Fraction Of Missing Information For A Scalar Estimate

Why a Fraction of Missing Information Is Useful

Multiple imputation produces several completed data sets, and the scalar estimate can vary across them. The fraction of missing information summarizes, for that estimate, how much the missing-data uncertainty contributes to the total uncertainty. It is an estimate tied to a particular scalar quantity and its pooled analysis—not a direct count of missing values.

A related quantity, the relative increase in variance, is often available first. It describes how much the pooled variance is increased by the between-imputation component relative to the within-imputation variance. The fraction of missing information is related to that quantity, but it also includes a correction involving the degrees of freedom. Keeping the two distinct matters, especially when the degrees of freedom are not large.

Definition and Conditions

Consider \(m\) completed data sets. Let \(Q_i\) be the estimate of the same scalar estimand from completed data set \(i\), and let \(U_i\) be its estimated variance. Define the pooled estimate \(\bar{Q}\), the average within-imputation variance \(\bar{U}\), and the between-imputation variance \(B\) by

\bar{Q}=\frac{1}{m}\sum_{i=1}^{m}Q_i,\qquad \bar{U}=\frac{1}{m}\sum_{i=1}^{m}U_i,\qquad B=\frac{1}{m-1}\sum_{i=1}^{m}(Q_i-\bar{Q})^2.

Rubin’s total variance estimate is \(T=\bar{U}+(1+1/m)B\). The relative increase in variance is

r=\frac{(1+1/m)B}{\bar{U}}.

For this tutorial, \(df\) is the degrees of freedom associated with the scalar Rubin pooling calculation. The standard estimated fraction of missing information is

Definition: For a scalar estimate with \(r\geq 0\) and \(df>0\), the finite-degrees-of-freedom estimate of the fraction of missing information is

\widehat{\lambda} =\frac{r+2/(df+3)}{r+1}.

The notation \(\widehat{\lambda}\) emphasizes that this is an estimate, not an exactly observed proportion of information that is missing.

The calculation presumes that the imputations and scalar analyses support Rubin pooling: each \(Q_i\) estimates the same target, the \(U_i\) estimates are valid within-imputation variances, and the imputations adequately represent uncertainty under the specified imputation and analysis models. In practice, this includes checking that the MICE chains have settled into their intended distributions and that the imputation models are appropriate for the analysis. A numerical fraction cannot repair poor imputation, incompatible estimands, or invalid variance estimates.

Why It Is Not the Same as Relative Increase in Variance

The variance decomposition makes the distinction clear. The between-imputation contribution to \(T\) is \((1+1/m)B\), and \(r\) is that contribution divided by \(\bar{U}\). Thus, the share of \(T\) represented by that contribution is

\frac{(1+1/m)B}{T} =\frac{(1+1/m)B}{\bar{U}+(1+1/m)B} =\frac{r}{1+r}.

This is the variance share obtained directly from the decomposition. The estimated fraction of missing information uses a finite-degrees-of-freedom adjustment. Writing \(c=2/(df+3)\), its formula is

\widehat{\lambda} =\frac{r+c}{1+r} =\frac{r}{1+r}+\frac{c}{1+r}.

The second term is the correction. It is positive for finite positive \(df\), so the estimated fraction is larger than \(r/(1+r)\). The correction decreases as \(df\) increases and tends to zero as \(df\) tends to infinity. This is a finite-sample adjustment in the standard estimate; it is not a second variance component added to \(T\).

The formula also gives useful checks. For \(r\geq 0\) and \(df>0\), \(0<c<1\). Therefore the numerator is positive and is smaller than the denominator:

0<r+c<r+1 \quad\Longrightarrow\quad 0<\widehat{\lambda}<1.

For fixed \(df\), the estimate increases with \(r\): its derivative with respect to \(r\) is \((1-c)/(1+r)^2>0\). For fixed \(r\), it decreases as \(df\) increases, because the correction \(c\) decreases. With very large degrees of freedom, \(\widehat{\lambda}\) approaches \(r/(1+r)\). These are checks on the formula, not claims that changing \(df\) alone changes the missingness in the data.

Worked Example: A Moderate Relative Increase

Suppose a hypothetical scalar pooling calculation gives \(r=0.50\) and \(df=18\). Calculate the estimated fraction of missing information and compare it with the variance share \(r/(1+r)\).

First calculate the finite-degrees-of-freedom correction:

\frac{2}{df+3}=\frac{2}{18+3}=\frac{2}{21}=0.09524.

Substitute into the fraction formula:

\widehat{\lambda} =\frac{0.50+0.09524}{1+0.50} =\frac{0.59524}{1.50} =0.3968.

The estimated fraction of missing information is about \(39.7\%\) for this scalar estimate. The variance share without the correction is

\frac{r}{1+r}=\frac{0.50}{1.50}=0.3333.

That share is about \(33.3\%\), not \(39.7\%\). The difference, about \(0.0635\), is the finite-degrees-of-freedom adjustment after division by \(1+r\).

Worked Calculations and Interpretation

A value of \(\widehat{\lambda}=0.40\) is commonly described as an estimated fraction of about 40% of the information for the scalar estimate being missing. This is an interpretation on the information scale, not a statement that 40% of the records or values are missing. Different estimates in the same data set can have different fractions because their relationships with incomplete variables, and their within- and between-imputation variances, differ.

