Division of Biostatistics, Washington University School of Medicine, St. Louis, MO, 63110, USA
Department of Ophthalmology & Visual Sciences, Washington University School of Medicine, St. Louis, MO, 63110, USA
Department of Ophthalmology, The Policlinico di Monza, University Bicocca of Milan, Milan, Italy
Laboratory of New Drugs Development Strategies, Mario Negri Institute, Milan, Italy
Abstract
Background
Primary openangle glaucoma (POAG) is one of the leading causes of blindness in the United States and worldwide. While lowering intraocular pressure (IOP) has been proven to be effective in delaying or preventing the onset of POAG in many largescale prospective studies, one of the recent hot topics in glaucoma research is the effect of IOP fluctuation (IOP lability) on the risk of developing POAG in treated and untreated subjects.
Method
In this paper, we analyzed data from the Ocular Hypertension Treatment Study (OHTS) and the European Glaucoma Prevention Study (EGPS) for subjects who had at least 2 IOP measurements after randomization prior to POAG diagnosis. We assessed the interrelationships among the baseline covariates, the changes of postrandomization IOP over time, and the risk of developing POAG, using a latent class analysis (LCA) which allows us to identify distinct patterns (latent classes) of IOP trajectories.
Result
The IOP change in OHTS was best described by 6 latent classes differentiated primarily by the mean IOP levels during followup. Subjects with high postrandomization mean IOP level and/or large variability were more likely to develop POAG. Five baseline factors were found to be significantly predictive of the IOP classification in OHTS: treatment assignment, baseline IOP, gender, race, and history of hypertension. In separate analyses of EGPS, LCA identified different patterns of IOP change from those in OHTS, but confirmed that subjects with high mean level and large variability were at high risk to develop POAG.
Conclusion
LCA provides a useful tool to assess the impact of postrandomization IOP level and fluctuation on the risk of developing POAG in patients with ocular hypertension. The incorporation of postrandomization IOP can improve the overall predictive ability of the original model that included only baseline risk factors.
Background
Ocular hypertension is a leading risk factor for the development of primary openangle glaucoma (POAG) which remains one of the major causes of blindness in the United States and worldwide
In recent years, one of the hot topics in glaucoma research has been the effect of IOP fluctuation (IOP lability), both within a single day (shortterm fluctuation) and from visit to visit (longterm fluctuation) on POAG
In this paper, we used LCA to model the postrandomization IOP in the OHTS. For each class, the change of IOP was characterized by 4 parameters: the initial IOP level (I), the linear (L) and quadratic (Q) trend over time, and the variance of IOP (V). We used data from the European Glaucoma Prevention Study (EGPS)
Methods
Study cohort
Our study used data from OHTS and EGPS, the two largest randomized trials to test safety and efficacy of topical hypotensive medication in preventing the development of POAG. In OHTS, 1636 subjects were randomized to either observation or treatment with ocular hypotensive medication and followed for a median of 78 months
In this paper, we excluded IOP values measured after POAG onset. The primary endpoint was time from randomization to the development of POAG. Those subjects who did not develop POAG were censored at the date of study closeout. In addition to the followup data, following 13 demographic and clinical characteristics at randomization were also included in this paper: treatment assignment (TRT, 0 for observation/placebo and 1 for treatment), male gender (Male), black race (Black), age at randomization (Age, decade), baseline IOP (IOP0, mmHg), central corneal thickness (CCT, μm), pattern standard deviation (PSD, dB), vertical cup/disc ratio (VCD), the use of systematic beta blocker (BB) or Calcium channel blockers (CHB), and the history of diabetes (DM), heart diseases (Heart), or hypertension (HBP). These baseline factors were identified
