The Application of Last Observation Carried Forward in the Persistent Binary Case

  

    
No Treatment    Effect Case
    
Parameter
    
Values (Group1, Group2)
    
Scenario
    
Total Scenario
  
  

    
With    equal dropout rate
      
 
    
Event    Rate (equal)
    
(0.2,0.2),    (0.5, 0.5), (0.8, 0.8)
    
3
    
3x3=9
      
 
  
  

    
Dropout rate    (equal)
    
(10%, 10%),    (25%, 25%), (40%, 40%)
    
3
  
  

    
With    unequal dropout rate
    
Event    Rate (equal)
    
(0.2,0.2),    (0.5, 0.5), (0.8, 0.8)
    
3
    
3x6=18
  
  

    
Dropout rate    (unequal)
    
(12.5%, 7.5%),    (15%, 5%)
    
6
  
  

    
(30%,    20%), (40%, 10%)
  
  

    
(45%, 35%),    (60%, 20%)
  
  

    
Note: “Group1” and “Group2”    represent Control and Treatment groups. Since event rates are equal, results    are the same regardless of which is considered Group1.
      
 
  

Table 1:  Parameters in Simulation.

Simulation of full dataset and missing dataset

SAS statistical software (version 9.4) was used to simulate data and perform statistical analyses. The simulation primarily used SAS Interactive Matrix Language.

The full dataset was simulated as follows. We assumed a sample size of 100 per group (which is large enough to detect a 20% difference in groups with power 80%). Based on the assumed true event rate, events at T3 were created according to a Bernoulli distribution. If at T3 no event was observed, the event could not have occurred at an earlier time point, so a no event marker was created for previous time points (T2 and T1). However, if at T3 an event was observed, the time of its first occurrence had to be determined. If the event could occur at T1, then its first occurrence must have been at T2 or T3; we assumed this was equally probable and used a Bernoulli random variable with probability 1/2 to assign first occurrence. Similarly, if the event could occur at T1, we assumed this was also equally probably and used a Bernoulli random variable with probability1/3 to determine the time of first occurrence.

Once the full dataset was generated in both groups, the possible missing outcome patterns were implemented. First, we determined which people would and would not have missing observations, using a Bernoulli random variable with parameter equal to the dropout rate. We also assumed that there was no dropout at T1, and that, for those who dropped out, the probability of dropping out at T2 and T3 would be equal. Therefore we would be able to know where the missing data started (which observation and which time point). Once the missing data was observed, we assumed monotone dropout. If dropout rate was random, equal dropout rate was assumed in Control and Treatment groups. If dropout rate was related to group, then unequal dropout rate was assumed.

According to different data imputation methods, LOCF dataset, Complete1 dataset, and Complete2 dataset were created. Again, the LOCF method replaces missing observations with the observation previous to the time point that the first missing observation appeared. The Complete1 method excludes all observations with missing data. For the Complete2 dataset, instead of excluding all the observations with missing data, if the event were observed at T2, but missing at T3, we assumed that the event also occurred and was observed at T3.

Methods and statistical analyses

The primary analysis focuses on testing for a treatment effect, which is the difference of event rates. The difference of estimated event rates between groups will be analyzed using the two sample proportion z-test.

$Z=\frac{{\text{R}}_{\text{C}}-{\text{R}}_{\text{T}}-0}{\sqrt{{\text{R}}_0(1-{\text{R}}_0)(\frac{1}{{\text{N}}_{\text{T}}}+\frac{1}{{\text{N}}_{\text{C}}})}}$

where

${\text{R}}_{\text{0}}=\frac{{\text{R}}_{\text{T}}\times {\text{N}}_{\text{T}}+{\text{R}}_{\text{C}}\times {\text{N}}_{\text{C}}}{{\text{N}}_{\text{T}}+{\text{N}}_{\text{C}}}$

R_T is the event rate of the treatment group, and R_C is the event rate of the control group, and R₀ is the pooled event rate for the entire sample. N_T is the number of subjects in the treatment group, and N_C is the number of subjects in the control group. In the full dataset and LOCF dataset, N_T=N_C. In the two Complete data sets, the sample sizes are not equal because observations are deleted depending on the missing value pattern. Each replication will have a Z test statistic calculated. Since the null hypothesis is no treatment effect the number of times that the absolute value of Z exceeds Z_0.05 (=1.96) over all replications will be counted. When the null hypothesis is true, the proportion will be the Type I error, which is expected to be around 5%.

