Background
Methods
Joint latent class model
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The multinomial logistic regression is defined by πig, the probability of individual i to belong to a given latent class g, conditional on a covariate vector Xi:$${} \pi_{ig}=P({ c_{i}}=g|\pmb{X}_{i})=\frac{\mathrm{e}^{\xi_{0g}+{\pmb{X}_{i}^{T}}\pmb{\xi}_{1g}}}{\sum_{l=1}^{{G}}\mathrm{e}^{\xi_{0l}+{\pmb{X}_{i}^{T}}\pmb{\xi}_{1l}}}, $$(1)where ci is the latent class for patient i, \(c_{i} \in (1, \cdots, G), \pmb {X}_{i}^{T}\) is a vector of explanatory variables for i necessarily independent of time, ξ1g the vector of coefficients associated to the covariates effects within class g. Note that ξ0G=0 and ξ1G=0 to assure the model identifiability. If no prior information about the latent class is available, it is possible to use the marginal probability of the class g, \( \frac {e^{\pmb {\xi }_{0g}}}{\sum _{l=1}^{G}e^{\pmb {\xi }_{0l}}}\) in Eq. (1).
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The mixed linear model for a trajectory of a longitudinal marker of an individual i over time points tij,Yij in a latent class g is defined as:$${}Y_{ij}|(c_{i}=g)={\pmb{X}_{1ij}^{T}}\pmb{\gamma}+{\pmb{X}_{2ij}^{T}}\pmb{\beta}_{g}+{\pmb{Z}_{ij}^{T}}\pmb{b}_{ig}+\epsilon_{ij}, $$(2)where \(\pmb {X}_{1ij}^{T}\) is the vector of explanatory variables common to all latent classes and γ the corresponding vector of coefficients, \(\pmb {X}_{2ij}^{T}\) is the vector of class-specific explanatory variables with βg the corresponding vector of coefficients, and Zij is the vector of explanatory variables associated with the random effects \(\pmb {b}_{ig} \sim \mathcal {N} (\pmb {\mu }_{g}, \pmb {B}_{g})\) (μg is a mean of random effects, Bg is a variance-covariance matrix of random effects, both of which can be common or specific to latent classes). Note that \(\pmb {X}_{1ij}^{T}\) and \(\pmb {X}_{2ij}^{T}\) have no variables in common.
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The survival model for an individual i over time is defined by its hazard function, αi(t), within each latent class as:$$ \alpha_{i} (t)|\left(c_{i}=g\right) = {\alpha_{0}}\left(t,\pmb{\zeta}_{g}\right)\exp\left(\pmb{X}_{1i}^{T}\pmb{\vartheta}+{\pmb{X}_{2i}^{T}}\pmb{\eta}_{g}\right) $$(3)with α0(·) the baseline risk function in latent class g, parametrized by vector \(\pmb {\zeta }_{g}, \pmb {X}_{1i}^{T}\) is the vector of explanatory variables and 𝜗 the associated parameters common to all latent classes, \(\pmb {X}_{2i}^{T}\) is the vector of class-specific explanatory variables and ηg the corresponding class-specific parameters of the model.We denote by Ti the observed time to a clinical event of interest for individual i. In the framework of JLCM, it is important to note that the measures of the longitudinal marker after Ti, if there exist, are excluded from the observed data. Indeed, the objective is to describe the link between the risk of the event and the marker change over time preceding the event. The observed duration \(T_{i}= \min (T^{\star }_{i},C_{i})\), where \(T^{\star }_{i}\) corresponds to the real time-to-event (possibly not observed) and Ci corresponds to the right-censored duration. The survival function corresponding to the hazard of Eq. (3), is defined as:$${} S(t)=\exp \left(-\int_{0}^{t} \alpha(u) du\right) $$(4)
