We analyzed a substantial amount of data on fibrosis progression using a new method that avoids many problems inherent in other methods that have been used to analyze such data. We found evidence that progression risk decreases after more time has been spent in a stage, which concords with an analysis of progression following liver transplant that used the same methods 
; methods previously used for analyzing fibrosis progression cannot assess such effects. This finding may reflect a dynamic nature of HCV infection, with recent progression indicating active disease and a higher risk of further progression. Older age increased progression, and this appeared to be driven by current age rather than being a fixed effect of age at HCV infection (the evidence for this is not conclusive, however, as shown by the wide confidence intervals in the model in that includes both age effects). This also accorded with the previous analysis of post-transplant progression, where progression increased with donor age. Other previous analyses have also found an age effect 
but were limited by their methodology to evaluating presumed age at HCV infection rather than current age; they also did not recognize or assess potential bias 
. A small simulation experiment indicated that little of our observed age effect appeared to be due to bias. We also found evidence for a protective effect of African American race, which has previously been suspected 
. There was a suggestion of increased risk due to HIV co-infection, particularly in the era of effective anti-retroviral therapy beginning in 1996. In prior years, HIV-infected potential participants who experienced accelerated HCV progression may have also had higher mortality from opportunistic infections, causing them to be excluded from our source studies. There was also a suggestion that slower progression through stage 0 tends to be followed by slower progression through stage 1, but this did not hold for later transitions between stages. A number of other factors may influence progression, as some estimates in may be large enough to be important (e.g., male sex, heavy alcohol consumption, and body mass index >30) and upper confidence bounds generally are not low enough to provide strong evidence against substantial increased risk.
The results here may seem to be less reliable than previous studies of similar data because of the complex methods and assumptions. The complexity, however, is inherent in the available data and the disease process. Previous studies only appear to avoid this by making strong simplifying assumptions that are implicit or not given strong emphasis. Consider, for example, the simple approach of obtaining a single fibrosis rate per year for each person by dividing current observed stage by the time since presumed infection 
. This implicitly assumes that infection is immediate at the reported time of first risk with no inaccuracy in those reported times, fibrosis is never misclassified, each progression between stages is numerically equivalent, and the observed stage was just reached at the time of biopsy. Each of these assumptions simplifies the statistical analysis and reduces the apparent statistical variation in resulting estimates, but each is also questionable or even known to be wrong. We have attempted to deal realistically with these complexities. Notably, using multiple imputation 
to address the unknown ages at HCV infection adds considerable complexity and results in wider confidence intervals than would have been produced by pretending that ages of HCV infection were all known, but this uncertainty really does exist. We have also used multistate models, which better match biopsy-based measurement of the disease process, and have evaluated departures from the Markov assumptions usually used in multistate modeling. We checked linearity and variation by stage for key predictors, and sensitivity analyses suggested that our main results did not rely on the particular size of time step, misclassification probabilities, or imputation model for age of HCV infection. We believe that all these facts add credence to our results.
Our methods permit analysis of time-varying covariates, which was important for HIV and age. The distinction between current age (time-varying) and age at HCV infection (fixed) may seem subtle, but they could have different implications for the biology of HCV disease and also for clinical prognosis. For example, detecting recent progression to higher fibrosis could be cause for alarm in an older patient even if original HCV infection was at a very young age. This is the second infectious disease for which one of us (PB) has found that careful consideration of both fixed and time-varying age effects points to a different conclusion than only considering fixed effects 
Despite the specialized analyses and other strengths, this study has a number of limitations. As for many other studies using liver biopsies, selection bias is a potential concern. Restricting study to clinic populations or those already known to be HCV infected can create selection bias toward more rapid progression 
, and the SFVA and HALS groups are clinic-based or partly clinic-based. For ALIVE, there should be little selection bias, because participants were selected from the community without respect to HCV status, enrolled participants were tested for HCV, and a random sample of those found to be chronically infected underwent biopsy 
. For all three studies, participants had to agree to undergo liver biopsy in order to be included; this could select for more severe disease if participants were more likely to agree if they had symptoms. On the other hand, the unavoidable restriction to participants who were alive at the time of recruitment could tend to exclude more rapid progressors. Statistical methods for dealing with this, known as left-truncation
or late entry
, are available but would require a model that includes death from fibrosis progression as an additional stage; calculations to deal with late entry are also not available in the mspath software. Our selection of only followup before treatment with interferon could also tend to exclude more rapid progressors, although this may be mitigated by the usual clinical desire to obtain a biopsy before starting treatment. Selection bias may be most important for estimation of overall rapidity of progression rates, which we have not emphasized. For the estimated effects of risk factors to be biased, selection for greater severity would have to differ according to the levels of the risk factors. Because selection likely did differ between source studies, we took steps to fully control for the effect of study (see next paragraph).
