Adaptive design clinical trials - now comes the FDA's draft guidance

For last several years, 'adaptive design' and 'adaptive clinical trials' are hot topics in biostatistics and clinical trial fields. The industry seems to think that 'adaptive design' is the solution for lengthy and costly drug development program. There are workshops, symposiums, many publications and books about the adaptive design in last several years. However, in reality, many examples, case studies are post hoc and  based on the simulation from the historical non-adaptive clinical trial data ("had we implemented the adaptive design, we would have saved xxx time/cost/sample size..."). Recently we do see some implementation of adaptive designs in real clinical trials, but

these are mainly in early stage studies.

This month, FDA issues its guidance for industry on "adaptive design clinical trials for drugs and biologicals". This lengthy guidance contains plenty of references (perhaps with most references for a FDA guidance). Overall tone of this guidance seems to advise the sponsors be cautious in adopting the adaptive design especially those not well established designs. If you don't want to read the entire guidance, slides by two of the FDA working group members summarize the key points in the guidance.

Ahead of FDA, EU has already issued its guidance on adaptive design (or flexible design) . Their opinions have been laid out in EMEA’s REFLECTION PAPER ON METHODOLOGICAL ISSUES IN CONFIRMATORY CLINICAL TRIALS WITH FLEXIBLE DESIGN AND ANALYSIS PLAN issued in 2006. 

There was also a well publicized workshop on adaptive design in 2006 in US. However, we have to wait for four years to see FDA's draft guidance. For documentation, all materials from 2006 workshop are now on the web.

Some of the topics are on technical sides, but there are also talks about the perspectives from FDA regarding the adaptive designs.

• Dr Robert Temple from FDA regarding “Myth Busting – Clinical”
• Dr Robert O’Neill from FDA regarding “Myth Busting – Statistical”

SAS procedures for group sequential design

Group sequential design has been proposed for a while perhaps since Pocock's paper in 1977. At least for last two decades, the group sequential design and interim analysis have been used in may major clinical trials.

Several years ago, I tried to purchase Cytel's EAST program. But I gave up due to the price and the infrequent use in day-to-day practices.

Now I am pretty happy to know that SAS version 9.2 includes two new procedures for group sequential design. Proc SEQDESIGN allows to calculate the boundaries and Proc SEQTEST allows to perform the tests during the interim analysis whether or not the boundaries have been reached. Proc SEQTEST can also be used to calculate the conditional power (Probability of observing a significant result at full information, given the current data and the specified alternative under the statistical design ) and predictive power.


Reference readings:

• Yang Yuan "Group Seuqential Analysis Using the New SEQDESIGN and SEQTEST procedures"

• Schulz and Grimes (2005) Multiplicity in randomized trials II: subgroup and interim analyses. Lancet

Cholesterol Drug Approved for People Without High Cholesterol

Are you willing to take Statin medication if you have normal or mild elevation of LDL-Cholesterol level, but with elevated CRP?

Last week, FDA approved new indication for Crestor -  one of the Statin class cholesterol lowering drug. This new indication has nothing to do with cholesterol level, but it is based on the CRP (c-reaction protein). Wall Street Jounal Health Blog had an article titled "Cholesterol drug approved for people without high cholesterol". It seems to me that we have now invented another disease - perhpas hyperCRP (instead of hypercholesterolemia). American are now perhaps inventing more diseases/indications than ever before.  In the end, both cholesterol level and CRP level are surrogate endpoint. While the relationship between high cholesterol level and the major cardiovascular events (mortality, MI, stroke,...) has been generally recognized, the relationship between high CRP and the major cardiovascular events mainly relies on one study - JUPITER study -  a study stopped early for efficacy.When the conflict of interest issue is considered, the purpose of the study is questioned and skeptical. According to the web blog, there are two conflict of interest issues in this study: "first, the study was funded by AstraZeneca, maker of the study drug, rosuvastatin (Crestor); second, the first author, Paul Ridker of Harvard, owns a patent on the high-sensitivity test for C-reactive protein, the test that would be widely used if the study results are accepted."

It will be inevitable that the next step for pharmaceutical companies like AstraZeneca is to push for routine testing of CRP (CRP screening) in clinical practice to identify the patients with normal or mild elevated LDL-Cholesterol level, but with elevated CRP. A cut point (or normal range) for CRP will be established. This is just how it works in capitalism world.

To find out how convincing of the evidences from JUPITER trial, you can read the publications by yourself. While the evidences seem to be convincing, I just don't want to take medications just for the elevated CRP. Perhaps, instead of the benefit from taking long-term treatment of Crestor, the treatment effect is really due to the detrimental effect of long-term treatment of Placebo. Perhaps, the treatment effect will be gone if we compare the Crestor with no-treatment (instead of Placebo).

• JUPITER hits New Orleans: landmark study shows statins benefit healthy individuals with high CRP levels
• Rosuvastatin to prevent vascular events in men and women with elevated c-reaction protein
• JUPITER study slides 
• The Jupiter Study, CRP Screening, and Aggressive Statin Therapy-implications for the Primary Prevention of Cardiovascular Disease

Now there is even a study claiming that the Statin prevention in patients with elevated c-reaction protein deems to be cost-effective.

Cost-benefit analysis: put a dollar value on human life?

Putting a Price Tag on Life: Today, companies and governments often use Jeremy Bentham’s utilitarian logic under the name of “cost-benefit analysis.” In Michael Sandel's lecture at Harvard, he presents some contemporary cases in which cost-benefit analysis was used to put a dollar value on human life. The cases give rise to several objections to the utilitarian logic of seeking “the greatest good for the greatest number.” Should we always give more weight to the happiness of a majority, even if the majority is cruel or ignoble? Is it possible to sum up and compare all values using a common measure like money?

 

Ford Pinto Case
One of the examples he used is the cost-benefit analysis in Ford Pinto case. According to Wikipedia,the Ford Pinto model became a focus of a major scandal when it was alleged that the car's design allowed its fuel tank to be easily damaged in the event of a rear-end collision which sometimes resulted in deadly fires and explosions. Critics argued that the vehicle's lack of a true rear bumper as well as any reinforcing structure between the rear panel and the tank meant that in certain collisions, the tank would be thrust forward into the differential, which had a number of protruding bolts that could puncture the tank. This, and the fact that the doors could potentially jam during an accident (due to poor reinforcement)allegedly made the car less safe than its contemporaries.
Ford allegedly was aware of this design flaw but refused to pay for a redesign. Instead, it was argued, Ford decided it would be cheaper to pay off possible lawsuits for resulting deaths. Mother Jones Magazine obtained the cost-benefit analysis that it said Ford had used to compare the cost of an $11 repair against the monetary value of a human life, in what became known as the Ford Pinto memo. The characterization of Ford's design decision as gross disregard for human lives in favor of profits led to significant lawsuits. While Ford was acquitted of criminal charges, it lost several million dollars and gained a reputation for manufacturing "the barbecue that seats four."

 

Repairing the Ford Pinto
Cost of Repairing                           Cost of Not Repairing
$ 11 per part                                 180 deaths x $200,000
x 12.5 million cars                          + 180 injuries x $67,000
                                                    + 2000 vehicles x $700
=====================================================================
$137 million                                   = $49.5 million
 (to improve safety)                          (to let it go) 

 

Philip Morris Czech Republic Cost-Benefit Analysis of Smoking

 

This is rather more recent case. The detail report can be found at this website.