Likelihood
Class prediction and goodness-of-fit
Results
Simulation study
Simulations design
Data generation
Normality assessment
Relative bias assessment
Longitudinal sub-model | Survival sub-model | ||||||||||
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n | τ | \(\hat {\sigma }_{b} \) | \(\hat {\sigma }_{\epsilon } \) | \(\hat {\beta }_{0g}\) | \(\hat {\beta }_{1g}\) | \(\hat {\zeta }_{1g}\) | \(\hat {\zeta }_{2g}\) | ||||
g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | ||||
100 | 5 | 0.1053 | 21.1797 | 0.0429 | 1.3974 | 0.0281 | 10.7052 | 0.3018 | 3.2362 | 2.5023 | 9.8304 |
10 | 0.0961 | 21.2395 | 0.1421 | 1.6954 | 0.0748 | 38.1642 | 0.2923 | 3.1830 | 2.4243 | 10.0367 | |
15 | 0.0997 | 21.3986 | 0.2846 | 1.9987 | 0.1936 | 59.0676 | 0.4148 | 3.6286 | 2.3160 | 10.5283 | |
25 | 0.0824 | 21.3810 | 0.4687 | 3.0567 | 0.4054 | 82.7586 | 0.4455 | 3.6646 | 2.2878 | 11.9084 | |
50 | 1.7125 | 21.8392 | 1.3486 | 5.3694 | 0.8271 | 120.1328 | 1.1201 | 7.5050 | 3.5401 | 14.0533 | |
500 | 5 | 0.2267 | 21.4488 | 0.1534 | 0.1412 | 0.0685 | 25.4826 | 0.0591 | 0.5932 | 0.0332 | 0.2021 |
10 | 0.2230 | 21.3820 | 0.2346 | 0.2261 | 0.1317 | 35.8174 | 0.1892 | 0.4996 | 0.1561 | 0.0727 | |
15 | 0.2062 | 21.4656 | 0.3380 | 0.4738 | 0.2027 | 48.1685 | 0.3062 | 0.0501 | 0.1232 | 0.0229 | |
25 | 0.2011 | 21.4412 | 0.5670 | 1.3390 | 0.4127 | 74.1988 | 0.2546 | 0.6707 | 0.4496 | 0.1703 | |
50 | 0.1945 | 21.6664 | 1.3876 | 3.7674 | 0.9627 | 116.6916 | 0.7377 | 1.6005 | 0.6217 | 1.4936 | |
1000 | 5 | 0.0004 | 20.1935 | 0.3260 | 0.7563 | 0.2925 | 8.4122 | 0.1684 | 0.1125 | 0.4160 | 0.2180 |
10 | 0.0015 | 20.2002 | 0.4197 | 1.0629 | 0.3599 | 19.7922 | 0.3399 | 0.0777 | 0.4994 | 0.0130 | |
15 | 0.0117 | 20.1928 | 0.5181 | 1.3744 | 0.4324 | 31.9018 | 0.3658 | 0.2015 | 0.3471 | 0.1781 | |
25 | 1.6848 | 20.2110 | 0.7436 | 2.1019 | 0.6690 | 60.0292 | 0.4954 | 0.6364 | 0.1559 | 0.4325 | |
50 | 0.0072 | 20.3934 | 1.5776 | 4.6004 | 1.0896 | 102.7405 | 0.8822 | 1.3124 | 0.5011 | 1.9393 | |
5000 | 5 | 1.6342 | 19.9957 | 0.0242 | 0.5270 | 0.2185 | 16.5831 | 0.1929 | 0.6064 | 0.0605 | 0.2997 |
10 | 1.6315 | 19.9945 | 0.1226 | 0.8526 | 0.3075 | 31.0770 | 0.2985 | 0.7797 | 0.0810 | 0.3320 | |
15 | 0.0380 | 19.9728 | 0.2215 | 1.2091 | 0.3905 | 44.6046 | 0.4162 | 0.8876 | 0.0823 | 0.5370 | |
25 | 0.0449 | 20.0112 | 0.4584 | 1.9183 | 0.5515 | 72.0024 | 0.6396 | 1.2441 | 0.0648 | 0.8037 | |
50 | 0.0400 | 20.2601 | 1.2738 | 4.3253 | 1.0406 | 113.4169 | 1.0629 | 1.8816 | 0.5238 | 1.7593 |
Longitudinal sub-model | Survival sub-model | ||||||||||
