Tetsuya Akaishi, Ichiro Nakashima

Evaluating the cluster formation of clinical attacks in chronic relapsing diseases is an important statistical issue because the presence of attack clusters may influence therapeutic strategies for relapse prevention.We recently reported the occurrence of unevenly clustered attacks in patients with anti-aquaporin-4 (AQP4) antibodypositive neuromyelitis optica spectrum disorder(NMOSD) (Akaishi et al., 2020a).The presence of attack clusters implies the necessity of implementing intensive relapse prevention with highly efficient drugs, such as the anti-complement C5 monoclonal antibody (eculizumab), during the cluster period (Pittock et al., 2019).This is especially true in diseases in which relapse is an established factor that causes severe and irreversible sequelae, such as NMOSD (Akaishi et al., 2020b).In this report, we present the following two statistical methods for evaluating the presence of uneven cluster formation of attacks in relapsing diseases including anti-AQP4-positive NMOSD:(1) the chi-square goodness-of-fit test and (2) the Kolmogorov-Smirnov (KS) test.

Distribution of interval between random attacks:Distribution of the frequency of random event occurrence during a specific time period can be described with discrete probability distributions from the Poisson model family (parameter:λ[expected rate of occurrence]) expressed as follows (Plan, 2014):

wherekis the frequency of event occurrence in a fixed time period and/or space (k= 0, 1, 2,...).Poisson distribution is a discrete probability that can describe the number of randomly occurring events in a fixed time interval.Different from the Poisson model family that are the count characterizations of random events, the distribution of the intervals between event occurrence of a Poisson process can be described with an exponential distribution (parameter:μ[expected waiting time until event occurrence] =1/λ) as follows (Frank, 2009):

The probability density function of an exponential distribution decreases monotonically, and the probability has the largest value at x = 0.In other words, a higher frequency of short intervals can be observed even when events occur randomly without cluster formation.Therefore, only a finding that a short attack interval ≤ 6 months from the last attack shows the highest frequency does not guarantee the presence of a cluster period in clinical attacks with an unevenly high relapse rate (Wang et al., 2021).To determine the presence of a relapse cluster period with a nonconstant relapse rate over time, a comparison between the distribution of the observed intervals(actual sample data) and that of the expected intervals derived from an exponential distribution(imaginary data of intervals assuming a random occurrence of events) is required.An important characteristic of the exponential distribution is its memoryless feature; that is, the expected time to the next event is independent of the elapsed time from the last event (Feng et al., 2019).

Individual difference in relapse rate: The Poisson process requires several conditions.For example, the occurrence of events is considered independent of each other, and the expected rate of event occurrence is considered constant over time (Faddy and Pettitt, 2022).Therefore, we evaluated whether the relapse rates among patients could be attributed to a single Poisson distribution.In our patient cohort,the 5-year relapse rates from the 39 patients showed a non-significant statistical level (P=0.26) with the chi-square goodness-of-fit test using a Poisson distribution (parameterλ= 1.55[relapses/5 years]), suggesting that interpersonal difference in relapse rate may be explained by randomness.Another required condition for using an exponential distribution to describe the distribution of relapse intervals is that the expected relapse rate should remain constant over time.This could be performed by comparing the annualized relapse rates obtained using the person-year method between different time periods (e.g., < 2 years from onsetvs.2–4 years from onset; Akaishi et al., 2022).

Chi-square goodness-of-fit test with an exponential distribution: After checking for the fulfillment of conditions using an exponential distribution for the relapse intervals (i.e.,independence between each event occurrence and consistency of relapse rate over time),a goodness-of-fit test with an exponential distribution to determine the randomness or nonrandomness of relapse intervals can be used for the observed data regarding the relapse intervals from the patients.Chi-square goodness-of-fit test with expected probability distributions can be performed by calculating thex2statistic after dividing the relapse interval into several ordinal scales with levels (e.g., 11 levels with 0–6 months,7–12 months, 13–18 months,...55–60 months,and ≥ 61 months).To calculate thex2statistic,observed (oi) and expected (ei) frequencies are obtained for each of theminterval ranges.The parameter (μ) of an exponential distribution for calculatingeiin each class of ordinal scale for attack interval can be provisionally obtained from the overall sample data by supposing that the population relapse rate is constant between patients and over time (although these may not be true in the real world).Thex2statistic can be calculated as follows:

We can decide the statistical significance of the goodness-of-fit test by calculating the upper-tail probability (P[X≥x2]) with a chi-square distribution of m-1 degrees of freedom (df=m-1).An example of evaluating the presence of a relapse cluster period (which cannot be explained by the random occurrence of relapse) using actual data from our patient cohort is shown in Figure 1.In this example, the parameterμof the exponential distribution was calculated by dividing the total follow-up period with relapse preventions (8189 months) with the total number of relapses (204 relapses), producing the parameterμ= 40.14 (i.e.,one relapse episode per every 40.14 months).As can be seen in the figure, the histogram of attack interval length skewed toward the short intervals≤ 12 months.The obtainedx2statistic (103.05)showed a statistical significance (P< 0.0001):

This suggests that the observed intervals (sample data) may not have been derived from randomly occurring events with a constant occurrence rate over time.

The statistical level of the goodness-of-fit test was still significant (P= 0.0063) even when the followup period after the last relapses in the 39 patients with anti-AQP4-positive NMOSD was excluded from the overall follow-up period (204 relapses in 4836 months with relapse preventions), producing the parameterμ= 23.71 and the resultantx2statistic of 24.52.Regardless of whether the follow-up period after the last relapse was included or excluded, the observed data from the 39 patients suggested the presence of an attack cluster period, although the presence of time dependency behind cluster formation remains uncertain.

Points to note in goodness-of-fit test: When performing the goodness-of-fit test using expected frequencies obtained from an exponential distribution, caution is needed due to the fact that the results will change with the length and number of the attack interval classes (e.g., 0–6 months, 7–12 months, …, 55–60 months, and ≥ 61 months).Owing to the nature of the exponential distribution, there is no upper limit for the random variable X (attack interval).Therefore, we can theoretically prepare as many attack interval classes as possible, if we wish to change the degrees of freedom according to the number of prepared attack interval classes.For example, if we divide the attack interval class of ≥ 61 months into 61–65 months and ≥ 66 months, then the calculatedχ2statistic may change and the degrees of freedom will also increase from 10 to 11,changing the statistical significance.Sensitivity analyses that prepare different attack interval class patterns for the goodness-of-fit test may provide information about the robustness of the goodness-of-fit test results.

In a goodness-of-fit test, each of the expected frequencies obtained from the ideal probability distribution should be 5 or more; otherwise, it may not be appropriate to fit the calculatedx2statistic to a chi-square distribution to obtain theP-value.If there is an attack interval class in which the expected frequency is less than 5, a possible countermeasure is to merge the class with the adjacent class to realize an expected frequency of 5 or more in the new class.In this case, the degrees of freedom of the chi-square distribution used for the goodness-of-fit test will decrease with the merged ordinal scale classes.

Figure 1 ｜Evaluation of cluster formation of clinical attacks with chi-square goodness-of-fit test.In this figure, an idea of statistically evaluating the presence of cluster formation in clinical attacks of relapsing diseases such as anti-AQP4-positive NMOSD by performing a chi-square goodness-of-fit test is shown.If attacks occur randomly and independently with a constant relapse rate, the expected distribution of the attack interval follows an exponential distribution.The exponential distribution parameter is estimated based on the observed relapse frequencies in all patients.By calculating the x2 statistic from the observed and expected relapse frequencies, we can perform the test of goodness of fit to an exponential distribution with the observed attack intervals from the patients.Unpublished data.Created with Microsoft PowerPoint 2016.AQP4: Aquaporin-4; df: degree of freedom; NMOSD: neuromyelitis optica spectrum disorder; Pr: probability; Pt: patient.