Worked Example: A Larger Correction at Lower Degrees of Freedom

For another hypothetical scalar analysis, suppose \(r=1.00\) and \(df=4\). Find the estimated fraction of missing information and compare it with the unadjusted variance share.

The correction is

\frac{2}{df+3}=\frac{2}{4+3}=\frac{2}{7}=0.2857.

Then

\widehat{\lambda} =\frac{1.00+0.2857}{1+1.00} =\frac{1.2857}{2.00} =0.6429.

The estimated fraction is about \(64.3\%\). By contrast, \(r/(1+r)=1/2=0.50\), or \(50.0\%\). Here the finite-degrees-of-freedom correction is substantial. Interpreting \(r=1\) itself as an FMI of 100% would be incorrect: \(r\) is a relative increase, not a fraction bounded above by one.

Worked Example: Same Relative Increase, Different Degrees of Freedom

Consider two hypothetical pooling calculations with the same \(r=0.25\), but with \(df=25\) in one and \(df=100\) in the other. Compare their estimated fractions.

For \(df=25\), the correction is \(2/(25+3)=2/28=0.07143\), so

\widehat{\lambda}_{25} =\frac{0.25+0.07143}{1.25} =\frac{0.32143}{1.25} =0.2571.

For \(df=100\), the correction is \(2/(100+3)=2/103=0.01942\), giving

\widehat{\lambda}_{100} =\frac{0.25+0.01942}{1.25} =\frac{0.26942}{1.25} =0.2155.

Both calculations have the same relative increase in variance, but their estimated fractions differ: approximately \(25.7\%\) and \(21.6\%\). The lower degrees of freedom produce the larger correction. This comparison isolates the formula’s finite-df effect; it does not imply that these two values of \(df\) can be chosen independently of the underlying pooling results.

Diagnostics, Alternatives, and Limitations

A useful check is to calculate both \(r/(1+r)\) and \(\widehat{\lambda}\). The first should be between zero and one when \(r\geq0\); the second should also be between zero and one for \(df>0\), and should be at least as large as the first. A negative \(r\), a negative or zero \(df\), or a fraction outside the stated range usually signals an input, coding, or definition problem. Round only after carrying out the calculation, since early rounding can matter when the correction is small.

The common alternatives answer different questions. Relative increase in variance, \(r\), compares the between-imputation contribution with the within-imputation variance and is not itself a fraction. The transformed quantity \(r/(1+r)\) is the between-imputation share of the total variance. The finite-df \(\widehat{\lambda}\) is the adjusted estimate of missing information. The percentage of cases with missing values is a property of the data, not of a particular scalar estimand. These measures should not be substituted for one another.

The estimated fraction is specific to the scalar estimate and the pooling calculation. It does not identify which records or variables caused the information loss, establish that the imputation model is correctly specified, or prove that missingness is ignorable. It is also sensitive to the quality of \(B\) and \(\bar{U}\), which depend on the imputations and analyses. When comparing estimates, report the target and the pooling quantities alongside the fraction when context requires it.

Common Mistakes

Calling \(r\) the fraction of missing information. \(r\) is a relative increase and may exceed one. Use \((r+2/(df+3))/(r+1)\) for the finite-df estimate.
Dropping the correction without qualification. \(r/(1+r)\) is the variance share from the decomposition; it is not the stated finite-df estimate. The two approach one another as \(df\) grows.
Equating FMI with the proportion of incomplete records. FMI describes information for a scalar estimate, while the missing-record percentage describes the data.
Interpreting the number as a probability. An estimated fraction of 0.40 does not mean there is a 40% probability that a value is missing or that an estimate is wrong.

Key takeaway: For a scalar Rubin pooling calculation, use \(\widehat{\lambda}=(r+2/(df+3))/(r+1)\). The correction makes the estimated fraction at least as large as the unadjusted variance share \(r/(1+r)\), with the difference shrinking as \(df\) increases.

Check Your Understanding

Answer each question, then check the worked solutions below.

A hypothetical scalar estimate has \(r=0.40\) and \(df=22\). Calculate \(\widehat{\lambda}\).
For \(r=0.75\) and \(df=17\), calculate the finite-df estimate and the unadjusted variance share. Which is larger?
An analyst reports that 30% of records are incomplete and concludes that the FMI for every scalar estimate must be 30%. Explain the error.
If \(r\) is fixed and \(df\) increases, what happens to the finite-df correction and to \(\widehat{\lambda}\)?

Solutions. (1) The correction is \(2/(22+3)=0.08000\), so \(\widehat{\lambda}=(0.40+0.08)/1.40=0.3429\), or about \(34.3\%\). (2) The correction is \(2/20=0.1000\). Thus \(\widehat{\lambda}=(0.75+0.10)/1.75=0.4857\), while \(r/(1+r)=0.75/1.75=0.4286\). The finite-df estimate is larger. (3) The incomplete-record percentage describes the data, whereas FMI is specific to a scalar estimate and depends on its \(r\) and \(df\); the two quantities need not match. (4) Since \(2/(df+3)\) decreases as \(df\) increases, the correction decreases. With \(r\) fixed, \(\widehat{\lambda}\) also decreases toward \(r/(1+r)\).