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n | τ | \(\hat {\sigma }_{b} \) | \(\hat {\sigma }_{\epsilon } \) | \(\hat {\beta }_{0g}\) | \(\hat {\beta }_{1g}\) | \(\hat {\zeta }_{1g}\) | \(\hat {\zeta }_{2g}\) | ||||
g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | ||||
100 | 5 | 0.2157 | 22.3405 | 0.9001 | 1.1061 | 0.6187 | 77.4912 | 0.1821 | 5.5733 | 2.6803 | 14.4064 |
10 | 3.3939 | 22.1855 | 1.1485 | 0.5399 | 0.2169 | 129.4999 | 0.7945 | 5.9039 | 2.4429 | 14.9468 | |
15 | 0.2315 | 22.2708 | 1.3985 | 0.0946 | 0.4619 | 175.0008 | 1.4886 | 6.5439 | 2.4417 | 16.5623 | |
25 | 1.5321 | 22.1504 | 1.8233 | 1.7052 | 1.6413 | 273.7677 | 3.1293 | 6.4496 | 3.1511 | 20.3027 | |
50 | 1.3699 | 21.7546 | 3.6344 | 5.3290 | 4.6976 | 532.4839 | 8.3763 | 9.3616 | 7.9156 | 30.6630 | |
500 | 5 | 0.2556 | 21.5565 | 0.2726 | 0.0707 | 0.6062 | 36.0688 | 0.4649 | 0.1000 | 0.0440 | 0.5662 |
10 | 1.9185 | 21.3569 | 0.4126 | 0.8612 | 1.1048 | 79.5267 | 1.1380 | 0.1392 | 0.0640 | 0.9334 | |
15 | 3.4729 | 21.2975 | 0.6091 | 1.3911 | 1.5773 | 120.2866 | 1.6905 | 0.6533 | 0.1375 | 1.3348 | |
25 | 0.1728 | 20.9193 | 0.9847 | 3.0168 | 2.8914 | 212.8892 | 3.0037 | 1.7108 | 0.1707 | 2.2996 | |
50 | 1.7796 | 19.8756 | 2.3534 | 7.4185 | 7.8224 | 444.8984 | 8.3268 | 2.8825 | 0.9417 | 4.7449 | |
1000 | 5 | 0.0184 | 20.1206 | 0.5111 | 0.9550 | 0.8944 | 44.8903 | 0.6416 | 0.3375 | 0.6727 | 0.2864 |
10 | 0.0302 | 20.0392 | 0.6927 | 1.5469 | 1.3819 | 87.2876 | 1.2630 | 0.2644 | 0.8368 | 0.0338 | |
15 | 0.0424 | 19.8918 | 0.8955 | 2.1650 | 1.8632 | 139.1406 | 1.8765 | 0.0905 | 0.9058 | 0.2346 | |
25 | 0.0763 | 19.6487 | 1.3465 | 3.5027 | 3.2279 | 240.5047 | 3.3728 | 0.7920 | 1.0379 | 0.6998 | |
50 | 0.2146 | 18.4371 | 2.7442 | 8.0212 | 7.5645 | 482.3023 | 8.8312 | 2.2757 | 1.4414 | 3.6641 | |
5000 | 5 | 1.6360 | 19.8545 | 0.0630 | 0.8358 | 0.6849 | 48.0499 | 0.6231 | 0.9167 | 0.0643 | 0.5431 |
10 | 0.0529 | 19.7483 | 0.2447 | 1.4861 | 1.2312 | 95.3766 | 1.2469 | 1.1418 | 0.1531 | 0.6085 | |
15 | 1.5950 | 19.6407 | 0.4366 | 2.1381 | 1.7922 | 142.7663 | 1.9299 | 1.3603 | 0.2632 | 0.9193 | |
25 | 0.1003 | 19.3803 | 0.8741 | 3.5248 | 3.0292 | 240.2600 | 3.4624 | 1.9415 | 0.5656 | 1.3986 | |
50 | 0.2294 | 18.2465 | 2.2807 | 7.9142 | 7.5368 | 487.8892 | 8.8603 | 3.0965 | 0.9679 | 3.1725 |
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As for the absolute values, in the high separation setting (Fig. 6 and Table 1), the RB is the most important for two parameters of class 2: 1) the survival sub-model Weibull shape parameter (RB over 10% for small number of individuals) and 2) the mixed sub-model slope parameter (RB varies from 10% to 120% depending on number of individuals and on the censoring rate, the mean number of longitudinal markers in the worse case (100 patients and a censor of 50%) is 5.1). For the remaining parameters the RB does not exceed 10%. The trend is quite similar for the low separation setting (Fig. 6 and Table 2), but to a higher extent: the RB varies from over 30% to 530% in the worst setting (small n and high τ).