Kolmogorov-Smirnov test: Another choice for evaluating whether the relapse interval followed an exponential distribution is the KS test.With this nonparametric method using the raw data of measurement, dividing the interval length into several categories is not needed, and the raw data of the measured intervals can be directly used.By performing the KS test, we can evaluate whether the distribution of some observations can be regarded as being drawn from a specific reference probability distribution, such as a normal, Poisson, or exponential distribution.Using the R Statistical Software (R Foundation,Vienna, Austria), the KS test was performed with the “ks.test” function.A statistical significance level similar to that of the chi-square goodnessof-fit test was obtained (D= 0.157,P< 0.0001).One female patient had a remarkably high relapse frequency with more than 20 relapses during the relapse prevention treatment period.To exclude the potential bias derived from this patient, the KS test was performed after excluding data from this patient; however, the KS test still showed statistical significance (D= 0.138,P= 0.0024).Furthermore,the statistical significance did not change even when the follow-up periods from the final relapse in the evaluated 39 patients were included as the raw data of interval (D= 0.202,P< 0.0001, KS test).

The KS test utilizes the cumulative distribution function of a theoretical continuous distribution,and ties within the measurements are considered to be better avoided.Therefore, strictly speaking,the KS test is not usually applied to discrete data, although it is technically implementable by regarding the number of events as continuous data.However, treating a discrete variable as though it is a continuous variable should be carefully considered in many situations, such as evaluations of count data.The KS test can be used to evaluate whether a distribution of intervals follows an exponential distribution (i.e.,a continuous distribution), but it is not typically used to evaluate whether a distribution of the numbers of events during a fixed time follows a Poisson distribution (i.e., a discrete distribution).For discrete data, the chi-square goodness-of-fit test would be better than the KS test for checking the goodness-of-fit.

Limitations of using exponential distributions:In this article, the parameter of the exponential distribution was estimated from the observed frequencies from all patients, because the followup period in each patient was not enough long to statistically evaluate the presence of attack cluster formation in each of them.Therefore,although we have demonstrated that the 5-year relapse rate from the patients could be regarded to follow a Poisson distribution, the justification for applying an exponential distribution to the data regarding attack interval might be limited.As another limitation, the relatively small number of patients could have reduced the power to detect the difference between the observed (oi) and expected (ei) distributions of the 5-year relapse rate.Furthermore, the different follow-up periods among the 39 patients could have biased the findings of this study.To deal with this problem,we have performed a sensitivity analysis after excluding a female with the highest relapse rates,and the conclusions of this study did not change.Finally, the finding of an unevenly clustered attack occurrence period in the patient cohort does not necessarily imply that all patients with NMOSD present a non-random occurrence of relapses.Some patients with NMOSD experience clustered attack occurrence periods with abnormally short attack intervals, while other patients may not experience such cluster formation of clinical attacks.

Summary: In this report, we presented statistical methods for evaluating the presence of uneven cluster formation in the clinical attacks of relapsing diseases.Both goodness-of-fit and KS tests revealed a significant difference between the observed and expected frequencies, suggesting that relapse occurrence in anti-AQP4-positive NMOSD cannot be considered a Poisson process.Further studies are warranted to determine the benefit of distinguishing the attack cluster period from other non-cluster periods to optimize strategies for relapse prevention.

Tetsuya Akaishi*, Ichiro Nakashima

Department of Neurology, Tohoku University,Sendai, Japan (Akaishi T)

Department of Neurology, Tohoku Medical and Pharmaceutical University, Sendai, Japan(Nakashima I)

*Correspondence to:Tetsuya Akaishi, MD, PhD,t-akaishi@med.tohoku.ac.jp.

https://orcid.org/0000-0001-6728-4966(Tetsuya Akaishi)

Date of submission: August 21, 2023

Date of decision: October 26, 2023

Date of acceptance: November 8, 2023

Date of web publication: December 15, 2023

https://doi.org/10.4103/1673-5374.390980

How to cite this article:Akaishi T, Nakashima I(2024) Statistical evaluation of cluster formation of relapse in neuromyelitis optica spectrum disorder.Neural Regen Res 19(9):1888-1889.

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