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: Sample size pain reliefSample Size Estimation for a Non-inferiority Pain Management Trial | Additive Low-cost morning eats multiplicative effects by dataset. If relative MCIDs Sajple Sample size pain relief, then we would expect sizs y -intercept of 0 and a slope equal to the MCID. Senn S. However, there are also larger issues that warrant addressing. Loudness production. Hawker GA, Mian S, Kendzerska T, French M. |

Pain Relief Sample Pack – Chiki Buttah Organic Body & Skin Care Products | Br Dent J ; 3 : Each patient's pre- and post-intervention scores were calculated using the mean of x points. Cook JA, Julious SA, Sones W, Hampson LV, Hewitt C, Berlin JA, et al. If authors feel the linearity and ratio assumptions are too strict, there are other models that can be used; e. Data were collected at five separate visits using the short form of the McGill Pain Questionnaire MPQ. Nat Commun. Since the authors used unidimensional measures and a Rasch model, this conclusion is based on stationarity assumptions and ratings' reliability, which are not necessary conditions for interval or ratio scales. |

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MENU HOME SHOP ABOUT WHOLESALE BLOG CONTACT. Doing so allows us to substitute σ 2 with the variance of the sample mean, σ 2 n , giving us an ICC that is a function of the number of data points sampled from each patient,. Importantly, the above concepts generalize to post-intervention scores as well. If we assume τ 2 and σ 2 do not change, and instead, there is a simple shift in mean scores without ceiling and floor effects, then the ICC also defines the Pearson correlation between pre- and post-intervention scores. The Pearson correlation is useful because it gives us direct insight into RTM—the slope between the pre-intervention scores and change scores approaches zero as the correlation between pre- and post-intervention scores approaches 1 Figure 3. Figure 3. Simulations of additive and multiplicative changes reveal the effect of different intraclass correlation coefficients on the slope between change scores and pre-intervention scores. Additive effects have slopes that trend toward zero with increasing ICC's, while multiplicative effects always have a negative slope no matter their ICC. All of these properties come together and should be considered when statistically modeling pain relief and the effect of an intervention. The multiplicative model is still mathematically simple but its implications are more complex. If pain relief is multiplicative, then it can be modeled as a relative reduction; i. This would imply that each person's post-intervention pain y i is a fraction of their starting pain x i ; i. However, ratios and relative reductions have unfavorable statistical properties. Instead, it is preferable to work on the log scale 7 — 9. Similarly, from this, one may realize that it is natural to model multiplicative effects as being generated from log -normal distributions rather than normal distributions Appendix A2. The implications of the log-normal distribution and its multiplicative properties are shown and described in Figures 2 , 3. Note that the multiplicative pain reductions follow a different distribution than additive effects owing to their errors compounding rather than adding. This is a hallmark of multiplicative processes that can be evaluated empirically. In addition to this fanning, it is quickly apparent that even with zero measurement error Figure 3 , multiplicative effects can look like RTM since greater pre-intervention scores will result in greater decreases in pain Figure 2B. However, as opposed to additive processes in which greater pre-intervention scores are attributable to RTM i. The multiplicative nature does not only apply to the relationship between pre- and post-intervention pain, but also the effect of a treatment. This is described in further detail in the next subsection. Randomized controlled clinical trials aim to compare pain between two groups. To do so, investigators commonly compare the absolute or percent pain relief itself e. However, such analyses are ill-conceived. Instead, especially for studies that record one or few follow-up measures as opposed to time-series , it is recommended that the data-generating process be modeled using an analysis of covariance ANCOVA with pre-intervention scores as a covariate 8 , The reasons for this are manifold:. The response variable in a statistical model should be the result of an experiment. Because patients enter studies with their baseline score, it is not the result of the experiment so it should not be treated as a dependent variable e. Accounting for RTM. Instead of a group × time analysis of variance, one could perform a simple t -test on the change scores. However, such an analysis ignores RTM, and, especially in the case of baseline imbalances, can produce biased estimates. ANCOVA can adjust for such effects. Improving statistical efficiency. ANCOVA has greater statistical efficiency, resulting in greater power and more precise intervals. Post-intervention scores are arguably more interesting than change scores. Patients must live with the pain following the intervention, not the change in pain. However, regressing post-intervention pain or change in pain produces the same group effect 8. These statistical and philosophical advantages are well-established in the biostatistics literature 8 , 10 — Note, the benefits of ANCOVA primarily apply to randomized studies, as ANCOVA may produce biased estimates in non-randomized studies depending