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As for the impact of the censoring rate, the RB increases linearly for a given number of individuals according to the decreasing number of events (increasing censoring rate). This trend is the same for both settings in terms of degree of class separation, but, in the same manner that the RB absolute values, in a higher extent for the low separation setting. Precisely, in the high separation setting the RB decreases by around 1% for the parameters of class 1 (2-8% in the low separation case) and for around 3-5% (2-15% in the low separation case) for the parameters of class 2, for the exception of the mixed model slope: 100% decrease in the RB in the high separation (respectively 400% in the low separation setting) for τ decreasing from 50% to 5%). Note that the linear trend for RB evolution in terms of τ is not always respected for small n.
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As for the impact of the number of individuals, the increasing n does not seem to strongly impact the RB. Moreover, the Weibull shape parameter is more influenced than the Weibull scale. Also, the low separation setting is more influenced than the high separation setting.
Coverage rate assessment
n | τ | \(\hat {\beta }_{0g}\) | \(\hat {\beta }_{1g}\) | \(\hat {\zeta }_{1g}\) | \(\hat {\zeta }_{2g}\) | ||||
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g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | ||
100 | 5 | 0.9664 | 0.9496 | 0.9328 | 0.9496 | 0.8824 | 0.8655 | 0.9580 | 0.9160 |
10 | 0.9664 | 0.9496 | 0.9328 | 0.9412 | 0.8571 | 0.8655 | 0.9496 | 0.9076 | |
15 | 0.9664 | 0.9496 | 0.9412 | 0.9496 | 0.8824 | 0.8908 | 0.9412 | 0.9328 | |
25 | 0.9580 | 0.9496 | 0.9412 | 0.9580 | 0.8487 | 0.7899 | 0.9496 | 0.9076 | |
50 | 0.9328 | 0.9076 | 0.9748 | 0.9328 | 0.8403 | 0.7899 | 0.8824 | 0.8571 | |
500 | 5 | 0.9667 | 0.9667 | 0.9833 | 0.9500 | 0.9250 | 0.9167 | 0.9333 | 0.9500 |
10 | 0.9667 | 0.9750 | 0.9833 | 0.9417 | 0.9583 | 0.9250 | 0.9250 | 0.9500 | |
15 | 0.9667 | 0.9667 | 0.9750 | 0.9750 | 0.9250 | 0.9083 | 0.9417 | 0.9333 | |
25 | 0.9583 | 0.9500 | 0.9750 | 0.9417 | 0.8667 | 0.7750 | 0.9083 | 0.9167 | |
50 | 0.8750 | 0.9167 | 0.9333 | 0.9083 | 0.9000 | 0.8333 | 0.9333 | 0.9250 | |
1000 | 5 | 0.9833 | 0.9333 | 0.9500 | 0.9250 | 0.8667 | 0.8750 | 0.9500 | 0.9417 |
10 | 0.9750 | 0.9167 | 0.9500 | 0.9333 | 0.8833 | 0.9417 | 0.9583 | 0.9417 | |
15 | 0.9750 | 0.9167 | 0.9333 | 0.9333 | 0.8833 | 0.8917 | 0.9833 | 0.9583 | |
