Sunday, January 17, 2021

Arithmetic mean, geometric mean, harmonic mean, least square mean, and trimmed mean

In statistics, a central tendency is a central or typical value for data distribution. Mean (or average) is commonly used to measure the central tendency. However, depending on the data distribution or the special situation, different types of Mean may be used: arithmetic mean, geometric mean, least-squares mean, harmonic mean, and trimmed mean.

The most common Mean is the arithmetic mean. If we say ‘Mean’, it is the default for arithmetic mean.

Arithmetic Mean is calculated as the sum of all measurements (all observations) divided by the number of observations in the data set.

Geometric Mean is the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale. Geometric mean is often used in the data that follows the log-normal distribution (for example, the pharmacokinetics drug concentration data, the antibody titer data...). 

In practice, geometric mean is usually calculated with the following three steps:
  • log-transform the original data
  • calculate the arithmetic mean of the log-transformed data
  • back transform the calculated value to the original scale
Harmonic Mean is the reciprocal the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.

The harmonic mean are calculated with the following steps:

  • Add the reciprocals of the numbers in the set. To find a reciprocal, flip the fraction so that the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of 6/1 is 1/6.
  • Divide the answer by the number of items in the set.
  • Take the reciprocal of the result.
The harmonic mean is not often used in day-to-day statistics but is quite often used in some statistical formula. For example, for two-group t-statistics with unequal sample size in two groups, the t value can be calculated using the following formula with harmonic mean to measure the average sample size.

Least Squares Mean is a mean estimated from a linear model. Least squares means are adjusted for other terms in the model (like covariates), and are less sensitive to missing data. Theoretically, they are better estimates of the true population mean.

In a previous post "Least squares means (marginal means) vs. means", the calculation of least squares mean is compared with the arithmetic mean.
In analyses of clinical trial data, the least-squares mean is more frequently used than the arithmetic mean since it is calculated from the analysis model (for example, analysis of variance, analysis of covariance,...). The difference between two least-squares means is called the ratio of geometric least-squares means (or geometric least-squares mean ratio) - along with its 90% confidence intervals - is the common approach for assessing the bioequivalence. 

Trimmed Mean may also be called truncated mean and is the arithmetic mean of data values after a certain number or proportion of the highest and/or lowest data values have been discarded. The data values to be discarded can be one-sided or two-sided. 

The key for trimmed mean calculation is to determine the percentage of data to be discarded and whether or not the data to be discarded is one-sided or two-sided. The percentage of data to be discarded may be tied to the percentage of missing data. 

Trimmed mean can be calculated and then used to fill in the missing data - a single imputation method for handling the missing data. Trimmed mean as a single imputation method for missing data has its limitations, but it is still used in analyses of clinical trials - usually for sensitivity analyses.

In ICH E9-R1 "Addendum on Estimands and Sensitivity Analysis in Clinical Trials" training material, about the composite strategy to handle the intercurrent event, trimmed mean is mentioned to be an approach in handling the intercurrent event. 

Monday, January 11, 2021

Single Imputation Methods for Missing Data: LOCF, BOCF, LRCF (Last Rank Carried Forward), and NOCB (Next Observation Carried Backward)

The missing data is always an issue when analyzing the data from clinical trials. The missing data handling has been moved toward the model-based approaches (such as multiple imputation and mixed model repeated measures (MMRM)). The single imputation methods, while being heavily criticized and cast out, remain as practical approaches for handling the missing data, especially for sensitivity analyses.

Single imputation methods replace a missing data point by a single value and analyses are conducted as if all the data were observed. The single value used to fill in the missing observation is usually coming from the observed values from the same subject - Last Observation Carried Forward (LOCF), Baseline Observation Carried Forward, and Next Observation Carried Backward (NOCB, the focus of this post). The single value used to fill in the missing observation can also be derived from other sources: Last Rank Carried Forward (LRCF), Best or Worst Case Imputation (assigning the worst possible value of the outcome to dropouts for a negative reason (treatment failure) and the best possible value to positive dropouts (cures)), Mean value imputation, trimmed mean,…Single imputation approaches also include regression imputation, which imputes the predictions from a regression of the missing variables on the observed variables; and hot deck imputation, which matches the case with missing values to a case with values observed that is similar with respect to observed variables and then imputes the observed values of the respondent.

In this post, we discussed the single imputation method of LOCF, BOCF, LRCF, and NOCB (the focus of this post). 

Last Observation Carried Forward (LOCF): A single imputation technique that imputes the last measured outcome value for participants who either drop out of a clinical trial or for whom the final outcome measurement is missing. LOCF is usually used in the longitudinal study design where the outcome is measured repeatedly at pre-specified intervals. LOCF usually requires there is at least one post-baseline measure. The LOCF is the widely used single imputation method.

Baseline Observation Carried Forward (BOCF): A single imputation technique that imputes the baseline outcome value for participants who either drop out of a clinical trial or for whom the final outcome measurement is missing. BOCF is usually used in a study design with perhaps only one post-baseline measure (i.e., the outcome is only measured at the baseline and at the end of the study).

Last Rank Carried Forward (LRCF): The LRCF method carries forward the rank of the last observed value at the corresponding visit to the last visit and is the non-parametric version of LOCF. However, unlike the LOCF that is based on the observation from the same subject, for the LRCF method, the ranks come from all subjects with non-missing observations at a specific visit.  From the early visits to the later visits, the number of missing values will be different, the constant ranking, carried forward, and re-ranking will be needed. Here are some good references for LRCF:

LRCF is thought to have the following features:

In a paper by Jing et al, the LRCF was used for missing data imputation: 

"...The last rank carried forward or last observation carried forward was assigned to patients who withdrew prematurely from the study or study drug for other reasons or who did not perform the 6-minute walk test for any reason not mentioned above (eg, missed visit), provided that the patient performed at least 1 postbaseline 6-minute walk test.
Next Observation Carried Backward (NOCB): NOCB is a similar approach to LOCF but works in the opposite direction by taking the first observation after the missing value and carrying it backward. NOCB may also be called Next Value Carried Backward (NVCB) or Last Observation Carried Backward (LOCB).