Variables
OHTS (N = 1600)
EGPS (N = 971)
Baseline predictors
TRT
795 (49.7%)
487 (50.2%)
Male
687 (42.9%)
445 (45.8%)
Black
396 (24.8%)
1 (0.1%)
AGE (decades)
5.56 (0.96)
5.70 (1.02)
IOP0 (mmHg)
24.9 (2.69)
23.4 (1.62)
CCT (μm)
572.6 (38.5)
573.3 (37.5)
PSD (dB)
1.91 (0.21)
2.00 (0.52)
VCD
0.39 (0.19)
0.32 (0.14)
BB
71 (4.4%)
64 (6.6%)
CHB
190 (11.9%)
66 (6.8%)
DM
188 (11.8%)
55 (5.7%)
Heart
99 (6.2%)
109 (11.2%)
HBP
606 (37.9%)
279 (28.7%)
Postrandomization IOP
Mean (mmHg)
21.44 (3.45)
19.73 (2.57)
SD (mmHg)
2.27 (1.04)
2.22 (1.03)
Median #visits (minmax)
13 (2–16)
9 (2–10)
POAG
146 (9.1%)
107 (11.0%)
Statistical analysis
Unconditional LCA
Suppose there were
where
• The mixing probability
• The specification of
Diagrams for unconditional (1A) and conditional (1B) latent class analysis (LCA) for OHTS data, where C denoted the latent classes
Diagrams for unconditional (1A) and conditional (1B) latent class analysis (LCA) for OHTS data, where C denoted the latent classes. The trajectory of postrandomization IOP (Y) in each class was described by 4 classspecific parameters: the initial IOP level (I), the systematic linear (L) and quadratic (Q) trend over time, and the variance of IOP (V).
• Given the estimated parameters
•
The best unconditional LCA was selected by enumerating and comparing a set of competing models differing only in the number of classes. In this paper, the model comparison was based primarily on the log likelihood values, including the Bayesian Information Criteria (BIC, with a smaller BIC indicating a better fit) and the LoMendellRubin adjusted likelihood ratio test (LMRLRT)
Conditional LCA
Since patterns of IOP change were found to be associated with the risk of POAG in an unconditional LCA, a conditional LCA was built by adding baseline covariates as predictors to the IOP change and adding time to POAG as an outcome due to IOP change (Figure
• Similar to Model (1), the term
• The term
The conditional LCA facilitated a better understanding of ocular hypotensive treatment on the risk of developing POAG. This model allowed us to determine whether the predictive accuracy on POAG can be improved by adding postrandomization IOP. For example, the survival probability (cumulative POAGfree rate) at any time t can be readily calculated as the average of the classspecific survival weighted by the posterior class probabilities,
where
Results
Unconditional LCA
Table
# latent classes (G)
OHTS
EGPS
BIC
LMRLRT*
Minimal class size
BIC
LMRLRT
Minimal class size
* LoMendellRubin likelihood ratio test, with a smaller pvalue favoring the Gclass model over the model with G1 classes (null hypothesis).
2
97097
<0.001
47%
39235
<0.001
44%
3
94219
0.002
24%
38395
0.001
14%
4
92922
0.609
14%
38109
0.005
11%
5
92107
0.003
13%
37870
0.009
12%
6
91644
0.042
10%
37760
0.452
9%
7
91289
0.147
7%
37682
0.060
5%
8
91045
0.406
6%
37608
0.011
4%
Postrandomization IOP values for 50 subjects randomly selected from each of the 6 latent classes indentified in OHTS, where red lines represented subjects who developed POAG and the black lines were for those without POAG
Postrandomization IOP values for 50 subjects randomly selected from each of the 6 latent classes indentified in OHTS, where red lines represented subjects who developed POAG and the black lines were for those without POAG. The class membership was based on the posterior probabilities from the optimal unconditional LCA, and the 4 parameters (I, L, Q, and V) in the plots represented the estimated initial level, the systematic linear and quadratic trend over time, and the variance of postrandomization IOP respectively.
Latent class
OHTS
EGPS
POAG%
HR
95% CI
POAG%
HR
95% CI
1
5.9%
1.00

