Appendix
Flow Diagram
Participants included (N=624)
in NMAPS 1993-2001
Excluded based on (N=26):- No assessment of
3MS or of gait speed
Participants available
(N=598)
Excluded based on (N=77):
- No consecutive 1-year assessments of 3MSE
or of gait speed
Objectives 1) & 2): To
estimate transition
probabilities and the effect
of covariates
(Participants N=521)
2129 Transitions
Excluded based on (N=137):
- Never normal 3MSE and normal gait speed,
never in normal-normal state
Objective 3): To determine
which (cognitive or physical
defect) happens first
(Subjects N=384)
Figure A1. Study Flow Diagram
1) Demographic summaries are for available participants (N=598).
2) Exclusions for Objective 1) & 2) include subject with only one assessment.
3) Exclusions for Objective 3) allow direct, paired comparison of times to low
cognition and to low gait speed.
1
Appendix Overview
This appendix consists of two parts; the second part includes the parallel analyses of worsening
transitions (parallel to the presentation of improving transitions in the text). The first part begins
with a Socratic discussion of important features of a Markov-like analysis and why we are using
it. This part gives a theoretical justification for defining transitions in a binary fashion. There is
also a detailed argument justifying for our division between younger-old and older-old.
There is a detailed presentation of the multi-state transition probabilities (Table A1) and the
effect of clinical variables on these transition probabilities (as % in Figure A3). The Kaplan-Meier
‘survival’ curves comparing the times to low gait speed and low cognition are given in Figure 4.
Methods
Multistate Markov-like Model
A multistate Markov-like model is being used here to study the natural history of worsening
physical function (gait speed) and cognitive function in this study, but this approach is not
commonly used in the literature and therefore not easily readily understood. The following
discussion addresses the less well-understood concepts in this analytical approach.
1. The going from normal (prior state) to normal (current state) shouldn’t be called a
transition and why do we count it as improving.
A. In the Markov chain literature this and all other changes and lack of changes are called
transitions. An illustration of a simple model with two states will help; one must show all
transitions otherwise the transition probabilities out of each state would not add up to 1.0; here
normal-normal is one of the transitions with probability of 0.4.
0.5
0.4
0.5
AbNormal
Normal
0.6
B. The normal-normal transition compared to the other possible transitions is as good as it can
be; thus it is stable. Backing up this point of view, the goal of primary prevention is a normalnormal transition; thus, this is the favourable outcome and should be classified as an improving
transition.
2. It is difficult to give a clinical interpretation for Markov chain results.
A. This problem begins with the conviction that age-related diseases are progressive going from
state to a worse state until one reaches an absorbing state from which there is no return. This
view can be quite wrong; and yet this may be the gestalt that a health care provider has
adopted. One should begin by destroying this belief; after all this is a dismal prospect if one had
to explain it to their patient who potentially has this disease. What probability of reversal is worth
mentioning to the patient; shouldn’t these probabilities be known by health care providers?
If the disease is progressive in the way mentioned above, then standard statistical analyses
can be applied to collected data. One could make the time to reaching the absorbing state the
outcome and conduct a survival analysis; this is often done with mortality data. Such analyses
are more difficult to apply if reversals are important that make reaching an absorbing state a
distant event, the more appropriate analysis would be Markov modelling. (Let us add that it may
2
not be necessary to include death as an absorbing state in such a Markov model; rare event
death can be treated as a separate, secondary analysis. The transition from each of the states
to death will be interesting but they probably do not substantially change the Markov modelling.)
B. We believe understanding transitions and transition probabilities give an elevated platform to
discuss age-related and/or chronic disease intervention/prevention. Most think about treating
the state and not the transition; we agree that these are difficult to separate. However, note that
it is theoretically possible that a treatment eliminates a state occupancy probability but leaves
the transition probabilities unchanged. This would be an ethereal victory; the system would soon
return to its original state (maybe in 3 years). That is the power of the transition probabilities.
