Observed-Data Likelihood Contributions By Pattern

Why Likelihood Contributions Depend on the Pattern

With repeated outcomes, different subjects can contribute different amounts of direct outcome information: one may have every measurement recorded, another only some, and another none. The response pattern is therefore part of the observed data, not a label to discard. A likelihood contribution must account for the probability of that pattern and for the outcomes actually observed under it.

The pattern-mixture factorization writes the joint law in terms of the response-pattern distribution and the outcome distribution conditional on that pattern. The earlier material on observed-data likelihood by marginalization establishes the general principle: unobserved quantities are summed or integrated out. Here we apply that principle pattern by pattern, making explicit which outcomes remain in the contribution.

Definition: For subject $i$, let $Y_i=(Y_{i1},\ldots,Y_{iT})$ be the outcome history and $R_i=(R_{i1},\ldots,R_{iT})$ the response indicators, where $R_{it}=1$ means $Y_{it}$ is observed and $R_{it}=0$ means it is missing. Write $r_i$ for the realized response pattern, $O(r_i)=\{t:r_{it}=1\}$ for the observed indices, and $M(r_i)=\{t:r_{it}=0\}$ for the missing indices. Let $X_i$ denote baseline covariates and $A_i$ treatment. Under the pattern-mixture factorization, let $\pi_r(x,a;\theta)=P(R=r\mid X=x,A=a;\theta)$ and $f_r(y\mid x,a;\theta)$ be the conditional joint density or mass function of $Y$ given $R=r,X=x,A=a$. The observed-data contribution is $$ L_i(\theta)=\pi_{r_i}(X_i,A_i;\theta) \int f_{r_i}(y_{O(r_i)},y_{M(r_i)}\mid X_i,A_i;\theta)\,d y_{M(r_i)}. $$ For discrete missing outcomes, the integral means a sum over all their possible values. If $M(r_i)$ is empty, the integral is absent; if $O(r_i)$ is empty, the outcome law is integrated over its entire support.

In this expression, the observed outcome values are held at the values recorded for subject $i$; only the missing outcomes are integrated out. The response pattern itself is observed, so its probability $\pi_{r_i}$ is included rather than summed over. The covariates and treatment are conditioned on here, as is usual when forming a likelihood conditional on their observed values.

If subjects are independent conditional on their covariates and treatment, the observed-data likelihood is the product of their contributions:

L(\theta)=\prod_{i=1}^{n} L_i(\theta).

This independence assumption concerns sampling across subjects; it does not require outcomes at different visits for the same subject to be independent. The within-subject joint distribution $f_r$ can represent their dependence. Parameters governing $\pi_r$ and $f_r$ may be distinct or shared, so the product form does not itself imply parameter separation.

Complete, Partial, and Fully Missing Histories

The same formula covers all response patterns. Its form changes only with the observed and missing index sets. For a complete history, all outcome values are observed and no outcome integration is needed. For a partial history, the observed values enter the conditional outcome law while the missing values are summed or integrated out. For a fully missing history, the outcome law is normalized over its whole support, leaving the probability of the observed response pattern.

History type	Observed indices	Outcome part of contribution
Complete	$O(r)=\{1,\ldots,T\}$	$f_r(y_1,\ldots,y_T\mid x,a;\theta)$
Partial	$O(r)$ is nonempty and not all indices	Joint law summed or integrated over $y_{M(r)}$
Fully missing	$O(r)=\varnothing$	$\sum_y f_r(y\mid x,a;\theta)=1$, or $\int f_r(y\mid x,a;\theta)\,dy=1$

Worked Example: A Partially Observed Binary History

Suppose a hypothetical study records three binary outcomes, $Y_1,Y_2,Y_3$. For a subject with pattern $r=(1,0,1)$, the observed values are $Y_1=1$ and $Y_3=0$, while $Y_2$ is missing. At the subject’s covariate and treatment values, suppose $\pi_r=0.20$. For this example only, assume the pattern-specific outcome law makes the three outcomes conditionally independent Bernoulli variables with success probabilities $0.60,0.40,0.25$, respectively. This specifies a normalized conditional law; independence is an example-specific assumption, not a requirement of the contribution formula.

The observed-data contribution sums over the two possible values of $Y_2$:

L_i =0.20\sum_{y_2\in\{0,1\}} P(Y_1=1,Y_2=y_2,Y_3=0\mid R=r)

Using the stated independence model, the probability of the observed outcomes is $P(Y_1=1)P(Y_3=0)=0.60(1-0.25)=0.60(0.75)=0.45$. Equivalently, the missing-outcome sum contributes $P(Y_2=0)+P(Y_2=1)=0.60+0.40=1$. Thus

$$ L_i=0.20(0.45)(1)=0.0900. $$

The factor $0.20$ accounts for observing this response pattern, and $0.45$ accounts for the recorded outcome values under that pattern. The missing $Y_2$ is not assigned a single value; both possible values are included in the sum.

