Fraction Of Missing Information For A Linear Contrast

Why Calculate Missing Information for a Contrast?

A regression contrast may combine several coefficients, such as the sum of two treatment effects or the difference between two slopes. Its uncertainty depends not only on the uncertainty in each coefficient but also on their covariance. For that reason, the fraction of missing information for a contrast must be calculated from the contrast’s own pooled variance components. It cannot generally be obtained by averaging the corresponding fractions for the coefficients that form it.

The fraction of missing information (FMI) summarizes how much the missing-data uncertainty contributes to uncertainty in a particular pooled scalar estimate. It is specific to the estimand: changing the linear contrast can change its within-imputation variance, its between-imputation variance, or both. The earlier material on linear contrasts and Rubin pooling supplies the setting; the focus here is how to calculate and interpret this contrast-specific quantity.

Definition: Let \(m\) completed data sets yield estimates \(q_j\) and estimated within-imputation variances \(u_j\) for the same scalar linear contrast, for \(j=1,\ldots,m\). Define \(\bar{q}=m^{-1}\sum_{j=1}^{m}q_j\), \(\bar{U}=m^{-1}\sum_{j=1}^{m}u_j\), and \(B=(m-1)^{-1}\sum_{j=1}^{m}(q_j-\bar{q})^2\). Rubin’s total variance is \(T=\bar{U}+(1+1/m)B\). The relative increase in variance is \(r=(1+1/m)B/\bar{U}\). A commonly used finite-degrees-of-freedom estimate of the scalar fraction of missing information is \(\lambda=(r+2/(\nu+3))/(r+1)\), where \(\nu\) is the degrees of freedom used for the pooled scalar inference.

What the Formula Measures

The factor \(1+1/m\) matters: it includes the finite number of imputations in Rubin’s total variance. The relative increase \(r\) compares the added variance associated with between-imputation variation to the average within-imputation variance. If \(\nu\) is large, the small-sample correction \(2/(\nu+3)\) becomes negligible, and the FMI approaches \(r/(1+r)\).

That limiting expression is also the between-imputation component’s share of total variance:

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

The finite-degrees-of-freedom FMI is not exactly this variance share. The adjustment \(2/(\nu+3)\) makes the FMI estimate slightly larger in the examples below. Always identify which quantity is being reported; calling both of them “the fraction” without qualification can obscure the distinction. The degrees of freedom and the exact small-sample procedure should be the ones used for the corresponding pooled scalar inference.

For a vector of coefficients \(\boldsymbol{\beta}\), let \(\mathbf{c}\) be a fixed contrast vector. In imputation \(j\), the scalar estimate is \(q_j=\mathbf{c}^{T}\hat{\boldsymbol{\beta}}_j\). If \(\mathbf{U}_j\) is that imputation’s estimated covariance matrix, then its within-imputation variance is \(u_j=\mathbf{c}^{T}\mathbf{U}_j\mathbf{c}\). Calculate \(\bar{U}\), \(B\), \(r\), and \(\lambda\) from these scalar \(q_j\) and \(u_j\). This makes the FMI correspond to the contrast itself, including the covariance among its coefficients.

A Short Derivation of the Contrast-Specific Calculation

The contrast is a scalar, so its between-imputation variance is the sample variance of its \(m\) scalar estimates, \(B\). Its average within-imputation variance is the average of the scalar variances \(u_j\), namely \(\bar{U}\). Substituting these two quantities into Rubin’s total-variance rule gives:

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

Thus, the between-imputation share of \(T\) is \(r/(1+r)\). The finite-degrees-of-freedom scalar FMI then adds the correction \(2/(\nu+3)\) to the numerator of that ratio, giving \(\lambda=(r+2/(\nu+3))/(r+1)\). This calculation also shows why the contrast’s pooled components must be used: the \(B\) and \(\bar{U}\) in the expression are those for the scalar contrast, not separate components for its coefficients.

The calculation relies on the same basic conditions as scalar Rubin pooling: every imputation must estimate the same contrast on the same scale; the within-imputation variances must validly estimate sampling variance under the completed-data analysis; and the imputations must adequately represent uncertainty about missing values. The scalar pooled estimate and its variance should also be sufficiently well approximated by the inference procedure used to supply \(\nu\). If the estimand changes across imputations, or if the variance estimates are invalid, an FMI computed from them does not repair the problem.

