Thomas J. Santner

Planning the duration of a comparative clinical trial with loss to follow-up and a period of continued observation.

Abstract The purpose of this paper is to assist in designing clinical trials which are to be analyzed with the Mantel-Haenszel (logrank) test for comparing survival curves. The results allow for loss to follow-up during the trial and let the planner reduce the total number of patients required by introducing a period of continued observation after the end of patient accrual. For simplicity, the actual calculation of the required trial length is based on use of the following parametric test. Assuming time to death is exponential, the ratio of the maximum likelihood estimators of the hazard rates may be used to test the null hypothesis that the hazard ratio is equal to one. A method is developed to compute the approximate trial length required to assure a desired statistical power for given significance level, hazard ratio, accrual rate, loss to follow-up rate, and length of the period of continued observation. Tables are included which give the required trial length for a wide range of cases. A figure is included which depicts the effects on the required trial length of loss to follow-up and demonstrates the practical advantages of a period of continued observation. Computer simulations verify that this method gives trial lengths which yield accurate power when used with the non-parametric Mantel-Haenszel (logrank) test. Further simulations establish the robustness of the calculated trial lengths. In particular, they demonstrate that the trial lengths yield approximately the desired power for the Mantel-Haenszel test when the death times are Weibull rather than exponential.

Arthrodesis as an Early Alternative to Nonoperative Management of Charcot Arthropathy of the Diabetic Foot

Background: This study was performed to evaluate the use of arthrodesis of the tarsal-metatarsal area for the treatment of Eichenholtz stage-I Charcot arthropathy in patients with diabetes. Currently, the standard treatment of stage-I Charcot arthropathy is the application of a non-weight-bearing total-contact cast. Although this treatment can be effective for allowing a patient to walk without undergoing an operation, a nonunion or malunion may still result. The subsequent deformities may lead to complications, including ulceration of the foot and the need for operative intervention. Recently, a group of patients who had had early operative intervention for a variety of reasons provided us with the opportunity to objectively evaluate the effects of such treatment. This analysis provided valuable information about whether this treatment is a reasonable alternative to current nonoperative approaches. Methods: Between January 1991 and December 1996, fourteen patients had an operation because of Eichenholtz stage-I diabetic neuropathy. The classification of the disease as Eichenholtz stage I (the developmental stage) was based on radiographic evidence of varying degrees of articular-surface and subchondral-bone resorption and fragmentation as well as joint subluxation or dislocation without evidence of coalescence or callus formation. The operative procedure consisted of extensive débridement, open reduction, and internal fixation of the tarsal-metatarsal region with autologous bone graft. Postoperative treatment consisted of immobilization of the limb in a non-weight-bearing cast for a minimum of six weeks. All of the patients returned for a final follow-up visit at a mean of forty-one months (range, 25.3 to 77.3 months) postoperatively, at which time clinical and radiographic evaluations as well as gait analysis (with measurement of plantar pressures) were performed. The gait-analysis data was compared with similar data from a group of fourteen patients with diabetic neuropathy who had had a below-the-knee amputation and with that from a group of fourteen patients with diabetic neuropathy who had no history of plantar ulceration. Results: All of the arthrodesis procedures were successful. Clinically, none of the patients had immediate or long-term complications postoperatively. No patient reported ulceration after the operation. The mean time to assisted weight-bearing was 10 ± 3.3 weeks (range, six to fifteen weeks), the mean time to unassisted weight-bearing was 15 ±8.8 weeks (range, eight to thirty-four weeks), and the mean time to return to the use of regular shoes was 27 ±14.4 weeks (range, twelve to sixty weeks). All of the patients regained the level of walking ability that they had had prior to the arthropathy. The calculated confidence intervals revealed no differences between the arthrodesis group and either of the two comparison groups with regard to the time-distance gait parameters of velocity, cadence, and stride length or with regard to the minimum, maximum, and total range of motion of each of the joints. In contrast to able-bodied subjects, all three groups showed a reduction in sagittal-plane ankle motion that was primarily related to loss of plantar flexion. The first metatarsal, great toe, and heel showed the highest peak plantar pressures, with little difference among the groups. Conclusions: To our knowledge, the present study is the first to demonstrate the potential for early operative treatment to restore anatomical alignment and improve function of diabetic patients with stage-I Charcot arthropathy.

Torque production in the shoulder of the normal young adult male. The interaction of function, dominance, joint angle, and angular velocity.

Shoulder torque measurements were obtained from 36 normal young adult males during flexion, abduction, internal rotation, and external rotation. Normative, an gle-specific torque measurements were determined un der isometric conditions and isokinetically at 48 deg/ sec. The effects of dominance, angular velocity, and joint position were determined and a method provided for determining the expected maximal torque at a spe cific speed and position based upon torque measure ments from the contralateral shoulder. This method can be used in cases of unilateral shoulder abnormalities to predict normative torque values for the affected side based on torque measurements from the unaffected shoulder.

