Overcoming Challenges in Survival Analysis Trial Design

# Challenges in Survival Analysis Trial Design: Complex Survival and Adaptive Designs Hello, and welcome to today’s webinar, **Challenges in Survival Analysis Trial Design: Complex Survival and Adaptive Designs as Special Topics**. Today’s webinar, brought to you by **Dotmatics nQuery**, will provide you with an overview of some of the common and less common challenges you might expect when designing a clinical trial involving a survival analysis endpoint. We’ll also go through more complex areas, such as non-proportional hazards and the specific issues that can occur when applying an adaptive design to a survival analysis endpoint. Before we get started, just some frequently asked questions. Firstly, is this webinar recorded? Yes. The recording and the slides will be sent to you after this webinar is complete. Can you ask questions? Yes. There’s a Questions option in your webinar tab. If time allows at the end, I’ll answer those questions live. Otherwise, I’ll answer them via email afterwards. So, you’ll get the slides and the recording after the day. If there are any other technical issues, let me know via the Questions tab, and we’ll get in touch and try to help you out there. ## Introduction Okay. So today’s webinar, **Challenges in Survival Analysis Trial Design**, is providing you with a hopefully useful primer on design challenges for survival analysis trials. We’ll then go into some more complex topics that you may be less familiar with and some of the unique issues that occur when doing survival analysis in the context of those designs. My name is Ronald Fitzpatrick. I’m the Head of Statistics and Research here at nQuery. We’ve worked on nQuery for over a decade now. We’ve made lots of changes, additions, and improvements to nQuery over that time, for survival analysis specifically as well as many other areas. If you have feedback on that, we very much welcome it, either in terms of our software or future webinars. All feedback is very welcome. ## Agenda In terms of today’s agenda, we’ll firstly talk about survival design challenges generally and some of the common things that you need to consider if you’re planning to do a survival analysis trial. Then we’ll talk about complex survival patterns and the non-proportional hazards area. Finally, we’ll talk about adaptive design and specific things that might occur, or that you might want to take into consideration, if you’re planning to use adaptive design in the context of a survival analysis trial compared with other types of endpoints. Then we’ll have some time for discussion at the end. As mentioned, questions are welcome through the Questions tab. ## About nQuery Today’s webinar is brought to you by **nQuery**, the complete solution for optimizing clinical trial designs, from preclinical research through to confirmatory trials. You can hear a flavor of some of the options available in nQuery, although there are many more beyond this. Just to note that nQuery is a tiered product, from Base to Expert. Base provides sample size and power calculations for fixed-term trials. Plus provides Base functionality along with additional sample size capabilities. Pro adds adaptive design functionality. Expert provides advanced simulation tools. You can see a URL here that you can use when the slides become available to compare these tiers and see the full range of options available in nQuery at the moment. With that out of the way, let’s get into today’s webinar, starting with survival analysis design challenges. ## What Is Survival Analysis? Before we get started, let’s provide an overview of what survival analysis is so we’re all on the same page. Survival analysis, often referred to as time-to-event analysis, is concerned with statistical inferences regarding the expected length of time until an event of interest occurs. We’re not just interested in whether an event occurred. We are specifically interested in inferences about how long after someone entered the study that event occurred, finding averages and making statistical inferences about what we expect to happen on average, and comparing that between multiple groups. The event of interest is usually something like death or disease progression in the context of a clinical trial. You’ll often hear these commonly referred to as overall survival as an objective or endpoint in the trial, or progression-free survival. Death or overall survival is fairly self-explanatory. Disease progression might mean moving from stage three to stage four cancer, for example, and there are other similar considerations in other diseases. Survival analysis is by far one of the most common approaches to statistical analysis in clinical trials, especially because it’s so prominent in oncology. Oncology has many specific guidances from regulators about what to do for survival analysis trials, or basically have a wide overlap with survival analysis aspects. I should note here that while a survival analysis trial, or time-to-event trial, is what we’re talking about here today, if you’re talking about earlier-stage trials, particularly Phase II, while you may still be using an event-based endpoint, it may often be analyzed as a binary variable rather than a survival variable or survival endpoint. Something like response rate, for example. Even if you are doing survival input, it might be a simplified version, like death after six months of follow-up. Obviously, because of the restrictions required for Phase II relative to Phase III in terms of time and resources and so on. So that’s something to keep in mind. If you’re doing a Phase III trial, you’ll almost certainly, if you have overall survival as an endpoint in oncology, be required to do a survival analysis. Even if that isn’t your primary analysis, it’s almost certain that overall survival is something that you’re required to track and have reasonable inferences about at the end of the trial. That could be immediately as you start your New Drug Application, or sometime down the road, either pre-approval or post-approval. There is guidance, for example, from the FDA regarding overall survival, specifically talking about how it needs to be included as part of the package, or what you’re planning to monitor, regardless of whether overall survival is your primary endpoint for your efficacy analysis, your p-value, and so on, in that particular study. ## Survival Analysis Design Challenges Survival analysis has a number of distinct design and statistical challenges compared with other endpoints, such as binary or continuous endpoints. We’ll talk about some of those during the rest of this webinar. We’ll talk about common challenges in survival analysis specifically, and then the impact of more complex design aspects such as non-proportional hazards and adaptive design. ### Trial Design In terms of some of those considerations for a survival analysis trial specifically, obviously there are some that are fairly shared across other endpoints. You need to think about your design: * Two-arm parallel trial * Paired analysis * Multi-arm trials * One-arm trials And so on. I would say that in survival analysis trials, due to the bias toward confirmatory Phase III trials, the vast majority will be two-arm trials. That’s what you’ll associate with the log-rank