Denoised MDS-UPDRS part-III scores yield new patterns of progression heterogeneity in early stage Parkinson’s disease

Abstract

Parkinson’s Disease (PD) Motor Scores (MDS-UPDRS Part III) are quite noisy. This paper proposes a new methodology for processing these scores by first denoising the scores to enhance the underlying progression signal, and then conducting a high-dimensional analysis which does not sum the scores into a total movement score. The analysis gives novel insights into PD progression heterogeneity: it reveals that the heterogeneity is continuously variable rather than clustered into “subtypes” and that the variability is along two easily understood axes. This analysis also resolves some of the discrepancies in previously reported progression subtypes. Finally, the analysis reveals that patient-specific progression cannot be predicted from baseline using only MDS-UPDRS Part III scores.

Introduction

The progression of Parkinson’s disease (PD) is known to be heterogeneous, i.e., the trajectory of the disease varies significantly from patient to patient. A quantitative understanding of this heterogeneity is important because it can shed light on the etiology of PD, predict PD progression at the individual level, and inform cohort selection and outcome monitoring in PD clinical trials.

PD heterogeneity is usually analyzed using PD subtypes. Each subtype represents a specific pattern of disease progression. Many different approaches for defining subtypes are used in the literature. One approach, based on clinical phenomenology, defines the following subtypes: tremor dominant (T-D), postural instability and gait disorder dominant (PIGD-D)^1–3, and akinetic-rigid dominant (AR-D)^4–6. A shortcoming of the subtyping approach is that subtypes do not seem to be stable. PD patients appear to change subtypes throughout the course of the disease, even over the course of a few years^7–10. One explanation offered for this change is that subtypes represent different stages of PD progression rather than differences in the underlying disease trajectories^7,11–13. Other subtyping approaches stratify PD by features such as age at onset^14–18, genetics^19–24, or the predominant non-motor symptom²⁵.

Data-driven subtypes obtained through cluster analysis have also been proposed^8,26–28. However, clusters identified by such methods are not consistent with each other, with some methods finding clusters that match T-D vs PIGD-D/AR-D subtypes while others finding clusters consistent with late-onset, fast progression and early-onset, slow progression²⁹. Yet other clustering methods suggest novel subtypes such as mild-motor predominant and diffuse malignant^30–32.

From a data-science point of view, the methods summarized above have some limitations: First, in many cases, subtypes are defined cross-sectionally rather than longitudinally^29,33–35. That is, they are defined by the state of the disease at a fixed point in time instead of by examining the longitudinal trajectory of the disease.

Second, most subtypes are defined using motor scores, which tend to be very noisy. This noise often masks the underlying progression of the disease, and can cause correlations between various observations to be smaller. Noise can also cause spurious clustering.

Third, motor scores are ordinal rather than numerical, and some statistical care is needed to deal with ordinality. Previous clustering-based attempts to produce subtypes do not account for ordinality of the scores nor do they use noise models that are appropriate for ordinal data^27,31,32,35.

Fourth, many (but not all) subtypes are defined by summing motor scores for different symptoms into a single Total Movement Score (TMS)^3,9,26,36. Using TMS obscures differential progression in the canonical motor symptoms (tremor, bradykinesia, rigidity, etc.). This is problematic because differential progression is likely to be critical to subtyping.

Finally, subtyping effectively places a patient in one of a finite number of categories without allowing for continuous gradation between categories⁸. But whether a finite category model of progression heterogeneity is more realistic than a continuously variable heterogeneity model is a priori unclear.

The goal of this study is to develop a new methodology for analyzing PD heterogeneity in motor scores using longitudinal scores while taking the above critique into account. We are especially interested in denoising the motor scores before drawing statistical conclusions. Denoising the scores improves their signal-to-noise ratio, and thus improves the reliability of statistical analysis. We denoise the Movement Disorder Society’s Unified PD Rating Scale (MDS-UPDRS) Part III scores using a recently reported method which accounts for the ordinal nature of the scores³⁷. We also do not sum motor scores into a TMS. Instead, we analyze the multivariate time series of all MDS-UPDRS Part III scores.

Visualizing the multivariate time series of MDS-UPDRS Part III scores (and the accompanying statistical analysis) shows the absence of clear clusters in the data. Given this, we propose a methodology to understand how the progression data are continuously distributed. The methodology finds and uses neighborhoods within which the data can be statistically summarized. A new understanding of PD progression heterogeneity emerges from this analysis.

In addition to quantifying heterogeneity, a goal of this paper is to examine whether longitudinal scores can be predicted from baseline. In data-science terms, a time series which is predictable from a fixed point in time is Markovian, or else it is non-Markovian. Thus we are interested in assessing whether the denoised time series of MDS-UPDRS Part III scores is Markovian or non-Markovian.

The data for our analysis come from the Parkinson’s Progression Markers Initiative (PPMI), a longitudinal study with the goal of finding biomarkers for PD progression (www.ppmi-info.org)³⁸. All PPMI subjects provided a signed informed consent form. The data were gathered in accordance with Institutional Review Board/Independenct Ethics Commitee approval. All data is available in anonymized form.

All PPMI patients were recruited at initial PD diagnosis. We refer to this time point as the baseline. Details about inclusion and exclusion criteria for PPMI can be found in ref. ³⁸. MDS-UPDRS scores are collected by PPMI for each patient at 3-6 month intervals over the first 5 years and then annually. However, missing visits are common. MDS-UPDRS scores are collected in an off-medication state.

We selected patients from PPMI who had at least 5 visits in the time span from baseline to baseline plus four years. This resulted in 386 patients being selected. Of these 96% were white, 2% were Asian, and the remaining participants were Black, Native American, or Pacific Islanders. They had a mean of 11.9( ± 3.3) visits. Their mean age was 61.2( ± 9.7) years and 62% were male. At baseline, the mean Hoehn and Yahr (H&Y) score was 1.8( ± 0.4) and the mean MDS-UPDRS Part III total movement score (TMS) was 21.3( ± 7.7).

Of the 386 patients, 66 (17.1%) had pathogenic LRRK2 variants and 20 (7.5%) had pathogenic GBA variants, while 84 (21.8%) had APOEε4 variants. DaTscan images were available at baseline and baseline plus year four for 198 patients.

Results

Because we chose not to analyze motor scores by summing them into a TMS, and because we do not wish to make any prior assumptions about the existence of well-formed clusters in the data, our analysis pipeline is quite different from analysis pipelines reported in the literature. Our pipeline is illustrated in Fig. 1. Figure 1a illustrates our ideas geometrically, while Fig. 1b illustrates the processing and analysis as a flow chart. Each step of Fig. 1b is summarized below.

Fig. 1. Overview of the analysis.

a Longitudinal 33-dimension MDS-UPDRS (Movement Disorder Society-Sponsored Unified Parkinson’s Disease Rating Scale) scores for a PD (Parkinson’s disease) patient form a noisy curve in a 33-D (33-dimension) space (solid-black trajectory). Denoised scores give a smooth trajectory (solid-green trajectory), which in turn can be efficiently approximated by a linear trajectory (linear-blue trajectory) from baseline to baseline plus year four. The trajectory is analyzed in terms of its baseline vector (dashed arrow) and progression vector (blue arrow). Baseline analysis is carried by statistical analysis of neighborhoods in the vectors are distributed. Progression analysis is carried by moving all progression vectors so that their tail is at the origin, and then statistically analyzing neighborhoods in which the vectors are distributed. b Flowchart of the analysis steps.

We begin by considering all MDS-UPDRS Part III scores, except the Hoehn-Yahr score, for every participant. The Hoehn-Yahr score is omitted because it represents an overall summary of the PD state, rather than a specific movement feature. The other 33 scores are movement specific.

For each patient, at any point in time, the 33 scores are arranged in a 33-dimensional (33-D) vector. As the disease progresses, this vector sweeps out a (noisy) curve in the 33-D space of scores (Fig. 1a, solid black trajectory). Each such curve represents the progression of the disease in a single patient.

We denoise the 33 scores to obtain a 33-D denoised progression curve for each patient (Fig. 1a, solid green trajectory). We refer to the pre-denoised scores as the raw scores and the post-denoised scores as the denoised scores. Thus, we obtain a single raw score trajectory and a single denoised score trajectory per patient.