8.3%
1.00

2
3.9%
0.59
0.37  0.88
10.2%
1.28
0.76  2.06
3
4.3%
0.83
0.57  1.14
8.7%
1.13
0.73  1.65
4
10.1%
1.87
1.32  2.57
10.5%
1.40
0.85  2.18
5
11.4%
1.93
1.50  2.46
19.4%
2.66
1.92  3.69
6
31.2%
5.61
4.46  7.08
In EGPS, the IOP change was best fit by a 5class LCA (Table
Postrandomization IOP values for 50 subjects randomly selected from each of the 5 latent classes indentified in EGPS, where red lines represented subjects who developed POAG and the black lines were for those without POAG
Postrandomization IOP values for 50 subjects randomly selected from each of the 5 latent classes indentified in EGPS, where red lines represented subjects who developed POAG and the black lines were for those without POAG. The class membership was based on the posterior probabilities from the optimal unconditional LCA, and the 4 parameters (I, L, Q, and V) in the plots represented the initial level, the systematic linear and quadratic trend over time, and the variance of postrandomization IOP respectively.
Table
Latent class
OHTS
EGPS
Observation
Treatment
Placebo
Treatment
1
15 (1.9%)
191 (24.0%)
113 (23.3%)
143 (29.4%)
2
67 (8.3%)
385 (48.4%)
69 (14.3%)
112 (23.0%)
3
226 (28.1%)
106 (13.3%)
162 (33.5%)
136 (27.9%)
4
55 (6.8%)
84 (10.6%)
64 (13.2%)
63 (12.9%)
5
279 (34.7%)
19 (2.4%)
76 (15.7%)
33 (6.8%)
6
163 (20.2%)
10 (1.3%)
Total
805 (100%)
795 (100%)
484 (100%)
487 (100%)
Conditional LCA
A conditional model was constructed for OHTS and EGPS separately by adding the baseline factors as predictors and the time to POAG as the outcome to the optimal unconditional LCAs (Figure
Predicted baseline cumulative incidence of POAG for each class, based on the conditional latent class analysis for the OHTS and EGPS data respectively
Predicted baseline cumulative incidence of POAG for each class, based on the conditional latent class analysis for the OHTS and EGPS data respectively.
Table
OHTS
Variables
Parameters for IOP change and the effects of covariates on class membership
Effects on POAG
Class 1
Class2 (Ref.)
Class 3
Class 4
Class 5
Class 6
*: p < 0.05; #: p < 0.001; **: Not estimable due to zero count of beta blocker use in the given class.
Class Size IOP Change
14.2%
27.1%
21.1%
9.1%
17.6%
10.9%
I
17.58(0.20)^{#}
19.83(0.21)^{#}
22.30(0.21)^{#}
22.82(0.79)^{#}
24.74(0.22)#
27.70(0.28)^{#}

L
−0.57(0.08)^{#}
−0.53(0.06)^{#}
−0.47(0.09)^{#}
−0.95(0.30)^{#}
−0.20(0.09)*
0.16(0.17)

Q
0.06(0.01)^{#}
0.05(0.01)^{#}
0.05(0.01)^{#}
0.07(0.04)
0.02(0.02)
−0.05(0.03)

V
4.36(0.27)^{#}
4.30(0.32)^{#}
4.80(0.25)^{#}
16.07(1.46)^{#}
4.66(0.31)#
12.15(1.15)^{#}

Covariates
Intercept
−2.64(0.63)^{#}
2.08(0.47)^{#}
−0.06(0.47)
2.35(0.54)^{#}
1.04(0.61)

TRT
1.60(0.66)^{*}

−3.25(0.35)^{#}
−2.19(0.63)^{#}
−5.98(0.56)^{#}
−6.46(0.70)^{#}
0.16(0.29)
MALE
0.25(0.23)

−0.99(0.24)^{#}
0.24(0.27)
−0.68(0.28)^{*}
−0.22(0.30)
0.23(0.19)
RACEB
−0.10(0.27)

−0.37(0.30)
0.75(0.31)^{*}
−0.91(0.34)^{*}
0.05(0.37)
−0.05(0.24)
AGE
0.06(0.12)

−0.01(0.12)
0.08(0.16)
−0.05(0.14)
0.13(0.15)
0.18(0.09) ^{*}
IOP0
−0.79(0.18)^{#}

0.21(0.22)
1.03(0.36)^{*}
0.89(0.18)^{#}
1.73(0.22)^{#}
−0.10(0.11)
CCT
−0.35(0.12)^{*}

0.18(0.11)
−0.08(0.17)
0.14(0.13)
0.09(0.16)
−0.64(0.13)^{#}
PSD
0.13(0.11)

0.04(0.11)
0.08(0.13)
0.17(0.13)
0.12(0.15)
0.23(0.10)^{*}
VCD
0.06(0.11)

−0.08(0.12)
−0.15(0.17)
−0.10(0.13)
−0.09(0.15)
0.60(0.10)^{#}
BB
−0.60(0.48)

−0.37(0.61)
 ^{**}
−0.30(0.62)
−1.21(0.77)
0.19(0.57)
CHB
−0.19(0.37)

−0.32(0.42)
0.47(0.45)
−0.42(0.44)
−0.50(0.49)
0.09(0.31)
DM
−0.23(0.35)