on the allocation mechanism where ϵ i ~ N 0 , σ 2 and g i is dummy-coded for group e. β 2 is the effect of interest: the average difference in post-intervention pain scores between groups after adjusting for pre-intervention scores. Of course, like any regression, one can add more covariates, especially those with prognostic value, which will further increase statistical efficiency. The ANCOVA can also be generalized to the multiplicative case. Since multiplicative effects can be linearized by taking the log-transform, we can write the model as. This model reveals a few things. First, in 1 , residuals will compound with increasing values of the predicted y i i. Indeed, this is consistent with what we observed in the simulations above, so this functional form can capture the compounding error. Since we are fitting β 2 rather than B 2 , the fit coefficient will be on the log scale, so exponentiating the coefficient will make it more interpretable despite the log scale having nicer mathematical properties. Note, even this multiplicative ANCOVA is more efficient than analyzing percent changes As a proof of principle, we assessed the properties of four separate datasets. Two of the datasets were collected in patients with subacute back pain and the other two consist of patients with chronic back pain. Ideally, data are analyzed using intention-to-treat. However, here, we included individuals for whom we had enough ratings to complete our analyses as the data are being used for illustrative purposes and we are not looking to draw inferences. The purpose of this study was to investigate factors associated with placebo analgesia in chronic pain patients This was the first trial designed to study chronic pain patients receiving placebo vs. no treatment. The total duration of the study lasted ~15 months. Protocol and informed consent forms were approved by Northwestern University IRB and the study was conducted at Northwestern University Chicago, IL, USA. To meet inclusion criteria, individuals had to be 18 years or older with a history of lower back pain for at least 6 months. This pain should have been neuropathic radiculopathy confirmed by physical examination was required , with no evidence of additional comorbid chronic pain, neurological, or psychiatric conditions. Individuals had to agree to stop any concomitant pain medications and had to be able to use a smartphone or computer to monitor pain twice a day. A total of 82 patients were randomized. Here, we include 18 participants from the no treatment group and 42 participants from the placebo group for whom we had complete rating data [cf. Supplementary Figure 1 in 16 ]. Data were collected using a custom pain rating phone app through which patients could rate their pain 0—10 NRS. For the purposes of demonstration, here we averaged pain ratings within a single day. The purpose of this study was to validate a prognostic model for classifying chronic pain patients based on their predicted improvement with placebo Protocol and informed consent forms were approved by Northwestern University IRB and the study was conducted at Northwestern University Chicago IL, USA. Individuals with chronic low back pain were recruited for this study. A total of 94 patients were randomized to no treatment, placebo, or naproxen. Here, we include 12 participants from the no treatment group, 33 participants from the placebo group, and 35 participants from the naproxen group for whom we had complete rating data [cf. Figure 1 in 17 ]. Data were collected using a custom pain rating phone app through which patients could rate their pain 0—10 NRS , as in Placebo I. The purpose of this trial was to investigate whether levodopa l-DOPA can block patients' transition from subacute to chronic back pain This week double-blind parallel group randomized controlled trial was conducted at Northwestern University Chicago, IL, USA. All enrolled participants provided written informed consent. The trial was registered on ClinicalTrials. gov , under registry NCT Individuals with a recent onset of low back pain were recruited. Figure 1B in 18 ]. The purpose of this study was to identify predictive biomarkers to identify individuals who will vs. will not recover from subacute back pain All participants were right-handed and were diagnosed by a clinician for back pain. Data were collected at five separate visits using the short form of the McGill Pain Questionnaire MPQ. The computed sensory and affective scores from the MPQ for each visit are used as individual pain scores for each subject. To evaluate whether each dataset was more compatible with an additive or multiplicative process, we conducted the same analyses from the Statistical Background section Figures 2 — 4 on these data. In particular, we investigated properties of the raw and log-transformed data, in addition to the properties of ANCOVAs fit to the data. To do so, all data were converted to a 0— scale. In doing so, we demonstrate how the aforementioned principles apply to real data. Figure 4. Simulations of additive and multiplicative changes reveal differential residual behavior for raw and log-transformed ANCOVA models. Left data generated with have an additive structure have homoscedastic residuals when fit with a standard ANCOVA top but heteroscedastic residuals when fit with a log-transformed ANCOVA bottom. Right data generated with a multiplicative structure have heteroscedastic residuals when fit on their raw scale top but homoscedastic residuals when log-transformed bottom. All datasets have positive relationships between pre- and post-intervention scores Figure 