The distribution of biopsy-measured fibrosis was heterogeneous across source studies. The strong influence of source study, and its variation by stage, likely result from non-biological influences. Differing misclassification of fibrosis due to differing readings of biopsies is one important possibility, but differing selection of participants is also likely to contribute to the study effects. For example, it is unlikely that the true rate of stage 4 (cirrhosis) was really the same in HALS as in the other studies; this would imply that about 10 biopsies showing cirrhosis were all misread in HALS. Because the studies were drawn from differing populations by differing methods, selection effects are bound to differ between them. The proportions of biopsies from different calendar time periods also differed between studies. The imputation models for age at HCV infection accounted for strong influences of calendar time 
; changes over time in progression rates may also be possible, but seem less likely with our focus on pre-treatment biopsies only. We minimized the potential for source study to confound other associations by fully controlling for its influence on every transition, using a full contingent of 8 parameters for study effects. This reduced the potential influence of selection bias on other estimates, but it also added complexity to the models and may have reduced the precision of other estimates. Using fewer parameters did not seem viable, because differences between the studies varied by stage, and because any oversimplification of study effects could increase concern about selection bias and confounding.
We accounted for possible misclassification of biopsy-measured fibrosis by factoring external estimates of misclassification rates into the estimation process. This increases the statistical uncertainty in our results, but rightly so. Along with multiple rather than single imputation of age at HCV infection, this results in a better assessment of precision and is a strength of this analysis. In addition, a sensitivity analysis showed that our main findings were insensitive to the exact misclassification assumptions. Nevertheless, an ideal approach would utilize information on biopsy quality to provide a customized matrix of misclassification probabilities for each biopsy. Unfortunately, we did not have such refined estimates available. Some multistate modeling methods, including the one used here, can estimate misclassification probabilities as part of the modeling process. This, however, would be computationally very challenging and seems likely to be less accurate than estimates from studies that were focused specifically on misclassification and therefore employed multiple readings of the same biopsy or multiple biopsies of the same liver. We have therefore used the best estimates from such studies that we could obtain 
. The median (interquartile range) biopsy length was 19mm (14–24) in HALS and 12 mm (9–19) in ALIVE, which are comparable to those in the studies used to estimate misclassification 
. (Biopsy length was not available for the SFVA study.)
We assumed no backward transitions to lower stages; our restriction to pre-treatment biopsies may make this assumption reasonable, and no participants had spontaneously cleared HCV infection at the time of any biopsy. This assumption implies that any apparent backward transitions must be ascribed to misclassification of at least one of the biopsies. Among the 159 participants with two or three biopsies, 21% had nominal backward transitions of one stage and 3% of two stages, rates that seem to be readily explainable by the substantial misclassification probabilities previously estimated 
. The progression model could in principle be extended to allow backward transitions, and the mspath software allows any specification of what transitions are possible. Unfortunately, allowing backward transitions would vastly increase the number of possible paths to be evaluated, making computations infeasible. In addition, parameters governing the four additional transitions would have to be added, and factors influencing regression could differ from those for progression, substantially complicating the modeling. We therefore cannot evaluate the effect of allowing backward transitions on our results, but we believe that excluding them is close enough to reality that it is unlikely to introduce serious bias.