0.22(0.32)
−0.71(0.45)
0.23(0.36)
0.64(0.40)
−1.67(0.53)^{*}
HEART
0.70(0.44)

0.10(0.49)
0.43(0.56)
−0.09(0.56)
−0.48(0.73)
0.71(0.29)^{*}
HBP
0.47(0.24)^{*}

0.40(0.27)
0.06(0.34)
0.41(0.30)
0.66(0.33)^{*}
0.08(0.22)
EGPS
Variables
Parameters for IOP change and the effects of covariates on class membership
Effects on POAG
Class1 (Ref.)
Class 2
Class 3
Class 4
Class 5
Class Size IOP Change
26.3%
20.1%
29.3%
12.9%
11.4%
I
18.66(0.25) ^{#}
18.24(0.18)^{#}
21.24(0.20)^{#}
21.85(0.47)^{#}
24.33(0.34)^{#}

L
−0.34(0.14)^{*}
−1.29(0.18)^{#}
−0.25(0.11)^{*}
−1.76(0.28)^{#}
0.06(0.26)

Q
0.05(0.03)
0.11(0.04)*
0.02(0.03)
0.02(0.07)
−0.08(0.07)

V
3.79(0.23)^{#}
3.75(0.25)^{#}
4.59(0.24)^{#}
7.12(0.85)^{#}
12.17(1.32)^{#}

Covariates
Intercept

−0.84(0.33)^{*}
0.30(0.28)
−0.85(0.58)
−0.71(0.34)^{*}

TRT

0.36(0.27)
−0.65(0.23)^{*}
−0.58(0.37)
−1.79(0.37)^{#}
−0.01(0.21)
MALE

−0.35(0.29)
0.14(0.23)
0.18(0.40)
0.38(0.33)
−0.24(0.22)
Black






AGE

−0.09(0.16)
0.01(0.13)
0.40(0.23)
0.45(0.20)^{*}
0.16(0.10)
IOP0

−0.61(0.24)^{*}
0.82(0.17)^{#}
1.24(0.24)^{#}
1.79(0.23)^{#}
0.11(0.13)
CCT

−0.33(0.13)^{*}
−0.14(0.12)
−0.43(0.15)^{*}
0.09(0.15)
−0.36(0.12)^{*}
PSD

0.27(0.16)
−0.23(0.14)
−0.18(0.24)
−0.41(0.23)
0.18(0.09)^{*}
VCD

−0.13(0.17)
−0.03(0.13)
0.72(0.26)^{*}
0.17(0.16)
0.46(0.12)^{#}
BB

−0.17(0.50)
−0.82(0.52)
 ^{**}
−0.58(0.69)
−0.07(0.41)
CHB

−0.10(0.56)
−0.97(0.52)
0.89(1.03)
−1.22(0.79)
−0.28(0.47)
DM

−0.46(0.52)
0.12(0.45)
−1.32(1.27)
0.82(0.81)
−0.18(0.54)
HEART

0.78(0.41)
0.11(0.44)
−0.83(0.77)
−0.79(0.61)
0.74(0.32)^{*}
HBP

0.02(0.36)
0.53(0.30)
−1.27(0.90)
0.11(0.54)
0.24(0.26)
Effects of the baseline covariates on IOP classification
• To identify baseline predictors for IOP classification, we only focused on factors that were significantly associated with the high risk groups (Classes 4, 5, 6 in OHTS, and Class 5 in EGPS), while treating the lowest risk group (Class 2 in OHTS and Class 1 in EGPS) as the reference. In OHTS, treatment assignment and baseline IOP were two most important predictors for IOP classification. Subjects randomized to treatment group had a much lower chance of inclusion in the high risk groups (with OR = 0.11, 0.003, and 0.002 for Classes 4, 5, and 6, respectively), while these with a higher baseline IOP were more likely to be in the Classes 4, 5, or 6 (with OR = 2.80, 2.44, and 5.64 respectively). The results also showed that male subjects were less likely to be in Class 5 (OR = 0.51), the black subjects were more likely to be in Class 4 (OR = 2.12) but with a lower chance in Class 5 (OR = 0.40), and subjects with a history of hypertension were more likely in Class 6 (OR = 1.93). In EGPS, the results confirmed that treatment assignment (OR = 0.17) and baseline IOP level (OR = 5.99) were important predictors for Class 5. The result also showed that older age (OR = 1.57) was significantly associated with Class 5.
Effects of the baseline covariates on the risk of POAG development
• As expected, the effects of baseline covariates on the risk of developing POAG from the conditional LCA reached consistent conclusions as previous analyses using Cox models
To explore the effect of followup IOP on the overall predictive accuracy of POAG, the 5year cumulative POAG incidence was calculated for each individual using the formula (3). The overall predictive accuracy was summarized as Cindex and calibration statistics (Model 1 in Table
Model
Model Features
C index
Calibration Chisquare
OHTS
EGPS
OHTS
EGPS
0
Cox mode with baseline factors only
0.778
0.706
5.0
2.1
1
LCA with a quadratic withinclass functional form
0.821
0.719
11.3
7.0
2
LCA with a linear withinclass functional form
0.825
0.720
10.2
4.9
3
LCA with a constant withinclass functional form
0.823
0.727
10.5
13.5
Sensitivity analyses
As in all the statistical models, LCAs were inevitably based on certain assumptions. One assumption of our LCA was that the trajectories of IOP followed a quadratic functional form. It is known that the parameter estimates, class sizes, and interpretation of latent classes could be heavily influenced by the withinclass distribution of longitudinal data