5. Interestingly and in contrast to the other studies, the variance of the post-intervention scores in the levodopa trial appears to increase with greater pre-intervention scores, consistent with a multiplicative effect. Finally, with the exception of the prospective cohort study, there are negative relationships between changes in pain and pre-intervention scores. These negative relationships may be explained by multiplicative effects or RTM. Further examination is needed to ascertain the nature of these data. Figure 5. Relationships between pre-intervention scores and change scores top and post-intervention scores bottom. Top Relationship between pre-intervention scores and change scores. Note that most of the studies have a negative relationship. Bottom Relationship between pre-intervention and post-intervention pain scores across all studies. Each study shows a positive relationship between pre- and post-intervention scores; however, the Levodopa study appears to have greater variance in post-intervention scores with greater pre-intervention scores. Including more points in the calculation of pre-intervention and post-intervention scores increases the ICC, thereby increasing the reliability and decreasing the effect of RTM Figure 3. Since three of the four datasets contained ecological momentary assessments of pain, we were able to sample and average more than one point from the beginning and end of each study. We averaged an increasing number of a pre- and post-intervention points and recalculated the slope between change score and pre-intervention score i. If the slopes strongly trend toward zero by increasing the number of points, this indicates that the data have additive properties. Slopes that stay negative regardless of increasing reliability number of points indicate that the data may be multiplicative. For the studies included in this analysis Placebo I, Placebo II, Levodopa Trial , Placebo I and Placebo II's slopes have slight upward trends: as the number of points in the calculation of pre-intervention and post-intervention scores increases, the negative slope due to RTM increases. In contrast, the Levodopa trial's negative slopes remain stable Figure 6. This again hints at the notion that the levodopa trial's data may be multiplicative, while Placebo I and Placebo II may be additive. Figure 6. Increasing the number of points used for each patient's pre- and post-intervention scores increases the slope between change scores and pre-intervention scores. Each patient's pre- and post-intervention scores were calculated using the mean of x points. By averaging over more points, we should increase the intraclass correlation coefficient. Negative slopes between change scores and pre-intervention scores are indicative of one of two things: 1 regression toward the mean or 2 multiplicative effects. In the datasets that show evidence of being additive, we see marked increases in slopes, indicating that we are decreasing regression toward the mean by including more points. However, because the Levodopa Trial displays multiplicative properties, it is only minimally affected by adding more points. Perhaps the most direct assessment of additive vs. multiplicative properties is to model the data and assess the model fits. When assessing and utilizing a model, one should ensure that the model's assumptions are met and that the model captures salient features of the data. Because multiplicative data-generating processes lead to compounding residuals, we can observe these effects when fitting ANCOVAs. In Figure 7 , we focus specifically on the variance observed in Figure 5 , illustrating the relationship between fitted values using the ANCOVA models from Figure 5 and the absolute value of the residuals. As shown in Figure 2 , multiplicative relationships possess higher variance as pre-intervention scores increase, compared to additive relationships which are homoscedastic. For this reason, we should observe a null correlation between fitted values and absolute residual error for data that have exhibited additive properties Placebo I, Placebo II, Prospective Cohort thus far, and observe a positive correlation between fitted values and absolute residual error for data that have exhibited multiplicative properties Levodopa Trial. As predicted, the Placebo I, Placebo II, and Prospective Cohort data all display this additive quality, as their residual error does not increase as fitted values increase. In contrast, the Levodopa Trial data display multiplicative properties, as its residual error increases as fitted values increase. The description and analyses of these data can be seen below Figure 7. Figure 7. Absolute values of residuals from additive ANCOVA models. We fit an ANCOVA to each dataset using pre-intervention score and group membership as covariates. From these models, we plotted the absolute values of the residuals as a function of the fitted value. Additive models should be homoscedastic, meaning the magnitudes of the residuals do not change as a function of the response variable. However, multiplicative models have compounding error, such that if you fit them using an additive model, greater predicted values will be associated with larger magnitudes of residual error. Placebo I, Placebo II, and the Prospective Cohort study all exhibit features of additive data. However, the Levodopa Trial exhibits multiplicative properties, as evidenced by the increasing error residual magnitude with increasing fitted values. From these plots, it is clear that the Placebo I, Placebo II, Prospective Cohort demonstrate additive properties while