Next, we visualize the 33-D trajectories by using multidimensional scaling (MDS) to embed the 33-D data into a 2-D space. Each 33-D data point from every trajectory is embedded as a point in the 2-D space. Connecting the embedded points from each trajectory gives an embedding of the trajectory in the 2-D space.

As discussed in detail in Section “Visualization with Multidimensional Scaling” below, the above MDS visualization reveals that the raw trajectories are extremely noisy and significantly mask the underlying progression. Progression is more clearly revealed by the denoised trajectories. Denoised trajectories have two properties: First, the denoised trajectories are almost linear, and second, the trajectories are non-Markovian (they cannot be predicted from the baseline). Consequently, we separately analyze each trajectory by decomposing it into its baseline vector (Fig. 1a, dashed vector from origin) and the changes from the baseline vector to the end (Fig. 1a, solid blue vector). We call the latter the progression vector. To be precise, suppose y_i, i = 1, ⋯ , T is the trajectory in the 33-D space, then the baseline vector is y₁, and the progression vector is y_T − y₁. The progression vector informs about the total progression of the disease from baseline.

After decomposing each trajectory into a baseline vector and progression vector we visualize the spread (distribution) of these vectors in 33-D space. Visualization clearly shows that the vectors do not cluster in any obvious way (Section “Visualization with Multidimensional Scaling”). Additional statistical analysis of the baseline and projection vectors shows that their distributions are not multimodal. That is, denoised baseline and projection data appear to be spread continuously without any evidence of a natural clustering. Because there is no natural clustering, we adopt the following strategy to statistically summarize the continuous spread of baseline and progression vectors: We find compact spherical neighborhoods in the 33-D space such that the neighborhoods cover all data (Fig. 1a). The neighborhoods are found by hierarchically clustering the vectors and finding a level in the hierarchy where the spheres seem to compactly fit the data. Methodological details of this step are in Section “ Hierarchical Clustering”. We emphasize that we do not take the clusters to represent subtypes. We simply consider each cluster to be a local data neighborhood within which the data are reasonably homogeneous so that local summarizing statistics can be calculated. To emphasize that the clusters are not subtypes but simply local regions for statistical analysis, we refer to them as data neighborhoods, or simply as neighborhoods. Baseline and progression vectors are both subjected to neighborhood analysis. Progression vectors are analyzed by translating them so that their tail is at the origin.

The vectors in each neighborhood are summarized statistically by grouping the scores into Bradykinesia and Rigidity subscores (B+R) (sum of MDS-UPDRS Part III scores: 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 (18 scores)), Tremor subscores (sum of MDS-UPDRS Part III scores 3.14, 3.15, 3.16, 3.17, 3.18 (10 scores), and Axial subscores (sum of MDS-UPDRS Part III scores 3.9, 3.10, 3.11, 3.12 (4 scores)). Bar plots of the means of these canonical subscores provide a summary of the data in each neighborhood. Statistical comparison of these subscores in different clusters provides an understanding of heterogeneity.

We further analyze the spread of the progression vectors by investigating the locations of their spherical neighborhoods in terms of the distance and the direction of the centers of the neighborhood.

Finally, we relate the statistics of baseline and progression vectors to other patient features such as demographics, genetic mutations, and DaTscan images. Each of the above steps is depicted in detail in Fig. 1.

Denoising

As mentioned above, we begin with the raw MDS-UPDRS Part III score time series for each patient (Fig. 1a) and denoise the time series of each score using the method reported in ref. ³⁷ and described in detail in the Methods Section “Denoising of MDS-UPDRS Part III Scores” of this paper. This denoising method takes into account the ordinality of data and the short duration of the time series. The method is intrinsically capable of handling irregularly spaced time series, so data interpolation for missing visits is not required. The method uses a hierarchical model to draw statistical power from the entire cohort of PPMI patients while denoising each patient-specific time series. We use this method to denoise MDS-UPDRS Part III scores from baseline to baseline plus 4 years (examples shown in Fig. 2). During denoising, scores for bilateral questions are rearranged by the more/less-affected side instead of left/right (described in Methods Section “MDS-UPDRS Part III Score Processing”).

Fig. 2.

Examples of raw and denoised time series of some MDS-UPDRS (Movement Disorder Society-Sponsored Unified Parkinson’s Disease Rating Scale) Part III (three) scores.

We note here that the publication³⁷ extensively validated the denoising method with simulated data that mimicked MDS-UPDRS Part III scores and with actual MDS-UPDRS Part III scores. The method was also compared with traditional smoothing. The publication³⁷ shows that the method outperforms traditional techniques, works well with simulated MDS-UPDRS part III data, and does not overfit actual MDS-UPDRS part III data.

Denoising can affect the signal along with the noise. To assess how much the method affects the underlying signal, we performed an additional principal component analysis (PCA) of the raw and denoised data (see Supplementary Figs. 1, 2). The first few principal components of noisy signals are usually sensitive to the signal, and a comparison of these shows that they are very similar for raw and denoised data. This suggests that denoising does not substantially change the underlying signal in the data.

Visualization with multidimensional scaling

Figure 3a shows the multidimensional-scaling embedding of the raw (non-denoised) 33-D MDS-UPDRS Part III trajectories to a plane (2-D) for 30 randomly chosen patients. Clearly, the raw trajectories are very noisy. Any attempt to statistically analyze these trajectories is likely to be strongly influenced by the noise.

Fig. 3. Visualizing sample trajectories by embedding them in 2-dimensions with multi-dimensional scaling.

a 15 sample raw trajectories. b Denoised trajectories. c Linear approximation of the denoised trajectories. d Progression vectors of the linear approximation.

Figure 3b shows the denoised trajectories of the same patients. Disease progression is now more clearly evident. Figure 3b also suggests that the denoised trajectories can be reasonably approximated as linear. Hence, we took the first and last time points of the denoised trajectories to create linear trajectories shown in Fig. 3c. Linear trajectories can be analyzed in terms of their baselines and progression vectors. The baseline vector is the vector from the origin to the denoised baseline score. The progression vector is the vector from the denoised baseline score to the baseline plus year 4 score. It represents the change in the scores over this span of time (examples shown in Fig. 3d). Because we analyze data over a fixed time period of 4 years, the progression vector can also be thought of as a rate of progression over this span of time.

Analysis of denoised baseline scores

Figure 4a shows the 33-D baseline scores of all subjects projected to 2-D using multidimensional scaling. Each point (dot) in the figure is the projection of a baseline score of a single patient. The colored contours indicate density of points obtained via a kernel-density estimate (described in Section “Kernel Density Estimation”). The contours of the kernel-density estimate suggest that baseline scores are unimodal. Unimodality implies that the heterogeneity at baseline is continuously variable and does not exhibit distinct clusters. Well-defined subtypes would imply a multimodal density.

Fig. 4. Visualization and hierarchical clustering of denoised baseline scores in 33-dimension space of MDS-UPDRS (Movement Disorder Society-Sponsored Unified Parkinson’s Disease Rating Scale) Part III (three) scores.

a Scatter plot of baseline scores with density projected onto 2-D using multi-dimensional scaling. b Dendrogram: four clusters (indicated by the dashed line) were chosen as data neighborhoods (referred to as B₁, ⋯ , B₄); c Silhouette Scores; d Gap Statistics. e Baseline neighborhoods projected on to 2-D. The population in each neighborhood is indicated. The mean ( ± stdev.) motor subscores of each neighborhood are shown as bar graphs. B+R (bradykinesia plus rigidity); Pop. (population).

Data modality

Motivated by Fig. 4a, we statistically evaluated baseline data for unimodality. The classic statistical test of unimodality is Hartigan’s dip test³⁹. However, this test only applies to univariate data. In the literature, several attempts have been made to extend Hartigan’s dip test to multivariate data, but none is generally accepted. In one approach, called the dip-dist test⁴⁰, the scalar Euclidean distances between each data point and all other data points are calculated and tested for unimodality using Hartigan’s dip test (see Methods Section “Tests for unimodality” for a longer description). Accepting the null hypothesis of unimodality of distances at each point amounts to testing for unimodality of the data. In another approach, called the folding test⁴¹, multivariate data are “folded” back on themselves and the original variance of the data is compared with the folded variance to test for unimodality (see Methods Section “ Tests for unimodality”).