test, hazard ratios, and similar statistics and analyses. But there are methods available for one-arm survival analysis trials, including parametric models and log-rank-type equivalents. There are equivalents for multi-arm trials, multiple comparisons, and other types of scenarios, even for paired analyses. It’s a little more complex to define what “paired” means for a survival endpoint than for other endpoints. Obviously, we can also talk about adaptive, complex, Bayesian-type features. All of that feeds into survival analysis trials as well. We’ll assume for most cases that we’re dealing with a two-arm trial. That’ll be our main assumption today. Although I would say that nQuery does have sample size tools for one-arm trials, multi-arm trials, and Bayesian approaches if you’re interested in those. ### Endpoint In terms of the endpoint, as I said, you might be considering overall survival. You might want to use progression-free survival. In the context of a particular disease, there might be other survival-type endpoints you might be considering. But even as I mentioned, if you’re not using overall survival as your primary efficacy endpoint, it might be something that you still need to track and include as part of your analysis plan as a harm or safety endpoint. That’s due to the guidance that exists for oncology, including, for example, the FDA guidance from 2025, which at the time of this webinar was still draft guidance. ### Recurring Events One thing I’ll mention briefly is that obviously we’re going to hear mostly about first-event time-to-event analyses. But there are cases where there might be recurring events. For example, in cardiology, think about cardiac arrests. You might have multiple cardiac events. How do you include an analysis that captures all of the inferential power of each of those cardiac events rather than just the first one? That’s not something we’re going to cover here today. It gets into recurrent-event models and that type of analysis. But I think it’s important to mention because it’s something that has been gaining additional traction in recent years. ## Primary Analysis and Statistic We also need to think about what our primary analysis is, what our primary statistic is, and our estimand strategy. Usually, the statistic is going to be the hazard ratio, but there are others, such as restricted mean survival time and other types of weighted hazard ratios. Those are mostly relevant in the context of non-proportional hazards, which we’ll talk about in more detail later. What’s your statistical test? Again, if proportional hazards is assumed and we’re using a hazard ratio, we’re probably going to use a log-rank test or a Cox proportional hazards regression model. You can actually show that a Cox regression model with a single independent variable is effectively equivalent to the log-rank test. You’ll get very small differences, but they’re basically the same. Obviously, with the Cox model, you have greater flexibility to add additional covariates and other considerations. In non-proportional hazards, you might have other methods such as: * Max-combo tests * Linear rank tests * Restricted mean survival time * Failure-time models * And so on But in a Phase III clinical trial, it’s probably still a log-rank test, possibly with a Cox model. ## Assumptions for Survival Trial Design Now let’s assume you have your design, you know your endpoint, and you’ve picked your particular analysis. Let’s get into what we need to think about from an assumption point of view to be able to do the design and sample size calculation for the survival analysis trial. First, accrual and censoring. ### Event-Driven Versus Fixed Follow-Up We need to decide whether our strategy is: * Event-driven * Fixed follow-up * Fixed study length The most common scenario in most Phase III clinical trials would be event-driven. Effectively, you have a target number of events and you keep running the trial until you reach that target. In reality, due to resourcing and other issues, you may or may not reach the exact number of events, if things go particularly badly. But that is the objective upfront when designing the study. Fixed follow-up is a slightly different approach. A subject enters the study and is followed up for a fixed period of time. After that period, they are administratively censored. They no longer contribute events after that point. For example, a subject could have the event of interest twelve months after entering the study, but if the protocol specifies twelve months of follow-up, they would be administratively censored after those twelve months. I would say event-driven is by far the most common strategy, but fixed follow-up is used in certain scenarios. Infectious diseases is one area where this might be more commonly used. Oncology and cardiology are usually event-driven. A fixed study length is more of a pragmatic idea. If we plan upfront for, say, a two-year or three-year study, and we reach three years but things have been slower than expected, perhaps because accrual was slower than expected or the hazard rate was lower than expected, that might actually be a good sign because the treatment hazard rate could be much lower than expected. But there are lots of different things you need to prioritize before doing the analysis. Are we going to stick with the two- or three-year timeline and stop then, or are we going to continue until we reach our target number of events? That’s something you want to keep in mind beforehand. You might also need to make adjustments to the study while it’s ongoing. Obviously, ad hoc adjustments are something you want to avoid. But in survival analysis, because things are much more uncertain, it’s not like a binary or continuous endpoint where you might have a straightforward six-month follow-up period. The actual event of interest itself is a random variable. So the end of the study is much more uncertain. Even within the accrual process, there’s uncertainty. Are we going to continue accrual until we reach our target sample size? Are we going to accrue for twelve or twenty-four months and then stop regardless of where we are? I’d say the latter is quite rare in a well-resourced trial, but in a more constrained trial, that may be something you have to consider. You also need to think about what adjustments you’ll make and how you will justify those adjustments if they are required. What’s your strategy to increase the number of sites or improve accrual? A huge number of trials fail to reach their recruitment goals or recruit more slowly than expected. That has knock-on implications for a survival trial because you have additional uncertainty coming from both the accrual process and the survival process in terms of when the trial will actually reach its target number of events. ### Accrual Pattern Within accrual, we also need to consider what we expect the accrual pattern to be. What you often see in survival trials, and in most trials to be fair, is an S-shaped or logistic-type pattern. It takes time to onboard sites and get recruitment going. The first period might have a slow amount of recruitment. Then things ramp up, and we reach our standard or uniform rate of recruitment. At the end, as we approach the maximum planned sample size, we might have fewer people remaining to recruit. At