The bias in the estimated treatment effect can be expressed as ${\stackrel{}{\text{(}\stackrel{\wedge }{\text{R}}}}_{\text{C}}-{\stackrel{}{\stackrel{\wedge }{\text{R}}}}_{{}_{\text{T}}})-{\stackrel{}{\text{(}\stackrel{\wedge }{\text{R}}}}_{\text{CO}}-{\stackrel{}{\stackrel{\wedge }{\text{R}}}}_{\text{TO}})$ the difference between the estimated and the true treatment effect, where R_C0 is the assumed true event rate in the control group, and R_T0 is the assumed true event rate in the treatment group. Since this simulation assumes equal event rates (R_C0- R_T0=0), bias in the estimation of treatment effect is equal the estimated treatment difference ${\stackrel{}{\text{(}\stackrel{\wedge }{\text{R}}}}_{\text{C}}-{\stackrel{}{\stackrel{\wedge }{\text{R}}}}_{\text{T}})$

Results and Application

Equal dropout rate

Three event rates (0.2, 0.5, and 0.8) and equal dropout rates (10%, 25%, and 40%) were assumed. The results (Figure 1) showed that the mean event rate was underestimated when LOCF analysis was used but was overestimated using Complete2 analysis, while Complete1 analysis had estimated mean event rate very close to the true event rate. Since the pattern for each event rate under different dropout rates was the same, only the highest event rate (0.8) was presented. When the dropout rate increased, the amount of underestimation of the mean event rate increased when the LOCF was used. The results also showed that the underestimation for LOCF was large while the overestimation for Complete2 analysis was relatively small. Furthermore, it was interesting to see that the estimated mean event rate was less underestimated in the LOCF analysis and more overestimated in the Complete2 analysis in the case where the event might occur at T1 as compared to the case when no events occur at T1. For example, when the event rates are 0.8, and the event rate cannot occur at T1, dropout rates of 10%, 25%, and 40% lead to estimated mean event rates of 0.74, 0.65 and 0.56 for LOCF, but 0.80, 0.81 and 0.82 for Complete2 respectively. Yet when the event could occur at T1, the estimated mean event rates were 0.76, 0.70 and 0.64 for LOCF, but 0.81, 0.82 and 0.84 for Complete2.

Figure 1: Estimated Mean Event Rate: Equal Event Rate (0.8) and Unequal Dropout Rate.

    

    

    Figure 1:  Estimated Mean Event Rate: Equal Event Rate (0.8) and Unequal Dropout Rate.
    
    

Close

Even though the mean event rate was underestimated when LOCF was used and overestimated when Complete2 was used in general, the difference in rates between two groups was near zero in both analyses. The results showed that the bias was close to 0 (Table 2). (Table 3) showed the proportion of times that the test of significant difference between the two event rates was rejected. Since no treatment effect was assumed, a rejection rate of 5% was expected. The result also showed that all type I error rate estimates in the equal drop-out cases were within +/- 2% of the nominal 5% level.

Table 2: Bias: Equal Dropout Rate.

  

    
Event Rate
    
Dropout Rate
    
Bias
  
  

    
No event occurred at T1 
    
Event might occur at T1 
  
  

    
LOCF
    
Complete1
    
Complete2
    
LOCF
    
Complete1
    
Complete2
  
  

    
20%
    
10%
    
0.002
    
0.002
    
0.002
    
0.002
    
0.002
    
0.002
  
  

    
25%
    
0.001
    
0.003
    
0.002
    
0.002
    
0.003
    
0.003
  
  

    
40%
    
0.002
    
0.004
    
0.003
    
0.002
    
0.004
    
0.002
  
  

    
50%
    
10%
    
0.002
    
0.003
    
0.002
    
0.003
    
0.003
    
0.003
  
  