NOCB may be useful in handling the missing data arising from the external control group, from Real-World Data (RWD), Electronic health records (EHRs) where the outcome data collection is usually not structured and not according to the pre-specified visit schedule. 

I can foresee that the NOCB may also be an approach in handing the missing data due to the COVID-19 pandemic. Due to the COVID-19 pandemic, subjects may not be able to come to the clinic for the outcome measure at the end of the study. The outcome measure may be performed at a later time beyond the visit window allowance. Instead of having a missing observation for the end of the study visit, the NOCB approach can be applied to carry the next available outcome measure backward. 

The NOCB approach, while not popular, can be found in some publications and regulatory approval documents. Here are some examples: 

In an article by Wyles et al (2015, NEJM) Daclatasvir plus Sofosbuvir for HCV in Patients Coinfected with HIV-1, "Missing response data at post-treatment week 12 were inferred from the next available HCV RNA measurement with the use of a next-value-carried-backward approach."

In BLA 761052 of Brineura (cerliponase alfa) Injection Indication(s) for Late-Infantile Neuronal Ceroid Lipofuscinosis Type 2 (CLN2)- Batten Disease, the NOCB was used to handle the missing data for comparison to the data from a natural history study. 

Because intervals between clinical visits vary a lot in Study 901, the agency recommended performing analyses using both the last available Motor score and next observation carried backward (NOCB) for the intermediate data points although the former one is determined as the primary. 

In FDA Briefing Document for Endocrinologic and Metabolic Drugs Advisory Committee Meeting for NDA 210645, Waylivra (volanesorsen) injection for the treatment of familial chylomicronemia syndrome, NOCF was used as one of the sensitivity analyses:

Similar planned (prespecified) analyses using different variables, such as slightly different endpoint definitions (e.g. worst maximum pain intensity versus average maximum pain intensity), or imputation methods for missing data (next observation carried backward versus imputation of zero for missing values) did not demonstrate treatment differences.

 Missing values were pre-specified to be imputed using Next Observation Carried Back (NOCB); i.e., if a patient did not complete the questionnaire for several weeks, the next value entered was assumed to have occurred during all intervening (missing) weeks.

 Missing data for any post-baseline visit will be imputed by using Next Observation Carried Back (NOCB) if there is a subsequent score available. Missing data after the last available score of each patient will not be imputed.

in NDA 212157 of Celecoxib Oral Solution for Treatment of acute migraine, the NOCB was used for sensitivity analysis

Headache Pain Freedom at 2 hours - Sensitivity Analysis

To analyze the missing data for the primary endpoint, Dr. Ling performed an analysis analyzing patients who took rescue medications as nonresponders and then also imputing missing data at the 2-hour time point using the next available time point of information (Next Observation Carried Backward (NOCB)) or a worst-case type of imputation (latter not shown in table).

Single imputation methods are generally not recommended for the primary analysis because of the following disadvantages (issues): 

  • Single imputation usually does no provides an unbiased estimate
  • Inferences (tests and confidence intervals) based on the filled-in data can be distorted by bias if the assumptions underlying the imputation method are invalid
  • Statistical precision is overstated because the imputed values are assumed to be true.
  • Single imputation methods risk biasing the standard error downwards by ignoring the uncertainty of imputed values. Therefore, the confidence intervals for the treatment effect calculated using single imputation methods may be too narrow and give an artificial impression of precision that does not really exist.  
  • the single imputation method such as LOCF, NOCB, and BOCF do not reflect MAR (missing at random) data mechanisms.

Further Readings:

Monday, January 04, 2021

Synthetic Control Arm (SCA), External Control, Historical Control

Lately, the term 'synthetic control' or 'synthetic control arm' or SCA, in short, is becoming popular - it is mainly driven by the desire to design more efficient clinical trials that are not traditional, the golden standard RCT (randomized controlled trials) with a concurrent control group. 

In a previous post, I compared historical control versus external control in clinical trials. The subtle difference is mainly in the time element. Historical control is one type of external control, but the reverse is not true. External control can be historical control or contemporaneous control. For example, in a clinical trial to assess the efficacy and safety of the donor lung preserved using ex-vivo lung perfusion (EVLP) technique, the EVLP lung transplantation cohort was compared to a contemporaneous (not concurrent) control cohort that was formed through the matched control from the traditional lung transplantation patients.   

Then what is 'synthetic control' or 'synthetic control arm'?

Synthetic control arm is the use of synthetic data as a control arm in clinical trials. According to an article "Synthetic data in the civil service" in the latest issue of SIGNIFICANCE, synthetic data is defined as "artificially generated data that are modelled on real data, with the same structure and properties as the original data, except that they do not contain any real or specific information about individuals. The goal of synthetic data generation is to create a realistic copy of the real data set, carefully maintaining the nuances of the original data, but without compromising important pieces of personal information."

Synthetic control arm is a control arm generated through existing data resources representing normal patient statistics. Synthetic control arm can serve as a comparator for a single-arm clinical trial or augment the smaller concurrent control group (for example with active:control ratio of 3:1 or 4:1) in RCTs. 

In a presentation by at Harvard Medical School Executive Education Webinar Series,  Mr. Chatterjee presented "Synthetic Control Arms in Clinical Trials and Regulatory Applications" and he defined the 'synthetic control arm' as the following:

In a paper by Thorlund et al "Synthetic and External Controls in Clinical Trials – A Primer for Researchers", they stated that synthetic control arms are external control arms - two terms can be used interchangeably:
External control arms are also called “synthetic” control arms as they are not part of the original concurrent patient sample that would have been randomized into the experimental or the control treatment arms as in a traditional RCT. External controls can take many forms. For example, external control arms can be established using aggregated or pooled data from placebo/control arms in completed RCTs or using RWD (Real World Data) and pharmacoepidemiological methods. Pooled data from historical RCTs can serve as external controls depending on the availability of selected “must have” data, similarity of patients, recency and relevancy of experimental treatments that were tested, availability and similarity of relevant endpoints (eg, operational definitions and assessments), and similarity of other important study procedures that were conducted in these historical trials. It is important to note that using control data from historical RCTs still results in a nonrandomized comparison but has the advantage of standardized data collection in a trial setting and patients who enroll in clinical trials may have more similar characteristics than those who do not.