that point, you might close some of the less successful sites or sites that want to move on to other studies. So there may be a tapering off in accrual at the end. It’s quite common in any trial, but it has particular implications for survival trials because you have much greater uncertainty about when the trial will end. ### Censoring Similarly, for censoring, what is your censoring strategy going to be? In an event-driven trial, you’re going to censor people who are still event-free at the end of the study. For fixed follow-up, you’ll be censoring people throughout the study after they reach their fixed follow-up period. There are many other censoring considerations that are beyond the scope of this webinar. There is obviously a lot of censoring methodology in survival trials. Interval censoring is something to keep an eye on. For example, if you’re only doing follow-ups every six months, three months, or one month, an event may occur between follow-up one and follow-up two. You don’t know exactly when it occurred between those two periods. Obviously, within a well-resourced trial, you might know within a day or so from the hospital or site. But for less-resourced trials, that’s something you might need to take into account when practically thinking about your trial and potentially in your design considerations. ## Survival and Hazard Models We then move to the survival or hazard model. You usually have some kind of parametric model. Exponential and Weibull are two of the most common, but there are also log-normal and gamma models. Even within exponential and Weibull models, you can have constant or piecewise approaches. The constant case is very commonly used because it’s a very easy assumption. You can summarize what you expect to happen using median survival. But you may, in some cases, expect the hazard rate to be lower or higher the further you get into the trial. You can retain proportional hazards while the underlying hazard rate still slows down or increases over time in a common way for both treatment groups. That’s something to keep in mind. A piecewise model is very flexible because you can essentially make an arbitrary survival pattern. But each additional period that you include in your piecewise pattern is another assumption you have to make when choosing your model. For accrual patterns, similarly, you can use a piecewise accrual pattern. There is a parametric model that can be used as well. So there are very similar considerations between piecewise accrual and piecewise hazard rates. ## Non-Proportional Hazards If we go beyond that, we get into complex survival patterns. Proportional hazards, by far, is the most common assumption, where we have a constant hazard ratio over time. But non-proportional hazards, or NPH, such as delayed effects, diminishing effects, and crossing effects, are becoming more common as we move beyond cytotoxic treatments toward targeted treatments such as immunotherapies. We’ll talk much more about that in the second section. ### Defining Survival Patterns We need to think about our statistic. Here, the statistic is really the statistic for defining the expected survival pattern in each group, compared with the statistic we use for comparing our two groups. For example, an RMST difference or hazard ratio is how we compare the groups. Then we need to define parametrically what each group’s survival pattern is going to look like. Common inputs include: * Hazard rate * Median survival * Landmark or milestone survival rate For example, what proportion of subjects do we expect to be alive after twelve months of follow-up? For more complex non-exponential models, you might have shape and scale parameters that need to be defined. Sometimes those map quite nicely to more commonly cited statistics. Sometimes they don’t, depending on the model. Usually, if you’re willing to make some assumptions, there are translations between these quantities. For example, many sample size formulas require a hazard rate, but you can translate median survival or survival rates into hazard rates under an exponential model. The hazard rate is effectively the instantaneous risk of having the event. ## Dropout and Crossover Then there are other considerations, such as dropout. Usually, in the context of design, we’re modeling dropout as a parallel survival process. Effectively, people are being removed from the population available to experience the event of interest. If you want to get into the weeds, there are considerations around different types of dropout and their implications for the estimand and intercurrent event strategy. Similarly, there is treatment crossover, noncompliance, and treatment switching. You may be modeling one or both of these, and the model is usually going to be similar to how you model dropout and hazard rates. There are also implications for the estimand strategy. Even if you think about the standard historical approach of intention-to-treat, many sample size formulas that include noncompliance and treatment switching don’t strictly use ITT. They may assign the event to the group the subject was in when the event occurred. For example, if a subject moves from treatment to control and then has an event, that event may be counted in the control group for powering purposes. That is obviously inconsistent with a strict ITT approach. So that’s something to keep in mind. There are also more complex topics, such as: * Time-varying effects and covariates * Competing risks * Cure effects * Multi-state models There are methods available if you have competing risks or cure effects and want to include those in the model, with implications for the test statistic and hazard ratio. ## Analysis Versus Estimation What happens if your test or analysis is different from your estimation strategy? For example, in an NPH scenario, you might use a max-combo test to obtain your p-value, but then use RMST for estimation. What are the implications of that? What if you have multiple intercurrent event policies? What if you have some kind of graphical multiplicity adjustment? For example, in a complex group sequential design, perhaps the interim analyses are based on PFS but the final analysis or post-stopping analysis is based on OS. What are the implications of that in terms of the study design? There are multi-state models and many other approaches beyond this in terms of what is available to actually do survival analysis. But in the context of Phase III clinical trials and most clinical trials, this is a fairly good survey of the types of things you need to consider once you get to the design stage. Obviously, design leads directly into power analysis and sample size calculation. ## Power and Events Power analysis should follow directly from the study design, the choice of statistic, and the analysis or test you’re going to use. If it’s a two-arm trial where you’re going to use a hazard ratio and analyze using the log-rank test, all of that feeds into the power analysis. Then we make assumptions about accrual rates, hazard rates, and so on. One thing I really want to emphasize, and I’m sure most people here know this, is that the power for a time-to-event analysis ultimately follows from the number of events, not the sample size. Imagine hypothetically that you recruit 100 people and nobody has an event. In that