Our study sample consisted of patients who, after providing informed consent, participated in a prospective THR registry at the Hospital for Special Surgery (HSS), New York, and accrued between April 30th, 2007 and October 3rd, 2008. The registry is approved by the HSS Institutional Review Board and conforms to the Helsinki Declaration. Primary total hip replacement patients filled out the Hip disability and Osteoarthritis Outcome Scores questionnaire (HOOS) [8
] as part of a battery of preoperative surveys. We calculated the WOMAC subscale scores of pain, stiffness and function from the HOOS questionnaire data. The HOOS includes the WOMAC in its complete form in addition to other questions about more strenuous activities. The 5 'Pain' questions in WOMAC were embedded with 5 additional questions in the HOOS pain subscale; 2 Stiffness questions of WOMAC embedded in 5 'Symptoms' related questions in HOOS; and the 17 'Physical Function' related questions were exactly the same in both questionnaires.
A univariate analysis was conducted on each individual WOMAC item in the attained data set to calculate the item-specific rate of missing values. An analytic data set was then created by deleting all subjects with any incomplete responses; this dataset thus contains only subjects with complete information whereby the true score was known for all subjects.
For each item in the analytic dataset, we introduced missing values at the same rate as observed in the attained dataset by randomly drawing subjects and deleting their data for that item. The goal was to recreate the same missing data percentage item by item as observed in original attained dataset. To compare methods, we first calculated the scores for the 3 subscales for complete cases (CC), i.e. the sum of item scores in each subscale for the subset of the analytic dataset that contains only cases with complete information. The purpose of this step is to set a baseline to which we compared the added value of the 2 imputation methods. Subsequently we employed the WOMAC imputation method, and the EM imputation method.
The WOMAC method is a variant of a standard mean imputation method. In the case of missing data, scores of the non-missing items for each case were added and the mean value was used to impute for the missing values. However, if the patient has not replied to one or two stiffness questions, one or two of the five pain questions or four or more of the 17 physical function questions were considered non-scorable.
The EM imputation method is a deterministic iterative algorithm that determines the maximum likelihood estimates of the parameters of the distribution which the complete (missing and observed) data are assumed to follow. We assumed that the data followed a multivariate normal distribution. At each iteration, in the first step (E-step), the conditional expectation of the log-likelihood of the complete data is evaluated, where the expectation is taken with respect to the distribution of the missing data conditional on the observed data and the parameters estimated at the previous iteration. In the second step (M-step), the expected log-likelihood evaluated in the E-step is maximized and new estimates for the parameters are obtained. The iterations are repeated until convergence is reached.
This exercise of creating missing data, calculating CC, WOMAC method, and EM subscale scores was repeated 1000 times in order to observe the variability and sensitivity of the results. For each time, the 3 subscale scores as well as the number of patients for which a score was successfully computed were recorded. Since the percent missing in our data was low and the cohort size was much larger than a conventional orthopedic study [3
], we re-analyzed our data using a sample of 200 randomly selected patients from our analytic dataset. We also created missing values at random that were 5 times the rates observed in our original attained dataset, however, capping the rate of missing values at 45% to account for the items which had high rates of missing values.
The results are displayed in a box-plot, which is a graphical way to depict summary statistics for data: a rectangle is drawn that extends from the first quartile to the third quartile, with a bar that identifies the median value. The height of the rectangle thus indicates the inter-quartile range (IQR). In addition, two whiskers are drawn: the upper whisker corresponds to the largest observed value within 1.5 times the IQR above the third quartile, and the lower whisker corresponds to the smallest observed value within 1.5 times the IQR below the first quartile. Observed values outside the two whiskers (outliers) are drawn as points.