25 | 0.9500 | 0.9000 | 0.9167 | 0.9083 | 0.8917 | 0.8917 | 0.9417 | 0.9000 | |
50 | 0.8667 | 0.8000 | 0.8917 | 0.9083 | 0.9000 | 0.7000 | 0.9083 | 0.9167 | |
5000 | 5 | 0.9750 | 0.9083 | 0.9333 | 0.8750 | 0.8917 | 0.9000 | 0.9667 | 0.9500 |
10 | 0.9833 | 0.9083 | 0.9417 | 0.8583 | 0.9000 | 0.9250 | 0.9500 | 0.9417 | |
15 | 0.9667 | 0.8667 | 0.9083 | 0.8333 | 0.8250 | 0.8750 | 0.9417 | 0.9333 | |
25 | 0.9333 | 0.7917 | 0.9000 | 0.8250 | 0.8167 | 0.8667 | 0.8917 | 0.9333 | |
50 | 0.5000 | 0.2833 | 0.8000 | 0.7167 | 0.7750 | 0.7750 | 0.8500 | 0.9000 |
n | τ | \(\hat {\beta }_{0g}\) | \(\hat {\beta }_{1g}\) | \(\hat {\zeta }_{1g}\) | \(\hat {\zeta }_{2g}\) | ||||
---|---|---|---|---|---|---|---|---|---|
g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | g=1 | g=2 | ||
100 | 5 | 0.9664 | 0.9496 | 0.9328 | 0.9496 | 0.8824 | 0.8655 | 0.9580 | 0.9160 |
10 | 0.9664 | 0.9496 | 0.9328 | 0.9412 | 0.8571 | 0.8655 | 0.9496 | 0.9076 | |
15 | 0.9664 | 0.9496 | 0.9412 | 0.9496 | 0.8824 | 0.8908 | 0.9412 | 0.9328 | |
25 | 0.9580 | 0.9496 | 0.9412 | 0.9580 | 0.8487 | 0.7899 | 0.9496 | 0.9076 | |
50 | 0.9328 | 0.9076 | 0.9748 | 0.9328 | 0.8403 | 0.7899 | 0.8824 | 0.8571 | |
500 | 5 | 0.9667 | 0.9667 | 0.9833 | 0.9500 | 0.9250 | 0.9167 | 0.9333 | 0.9500 |
10 | 0.9667 | 0.9750 | 0.9833 | 0.9417 | 0.9583 | 0.9250 | 0.9250 | 0.9500 | |
15 | 0.9667 | 0.9667 | 0.9750 | 0.9750 | 0.9250 | 0.9083 | 0.9417 | 0.9333 | |
25 | 0.9583 | 0.9500 | 0.9750 | 0.9417 | 0.8667 | 0.7750 | 0.9083 | 0.9167 | |
50 | 0.8750 | 0.9167 | 0.9333 | 0.9083 | 0.9000 | 0.8333 | 0.9333 | 0.9250 | |
1000 | 5 | 0.9833 | 0.9333 | 0.9500 | 0.9250 | 0.8667 | 0.8750 | 0.9500 | 0.9417 |
10 | 0.9750 | 0.9167 | 0.9500 | 0.9333 | 0.8833 | 0.9417 | 0.9583 | 0.9417 | |
15 | 0.9750 | 0.9167 | 0.9333 | 0.9333 | 0.8833 | 0.8917 | 0.9833 | 0.9583 | |
25 | 0.9500 | 0.9000 | 0.9167 | 0.9083 | 0.8917 | 0.8917 | 0.9417 | 0.9000 | |
50 | 0.8667 | 0.8000 | 0.8917 | 0.9083 | 0.9000 | 0.7000 | 0.9083 | 0.9167 | |
5000 | 5 | 0.9750 | 0.9083 | 0.9333 | 0.8750 | 0.8917 | 0.9000 | 0.9667 | 0.9500 |
10 | 0.9833 | 0.9083 | 0.9417 | 0.8583 | 0.9000 | 0.9250 | 0.9500 | 0.9417 | |
15 | 0.9667 | 0.8667 | 0.9083 | 0.8333 | 0.8250 | 0.8750 | 0.9417 | 0.9333 | |
25 | 0.9333 | 0.7917 | 0.9000 | 0.8250 | 0.8167 | 0.8667 | 0.8917 | 0.9333 | |
50 | 0.5000 | 0.2833 | 0.8000 | 0.7167 | 0.7750 | 0.7750 | 0.8500 | 0.9000 |
Class membership prediction assessment
n | τ | High separation | Low separation | Difference |