Cost                                             Benefits
Increased Health Care costs          Tax revenue from cigarette sales
(due to lung cancer)                       Health care savings
                                                         (from early deaths)
                                                     Pension savings
                                                     Savings in housing costs

 

Net gain if citizens smoke is $147 million
Saving from premature deaths is $1227.00 per person

 

 

The fundamental issue in both cases is whether or not we can put a dollar value on human life (in the first example, a $200,000 tag for each death; in the second example, early death from smoking to benefit the government or other people).

 

We can easily see what is wrong and what is right in  both of these examples. However, in our real life, it is not easy to distinguish the right and the wrong.

 

In UK, a famous government agency is called The National Institute for Health and Clinical Excellence (NICE). NICE's main function is to provide the appraisals that are based primarily on evaluations of efficacy and cost-effectiveness. They actually put a monetory tag on human life (not for each death, but for a good-quality year). The cost-effectiveness limit (or threshold) for NICE is £30,000 per good-quality year of life gained. In many occasions, the novice drug could be denied if the drug is too expensive (over the limit of the price tag). NICE is not nice to the pharmaceutical companies.

If we can not put a price tag on human life, can we put a price tag on 3 months or 6 months of human life? If we can not put a price tag on 3 months or 6 months of human life (saved) and curb the use of extremely expensive drug, how the medical cost will be controlled? I don't think there is an easy solution.

Further readings: 

• Smoking is cost-effective, says report
• Doctor's Fury as NICE bans heart drug that could help 40,000 patients
• U.K. Says Tykerb Isn’t Worth Cost, Even With 12 Free Weeks
• UK Says Kidney Cancer Drugs Aren’t Worth the Cost

 

Significant level of 0.00125

Recently, I found some materials related to a lawsuit against the Nuvelo Pharmaceuticals – a company who used to develop a clot-busting product in the indication of occluded central venous catheter. It is very interesting to see quite some arguments about the study design and the significance of a p-value. Since Nuvelo planned to conduct only one pivotal trial (instead of two), they had agreed with FDA to use an extraordinarily high threshold (0.00125). This stringent p-value required for a single pivotal trial, less than 0.00125, was not met and the development program was eventually terminated. The issue is that such a stringent p-value was not communicated to the investment community upfront – one of the reasons for the lawsuit.

There are scientific basis for this extraordinarily high significant level (0.00125). Here are the contexts extracted from one of the FDA’s NDA Statistical Review for United Therapeutics Corporation Drug: UniprostTM (treprostinol sodium) for pulmonary arterial hypertension.

“...A more important issue is the overall Type I error rate for the proposed analysis in this submission. First, consider the traditional standard for approval at the FDA based on two confirmatory trials. Even if the efficacy of a treatment is shown convincingly in one study, the agency likes to see replication in a second study because we will then be in a better position to infer that the results generalize to the entire population of patients with the disease. The overall Type I error rate (or false positive rate) is the chance that both studies will have a p-value less than 0.05 and the results of both studies are in the same direction. If the treatment effects in the two studies are identically 0, then the chance that both p-values will be less than 0.05 and both treatment effects are in the same direction is 0.001251. For this reason, the Division of Cardio-Renal Drugs has often advised sponsors that one study with a p-value less than 0.00125 may be sufficient for approval…”

While not very common, we do see quite some drug development programs with one-single pivotal trial. One such example is the trial called PROTECT where the sample size and power for primary composite endpoint was based on “90% power at two-sided 0.00125 significance level todetect a difference between a distribution of 33% failure, 35% unchanged and 32% success (placebo group) and 25 failure, 34% unchanged, and 42% success (rolofylline group), using the van Elteren extension of the Wilcoxon test”

Designing one single pivotal trial with a significant level of 0.00125 may not be a good strategy in comparing to the conventional two pivotal trials with a significant level of 0.05. A significant level of 0.00125 is forty times more stringent than a significant level of 0.05. Employing such a small significant level will typically require a large sample size and may be difficult to be successful.


FDA’s perspectives for clinical development of tropical microbicides indicated the followings for a single trial:

• No single site provides unusually large fraction of participants
• No single investigator or site provides a disproportionate favorable effect
• Consistency across study subset
• Statistically persuasive
  Single Multi-Center Trial Level of Evidence (p value, 2-sided)
  · P < 0.001 : persuasive, robust 2*[0.025^2]=0.00125
  · 0.05 > P > 0.01: inadequate
  · 0.01> p > 0.001: acceptable, if:
  
  - good internal consistency
  - low drop-out rates
  - Other supportive data

In the end, the evidence of efficacy should not purely rely on the p-values. There are other considerations in assessing the evidence of efficacy. This has been spelled out in FDA’s guidance for Industry: Providing Clinical Evidence of Effectiveness for Human Drug and Biological Products:
Tthe evidence of effectiveness could come from one single study with the following:

• Large multicenter study
• Consistency across study subsets
• Multiple studies in a single study
• Multiple endpoints involving different events
• Statistically very persuasive finding

Here "statistically very persuasive finding" means a very small p-value even though the guidance does not specifically specify how small the p-value should be. It may depend on the negotiation with the corresponding branches in FDA.

Additional reading:

• Lloyd Fisher (1999) ONE LARGE, WELL-DESIGNED, MULTICENTER STUDY AS AN
  ALTERNATIVE TO THE USUAL FDA PARADIGM. Drug information journal Vol. 33, pp. 265–271
• Boguang Zhen (2007) Consideration of Operational á Level With Different Approval Strategies. Drug Information Journal, Vol. 41, pp. 23–29, 2007 • 0092-8615
  FDA guidance
• Guidance for Industry: Evidence-Based Review System for the Scientific Evaluation of Health Claims – Final, 2009
• Shun Z, Chi E, Durrleman S, Fisher L. (2005) Statistical consideration of the strategy for demonstrating clinical evidence of effectiveness—one larger versus two smaller pivotal studies. Stat Med. 24:1619–1637.
  
  

Logistic regression: complete or quasi-complete separation of data points

When we perform the logistic regression, sometimes, we may run into an issue so called ‘complete or quasi-complete separation of data points’. In this situation, the maximum likelihood estimate does not exist. If we use SAS Proc Logistic, SAS log will give a warning message "WARNING: There is possibly a quasi-complete separation of data points. The maximum likelihood estimate may not exist. WARNING: The LOGISTIC procedure continues in spite of the above warning. Results shown are based on the last maximum likelihood iteration. Validity of the model fit is questionable." SAS will continue to report the Wald test results and odds ratios, however, these tests are no longer valid and results are not reliable (actually not accurate at all).

Complete separation data is something like below:
Y X
0 1
0 2
0 4
1 5
1 6
1 9

There is complete separation because all of the cases in which Y is 0 have X values equal to or less than 4, and the cases in which Y is 1 have X values equal to or greater than 5. In other words, Maximal value in one group is less than the minimal value in another group. When maximal value in one group is equal to the minimal value in another group, quasi-complete separation data may occur.