Next, two additional sensitivity analyses were performed in the OHTS data, one excluding participants with Black race and the other only using participants randomized to the observation group. The IOP change in the nonBlack was best described by 6 distinct classes, while the LCA in the untreated participants identified 5 classes. Figures
Sensitivity analyses of latent class models in the OHTS data
Sensitivity analyses of latent class models in the OHTS data. Plots A and B: the observed mean IOP during followup visits of the latent classes and the corresponding KaplanMeier POAGfree curves in the nonBlack participants; Plots C and D: the observed mean IOP during followup visits of the latent classes and the corresponding KaplanMeier POAGfree curves in the untreated participants.
Finally, our LCA also made an implicit assumption that the baseline covariates influenced the IOP change exclusively through their effects on the class membership (i.e., no direct effects on the withinclass growth parameters). The validity of this assumption can be checked by comparing the conditional LCAs with the unconditional models. The assumption violation is often signified by a dramatic shifting in the meaning and size of latent classes when the baseline predictors are added to the unconditional LCA
Discussion
In recent years, one of the hot topics in glaucoma research has been the effect of IOP fluctuation on POAG. Although more and more studies have confirmed that a decrease in the mean IOP level can reduce the risk of developing POAG, the findings from major prospective clinical trials about the impact of IOP fluctuation on POAG remain controversial
Conventionally the change of longitudinal data is described using linear mixed models with random coefficients
LCA also provides an attractive alternative for making prediction with timedependent covariates
Despite its advantages, the LCA has several limitations. First, the computational load of LCA can be high, especially for models with complexity structures. In OHTS data (N = 1600), for example, it ran less than 10 minutes for an unconditional 6class LCA, but it took more than 30 minutes to develop the full conditional model. Because of the exploratory nature of data analysis with LCA, the cumulative time can be substantial. For this consideration, in practice the best LCA model is often constructed taking a twostep approach as in this paper. Another issue in LCA is that the loglikelihood function may end up at local rather than global maxima. Fortunately this issue has been taken into consideration by the statistical package Mplus which automatically uses 10 sets of randomly generated starting values for estimation. The program also allows investigators to rerun and compare the estimates from user specified starting values if necessary
Conclusion
LCA provides a useful alternative to understand the interrelationship among the baseline covariates, the change in followup IOP, and the risk of developing POAG. The inclusion of postrandomization IOP can improve the predictive ability of the original prediction model that only included baseline risk factors.
Abbreviations
IOP: Intraocular pressure; POAG: Primary openangle glaucoma; LCA: Latent class analysis; OHTS: The ocular hypertension treatment study; EGPS: The european glaucoma prevention study; HR: Hazards ratio; OR: Odds ratio; CCT: Central corneal thickness; PSD: Pattern standard deviation; VCD: Vertical cup/disc ratio.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
FG, JPM, JAB and MOG conceived the study. FG and JAB carried out the data analysis. FG and MOG drafted the first version of the manuscript. All authors contributed to the critical review and approved the final version.
Acknowledgments
This study is supported by grants from the National Eye Institute of Health (EY091369 and EY09341) and Research to Prevent Blindness (RPB).
Prepublication history
The prepublication history for this paper can be accessed here: