We appreciate Riddle et al's (Riddle & Dumenci, 2021) interest in our article on pain profiles after total knee arthroplasty (TKA; Lindberg et al., 2020) and the opportunity to discuss this important clinical problem of common concern.

We agree that long‐term pain interference is an important outcome following TKA. Thus, we previously published a paper on the trajectory of pain‐related interference with walking. (Lindberg et al., 2016). Similar to our current article, we identified two subgroups of patients with distinct trajectories over time based on Growth Mixture Modeling (GMM) of pain interference scores during the first 12 months after TKA. Approximately 22% of the included patients had only temporary improvements up to 3 months following surgery, followed by an increase in pain‐related interference with walking, returning to preoperative levels 12 months after TKA. We also identified pre‐operative symptoms and psychological factors associated with recurrent pain interference with walking.

Studies have indicated that osteoarthritis patients describe two distinct pain experiences (Hawker et al., 2008): a dull, aching pain, assumed to be reflected in our ‘average pain’ measure, and a more intense, sharp, often unpredictable pain, which may be reflected in our worst pain measure. This was our reason for including both ‘lower average’ and ‘lower worst’ pain as outcomes.

We chose to use GMM with polynomial coefficients based on our understanding that the estimation of a curvilinear trajectory requires the joint estimation of the polynomial terms all being in the model at the same time. Because the quadratic component of the trajectory is estimated as a departure from linearity, the linear slope must also be included in the model. The single trajectory is a nonlinear slope with a single bend in the curve. In our case, the first part of both trajectories starts with a strong rate of change, followed by a more gradual descending trajectory beginning at month 3. This form for a quadratic curve is not unusual for the change in a measure immediately following an intervention like surgery, then followed over a long period of time. A piecewise model may have been a useful alternative to the linear/quadratic trajectory we employed, had there been more assessments across time. However, with assessments prior to surgery, at 4 days, 6 weeks, followed by the long interval of 3 months to 1 year, the possible pieces that could be fit are really only a couple of straight lines, which seems to us to be less sensitive as an estimation for the shape curves than a polynomial continuous curve.

We used Cohen's d as a standardized effect size to calculate and quantify the magnitude and clinical meaningfulness of the differences in the mean scores of preoperative covariates, between patients in the higher versus lower classes. We did not use Cohen's d to separate the classes. Based on recommendations by Muthén & Muthén (Muthén, 2021), (See B. Muthén, Discussion Topic ‘Goodness of Fit measures for mixture models’, ‘http://www.statmodel.com/discussion/messages/13/119.html?1453602211’, August 14th at 4:07 p.m.) entropy is not an appropriate measure to select the best fitting number of classes. Our use of entropy is merely descriptive and may be interpreted as a measure of consistency between the classifications of cases in the model‐based classes versus the predicted latent classes. Instead, they recommend that the Bayesian information criterion (BIC) be used as the primary criterion. Thus, as shown in the supplementary Tables 1 and 3 that accompany our paper, we used the BIC and Vuong‐Lo‐Mendell‐Rubin likelihood test to identify the model with the best fit and thus to determine the number of classes that best fit the data. Finally, we performed visual inspection of the plots of the predicted values against the observed values to further assess model fit.

In our relatively small sample (n = 202), including distal variables would increase the complexity of the model, which may make the results less generalizable to other populations. The more complex the model, the more sample‐dependent it is. Thus, we chose to use the latent variable analysis to identify the number of classes in MPLUS, and then used the predicted class membership in SPSS for further analysis of risk factors.

Lastly, we could have added more information regarding the proportion and handling of missing data. Using GMM estimated with full‐information maximum likelihood, it is possible to derive unbiased estimates despite some missing data provided that the missing at random assumption is fulfilled (Schafer et al., 2002). The proportion of missing data was relatively low in our material and did not exceed 8.9% at any time point. The Little's Missingness Completely at Random test indicated missingness at random for both our average and worst pain outcomes (p = .10).

Finally, we would like to thank Riddle et al for bringing up these questions, and thereby providing us the opportunity to further clarify the chosen methodologies in this study.