Worked Example: FMI for a Pooled Contrast

Suppose 20 hypothetical imputations give a linear contrast with \(\bar{U}=6.000\) squared units and \(B=0.95257\) squared units. Assume the scalar pooled inference uses \(\nu=196\) degrees of freedom. Calculate the relative increase, the variance-component share, and the finite-degrees-of-freedom FMI.

First, the total variance is:

$$ T=6.000+(1+1/20)(0.95257) =6.000+1.0001985 =7.0001985. $$

The relative increase is:

r=\frac{(1+1/20)(0.95257)}{6.000} =\frac{1.0001985}{6.000} =0.1667.

The between-imputation share of total variance is \(r/(1+r)=0.1667/1.1667\), or about 14.29%. The finite-degrees-of-freedom FMI is:

\lambda =\frac{0.1667+2/(196+3)}{1+0.1667} =\frac{0.1667+0.01005}{1.1667} =0.1515.

The estimated FMI is therefore about 15.15% for this contrast. This is not the claim that 15.15% of the observations are missing. It describes an uncertainty component for this particular pooled estimate under the stated pooling and degrees-of-freedom calculation.

Why a Contrast Can Differ from Its Coefficients

For a contrast \(\mathbf{c}^{T}\boldsymbol{\beta}\), both variance components depend on the contrast vector. For example, with two coefficients and \(\mathbf{c}=(1,1)^{T}\), the contrast variance includes twice the covariance term:

\operatorname{Var}(\hat{\beta}_1+\hat{\beta}_2) = \operatorname{Var}(\hat{\beta}_1) +\operatorname{Var}(\hat{\beta}_2) +2\operatorname{Cov}(\hat{\beta}_1,\hat{\beta}_2).

The same principle applies to the between-imputation covariance matrix. Positive covariance can increase the variance of a sum, while negative covariance can reduce it. A difference uses a minus sign on its covariance term. Consequently, neither the FMI for a sum nor the FMI for a difference is determined by the coefficient-specific FMIs alone.

Worked Example: Covariance Changes the FMI of a Sum

Consider a hypothetical two-coefficient model pooled over \(m=10\) imputations. Suppose the average within-imputation covariance matrix and between-imputation covariance matrix are:

\bar{\mathbf{U}}= \begin{pmatrix} 1.00 & 0.20\\ 0.20 & 1.00 \end{pmatrix}, \qquad \mathbf{B}= \begin{pmatrix} 0.25 & -0.20\\ -0.20 & 0.25 \end{pmatrix}.

Each coefficient has \(\bar{U}=1.00\) and \(B=0.25\). For either coefficient, the relative increase is \((1+1/10)(0.25)/1.00=0.275\), giving a between-imputation share of \(0.275/1.275=21.57\%\).

Now calculate the variance components for the sum, with \(\mathbf{c}=(1,1)^{T}\). The within-imputation component is \(1.00+1.00+2(0.20)=2.40\). The between-imputation component is \(0.25+0.25+2(-0.20)=0.10\). Therefore:

r_{\mathrm{sum}} =\frac{(1+1/10)(0.10)}{2.40} =0.04583, \qquad \frac{r_{\mathrm{sum}}}{1+r_{\mathrm{sum}}} =\frac{0.04583}{1.04583} =0.04382.

The sum’s between-imputation share is about 4.382%, much lower than either coefficient’s 21.57% share. The negative between-imputation covariance reduces \(B\) for the sum. If, for illustration, the contrast uses \(\nu=100\), its adjusted FMI is \((0.04583+2/103)/(1.04583)=0.06239\), or 6.239%. This calculation uses the contrast’s own components; averaging the coefficient-specific values would not reproduce it.

Worked Example: Applying the Finite-Degrees-of-Freedom Adjustment

Suppose another hypothetical contrast has \(m=5\), \(\bar{U}=2.00\) squared units, \(B=0.500\) squared units, and pooled degrees of freedom \(\nu=24\). The relative increase is:

r=\frac{(1+1/5)(0.500)}{2.00} =\frac{0.600}{2.00} =0.300.