If the explanatory variable is categorical, complete separation of data points could be something like this:
Response Failure Success
Predictor
0 25 0

1 0 21


Where There are no successes when the value of the predictor variable is 0, and there are no failures when the value of the predictor variable is 1.

For maximum likelihood estimates to exist, there must be some overlaps in the two distributions. Since logistic regression models uses maximum likelihood estimates, when there is no overlaps of data points between two groups, the results from logistic regression models are unreliable and should not be credited.

Starting from SAS version 9.2, Proc Logistic provides Firth estimation for dealing with the issue of quasi or complete separation of data points.

proc logistic;
model y = x /firth;
run;

However, even after Firth estimation, the results should still be interpreted with extreme caution. Complete separation and quasi-complete separation of the data points may occur when the sample size is small and number of data points is not large or in the situation the samples are determined by the outcome (i.e., response) rather than explanatory variables – we see many publications where the analysis is based on the responders vs. non-responders.

When complete separation or quasi-complete separation occurs, for multivariate regression, the explanatory variable causing this situation should be identified and preferably excluded from the model. For univariate regression, other alternative statistical tests (for example group t-test) should be used.

Further reading:

• Computation of the Odds Ratio with Small or Zero Cell Counts by Dr Robin High
• Convergence Failures in Logistic Regression by Paul Allison
• A tutorial on logistic regression by Ying So
• What is new in SAS 9.2?
  

Rasch Analysis

I recently noticed a new approach so called 'Rasch Analysis' when I worked on a paper in dealing with the MCID (minimal clinically important difference). I have not got chance to do any Rasch analysis on my own, but I have collected some information here for the future use.

Rasch analysis start to be used in education, survey area. In clinical trial, it is mostly used in psycometric, neurology areas where the outcome assessment relies on the instrument which typically contains certain number of items. These instruments are frequently used in CNS and neurology disease such as stroke, alzheimer, dementia. Traditionally, a scale or instrument will need to be validated through the reliability and validity tests. Recently, in addition to the reliability and validity tests, the Rasch measurement model has set new quality standards for outcome measures by appraising a broad range of measurement properties. You will not be surprised to see many papers if you user the search keyword "Rasch analysis" or "Rasch model" in Pubmed.gov.

There is no existing procedure within SAS to perform the Rasch analysis. However, there are some SAS macros for Rasch analysis on the internet developed by Karl Bang Christensen. The most popular software for Rasch analysis is Winsteps which provide a free download of a Ministep with capability of performing Rasch analysis for less items and less records.

• Rasch model from Wikipedia
• Rasch-analysis.com
• Rasch model using Winsteps

Winning the holiday gift

During the holiday season, it is very typical for a corporate to hold a party for its employees. During the party, one activity is to win the prizes with the raffle tickets.

Say each employee is distributed with 20 tickets in a raffle with 80 prizes. Which gives you a better chance of winning: putting all of your tickets in one of 80 baskets (your favorite item) or spreading them among 20 baskets with each ticket in one basket?

This seems to be a probability issue. There is an answer from AskMarilyn for the similar issue:

If you can see the baskets and tickets, you should wait until the last minute and then put all of your tickets in the basket that appears to contain the fewest tickets. If you can't see the tickets, put all your tickets in the basket for the least-desirable prize. But if you can't see the tickets and the prizes are equal, it doesn't matter what you do.

In previous year, I won nothing because I put all my raffles in a couple of hot items (there are thousands tickets in the boxes for these hot items). This year, I changed the strategy and put my tickets in the basket with the fewest tickets. I won a digital photo cube. Digital photo cube is not my favorite item, but I demonstrated how changing the strategy could increase the possibility of winning.

SAS SGPLOT for creating statistical graphs

For a long time, we have been using SAS GPLOT for creating graphs. Beginning from SAS version 9.2, there is a new procedure called SAS SGLPOT, which could be a good tool for statisticians.

The detail about this procedure is described in SAS onlind document. There are several white papers about using this procedure.

• SAS/GRAPH® Procedures for Creating Statistical Graphics
  in Data Analysis
• SAS blog talking about SGPLOT procedure
• Using PROC SGPLOT for Quick High Quality Graphs

Subject Diaries in Clinical Trials

Subject Diary, often called Patient Diary, is a tool used in the clinical trials. There could be three types of diary technologies. The traditional approach has been to use paper cards or booklets configured to help the subject follow directions from the clinical protocol. More recently, electronic means have been used, such as dial-in phone numbers with computer-driven questions to answer (interactive voice response systems, IVRS) and handheld devices with alarms and menu-driven prompts to guide the subject through the protocol requirements - e-diaries.

In 2006, FDA issued a draft Guidance for Industry "Patient-Reported Outcome Measures: Use in Medical Product Development to Support Labeling Claims" This guidance describes how the FDA evaluates patient-reported outcome (PRO) instruments used as effectiveness endpoints in clinical trials. Here the patient-reported outcome is typically collected through subject diary. However, the use of subject diary to collect the information on efficacy is just one of many usages. Subject diary can also be used to collect the information about:

• Daily symptoms, dialy activities
• Safety assessment (such as adverse events, exacerbations)
• Usage of the study medication to measure the compliance
• Usage of the concomitant medication
• Disease episodes on daily basis

Diary does not have to be filled out on daily basis. There are studies using diary and the subjects write down in a diary each time they take the medication,

FDA guidance listed three reasons to collect the data reported by the subject:

• Some treatment effects are known only to the patient;
• There is a desire to know the patient perspective about the effectiveness of a treatment;
• Systematic assessment of the patient’s perspective may provide valuable information that can be lost when that perspective is filtered through a clinician’s evaluation of the patient’s response to clinical interview questions.

The drawback of the diary data is the reliability of the data or the quality of the data. Subject could lose the diaries, forget to complete them or forget to complete the diaries in real time, and in worse situation, falsify information. To ensure that subject diaries comply with GCP standards, the followings are critical (adapted from Good Clinical Practice: A Question & Anwer reference)

"At the beginning of a study, site staff should explain to each subject (or parent) the importance of the diary and how the subject should record data within it. Site staff should review the diary at each visit; deficiencies and attempts to correct these deficiencies should be noted in source records. Site staff must ensure that the diaries are returned at the time designated in the trial protocol. If a patient diary is not returned, the site should make several attempts to retrieve it. These attempts should be documented in the subject’s medical record.

Although clinical auditors and FDA inspectors recognize that diaries often pose a source documentation problem, they expect to see documented efforts to minimize these problems. Diaries that are too neat, all look the same, or have been rewritten by the study coordinator are sure to raise suspicious. "

Subject diary data is often subjected to the audit findings. For example, in one of FDA's warning letters, it cited the violation about subject diary data.