A reviewer pointed out a potential bias that could impact our estimated effects of years in prior stages, at the bottom of . For example, if a participant were known to have reached stage 2 by a given time, with an unknown time of transition from stage 0 to 1, then a shorter time in stage 0 would mean a longer time in stage 1 before progressing to stage 2, while a longer time in stage zero would mean a shorter time in stage 1 before progressing to stage 2. Either way would contribute to an apparent effect of longer time in stage 0 increasing risk of progression from stage 1 to 2. Fortunately, we observed the opposite of what this bias would produce, instead estimating a protective effect of longer time in stage zero. Thus, the possibility of this bias only strengthens the evidence for this effect. For the 2 to 3 and 3 to 4 transitions, we did estimate increases in risk such as this bias would produce. This bias is directly analogous to the potential bias in the estimated effect of age at infection that we described, with time in previous stages playing the role of time before (i.e., age at) HCV infection, so a similar simulation-based investigation could be undertaken. We do not consider this to be worthwhile, however, because the issue is largely overshadowed by the very wide confidence intervals for these effects.
Most of our participants had only one biopsy, which made them less informative for estimation of the effects of years in current stage and years in previous stages. Fortunately, we had enough with multiple biopsies (15%) to obtain some useful estimates, notably the protective effects of years in current stage in for the 0 to 1 and 2 to 3 transitions. We also had relatively few participants with advanced fibrosis, which is reflected in extreme estimates or wide confidence intervals for most estimated effects that are specific to the 2 to 3 transition and the 3 to 4 transition. Some predictors may have been inaccurately measured, due to reliance on self-report and extrapolation of study values to the entire period of HCV infection, and some were missing for a considerable proportion of participants. We did not analyze HCV genotype or viral load as predictors, and complete history of alcohol consumption was only available in the HALS study. We did not evaluate the influence of measured immune status and antiretroviral treatment history directly for HIV-infected participants. This would be very complicated and could be distorted by self-selection of treatment and incomplete histories; we investigated differing HIV effects over calendar time as a feasible alternative. Our statistical methods model progression over the entire period of HCV infection, so we could not evaluate factors such as inflammation and steatosis grade that are known only at the time biopsy; these would have to be known at all times and treated as time-varying covariates, but are unlikely to have remained constant since infection. Our assumptions concerning fibrosis misclassification and our model for imputing age of HCV infection could be inaccurate, but sensitivity analyses using alternatives showed similar results. All of our fitted models had some parameters estimated to be effectively infinite. These could not be used in standard methods for multiple imputation analysis, so we used only the finite estimates and reported the infinite estimates separately. In some cases, standard errors for other parameters had to be obtained by re-estimating models with the effectively infinite parameters held fixed; those standard errors did not appear to differ substantially from the cases where standard errors were available despite estimation of some effectively infinite parameters. Finally, the computational burden of the method we used is substantial. We were only able to complete our analyses within about a month by often running 20 or more analyses simultaneously using a specialized parallel computing facility.
Ideally, prospective followup of persons known to be recently infected with HCV would prevent selection biases and maximize the value of information obtained from biopsies. Performing such studies, however, might be difficult and expensive. Steps to minimize misclassification (e.g., using multiple readings of each biopsy) could also make data more informative and potentially reduce the computational burden of our methods (if some misclassifications become impossible). The methods used here might provide more credible results and additional insights if applied to larger data sets with more repeat biopsies. Computational feasibility is a potential issue, but will improve with time. The non-Markov multistate modeling software that we used is freely available at http://cran.r-project.org/web/packages/mspath/index.html
An appealing alternative to biopsy is non-invasive measurement of fibrosis via imaging or biochemical analysis of peripheral blood samples 
. Some methods may already be as accurate as biopsy, but evaluation of alternatives has suffered from inappropriate use of biopsy as a gold standard 
. Studies with frequent non-invasive measurements could be less dependent on imperfectly known times of HCV infection, because they could better focus on observed trajectories. Evaluating how recent changes in non-invasive measures predict subsequent change would permit exploration of the phenomenon we found of decreasing progression risk with longer times already spent in a stage. If the invasiveness, risk 
, and expense of biopsy curtails its use in HCV research, the methods used here may still be useful for analysis of data for other diseases that progress through stages.