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100 | 5 | 0.9760 | 0.9418 | -0.0342 |
10 | 0.9767 | 0.9347 | -0.0420 | |
15 | 0.9748 | 0.9248 | -0.0500 | |
25 | 0.9689 | 0.9039 | -0.0650 | |
50 | 0.9556 | 0.8335 | -0.1221 | |
500 | 5 | 0.9790 | 0.9440 | -0.0350 |
10 | 0.9778 | 0.9376 | -0.0402 | |
15 | 0.9764 | 0.9321 | -0.0443 | |
25 | 0.9720 | 0.9148 | -0.0572 | |
50 | 0.9586 | 0.8458 | -0.1128 | |
1000 | 5 | 0.9814 | 0.9477 | -0.0337 |
10 | 0.9798 | 0.9419 | -0.0379 | |
15 | 0.9782 | 0.9354 | -0.0428 | |
25 | 0.9745 | 0.9186 | -0.0559 | |
50 | 0.9605 | 0.8488 | -0.1017 | |
5000 | 5 | 0.9817 | 0.9480 | -0.0337 |
10 | 0.9801 | 0.9417 | -0.0384 | |
15 | 0.9784 | 0.9348 | -0.0436 | |
25 | 0.9748 | 0.9189 | -0.0559 | |
50 | 0.9618 | 0.8504 | -0.1114 |
Real data application
Data collection
Model construction
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A model without covariates (Eq. 9) includes a random-intercept mixed model with a class-specific quadratic function of time specified for the longitudinal marker evolution Yij; the variances of the random effect (\(\sigma ^{2}_{b}\)) and of the error (\(\sigma ^{2}_{\epsilon }\)) were considered common to all classes. Survival curves are also considered as class-specific.The originally interval-censored survival times, collected at baseline and at months 1, 3, 6, 9, 12, 15 and 18, were imputed from a Weibull distribution of these interval-censored dates to obtain the exact event times. The imputation was carried out in order to obtain the setting close to that used in simulations. Specifically, a Weibull distribution was first fitted to the interval-censored dates, and then the exact event times were sampled from this distribution truncated by the limits of the observed intervals for each patient.$$ \left\{\begin{array}{ll} \pi_{ig}=\frac{\mathrm{e}^{\xi_{0g}}}{\sum_{l=1}^{{G}}\mathrm{e}^{\xi_{0l}}} \text{, from Eq.~(1)}\\ \\ Y_{ij}|(c_{i}=g)= \beta_{0g} + \beta_{1g} t_{ij} + \beta_{2g} t_{ij}^{2}+ b_{0i}+b_{1i} t_{ij}\\+b_{2i} t_{ij}^{2}+\epsilon_{i},& \\ \quad \quad \pmb{b}_{i} \sim \mathcal{N}\left(0, \pmb{B}\right), \epsilon_{ij} \sim \mathcal{N}\left(0, \sigma^{2}_{\epsilon}\right) \text{, from Eq.~(2)} \\ \\ S(t_{i})|(c_{i}=g) = \exp\left(-\left(\frac{t_{i}}{\zeta_{1g}}\right)^{\zeta_{2g}}\right),& \\ \quad \quad T^{\star}\sim \mathcal{W}eibull \left(\zeta_{1g}, \zeta_{2g}\right), \text{ from Eq.~(4)} \end{array} \right. $$(9)with B covariance matrix of random effects.