case, you have essentially no information for testing a treatment effect. The power is coming from the events, not the sample size. What you’ll often see is that people might use an approximation such as the Schoenfeld approximation to calculate the required number of events first, and only then calculate the sample size needed to obtain that number of events. Other approaches, such as the Freedman approximation, work similarly. Effectively, you have a term related to the required number of events and then a probability-of-event component. So it’s really important to emphasize that power comes from the events, not the sample size. Sample size is often a secondary model that tells you how many people you need in your study to achieve that number of events, given assumptions about accrual, hazard rates, censoring, and so on. In an event-driven trial, if we’re wrong about those assumptions, we may have to run the trial longer to reach the required number of events. But that hasn’t fundamentally changed the statistical analysis or the required information. We still need that number of events. We were simply wrong about how long it would take, or how many people we would need, to obtain those events. There are various approaches to doing this kind of modeling. You have formulas such as Freedman and Lakatos, and approaches using integration. There are also Markov-model approaches from Lakatos. All of these have different strengths and weaknesses. The latter approaches are probably better in some respects, but they’re less simple to implement if you’re just doing the calculation by hand. Some methods tie the events and sample size processes together more closely. Integration and Markov approaches probably do that more closely. But the fundamental idea that events drive the power is still true, even if it’s harder to separate the event and sample size components. ## Factors Affecting Events and Sample Size Sample size is affected by a much wider variety of factors than the required number of events. The number of events is primarily driven by: * Significance level * Hazard ratio * Desired power The sample size is then affected by additional factors, including: * Accrual period * Accrual pattern * Study length * Hazard rates * Survival patterns * Dropout * Censoring * Treatment switching * Noncompliance Median survival in each group gives you the hazard ratio under an exponential model, which affects the event calculation. But the individual median survivals can go up and down. As long as the ratio remains the same, that may have little or no effect on the required number of events, but it can have a substantial effect on the sample size required to obtain those events. ## Example: Everolimus in Advanced Pancreatic Cancer Let’s look at a simple log-rank test example for everolimus in advanced pancreatic cancer. You can see here that, for a 2.5% significance level, 92% power, and a 23% risk reduction, which corresponds approximately to a hazard ratio of 0.67, we require 282 events. Before they even talk about median survival or accrual rates, they’ve already given you the required number of events. That’s because, in this case, they likely used a Schoenfeld-type approximation. All the additional information after that first sentence is essentially used to calculate the sample size required to achieve those 282 events. We have median survivals of nine and six months in the treatment groups. Then we have a uniform accrual period of 74 weeks and a minimum follow-up period of 39 weeks. That gives us 352 patients, or 176 per group, to obtain 282 events under those assumptions. This is just making assumptions about how many people we think we need to obtain the required number of events. Those assumptions could be wrong. How you adjust for that after the fact is something you might want to think about upfront. It’s not a major emphasis here because we’re talking about the design stage, but it’s something you’ll probably deal with in practice if you’re running a clinical trial with a survival analysis endpoint. ## Implementing the Example in nQuery Let’s go into nQuery. When you open nQuery and click **New Table**, it opens the Table Select window. If you want to see what’s available for survival analysis, there’s a Survival option in the table selector on the left-hand side. You can see a wide variety of different options. If we open the two-group options, there are even more. If you’re looking for a specific analysis, this should hopefully guide you toward the appropriate analysis for your trial. For this example, we have: * Two groups * Log-rank test * Event-driven design * Constant hazard There is also a piecewise version available. When you open nQuery, you have the design table in the top-left-hand corner. On the right-hand side, you have a Help card that defines each of the rows. Each row is essentially a required input to perform a sample size, power, or other calculation. The yellow rows are things we can calculate. The white rows are things that we need to input. The gray rows are read-only and provide supplementary information. In this case, that includes the total number of events. We can’t edit that directly. nQuery calculates it. ### Time Units We first enter the significance level of 0.025 for a one-sided test. We have an accrual period of 74 weeks. But remember, in survival analysis, all time-dependent units must use the same time scale. They must all be expressed in months, weeks, days, years, or whatever unit you choose. In this case, the median survivals are in months, but the accrual period is in weeks. Therefore, we need to convert one to the other. While that’s fairly obvious for median survival, it may not be as obvious if, for example, you have an exponential hazard rate. The hazard rate is dependent on the time scale. For a Weibull model, the shape and scale parameters can also be dependent on the time scale. So you need to be careful, particularly with more complex analyses, that all time-dependent inputs are on the same time scale. In this case, we’ll convert the accrual period and maximum follow-up period to months rather than weeks. Using four weeks per month as a reasonable approximation, 74 weeks becomes 18.5 months. The maximum length of follow-up is the accrual period plus the minimum follow-up period. So: 74 weeks + 39 weeks = 113 weeks. Dividing by four gives us approximately 28.25 months. ### Median Survival and Hazard Rates We then have our exponential parameters, which you might often hear referred to as exponential hazard rates. In this case, we don’t have those directly. We have median survival, which is very commonly cited in sample size calculations. You need to be able to convert between these. We can set the median survival in the side table to nine and six months. nQuery then calculates the corresponding hazard rates. There is also an option in the Assistance menu called the Survival Parameter Converter. That’s a separate table where you can perform these calculations side by side. For example, if we enter nine months and six months, we get the corresponding hazard rates. You could also use survival probabilities. For example, if after twelve months the survival proportions were 50% and 33%, we could use those to obtain the corresponding parameters. In the simpler tables, there’s a side table that lets