Why Carrying a Value Forward Can Be Misleading
When participants stop attending visits, a familiar shortcut is to fill in their later missing biomarker measurements with their most recent observed value. This is called last observation carried forward, or LOCF. It creates a complete-looking data set, but the filled-in number is not a later measurement. It is an assumption: after dropout, the participant's biomarker remained unchanged.
That assumption is especially questionable when health changes after follow-up ends. A participant may stop attending because their condition is worsening, for example, and their biomarker may continue to worsen as well. In that situation, carrying an earlier, less severe value forward can make later health appear better than it is. The bias can go in the opposite direction when the biomarker improves after dropout.
The target matters. If the question concerns the cohort's mean biomarker at week 12, a week-4 measurement is not automatically a valid substitute for a week-12 measurement. The two values represent different times, and the participant's health may have changed between them. The earlier material on informative follow-up loss and selection bias explains why the participants still observed can also differ systematically from those who have dropped out.
The Assumption Behind LOCF
LOCF is not merely a way of organizing data. It assigns a specific value to every missing outcome. For a participant who has dropped out before the target time, it acts as though their biomarker at that target time equals their last observed value. This amounts to assuming no change after the last measurement, at least for the purpose of the imputation.
This assumption is different from saying that dropout is missing at random (MAR). MAR concerns how missingness relates to observed and unobserved information, under a specified model. LOCF instead fixes each missing value at an earlier measurement. It does not become valid just because dropout is unrelated to the last observed biomarker, and it does not model how the biomarker may have changed after dropout.
A useful way to examine the consequence is to compare the imputed value with the outcome that would actually have been measured at the target time. The latter is unobserved for a dropout in a real data set, but it can be specified in a hypothetical example to reveal what the imputation is assuming.
How the Mean Bias Arises
Suppose a cohort has \(n\) participants, and \(R_i=1\) if participant \(i\)'s biomarker is observed at the target time and \(R_i=0\) if that measurement is missing because the participant dropped out earlier. Let \(\widetilde{Y}_i(t)\) be the value after LOCF imputation. For observed outcomes, \(\widetilde{Y}_i(t)=Y_i(t)\); for missing outcomes, \(\widetilde{Y}_i(t)=Y_i(L_i)\).
This identity follows by considering both possibilities. If \(R_i=1\), then \(1-R_i=0\) and the imputed and actual values agree. If \(R_i=0\), the imputed value is \(Y_i(L_i)\), so its difference from the actual target-time value is \(Y_i(L_i)-Y_i(t)\). Averaging across the cohort gives the discrepancy between the LOCF mean and the target-time mean:
In a population formulation, let \(q=P(R=0)\) be the proportion missing at the target time. Taking expectations of the identity gives
Thus, the bias depends on both how many outcomes are missing and the average difference between the carried-forward and actual target-time values among those missing. If higher biomarker values indicate worse health and health worsens after dropout, then \(Y(t)>Y(L)\) for many dropouts. The difference \(Y(L)-Y(t)\) is negative, so LOCF tends to underestimate the cohort mean. If higher values indicate better health, the direction may reverse.
Worked Example: A Symptom Score Worsens After Dropout
Hypothetical data. Consider four participants and a symptom score measured at week 12, where higher scores indicate more severe symptoms. Two participants drop out before week 12. For this illustration, suppose their later, otherwise unobserved week-12 scores are known. This lets us compare LOCF with the target values; in a real analysis, those post-dropout values would not be available.
| Participant | Week-12 status | Last observed score | Actual week-12 score | LOCF week-12 value |
|---|---|---|---|---|
| A | Observed | 4 | 4 | 4 |
| B | Observed | 6 | 6 | 6 |
| C | Dropped out at week 4 | 2 | 8 | 2 |
| D | Dropped out at week 8 | 5 | 9 | 5 |
The target-time cohort mean is
LOCF substitutes the last observed scores for C and D, giving
The LOCF mean is \(4.25-6.75=-2.50\) score points below the target-time mean. The same result follows from the two missing outcomes: C contributes \(2-8=-6\) points of discrepancy, and D contributes \(5-9=-4\). Their combined discrepancy divided by four participants is \((-6-4)/4=-2.50\) points. The example shows how carrying forward less severe earlier scores understates later symptom severity when the score worsens after dropout.
The Direction of Bias Depends on the Biomarker
“Worsening health” does not always mean an increasing biomarker. A symptom scale may rise as symptoms worsen, while a measure of physical function may fall as health declines. Therefore, a statement such as “LOCF biases the outcome downward” is incomplete unless the biomarker's direction is specified.
Worked Example: A Health Measure That Falls After Dropout
Hypothetical data. A functional score is measured in points, with higher scores indicating better function. For four participants, two have observed week-10 scores and two drop out earlier. Suppose the values that would have been recorded at week 10 are as shown.
| Participant | Week-10 status | Last observed score | Actual week-10 score | LOCF value |
|---|---|---|---|---|
| A | Observed | 70 | 70 | 70 |
| B | Observed | 60 | 60 | 60 |
| C | Dropped out at week 4 | 65 | 45 | 65 |
| D | Dropped out at week 7 | 55 | 35 | 55 |
The target-time mean is \((70+60+45+35)/4=210/4=52.50\) points. The LOCF mean is \((70+60+65+55)/4=250/4=62.50\) points. Here LOCF overestimates the cohort's week-10 functional score by \(62.50-52.50=10.00\) points, making the cohort appear to function better than it does. For the dropouts, the individual imputation differences are \(65-45=20\) and \(55-35=20\) points; their total contribution divided by four participants is \(40/4=10.00\) points.