However, I think that there are subtle differences between these two terms. With 'synthetic' control arms, the term 'synthetic' implies there are some selection, manipulation, derivation, matching, pooling, borrowing from the source data. Just like the meta-analysis is also called research synthesis and requires the statistical approaches to combine the results from multiple scientific studies, the 'synthetic' control also requires the use of statistical approaches to process the data from multiple sources to form a control group to replace the concurrent control in traditional RCT clinical trials. 

The source data for constructing synthetic control can be the data from previous RCT clinical trials, real-world data, registry data, data from natural history studies, electronic health records, ... The source data must be the subject-level data, not the summary or aggregate data. 

ICH E10 "CHOICE OF CONTROL GROUP AND RELATED ISSUES IN CLINICAL TRIALS" included "External Control (including Historical Control)" as one of the options as the control groups in clinical trials. The external control here is not the same as synthetic control. 

1.3.5 External Control (Including Historical Control)
An externally controlled trial compares a group of subjects receiving the test treatment with a group of patients external to the study, rather than to an internal control group consisting of patients from the same population assigned to a different treatment. The external control can be a group of patients treated at an earlier time (historical control) or a group treated during the same time period but in another setting. The external control may be defined (a specific group of patients) or nondefined (a comparator group based on general medical knowledge of outcome). Use of this latter comparator is particularly treacherous (such trials are usually considered uncontrolled) because general impressions are so often inaccurate. So-called baseline controlled studies, in which subjects' status on therapy is compared with status before therapy (e.g., blood pressure, tumor size), have no internal control and are thus uncontrolled or externally controlled.  

How to Create a Synthetic Control Arm? 

The first step of creating a synthetic control arm is to harmonize the source data. The data from different sources or from different clinical trials should be standardized so that they can be used for the synthesis process. 

Various statistical approaches can be used to create a synthetic control arm. In an audiobook on synthetic control arms by Cytel, propensity scoring and Bayesian Dynamic Borrowing methods were discussed. 

The synthetic control arm can be considered as an approach of 'borrowing control' - i.e., some controls are borrowed from historical data. There are numerous options for borrowing controls: 

  • Pooling: adds historical controls to randomized controls 
  • Performance criterion: uses historical data to define performance criterion for current, treated-only trial to beat 
  • Test then pool: test if controls sufficiently similar for pooling 
  • Power priors: historical control discounted when added to randomized controls
  • Hierarchical modeling: variation between current vs. historical data is modeled in Bayesian fashion 

In the article by Thorlund et al, the pros and cons of different methods for generating synthetic control arms were discussed. 

In Mr Chatterjee presentation, "Synthetic Control Arms in Clinical Trials and Regulatory Applications", there is a diagram to describe the process for creating a synthetic control arm. 

Even though the synthetic control arms, the use of real-world data, conducting the single-arm clinical trials are very appealing, the challenges are ahead and the regulatory acceptance is uncertain. There may be limited use in special cases (such as ultra-rare diseases, pediatric clinical trials) and for post-marketing activities (such as label expansion, label modification, post-marketing studies), but not in prime time to replace the concurrent control in traditional RCTs. 

In an article at "Synthetic control arms can save time and money in clinical trials", 

Even with the FDA making the use of real-world data a strategic priority, synthetic control arms can’t be used across the board to replace control arms. Synthetic control arms require that the disease is predictable (think idiopathic pulmonary fibrosis) and that its standard of care is well-defined and stable. That certainly isn’t the case for every disease.

It’s also important to consider that even when information is available from real-world data sources, it may be difficult to extract or of low quality. Routinely captured health care data, such as electronic health records, are typically siloed, fragmented, and unstructured. They are also often incomplete and difficult to access. New tools and methodologies are needed to consolidate, organize, and structure real-world data to generate research-grade evidence and ensure that confounding variables are accounted for in analyses. Analytic techniques such as natural language processing and machine learning will be needed to extract relevant information from structured and unstructured data.

The same view is also expressed in a Pink Sheet article "External Control Arms: Better Than Single-Arm Studies But No Replacement For Randomization".

Synthetic control group derived from historical clinical trial data could augment smaller randomized trials and yield better information than single-arm studies, but this approach should not be viewed as a substitute for randomized trials where feasible


Monday, December 28, 2020

120-Day Safety Update or 4-Month Safety Update - The Requirement for NDA/BLA

After the new drug application (NDA) or biological license application (BLA) is submitted by the sponsor and is accepted by FDA, FDA reviewers will take 10 months (regular review) or 6 months (expedited review) to review the submission package and issue a decision on or before the decision date (PDUFA date). FDA reviewers will evaluate marketing applications for efficacy and safety and consider benefit and risk and will expect to receive a complete application at the time of filing (exclusive of the 120-day safety update).

It is very possible that during the 6-10 month review time, the sponsor will have additional data to supplement the already submitted NDA/BLA package. The regulatory pathway for providing additional data to the FDA is through so-called ‘120-Day Safety Update’, also referred to as ‘4-Month Safety Update, 4MSU).

The ‘120-Day Safety Update’ or ‘4-Month Safety Update” is specified as a requirement in Code of Federal Regulations - 21CFR314.50.


The 120-Day Safety Update contains any new safety information learned about the drug that may reasonably affect the statement of contraindications, warnings, precautions, and adverse reactions in the draft drug labeling.

The report must be received by the FDA within 120 days of drug approval submission (receipt by the FDA of the New Drug Application (NDA), comprising the CTD/Integrated Summary Report) to avoid triggering an extension of the review clock.

The 120-Day Safety Update Report is mandated for submission to the FDA 120 days after submission of the NDA/BLA, and is intended to provide a summary update of any new safety data gathered by the sponsor since the data cut-off for the NDA submission documents, which could have been as far back as 6 months prior to the NDA submission date. In effect, the 120-Day Safety Update report could represent almost 1 year’s worth of new safety data, which needs to be reviewed by the authorities to ensure there has been no change in the product’s recorded safety profile. This is particularly important for medications intended for long-term treatment.

