection and pattern-mixture expressions factor the full joint law, while the observed-data law is a marginal distribution induced by it. A related error is to use \(p(Y=y,R=10)\) as the probability of observing only \(Y_1=y_1\) with pattern \(10\). The correct observed-data probability is \(\sum_{y_2}p(Y_1=y_1,Y_2=y_2,R=10)\), because \(Y_2\) is unobserved and either value is compatible with that record.

The selection representation organizes the joint law by the outcome distribution and response probabilities conditional on the complete outcome. The pattern-mixture representation organizes it by pattern frequencies and outcome distributions within patterns. The observed-data law retains only distributions of recorded quantities. These are complementary descriptions, but the full-data factors involving unobserved outcomes are not necessarily identifiable from observed data alone. The example calculates them because a complete numerical joint distribution was stipulated; it does not show that a real dataset uniquely determines them.