    
25%
    
0.002
    
0.002
    
0.003
    
0.003
    
0.002
    
0.003
  
  

    
40%
    
0.003
    
0.003
    
0.003
    
0.002
    
0.003
    
0.003
  
  

    
80%
    
10%
    
0.000
    
0.001
    
0.001
    
0.000
    
0.001
    
0.001
  
  

    
25%
    
0.001
    
0.001
    
0.001
    
0.000
    
0.001
    
0.000
  
  

    
40%
    
0.002
    
0.001
    
0.001
    
0.002
    
0.001
    
0.001
  

Table 2:  Bias: Equal Dropout Rate.

Close

Table 3: Type I Error: Equal and unequal dropout rates.

  

    
Event Rate
    
Dropout Rate
    
Type I error 
  
  

    
No event occurred at T1 
    
Event might occur at T1 
  
  

    
Ave.
    
Gp1
    
Gp2
    
LOCF
    
Complete1
    
Complete2
    
LOCF
    
Complete1
    
Complete2 
  
  

    
20% 
    
10%
    
10
    
10
    
0.050
    
0.055
    
0.052
    
0.051
    
0.055
    
0.049 
  
  

    
12.5
    
7.5
    
0.051
    
0.050
    
0.050
    
0.050
    
0.050
    
0.050 
  
  

    
15
    
5
    
0.056
    
0.049
    
0.049
    
0.054
    
0.049
    
0.054 
  
  

    
25%
    
25
    
25
    
0.046
    
0.046
    
0.048
    
0.049
    
0.046
    
0.050 
  
  

    
30
    
20
    
0.058
    
0.049
    
0.053
    
0.052
    
0.049
    
0.060 
  
  

    
40
    
10
    
0.128
    
0.047
    
0.064
    
0.085
    
0.047
    
0.104 
  
  

    
40%
    
40
    
40
    
0.044
    
0.046
    
0.045
    
0.049
    
0.046
    
0.045 
  
  

    
45
    
35
    
0.051
    
0.044
    
0.051
    
0.052
    
0.044
    
0.061 
  
  

    
60
    
20
    
0.217
    
0.042
    
0.094
    
0.111
    
0.042
    
0.190 
  
  

    
50% 
    
10%
    
10
    
10
    
0.054
    
0.052
    
0.051
    
0.056
    
0.052
    
0.049 
  
  

    
12.5
    
7.5
    
0.057
    
0.055
    
0.055
    
0.053
    
0.055
    
0.056 
  
  

    
15
    
5
    
0.082
    
0.055
    
0.053
    
0.061
    
0.055
    
0.059 
  
  

    
25%
    
25
    
25
    
0.053
    
0.052
    
0.050
    
0.057
    
0.052
    
0.054 
  
  

    
30
    
20
    
0.087
    
0.051
    
0.054
    
0.066
    
0.051
    
0.054 
  
  

    
40
    
10
    
0.361
    
0.047
    
0.066
    
0.187
    
0.047
    
0.114 
  
  

    
40%
    
40
    
40
    
0.041
    
0.044
    
0.044
    
0.052
    
0.044
    
0.047 
  
  

    
45
    
35
    
0.074
    
0.048
    
0.046
    
0.065
    
0.048
    
0.057 
  
  

    
60
    
20
    
0.598
    
0.049
    
0.101
    
0.301
    
0.049
    
0.241 
  
  

    
80% 
    
10%
    
10
    
10
    
0.055
    
0.051
    
0.048
    
0.054
    
0.051
    
0.050 
  
  

    
12.5
    
7.5
    
0.080
    
0.053
    
0.051
    
0.067
    
0.053
    
0.056 
  
  

    
15
    
5
    
0.151
    
0.054
    
0.053
    
0.096
    
0.054
    
0.058 
  
  

    
25%
    
25
    
25
    
0.047
    
0.054
    
0.050
    
0.055
    
0.054
    
0.053 
  
  

    
30
    
20
    
0.145
    
0.050
    
0.050
    
0.098
    
0.050
    
0.055 
  
  