The 120-Day Safety Update is focused on additional safety data. If additional data is collected for efficacy variables, the efficacy information can also be included - but in general, it is for the summary propose and there is no inferential statistics needed. 

The data to be included in the 120 Day Safety Update can include:

  • The long-term follow-up data from the on-going clinical trials
  • Open-label extension studies with patients rolled over from the pivotal studies (usually the double-blinded controlled studies)
  • Additional data from later time points and from newly enrolled patients
  • Newly initiated clinical trials

Depending on the type of data to be included in the 120 Day Safety Update, the submission package could be just a written report or a full submission package (including the report; post-text tables, listings, figures; the data sets; define documents; SDRG/ADRG, etc.).

There are a lot of examples of 120 Day Safety Update from market applications. Here are some examples:

Briefing Document for Advisory Committee Meeting on Novo Nordisk’s Insulin degludec/liraglutide (IDegLira) for Treatment to Improve Glycemic Control in Adults with Type 2 Diabetes Mellitus. The NDA submission was based on two pivotal trials. Two pivotal trials (Trial 3697 in patients inadequately controlled on OAD treatment and Trial 3912 in patients inadequately controlled on basal insulin treatment) were designed to assess the contribution of the individual components of the combination to its primary efficacy effect (i.e., overall glycemic control). Additional data from other ongoing studies and the studies initiated after the data cut for NDA submission were submitted to the NDA as ‘120 Day Safety Update’:

The NDA submitted to the FDA had a data cut-off of 31 March 2015. Additional blinded safety data from two phase 3 trials that were ongoing at the time of NDA submission (Trials 4119 and 4056) as well as from a trial that was subsequently initiated (Trial 4185) was submitted to the FDA in a 120 Day Safety Update with a cut-off date of 30 September 2015. A brief overview of the ongoing trials included in the 120-day safety update is provided in Table 1–1. The 120-day safety update included available blinded safety data from these trials on deaths, other serious adverse events, pregnancies (including updates for pregnancies reported as ongoing in the NDA) and adverse events leading to withdrawal. 

Sunovion Pharmaceuticals NDA of Latuda for treatment of major depressive episodes associated with bipolarI disorder in pediatric patients aged 10 and older. The NDA submission was mainly based on the pivotal study (Study D1050326). Subjects who completed Study D0150326 was recruited into an open-label study (D1050302). As a 120-day safety date, the date from the open label study was submitted to support the NDA.

Study D1050302 is a 104-week open-label trial designed to assess the long-term safety profile of lurasidone (dosed 20-80 mg per day) in pediatric patients recruited from the pediatric schizophrenia (Study D1050301), bipolar depression (Study D1050326), and autism trials. This study was scheduled for completion last December, 2017. The Applicant submitted preliminary data for 619 patients participating in this trial with a cutoff date of October, 2016. Additionally, the 120-day safety update submitted with this application focused on the available safety data from 305 patients recruited from Study D1050326 with a cutoff date of May, 2017. It should be noted that although the final report for Study D1050302 has not been submitted for review, the Applicant presented acceptable long-term data to make an approval determination for this sNDA, including lurasidone exposure of 153 patients for ≥ 52 weeks.

Clinical Review for BLA for Mepolizumab for Add-on maintenance treatment of severe asthma. The data from long-term open-label studies were not available at the time of BLA preparation but was submitted to FDA as a 120-Day Safety Update.

This safety review primarily relies on data from three placebo-controlled studies in a severe asthma population: MEA112997 (Study 97), MEA115588 (Study 88) and MEA115575 (Study 75) as these studies most closely approximate the patient population to receive mepolizumab in the clinical practice. Within this review, the pooled database for these studies is referred to as the Placebo-Controlled Severe Asthma Studies (PCSA). Longer term safety data are provided by two open-label studies, MEA115666 (Study 66), MEA115661 (Study 61). These studies were ongoing at the time of the BLA submission with updated data provided to the Division in a 120-day safety update. The data from this safety update used a cutoff date of October 27, 2014 and provides cumulative review of the data for the studies ongoing at the time of BLA submission25 .

Tuesday, December 08, 2020

COA (Clinical Outcome Assessment): PRO, ClinRO, PerfRO, and ObsRO

Clinical Outcome Assessment (COA) has triggered multiple acronyms: PRO, ClinRO, PerfRO, and ObsRO

According to FDA's website, these acronyms are defined as the following: 

PRO - patient-reported outcome
A type of clinical outcome assessment. A measurement based on a report that comes directly from the patient (i.e., study subject) about the status of a patient’s health condition without amendment or interpretation of the patient’s response by a clinician or anyone else. A PRO can be measured by self-report or by interview provided that the interviewer records only the patient’s response. Symptoms or other unobservable concepts known only to the patient can only be measured by PRO measures. PROs can also assess the patient perspective on functioning or activities that may also be observable by others. PRO measures include:
  • Rating scales (e.g., numeric rating scale of pain intensity or Minnesota Living with Heart Failure Questionnaire for assessing heart failure)
  • Counts of events (e.g., patient-completed log of emesis episodes or micturition episodes)
Specifically for PRO, FDA has a guidance "Patient-Reported Outcome Measures: Use in Medical Product Development to Support Labeling Claims". PRO can be further separated into generic (such as SF-36, EQ-5D, ...) and disease-specific PROs (SGRQ for COPD, PAH-SYMPACT for pulmonary arterial hypertension, ...). 