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A model with covariates (Eq. 10, the hazard function is specified for easier interpretation) was specified based on clinical expertise and a preliminary unpublished study. This model includes baseline individual characteristics in the random intercept mixed sub-model and in the survival sub-model; the impact of these characteristics is specified common to all classes, following the clinical considerations. The quadratic term of time for the mixed sub-model appeared to be not significantly different from 0 for this model and is thus removed. Baseline covariates and their interactions with time were as well chosen from clinical expertise.The following abbreviations are used: AO (Age at onset), SO (Symptom Onset), BMI (Body Mass Index), MUSC (Muscular capacity), SVC (Slow vital capacity), MCV (Mean corposcular volume).$$ {}\left\{ \begin{array}{lll} \pi_{ig}&=\frac{\mathrm{e}^{\xi_{0g}}}{\sum_{l=1}^{{G}}\mathrm{e}^{\xi_{0l}}} \text{, from Eq.~(1)}\\ \\ Y_{ij}|(c_{i}=g)&= \beta_{0g} + \beta_{1g}t_{ij} + \gamma_{1}SO_{i}+ \gamma_{2}BMI_{i} + &\\& \gamma_{3}MUSC_{i}+ \gamma_{4}SVC_{i}+ \gamma_{5}MCV_{i} + &\\& t_{ij} \times \left(\gamma_{6}SO_{i}+ \gamma_{7}MUSC_{i} + \gamma_{8}SVC_{i} \right)+ &\\& b_{0i}+b_{1i} t_{ij}+\epsilon_{ij}, &\\& b_{i} \sim \mathcal{N}\left(0, \sigma^{2}_{b}\right), &\\& \epsilon_{ij} \sim \mathcal{N}\left(0, \sigma^{2}_{\epsilon}\right) \text{from Eq.~(2)} \\ \\ \alpha_{i}(t)|(c_{i}=g) &=\underbrace{\zeta_{1g}^{\zeta_{2g}}\zeta_{2g} t^{\zeta_{2g} - 1}}_{\alpha_{0} (t)} \exp(\vartheta_{1}{SO}_{i} +\vartheta_{2}{BMI}_{i} + &\\& \vartheta_{3}{MUSC}_{i} + \vartheta_{4}SVC_{i} + &\\&\vartheta_{5}AO_{i}) \text{, from Eq.~(3)} \\ \end{array} \right. $$(10)
Real data analysis results
number of observations | 2591 | ||
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number of patients | 511 | ||
average number of longitudinal measure | 5 | ||
number of events | 132 | ||
censoring rate | 0.74 | ||
Sub-model | Parameter | Estimate (se) | p-value |
Multinomial logistic regression | ξ01 | -0.29 (0.49) | 0.55 |
ξ02 | 2.26 (0.44) | <0.001 | |
ξ03 | 1.21 (0.53) | 0.022 | |
Weibull survival model | ζ11 | 0.37 (0.02) | <.001 |
ζ21 | 1.52 (0.13) | <0.001 | |
ζ12 | 0.12 (0.02) | <0.001 | |
ζ22 | 1.25 (0.14) | <0.001 | |
ζ13 | 0.24 (0.01) | <0.001 | |
ζ23 | 1.56 (0.12) | <0.001 | |
ζ14 | 0.26 (0.01) | <0.001 | |
ζ24 | 2.02 (0.31) | <0.001 | |
Linear mixed model : fixed effects | β01 | 35.22 (1.11) | <0.001 |
β11 | -5.29 (0.32) | <0.001 | |
β21 | 0.31 (0.04) | <0.001 | |
β02 | 39.37 (0.34) | <0.001 | |
β12 | -0.52 (0.06) | <0.001 | |
β22 | -0.01 (0.00) | 0.007 | |
β03 | 37.44 (0.66) | <0.001 | |
β13 | -1.92 (0.16) | <0.001 | |
β23 | 0.04 (0.01) | <0.001 | |
β04 | 39.00 (1.30) | <0.001 | |
β14 | -0.36 (0.20) | <0.001 | |
β24 | -0.10 (0.02) | <0.001 | |