    
40
    
10
    
0.761
    
0.048
    
0.059
    
0.468
    
0.048
    
0.086 
  
  

    
40%
    
40
    
40
    
0.054
    
0.053
    
0.056
    
0.049
    
0.053
    
0.059 
  
  

    
45
    
35
    
0.140
    
0.051
    
0.052
    
0.095
    
0.051
    
0.058 
  
  

    
60
    
20
    
0.944
    
0.050
    
0.080
    
0.664
    
0.050
    
0.159 
  
  

    
Note: “Ave.” represents average dropout rate.  “Gp1” and “Gp2” represent Group1 and    Group2.  Since event rates are equal,    results are the same regardless of which is considered the Control group.
      
 
  

Table 3:  Type I Error: Equal and unequal dropout rates.

Close

Unequal dropout rate

The mean event rates were estimated under the assumption of equal event rate (0.2, 0.5 or 0.8) and unequal dropout rate. Note that since event rates are equal, results are the same regardless of whether the dropout rate is higher in the Treatment or Control group, thus tables and figures simply refer to Group1 and Group 2. Similar to the results under the condition of equal dropout rate, mean event rates were underestimated using LOCF analysis and overestimated using Complete2 analysis; Complete1 analysis still had estimates of mean event rate close to the true rate (Figure 2). (Figure 3) displayed the bias in estimated treatment difference for equal event rates of 0.8 and unequal dropout rate. In LOCF and Complete2 analysis, for each equal event rate case the absolute bias increased as the difference between dropout rates increased under a fixed average dropout rate. For example, when no event can occur at T1, and the average dropout rate was 40%, the absolute bias with LOCF analysis increased from 0.058 to 0.241 as the dropout difference increased from 10% to 40%, while with Complete2 analysis the absolute bias increased from 0.01 to 0.038. It was interesting to see that under any fixed equal event rate and fixed difference of dropout rate the bias was stable in LOCF analysis (Figure 3a). For example, under the assumption of equal event rate of 0.8, if the dropout difference was 10%, regardless of average of dropout rates the absolute bias was 0.06. However, in Complete2 analysis instead of observing similar bias, we observed that under the same dropout difference the bias was slightly increased as the average dropout rate increased (Figure 3a). For instance, under the assumption of equal event rate of 0.8 and dropout difference of 10%, the bias increased from 0.006 to 0.007 to 0.01 as the average dropout rate increased from 10% to 25% to 40%. (Figure 3b) showed the same pattern as (Figure 3a).

Figure 2: Estimated Mean Event Rate: Equal Event Rate (0.8) and Unequal Dropout Rate.

    

    

    Figure 2:  Estimated Mean Event Rate: Equal Event Rate (0.8) and Unequal Dropout Rate.
    
    

Close

Figure 3a: No event occurred at Time 1.

    

    

    Figure 3a:  No event occurred at Time 1.
    
    

Close

Figure 3b: Bias: Equal Event Rate (0.8) and Unequal Dropout Rate. Event might occur at Time 1.

    

    

    Figure 3b:  Bias: Equal Event Rate (0.8) and Unequal Dropout Rate. Event
might occur at Time 1.
    
    

Close

(Table 3) has the estimated Type I error rates. If the event rate was low (0.2) and dropout rate was very low (average 10%), the dropout rate did not have much effect on Type I error in each analysis, which was close to 0.05. In contrast to results for equal dropout rates, however, the Type I error in LOCF analysis increased when the difference of the dropout rates increased. When the equal event rate was 0.2 and dropout difference was 40%, the Type I error was 0.217 and it was 0.944, when equal event rate was 0.8 with 40% dropout difference. With an estimated Type I error rate as high as 0.944 (rather than the desired 0.05), LOCF analysis should not be used in cases of unequal dropout rates. Complete2 analysis also behaved poorly for higher event rate with bigger difference of dropout rate (Table 3). The Type I error of 0.101 was observed under equal event rate of 0.5 and dropout difference of 40%. This result was not as extreme as that of the LOCF analysis.