We applied the dip-dist test and the folding test to the baseline scores. The dip-dist test, with Bonferroni and Benjamini-Yekutieli corrections for multiple testing, applied to the 33-D denoised baseline scores of 386 patients, could not reject the null hypothesis of unimodality at p = 0.05. The folding test also accepted the hypothesis of unimodality at p = 0.0001 (Φ = 5.10). We also carried out additional tests of unimodality vs multimodality using the SigClust simulation method (available in R). The result, which is available in the Supplemenatry Material, also accepts the hypothesis of unimodality.

Finally, we evaluated whether there were any linearly separable clusters in the data. This was done by projecting the 33-D data repeatedly onto 1-D with 10, 000 random Gaussian projections. Rejecting the null hypothesis of uni-modality in any projection would indicate the presence of linearly separable clusters in the data. We applied Hartigan’s dip test with Bonferroni and Benjamini-Hochberg corrections for multiple testing to evaluate unimodality of the projection. We were unable to reject unimodality in any of the projections at p = 0.05.

Neighborhoods for analyzing data

After establishing that the 33-D denoised baseline scores were unimodal, we sought to characterize the heterogeneity by creating different data neighborhoods to summarize the population. To generate these neighborhoods, we hierarchically clustered the baseline scores using Agglomerative Clustering (see Section “Hierarchical Clustering” for details). To select the number of clusters, we looked at a dendrogram drawn with Ward linkage method, silhouette scores, and the gap statistics for different numbers of clusters (see Fig. 4b–d). These three statistics did not agree on a clear choice of the number of clusters, which is expected as the data were shown to be unimodal (see Section “Data Modality”). We chose four clusters as a compromise between having sufficiently many neighborhoods to capture heterogeneity, while keeping the neighborhoods large enough to give reliable statistics, and being a reasonable choice of clusters according to all three metrics (see Section “Other differences in B₁,···, B₄”).

We label these neighborhoods B₁,⋯, B₄. The multidimensional scaling projections of the neighborhoods along with the projection of the data points is shown in Fig. 4c. The overlap in the B₁,⋯, B₄ neighborhoods in the 2-D visualization occurs because the clustering is performed on the full 33-D baseline scores, but projected in 2-D. Figure 4c also shows the population of B₁, ⋯ , B₄. The neighborhoods B₂, B₃ contain the majority of the data (58%).

Heterogeneity in baseline motor scores

Figure 4c shows the mean baseline B+R, Tremor and Axial subscores in each neighborhood as bar graphs. These bar graphs suggest that the subscore means are different in each neighborhood. To assess this more formally, we calculated the means and standard deviations of TMS, H&Y, and B+R, Tremor, and Axial subscores of baseline data in B₁, ⋯ , B₄ (displayed as the BL rows in Table 1). We used two-sided t-tests (unequal variances, unequal data sizes) to assess whether the means were similar. After adjusting for multiple comparisons (Benjamini-Hochberg correction), none of the hypotheses of equality of means were significant at p = 0.05. This suggests that each of B₁, ⋯ , B₄ contains baseline scores with distinct mean values, confirming heterogeneity at baseline.

Table 1.

Comparison of mean TMS, B+R, Tremor, Axial, and H&Y scores in baseline neighborhoods B₁, ⋯ , B₄

		B1	B2	B3	B4
B+R	BL	23.03 (0.53)	9.58 (0.34)	12.41 (0.4)	18.82 (0.45)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	4.96 (2.69)	5.13 (2.05)	5.4 (2.77)	5.02 (2.76)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
Tremor	BL	5.74 (0.56)	2.88 (0.26)	6.55 (0.69)	2.02 (0.14)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	0.78 (1.39)	0.89 (1.3)	1.17 (1.34)	0.77 (1.16)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
Axial	BL	2.24 (0.46)	1.07 (0.21)	1.31 (0.27)	1.59 (0.33)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	1.07 (0.84)	0.82 (0.67)	0.89 (0.76)	0.95 (0.86)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
TMS	BL	31.8 (6.05)	13.9 (4.07)	20.75 (4.87)	23.12 (4.78)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	7.0 (3.91)	7.05 (2.91)	7.63 (3.89)	6.92 (3.71)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
H&Y	BL	1.88 (0.32)	1.43 (0.38)	1.54 (0.36)	1.8 (0.31)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	0.23 (0.29)	0.35 (0.28)	0.37 (0.24)	0.23 (0.23)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4

Each of TMS, B+R, Tremor, Axial parts, and H&Y have two rows. The BL row shows the mean and std. dev. of the score for the neighborhood. The Δ row shows the mean and std. dev. of the score progression vector for the neighborhood. Bold-faced neighborhoods below each entry indicate the neighborhood that has a significantly different mean.

Progression vectors in B₁, ⋯ , B₄

Next, we analyzed how baseline neigborhoods relate to progression vectors. To this end, we compared the mean progression vectors for TMS, B+R, Tremor, and Axial subscores in the neighborhoods B₁, ⋯ , B₄ (displayed as the Δ rows in Table 1) with a two-sample t-test. None of the null hypotheses of equality of means could be rejected at p = 0.05 after adjusting for multiple comparisons (Benjamini-Hochberg correction). This suggests that progression in all baseline neighborhoods was similar over 4 years. That is, knowing the baseline neighborhood does not contribute to predicting the progression vector.

Predicting progression vectors

We further explored whether baseline vectors can predict progression vectors by using Partial Least Squares Regression (PLS) and Principal Component Regression (PCR). The goal of these analyses is to evaluate whether the entirety of the progression vector, i.e., each of its 33 components, can be estimated from the entirety of the baseline vector.

PLS was used to estimate each component of the progression vector from the entire 33-D baseline vector. When analyzing a progression vector component, the baseline and progression vector (component) data were split into a training (80%) and test (20%) set. The PLS model with k components was used to fit the training data set, and the learned model was used to predict the progression vector component using the test set. The R² (coefficient of determination) value of the test set prediction was used to assess the quality of the prediction. The R² value is the fraction of the progression vector component variance explained by PLS. Values of R² less than 0.2 are generally taken to indicate no or poor prediction.

The number of PLS components (k_opt) which gave the highest R² value for the test set was chosen, and is reported in Table 2. Next, the entire was dataset was repeatedly repartitioned into a training (80%) and test set (20%) for 1000 times. For each partition, PLS was fit to the training set using k_opt components, and evaluated on the test set using R². Table 2 shows the median ± median-absolute-deviation values of R² for the training and test set over the 1000 trials for each progression vector component. As Table 2 shows, the largest R² value for the test set occurs for the fourth component (arm rigidity on the less affected side). This value, and in fact all R² values for the test data, are less than 0.2, many significantly so, suggesting no or very poor ability to predict the progression vector from the baseline.

Table 2.

Partial Least Squares Regression (PLS) and Principal Component Regression (PCR) of denoised baseline vector with each component of denoised progression vector

