Figures & data
Figure 1. Four displacement vectors are shown in the x–y plane. Each is symmetrically distributed about the x-axis and the y-axis. The mean of the vector components in the x- or y-directions would be reported as zero even though each individual vector is much greater than zero.
Figure 2. Diagram showing how the direction of the mean migration has been reported by others. The length of the line at each landmark signifies the mean migration and its angle gives the direction of the mean (see text). The magnitude of the ellipse is related to the standard error.
Figure 3. Diagram showing why the resultant of several vectors cannot be used to determine the mean direction of the vectors. The mean direction calculated using the resultant (i.e. 29 degrees) is dominated by the largest (6-mm) displacement vector. The true mean direction angle is 40 degrees.
Figure 4. (a) The mean direction cannot be determined as the mean of the vector angles from the positive x-axis; here, this would be 100 degrees, but the mean direction should be in the bottom right quadrant. (b) If the angle is measured in positive and negative (clockwise) directions with respect to the positive x-axis, the mean can still not be determined; here, it would be zero degrees, which is incorrect.
Figure 5. Vector plane chart (created using RSA DataViewer: www.orthomech.co.uk) showing all the vectors in the transverse plane at the 6-month stage of a cemented hip stem study. The ellipse shows the variation of the resultant of the repeatability (precision) error in the transverse (x–z) plane. The intersection of the ellipse with the x-axis (mediolateral) and z-axis (anteroposterior) is the precision (determined from double examinations) in the x- and z-directions, respectively. A summary of the vector directions is given by the percentage of vectors (beyond the repeatability error, i.e. outside the ellipse) in each direction quadrant. In this case, 36% of the vectors were within the ellipse and they were not therefore considered in the calculations of the proportions in each quadrant. The mean scalar value of all the vectors in that plane is shown at the bottom of the chart.
Figure 6. Variation of the upper confidence limit of the population percentage failure rate with sample size for different numbers of failures in the sample. The horizontal line indicates a 10% threshold of acceptability for the population failure rate. Even if there were no predicted failures in the RSA sample, a minimum sample size of 36 would be required for the upper confidence limit to fall within the 10% threshold. If there was just 1 predicted failure, the sample size would have to be at least 54.
Table 1. This table shows that even a well-performing femoral stem design (as determined from arthroplasty registers) could have a high probability of being designated as a possible future failure by an RSA study. Left column: here, a well-performing femoral stem design is considered to have a population failure rate of up to 5% (aseptic survival rate: 95%). The probabilities of RSA revealing possible failures (excessively migrating stems) in samples of 36 and 54 of such stem designs are shown in columns 2 and 3.
Table 2. In order to have an 80% chance that RSA would not reveal possible failures (excessively migrating stems) in a test sample of well-performing femoral stems (as denoted by the low failure rates in the left column), the RSA sample size would have to be as shown. Columns 2 and 3 show the minimum sample size required when the upper 95% confidence limit of the failure rate is 10% or less (the generally accepted failure rate benchmark for hip prostheses at 10 years). Columns 4 and 5 show the sample size when this confidence limit is raised to 15%. The effect of using a 1-tailed or 2-tailed test for the sample size calculation is also shown