you perform the conversion directly. Once we enter our median survivals, we get a hazard ratio of approximately 0.667. Under the constant exponential proportional hazards model, the hazard ratio is directly related to the ratio of the exponential hazard rates. We enter 92.6% power. We get 176 subjects per group, or 352 in total, and 282 events. The main thing we’re concerned about is the 282 events. The calculation is effectively saying that if you require 282 events for this power, significance level, and hazard ratio, and you make these assumptions about the hazard rates, accrual pattern, and study length, then you’ll need 176 people per group. ## More Complex Log-Rank Designs We can do much more complex dropout and accrual modeling. Let’s look at a more complex log-rank test table. This is effectively a maximal version of the log-rank approach under proportional hazards. We can replicate the original example. The event count might be off by one because of rounding differences, but it’s effectively the same calculation. Here, we can specify multiple periods. For example, we might have an accrual period followed by a follow-up period. We can have different assumptions within each period. We can also choose between event-driven and fixed follow-up designs. Let’s assume, for example, that we have a fixed follow-up period of twelve months. This means that after twelve months, regardless of where the study is, the subject is administratively censored. We can then set the hazard ratio to 0.666 as before. We can use an alternative version of the Schoenfeld approximation where the sample size ratio is incorporated. It’s not as commonly used, but we keep it for completeness. We’ll use a sample size ratio of one and power of 92.6%. ### Piecewise Accrual and Hazard Now let’s use the side table to introduce a more complex scenario. Suppose we have a six-month period followed by the remaining study periods. We could set up three periods. Let’s assume there’s a ramp-up period during the first six months. Only 10% of the total accrual occurs during those first six months. The remainder occurs uniformly after that. Let’s also assume the hazard rate during the first six months is lower than expected. For example, we might use 60% of the original hazard rate. After six months, we return to the original hazard rate. The treatment-group values are automatically calculated based on the specified hazard ratio. ### Dropout We can also introduce a dropout process. Suppose that after twelve months we expect a 5% dropout rate. We can specify the time as twelve months and the survival proportion as 0.95. This is the surviving proportion for the dropout process, not the event proportion. nQuery converts this to the corresponding exponential dropout parameter. It’s important to understand why that parameter isn’t simply 0.05. The relationship depends on the time scale. If the time scale were one month, then the parameter would be interpreted as a monthly rate. But if we’re talking about a twelve-month period, we need the parameter corresponding to a 5% dropout probability over twelve months. So the conversion is important. We need to make sure that all the timing scales are consistent. When we put all of this together, we can see what happens. Under the original assumptions, we might have required 176 subjects per group. After introducing the slower accrual, lower event rate, and dropout process, the required sample size might increase substantially, for example to around 244 per group. But the required number of events remains almost exactly the same. That’s because the events are still driving the power. We’ve changed the assumptions about how we get those events. We’ve changed the accrual pattern. We’ve changed the hazard pattern. We’ve introduced dropout. All of that changes the sample size required to achieve the same information. ## Complex Survival Patterns and Non-Proportional Hazards Let’s move to the second section: complex survival patterns and non-proportional hazards. The most common assumption in clinical trials is proportional hazards. This is the assumption underlying the log-rank test and Cox proportional hazards regression model. In practical terms, it means that the hazard ratio, the ratio of the instantaneous risks between the two groups, is constant over time. The two survival curves may diverge over time, but the relative treatment effect is assumed to remain constant. ### Why Non-Proportional Hazards Matter In real survival trials, the proportional hazards assumption can break down. There is growing interest in methods that can handle these scenarios because newer treatment modalities, particularly immunotherapies, can have very different and often more complex survival effects than traditional cytotoxic treatments. Radiotherapy and chemotherapy often have relatively immediate treatment effects that may be reasonably approximated by proportional hazards. Immunotherapies can produce more complex patterns. There are several common examples. #### Delayed Effects There may be little difference between treatment groups initially, followed by separation later. #### Diminishing Effects The groups may diverge initially, but then converge later. #### Crossing Hazards One treatment may be better initially while the other becomes better later. The interpretation of such patterns from a clinical perspective can be complex. Is a treatment beneficial if it produces worse survival initially but better survival later? That is ultimately a clinical and regulatory question. The statistician needs to provide an appropriate summary of the observed survival experience, but the clinical meaning of that pattern needs to be considered separately. ### Statistical Methods for NPH Once we move beyond proportional hazards, the standard log-rank test and average hazard ratio may be underpowered relative to methods that are better suited to the specific survival pattern. Multiple methods have been proposed. Ideally, these methods should: * Improve power under plausible NPH scenarios * Be robust to different types of NPH * Provide interpretable estimates * Maintain appropriate type I error control There is ongoing debate about the best methods for analysis and testing in NPH clinical trials. In practice, the log-rank test and Cox model are still commonly used, even though they may be suboptimal under NPH. The resulting hazard ratio can be viewed as a weighted average treatment effect over the follow-up period. But whether that is a meaningful summary depends on the survival pattern. For example, with crossing hazards, a hazard ratio close to one may obscure important differences between the groups over time. There are several review articles and meta-analyses that summarize the proposed statistical methods for NPH trials. ## Methods for Non-Proportional Hazards The main methods include: ### Log-Rank Test and Cox Model As mentioned, these are essentially equivalent for a simple two-arm comparison with a single covariate. The Cox model becomes more flexible when additional covariates or time-varying effects are incorporated. ### Weighted Log-Rank Tests Weighted log-rank tests assign different weights to events depending on when they occur. The standard log-rank test gives approximately equal weight to events regardless of when they occur. A weighted test can assign