ObsRO - observer-reported outcome

A type of . A  based on a report of observable signs, events or behaviors related to a patient’s health condition by someone other than the patient or a health professional. Generally, ObsROs are reported by a parent, caregiver, or someone who observes the patient in daily life and are particularly useful for patients who cannot report for themselves (e.g., infants or individuals who are cognitively impaired). An  measure does not include medical judgment or interpretation. ObsRO measures include:

  • Rating scales, such as:
    • Acute Otitis Media Severity of Symptoms scale (AOM-SOS), a measure used to assess signs and behaviors related to acute otitis media in infants
    • Face, Legs, Activity, Cry, Consolability scale (FLACC), a measure used to assess signs and behaviors related to pain
  • Counts of events (e.g., observer-completed log of seizure episodes)

ObsRO is often used in rare diseases, in pediatric diseases, or in diseases that the patients may lose self-control and patients can not detect the signs/symptoms on their own (such as seizure, stroke).  For patients who cannot respond for themselves (e.g., infants or cognitively impaired), observer reports should include only those events or behaviors that can be observed. As an example, observers cannot validly report an infant’s pain intensity (a symptom) but can report infant behavior thought to be caused by pain (e.g., crying). For example, in the assessment of a child’s functioning in the classroom, the teacher is the most appropriate observer. Examples of ObsROs include a parent report of a child’s vomiting episodes or a report of wincing thought to be the result of pain in patients who are unable to report for themselves.

Additional examples are OsRO-Celiac Disease Daily Symptom Diary (ObsRO-CDSD©), the Pediatric Quality of Life Inventory™,  the Edmonton Symptom Assessment System Revised (ESAS-r), ....

ClinRO - clinician-reported outcome

A type of . A  based on a report that comes from a trained health-care professional after observation of a patient’s health condition. Most  measures involve a clinical judgment or interpretation of the observable signs, behaviors, or other manifestations related to a disease or condition. ClinRO measures cannot directly assess symptoms that are known only to the patient. ClinRO measures include:

  • Reports of particular clinical findings (e.g., presence of a skin lesion or swollen lymph nodes) or clinical events (stroke, heart attack, death, hospitalization for a particular cause), which can be based on clinical observations together with  data, such as electrocardiogram (ECG) and creatine phosphokinase (CPK) results supporting a myocardial infarction
  • Rating scales, such as:
    • Psoriasis Area and Severity Index (PASI) for  of severity and extent of a patient’s psoriasis
    • Hamilton Depression Rating Scale (HAM-D) for  of depression

The majority of neurological assessment tools are falling into this category. additional examples are INCAT (Inflammatory Neuropathy Cause and Treatment), Guillian-Barre Syndrome disability score, MRC sum score...

PerfRO - performance outcome

A type of clinical outcome assessment. A  based on standardized task(s) actively undertaken by a patient according to a set of instructions. A  assessment may be administered by an appropriately trained individual or completed by the patient independently. PerfO assessments include:

  • Measures of gait speed (e.g., timed 25 foot walk test using a stopwatch or using sensors on ankles)
  • Measures of memory (e.g., word recall test) 
For example, the frequently used outcome measures such as the six-minute walking test (6MWT), cardio-pulmonary exercise test (CPET), Grip strength, ... are falling into PerfRO. 

FDA created a division "Division of Clinical Outcome Assessment (CDOA)" with a mission of Integrating the patient voice into drug development through COA endpoints that are meaningful to patients, valid, reliable and responsive to treatment."

For a scale that has not bee validated before and is intended to be used as the primary efficacy outcome measure in a clinical development program, CDER has two pathways for reviewing COAs:
  • The CDER COA Qualification Program or
  • Under an individual drug development program

Sunday, December 06, 2020

Multiple Imputation: Imputation Model versus Analysis Model

Multiple imputation has become more and more popular in handling the missing data in clinical trials. Multiple imputation inference involves three distinct phases:

  • The missing data are filled in m times to generate m complete data sets. This step is through the imputation model and can be implemented using SAS Proc MI
  • The m complete data sets are analyzed by using standard procedures. This step is through the analysis model – depending on nature of the outcome variable, the analysis model can be ANCOVA (analysis of covariance), MMRM (mixed model repeated measures), Logistic regression, GEE (generalized estimating equation), GENMOD (generalized linear model),… The analysis model is also the primary model for analyzing the corresponding outcome variable.
  • The results from the m complete data sets are combined for the inference. This step is using Robin’s rule and can be implemented with SAS Proc MIANALYZE

For both the imputation model and the analysis model will need to include a list of explanatory or independent variables, but for different purposes. The list of explanatory or independent variables in the imputation model is to impute the missing values; the list of explanatory or independent variables in the analysis model are covariates as part of the standard statistical models. Here are some comparisons for the variables used in the imputation model and analysis model:

  • The covariates included in the analysis model must also be included in imputation model
  • The imputation model can include additional auxiliary variables including those variables that are not used as covariates in the analysis model
  • The number of variables used in imputation model is greater than or equal to the number of variables in analysis model
  • The imputation model can include variables measured after the randomization (such as secondary outcomes, concomitant medication use, compliance data). However, for analysis model, “variables measured after randomisation and so potentially affected by the treatment should not be included as covariates in the primary analysis.”
  • For longitudinal data or repeated measures, the outcome measures at early time points will be included in the imputation model.
  • If the variables used in the analysis model are transformed, the transformed variable should also be used in the imputation model
  • If the interaction term is used in the analysis model, it should also be included in the imputation model - this can make the imputation model pretty complicated though. 

In many publications, multiple imputation was stated as the method for handling the missing data, however, the details about the imputation model (i.e., which variables are included in the imputation model) were not usually described. 

While there is no clear guidance about the variables included in the imputation model, it is important to pre-specify the list of variables included in the imputation model especially if the auxiliary variables or variables not included in the analysis model. 

Below are some excerpts from the literature about the imputation model and analysis model.


Imputation Model, Analytic Model and Compatibility :

When developing your imputation model, it is important to assess if your imputation model is “congenial” or consistent with your analytic model. Consistency means that your imputation model includes (at the very least) the same variables that are in your analytic or estimation model. This includes any transformations to variables that will be needed to assess your hypothesis of interest. This can include log transformations, interaction terms, or recodes of a continuous variable into a categorical form, if that is how it will be used in later analysis. The reason for this relates back to the earlier comments about the purpose of multiple imputation. Since we are trying to reproduce the proper variance/covariance matrix for estimation, all relationships between our analytic variables should be represented and estimated simultaneously. Otherwise, you are imputing values assuming they have a correlation of zero with the variables you did not include in your imputation model. This would result in underestimating the association between parameters of interest in your analysis and a loss of power to detect properties of your data that may be of interest such as non-linearities and statistical interactions. 