"Protocol (b)(4) specified that subjects were to be distributed patient daily diary cards every 4 weeks, beginning with Week 0, and ending on Week 53. Per the protocol, patients should only have received one set of diary cards for each four-week period. However, during its inspection, FDA discovered that your records contained multiple diary entries for the same subjects on the same dates. Furthermore, the information you reported to the CRF excluded some of the data from one or more sets of patient diaries. This is a violation of 21 CFR 312 .70.
Protocol (b)(4) specified that the study coordinator was to review the patient diary cards with the patient during each scheduled visit and, if possible, query the patient to obtain any missing information at the scheduled visit. Thus, patient diaries should have only contained dated entries for dates that occurred between two scheduled visits.
The daily diary cards included information concerning whether the subject took the doses of medication in the morning and the evening (diary question #1), whether any concomitant medications were taken (diary question #2), the name of the concomitant medication taken (diary question #3), usual daily activity interruptions due to (b)(4 ) pain (diary question #4), whether any medical facility was visited due to ( b) (4) pain (diary question #5), the name of the medical facility visited (diary question 7T6), and the daily pain level experienced on a scale of 0 to 10 (diary question #7). Per the protocol, information in patient diary cards was to be recorded onto the appropriate case report form (CRF).
FDA's audit identified that 2 of 12 subjects enrolled in the study, Subjects #003 and #0 10, had two sets of patient diaries which contained different information for the same dates . From our review of the two sets of patient diaries, we were unable to determine which diary provided the correct information. In addition, we note that the information reported to the CRF was either (1) obtained from one diary but not the other, or (2) could not be verified in our review of either version of the duplicate diaries' entries for specified dates. The discrepancies we observed included, but were not limited to, the following: ...."

In terms of the handling of the subject or patient diary data, there are many different ways depending on the diary technology and the purpose of the diary. Subject diary data could be directly transmitted to the data management group without review by the site investigator or study coordinator. The data clarification (query) process is omitted. For example, in a study with Irritable Bowel Symdrome, diary data collected through IVRS was directly transferred to teh data management group and biostatistics group for analysis. Subject diary data could also be collected by the site and reviewed by the site staff (investigator or study coordinator). In this situation, the subject diary is a tool to assist the site staff in evaluating the subject about the disease status, any significant event, drug compliance,... For example, the subject diary could used to collect the COPD exacerbation (subjects record the dialy symptoms, antibiotic use, steroid use,...). in this case, the subject diary should be periodically reviewed by the site staff.

Box-Cox Transformaton

In statistical / biostatistical analysis, it is pretty common to apply the data transformation technique. The reason is to achieve the normality assumption. data transformation refers to the application of a deterministic mathematical function to each point in a data set — that is, each data point zi is replaced with the transformed value yi = f(zi), where f is a function.

The typical data transformations include logarithm, square root, Arcsine transformation. Log transformation is suitable for variables with log-normal distributions. The square-root transformation is commonly used when the variable is a count of something. For arcsin transformation, the numbers to be transformed must be in the range −1 to 1. This is commonly used for proportions, which range from 0 to 1.

Another popular data transformation technique Box-Cox transformation, which we may not use frequently in clinical trials. Box-Cox transformation belongs to the so-called 'power transform'. The Box-Cox family of transformations has two useful features: first, it includes linear and logarithmic transformations as special cases; and, second, it possesses strong scale equivariance properties, including the property that the transformation parameter is unaffected by the rescaling. Application of the Box-Cox transformation algorithm reduced the heterogeneity of error and permitted the assumption of equal variance to be met. Its main disadvantage is that both the domain and the range of the transformation are, in general, bounded.

Box-Cox transformation can be easily implemented with SAS Proc Transreg.

Further readings:

• Box-Cox Transformation: an Overview by Pengfei Li

• Basic Box-Cox Transformations using SAS Proc Transreg
• A Box-Cox Transformation using SAS Stat Studio
• What to do when data are non-normal?

Dealing with the paired data

Paired data contains values which fall normally into pairs and can therefore be expected to vary more between pairs than within pairs. The pairing is to reduce the variability. After the pairing, The between-subject variability will be eliminated. If pairing is effective it will reduce variability enough to justify the effort involved to obtain paired data.

There are many practical examples of paring. In clinical trial, crossover design is a special case of the pairing where the same subject receive more than one treatment. If all subjects receive treatment A, then treatment B, it can still be called crossover design (single sequence cross over design). In Epidemiology field, the case-control study is typically paring. There are terms 1:1 matched case-control, and 1:m matched case-control. In education, we can do the paring to compare the scores before and after the training;......

When outcome measures are continuous variable (such as drug concentration), without considering the covariates, analysis of paired data can be implemented by using paired t-test which can be easily performed using SAS PROC UNIVARIATE (calculate the difference for each pair, then run PROC UNIVARIATE) or SAS PROC TTEST (without calculating the difference first). Suppose x1 and x2 are paired variables,

proc ttest;
 paired x1*x2;
run;

If the normality assumption is questionable, the non-parametric tests (sign test and Wilcoxon signed rank sum test) can be used. UCLA's Statistical Consulting Services web site provided examples for these tests.

In more complicated situation (such as crossover design) or if we have to do the modeling to include the covariates, mixed model needs to be used. SAS PROC MIXED can implement the mixed model easily. See SAS/Stat User's Manual for PROC MIXED. In a research paper titled "Detection of emphysema progression in alpha 1-antitrypsin deficiency using CT densitometry; Methodological advances", I actually dealt with the paired data using so called 'random coefficient model'.


When outcome variable is discrete data, the easiest example is McNemar test. McNemar's test is performed if we are interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a case-control study) or two outcome variables from a single group.

In more complicated situation or if the covarites need to be included in the model, 'conditional logistic regression' needs to be employed. 'Conditional logistical regression' can be implemented using SAS Proc Logistic or SAS Proc PHREG. See following links for detail descriptions.

• A Tutorial on Logistic Regression by Ying So in SAS

• Condition Logistic Regression using SAS Proc PHREG procedure by David Brown

Pediatric use and geriatric use of drug and biological products

In the United States, every marketed drug or biological product needs to have its product label or package insert. The product label contains the use in special populations including pediatric and geriatric population. Here is a paragraph from FDA guidance on "Labeling for Human Prescription Drug and Biological Products — Implementing the New Content and Format Requirements"

Use in Specific Populations (§ 201.57(a)(13))
Information under the Use in Specific Populations heading includes a concise summary
of any clinically important differences in response or recommendations for use of the
drug in specific populations (e.g., differences between adult and pediatric responses, need
for specific monitoring in patients with hepatic impairment, need for dosing adjustments
in patients with renal impairment). Typically, information under this heading includes
limitations or precautions for specific populations or established differences in response.

Absence of the clinical study data in pediatric and geriatric population could sometimes cause problems in product label or in the drug approval process. During the drug development process, it is prudent to consider the inclusion/exclusion of patient population in terms of the age limit. In the study protocol, the inclusion criteria pertinent to the age limits (upper and lower limits) should be carefully considered. In the statistical analysis, when data for pediatric and/or geriatric population is available, subgroup analysis should always be performed.

In regulatory environment, the classification of the pediatric and geriatric population are defined as:

Pediatric population: according to ICH guidance E11 "Clinical Investigation of Medicinal Products in the Pediatric Population", the pediatric population contains several sub-categories:

• preterm newborn infants
• term newborn infants (0 to 27 days)
• infants and toddlers (28 days to 23 months)
• children (2 to 11 years)
• adolescents (12 to 16-18 years (dependent on region))

Notice that in FDA's guidance "General Considerations for Pediatric Pharmacokinetic Studies
for Drugs and Biological Products", the age classification is a little bit different. I am assuming that the ICH guidance E11 should be the correct reference.
Geriatric population:
Geriatric population is defined as persons 65 years of age and older. There is no upper limit of age defined. The Food and Drug Administration has regulations governing the content and format of labelling for human prescription drug products, including biological products, to include information pertinent to the appropriate use of drugs in the elderly and to facilitate access to this information by establishing a “Geriatric use” subsection in the labelling.