Application

In this section we illustrate the missing imputation methods using an example from a study of breast cancer screening - the WISER study [7]. The purpose of WISER was to assess whether a simple, tailored health promotion intervention would increase mammography screening rates. Subjects were randomly assigned to an Intervention group (449 participants) or a Control group (450 participants).Risk for breast cancer was assessed at baseline using the Gail Model [11]. The intervention group received, at baseline, recommendations tailored to their individual risk category, including information based on the Health Belief Model (HBM) such as barriers to mammography, the seriousness of breast cancer, and benefits of yearly mammograms. The control group only received general information about breast cancer prevention practices, but no individual recommendations, no HBM report, and no risk report. Subjects were followed at 1 month, 6 months and 18 months after randomization and intervention. They were asked at each follow-up time point if they had gotten a mammogram since enrolling in the study. The primary question was whether there was a difference of mammogram rates at the 18th month follow-up between intervention and control groups.

If subjects responded at all the follow-ups, analysis would be straightforward. Only a z-test was needed to be performed in analyzing the 18th months’ follow-up on mammogram. However, as is typical, there was missing data. The question became how to analyze data when the dataset was incomplete. Our simulation investigated 3 methods, which we applied here.

No mammogram was assumed at baseline. Occasionally a subject who did not respond at an earlier follow-up time point responded to a later follow-up. This violates the assumption of monotone dropout. For purposes of this paper, though, the data were altered to force the assumptions to hold by ignoring later responses (considering them to be missing). In addition to the self-reported mammography data, information was obtained from medical records and the health information system. It was assumed that if there was no report of a mammogram from self-report, medical records or the information system, and then no mammogram had been done. This was used to construct a gold standard or “full dataset”.

Participants in the WISER study were between 40 and 82 years old, with 56% under the age of 50. Half of the participants were Caucasian and 59% had at least some college education. The dropout rate in the control group increased from 21% at month 1 to 35% at month 6, and 41% at month 18, while in the intervention group the dropout rate was 17%, 37% and 43% at month 1, 6 and 18, respectively. It appeared reasonable to expect results to be similar to those presented in our simulation with equal dropout.

For analysis of the mammogram data, the final event rates were different in the four datasets (Table 4). At 18 months, in the full dataset 74.89% and 73.05% of the sample had a mammogram in the control group and the intervention group respectively - a scenario of equal event rate. In the LOCF analysis, the estimated mammogram rates (58.22% and 56.79% in the control group and the intervention group respectively) appeared to be underestimated. In the Complete2 analysis, much higher event rates (>84%) were observed in both groups. The sample sizes were decreased almost 40% in the Complete1 and Complete2 analyses (Table 4).

Table 4: Analysis sample size and event rates (%).

  

    
 
    
Full
    
LOCF
    
Complete1 
    
Complete2 
  
  

    
 
    
Con.
    
Int.
    
Con.
    
Int.
    
Con.
    
Int.
    
Con.
    
Int. 
  
  

    
Analysis Sample Size
    
450
    
449
    
450
    
449
    
259
    
256
    
298
    
302 
  
  

    
Event rate
    
74.9
    
73.0
    
58.2
    
56.8
    
86.1
    
81.6
    
87.9
    
84.4 
  
  

    
Note: “Con.” represents Control group; “Int.” represents    Intervention group.
      
 
  

Table 4:  Analysis sample size and event rates (%).

Close

The Z-test was performed for each analysis method to test if there was an intervention effect. None of the p-values of the test statistics were less than 0.05 (Table 5), so the null hypothesis could not be rejected, implying that we could not find evidence to show significant difference under a=0.05. Based on the simulation results when there was no treatment effect, while the actual event rates may be biased, the expected bias for treatment effect using LOCF, Complete1, and Complete2 should be close to 0, which could be ignored.

Table 5: Analysis Results: Mammogram data.

  

    
 
    
Z-test Statistic for Mammogram 
  
  

    
Group
    
Full
    
LOCF
    
Complete1
    
Complete2 
  
  

    
Event Rate Difference
    
0.0184
    
0.0143
    
0.0446
    
0.0348 
  
  

    
Z-test Statistic
    
0.629
    
0.434
    
1.376
    
1.235 
  
  

    
p-value (two-tailed)
    
0.529
    
0.664
    
0.169
    
0.217 
  
  

    
Significance
    
No
    
No
    
No
    
No 
  

Table 5:  Analysis Results: Mammogram data.