Auxiliary variables are variables in your data set that are either correlated with a missing variable(s) (the recommendation is r > 0.4) or are believed to be associated with missingness. These are factors that are not of particular interest in your analytic model , but they are added to the imputation model to increase power and/or to help make the assumption of MAR more plausible. These variables have been found to improve the quality of imputed values generate from multiple imputation. Moreover, research has demonstrated their particular importance when imputing a dependent variable and/or when you have variables with a high proportion of missing information (Johnson and Young, 2011; Young and Johnson, 2010; Enders , 2010).

You may a priori know of several variables you believe would make good auxiliary variables based on your knowledge of the data and subject matter. Additionally, a good review of the literature can often help identify them as well. However, if your not sure what variables in the data would be potential candidates (this is often the case when conducting secondary data analysis), you can uses some simple methods to help identify potential candidates.

In a presentation of “multiple imputations” by Adrienne D. Woods

Which variables should you include as predictors in the imputation model?

  • Any variables you plan to use in later analyses (including controls)
  • General advice: use as many as possible (could get unwieldy!)
  • Although, some (i.e., Kline, 2005; Hardt, Herke, & Leonhart, 2012) believe that this introduces more imprecision, especially if the auxiliary variable explains less than 10% of the variance in missingness on Y… thoughts?
  • Know your analysis model beforehand and include at least all analysis variables in imputation model (including interaction terms)

FDA’s Statistical Review for Vantrela (hydrocodone bitartrate) extended-release tablets in Management of pain severe

Analysis model:

"The primary efficacy endpoint of trial 3103 was change from baseline to week 12 in the weekly average of worst pain intensity (WPI). The primary analysis was ANCOVA model with baseline WPI, randomized treatment, opioid status, and center as covariates. The intent-to-treat analysis population, defined as all randomized patients, was used for the primary efficacy analysis."

Imputation model:

"The applicant performed multiple imputation on the week 12 missing data for the primary analysis. The imputation model included randomized treatment, opioid status, baseline and postbaseline WPI values while subjects in the active-drug treatment group who discontinued study drug because of an adverse event, were treated as if they were in the placebo group and their missing data were imputed based on the observed placebo subjects' data."

FDA's Statistical Review for EUCRISA™ (crisaborole) topical ointment, 2% for Atopic Dermatitis mentioned the imputation model for missing dichotomized outcome variable. 

The protocol specified the primary imputation method to be the multiple imputation (MI) approach. For each treatment arm separately, missing data was imputed using the Markov Chain Monte Carlo (MCMC) method. The protocol specified the following two sensitivity analyses for the handling of missing data:

· Repeated-measures logistic regression model (GEE), with dichotomized ISGA success as the dependent variable and treatment, analysis center, and visit (i.e., Days 8, 15, 22, and 29) as independent factors. In this analysis, data from all post-baseline visits will be included with no imputation for missing data.

· Model-based multiple imputation method to impute missing data for the dichotomized ISGA data. The imputation model (i.e., logistic regression) will include treatment and analysis center.

Kaifeng Lu et al (2010) Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data pointed out the issue if the analysis model is different from the imputation model. 

Despite its conceptual simplicity and flexibility, the above MI procedure is not valid for the analysis of dichotomized responses because Rubin’s variance estimator is biased when the analysis model is different from the imputation model (Meng, 1994; Robins and Wang, 2000). This is true even when the imputation and analysis models are compatible, e.g. when the treatment is the only effect in the logistic regression model.

Ian R. White  et al (2012) Including all individuals is not enough: lessons for intention-to-treat analysis

In some cases, an MI procedure can be improved by including in the imputation model ‘auxiliary variables’ that are not in the analysis model [36, Chapter 4]: auxiliary variables in a randomised trial might be secondary outcomes or compliance summaries. MI then produces estimates of the treatment effect that are genuinely different from a likelihoodbased analysis, by incorporating information on individuals with missing outcome but observed values of auxiliary variables. However, in our experience, the contribution to such an analysis of individuals missing the outcome of interest is moderate unless correlations between the outcome and one or more auxiliary variables are substantial [37].

Michael Spratt et al (2010) Strategies for Multiple Imputation in Longitudinal Studies

Where there are nontrivial amounts of missing data in covariates, both preliminary analyses and imputation models will become more complex. An MAR assumption may often become more plausible after the inclusion in the imputation model of additional variables that are not in our analysis model (because they are on the causal pathway, for example). Thus, multiple imputation models should typically be more complex than the analysis model. Including variables that are not related to the variable being imputed in the imputation models may slightly decrease efficiency but should not cause bias (29, 31). Model diagnostics should be used to highlight any implausibility in the imputed values. For example, the distributions of observed and imputed data should be compared and the plausibility of any differences examined. Imputation models should also preserve the structure of the analysis model (32). For example, where the substantive analysis exploits the hierarchical nature of longitudinal data (e.g., using a multilevel model), the imputation model should be similarly structured. Here, the longitudinal nature of the data allowed us to include variables (previous wheezing) that predicted the values of the variable with the most missing data (wheeze at 81 months) in imputation models.

Jochen Hard et al (2012) Auxiliary variables in multiple imputation in regression with missing X: a warning against including too many in small sample research

  • An additional advantage of MI over CC (complete-case analysis) is the possibility of including information from auxiliary variables into the imputation model. Auxiliary variables are variables within the original data that are not included in the analysis, but are correlated to the variables of interest or help to keep the missing process random [MAR: 1]. Little [6] has calculated the amount of decrease in variance of a regression coefficient Y on X1 when a covariate X2 is added that has no missing data. White and Carlin [7] have extended this proof to more than one covariate. In practice however, it is likely that auxiliary variables themselves will have missing data.

EMA Guideline on Missing Data in Confirmatory Clinical Trials mentioned the multiple imputation as an approach to handle the missing data with MAR assumption, however, it did not mention anything about the imputation model.   