Further readings:

• Regulatory requirements for the development of medicinal products for pediatric use by Dr. Tom Sam
• FDA good review process "Labeling for Human Prescription Drug and Biological Products — Determining Established Pharmacologic Class for Use in the Highlights of Prescribing Information"
• FDA's Pediatric Drug Development
• Pediatric Trials: a Worldview

GxP: a collection of quality guidelines in clinical trial

GxP is now used to represent a collection of quality guidelines in clinical trial. The titles of these good practice guidelines usually begin with "Good" and end in "Practice", with the specific practice descriptor in between. A "c" or "C" (stands for 'current') is sometimes added to the front of the acroynm to form cGxP. For example, cGMP is an acronym for "current Good Manufacturing Practices."

Professionals who are working in the pharmaceutical or biotechnology industry should be very familiar with three common GxPs: GCP, GMP, and GLP.

GCP: Good Clinical Practices is an international ethical and scientific quality standard for designing, conducting, recording and reporting trials that involve the participation of human subjects. The GCP is governed by ICG guideline E6. To learn more about GCP, watch the GCP 101: An introduction at FDA website.

http://www.fda.gov/Training/CDRHLearn/ucm176411.htm

GLP: Good Laboratory Practice. Refer to Wikipedia for detail. GLP is the guidance for laboratory tests, pre-clinical tests, bioanalytical assays/measures, toxicology tests,...

cGMP: current Good Manufacturing Practice regulations for drugs contain minimum requirements for the methods, facilities, and controls used in manufacturing, processing, and packing of a drug product. FDA has many guidance on cGMP.

Recently, many other GxP terms have surfaced. It looks like that each functional area in clinical trial will have its own GxP. Below are some examples: GRP, GPP, GSP, and GCDMP.

GRP: Good Reprint Practices. In January 2009, FDA issued its final version of the guidance "Good Reprint Practices for the Distribution of Medical Journal Articles and Medical or Scientific Reference Publications on Unapproved New Uses of Approved Drugs and Approved or Cleared Medical Devices"

GPP: Good Pharmacovigilance Practices. In 2005, FDA issued its guidance on "Good Pharmacovigilance
Practices and Pharmacoepidemiologic Assessment" to provide guidance on (1) safety signal identification, (2) pharmacoepidemiologic assessment and safety signal interpretation, and (3) pharmacovigilance plan development.

While GRP and GPP are proposed by the regulatory agencies, there is no officially issued guidance on GSP (Good Statistical Practices) and GCDMP (Good Clinical Data Management Practices). However, the principles of these two good practices have been largely covered in ICH guidances, specifically, E9 (Statistical Principles for Clinical Trials) and E6 (Good Clinical Practice).

PSI Professional Standards Working Party developed a GUIDELINES FOR STANDARD OPERATING PROCEDURES for Good Statistical Practice in Clinical Research.

In several DIA presentations, Good Statistical Practices were said to include the following components:

Science:

• Protocol – Minimize bias – Maximize precision
• Analysis plan
• Presentation of results
• Leadership

Operational Processes

• Controlled statistical environment
  
• SOPs • Productivity tools
  
• Data standards
  
• Training
  

Credibility Results

• Reproducible research
  
• Transparent and efficient processes
  
• Validated analysis
  
• Data integrity assurance
  

The Good Clinical Data Management Practices (GCDMP) is developed by the SCDM (Society of Clinical Data Management). It provides assistance to clinical data managers in their implementation of high quality clincal data management processes and is used as a reference tool for clinical data managers when preparing for CDM training and education.

Biostatistics conferences/workshops

When I started my career in biostatistics, I joined the American Statistical Association (ASA) and attended its annual meeting (Joint Statistical Meeting) rotated in different large cities in North America (US and Canada). I have enjoyed the atmoshpere of the conference and networked with friends and professors in the statistical field.

For two consective years, I have skipped the meeting. Instead, I attended the FDA/Industry Statistical Workshop. JSM may be good to the students, but may not be good for professions (especially for statisticians who are working in drug development area). JSM has a lot of sessions/presentions that are unfiltered and too theoretic. A lot of stuff may never have the value in application. Even though it may be applicable one day, it may not be acceptable to the regulatory agencies.

The statistical conferences, symposiums, workshops with focus on clinical trial and drug development have thrived in recent years. Twice a year, FDA holds its workshops: one with Drug InformationAssociation "FDA/industry statistical forum" and one with ASA "FDA/industry statistical workhop". These conferences are more specific to the biopharmaceutical field and the topics are more relevant to the daily work of biostatisticians.

There are also several societies with focus on biostatics, for example, the International Society for Biopharmaceutical Statistics (ISBS) and the International Society for Clinical Statistics (ISCS). The International Chinese Statistical Association (ICSA) is also adjusting its focus to the biopharmaceutical field. Within ASA, biopharmaceutical network has been formed.

To get a flavor of the topics in these meetings, see the following links:

• 2009 FDA/Industry Statistical Workshop online program and presentations (The password to view presentations is Capitol
• Biopharmaceutical network's webinar topics and handouts

Positive Psychology - science to find happiness

When I see a news headline about "what is the most popular course in the Harvard University?", my curiosity drives me to find out what the course is. This leads me to the concept of "Positive Psychology". The most popular class is the Psychology 1504 (ie, positive psychology) taught by Dr. Ben-Shahar.

As mentioned in NPR news, "almost every semester for the past ten years, the most popular class at Harvard has been Intro to Economics, or as Tal Ben-Shahar likes to call it, how to get rich, but today there's an even bigger class on campus. It's Ben-Shahar's course on what he calls, how to get happy."

According to Wikipedia, Positive psychology is a recent branch of psychology that "studies the strengths and virtues that enable individuals and communities to thrive". Positive psychologists seek "to find and nurture genius and talent", and "to make normal life more fulfilling", not simply to treat mental illness. In other words, the positive psychology deals with love, happiness, job satisfaction, ...

In contrary to the Positive psychology, there should be a concept of negative psychology. However, even though the current psychology is so focused on the negative side (depression, fear, anxiety, mental illness,...), there is no formal definition of negative psychology.

Further readings about the negative psychology:

• The website of Tal Ben-Shahar
• Tal Ben-Shahar's video on Youtube
• Dr Seligman's vedio on Youtube
  
• NPR report on finding happiness
• Harvard psychology course information (including course slides)
• UPenn's Positive Psychology Center
• Good Life: Happiness on Youtube
  

Unlike the negative psychology which belongs to the medical science, the positive psychology has its applications in corporate business. It could be used to promote the positive culture, attitudes, employee's job satisfaction,...

However, there is also negative side about the positive psychology. See Dr. BARBARA S.HELD's argument.

In practice, Positive psychology encompass a variety of techniques that encourage people to identify and further develop their own positive emotions, experiences, and character traits. In many ways, positive psychology builds on key tenets of humanistic psychology. Whether or not the positive psychology techniques work will eventually rely on the evidence from the clinical trials. Since the psychology measures are typically intangible, how to design a trial or intervention, what to measure, how long to measure, what instrument to use,... could be challenging even more than the typically psychology measures (with the focus on disease or negative psychology). The following paper discussed this issue.