the Levodopa Trial demonstrates multiplicative properties. An understanding of these concepts and model assumptions have real implications. In Table 1 , we include the average absolute additive and log-transformed multiplicative change in pain scores for each dataset. Pain relief is a ubiquitous clinical trial outcome with direct treatment implications. Treatments that yield appreciable pain relief will be employed in the clinic, and findings from these trials may be communicated to patients. However, if data from trials are not properly modeled, then the resulting treatment effects may be both biased and highly variable, which in turn may mislead researchers, clinicians, and patients. In this theory-based paper, we have emphasized the difference between additive and multiplicative treatment effects from mathematical, statistical, and empirical perspectives. It is clear that the assumptions behind these effects are not interchangeable and thus should be more thoughtfully considered when planning and analyzing clinical trial data. Moreover, how pain relief is conceptualized will propagate into the interpretation of effects, which we briefly discuss herein. Pain intensity ratings can be difficult to interpret—they are a reductionist, unidimensional measurement intended to capture a single aspect of a private, complex, incommunicable experience 20 , To help make sense of improvements, researchers and clinicians commonly rely on minimal clinically important differences MCID. In clinical pain research, MCIDs are commonly derived by mapping changes in pain ratings onto a different scale, such as global impression of change This mapping is then commonly used as a guidepost for interpreting other studies, and in some cases, individual patient changes Although commonly derived and used without justification, absolute and relative MCIDs are not interchangeable since they are mathematically incompatible with one another. Farrar et al. Much attention has been and continues to be given to both additive and multiplicative MCIDs without considering the conceptual difference between the two. This conceptual incompatibility needs to be reconciled if MCIDs are to be used in a meaningful way. However, there are also larger issues that warrant addressing. This relationship calls into question both absolute and relative MCIDs. If absolute MCIDs were valid, then we would expect the MCID to be constant across all baseline pain scores. If relative MCIDs were valid, then we would expect a y -intercept of 0 and a slope equal to the MCID. If true, this would be consistent with the idea that it is a patient's pain, not change in pain, that is important. More generally, MCIDs arguably represent a conflation of constructs. For example, researchers may threshold and dichotomize changes in VAS into improvement vs. non-improvement using the global impression of change scale This dichotomization of pain scores is then applied to other studies. Yet, such an approach is curious—it implies we are actually interested in global impression of change but use pain scores as a noisy proxy. If a researcher is interested in global impression of change, they should measure global impression of change as an outcome in their sample. Further, the ontological basis for dichotomous change scores is arguably ill-conceived. The insipid use of MCIDs in pain research and practice deserves greater scrutiny. From this perspective, it has been argued that greater context is needed in deriving metrics of clinical importance 26 , 27 for which decision theory may provide a rigorous foundation. However, such analyses have undesirable properties on both the individual and group levels. On the individual level, inferences cannot be made regarding response magnitude for several reasons. First, individual counterfactuals are not observed in parallel group trials; for example, we do not know what an individual's pain would have been had they been randomized to the placebo group instead of the drug group. An individual's observed improvement or worsening may have been due to the intervention or alternatively, RTM, natural history, or some other unmeasured, stochastic process. These issues have been previously discussed in great detail 28 — Thus, the dichotomization of improvements is arguably unethical since it discards information, effectively decreasing the sample size 32 and, in turn, the ability to quantify or rule out meaningful intervention effects. Rather than being treated as an analytical tool, MCIDs are perhaps better viewed from an interpretive and decision-making perspective. Notwithstanding MCID's limitations, it is perhaps most useful at the planning stage of clinical research. A clinically important difference is just one approach to justifying an effect size of interest for a study 34 , which may be used for sample size calculations or stopping rules in adaptive trials. However, beyond planning, dichotomizing trial and especially individual patient outcomes using an MCID is a questionable practice that commonly ignores context and variability 9. Psychological measurement scales have a rich history across the fields of psychometrics and psychophysics Anchors determine the extremes within which a participant must rate their experience, ultimately constraining the measurement construct and how accurately participants understand what they are rating Bounded by these anchors, the measurements themselves can be on one of a number of scales: nominal, ordinal, interval, ratio, and absolute. Several renowned psychophysicists have argued—not without criticism 38 , 39 —that perceptual ratings are