more weight to early or late events. For example, if you expect a delayed treatment effect, you may want to downweight early events and give more weight to later events. The Fleming-Harrington family provides a flexible framework for this. ### Combination Tests The max-combo test builds on the weighted log-rank approach. Instead of choosing a single weighting scheme in advance, you select several candidate tests. The max-combo procedure then effectively identifies the most favorable test while maintaining overall type I error control. ### Restricted Mean Survival Time Another approach is to use the log-rank or another test for hypothesis testing and then use restricted mean survival time for estimation. RMST can be interpreted as the expected survival time up to a specified time horizon. The RMST difference is essentially the difference between the areas under the two Kaplan-Meier curves up to that time point. ### Other Models There are also: * Time-varying coefficient models * Change-point models * Spline models * Failure-time models * Relative-time models * Frailty models * Piecewise parametric models * Multi-state models * Competing-risk approaches * Cure models There are many different proposals in this area, although some are less common in confirmatory clinical trials. ## Max-Combo Tests The method I’m going to cover here, because we don’t have time to go through all of these in complete detail, is the maximum combination test, or max-combo test. The max-combo test combines multiple proposed linear rank tests. Commonly, this includes the log-rank test combined with different Fleming-Harrington tests. Usually, two to four tests are considered. There have been a number of proposals from different authors over the years. The weighted linear rank tests weight events differently depending on when they occur during follow-up. Some approaches weight early events more heavily. Others weight later events more heavily. The Fleming-Harrington family is particularly flexible. For example: * FH(0,0) approximates the log-rank test * FH(0,1) gives greater weight to later events * FH(1,0) gives greater weight to earlier events This means that if you select several tests, you can cover a range of possible NPH patterns. You can include a test appropriate for proportional hazards, one appropriate for delayed effects, one for diminishing effects, and potentially another for more complex patterns. The max-combo procedure effectively allows the data to determine which of these tests provides the strongest evidence, while retaining type I error control. Obviously, if you simply ran four tests independently, you would inflate the type I error. The max-combo procedure addresses that multiple-testing issue. ## Concerns With Max-Combo There are also criticisms of max-combo methods. Some authors have shown that under certain extreme survival patterns, a max-combo test can reject the null even when the survival curves seem to indicate a counterintuitive treatment effect. This can occur when events in one part of the follow-up period are heavily downweighted. For example, imagine a situation where the treatment group has consistently worse survival than the control group, but there is a very large number of early events. If the early events are heavily downweighted, a weighted test could potentially favor the treatment despite the treatment survival curve remaining below the control curve. This illustrates an important issue with weighted log-rank methods. The problem is not necessarily unique to the max-combo procedure itself. It can be inherited from the weighting functions used in the component tests. Several proposals have been made to address this. One approach is to use less aggressive Fleming-Harrington weights, such as 0.5 instead of 1. Another approach is the modestly weighted log-rank test, where the weighting is gradually increased during an expected delayed-effect period and then becomes constant. More recently, there have been proposals combining the modestly weighted log-rank test with the standard log-rank test. The goal is to retain good power under proportional hazards while improving robustness to delayed effects. So max-combo is not limited to the original proposals using Fleming-Harrington tests. You can combine a variety of component tests. The general principle is that the max-combo procedure may provide useful robustness, but the choice of component tests remains important. ## Max-Combo Example in nQuery Let’s look at a max-combo example. This example is based on work from a Cross-Pharma working group. The original proposal used a procedure involving individual test power calculations and simulation to estimate correlations. nQuery implements a more recent approach that allows this calculation to be performed directly. We can open the Max-Combo table in nQuery. We have: * One-sided significance level of 0.025 * Four component tests * Two periods The default tests include the log-rank test and Fleming-Harrington tests. Each component test can be edited individually. For example, if we wanted to use a weight of 0.5 instead of 1, that can be changed directly. nQuery also allows max-combo tests using a variety of other weighted log-rank methods, including Tarone-Ware, Peto-Peto, threshold lag, and modestly weighted log-rank tests. There is also the option to use the modestly weighted approach if that is the methodology you want to follow. We can specify an initial proportion of accrual, an accrual period, an additional accrual rate, and a follow-up period. For this example, we have an accrual period of fifteen months and a follow-up period of thirty-two months. We have a sample size ratio of one and a total sample size of 442. We then define the survival pattern. Suppose there is a six-month delayed-effect period. For the first six months, the hazard ratio is 1. After that, the hazard ratio becomes 0.56. We can enter those values directly into the side table. We can also model dropout. Suppose we have a dropout rate of 0.001 per month. Using the survival parameter converter, we can convert this into the corresponding exponential parameter. Similarly, we can convert the median survival parameters into the corresponding exponential hazard rates. When we put these values together, we obtain approximately 90% power for a total sample size of 442. The required number of events is also very similar to the original calculation. The small difference comes from the different calculation method and rounding. We can also see the power of each individual component test. The log-rank test might provide approximately 77% power. The Fleming-Harrington FH(0,1) test might provide approximately 94% power. The other component tests may provide lower power under this particular delayed-effect scenario. The advantage of max-combo is that we retain the ability to detect alternative NPH patterns without having to select a single test in advance. ## Comparing Different NPH Approaches We can summarize several proposed approaches under the same assumed delayed-effect scenario. The log-rank test might have around 77% power. FH(0,1) might have around 94%. A less aggressively weighted FH(0,0.5) might have around 91%. A threshold-lag test might have very high power because it is specifically targeted at the six-month delayed effect. But that comes at the cost