Overtreated, excess care

"Overtreated", "Overdiagnosed", and "Overdosed",... these are the terms I have used in one of seminars several years ago. By comparing the health care system between the United States and the China, you could easily think of these terms, especially when I heard the new medical conditions "ADHD - Attention-Deficit Hyperactivity Disorder", "M-IBS - Mixed Irritable Bowel Syndrome", "Chronic Fatigure Syndrome (CFS)", "fibromyhalia"; when I saw the images how many pills a patient took regularly.

Driven by the NPR interview with Shannon Brownlee (Are Today's Hospital Patients "Overtreated"?), I went to the local library to borrow her book "Overtreated: why to much medicine is making us sicker and poorer". I enjoyed very much in reading this book.

I intended to write a blog about this book, then found that many people had already expressed their opinion about this book. See Book Reviewer's comments from Amazon

Even though this book was written two years ago (in 2007), the arguments, the facts, the reasoning described in this book is very much relevant to the situation today (when the debate on the health care reform heats up). Below is a list of chapters:

• One: Too Much Medicine
• Two: The Most Dangerous Place
• Three: Your Local Hospital
• Four: Broken Hearts
• Five: The Desperate Cure
• Six: The Limits of Seeing
• Seven: The Persuaders
• Eight: Money, Drugs, and Lies (my favorite chapter)
  
• Nine: The Doctor Isn't In
• Ten: Less is More

Instead of going to detail, I would just cite some sentences from the book:

• "Doctors have a saying: Never get admitted to a teaching hospital in July, because that's when all the new interns arrive fresh from medical schools."
• "As research would show over the coming decades, stunningly little of what physicians do has ever been examined scientifically, and when many treatments and procedures have been put to the test, they have turned out to cause more harm than good."
• "Every patient admitted to a hospital risks being hurt or even killed by the very people who wish to help her."
• "Even as the number of [medical] imaging tests [X-ray, CT, MRI] is going up, numerous studies suggest that all those pictures are not nearly as effective at improving diagnosis as many doctors--and patients--tend to think."
• "The drug company representative, or drug rep, usually [is] a handsome young man or shapely young woman who has been recruited more for his or her good looks and outgoing personality than for his or her aptitude for science or medicine."
• "Among drug reps the unofficial name for thought leaders who work for multiple companies is 'drug whores'"
• "The more specialists involved in your health, the more likely it is that you will suffer from a medical error, that you will be given care you don't need and be harmed by it."
• "The Institute of Medicine estimates that only 4 percent of treatments and tests are backed up by strong scientific evidence; more than half have very weak evidence or none."
• "In the view of Richard Horton, a British physician and editor of the prestigious medical journal the Lancet, 'Journals have devolved into information-laundering operations for the pharmaceutical industry'"
• Says John abramson "The primary mission of medical research has been transformed. It used to be all about gathering information to improve health. Now clinical research is aimed at gathering information that will maximize return on investment"
  

Further readings:

• Are Americans Seeing Their Doctors Too Often? by Shannon Brownlee
• Overtreated blog by Shannon Brownlee
• Getting Well: It's About Time by Melinda Beck "Americans spend billions of dollars a year to treat things that would go away on their own."
• HARLOT plc: an amalgamation of the World's Two Oldest Professions by David Sackett and Andrew Oxman. BMJ
• Overdosed America: The Broken Promise of American Medicine by John Abramson

Reporting pregnancies during clinical trials

Unless a clinical trial is designed for the pregnancy women, the typical clinical trial will exclude the females with pregnancy and lactating. In either the inclusion or exclusion criteria, there will be one criterion related to the exclusion of female subjects with pregnancy. The wording for inclusion or exclusion criteria varies. Here are some examples:

Inclusion criteria:

"Women of childbearing age must have a negative pregnancy test and must useadequate contraception during the treatment phase of the study and for 9months afterwards. Women who wish to breast feed are not eligible for thestudy"

"Females must be of non-childbearing potential. Women of non-childbearing potential are defined as those who have no uterus, ligation of the fallopian tubes, or permanent cessation of ovarian function due to ovarian failure or surgical removal of the ovaries. Documentation of surgical procedure or physical examination is required for subjects who have had a hysterectomy or tubal ligation. In the absence of such documentation, a urine pregnancy test is required for inclusion into the study. A woman is also presumed to be infertile due to natural causes if she has been amenorrheic for greater than 12 months and has an FSH greater than 40 IU/L"

Exclusion criteria:
"Pregnant or nursing (lactating) women, where pregnancy is defined as the state of a female after conception and until the termination of gestation, confirmed by a positive hCG laboratory test (>= 5 mIU/mL) "

"Pregnant or breast-feeding patients. Women of childbearing potential must have
a negative pregnancy test performed within seven days prior to the start of study
drug. Both men and women enrolled in this trial must use adequate birth control"

The females that are childbearing potential are typically allowed to be enrolled in the clinical trials as long as they are willing to practice a highly effective method of contraception (oral, injectable or implanted hormonal methods of contraception, placement of an intrauterine device [IUD] or intrauterine system [IUS] condom or occlusive cap with spermicidal foam/gel/film/cream/suppository, male sterilization, or true abstinence) throughout the study.

However, it is not uncommon to have female subjects who become pregnant during a clinical investigation. In such instances, should the pregnancy be reported as AE or SAE?

The answer depends on the outcome of the pregnancy (either on mother side or on fetus side).

Pregnancy occurring during a patient’s participation in a clinical trial, although not
typically considered an SAE, must be notified to the sponsor within the same timelines as an
SAE (within one working day) on a Pregnancy Monitoring Form. The
outcome of a pregnancy should be followed up carefully and any abnormal outcome
of the mother or the child should be reported. This also applies to pregnancies
following the administration of the investigational product to the father prior to
sexual intercourse.

Based on the outcome and the timing of the delivery, the pregnancies during the clinical trial can be categoried into the followings:
Female is study participant and becomes pregnant during study participation:

1. Normal outcome before end of study
2. Abnormal outcome before end of study
3. Normal outcome after end of study
4. Abnormal outcome after end of study

Female is partner of study participation and becomes pregnant during study:

5. Normal outcome before or after end of study

6. Abnormal outcome before or after end of study

In all of these situations, the Pregnancy Monitoring Form should always be filled out. However, only for situation #2, a SAE needs to be reported.

Since the typical clinical trials do not include the pregnancy women, the potential impact of the drug on pregnancy women is not obtained during the pre-market studies. A lot of drug labels contain a statement in the contradiction section about the pregancy women. Drug exposure could also have impact on fetus - a term called 'Teratogenicity'. Teratogenicity refers to the capability of a drug to cause fetal abnormalities when administered to the pregnant mother. One of the best-known examples of such a drug- induced birth defect is the Thalidomide disaster. The drug was prescribed on a wide scale to pregnant mothers to ease the anxiety associated with it. The large-scale consumption of the drug resulted in children born with seal like limbs, often referred to as phocomelia. The drug was banned for prescription in 1961.

The potential impact of the drug on pregnancy could be obtained from observational studies - pregnancy registries. Refer to FDA's website about "General Information about Pregnancy Exposure Registries".