Worked Example: A Complete Binary History

In another hypothetical setting with three binary outcomes, suppose a subject has complete pattern $r=(1,1,1)$, observed history $(Y_1,Y_2,Y_3)=(1,0,1)$, and pattern probability $\pi_r=0.50$. Suppose the conditional outcome law given this pattern consists of independent Bernoulli outcomes with success probabilities $0.60,0.40,0.70$. As in the previous example, this independence is stipulated for calculation; a general model may instead use a joint distribution.

No outcome is missing, so there is no sum or integral. The conditional probability of the recorded history is $0.60(1-0.40)(0.70)=0.60(0.60)(0.70)=0.252$. Therefore,

$$ L_i=0.50(0.252)=0.126. $$

This is a probability contribution for the recorded response pattern and recorded binary history. It is not a density per unit of measurement, because all outcomes in this example are discrete.

Worked Example: A Fully Missing Binary History

Suppose a third hypothetical subject has $r=(0,0,0)$, so none of the three binary outcomes is recorded. Let the probability of this pattern at the subject’s covariate and treatment values be $\pi_r=0.15$. The conditional outcome distribution given this pattern may be complex, but it must be a normalized probability distribution over the eight possible histories.

For example, if $q(y_1,y_2,y_3)=f_r(y_1,y_2,y_3\mid X_i,A_i;\theta)$, then the outcome part of the contribution is $\sum_{y_1=0}^1\sum_{y_2=0}^1\sum_{y_3=0}^1 q(y_1,y_2,y_3)=1$. Hence

$$ L_i=0.15(1)=0.150. $$

The contribution contains no observed outcome value. It still contains information about the probability of the all-missing pattern. The conditional outcome distribution does not supply an observed outcome for this subject.

Writing the Integral for Continuous Outcomes

For continuous outcomes, the same bookkeeping applies, but the contribution is a density for the observed values. For example, suppose $Y_1,Y_2$ are continuous and only $Y_1$ is observed. Under pattern $r=(1,0)$, the outcome part is $\int f_r(y_1,y_2\mid x,a;\theta)\,dy_2$, evaluated at the recorded $y_1$. The result has units reciprocal to the units of $Y_1$, as a density should.

Worked Example: Integrating a Missing Continuous Outcome

Consider a hypothetical two-visit study. A subject has pattern $r=(1,0)$, with $Y_1=10$ recorded in the outcome’s measurement units and $Y_2$ missing. Suppose $\pi_r=0.30$, and under this pattern the conditional law factors as $f_r(y_1,y_2\mid x,a)=f_r(y_1\mid x,a)f_r(y_2\mid y_1,x,a)$. Let $Y_1\mid r,x,a\sim N(12,4^2)$, and suppose $Y_2\mid Y_1,r,x,a$ has a normal density for every $Y_1$. Each such conditional density integrates to one over $y_2$, so the required integration is justified by normalization.

The contribution is

L_i =0.30\int f_r(10,y_2\mid x,a)\,dy_2 =0.30 f_r(10\mid x,a)\int f_r(y_2\mid 10,x,a)\,dy_2.

Using the normal density and the fact that the conditional density integrates to one,

\begin{aligned} f_r(10\mid x,a) &=\frac{1}{4\sqrt{2\pi}} \exp\left[-\frac{1}{2}\left(\frac{10-12}{4}\right)^2\right]\\ &=\frac{1}{4\sqrt{2\pi}}\exp(-0.125) \approx 0.0880. \end{aligned}

Thus $L_i=0.30(0.0880)(1)\approx0.0264$ per unit of $Y_1$. The observed-data contribution is a density, not a probability for the exact event $Y_1=10$. The value of the contribution depends on the measurement scale for $Y_1$.

Derivation and Checks

The pattern-specific contribution follows by starting with the joint law of the response pattern and complete outcome history, conditional on $X_i,A_i$. Under the pattern-mixture factorization that law is $\pi_{r_i}(X_i,A_i;\theta)f_{r_i}(y\mid X_i,A_i;\theta)$. The observed record specifies $r_i$ and $y_{O(r_i)}$, but not $y_{M(r_i)}$. To obtain the probability or density of the observed record, sum or integrate the joint law over every complete history consistent with the observed values.

\begin{aligned} L_i(\theta) &=\int \pi_{r_i}(X_i,A_i;\theta) f_{r_i}(y_{O(r_i)},y_{M(r_i)}\mid X_i,A_i;\theta) \,dy_{M(r_i)}\\ &=\pi_{r_i}(X_i,A_i;\theta) \int f_{r_i}(y_{O(r_i)},y_{M(r_i)}\mid X_i,A_i;\theta) \,dy_{M(r_i)}. \end{aligned}

The second equality holds because the pattern probability does not depend on the outcome values being integrated in this pattern-mixture factorization. For discrete missing outcomes, replace the integral by a sum. If no outcomes are missing, the integral over an empty set leaves the joint conditional density or mass at the complete history. If all outcomes are missing, normalization of the conditional outcome law makes its integral or sum equal to one.