In both examples, the issue is not the arithmetic. The calculations are straightforward, and the imputed means are easy to produce. The problem is that the carried values fail to represent what happens to participants after their last observed visit. If change after dropout is related to health, the direction and size of the resulting distortion depend on that change and on how the biomarker is coded.
LOCF Can Also Distort Variability
An analysis should not be judged only by its estimated mean. LOCF can also make the data appear less variable by repeating earlier values at later times, even when participants' health continues to change. Standard errors and confidence intervals calculated as if the filled-in values were observed measurements may then give a misleading impression of precision. The imputed values carry uncertainty precisely because the post-dropout outcomes were not measured.
Worked Example: Repeated Values Can Hide Later Spread
Hypothetical data. Four participants have week-2 biomarker values of 2, 4, 2, and 4. The first two remain observed at week 8, with values 2 and 4. The other two drop out after week 2; suppose their week-8 values would have been 8 and 10. Compare the observed target-time cohort values with the LOCF values.
The target-time values are \(2,4,8,10\), with mean \(6\). Their squared deviations from the mean sum to
The sample standard deviation is therefore \(s=\sqrt{40/(4-1)}=\sqrt{13.33}=3.651\) points. LOCF instead produces \(2,4,2,4\), with mean \(3\). Its squared deviations sum to
The LOCF sample standard deviation is \(s=\sqrt{4/3}=1.155\) points. In this illustration, the carried-forward data show much less spread than the target-time values. A real analysis cannot compare to the unobserved values in this way, but the calculation demonstrates why treating imputed values as if they were measured can misrepresent uncertainty.
Alternatives, Diagnostics, and Limitations
There is no single replacement that is correct for every follow-up problem. The choice depends on the scientific target, the observed data, and assumptions about why measurements are missing. The main distinction is that alternatives should make their assumptions explicit rather than silently equating a later outcome with an earlier one.
| Approach | What it uses | Important limitation |
|---|---|---|
| LOCF | Each participant's last observed value for later missing measurements | Assumes no change after the last observation and can distort means and variability |
| Complete-case analysis | Only participants observed at the target time | May describe a selected subset rather than the full cohort when retention is health-related |
| Likelihood-based longitudinal analysis | Observed repeated measurements under a specified outcome model | Its conclusions depend on the model and missingness assumptions, commonly MAR conditional on modeled observed information |
| Multiple imputation | Repeated draws for missing outcomes from a specified imputation model | Requires a defensible model; it does not make unobserved outcomes known or remove dependence on assumptions |
| Sensitivity analysis | Results under a range of assumptions about unobserved outcomes | Shows how conclusions depend on assumptions rather than identifying the missing values from data alone |
A practical diagnostic starts by plotting observed biomarker values over time, including the visits before dropout, and checking whether the observed patterns differ by follow-up status or reason for loss. If values were already changing rapidly before participants stopped attending, assuming a flat trajectory afterward is difficult to defend. Clinical information about the reasons for dropout can also help assess whether continued health change is plausible.
These checks cannot reveal the actual biomarker values after dropout: those are missing. Similar observed histories do not prove that participants' unobserved later outcomes were stable. It is therefore important to distinguish a diagnostic that raises concern from evidence that establishes the post-dropout trajectory. Sensitivity analyses can examine how conclusions change under plausible worsening or improvement scenarios.
LOCF may be defensible in a narrow setting if carrying the last value forward genuinely matches the scientific definition of the outcome or if stability after dropout is supported by strong subject-matter knowledge. It is not justified merely because it is simple, preserves the sample size, or produces a complete table. In particular, making every row complete does not restore the information that was never collected.
Common Mistake
- Wrong reasoning: “The last recorded biomarker is the best available value, so using it at every later visit is conservative.”
- Correction: A last observed value is a measurement at an earlier time, not direct evidence of the later outcome. Whether LOCF is conservative depends on the outcome's direction, the post-dropout changes, and the quantity being estimated. It can overstate or understate the later cohort mean and can distort variability.
- Wrong reasoning: “LOCF solves attrition because no values remain missing.”
- Correction: Filling a missing cell does not recover the missing outcome. It replaces it with a value determined by the imputation rule, so the rule's assumptions must be assessed.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A symptom score increases as health worsens. A participant's last observed score is 3, but their later score would have been 7. What is the LOCF imputation error, defined as imputed value minus actual value?
- Suppose 25% of a cohort is missing at the target time, and among those participants the average last-observed value is 5 points lower than the average target-time value. What is the population-formula LOCF mean bias?
- A biomarker is higher when health is better. Participants' biomarker values decline after dropout. Does LOCF tend to overestimate or underestimate the target-time cohort mean?
- Why can LOCF produce misleading standard errors even if its imputed mean is close to the target-time mean?
- Does the fact that LOCF creates a complete data set establish that the missing outcomes are MAR? Explain.
Solutions. (1) The error is \(3-7=-4\) score points, so LOCF understates the actual later score by 4 points. (2) The average difference \(Y(L)-Y(t)\) among those missing is \(-5\) points and \(q=0.25\), so the mean bias is \(0.25(-5)=-1.25\) points. (3) LOCF tends to overestimate the mean: it carries forward higher, healthier values instead of the lower later values. (4) Repeating earlier values can reduce the apparent spread, and treating imputations as observed can fail to account for uncertainty about the missing outcomes. (5) No. LOCF fills in missing cells by a rule; it does not establish any missingness assumption, including MAR.