In ICH E2D "Post-Approval Safety Data Management: Definitions and Standards for Expedited Reporting", the following paragraph is stated:

"5.4.1 Pregnancy Exposure
MAHs (market authorization holders) are expected to follow up all pregnancy reports from healthcare professionals or consumers where the embryo/foetus could have been exposed to one of its medicinal products. When an active substance, or one of its metabolites, has a long half-life, this should be taken into account when considering whether a foetus could have been exposed (e.g., if medicinal products taken before the gestational period should be considered). "

A sample pregnancy registry form can be found from GSK website.

Conficence Interval vs. Credible Interval

I recently participated in a project to compare two different ways to do the meta analysis: the traditional way to pool the database directly (sort of the integrated analysis) and the Bayesian approach (prior distribution + likelihood function -> posterior distribution). When we try to compare the results from two different approaches, we run into the issue of comparing 'confidence interval' and 'credible interval'. While these two terms have some similarities, the interpretations are quite different.

The "confidence interval" is a term used by frequentist - I am a frequentist. If we say an estimate has its 90% confidence interval of 35-45, it means that with a large number of repeated samples, 90% of times, the true value of the parameter will fall within the range of 35-45.

The term 'credible interval' is used by Bayesian statisticians and it may also be called 'Bayesian Posterior Interval'. In Bayesian statistics, a credible interval is a posterior probability interval, used for purposes similar to those of confidence intervals in frequentist statistics. Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ...

The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... In statistics, a confidence interval (CI) is an interval between two numbers, where there is a certain specified level of confidence that a population parameter lies. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ...

For example, a statement such as "following the experiment, a 95% credible interval for the parameter t is 35-45" means that the posterior probability that t lies in the interval from 35 to 45 is 0.9.

A Bayesian credible interval incorporates information from the prior distribution into the estimate, while confidence intervals are based solely on the data.

Like 'confidence interval' vs 'credible interval, there is also 'confidence region' vs 'credible region'.

Here are some links for further reading:

• Credible interval at wikepedia
• Bland and Altman (1998) Bayesians and frequentists. BMJ
• Fisher (1996) Comments on Bayesian and Frequentist Analysis and Interpretation of Clinical Trials. Controlled Clinical Trials
• Credible Interval
• Credible Intervals vs. Confidence Intervals

Placebo and Sham treatment: are they really inactive?

According the ICH guidance E10 (CHOICE OF CONTROL GROUP AND RELATED ISSUES IN CLINICAL TRIALS), "A placebo is a "dummy" treatment that appears as identical as possible to the test treatment with respect to physical characteristics such as color, weight, taste and smell, but that does not contain the test drug."

We typically use the term Placebo, but sometime, the term 'sham treatment' is used. The word 'sham' means something that is a fake or an imitation that purports to be genuine.

In practice, a placebo is often defined as an inactive substance made to appear like a medication or a sham procedure or device imitating a known treatment. Sometimes, the 'inactive' substance used in the clinical trial may not be totally 'inactive'. One example is albumin. On the one hand, the albumin may be treated as inactive substance for Placebo; on the other hand, there are studies to study the effect of albumin in certain diseases. In a study about "the Effectiveness of Intravenous Immune Globulin (10%) for the Treatment of Multifocal Motor Neuropathy", 0.25% human albumin solution was used as Placebo. But there are also plenty of clinical trials to study the efficacy of albumin in sepsis, renal impairment, acute stroke,...

In a lot of publications, the author did not disclose what the placebo is. You can use the same term 'placebo', but the 'placebo' could be sugar pill, saline, albumin,...

Even more...

In article web article by BJ Appelgren titled "The Placebo as Medicine Viewed as Sham, Placebo Itself May Be the Most Significant", the following examples are cited as other type of placebos.

• Having an interaction with a health care provider
• The presence of something symbolic in the encounter such as contact with a person wearing a “white coat,” perceived as a provider of healing

The significance of symbols cannot be measured objectively and, for that reason, is not valued. Researchers also have an additional puzzle when non-treatment causes positive results. Too often, when the effect of symbolism is recognized by conventional medicine, it is removed from a context of positive meaning and denigrated as “being all in the mind,” as if that makes it illusory.

SAS IQ and OQ

I am not sure how many people really know these abbreviations: DOE, IQ, OQ, PQ, PV. These are the terms used in the validation of a software or computerized systems.

DOE = Design of Experiment
IQ = Installation Qualification
OQ = Operational Qualification
PQ = Performance Qualification
PV = Process Validation

For off-the-shelf software, I have never really thought about the validation or qualification issue. I always think that the software like SAS (we use it almost daily) is just like the Miscrosoft Office. Once you install on your PC, you are ready to use.

I really learned that if the software is used for regulatory submission, certain level of validation (precisely qualification) need to be performed. There is no exception for SAS.

For SAS software, the IQ and OQ need to be performed. The instructions for the SASInstallation Qualification (IQ) and Operational Qualification (OQ) tools can be found at SAS website and at http://support.sas.com/kb/17/046.html. This was also mentioned in SAS quality document. SAS actually has a SOP for IQ and OC.

By digging in this issue further, the verification and validation of a software or computerized system is not a trivial task. Wikipedia has a topic discussing about verification and validation. The recent issue of DIA Global forum has an article by Chamberlain and they discussed qualification (vs validation) of the infrastructure.

However, I think that the validation for off-the-shelf software should be much simpler than a self-developed computerized system (such as an internal EDC system). FDA has a pertinent guidance about the computerized systems: Computerized systems used in clinical investigations and its old version titled "Computerized systems used in clinical trials".

CDRH also had a guidance titled "General Principles of Software Validation; Final Guidance for
Industry and FDA Staff". This guidance indicated that the software used in medical device need to go through a full-scale validation process.

Hochberg procedure for adjustment for multiplicity - an illustration

In a May article, I discussed several practical procedures for multiple testing issue. One of the procedures is Hockberg's procedure. The original paper is pretty short and published in Biometrika.

Hochberg (1988) A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75(4):800-802

Hochberg's procedure is a step-up procedure and its comparison with other procedures are discussed in a paper by Huang & Hsu.

To help the non-statisticians to understand the application of Hochberg's procedure, we can use the hypothetical examples (three situations with three pairs of p-values).

Suppose we have k=2 t-tests
Assume target alpha(T)=0.05

Unadjusted p-values are ordered from the largest to the smallest

Situation #1:
P1=0.074
P2=0.013

For the jth test, calculate alpha(j) = alpha(T)/(k – j +1)

For test j = 2,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(2 – 2 + 1)
= 0.05

P1=0.074 is greater than 0.05, we can not reject the null hypothesis. Proceed to the next test

For test j = 1,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(2 – 1 + 1)
= 0.025

P2=0.013 is less than 0.025, reject the null hypothesis.

Situation #2:
P1=0.074
P2=0.030
For the jth test, calculate alpha(j) = alpha(T)/(k – j +1)

For test j = 2,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(2 – 2 + 1)
= 0.05

P1=0.074 is greater than 0.05, we can not reject the null hypothesis. Proceed to the next test

For test j = 1,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(2 – 1 + 1)
= 0.025

P2=0.030 is greater than 0.025, we can not reject the null hypothesis.