	PLS			PCR
	Num.	R2 Train	R2 Test	Num.	R2 Train	R2 Test
	Comp.	med (mad)	med (mad)	Comp.	med (mad)	med (mad)
	(kopt)			(kopt)
1	3	0.145(0.058)	− 0.014(0.058)	1	0.003(0.013)	− 0.012(0.013)
2	4	0.156(0.055)	− 0.013(0.055)	1	0.007(0.015)	− 0.004(0.015)
3	2	0.166(0.056)	0.020(0.056)	4	0.001(0.012)	− 0.014(0.012)
4	3	0.291(0.055)	0.156 (0.055)	3	0.015(0.019)	− 0.003(0.019)
5	1	0.176(0.052)	0.136(0.052)	4	0.106(0.038)	0.090(0.038)
6	3	0.137(0.062)	− 0.037(0.062)	1	0.010(0.019)	− 0.005(0.019)
7	2	0.172(0.069)	0.023(0.069)	4	0.017(0.018)	0.003(0.018)
8	1	0.153(0.048)	0.107(0.048)	3	0.032(0.024)	0.016(0.024)
9	2	0.150(0.048)	0.034(0.048)	2	0.061(0.030)	0.046(0.030)
10	1	0.098(0.044)	0.045(0.044)	1	0.026(0.024)	0.010(0.024)
11	1	0.058(0.036)	0.011(0.036)	4	0.024(0.023)	0.007(0.023)
12	1	0.068(0.039)	0.014(0.039)	1	0.010(0.019)	− 0.007(0.019)
13	4	0.248(0.073)	0.065(0.073)	4	0.066(0.031)	0.047(0.031)
14	1	0.060(0.036)	− 0.006(0.036)	2	0.006(0.015)	− 0.010(0.015)
15	4	0.240(0.064)	0.069(0.064)	3	0.039(0.026)	0.022(0.026)
16	1	0.042(0.031)	− 0.020(0.031)	1	0.008(0.017)	− 0.009(0.017)
17	4	0.160(0.056)	− 0.024(0.056)	4	0.007(0.015)	− 0.008(0.015)
18	1	0.145(0.057)	0.079(0.057)	3	0.070(0.032)	0.049(0.032)
19	4	0.264(0.078)	0.073(0.078)	2	0.002(0.012)	− 0.014(0.012)
20	2	0.248(0.139)	− 0.005(0.139)	4	0.020(0.023)	0.007(0.023)
21	2	0.309(0.127)	0.173(0.127)	4	0.037(0.026)	0.014(0.026)
22	1	0.037(0.034)	− 0.034(0.034)	2	0.006(0.016)	− 0.008(0.016)
23	4	0.305(0.065)	0.140(0.065)	4	0.024(0.024)	0.010(0.024)
24	3	0.136(0.051)	− 0.019(0.051)	4	0.044(0.031)	0.031(0.031)
25	3	0.215(0.069)	0.082(0.069)	4	0.044(0.034)	0.027(0.034)
26	4	0.261(0.079)	0.097(0.079)	4	0.008(0.017)	− 0.011(0.017)
27	2	0.225(0.073)	0.106(0.073)	4	0.073(0.037)	0.053 (0.037)
28	2	0.132(0.048)	0.019(0.048)	4	0.025(0.022)	0.009(0.022)
29	2	0.087(0.046)	− 0.011(0.046)	3	0.001(0.011)	− 0.012(0.011)
30	1	0.087(0.042)	0.030(0.042)	3	0.037(0.028)	0.024(0.028)
31	4	0.138(0.073)	− 0.068(0.073)	4	0.007(0.017)	− 0.010(0.017)
32	4	0.088(0.068)	− 0.281(0.068)	1	0.002(0.015)	− 0.016(0.015)
33	2	0.081(0.042)	− 0.050(0.042)	4	0.003(0.013)	− 0.011(0.013)

The first column shows the identity of the progression vector component. Within PLS and PCR, the first column shows the optimal number of components obtained by cross-validation, and the second and third columns show the R² value of the regression for training and test data. The largest R² value for test data occurs for the 4th progression vector component for PLS and the 26th component for PCR. Both are displayed in bold font.

We also analyzed the relation between baseline vectors and progression vectors using PCR. The motivation for using PCR is that the components of baseline vectors are likely to be correlated, and this correlation can adversely affect the regression of baseline and progression vectors. PCR uses principal component analysis to eliminate the correlation and provide a more robust regression. We evaluated PCR in the same manner as we evaluated PLS – an initial training- test split and the R² value was used to find the optimal number of components. Then, as above, a 1000-fold repartitioning of the data was used to calculate the R² values of for training and test data for each progression component. The results are also shown in Table 2. The progression vector component with the highest R² value is “hand kinetic tremor on the less-affected side”. This R² value, and in fact all R² values, are significantly less that 0.2, suggesting no or very poor ability to predict the progression vector from the baseline.

Other differences in B₁, ⋯ , B₄

In a further attempt to investigate what was different about the baseline neighborhoods, we compared age, sex, genetic mutations, and DaTscan striatal binding ratios (SBR, calculated as described in Section “DaTscan Analysis”) in B₁, ⋯ , B₄. Table 3 shows the mean values ( ± stdev.) for age, the male/female ratio, ratios of pathogenic mutations for genetics, and mean ( ± stdev.) baseline SBR (BL row of DaTscan entry) and SBR change in (baseline-year 4, Δ row of DaTscan entry) in each neighborhood.

Table 3.

Comparison of age, sex, genetics, and DaTscan striatal binding ratios (SBR)s for baseline neighborhoods

Neighborhood		B1	B2	B3	B4
Age		62.9 (8.6)	60.0 (9.8)	62.5 (9.4)	59.8 (10.1)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
Male/Female		41/24	65/44	72/43	61/36
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
LRRK2		5/60	30/79	11/104	20/77
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
GBA		7/58	11/98	4/111	6/90
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
APOEε4		13/52	24/85	19/96	27/69
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
DaTscan
	BL	2.52 (0.62)	2.92 (0.73)	2.96 (0.78)	2.52 (0.74)
Total SBR		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
	Δ	− 0.68 (0.38)	− 0.67 (0.36)	− 0.75 (0.41)	− 0.7 (0.47)
		B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4	B1 B2 B3 B4
		31 (57%)	52 (46%)	69 (60%)	46 (47%)

For numerical variables (age, SBR): means and standard deviations in each neighborhood are shown. For genetics, the numbers indicate YES/NO counts. For LRRK2 and GBA, YES indicates the presence of a pathogentic mutation. For APOEε4, YES indicates the presence of the APOEε4allele. Bold-faced neighborhoods below a value indicate that they are significantly different from the above value.

Statistical tests (two-sided t-tests with differing variance and counts for age and SBR baseline and change, two sample Z-tests for equality of proportions for male/female ratio and genetic mutations) with Benjamini-Hochberg correction for multiple comparisons showed that only pathogenic LRRK2 mutations were significantly different. The occurrence of the mutations in B₂, B₄ had a significantly higher frequency than in B₁, B₃.

Summary of baseline results

In summary, the following picture emerges for the heterogeneity of denoised baseline scores:

1. Denoised baseline scores do not show evidence of clustering. They seem to be continuously spread in the data space.

2. Denoised baseline scores are heterogeneous. Different regions of the data space, as represented by the different baseline neighborhoods, have statistically different subscores.

3. Denoised baseline scores do not predict, or very poorly predict, progression.

4. LRRK2 mutations have a higher rate in B₂, B₄. However, there are no differences in age, male/female ratios, other genetic mutations, and SBRs.

Heterogeneity of progression vectors

Next, we analyzed progression vectors. This analysis was carried out by translating all progression vectors so their tails were at the origin. Then multidimensional scaling was used to visualize the tips of the progression vectors, see Fig. 5a. Figure 5a is analogous to Fig. 4a; each point (dot) in Fig. 5a represents the progression vector of a participant. The colored contours represent the density of points (i.e., density of progression vectors). The baseline scores, progression vectors are spread continuously rather than being clustered.

Fig. 5. Visualization of denoised progression vectors in 33-D (33-dimensional) space of MDS-UPDRS (Movement Disease Society-Sponsored Unified Parkinson’s Disease Rating Scale) Part III (three) scores.

a Two-dimensional scatter plot and density of progression vectors after multidimensional scaling. b Dendrogram; c Silhouette Scores; d Gap Statistics. e Six neighborhoods in which progression vectors are visualized in 2 dimensions after multidimensional scaling. The bar graphs in each neighborhood show mean progression of B+R (bradykinesia plus rigidity), Tremor, and Axial features. Pop. (population).

Data modality

We tested for unimodality of progression vectors using the dip-dist and folding tests mentioned in Section “Data Modality”. The dip-dist test could not reject unimodality at p = 0.05. The folding test accepted unimodality at p = 0.0001 (Φ = 29.06). Further 10, 000 random Gaussian projections of the progression vectors onto 1-D were all unimodal at p = 0.05 with Bonferroni and Benjamini–Hochberg corrections. As we did for baseline scores, we also tested progression vectors for unimodality using SigClust. The results, available in Supplementary Materials, support the hypothesis of unimodality.