Close

In the mammogram data analysis, LOCF underestimated the final event rate, and Complete2 overestimated final event rate for mammography (assuming that the “full dataset” represents the truth). In this dataset, Complete1 also overestimated the final event rate and the estimated event rates in Complete1 and Complete2 analyses were very close. However, the previous simulation study showed that Complete1 tended to have estimates similar to the full dataset. One possible explanation for the conflict might be that the “full” dataset in the mammogram was probably not accurate. It was based on the assumption that if there were missing self-reported data, and no other sources (such as the chart or info systems) indicated that the subject had a mammogram, and then they did not have had a mammogram. Mammograms at an outside institution would not have been noted.

When the “full” dataset was used as a reference for the mammography data, it was reasonable to think that this was an application of the simulation study under the scenario of equal dropout rate and equal event rate. Even though the z-test in the LOCF analysis showed that there was no significant effect between the control group and the intervention group in this mammogram data analysis, it was not a good imputation method to analyze missing data because it also underestimated the individual event rates.

Discussion

This paper evaluated the application of the LOCF method with persistent binary data, assessing Type I error, estimated event rate, treatment effect, and bias. The simulation study showed that regardless of whether the dropout rate was equal or unequal, the mean event rate was underestimated using LOCF analysis, slightly overestimated using Complete2 analysis, and unbiased using Complete1 analysis. It was useful to try to understand why in LOCF analysis fewer events were observed at T3 than in the full dataset. Whether it was assumed that events could or could not occur at T1, if missing data occurred at T2, the outcome of no-event would be carried forward to T3. Therefore, we would observe lower event rates at T3 in LOCF analysis. Thus the phenomenon of underestimated mean event rate appeared in LOCF analysis, regardless of dropout rate. In Complete2 analysis, the missing data was excluded when the event hadn’t been observed to occur, but if the event was observed to occur, the data was kept and the observation was carried forward. Since it only excluded the entire missing observations with no event occurred in the last observation, Complete2 analysis had a higher percentage of events observed at the end of the study than the full dataset, which caused the mean event rate at T3 to be slightly overestimated. It was interesting to notice that compared to the situation when no events could occur at T1, the estimated mean event rate was underestimated less in LOCF analysis and overestimated more in Complete2 analysis when the an event could occur at T1.

The results also showed that neither LOCF analysis nor Complete2 analysis was a good choice for missing data imputation in the longitudinal binary data analysis due to poorly estimated event rates and the bias involved. If finding the treatment effect was the only interest, LOCF analysis could be used in the case of equal event rate and equal dropout rate. This was the only case that LOCF analysis showed proper Type I error around 0.05.But in practice we will not know whether event rates are equal. However, if we were interested in estimated mean event rates, Type I error, LOCF analysis and Complete2 analysis were both bad choices. Complete1 analysis seemed to behave well, but it was not practical. At the beginning of the study, we assumed that once the event happened, it persisted. Complete1 analysis excluded all the random missing observations, even though we observed the event in a previous time point. This action would likely not be considered appropriate by most users, although it gave the least biased results. Naturally, when we choose the method to analyze the data without formal imputation, we tend to use Complete2, which carried the event from the previous time point to the following time points if the observation was missed at this point. This method sounded logical, but unfortunately, it produced a somewhat biased result. Given that the analysis sample size of Complete2 is larger than Complete1, if mean squared error is important, though, the tradeoff between bias and variance may make Complete2 preferred after all.

Currently, in this simulation study, we only focused on random dropout and dropout related to treatment arm. If missing data was related to outcomes or event occurrence, the conclusion might be different. In the future, we might consider increasing the number of time points, or consider the time point that the first missing started to occur.

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Citation: He J and McClish D. The Application of Last Observation Carried Forward in the Persistent Binary Case.Austin Biom and Biostat. 2015;2(2): 1018. ISSN: 2378-9840

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