我们感谢Riddle等人（Riddle＆Dumenci，  2021）对我们关于全膝关节置换术后疼痛状况的文章（TKA; Lindberg等人，  2020）的关注，以及有机会讨论这个共同关注的重要临床问题。

我们同意，长期疼痛干预是TKA术后的重要结局。因此，我们之前曾发表过一篇关于与步行相关的疼痛干扰轨迹的论文。（Lindberg et al。，  2016）。与我们目前的文章类似，我们根据TKA术后最初12个月的疼痛干扰评分的生长混合模型（GMM），确定了随时间推移具有不同轨迹的两个患者亚组。大约22％的患者在手术后3个月内只有暂时的改善，随后疼痛相关的行走干扰增加，在TKA术后12个月恢复到术前水平。我们还确定了与行走反复发作的疼痛相关的术前症状和心理因素。

研究表明，骨关节炎患者描述了两种截然不同的疼痛经历（Hawker等人，  2008年）：一种钝痛，疼痛，被认为反映在我们的“平均疼痛”指标中；一种更为剧烈，尖锐，通常无法预测的疼痛，可能反映在我们最糟糕的疼痛评估中。这就是我们将“较低的平均疼痛”和“较低的最严重疼痛”纳入结果的原因。

我们基于对曲线轨迹的估计需要对多项式项进行联合估计的理解，因此选择将GMM与多项式系数一起使用，所有这些项都同时在模型中。因为轨迹的二次分量被估计为偏离线性，所以线性斜率也必须包含在模型中。单个轨迹是曲线上具有单个弯曲的非线性斜率。在我们的案例中，两条轨迹的第一部分均以较大的变化率开始，然后从第3个月开始以逐渐变化的下降轨迹开始。对于二次曲线而言，这种形式的二次曲线对于在干预之后立即变化的量度变化并不罕见。手术，然后经过很长一段时间。如果跨时间进行更多评估，则分段模型可能是我们采用的线性/二次轨迹的有用替代方法。但是，在手术前，第4天，第6周进行评估，然后间隔3个月至1年的较长时间，可能适合的部件实际上只是几条直线，在我们看来，这要少得多比多项式连续曲线更灵敏，可以作为形状曲线的估计。

我们使用Cohen d作为标准化的效应量来计算和量化高级别和低级别患者之间术前协变量平均评分差异的大小和临床意义。我们没有使用Cohen的d来分隔类。根据Muthén＆Muthén（Muthén，2021年的建议））（参见B.Muthén，讨论主题“混合模型的拟合优度”，“ http://www.statmodel.com/discussion/messages/13/119.html?1453602211”，8月14日4:07） pm）熵不是选择最佳拟合类别数的适当方法。我们对熵的使用仅是描述性的，可以解释为衡量基于模型的类别与预测的潜在类别之间的案例分类之间一致性的一种度量。相反，他们建议将贝叶斯信息标准（BIC）用作主要标准。因此，如本文随附的补充表1和3所示，我们使用了BIC和Vuong-Lo-Mendell-Rubin似然检验来确定最适合的模型，从而确定最适合该模型的类别数量。数据。最后，

在我们相对较小的样本中（n  = 202），包括远侧变量会增加模型的复杂性，这可能会使结果难以推广到其他人群。模型越复杂，则样本越依赖。因此，我们选择使用潜变量分析来识别MPLUS中的类别数量，然后使用SPSS中的预测类别成员资格来进一步分析风险因素。

最后，我们可以添加有关丢失数据的比例和处理的更多信息。使用具有最大信息最大可能性的GMM估计，尽管满足了随机假设下的遗漏，但尽管有一些遗漏的数据，仍可以得出无偏估计（Schafer等，  2002）。在我们的资料中，缺失数据的比例相对较低，在任何时间点均不超过8.9％。完全随机的Little's失踪测试表明，我们的平均和最严重的疼痛结局均是随机失踪（p  = .10）。

最后，我们要感谢Riddle等人提出了这些问题，从而为我们提供了进一步阐明本研究中选择的方法的机会。