Neighborhoods for progression vectors

Using the same methodology as used for baseline scores, we hierarchically clustered the 33-D progression vectors. Similar to baseline vectors, Ward’s metric, silhouette scores, and gap statistics (see Fig. 5b–d) did not reveal any natural clusters. Using these metrics, we selected six neighborhoods named P₁ − P₆ as sufficiently high in number to examine different regions of the data but still a reasonable choice according to the previously mentioned statistics (Fig. 5b shows the dendrogram; details of the choice of six clusters are in the Section “Hierarchical Clustering”). The multidimensional scaling projections of P₁ − P₆ are shown in Fig. 5c. The figure also shows the population in each neighborhood, as well as bar graphs for the mean values of the B+R, Tremor, and Axial progression subscores.

Table 4 shows the mean and std. dev. of TMS scores, B+R, Tremor, and Axial subscores in P₁, ⋯ , P₆. We carried out paired t-tests comparing the TMS, B+R, Tremor, and Axial subscore progression vectors in neighborhoods P₁ − P₆. The lines below the mean (std. dev.) numbers show in bold which neighborhood had significantly different means (at p = 0.05 with Benjamini-Hochberg correction for multiple comparisons). Table 4 shows that each neighborhood differs from every other neighborhood on some subscore. For example, P₂ differs from P₁, P₃, P₄, P₆ in the B+R subscore, and from P₄ in the Tremor subscore. This suggests that progression vectors are significantly heterogeneous.

Table 4.

Comparison of the 6 neighborhoods identified from the projection vectors

Neighborhood	P1	P2	P3	P4	P5	P6
ΔH&Y	0.15(0.2)	0.29(0.25)	0.37(0.24)	0.3(0.26)	0.44(0.23)	0.59(0.3)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
MDS-UPDRS Part 1
ΔTotal	2.98(4.06)	2.75(4.84)	4.43(4.88)	3.24(4.25)	3.96(5.0)	4.71(6.21)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
MDS-UPDRS Part 2
ΔTotal	0.69(2.32)	1.95(2.91)	2.78(3.82)	1.5(2.51)	3.7(3.02)	4.9(4.37)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
MDS-UPDRS Part 3
ΔTMS	2.65(1.42)	6.07(1.77)	10.07(1.63)	7.44(2.16)	13.06(2.18)	11.59(3.96)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
ΔB+R	2.15(1.32)	4.72(1.32)	8.06(1.19)	4.8(1.55)	8.84(2.03)	7.83(3.29)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
ΔTremor	-0.16(0.94)	0.42(0.66)	0.41(0.84)	1.78(1.04)	2.97(1.12)	0.28(0.7)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
ΔAxial	0.53(0.46)	0.75(0.47)	1.35(0.5)	0.68(0.38)	1.0(0.49)	3.24(0.79)
	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
Progression Vector Length	1.28	1.58	2.39	1.87	2.90	3.05

For numeric variables: means and standard deviations of participants in each neighborhood. Bold-faced neighborhoods below a mean indicate that the above mean is significantly different from the mean of the bold-faced neighborhood.

Heterogeneity of progression vector length and direction

The analysis in the previous section is based on movement subscores. A different perspective is obtained if we take each 33-D progression vector and analyze its length and direction in the 33-D space. We compared the length of the progression vectors in each neighborhood. The mean length of the progression vectors in each neighborhood is shown in the last row of Table 4. A paired t-test of the lengths of progression vectors in P₁, ⋯ , P₆ reveals that the mean lengths are all different at p = 0.05 (with Benjamini-Hochberg correction). P₆ has the largest mean length, while P₁ has the smallest. Further, P₃ and P₄ have larger progression lengths than P₂ and P_5, respectively.

Dividing the progression vector by its length projects it onto the unit sphere (see Fig. 1a5 for an illustration of this idea). This unit length vector represents the direction of the progression vector. The means of the unit vectors in each neighborhood are visualized in 2-D in Fig. 6 using multidimensional scaling. The visualization in Fig. 6 suggests that P₂, P₃, and separately P₄, P₅, are similar in direction. To test this statistically, we used the integrated likelihood ratio test (ILRT) for mean directions⁴². The ILRT for mean directions was selected because it is known to perform well in high dimensions⁴². The ILRT for mean direction compares two groups of unit vectors for equality of mean. Using the ILRT to compare equality of mean in every pair of neighborhoods shows that the mean unit vectors in P₂, P₃ have a similar directions (p = 4.6e − 11). The same is true for P₄, P₅ (p = 3.4e − 11).

Fig. 6. Directions of neighborhoods.

Mean directions of each neighborhood visualized in 2-D (2-dimension) using multidimensional scaling.

Additional characteristics of P₁, ⋯ , P₆

We compared the age and sex of the participants in P₁, ⋯ , P₆ (Table 5). Participants in P₆ are nearly a decade older (mean age of 70.2) than participants in the other groups (mean age of 60.6 for P1 − P5). There are no statistically significant differences in age among P₁ − P₅. P₂ − P₅ have more men than women. P₁, however, has more women than men (31 men/34 women) but comparing P₁ to the rest of the population P2 − P6 using a two-sample z-test for proportions does not yield a statistically significant difference after correcting for multiple comparisons (p = 0.009 > p = 0.0083).

Table 5.

Comparison of age, sex, genetics, and DaTscan imaging in P₁ − P₆ identified from the projection vectors

Neighborhood		P1	P2	P3	P4	P5	P6
Age		62.0(8.2)	59.4(9.4)	60.8(11.7)	61.1(9.7)	60.5(8.1)	70.2(8.8)
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
Male/Female		31/34	76/44	22/12	72/38	25/9	12/10
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
LRRK2		12/53	19/101	8/26	17/93	6/28	4/19
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
GBA		4/61	10/110	4/30	9/101	0/34	2/21
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
APOEε4		12/53	23/97	8/26	25/85	10/24	6/17
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
DaTscan
	Bl	2.65(0.66)	2.71(0.59)	2.67(0.76)	3.04(0.82)	2.9(0.65)	2.09(0.83)
Total SBR		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
	Δ	-0.71(0.4)	-0.63(0.46)	-0.77(0.39)	-0.73(0.37)	-0.97(0.41)	-0.57(0.31)
		P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6	P1 P2 P3 P4 P5 P6
Participants with DaTscans		37 (56%)	55 (46%)	19 (56%)	67 (64%)	8 (24%)	12 (55

For numeric variables: means and standard deviations of participants in each neighborhood. For genetics: numbers indicate YES/NO counts. For LRRK2and GBA, this is the presence of a pathogentic mutation. For APOEε4, this is the presence of APOEε4. Bold-faced neighborhoods below a value indicate that the above value is significantly different from the value of the bold-faced neighborhood.

We next compared the prevalence of pathogenic genetic variants for LRRK2 and GBA and labels of APOEε4 alleles for each participant (Table 5). Rates of pathogenic genetic mutations were not found to be significantly different between any progression neighborhoods.

Finally, we compared DaTscan images of participants in each neighborhood. DaTscan images were compared using the total striatal binding ratio (SBR) as described in Methods Section “DaTscan Analysis”. The SBR was calculated at baseline and at year four for 198 of the 386 participants who had DaTscan images from both times. The fraction of subjects in each neighborhood for whom DaTscans were available is shown in the last row of Table 5. Except for P₅ the neighborhoods have a more or less similar fraction of subjects with DaTscans.

To account for variation in the actual time between the baseline imaging and the 4-year imaging, the 4-year change in SBR was linearly interpolated. At baseline, we found that the total SBR in P₆ was significantly lower than in many other neighborhoods (the mean total SBR in P₆ was 2.09 while the mean total SBR for P₁ − P₅ ranged from 2.65 − 3.04. See Table 5).

Summary of progression vector heterogeneity in movement scores

In summary, the following picture emerges of progression vector heterogeneity (see Fig. 7a for a multidimensional scaling visualization of the progression vectors in the neighborhoods.):

1. Neighborhoods P₂, P₄ contain over 60% of the participants. When additionally combined with P₃, P₅, these neighborhoods contain 88% of the population. Thus, these four neighborhoods contain majority of the population. P₂, P₃, P₄, P₅ have the following characteristics:

   • P₂ and P₃ have similar mean progression directions, but the mean progression vector in P₃ is bigger than the mean progression vector in P₂. That is, P₃ is a faster-progressing version of P₂. A similar argument holds for P₄ and P₅; P₅ is a faster version of P₄.
   • P₂ has moderate progression of B+R subscores while tremor and axial subscores progress relatively slowly. P₄ also exhibits moderate progression of B+R subscores, but in addition, it exhibits moderate progression of the tremor subscore.
   • All of the above can be summarized by saying that progression for most of the patients can be explained as variability along two “axes” (see Fig. 7a). The magnitude of tremor progression varies along one axis, while progression vector length (i.e the speed of progression) varies along the other.