(We should stop to explain that in a Markov model there are two sets of probabilities, the
transition probabilities and the occupancy probabilities. The occupancy probabilities give the
current distribution of patients among the states in the model.) To us it is clear that more
attention needs to be given to treatments that might affect transition probabilities and the
mechanisms involved. Primary prevention and maybe secondary prevention are about
transitions, mainly preventing transitions. It is more difficult to come up with a valid, illustrative
intervention example; let us try. Consider a chronic disease, say DM, where one tracts the A1c.
A drug might lower the A1c to an acceptable level, but the patient is not considered cured.
Since transition back to high levels of A1c is not good, the patient takes the drug regularly in
order to modify (lower) this transition probability. So am I treating the disease (evidently not),
the high A1c (maybe), or the transition probability (probably)?
In biochemistry and physiology, one can think about states but one is equally likely to think
about processes that cause transitions. So there is hope of making detailed, fully developed
models of what is being said here. One needs to sort out disease states, indicators/markers of
disease state and transition among states. This will require Markov –like modelling. Such
modelling is not well represented in the literature; in our opinion, this is a large gap.
3. The lack of memory requirement disqualifies a Markov model from being used.
It is true that our transition data do not satisfy the Markov property of lack of memory; thus the
analytical model here is a Multi-State Model instead of a Markov chain process. The Markov
property requires the probability of the transition state j -> state I dependent on j but not on any
prior transition history – the lack of memory requirement. Thus the transition probabilities in a
Multi-State process are conditional probabilities that may depend on the prior transition history;
the estimate of the transition probability that we provide is the (weighted) average of those
conditional probabilities over the prior history:
P( i / j ) =Σ P( ϕ ) P( i / j ϕ), where ϕ represents the prior transition history.
This result is based the Theorem of Total Probabilities.
Thus, the probabilities that we compute are transition probabilities. A main concern that one
might have is about projections based on repeated multiplication by the transition matrix and the
resulting limiting stationary distribution among the states. We don’t use these ideas here.
Our concern is about whether these computable transition probabilities are functions of clinical
variables.
4. Are findings due to misclassification?
There is no doubt that there will be misclassification of which state a participant may be in.
(This is also true for every diagnostic test based on a cut score used in medical practice.)
In order to discussion the effect of noise on transitions, consider the follow probabilistic model
for a beneficial transition. Let X1 be a continuous random variable representing the underlying
construct (say gait speed) at time 1, X2 be the random variable at the next time 2, and c be the
threshold above which the participant is impaired in the underlying construct. Then the event X1
> c and X2 < c is the form of a beneficial transition. Let X1, X2 be bivariate normally distributed
with means µ1, µ2 and standard deviations σ1, σ2 , respectively. It is reasonable that σ1=σ2 = σ
3
and let the correlation coefficient be ρ= .5; it can be shown that a higher correlation is less of a
problem in the following argument. Now the likelihood of a beneficial transition defined above
can be computed in terms of probabilities for the bivariate normal; there is a function that does
this in SAS. To be clear, consider the condition 1) µ1= µ2 > c, a null hypothesis that there is no
real transition. One is concerned about hypothesis 1), namely, how likely are false transitions
and does this likelihood increase with increasing measurement error? In the language of
diagnostic tests, the concern is about specificity.
Proposition. Under hypothesis 1) that there is no real transition, the probability of such a false
transition, P[X1 > c and X2 < c] , increases as the size (standard deviation σε) of measurement
error, ε, increases.
Proof. True and relatively straight forward to show.
Though the intuition was correct, it does not provide an effect size (as is the usual case with
intuition). This brings me to my argument.
Argument. It is reasonable that the measurement error is 10 % or less of the basic variability of
the underlying variables X1, X2. That is, measurement error of gait speed or 3MSE ascertained
for the same participant under identical conditions (ideally the participant has not changed in
any way, the error is just measurement error) should be small. The usual test-retest error
includes this measurement error but is also due to real changes in the participant at the time of
the repeated test. Perhaps, the participant just feels better or worse on that day. The variance
of X1 + ε1 = σ2 + σε2 , since the measurement error is assumed to be independent. So the
standard deviation of the measured X1 + ε1 = σ √1 + 0.102 = σ 1.005. In other words, the 10%
error has an 0.5% effect. The change in the probability of a false transition, computed at several
values, is an increase of less than 0.3%. A measurement error of 20% yields an effect of 2%
and an increase in the probability of a false transition of 1%.