Panel on Handling Missing Data in Clinical Trials; National Research Council  (2010) The Prevention and Treatment of Missing Data in Clinical Trials

Multiple imputation methods address concerns about (b) “simple imputation is generally not true because the methods do not always yield conservative effect estimators, and standard errors and confidence interval widths can be underestimated when uncertainty about the imputation process is neglected.”  and enable the use of large amounts of auxiliary information.

An important advantage of multiple imputation in the clinical trial setting is that auxiliary variables that are not included in the final analysis model can be used in the imputation model. For example, consider a longitudinal study of HIV, for which the primary outcome Y is longitudinal CD4 count and that some CD4 counts are missing. Further, assume the presence of auxiliary information V in the form of longitudinal viral load. If V is not included in the model, the MAR condition requires the analysis to assume that, conditional on observed CD4 history, missing outcome data are unrelated to the CD4 count that would have been measured; this assumption may be unrealistic. However, if the investigator can confidently specify the relationship between CD4 count and viral load (e.g., based on knowledge of disease progression dynamics) and if viral load values are observed for all cases, then MAR implies that the predictive distribution of missing CD4 counts given the observed CD4 counts and viral load values is the same for cases with CD4 missing as for cases with CD4 observed, which may be a much more acceptable assumption.

Meyer et al (2020) Statistical Issues and Recommendations for Clinical Trials Conducted During the COVID-19 Pandemic

Multiple imputation (MI) methodology (Rubin, 1987) may be helpful in this respect as it allows inclusion of auxiliary variables (both pre- and post-randomization) in the imputation model while utilizing the previously planned analysis model. Multiple imputation with auxiliary variables may be used for various types of endpoints, including continuous, binary, count, and time-to-event and coupled with various inferential methods in the analysis step.

Thomas R Sullivan et al (2018) Should multiple imputation be the method of choice for handling missing data in randomized trials?

In the first stage of MI, multiple values (m > 1) for each missing observation are independently simulated from an imputation model. For missing data restricted to the outcome, the imputation model would typically regress observed values of Y on X and T. Additional auxiliary variables that are not in the analysis model can also be added to the imputation model to improve the prediction of missing values.

In applying MI, the repeated measurements of the outcome are usually treated as distinct variables in the imputation model. Where interest lies in the treatment effect at the final time point, the analysis model need not include the intermediate outcome measures; following imputation a comparison of final time point results is sufficient. In this case, the intermediate measures operate as auxiliary variables, assisting with the prediction of missing values at the final time point and making the MAR assumption more plausible. Other auxiliary variables, for instance measures of compliance or related outcomes, can also be added to the imputation model as required. If data are collected but more likely to be missing following treatment discontinuation, an indicator variable for discontinuation may also be valuable as an auxiliary variable. The ability to incorporate auxiliary variables, both for univariate and multivariate outcomes, is considered one of the key strengths of MI.

Thus in settings where MI is adopted, we recommend imputing by randomized group; compared to MI overall, this approach offers greater robustness at little cost. The approach is also consistent with general recommendations for over- rather than under-specifying imputation models. It should be noted that imputing by group only protects against bias in estimating the ATE if effect modifiers are included in the imputation model.

One of the strengths of MI is its ability to easily incorporate variables of different types (e.g. continuous, binary) in the imputation model, whether for univariate or multivariate data. An added benefit of including all outcomes in a single imputation model is that associations between related outcomes can aid imputation. Another appealing feature of MI is its ability to be implemented under an assumption that data are MNAR. This property makes MI well suited to undertaking sensitivity analyses around a primary assumption that data are MAR, and as a primary method of analysis in settings where data are believed to be MNAR. One such setting is RCTs where participants cannot followed up after discontinuing treatment. If all observed data are ‘on-treatment’, a MAR assumption entails estimating the effect of treatment had all participants remained on their assigned treatment.27 However, for a de facto type estimand (such as ITT), it may be more appropriate to assume that data are MNAR. In this situation, reference based sensitivity analyses have been proposed, which at present require the use of MI.2

Interaction terms are not suggested.

Although the bias of MI overall could be eliminated by including the interaction term in the imputation model (results not shown), this may not be an obvious strategy if subgroup analyses are not of interest.

Simon Grund et al (2018) Multiple Imputation of Missing Data for Multilevel Models: Simulations and Recommendations

A crucial point in the application of MI to multilevel data is that the imputation model not only includes all relevant variables, but also that it “matches” the model of interest (i.e., the substantive analysis model; see Meng, 1994; Schafer, 2003). In other words, the imputation model must capture the relevant aspects of the analysis model, making the imputation model at least as general as (or more general than) the analysis model. If the imputation model is more restrictive than the analysis
model, then imputations are generated under a simplified set of assumptions, and the results of subsequent analyses may be misleading.

Protocol for: Hatemi G, Mahr A, Ishigatsubo Y, et al. Trial of apremilast for oral ulcers in Behçet’s syndrome. N Engl J Med 2019;381:1918-28. DOI: 10.1056/NEJMoa1816594


Sunday, November 29, 2020

Handling of Missing Data: Comparison of MMRM (mixed model repeated measures) versus MI (multiple imputation)

Longitudinal study has become one of the most commonly adopted designs in clinical trials. Since the outcome measures are performed at various visits, it is usually the case that for some subjects in the study, the outcome measures will not be available at some visits (for example after subjects drop out from the study or lost-to-follow-up) - this is where the missing data issue arises. If the outcome measure is a continuous variable, the missing data issues can be handled implicitly through using the mixed-effects repeated measure (MMRM) models or explicitly through multiple imputations (MI).

Both MMRM and MI methods are based on the assumption of missing at random (MAR) and are model-based approaches suggested by EMA's Guideline on Missing Data in Confirmatory Clinical Trials and US National Research Council: The Prevention and Treatment of Missing Data in Clinical Trials. US FDA has not issued any guidance on handling the missing data in clinical trials, but generally follows the guidelines from the National Research Council. 

In terms of MMRM and MI, which one should be the primary method for handling the missing data? For a long time, it seems that in the US, the MMRM is the preferred method in handling the missing data and analyzing the longitudinal data with continuous outcome measures. The MI methods are generally used as sensitivity analyses to check the robustness of the primary analyses against the deviation from the MAR assumption. This can be observed by the article by Dr. Siddiqui in FDA "MMRM versus MI in Dealing with Missing Data - a Comparison Based on 25 NDA data sets" and many NDA / BLA reviews (listed below). 