2. P₆ and P₁ are the exceptional neighborhoods. Each of these neighborhoods contains less than 10% of the population.

3. P₆ has the longest progression vectors (fastest progression speed) of all participants. Progression vectors in P₆ show large changes in B+R and axial subscores. However, tremor subscores are virtually unchanged. P₆ also has the oldest population, and significantly lower SBRs.

4. P₁ has the smallest progression vectors (slowest speed). P₁ has mildly changing B+R subscores, while other subscores do not change significantly.

5. There are no genetic mutation differences between the neighborhoods.

Fig. 7. Visualization of progression vectors and their neighborhoods.

a Multi-dimensional scaling (MDS) projections of progression vectors in the neighborhoods. b An illustration of the relative locations of progression vector neighborhoods. The means of P₂, P₃ and P₄, P₅ lie along lines through the origin. PC (principle component).

Discussion

Parkinson’s disease presents and progresses heterogeneously, yet there is no clear consensus on a taxonomy for this heterogeneity. Possible reasons for this are that studies may be underpowered for the amount of noise in the data, or that the ordinal nature of PD clinical scores is not taken into account. In some cases, the non-Markovian nature of progression may hinder the analysis. We attempted to overcome these limitations by using data from a large study, by explicitly denoising the longitudinal motor scores of the participants, and analyzing baseline scores and progression vectors. The analysis of progression vectors could be simplified to a linear analysis because PD progression is slow in the early stage and can be approximated linearly as discussed in the Results section. Extending our analysis to longer time durations may require a more sophisticated approach.

Visualizing baseline and progression scores shows that both are heterogeneous, but there do not appear to be any obvious and distinct clusters in either. As mentioned earlier, there is no generally acceptable statistical test of unimodality, but the dip-dist, the folding test, and the random projections strongly suggest that baseline scores and progression vectors may be better thought of as continuously variable rather than as subtypes. This conceptualization is consistent with a recent call for analyzing PD heterogeneity without explicit subtypes⁸.

The analysis of denoised baseline and progression vectors shows that the latter are not linearly predictable from the former (PLS and PCR are linear models). That is, motor-score progression is not linearly Markovian from baseline. In principle, supplementing baseline motor scores with other information (e.g., demographics or DaTscans) may make progression predictable. It is also possible that a nonlinear model might demonstrate increased predictability. A third and intriguing possibility is that while the entire 33-D progression vector is not predictable, some meaningful low-dimensional function of it might be. This possibility is suggested by the study⁴³ where gait, freezing of gait, and instability scores are summed into an ambulatory capacity measure, which is then shown to be predictable from baseline.

Finally, different baseline score neighborhoods have a very similar distribution of age, sex, and genetic mutations with the exception of LRRK2. LRRK2 mutations have a higher rate in B₂, B₄ compared to B₁, B₃. B₂, B₄ also appear to have lower tremor scores than B₁, B₃. One possible explanation for this is the slower progression of PD for LRRK2 mutation as noted in ref. ⁴⁴.

In contrast to baseline scores, progression vectors offer more meaningful information: most of the PD participants are contained in neighborhoods P₂ − P₅. The progression in these neighborhoods can be explained by using two axes. The progression in Tremor subscores varies along one axis, while the overall speed of progression varies along the other.

Two smaller subsets of the population are contained in P₁ and P₆. The relatively high baseline scores and very gradual progression of P₁ do not match any previously reported subtype of PD progression. P₁ does have the highest percentage of women (although not statistically significant), a factor that might be related to the slower progression. But note that previous comparisons of differences in PD progression in men and women have not yielded conclusive results^45,46. Interestingly, despite slow progression in P₁, the DaTscans SBRs for this group showed no significant differences when compared to P₂ − P₅. Participants such as these—with very slow symptom progression but with decreasing total SBR typical of PD—may partially explain why some studies^47–50 found inconsistent and low correlations between motor symptoms and DaTscans.

P₆ is perhaps the most distinct of the six neighborhoods, with the oldest population and the fastest progression of axial symptoms and H&Y stage. P₆ bears similarity to the late-onset subtype proposed previously^14,15,51. P₆ is consistent with the idea that late onset is more severe²⁶. P₆ is also comparable to the PIGD-D clinical subtype^2,3,13, which is characterized by higher initial axial symptoms and more rapid progression. However, we do not find baseline axial subscores to be higher in P₆ compared to P₁ − P₅. Additionally, higher initial H&Y scores in P₆ suggest that patients in P₆ may also be further along in their disease trajectory at the time of enrollment in the study. While P1 and P6 are intriguing subpopulations, they do contain only a small number of subjects. It is possible that with a larger study, the characteristics of P1 and P6 might change. In this context, we also want to point out a possible selection bias in P₆. As we noted above, P₆ contains the oldest population, and given our selection criterion (at least 5 visits in 4 years) it is possible that some subjects who might otherwise fall into category are excluded (members of the older population may not have had 5 visits in 4 years).

Regarding the relation of genetic mutations and PD progression, previous literature has shown a weak protective relationship between LRRK2 mutations and speed of motor symptom progression^21,23,44. But there are conflicting results⁵² which suggest that LRRK2 mutations may be associated with more severe PIGD symptoms. Previous research comparing GBA mutations and motor symptoms also have mixed results, some²³ finding no relationship between the two while others²² show that GBA was associated with more rapid motor-symptom progression. Finally, some have found APOEε4to have no effect on motor symptom progression²⁰, while others have found a correlation with faster progression but only after 5+ years¹⁹.

For all three of these genetic risk factors (LRRK2, GBA, and APOEε4), we found no significant differences between any of the progression-vector neighborhoods. In part, this may be due to the relatively small number of participants with pathogenic versions of each mutation (29(7.5%) for GBA, 66(17.1%) for LRRK2, 84(21.8%) for APOEε4) and also due to the fact that we only used data from 4 years of early stage progression. Note that Tan et al.⁵³ also found no significant relationship between any genetic mutations and motor-symptom progression.

One limitation of our study is that it focuses on relatively early-stage PD. It is possible that later stages of the disease may be more Markovian and less heterogeneous. Moreover, all of our analysis is retrospective, and some effort is needed to make it predictive. Another limitation of the study is that over 96% the participants in the study are Caucasian and all are US-based. There is evidence that ethnicity and nationality have an effect on PD progression. For example, at first diagnosis, the African American population in the US appears to present with a later stage of the disease^54–56. It is possible that the longitudinal progression and the heterogeneity of the disease may also differ in this subpopulation. Other mono-ethnicity studies show differences in the prevalence of various PD subtypes around the world⁵⁷. Turning to nationality, there is evidence to suggest that phenotyping of PD in closely related ethnicities differs based on geographic location⁵⁸. Currently, there is no data set comparable to the size and complexity of PPMI to address these issues. However, we note that our methodology would be applicable without change to such datasets.

In conclusion, our data-driven denoising and clustering approach provides novel insights into the heterogeneity of baseline and progression characteristics of PD and sheds light on the root causes of inconsistencies in previous reports. Our methodology can serve as an analytical linchpin tying together the biological and clinical characteristics of the disease anchored in an in-depth quantitative understanding of its inherent heterogeneity. Our findings also have implications for recent efforts to develop biological staging frameworks of PD for research purposes^59,60 as well as for outcome assessments in clinical trials using disease-modifying therapies. For example, our results suggest that denoising is important in assessing PD progress (staging) and the effect of therapy on it. Our results also suggest that any clustering of baseline or of progression measures should be accompanied by clear statistical evidence that the clustering is statistically valid.