A practical way to check a proposed contribution is to identify the pattern first, write down $O(r_i)$ and $M(r_i)$, and then verify that every observed outcome is evaluated at its recorded value while each missing outcome is integrated over its full support. Also check that the pattern probability refers to the observed pattern and that the conditional outcome distribution is normalized. A pattern with probability zero gives a zero contribution; a recorded outcome outside the support of its conditional law also gives a zero density or mass.

Common Errors and Alternatives

Integrating out an observed outcome. This changes the data being represented. Hold every recorded outcome fixed at its observed value; integrate only over the missing outcomes.
Plugging in a value for a missing outcome. Replacing a missing value by a mean or another single imputation is not the likelihood contribution. The likelihood sums or integrates over all values allowed by the model.
Dropping the pattern probability. The response pattern is observed and has a probability under the model, so $\pi_{r_i}$ belongs in the contribution.
Dropping the outcome factor for a partial history. Only a fully missing history leaves the normalized outcome law equal to one. Partial histories retain a marginal density or mass for their observed outcomes.
Calling a continuous density a probability. For a continuous observed history, the contribution is a density with units; the probability of an exact value is generally zero.

The complete-data likelihood would use the joint pattern-and-outcome law at a fully specified $y_i$. The observed-data likelihood instead sums or integrates that law over values not recorded. A selection-factorization expression can also represent the joint law, but its factors are organized differently; the pattern-mixture expression used here makes the conditional outcome law within each observed pattern explicit. These are alternative factorizations of a joint distribution, not permission to use different observed records.

The likelihood contribution is only as appropriate as its specified response-pattern probabilities and conditional outcome laws. In particular, a fully missing history contributes the pattern probability but supplies no directly observed outcome value. For a complicated conditional distribution, the required integral may have no closed form and may need numerical evaluation. Neither difficulty changes which quantities must be integrated out.

Key takeaway: For each subject, retain the observed response pattern, evaluate the conditional outcome law at observed outcomes, and sum or integrate over missing outcomes only. Complete histories require no outcome marginalization; fully missing histories leave the normalized outcome law equal to one.

Check Your Understanding

For each question, determine the contribution or identify the correct integration. Solutions follow each problem.

A subject has three discrete outcomes, pattern $r=(1,0,1)$, and recorded values $Y_1=a,Y_3=c$. Write the observed-data contribution. Solution: It is $\pi_{(1,0,1)}(x,a_0;\theta)\sum_{y_2}f_{(1,0,1)}(a,y_2,c\mid x,a_0;\theta)$, where $a_0$ denotes the treatment value. Sum over the support of the missing $Y_2$; do not integrate out $Y_1$ or $Y_3$.
If a subject’s entire outcome history is missing and the pattern probability is $0.08$, what is the contribution, assuming the conditional outcome law is normalized? Solution: The outcome law sums or integrates to one, so the contribution is $0.08(1)=0.08$.
For continuous $Y_1,Y_2$, only $Y_2=b$ is observed under pattern $r=(0,1)$. Which variable is integrated out? Solution: Integrate over $Y_1$, holding $Y_2=b$ fixed: $\pi_{(0,1)}(x,a;\theta)\int f_{(0,1)}(y_1,b\mid x,a;\theta)\,dy_1$.
A proposed contribution for a partial history includes $\pi_r$ but no outcome term. When would that be correct? Solution: Only when no outcomes are observed and the conditional outcome law is normalized. For a partial history, the recorded outcome values must contribute their marginal density or mass after summing or integrating over the missing outcomes.

History type	Observed indices	Outcome part of contribution
Complete	\(O(r)=\{1,\ldots,T\}\)	\(f_r(y_1,\ldots,y_T\mid x,a;\theta)\)
Partial	\(O(r)\) is nonempty and not all indices	Joint law summed or integrated over \(y_{M(r)}\)
Fully missing	\(O(r)=\varnothing\)	\(\sum_y f_r(y\mid x,a;\theta)=1\), or \(\int f_r(y\mid x,a;\theta)\,dy=1\)