Small-Sample Confidence Intervals for p 1– p 2 and p 1/p 2 in 2 × 2 Contingency Tables

Abstract Consider two binomial populations II1 and II2 having “success” probabilities p 1 in (0, 1) and p 2 in (0, 1), respectively. This article studies the problem of constructing small-sample confidence intervals for the difference of the success probabilities, Δ = p 1 – p 2 and their ratio (the “relative risk”), ρ = p 1/p 2 based on independent random samples of sizes n 1 and n 2 from II1 and II2, respectively. These are nuisance parameter problems; hence the proposed intervals achieve coverage probabilities greater than or equal to their nominal (1 – α) levels. Three methods of constructing intervals are proposed. The first one is based on the well-known conditional intervals for the odds ratio ψ = p 1(1 – p 2)/p 2(1 – p 1). It yields easily computable Δ and ρ intervals. The second method directly generates unconditional intervals of the desired size. An algorithm is given for producing the intervals for arbitrary n 1 and n 2. The 2 × 2 case is given as an illustrative example. The third method const...

Design and Analysis of Robust Total Joint Replacements: Finite Element Model Experiments With Environmental Variables

Computer simulation of orthopaedic devices can be prohibitively time consuming, particularly when assessing multiple design and environmental factors. Chang et al. (1999) address these computational challenges using an efficient statistical predictor to optimize a flexible hip implant, defined by a midstem reduction, subjected to multiple environmental conditions. Here, we extend this methodology by: (1) explicitly considering constraint equations in the optimization formulation, (2) showing that the optimal design for one environmental distribution is robust to alternate distributions, and (3) illustrating a sensitivity analysis technique to determine influential design and environmental factors. A thin midstem diameter with a short stabilizing distal tip minimized the bone remodeling signal while maintaining satisfactory stability. Hip joint force orientation was more influential than the effect of the controllable design variables on bone remodeling and the cancellous bone elastic modulus had the most influence on relative motion, both results indicating the importance of including uncontrollable environmental factors. The optimal search indicated that only 16 to 22 computer simulations were necessary to predict the optimal design, a significant savings over traditional search techniques.

Prediction for Computer Experiments Having Quantitative and Qualitative Input Variables

This article introduces a Bayesian methodology for the prediction for computer experiments having quantitative and qualitative inputs. The proposed model is a hierarchical Bayesian model with conditional Gaussian stochastic process components. For each of the qualitative inputs, our model assumes that the outputs corresponding to different levels of the qualitative input have “similar” functional behavior in the quantitative inputs. The predictive accuracy of this method is compared with the predictive accuracies of alternative proposals in examples. The method is illustrated in a biomechanical engineering application.

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.

Robust Optimization of Total Joint Replacements Incorporating Environmental Variables

Direct search techniques for the optimal design of biomechanical devices are computationally intensive requiring many iterations before converging to a global solution. This, along with the incorporation of environmental variables such as multiple loading conditions and bone properties, makes direct search techniques infeasible. In this study, we introduced new methods that are based on the statistical design and analysis of computer experiments to account efficiently for environmental variables. Using data collected at a relatively small set of training sites, the method employs a computationally inexpensive predictor of the structural response that is statistically motivated. By using this predictor in place of the simulator (e.g., finite element model), a sufficient number of iterations can be performed to facilitate the optimization of the complex system. The applicability of these methods was demonstrated through the design of a femoral component for total hip arthroplasty incorporating variations in joint force orientation and cancellous bone properties. Beams on elastic foundation (BOEF) finite element models were developed to simulate the structural response. These simple models were chosen for their short computation time. This allowed us to represent the actual structural response surface by an exhaustive enumeration of the design and environmental variable space, and provided a means by which to validate the statistical predictor. We were able to predict the structural response and the optimal design accurately using only 16 runs of the computer code. The general trends predicted by the BOEF models were in agreement with previous three-dimensional finite element computer simulations, and experimental and clinical results, which demonstrated that the important features of intramedullary fixation systems were captured. These results indicate that the statistically based optimization methods are appropriate for optimization studies using computationally demanding models.

Maximum likelihood estimation of the survival function based on censored data under hazard rate assumptions

Kiefer-woifowitz (1956) maximum likelihood estimators of the survival function are derived under the model assumptions of (a) increasing, (b) decreasing, and (c) U-shaped hazard rate based on right censored data. A strong consistency theorem is proved. Monte Carlo small sample comparisons are made with the maximum likelihood estimators based on the correct parametric model and the fully nonparametric maximum likelihood (product limit) estimator. Applications to covariate analysis are discussed.

Acetabular cup geometry and bone-implant interference have more influence on initial periprosthetic joint space than joint loading and surgical cup insertion

Environmental variations in patient-dependent and surgical factors were modeled using robust optimization with a finite element acetabular cup-pelvis model. A previously developed statistical optimization scheme was used to: (1) determine the cup geometry and the optimal cup-bone interference that maximized bone-implant contact areas and minimized changes in the gap volume between the implant and bone surface during gait loading and unloading; and (2) determine the relative contributions of design, patient-dependent, and surgical factors to variations in bone-implant contact areas and a change in gap volume. The statistical analyses indicated that the design variables, namely the equatorial diameter and eccentricity, explained most of the variations in the performance measures. Further, the hemispherical designs performed better than the nonhemispherical designs. The 58 mm hemispherical cup, with 2 mm diametral interferences, minimized the change in gap volume and attained 82% and 81% of the maximum predicted total and rim contact areas, respectively. The equatorial diameter and eccentricity, not the patient-dependent and surgical factors, explained most of the variations in the performance measures. Perfect surface apposition was not attained with any of the cup designs.