Methods

Inclusion and exclusion of participants

Data were obtained by the Parkinson’s Progression Markers Initiative (PPMI, http://www.ppmi-info.org)³⁸. The criteria for inclusion and exclusion in the PPMI PD cohort are available in ref. ³⁸. MDS-UPDRS Part III assessments were conducted by PPMI at the following monthly intervals: 0, 3, 6, 9, 12, 18, 24, 30, 36, 42, 48, 54, 60 and then annually. We excluded MDS-UPDRS assessments that were done in the “ON medication” state as defined by PPMI ( < 12 h since last dose of LDOPA or LDOPA equivalent). Assessments which were conducted remotely, or which only had partial data, or which occurred after a participant had received deep brain stimulation surgery were also excluded. From the remaining, we selected participants who satisfied all of the following criteria: a clinical assessment within the first six months of their participation, a clinical assessment at least 4 years or longer after baseline, and at least 4 clinical assessments in total. This resulted in a pool of 386 participants, which we used in our analysis.

Genetics

Genetic testing for LRRK2, GBA, and APOEε4 was performed as part of the PPMI study. Labels of pathogenic mutations for both LRRK2 and GBA provided for each participant by PPMI were used. Participants with either one or two APOEε4 alleles as determined by PPMI were considered positive for APOEε4 in our analyses.

Denoising of MDS-UPDRS part III scores

The denoising method is specifically designed for use with MDS-UPDRS Part III questions³⁷. The method is implemented in MATLAB and is available for download (https://github.com/tagarelab/Ordinal-Time-Series-Denoising). As explained below, the method is a hierarchical, maximum-likelihood procedure.

The main idea behind the method is illustrated in Fig. 8. Suppose there are N subjects, with each subject represented by an ordinal time series of measurements. An ordinal regression model (regression with time) can be fit to each time series. The ordinal regression models the probability distribution of ordinal scores and their evolution in time. The regression model is explained in more detail below. We assume that the mean of the ordinal distribution represents the underlying “signal”, and the variability around the mean represents “noise”. The temporal change in the mean thus indicates any progression of the score. Note that the mean of an ordinal distribution is continuous-valued and not discrete, and thus the progression of the mean can be viewed as a smooth “interpolation” of the noisy discrete ordinal scores.

Fig. 8. Ordinal time series denoising model.

Given N time series the model uses N ordinal regression models, one for each time series. The parameters of the regression model come from a k-component Gaussian mixture prior. Each component (mode) of the prior represents similarly evolving time series. The parameters of the components of the prior share a conjugate hyperprior, representing the overall progression of all time series.

When the ordinal time series is short, the estimate of the probability distribution may not be reliable. However, if there are sufficiently large number of subjects, it is likely that several subjects have similar (but not identical) time series. These subjects can “pool” their data by assuming that the coefficients of their ordinal regression come from a single prior. More generally, assuming a mixture prior with k components and associating each subject’s regression with one component enables the regressions to draw statistical power from each other (see Fig. 8). All time series whose models share a component in the prior are similar. They are also distinct from time series which share any other component. By changing the variance of the Gaussian components, the model adapts to tightly clustered time series heterogeneity (using small component variance) or more continuously varying time series heterogeneity (using large component variance). The prior components themselves have parameters and these parameters are assumed to come from a single hyperprior which represents the overall progress of the disease in the entire population.

To be specific, the ordinal regression model we use is the a variant of the proportional-odds model⁶¹. Briefly, the model is as follows (this description is adapted from³⁷):

Suppose y is an ordinal random variable taking values in ordinal categories {0, ⋯ , C − 1}. A model for the probability distribution of y is

$$\begin{array}{c}\hfill p\left(y\mid \gamma \right)=\left\{\begin{array}{cc}\hfill F\left({\gamma }_1\right)\hfill & \mathrm{for}y=0\hfill \\ \hfill F\left({\gamma }_2\right)-F\left({\gamma }_1\right)\hfill & \mathrm{for}y=1\hfill \\ \hfill \cdots \hfill & \hfill \\ \hfill F\left({\gamma }_k\right)-F\left({\gamma }_{k-1}\right)\hfill & \mathrm{for}y=k\hfill \\ \hfill \cdots \hfill & \hfill \\ \hfill 1-F\left({\gamma }_{C-1}\right)\hfill & \mathrm{for}y=C-1,\hfill \end{array}\right)\end{array}$$ 1

where F is the cumulative distribution function (cdf) of the standard normal density and γ = (γ₁, ⋯ , γ_C−1) ∈ R^C−1 is a parameter of the distribution subject to the constraint γ₁≤γ₂≤γ₃≤ ⋯ ≤γ_C−1. It is useful to implicitly impose these constraints by a change of variables. Define a map Γ which maps u = (u₁, ⋯ , u_C−1) to γ = (γ₁, ⋯ , γ_C−1) as follows:

$$\begin{array}{c}{\gamma }_1=u_1\hfill \\ {\gamma }_k=u_1+\sum _{i=2}^k\log (1+e^{u_i}),\mathrm{for}k=2,\cdots ,C-1,\hfill \end{array}$$ 2

Then, the ordinal probability model of equation (1) can be written as p(y∣Γ(u)) with no constraints on u.

The above model extends to a time-series model as follows: suppose Y = (y₁, ⋯ , y_T) is a discrete ordinal time series with y_i ∈ {0, ⋯ , C − 1} for observed at time t_i ∈ [0, 1] for i = 1, ⋯ , T, with t_i+1 > t_i. Note that no assumption is made about the time series being uniformly sampled in time (i.e., t_i need not be equal to (i − 1)/(T − 1)). Then,

$$\begin{array}{c}\hfill p\left(Y\mid u,v\right)=\prod _{i=1}^{T}p\left(y_i\mid \mathrm{\Gamma }\left(u\right)\left(1-t_i\right)+\mathrm{\Gamma }\left(v\right)t_i\right),\end{array}$$ 3

where $u,v\in {\mathbb{R}}^{C-1}$ generate the thresholds γ that divide the probabilities at the start and end of the time interval [0, 1]. This model effectively assumes a linear change over time of the u value used in p(y∣Γ(u)). The model can be made more complex by making the change non-linear, but for slowly progressing diseases such as PD, a linear change is sufficient. To proceed, let θ = (u, v).

Suppose there are N subjects, with time series $Y_i=\left(y_{i,t_{i,1}},\cdots ,y_{i,t_{T_i}}\right)$, i = 1, ⋯ , N_i. Assuming that the ith subject has the subject’s own parameter θ_i = (u_i, v_i),

$$\begin{array}{c}\hfill p\left(Y_i\mid {\theta }_i\right)=\prod _{j=1}^{T_i}p\left(y_{i,t_{1,j}}\mid \mathrm{\Gamma }\left(u_i\right)\left(1-t_{i,j}\right)+\mathrm{\Gamma }\left(v_i\right)t_{i,j}\right).\end{array}$$ 4

Note that this formulation allows each subject to have a time series with different number of points (number of points in the time series = T_i), and the data in the series need not be uniformly sampled in time.

As mentioned above, we set the prior for θ_i to a normal mixture with K components:

$$p\left({\theta }_i\mid \left\{{\alpha }_k\right\},\left\{{\mu }_k\right\},{\rho }_{\mu }I\right)=\sum _{k=1}^K{\alpha }_k\mathcal{N}\left({\mu }_k,{\rho }_{\mu }I\right),$$ 5

where μ_k are the means of the components, and ρ_mu is the variance (assumed diagonal and equal for all components). The μ_k’s and ρ_μ are hyperparameters, and we use the following hyperprior for them:

$$p\left({\mu }_k\mid \zeta ,{\rho }_{\zeta }\right)=\mathcal{N}\left(\zeta ,{\rho }_{\zeta }I\right),$$ 6

$$p\left({\rho }_{\mu }\mid a,b\right)=IG\left(a,b\right)=\frac{b^a}{\mathrm{\Gamma }\left(a\right)}\frac{1}{{\left({\rho }_{\mu }\right)}^{\left(a+1\right)}}{e}^{\frac{-b}{{\rho }_{\mu }}},$$ 7

where IG is the inverse-gamma distribution. The hyperprior parameters a, b, ρ_ζ are fixed, by a cross-validation procedure described in detail in³⁷. The cross-validation procedure also establishes, K, the number of clusters in the prior of equation (5).