Situation #3:
P1=0.013
P2=0.001
For the jth test, calculate alpha(j) = alpha(T)/(k – j +1)

For test j = 2,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(2 – 2 + 1)
= 0.05

P1=0.013 is less than 0.05, we reject the null hypothesis.
Since the all p-values are less than 0.05, we reject all null hypothesis at 0,05.

More than two comparisons
If we have more than two comparisons, we can still use the same logic

For the jth test, calculate alpha(j) = alpha(T)/(k – j +1)

For example, if there are three comparisons with p-values as:
p1=0.074
p2=0.013
p3=0.010

For test j = 3,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(3 – 3 + 1)
= 0.05

For test j=3, the observed p1 = 0.074 is less than alpha(j) = 0.05, so we can not reject the null hypothesis. We proceed to the next test.


For test j = 2,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(3 – 2 + 1)
= 0.05 / 2
= 0.025

For test j=2, the observed p2 = 0.013 is less than alpha(j) = 0.025, so we reject all remaining null hypothesis.



For example, if there are three comparisons with p-values as:
p1=0.074
p2=0.030
p3=0.010

For test j = 3,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(3 – 3 + 1)
= 0.05

For test j=3, the observed p1 = 0.074 is less than alpha(j) = 0.05, so we can not reject the null hypothesis. We proceed to the next test.


For test j = 2,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(3 – 2 + 1)
= 0.05 / 2
= 0.025

For test j=2, the observed p2 = 0.030 is greater than alpha(j) = 0.025, so we can not reject the null hypothesis. We proceed to the next test.

For test j = 1,
alpha(j) = alpha(T)/(k – j +1)
= 0.05/(3 – 1 + 1)
= 0.05 / 3
= 0.017

For test j=2, the observed p2 = 0.010 is less than alpha(j) = 0.017, so we can reject the null hypothesis.
.

Poisson regression and zero-inflated Poisson regression

Poisson regression is a method to model the frequency of event counts or the event rate, such as the number of adverse events of a certain type or frequency of epileptic seizures during a clinical trial, by a set of covariates. The counts are assumed to follow a Poisson distribution with other variables that are modeled as a function of the covariates. The Poisson regression model is a special case of a generalized linear model (GLM) with a log link - this is why the Poisson regression may also be called Log-Linear Model . Consequently, it is often presented as an example in the broader context of GLM theory.

Poisson regression is the simplest regression model for count data and assumes that each observed count Yi is drawn from a Poisson distribution with the conditional mean ui on a given vector Xi for case i. The number of events follows the Poisson distribution that is described blow:

where

• e is the base of the natural logarithm (e = 2.71828...)
• k is the number of occurrences of an event - the probability of which is given by the function
• k! is the factorial of k
• λ is a positive real number, equal to the expected number of occurrences that occur during the given interval, the interval could be a time interval or other offset variables (denominators).

The most important feature of the Poisson regressin is that the parameter λ is not only the mean number of occurrences, but also its variance. In other words, to follow the Poisson distribution, the mean equals to the variance. However, with the empirical data (observations), this feature may not always fit - a situation called overdisperse or underdisperse. When the observed variance is higher than the variance of a theoretical model (or for Poisson distribution, the observed variance is higher than the observed mean), overdispersion has occurred. Conversely, underdispersion means that there was less variation in the data than predicted.

When overdisperse occurs, an alternative model with additional free parameters may provide a better fit. In the case of the count data, an alternaitve model such as negative binomial distribution may be used.

In practice, we often see the count data with excessive zero counts (no event), which may cause the deviation from the Poisson distribution - overdispersion or underdispersion. If this is the case, zero-inflated Poisson regression may be used.

In SAS, several procedures in both STAT and ETS modules can be used to estimate Poisson regression. While GENMOD, GLIMMIX (from SAS/Stat), and COUNTREG (from SAS/ETS) are easy to use with standard MODEL statement, NLMIXED, MODEL, NLIN provide great flexibility to model count data by specifying the log likelihood function explicitly.

• Excellent Discussion about Zero-Inflated Poisson Regression by UCLA Academic Technology Services
• Count Data Model using SAS by Liu and Cela
• Zero-Inflated Poisson and Zero-Inflated Negative Binomial Models Using the COUNTREG Procedure by Erdman et al
• Design and Analysis of Count Data by Lakshminarayanan
• Proc Countreg user's guide from SAS
• Course notes about negative binomial distribution by Dr.Preisser in UNC
  
  

Epidemiology terms, but used in clinical trials

Some terms often used in epidemiology fields are actually pretty commonly used in clinical trials. At least for me, I first learned these terms in epidemiology classes. Some of these terms are very close, but different.

Ratio, Proportion, Rate

• Ratio: Division of two unrelated numbers
• Proportion: Division of two related numbers; numerator is a subset of denominator
• Rate: Division of two numbers; time is always in denominator

For example, 'sex ratio' is a ratio (the number of males divided by the number of females). Rising sex-ratio imbalance is a danger in China.
In a clinical trial, there are males subjects and female subjects. We summarized the data using the percentage of male subjects among the total - this is proportion. The # of male subjects (numerator) is a subset of the total subjects (denominator).
A rate is also one number divided by another, but time is an integral part of the denominator. For example, the speed limit is a rate (65 miles per hour). In clinical trial, rate is often used to describe the incidence of adverse events.

Incidence and prevalence

• Incidence: measures the occurrence of new disease; deals with the transition from health to disease; defined as the occurrence of new cases of disease that develop in a candidate population over a specified time period
• Prevalence: measures the existence of current disease; focuses on the period of time that a person lives with a disease; measures the frequency with which new disease develops; defined as the proportion of the total population that is diseased.
  

Incidence is typically used for describing the acute disease while the prevalence is typically used for describing the chronic disease.

Both incidence and prevalence are rate, not ratio, not proportion. In practice, it comes the terms such as incidence rate and prevalence rate.

While prevalence is purely an epidemiology term, 'incidence' or 'incidence rate' is commonly used in the analysis of clinical trial data, especially the adverse event data. Clinical trial design is always prospective and can be considered as a special case of cohort study in epidemiology term.
In statistics and demography, a cohort is a group of subjects who have shared a particular experience during a particular time span. Cohorts may be tracked over extended periods of time in a cohort study. Notice that we also use the term 'cohort' in dose-escalation clinical studies where the cohort refers to a group of subjects who receive the same level of dose (this is contrary to the term 'arm' used in parallel design).

Incidence vs. Incidence Rate

In clinical trial, when we summarize the adverse event data, should we say "incidence of adverse events" or "incidence rate of adverse events"?

While both terms may be used, "incidence of adverse events" should be more accurate and is more frequently used. This can be easily seen in FDA guidance documents, for example:

• Formatting of Bioequivalence Summary Tables
• Guidance on Immune Globulin Intravenous (Human) as Replacement Therapy for Primary Humoral Immunodeficiency
• Guidance on FDA’s Transition Plan for Existing Postmarket Surveillance Protocols
  

"Incidence rate" may be more appropriate when the # of adverse events are normalized by the person year or patient year or normalized by the # of infusions, for example, the following terms may be used: "incidence rate of adverse events per infusion" "The incidence rates per 100 patients-year"