The total variance is \(T=2.00+0.600=2.60\), so the between-imputation share is \(0.600/2.60=23.08\%\). The finite-degrees-of-freedom FMI is:

\lambda =\frac{0.300+2/(24+3)}{1.300} =\frac{0.300+0.07407}{1.300} =0.2877.

The adjusted FMI is about 28.77%, compared with a variance-component share of 23.08%. The difference arises from the finite-degrees-of-freedom correction, not from a change in \(B\) or \(\bar{U}\).

Diagnostics, Alternatives, and Limitations

A reliable calculation begins by checking the scalar contrast and its variance components before interpreting the result. Useful checks include:

Confirm that every \(q_j\) represents the same contrast, in the same units and on the same scale.
Confirm that each \(u_j\) is the variance of that contrast, including all covariance terms, rather than a standard error or a coefficient’s variance alone.
Check that \(\bar{U}>0\), \(B\geq 0\), and \(T=\bar{U}+(1+1/m)B\). A negative computed \(B\) usually signals a calculation or data-handling error.
Use the degrees of freedom associated with the pooled scalar inference when applying the finite correction. Do not silently substitute a different degrees-of-freedom convention.
Investigate whether a large \(B\) is driven by a few unusual imputation-specific contrast estimates. The FMI summarizes variance components; it does not diagnose why they differ.

The main alternative summaries answer related but distinct questions. The relative increase \(r\) reports added variance relative to \(\bar{U}\); \(r/(1+r)\) reports the between-imputation share of total variance; the finite-df FMI \(\lambda\) includes a degrees-of-freedom adjustment. These are not interchangeable labels for a single number. Nor is FMI a raw percentage of records with missing data, which is determined by the missingness pattern and can differ substantially from the information loss for a particular estimand.

FMI is also contrast-specific and model-dependent. It does not by itself establish that the imputation model is appropriate, that the missingness assumptions are credible, or that the chosen contrast is scientifically useful. It is a summary based on the estimates and variance components supplied to it. When covariance is important, reporting only coefficient-specific FMIs can conceal how much information is available for a scientifically central contrast.

Common Mistakes

Using coefficient FMIs for a contrast. This ignores covariance. Form the contrast’s \(q_j\) and \(u_j\), then pool those scalar quantities.
Dropping the \(1+1/m\) factor. The relative increase and total variance must include the finite-imputation factor in Rubin’s rule.
Calling \(r/(1+r)\) the adjusted FMI. It is the between-imputation share of total variance; the finite-df FMI additionally includes \(2/(\nu+3)\).
Interpreting FMI as the percentage of cases missing. FMI is about uncertainty for an estimand, not a count or proportion of incomplete records.

Key takeaway: Calculate FMI for a linear contrast from the contrast’s pooled within- and between-imputation variance components. Its covariance terms can make its information fraction very different from the values for its individual coefficients.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

A scalar contrast has \(m=4\), \(\bar{U}=3.00\), \(B=0.600\), and \(\nu=18\). Calculate \(r\), the between-imputation share, and the adjusted FMI.
For the sum of two coefficients, their between-imputation covariance is negative. How does that covariance enter the contrast’s \(B\), and what can it do to the sum’s FMI?
A report states that a contrast’s FMI is 30%. Does this mean 30% of participants had missing values? Explain.
Why is averaging the two coefficient-specific FMIs generally not a valid way to obtain the FMI for their difference?

Solutions. (1) \(r=(1+1/4)(0.600)/3.00=0.250\). The between-imputation share is \(0.250/1.250=0.2000\), or 20.00%. The adjusted FMI is \((0.250+2/21)/1.250=0.2762\), or 27.62%. (2) For a sum, the between-imputation variance includes twice the covariance: \(B_1+B_2+2B_{12}\). A negative \(B_{12}\) reduces that component and may reduce the contrast’s FMI, though its within-imputation variance and degrees of freedom also matter. (3) No. It is an uncertainty summary for that estimand under the pooling calculation, not the proportion of participants with missing data. (4) The difference’s within- and between-imputation variances include covariance terms with minus signs. Those contrast-specific components need not correspond to the average of the coefficient-specific fractions.