of a strong assumption about when the delayed effect occurs. If the actual delayed effect occurs at a different time, that test could be substantially less powerful. The modestly weighted log-rank test might have around 85% power. That is lower than the highly optimized delayed-effect test, but it provides better robustness across a wider range of scenarios. These are the trade-offs you need to consider. If we convert these power values into sample sizes for 90% power, we can see the implications more directly. The log-rank test may require a substantially larger sample. The highly targeted delayed-effect methods require fewer subjects under the assumed scenario. The modestly weighted approach sits somewhere in between. So the question is not simply which test gives the highest power. You need to consider how credible the assumed survival pattern is and how robust you need the analysis to be if the actual pattern differs. I also ran a small simulation here to verify the expected power under the specified scenario. The simulation gives a result close to the analytical calculation. The simulation files for these examples will be sent with the slides and recording after the webinar. ## Adaptive Survival Analysis Trials Let’s move on to adaptive survival analysis trials. At a high level, an adaptive clinical trial is a trial design that allows prospectively planned modifications to one or more aspects of the trial based on interim analysis of accumulating data. That definition is from the ICH E20 draft guidance on adaptive designs. Examples include: * Early stopping * Group sequential designs * Sample size re-estimation * Adaptive allocation * Arm selection * Multi-arm multi-stage designs There has been increasing interest in adaptive designs as regulatory guidance and methodological expertise have improved. The FDA guidance from 2019 was particularly important in increasing acceptance of adaptive designs. The ICH E20 draft guidance also represents an important development. ### Why Survival Trials Are Natural Candidates for Adaptive Design Survival analysis is a natural target for adaptive design. We often have: * Long study durations * Large sample sizes * Large numbers of events * Natural information checkpoints We are already tracking the number of events because we need to know when the trial will end. So it’s natural to consider whether we can use those checkpoints for interim analyses. For example: * Can we stop early for efficacy? * Can we stop early for futility? * Should we increase the sample size? * Should we drop an arm? However, survival analysis creates some unique challenges for adaptive design. ## Timing of Interim Analyses One obvious but important issue is that we don’t know exactly when the final analysis will occur. The same applies to interim analyses. We don’t know exactly when an interim analysis will occur in calendar time. We can make assumptions based on expected accrual and event rates. But those assumptions may be wrong. A ramp-up period, for example, can have a disproportionate impact on an early interim analysis. This is much more important when you’re planning an interim after 50 or 100 events than when you’re planning a final analysis after 300 events. Therefore, you need a plan to deal with deviations in timing. ### Information Time Under proportional hazards, you can show that for many group sequential designs, the number of events and information time are directly related. If 50% of the required events have occurred, you have approximately 50% of the information. But that doesn’t mean you’re 50% of the way through the calendar duration of the study. The relationship between events, calendar time, and information needs to be considered carefully. ### Spending Functions If you’re doing a group sequential design, I would generally recommend a flexible method such as an alpha-spending approach. That allows you to make adjustments if the actual information time differs slightly from the original plan. For example, you might originally plan an interim analysis at 100 events but actually perform it at 95 or 102 events. A spending-function approach gives you flexibility to accommodate that without compromising the overall type I error control. ## NPH and Adaptive Design The challenge becomes greater once you move beyond proportional hazards. The direct proportionality between information and events no longer necessarily holds. The relationship becomes more complex depending on the NPH pattern and the analysis method. If you use an average hazard ratio, max-combo test, Fleming-Harrington test, RMST, or another statistic, the amount of information represented by a given number of events can differ. For example, with a delayed effect, an early interim analysis could occur before the treatment effect has had time to emerge. The hazard ratio might be close to one at the interim analysis even though there is a clinically meaningful treatment effect later. You could therefore stop too early and miss the eventual treatment effect. This creates both practical and statistical challenges. If you expect NPH, you might consider: * Later interim analyses * Flexible stopping criteria * Alpha-spending approaches * Calendar-time constraints * Information-time constraints * Simulation-based operating-characteristic evaluation Rather than simply saying, “We’ll stop after X events,” you might want a more flexible rule. For example, you might specify a calendar-time minimum and then require a minimum information fraction before making the decision. The exact approach depends on the trial. ## Operational Concerns There are also operational concerns. Overall survival may need to be measured regardless of whether PFS is the primary endpoint. You therefore need to consider how interim decisions based on PFS affect subsequent OS analysis. For example, perhaps interim analyses are based on PFS, while OS remains the ultimate primary criterion. You might stop early if PFS is sufficiently compelling but continue to follow subjects for OS. This creates questions around multiplicity and the sequential design strategy. ### Overrunning Survival trials also create a particularly important issue with overrunning. Because the trial is event-driven, you may have many subjects enrolled who have not yet experienced an event when an interim analysis occurs. If you stop early, what happens to those subjects? Do you include them in the final analysis? Do they continue to be followed? How does that affect the final inference? The number of subjects in this overrunning cohort can be substantial. That needs to be considered in the design. ## Sample Size Re-Estimation For sample size re-estimation, it’s more accurate in a survival trial to think in terms of **event re-estimation**. Power is driven by events, not sample size. If you decide to increase the number of events, you need to think about how those additional events are generated. Suppose you increase the sample size by recruiting more subjects. Those new subjects will generally contribute events later in follow-up. Alternatively, if you simply extend the study duration, you’ll obtain more events from subjects already in the trial. Those two approaches can create different distributions of event times. Under NPH, that can have implications for operating characteristics. If you knew there was a