Linear mixed model : random effects | \(\sigma ^{2}_{b_{0}}\) | 22.93 | |
\(\sigma ^{2}_{b_{1}}\) | 0.20 | ||
\(\sigma ^{2}_{b_{2}}\) | 0.00 | ||
\(\sigma ^{2}_{\epsilon,1}\) | 1.67 |
number of observations | 2525 | ||
---|---|---|---|
number of patients | 497 | ||
average number of longitudinal measure | 5 | ||
number of events | 129 | ||
censoring rate | 0.74 | ||
Sub-model | Parameter | Estimate (se) | p-value |
Multinomial logistic regression | ξ01 | 2.22 (0.31) | <0.001 |
Weibull model | ζ11 | 0.48 (0.17) | 0.004 |
ζ21 | 1.48 (0.08) | <0.001 | |
ζ12 | 0.68 (0.21) | 0.001 | |
ζ22 | 1.64 (0.12) | <0.001 | |
𝜗1 | -0.05 (0.01) | 0.008 | |
𝜗2 | -0.05 (0.03) | 0.079 | |
𝜗3 | -0.03 (0.01) | <0.001 | |
𝜗4 | -0.41 (0.12) | <0.001 | |
𝜗5 | 0.04 (0.01) | <0.001 | |
Linear mixed model : fixed effects | \(\hat {\beta }_{01}\) | 9.79 (4.02) | 0.015 |
β11 | -2.32 (0.27) | <0.001 | |
β02 | 7.83 (4.12) | 0.057 | |
β12 | -4.06 (0.39) | <0.001 | |
γ1 (SO) | -0.06 (0.02) | <0.001 | |
γ2 (BMI) | -0.13 (0.05) | 0.009 | |
γ3 (MUSC) | 0.16 (0.00) | <0.001 | |
γ4 (SVC) | 1.04 (0.18) | <0.001 | |
γ5 (MCV) | 0.10 (0.04) | 0.007 1 | |
γ6 (SO ×tj) | 0.02 (0.00) | <0.001 | |
γ7 (MUSC ×tj) | 0.01 (0.00) | <0.001 | |
γ8 (SVC ×tj) | 0.06 (0.02) | 0.018 | |
Linear mixed model : random effects | \(\sigma ^{2}_{b_{0}}\) | 14.10 (0.00) | |
\(\sigma ^{2}_{b1}\) | 0.18 | ||
\(\sigma ^{2}_{\epsilon,1}\) | 1.97 |
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Classes 1 and 4 from the model without covariates are each composed of 5.1% of population. They represent patients with the most rapid decrease of ALSFRS and the highest risk of death, with a median survival around 7 months and 14 months for class 1 and 4 respectively.
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Class 2 is the largest (68.5% of patients) and is characterized by the slowest evolution of ALSFRS and the highest survival rate (median survival over 20 months).
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Class 3 is composed of 21.3% of population and represents an “average” class with an ALSFRS progression similar to that in class 1 but with a lower baseline value: from Table 6 we observe the baseline value of 37 in class 3 vs 39 for class 2. The survival probability in class 3 is lower than that in class 2, with a median survival around 15 months.
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Class 1 is the largest (92.6% of patients), is characterized by a moderate ALSFRS progression (-2.3 point by months) and by a better survival prognosis (over 20 months median survival compared to around 8 months for class 2, for a patient with the average covariates vector).
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Class 2 is composed only of 37 patients (7.4%) and describes a specific patient profile, worsening and dying very quickly.