Grouping together all parameters to be estimated as ϕ = ({θ_i}, {μ_k}, ρ_μ, {α_k}, ζ), we get:

$$\begin{array}{ccc}\hfill p\left(\left\{Y_i\right\}\mid \varphi \right)& \hfill =\hfill & \prod _{i=1}^{N}p\left(Y_i\mid {\theta }_i\right)p\left({\theta }_i\mid \left\{{\alpha }_k\right\},\left\{{\mu }_k\right\},{\rho }_{\mu }\right)\hfill \\ \hfill & \hfill \hfill & \times \left(\prod _{k=1}^{K}p\left({\mu }_k\mid \zeta ,{\rho }_{\zeta }\right)\right)\times p\left({\rho }_{\mu }\mid a,b\right)\hfill \end{array}$$ 8

from which the maximum-likelihood estimates of ϕ are obtained as $\widehat{\varphi }=\arg \underset{\varphi }{\max }\log p\left(\left\{Y_i\right\}\mid \varphi \right)$. The maximum-likelihood estimates are easily calculated by the EM-algorithm whose details are given in³⁷.

The maximum likelihood estimates of ${\widehat{\theta }}_i=\left({\widehat{u}}_i,{\widehat{v}}_i\right)$ give the estimated mean for the ith subject at time t as:

$${\widehat{m}}_i\left(t\right)=\sum _{y=0}^{C-1}yp\left(y_t\mid \mathrm{\Gamma }\left({\widehat{u}}_i\right)\left(1-t\right)+\mathrm{\Gamma }\left({\widehat{v}}_i\right)t\right)$$ 9

The estimated mean is the denoised signal.

Mathematical details of the model are available in³⁷. That publication also evaluates the method with simulations and real MDS-UPDRS Part III scores. Simulations with data representing MDS-UPDRS Part III scores show that the method is superior to alternative filtering methods. Further experiments with real MDS-UPDRS Part III data shows that the method does not overfit: the distribution of the log-likelihood of training and test data is very similar.

MDS-UPDRS part III score processing

For each participant, the more severely affected side was determined by comparing sums of the denoised scores for all left-side and all right-side scores at baseline. The more severely affected side at baseline was considered more severely affected for all future visits.

To conveniently summarize results in this paper, we grouped and summed MDS-UPDRS Part III questions into three categories according to the cardinal signs the questions address as follows:

• Bradykinesia and Rigidity (B+R): 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 (18 scores)

• Tremor: 3.14, 3.15, 3.16, 3.17, 3.18 (10 scores)

• Axial: 3.9, 3.10, 3.11, 3.12 (4 scores)

MDS-UPDRS question 3.1, which addresses speech did not fit any of these categories, was not included in any of the sums, but was included in the vector representing baseline and progression scores.

Dimensionality reduction and statistical analyses

Multidimensional scaling (MDS) was performed in Python using the Scipy and Scikit-learn libraries. Multidimensional scaling (MDS) was then used to reduce the 33-D progression vectors to 2-D for visualization.

Kernel density estimation

Kernel density estimation was used to visualize the density of points in 2-D space. We used the gaussian_kde function from the Scipy Python package.

Tests for unimodality

Hartigan’s dip test is the classic unimodality test³⁹. We used the “diptest” package in R for this test.

The dip-dist test⁴⁰ calculates Euclidean distances from each point to all other points. Dip-dist tests the unimodality of the distances at each point by using Hartigan’s dip test. Thus, Hartigan’s test is applied multiple times, once at every point. We correct for multiple testing by using the more conservative Bonferroni correction and the less conservative Benjamini-Yekutieli correction. The latter is used because distances from a point to all other points is likely to be correlated.

The folding test⁴¹ folds data with respect to a center. The variance of the folded data is smaller than the variance of the original data. When the center is appropriately chosen (see⁴¹ for details) the reduction in the variance is larger for multimodal data compared to unimodal data. The folding test is based on this idea. The test is available as an R package (Rfolding). Our tests were carried out using this package.

If the data are clustered and the clusters are linearly separable, then an appropriate projection of the data onto a 1-D subspace can be multimodal. We projected data onto 10,000 random subspaces and applied Hartigan’s dip test to each projection, while correcting for multiple comparisons using Benjamini-Hochberg and Bonferroni corrections.

PLS and PCR

Partial Least Squares Regression (PLS) and Principal Component Regression are classic methods in statistics. We used implementations of both methods in the Python package Scikit-learn.

Hierarchical clustering

Baseline and progression vectors were clustered using Scikit-learn’s Agglomerative Clustering⁶². Agglomerative clustering begins by treating each data point as a cluster and then sequentially merging clusters so as to reduce the within-cluster variances. The process ends when all data have been merged into a single cluster. The merging process is represented as a tree (the dendrogram), with the individual data points as leaf nodes, and a single cluster as the root node. At any level in the tree, the effectiveness of the clustering is measured by Ward’s metric, which is the sum of the within-cluster variances of the data. Figure 9a shows this quantity as a function of number of clusters for baseline scores. Fig. 9b shows the same for progression vectors. Since the baseline and progression vector data are shown to be unimodal, there are no natural clusters. There is no single, generally accepted method for selecting the optimal number of clusters so we examined several different metrics. The first was Ward’s metric which is shown in the dendrogram. We also calculated the silhouette coefficients (Fig. 4c and Figure 5c) and gap statistics (Fig. 4d and Fig. 5d) for different numbers of clusters⁶³. Both were calculated using the agglomerative clustering results and the Python package, Scikit-learn. For the silhouette score, it is common to pick the number of clusters at the “elbow” of the plot—the number of clusters before the values plateau. Although neither graph has a clear “elbow”, 4 clusters and 6 clusters occur before the silhouette scores plateau. For the gap statistic, it is common to select the number of clusters for which there is a large gap between the gap statistic for n and n − 1 clusters.

Fig. 9. Within-cluster variances.

Plots of the total within-cluster variances for each number of clusters using agglomerative clustering for a baseline scores and b progression vectors. These values correspond to the heights of divisions in the dendrogram.

Neighborhoods were identified with Scikit-learn’s Agglomerative Clustering⁶². Ward’s method was used to form linkages. We selected the number of clusters by examining the dendrogram and selecting a cut-point with large distances between clusters. More specifically, this method stops clustering at a point where further merging of clusters would substantially increase the total within-cluster variance (the metric used for Ward’s method). To make meaningful statistical comparisons, we only considered sets of clusters where all clusters had populations above 10. This resulted in 4 clusters for baseline vectors and 6 for the progression vectors.

Pairwise comparisons between neighborhoods

Means of numerical quantities between neighborhoods were assessed with the Student’s t-test with a significance threshold of p = 0.05. Binary labels such as sex and genetic variants were compared between neighborhoods with a two sample Z-test for equality of proportions. All pairs of neighborhoods were compared. To account for multiple comparisons, the Benjamini-Hochberg correction was used with a 5% false discovery rate.

Integrated likelihood ratio test (ILRT)

We used the variant of the integrated likelihood ratio test (ILRT) for comparing mean directions of groups of vectors described in ref. ⁴². This test was chosen for its performance in high-dimensional cases (d > > 3) and its ability to deal with groups with low counts. Direction was defined as the 33-D unit vector with the same direction as the progression vector for all 33 MDS-UPDRS Part III scores.

DaTscan analysis

DaTscan images are available for PPMI participants at baseline and then annually for up to the next 5 years, with possible missing scans. Our analysis included participants who had DaTscans at baseline and at year 4. This criterion was satisfied by 199 of the 386 participants. DaTscans are already aligned by PPMI to the Montreal Neurological Institute’s (MNI) Atlas. We obtained regions for the bilateral caudates and putameni from the MNI Atlas. The DaTscans were normalized using the occipital region and then mean striatal binding ratios (SBRs) were calculated for the two caudates and two putameni. These four values were added together to get the total SBR. Since not all scans were exactly 4 years apart, SBR values were linearly interpolated.

Author contributions

J.D.K., H.D.T., and S.T. jointly developed the data analysis methodology. JDK and HDT analyzed the PPMI data. S.T. helped with the interpretation of the results. All authors contributed to the writing. All authors have read and approved the final manuscript.

Data availability

The data used in this article is publicly accessible and available from PPMI (www.ppmi-info.org). The denoising software used in this article is available from GitHub (https://github.com/tagarelab/Ordinal-Time-Series-Denoising).

Code availability

The denoising software used in this article is available from GitHub (https://github.com/tagarelab/Ordinal-Time-Series-Denoising).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41531-026-01369-w.