diminishing or delayed treatment effect, choosing one approach over another could potentially create operational bias. That is why the approach needs to be prespecified. You should not choose the method after seeing the emerging survival pattern in a way that could favor the desired outcome. ## Example: MONALEESA-3 Let’s look at an example from MONALEESA-3, evaluating ribociclib combination therapy. There are several complicated aspects to this example. We can first replicate the original design. We have: * Significance level * Accrual period * Maximum follow-up period * Exponential parameters * Hazard ratio * Dropout assumptions * Spending function As before, we can obtain the exponential parameters using the survival parameter converter. In this example, the median survival is nine months and the hazard ratio is 0.67. Notice that the treatment-group median survival was calculated using the control median survival and the hazard ratio rather than simply taking the ratio of two independently specified medians. That is an important distinction. ### Group Sequential Design We can specify the spending function. There are several options, including: * Lan-DeMets-type approaches * Hwang-Shih-DeCani * Beta spending * Gamma-type spending functions * Custom boundaries You can also specify your own custom boundaries on a p-value, Z-score, or other scale. For futility, conditional power or predictive power can be used. In this example, the information times are 0.5 and 0.8. That means the interim analyses are based on approximately 50% and 80% of the required events. It does not mean 50% and 80% of the calendar duration. The first analysis is a futility analysis. The second is an efficacy analysis. The final analysis occurs after the target number of events. The boundaries can be expressed on a Z-score scale. For example, under this particular parameterization, a sufficiently negative Z-score could trigger efficacy stopping. Alternatively, we can view the same boundaries on the p-value scale. This can make the decision criteria easier to communicate. ## Does the Adaptive Design Actually Add Value? The important question is whether the adaptive design actually adds value compared with a fixed design. The group sequential report gives us expected event counts and stopping probabilities. Under the null hypothesis, there may be a reasonably high probability of stopping early for futility. That’s useful. Under the alternative, there may be a high probability of stopping early for efficacy. That’s also useful. But we need to look at the calendar time and number of subjects involved. For example, under the null hypothesis, the futility analysis might occur around fourteen months into the study. By that point, perhaps 466 subjects have already been recruited. If we stop, we might only save around 130 subjects. That still has value, but perhaps not as much as initially expected. At the efficacy analysis, perhaps most or all of the planned subjects have already been recruited. If that’s the case, what value do we gain by stopping early? We might only save a few months of follow-up. That may or may not justify the complexity of the interim analysis. This is a useful example because it demonstrates that adaptive designs need to be evaluated based on their actual operating characteristics. You shouldn’t assume that an adaptive design automatically improves efficiency. You need to quantify the efficiency gain. Interestingly, in a later version of the protocol, the efficacy interim analysis was removed. The authors reported that additional information changed their perspective. But there is also a practical argument: if the full sample has already been recruited by the time the efficacy analysis occurs, the benefit of stopping early may be relatively limited. This is a good example of why adaptive designs should be evaluated carefully before implementation. ## Simulation nQuery also includes simulation tools. I won’t have time to go through the simulation in detail, but I’ve prepared two examples. One assumes a hazard ratio of 0.7. The other uses the original target hazard ratio of 0.67. These simulations can be compared against the published protocol and supporting materials. The simulation results reproduce the key operating characteristics reported in the original example. The saved files for these simulations will also be provided with the slides and recording. ## Conclusions Let’s summarize. Survival analysis is one of the most common endpoints in clinical trials and has distinct design and statistical challenges. Sample size determination requires first determining the required number of events and then determining the sample size needed to obtain those events given assumptions about accrual, hazard rates, censoring, and study duration. Some methods tie the event and sample size processes together more closely, but in practical terms, events are what drive the statistical information. Non-proportional hazards is an increasingly important area due to the emergence of immunotherapies and other treatments with delayed or otherwise complex effects. There is now a wide range of statistical approaches for dealing with NPH. Tools such as those available in nQuery make these approaches more accessible during trial design. Adaptive design offers opportunities to improve efficiency in survival trials, but there are additional practical and statistical challenges compared with binary or continuous endpoints. The timing of interim analyses is less predictable because it depends on the accrual process and the event process. This becomes even more complicated under NPH because the relationship between events and information is no longer straightforward. It’s therefore important to make sure that your adaptive design actually adds value compared with an equivalent fixed design. You should also consider the implications of overrunning, sample size or event re-estimation, multiplicity, and potential operational bias. Simulation is particularly useful for understanding how these designs behave in practice. ## Using nQuery If you want to use nQuery, or if you want to explore all the tiers of nQuery, including the adaptive design features shown toward the end of the webinar, those features are available in the Pro tier. You can go to the nQuery demo page for a fourteen-day trial. No additional setup is required, and you can use nQuery directly within your browser. ## Further Information If you want to get in contact with me, my email is available on the slide. The nQuery website also includes a repository of previous webinars covering topics such as Phase II designs, survival analysis, and ICH E20. I’ve also provided links to some relevant webinars and references, including the references used for the examples and the key literature on survival analysis, non-proportional hazards, and adaptive design. ## Questions We’ve run a little over time, so I’m afraid I won’t have time to answer the questions live. I have seen the questions in the Questions tab, and I’ll answer them via email afterwards. The slides, recording, and example files will also be provided following the webinar. Thank you for joining today’s webinar. I hope it was useful. If you have any additional questions, please get in touch using the contact details provided. Thank you very much, and have a good day.