Disease progression model for Clinical Dementia Rating–Sum of Boxes in mild cognitive impairment and Alzheimer’s subjects from the Alzheimer’s Disease Neuroimaging Initiative

Figures & data

Table 1 Summary of structural models

Model description	Akaike information criterion
Linear progression	2,883.8
Linear progression with shape parameter θ	2,814.1
Linear progression with shape parameter θ1	2,814.3
Linear progression with shape parameter θ2	2,814.8
Linear progression with two shape parameters	2,814.8
Linear progression with power relationship	2,812.8
Exponential progression	2,551.4
Verhulst logistic model	2,571.5
Richard’s logistic model	2,496.7
Three-parameter logistic model	2,496.2

Figure 1 (A) Distribution of CSF p-tau_181P/Aβ_1–42 on log scale in CN, LMCI, and mild AD subjects. The curves and the vertical line are the density of the subpopulations and threshold of 0.147 (untransformed value), based on the CSF mixture model. (B) The observed density for log CSF p-tau_181P/Aβ_1–42 in CN, LMCI, and AD subjects is overlaid on top of the estimated threshold of −1.92 (natural log transformed value). (C) Mean CDR–SB profiles by biomarker status, in LMCI subjects. (D) Mean CDR–SB profiles by biomarker status, in mild AD subjects.

Note: The CSF p-tau_181P/Aβ_1–42 threshold of 0.147 separating the two subgroups was used to dichotomize the population for Panels (C) and (D).
Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; AD, Alzheimer’s disease; CN, cognitively normal; CSF, cerebrospinal fluid; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; SE, standard error.

Figure 2 Baseline covariates influencing CDR–SB disease progression rate in subjects with CSF p-tau_181P/Aβ_1–42>0.147.

Notes: (A) delayed logical memory in LMCI; (B) delayed logical memory in mild AD; (C) Trails A test in LMCI; and (D) Trails A test in mild AD. Covariates were dichotomized (>median and ≤median) to create roughly two equal groups in each panel. Median delayed logical memory scores for the ADNI-1 mild AD and LMCI biomarker-positive population at baseline were 0 and 3, respectively. Median Trails A test scores for the ADNI-1 mild AD and LMCI biomarker-positive population at baseline were 57 and 43 seconds, respectively.
Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; AD, Alzheimer’s disease; ADNI, Alzheimer’s Disease Neuroimaging Initiative; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; CSF, cerebrospinal fluid; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; SE, standard error; Trails A test, Trail-Making Test part A.

Table 2 Final CDR–SB disease progression model parameters

Parametera	Estimate	90% CIb
CDR–SB baseline score LMCI population (θRo)	1.53	1.40 to 1.66
Hippocampal volume (θRo_HVOL)	−0.916	−1.18 to −0.65
Mild AD (θRo_DIS)	1.22	1.02 to 1.42
Zero comedication (θRo_COMED_0)	−0.117	−0.22 to −0.02
Two comedications (θRo_COMED_2)	0.169	0.05 to 0.28
Progression rate parameter pathologic CSF population (θα)	0.306	0.23 to 0.39
Delayed logical memory (θα_DLM)	−0.057	−0.09 to −0.02
Trails A test (θα_TAT)	0.411	0.24 to 0.58
Shape parameter βc	1.22	1.02 to 1.42
IIV, progression rate parameter (%CV)d	62.2	39 to 86
IIV, baseline CDR–SB (%CV)d	39.0	35 to 43
Residual variability, SD (pathologic CSF population)	0.422	0.38 to 0.46
Residual variability, SD (nonpathologic CSF population)	0.551	0.48 to 0.62

Notes:

a Relationships between covariates and the TV of final parameter estimates: TV Ro =θRo · (HVOL/2954)^θRo_HVOL · (1+θRo_DIS)^DIS · (1+θRo_COMED_0)^C_0 (1+ θRo_COMED_2)^C_2 and TV α= θα · CSFFLAG · (TAT/44)^θα_TAT · exp(θα_DLM · (DLM-2)), where; DIS is a 0/1 flag for LMCI/mild AD respectively; C_0 is a 0/1 flag for subjects taking vs not taking comedications respectively; C_2 is a 0/1 flag for subjects taking less than two comedications – vs those taking two comedications respectively; and CSFFLAG is a 0/1 flag for subjects without/with pathologic CSF respectively (CSF p-tau_181P/Aβ_1–42 ratio>0.147 is considered pathologic).

b Confidence interval represents precision of parameter estimates.

c Estimated inflection based on the formula β 18/(1+β) is 10.

d Between the base model and final covariate model, the IIV estimates improved from 99.0% and 65.5% to 62.2% and 39.0%CV, respectively.

Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; AD, Alzheimer’s disease; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; CI, confidence interval; COMED, comedication use; CSF, cerebrospinal fluid; CV, percent coefficient of variation; DIS, disease state; DLM, delayed logical memory; HVOL, age and head size–adjusted hippocampal volume; IIV, interindividual variability; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; SD, standard deviation; TAT, Trails A test; TV, typical value; vs, versus.

Table 3 Parameter estimates from the dropout model

	Estimate	90% confidence intervala
Baseline hazard (α parameters)
Period 1 in years [0,0.5)b	−4.020	−4.58 to −3.46
Period 2 in years [0.5,1)	−3.602	−4.08 to −3.13
Period 3 in years [1,1.5)	−3.349	−3.77 to −2.93
Period 4 in years [1.5,2)	−3.036	−3.49 to −2.58
Period 5 in years [2,3)	-1.352	−1.63 to −1.08
Coefficients (β parameters)c
CDR–SB score prior to drop out	0.189*	0.15 to 0.23

Notes:

* P<0.0001.

a Confidence interval represents precision of parameter estimates.

b [lower boundary, upper boundary) indicates that the range includes the lower boundary but not the upper boundary.

c Hazard ratio can be obtained by exponentiating the parameter estimate, indicating there is approximately a 21% increase in the hazard of dropping out (ie, data being missing) with 1-point increase in CDR–SB.

Abbreviation: CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes.

Figure 3 Stratified visual predictive check.

Notes: The upper, middle, and lower profiles indicated by the open circles represent the 95th, 50th, and fifth percentiles of the observed data, respectively. The upper, middle, and lower curves indicated by the lines are the median model-based predictions for the 95th, 50th, and fifth percentiles, respectively, and these predictions account for missing data. The shaded areas are the 90% confidence intervals of the corresponding percentiles of the simulations based on the model.
Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; AD, Alzheimer’s disease; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes.

Figure S1 Overview of the model building process.

Note: δ is a small noise used for rescaling all the data to move the boundary observations slightly within the edges.

Abbreviation: VPC, visual predictive check.

Figure S2 (A) 1-year change scores vs disease severity on CDR–SB scale for mild AD and LMCI subjects. The line and gray areas represent the loess smoother and the 95% confidence bands. (B) The loess smoother and confidence bands are shown alone to allow illustration of the inverted U-shaped relationship between progression rate and disease severity.

Notes: The calculations are based on the 12-month interval method. The difference in scores between a pair of points separated by 12 months was computed (later score minus the earlier baseline score) and plotted against the baseline score for that pair. If subjects contributed more than one pair of scores separated by 12 months, those subjects generated more than one data point for the plot (the legend in Panel A indicates the year of the study during which the pair of scores were generated). Due to the statistical issues associated with subjects contributing different numbers of observations and the correlations among data points from one subject, these graphs are used only for diagnostic/exploratory purposes.

Abbreviations: AD, Alzheimer’s disease; LCMI, late mild cognitive impairment; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; vs, versus.

Figure S3 Results of the exploratory analysis for missing data: mean observed scores as a function of different study discontinuation times.

Abbreviations: AD, Alzheimer’s disease; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; LMCI, late mild cognitive impairment.

Figure S4 Goodness-of-fit plots.

Notes: The observed data versus the population and individual predictions are shown in the top panels. The bottom left panel shows the absolute values of the individually weighted residuals, and the bottom right panel shows the population conditional weighted residuals. Individual data points are indicated by blue circles; the red lines are loess smoothers, the black diagonal lines (top panels) represent the lines of identity, and the black horizontal line (bottom right) indicates the ordinate value of 0.