FDA Statistical Review for NDA 210655 in the indication of Schizophrenia:
"The primary analysis was conducted on the change from baseline in the total PANSS score at Day 57 (primary time point) based on the ITT population. A mixed-effects model for repeated measures (MMRM) was used with treatment, visit, interaction of treatment and visit as fixed effects and the baseline total PANSS score as a covariate. Data from Days 15, 29, 43, and 57 were used. The unstructured covariance matrix was be used to model the within-subject variance-covariance errors."

"In addition to the model-based missing data approach of the MMRM model, the primary efficacy analysis was also analyzed using a pattern mixture model (PMM) and a multiple imputation approach as sensitivity analyses. "

FDA BLA 761037 Kevzara (sarilumab) in Treatment of rheumatoid arthritis
"The continuous HAQ-DI change from baseline at Week 16 was analyzed with a mixed model for repeated measures (MMRM). The repeated-measures analysis was based on the restricted maximum likelihood method assuming an unstructured covariance structure to model the within-subject errors. The model, including treatment, region, prior biologic use, visit (all visits from week 2 to week 16), and treatment-by-visit interaction as fixed effects and baseline as a covariate, was used to test the difference between each active treatment group versus placebo in the change from baseline in HAQ-DI at Week 16. The data collected after treatment discontinuation or rescue were set to missing. Therefore, the MMRM analysis assumed a missing-at-random (MAR) mechanism for missing data due to dropout and post-rescue data."
FDA NDA 203313/203314S-2 /S-3Tresiba;Ryzodeg 70/30Glycemic Control in Patients with Diabetes
The applicant used a mixed effect model for repeated measure (MMRM) to assess the efficacy of IDegAsp compared with IDet. The MMRM model included treatment, sex, region, age group and visits as factors and baseline as covariate, and interactions between visits and all factors and covariate. An unstructured covariance matrix was utilized for model fitting.

Multiple imputation was performed as sensitivity analysis
SNDA for Merck's Dulera in the treatment of asthma (2019)

"Missing Data Handling and Sensitivity Analyses The primary analysis incorporated a control-based multiple imputation of missing data. Missing data for subjects who discontinued treatment early were estimated using the MF group; that is, the change from baseline AM post-dose ppFEV1 in patients who discontinued treatment and missed study visits was assumed to be similar to the change from baseline in patients who continued study visits through Week 12 in the MF treatment group. The dataset was first multiply imputed to have monotone missing patterns, then for each visit, a regression method was used to impute for missing data on both study drug arm and the control arm based on trend from the control arm. After applying the control-based multiple imputation, the cLDA analysis was performed. MF/F 100/10 mcg BID was considered superior to MF 100 mcg BID with a p-value less than 0.05. "

EMA seems to have a different opinion about missing data handling using MMRM or MI. On several occasions, we have heard that EMA prefers the MI approach in handling the missing data especially the reference-baseline multiple imputation. They are moving towards developing the reference-based multiple imputation into the new standard missing data approach. 

Here is a table summarizing some comparisons between the MMRM and MI in handling the missing data. 




Missing data mechanism

MAR (missing at random)

Missing data imputation

Not imputed for individual missing values

But missing data is implicitly imputed

Individual missing values are explicitly imputed

# of steps for calculations

One step

At least three steps:

Imputation model to create multiple data sets with missing values filled in

Analysis model to analyze each imputed data set

Using Robin’s rule to combine results for inference

Analysis Model

Mixed model with Maximum likelihood-based method

Analysis of Covariance or Mixed model using maximum likelihood-based method

Data points used in analyses

Utilized all observed data points from all visits

Usually, with ANCOVA, only the data points for the corresponding visits (with imputed values) are used.

SAS procedure(s)

Proc Mixed

Imputation model: Proc MI

Analysis model: Proc Mixed, Proc GLM, Proc Genmod,…

Robin’s rule: Proc MIANALYZE


The two approaches will be approximately equivalent, provided the variables used in the imputation model are the same as those included in the analysis model, and conditionals are accommodated by a single joint model. In such settings, MI essentially provides an approximation to the observed likelihood analysis. If an infinite number of imputations could be performed, then the two approaches would be equivalent. In practice, the level of equivalence will depend on the number of imputations due to the Monte Carlo (simulation) sampling variability of the imputation process (described in more detail below), thus will be stronger for a larger number of imputations.

Auxiliary variables

Can not be used

Auxiliary variables can be used in the imputation model to improve the accuracy of the missing data prediction

Information observed post-randomization

Can not be included in the MMRM model

Can be included in the imputation model to improve the accuracy of the missing data prediction and can’t be included in the analysis model (MI approach allows the differences in the covariates used in the imputation model and in analysis model

Justification of MAR assumption

Not available through MMRM model

Justification of MAR assumption can be performed through the tipping point approach or delta-based imputation

Handling the MNAR (missing not at random)

Not directly available through MMRM

Can be performed through PMM (pattern mixed model), reference-based or control-based multiple imputation

For studies with only one post-baseline measure

Not appropriate

Appropriate to use MI to impute the missing data and then run analysis of covariance model as the analysis model

For outcome measures that are not continuous variables

Like MMRM, there are statistical approaches that handle missing data without employing explicit imputation. As mentioned in the EMA guideline “For categorical responses and count data, the so-called marginal (e.g. generalized estimating equations (GEE)) and random-effects (e.g. generalized linear mixed models (GLMM)) approaches are in use. Likelihood-based methods (MMRM and GLMM) and some extended GEE (i.e. weighted GEE) models are applicable under MCAR and MAR assumptions.”

MI approach can be easily applied to the outcome measures that are categorical responses or count data with missing data. The analysis model may need to be PROC Logistics; PROC GLIMMIX, PROC NLMIXED, or


Preferred by regulatory agencies


but with multiple imputation approaches as sensitivity analyses (for example, reference-based MI, PMM, tipping point)