or can easily be converted to ratio scale 35 , Importantly, the additive and multiplicative models rely on interval and ratio assumptions, respectively. Thus, the validity of these assumptions for clinical pain must be considered. The numerical nature of clinical pain is an open, controversial, and perhaps unanswerable question. Early psychophysics work argues that VAS and NRS pain scales are ratio for both experimental and clinical pain. Price et al. However, by mapping clinical pain onto heat pain, this finding is arguably tautological—they assessed whether clinical pain-matched heat pain follows the same power law as heat pain. Others have used item-response theory to argue that pain ratings are ordinal scale nonlinear rather than ratio or interval scale Since the authors used unidimensional measures and a Rasch model, this conclusion is based on stationarity assumptions and ratings' reliability, which are not necessary conditions for interval or ratio scales. Although the perceptual ratings from psychophysics are undoubtedly related to clinical pain, assessing the measurement properties of clinical pain is much more complex since we cannot precisely control the sensory input. Thus, clinical pain measurement scale assumptions arguably cannot be rigorously evaluated, reinforcing that they are indeed assumptions. However, the strength of assumption varies, with interval scales additive having weaker assumptions than ratio scales multiplicative. The assumptions a researcher makes directly affects the model they should choose. The choice of a statistical model can greatly affect the inferences drawn from the same dataset. Here, we observed that applying a multiplicative model to a dataset that exhibits additive properties can create wide CIs, making it difficult to interpret the results of an experiment Table 1. This is consistent with the idea that a properly specified model will be more statistically efficient 12 , and perhaps most importantly, it will better represent the underlying data. We presented two ways of modeling data: additively and multiplicatively. Both rely on ANCOVA, with the former using raw pain scores and the latter using log-transformed pain scores. These models have different assumptions about the underlying data and, as a result, have different interpretations. If authors feel the linearity and ratio assumptions are too strict, there are other models that can be used; e. Indeed, there are good examples in the pain literature of ANCOVA-type models being implemented with more complicated data structures [e. In any case, researchers should be aware of the assumptions of their statistical models of the properties of their data, and of course, researchers are encouraged to collaborate with statisticians We have clearly demonstrated the mathematical, conceptual, and interpretive differences between additive and multiplicative effects. From this explication, there are tangible takeaways and recommendations for clinical researchers. Specifically, we suggest that researchers include and consider the following:. When deciding which metric to use—absolute pain decreases or percent pain decreases—use the data as a guide unless there is a principled reason to choose one or the other. Since it is unclear what influences the presence of additive or multiplicative characteristics in pain data, it is safer to use the metric that accurately represents the properties of the data. Table 2 summarizes the differences between additive and multiplicative properties. In time, we may develop a better understanding of pain conditions and improvements such that more general recommendations can be provided. We view this data-driven approach as being no different than checking statistical model assumptions. When reporting descriptive statistics, use the arithmetic mean to calculate between-subject average intervention for additive data; conversely, use geometric mean for multiplicative data. Ensure that patients' pre-intervention scores are heterogeneous for drawing conclusions about the nature of the data. By including a wide range of pre-intervention scores, it makes the additive or multiplicative properties more apparent. If the data are not heterogeneous, false conclusions may be made about the data's additive or multiplicative properties. The properties of changes in self-reported pain are commonly implicitly assumed to be additive, multiplicative, or are conflated. Ignoring the properties of pain relief can result in model mis-specification, in turn leading to bias and statistical inefficiency. These errors further propagate into metrics such as minimal clinically important differences. We contend that more attention should be paid to the statistical properties of pain relief to ensure model assumptions are met. By paying closer attention to these properties, we can gain more insight from and make better use of data from pain clinical trials. Publicly available datasets were analyzed in this study. This data can be found at: OpenPain. The studies involving human participants were reviewed and approved by Northwestern University. AV, ST, and AVA conceptualized the paper. AV drafted the paper. AV and ST produced the figures. AV, ST, JG, and AVA read, provided feedback on, approved the final version of, and agree to be accountable for the contents of the manuscript. All authors contributed to the article and approved the submitted version. This work was funded by the National Institutes of Health 1P50DAA1. This material was based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. |

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