Signature
Delta_tilde = (tau_T * mu_T_last + (1 - tau_T) * mu_T_end) - (tau_C * mu_C_last + (1 - tau_C) * mu_C_end); Delta = mu_T_end - mu_C_end; bias_Delta = tau_T * (mu_T_last - mu_T_end) - tau_C * (mu_C_last - mu_C_end)
| Inputs | Definition | Unit |
|---|---|---|
tau_T | Fraction of the treatment arm whose final-visit value is carried forward | proportion from 0 to 1 |
mu_T_last | Treatment arm mean at the earlier visit whose values are carried forward | the outcome's unit |
mu_T_end | Treatment arm mean at the final visit with complete follow-up | the outcome's unit |
tau_C | Fraction of the control arm whose final-visit value is carried forward | proportion from 0 to 1 |
mu_C_last | Control arm mean at the earlier visit whose values are carried forward | the outcome's unit |
mu_C_end | Control arm mean at the final visit with complete follow-up | the outcome's unit |
Delta_tilde | Expected treatment minus control difference in the observed and imputed final-visit values | the outcome's unit |
|---|---|---|
Delta | Treatment minus control difference in true final-visit means, as it would be with complete follow-up | the outcome's unit |
bias_Delta | Analysed difference minus the true difference. A positive value overstates the treatment effect when higher values are better | the outcome's unit |
Function
Last observation carried forward imputation and the bias it adds to an analysed mean
Maps a participant's last observed value to every later scheduled visit, so that the value used at visit k is the value recorded at the last completed visit d whenever k is after d, and then maps the fraction of each arm imputed this way, with the true means at the last observed visit and at the final visit, to the expected mean and between-arm difference that the completed data return. The difference between those expected values and the true final-visit values is the bias that the imputation adds. Trapezoidal QALYs reuse HE-FM-QALY-002 and the cost-effectiveness ratio reuses HE-FM-ICER-001; this package adds the mixture mean, the between-arm bias under unequal dropout and a function that carries both into incremental QALYs and the ICER.
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Implementations
Excel
LOCF between-arm difference and its bias from named cells
With named cells TauT, TauC, MuTLast, MuTEnd, MuCLast and MuCEnd, the first formula, in a cell named DeltaTilde, returns the analysed difference and the second, in a cell named BiasDelta, its bias.
=(TauT*MuTLast+(1-TauT)*MuTEnd)-(TauC*MuCLast+(1-TauC)*MuCEnd); =TauT*(MuTLast-MuTEnd)-TauC*(MuCLast-MuCEnd)
Assumptions
Dropout completely at random within each arm for the LOCF difference
Within each arm, missingness does not depend on the earlier or the final value, although the fraction missing may differ between arms, for example because one treatment is less well tolerated. The arms are independent and each arm's dropouts were last observed at the same earlier visit.
Arm means taken from complete follow-up for the LOCF bias
mu_T_end and mu_C_end are the means that complete follow-up would give, which the analyst cannot observe in a real trial. The formula is used to show the direction and size of the bias under stated scenarios, not to correct an LOCF estimate.
Worked examples
LOCF overstates a widening utility difference under four-fold dropout
In the article's illustrative trial, 40% of the treatment arm and 10% of the control arm are imputed from 6-month means of 0.68 and 0.64, against true 12-month means of 0.64 and 0.58. The true difference is 0.06 but LOCF returns 0.07, an overstatement of 0.010, about 17%, although the true difference widens over the second half-year.
tau_T = 0.4; tau_C = 0.1; mu_T_last = 0.68; mu_T_end = 0.64; mu_C_last = 0.64; mu_C_end = 0.58; Delta_tilde = 0.07; Delta = 0.06; bias_Delta = 0.01
LOCF understates the same utility difference when the control arm drops out more
Reversing the dropout fractions, 10% in the treatment arm and 40% in the control arm, with the same means, gives an analysed difference of 0.04 against the true 0.06, an understatement of 0.02 (computed here for illustration). The direction of the bias depends on who leaves.
tau_T = 0.1; tau_C = 0.4; mu_T_last = 0.68; mu_T_end = 0.64; mu_C_last = 0.64; mu_C_end = 0.58; Delta_tilde = 0.04; Delta = 0.06; bias_Delta = -0.02
Equal LOCF dropout with a larger earlier difference
Lachin's Table 2 case 8: 30% of each arm imputed, a true final difference of 1 and an earlier difference of 2. The analysed difference is 1.3, biased upwards because the earlier difference exceeds the final one.
tau_T = 0.3; tau_C = 0.3; mu_T_last = 1; mu_T_end = 1; mu_C_last = -1; mu_C_end = 0; Delta_tilde = 1.3; Delta = 1; bias_Delta = 0.3
Equal LOCF dropout with an earlier difference of the opposite sign
Lachin's Table 2 case 5: with 30% of each arm imputed, an earlier difference of minus 1 and a true final difference of 1 give an analysed difference of 0.4, biased downwards by 0.6.
tau_T = 0.3; tau_C = 0.3; mu_T_last = 1; mu_T_end = 1; mu_C_last = 2; mu_C_end = 0; Delta_tilde = 0.4; Delta = 1; bias_Delta = -0.6
Common errors
Applying the equal-dropout direction rule when dropout differs by arm
Lachin's rule, an LOCF difference too large when the earlier difference exceeds the final one, assumes the same fraction missing in both arms. In the article's example the true difference widens from 0.04 to 0.06, so the rule predicts an understatement, yet with 40% against 10% dropout LOCF overstates the difference by 0.010.
Using a pooled dropout fraction for both arms in the LOCF bias
Replacing the arm-specific fractions with a pooled 25% for the article's example, with arms of equal size, gives a bias of 0.25 times (0.04 minus 0.06), or minus 0.005, an apparent understatement, instead of the overstatement of 0.010 (computed here for illustration). Each arm's error has to use its own fraction.
Sources
Lachin two-sample bias of LOCF-imputed group differences
Lachin JM. Fallacies of last observation carried forward analyses. Clinical Trials. 2016;13(2):161-168. Section on two-sample biases, equations 4 and 5 and Table 2: arm means after LOCF as mixtures, the analysed difference as tau times the earlier difference plus one minus tau times the final difference when the fraction missing is the same in both groups, and the direction of the bias. The unequal-fraction form applies equation 4 to each arm with its own fraction.
EMA guideline on the direction of LOCF bias by disease course and dropout
European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on Missing Data in Confirmatory Clinical Trials. EMA/CPMP/EWP/1776/99 Rev. 1. London: EMA; 2010. Section 6.3.1 on single imputation methods: LOCF unbiased only under restrictive assumptions, likely to be overly optimistic in a deteriorating condition such as Alzheimer's disease, and possibly conservative in a condition expected to improve, depending on when and in which arm participants withdraw.
Canonical Identity
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