Abbreviation: iWRES, the individually weighted residuals.

Figure S5 VPC with and without dropout.

Notes: In the left panel, simulations were performed with the combined disease progression plus dropout model. In the right panel the simulations were performed without the dropout model. The black arrow in the right panel indicates that predictive performance is hampered at later times (where scores are high and probability of dropout higher) when dropouts are not accounted.

Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; VPC, visual predictive check.

Table S1 Boundary data handling: results of sensitivity analysis*

5	R0	α	β	RUV SD	R0 IIV%CV	α IIV%CV
Reference parameters
N/Aa	1.94 (4.1%)	0.259 (11.2%)	1.25 (6.4%)	0.385 (3.2%)	68.3% (7%)	74% (27.4%)
Comparable parameter estimatesb based on the sensitivity analysis
0.001	1.87 (4.4%)	0.336 (13.6%)	1 (10.9%)	0.672 (8%)	69.5% (10.3%)	115% (27.7%)
0.0018	1.89 (4.3%)	0.316 (13.9%)	1.05 (10.6%)	0.614 (7.4%)	69.1% (9.6%)	112% (29.3%)
0.003	1.91 (4.2%)	0.298 (14%)	1.09 (10.2%)	0.566 (6.7%)	68.5% (9.1%)	108% (31.1%)
0.005	1.94 (4.1%)	0.278 (14%)	1.14 (9.6%)	0.52 (6%)	67.6% (8.6%)	105% (32.2%)
0.0075	1.97 (4%)	0.262 (13.9%)	1.18 (9.2%)	0.485 (5.3%)	66.6% (8.3%)	102% (33.1%)
0.01	2.00 (4%)	0.25 (12.5%)	1.21 (8.4%)	0.46 (4.9%)	65.5% (8.1%)	99% (32.2%)
0.02	2.11 (3.8%)	0.219 (13.1%)	1.3 (7.5%)	0.404 (3.9%)	62% (7.6%)	93% (32.6%)
0.025	2.16 (3.6%)	0.207 (13.2%)	1.33 (7.3%)	0.386 (3.7%)	60.5% (7.5%)	91% (32.9%)
0.03	2.2 (3.5%)	0.197 (12.5%)	1.35 (6.7%)	0.372 (3.4%)	59% (7.5%)	89% (31.4%)
0.04	2.3 (3.4%)	0.18 (12.8%)	1.4 (6.5%)	0.348 (3.2%)	56.3% (7.4%)	87% (31.9%)
0.05	2.39 (3.2%)	0.165 (12.9%)	1.44 (6.2%)	0.33 (3.1%)	53.9% (7.3%)	85% (31.8%)
0.1	2.8 (2.7%)	0.112 (13.9%)	1.6 (5.5%)	0.27 (3%)	44.5% (7.3%)	80% (32.3%)
0.33	4.5 (1.3%)	0.019 (27.4%)	2.26 (6.5%)	0.152 (3.5%)	22.4% (8.6%)	79% (57.8%)

Notes:

a δ was not used, ie, Model estimation without rescaling data and excluding data on boundaries;

b sets of parameter estimates (parameter imprecision, expressed as percent coefficient of variation) are reported for each tested value of δ. δ is a small noise used for rescaling all the data to move the boundary observations slightly within the edges.

* Parameter estimates (with parameter precision) reported for each value of δ that was tested.

Abbreviations: α, progression rate parameter; β, shape parameter in the logistic structural model; CV, coefficient of variation; IIV, interindividual variability; N/A, not applicable; R₀, baseline CDR–SB score; RUV SD, standard deviation of the residual unexplained variability.

Table S2 Parameters of the mixture models fitted to the baseline CSF biomarker data and hippocampal volume in the ADNI-1 CN, LMCI, and mild AD subjects

Baseline biomarker	Mean of subpopulation with pathologic biomarker*	SD of subpopulation with pathologic biomarker*	Mean of subpopulation without pathologic biomarker*	SD of subpopulation without pathologic biomarker*	Threshold between the two subpopulations	Mixing proportion: pathologic* group in CN	Mixing proportion: pathologic* group in LMCI	Mixing proportion: pathologic* group in mild AD	AIC for CDR–SB disease progression model***
log t-tau	4.79	0.43	4.15	0.37	4.5‡	4%	55%	83%	−3,879.0
log p-tau181P	3.69	0.39	2.89	0.29	3.2§	25%	69%	91%	−3,879.7
Hippocampal volume (mm^3)	2,810	397	3,570	369	3,233	0%	64%	97%	−3,902.1
log t-tau/Aβ1–42 ratio	−0.14	0.47	−1.29	0.39	−0.78**	15%	66%	89%	−3,910.8
Aβ1–42 (pg/mL)	136	27	241	30	192	34%	74%	92%	−3,919.6
log p-tau181P/Aβ1–42 ratio	−1.19	0.47	−2.46	0.39	−1.92†	20%	68%	91%	−3,921.1

Notes: The LMCI and AD populations were dichotomized based on the thresholds reported below, and AIC values are reported in the last column for the CDR–SB disease progression model, where the biomarker was introduced as a dichotomized covariate.

* Pathologic biomarker (low CSF Aβ_1–42 low hippocampal volume, high p-tau_181P, high p-tau_181P/Aβ_1–42, high t-tau, or high t-tau/Aβ_1–42).

‡ This corresponds to a cutoff value of 90 pg/mL on the untransformed scale for baseline CSF t-tau.

§ This corresponds to a cutoff value of 24.5 pg/mL on the untransformed scale for baseline CSF p-tau_181P.

** This corresponds to a cutoff value of 0.458 on the untransformed scale for baseline CSF t-tau/Aβ_1–42 ratio.

† This corresponds to a cutoff value of 0.147 on the untransformed scale for baseline CSF p-tau_181P/Aβ_1–42 ratio.

*** As a comparison, the base model for CDR–SB disease progression, which included data at the boundaries, had an AIC value of −3,819.1.

Abbreviations: Aβ_1–42, 42 amino acid isoform of amyloid beta peptide; AD, Alzheimer’s disease; ADNI, Alzheimer’s Disease Neuroimaging Initiative; AIC, Akaike information criterion; CDR–SB, Clinical Dementia rating Scale–Sum of Boxes; CN, cognitively normal; LMCI, late mild cognitive impairment; p-tau_181P, tau protein phosphorylated at position threonine 181; SD, standard deviation; t-tau, total tau protein; CSF, cerebrospinal fluid.

Table S3 Mean baseline CDR–SB for the subgroups, based on the statistically significant covariates for the baseline parameter in the disease progression model

Disease status	Stratification variable	N	Mean baseline CDR–SB ± SD
Disease group
LMCI	N/A	199	1.55±0.9
Mild AD	N/A	102	4.25±1.6
Comedication status
LMCI	0 comedication	90	1.39±0.7
LMCI	1 comedication	86	1.64±0.9
LMCI	2 comedications	23	1.85±1.1
Mild AD	0 comedication	7	3.50±1.1
Mild AD	1 comedication	54	4.08±1.6
Mild AD	2 comedications	41	4.61±1.6
Hippocampal volume
LMCI	Normal hippocampi	66	1.28±0.6
LMCI	Shrunken hippocampi	133	1.69±1.0
Mild AD	Normal hippocampi	20	3.15±1.1
Mild AD	Shrunken hippocampi	82	4.52±1.6

Notes: Normal hippocampi (hippocampal volume >3,233 mm³); Shrunken hippocampi (hippocampal volume ≤3,233 mm³). The hippocampal volume threshold of 3,233 mm³ was based on the biomarker mixture model (Table S2). The threshold of 3,233 mm³ is also very close to the threshold of 3,226 mm³, which is 1 SD below the mean hippocampal volume for the elderly normal controls.

Abbreviations: AD, Alzheimer’s disease; CDR–SB, Clinical Dementia Rating Scale–Sum of Boxes; LMCI, late mild cognitive impairment; N/A, not applicable; SD, standard deviation.