Though the intuition that measurement error increases the probability of a false transition is
correct, the effect size is quite small and does not explain any of our positive findings.
5. Here, the multi-state model has 4 states; why isn’t the analysis based on the
multinomial outcome and the multinomial distribution?
We simplify the analysis by defining improving transitions as a binary outcome. This allows
modelling to be in terms of logistic regression instead of multinomial modelling. We are then
thinking about transition probabilities instead of occupancy probabilities and binomial transitions
instead of multinomial ones. Thus, we can say that the unit of analysis for this study is the
transition.
These analyses have an illustrious history.
First, the multinomial transitions are defined in a 4x4 transition matrix where the 4 rows
represent the initial or the “from” states and the 4 columns represent the final or “to” states.
There are 2119 such 1-year transitions consisting of an initial and final state that can be defined
in the New Mexico Aging Process Study data for years 1993-2001. (The definitions of the 4
states are given in the text.) Counts of these transitions form a 4x4 frequency table; for a given
row, the 1x4 frequency table of transitions counts the number of transition into the four final
states. These 4 frequencies can be modelled as multinomial counts with transition probabilities
4
p1, p2, p3, and p4 whose sum is 1.0. These transition probabilities are estimated from the
frequencies fi by fi/ (f1+f2+f3+f4).
Second, each transition probability can be considered as a function of age and the other
covariates; and the analysis will maintain the sum pi = 1.0 at ever covariate combination.
Following the 1950 suggestion of R.A. Fisher1, this is achieved in a model where the cell
frequencies are considered as Poisson counts and Poisson regression models with the
dependencies on covariates as well as states. The link between the multinomial frequencies
and the Poisson counts was also provided earlier by R.A. Fisher; in a footnote of his 1922
paper2.
A statement of the Fisher’s 1922 result. Suppose the four cell counts X1 – X4 have Poisson
distributions and are independent random variables, which is guaranteed if the classifications
into the four cells were independent. Let the Poisson means be λ1 - λ4, and then the four
relative frequencies Xi/(X1+X2+X3+X4) have a multinomial distribution with probabilities pi = λi /
(λ1 + λ2 + λ3 + λ4). Here the sum of pi always = 1.0 and λ1 + λ2 + λ3 + λ4 = X1 + X2 + X3 +
X4. End of Result.
The whole transition matrix then could be modelled for the initial states, and final states, using
Poisson regression and ANOVA syntax. With the argument that transition probabilities are
conditional on row, we will analysed each row of the transition matrix separately.
However, the definition of improving transitions reduces the problem to a binomial distribution.
Logistic regression is now appropriate; the only remaining difficult is that the transitions are
repeated measurements within participants (more than one transition per participant). We will
use SAS’s PROC GENMOD for logistic regression; this procedure deals with repeated
measures as well as the categorical predictor defined for the four initial states of the transitions.
5
Justification of Age Cut-Score
100
80
Markov state probability
In order to determine binary age
categories (with a cut score), we
plot the state 3 (low-low)
probabilities (red circles in
Fig.A2) as a function of age and
use non-linear regression to fit a
piece-wise linear functions with
varying knots. The choice of cutscore is based the sum of
squared errors (SSE) of the
piece-wise linear model fits
plotted as a function of varying
change points (knots, say ages
65 to 85 with a step size of 1).
The minimum SSE for state 3
was 78 years with a 95% interval
of (76,79). In a similar analysis
for Markov state 0 (normal-
60
40
20
0
60
70
Age
80
90
normal, the blue circles in Fig. A2
Figure A2. The slow gait – low cognition (state 3)
the optimal knot was 76 (75-79).
probabilities (red circles) and nonlinear fit (red line) with knot
The choice of cut-score 78
(age = 78) versus age showing non-linearity (non-linear
provides adequate sample sizes
regression, P < 0.001). Normal gait-normal cognition (state
for the computation of
0) probabilities (blue circles) versus age with piece-wise
probabilities in each cell of the
linear fit with knot (age78), showing non-linearity (dark blue
transition matrices. The age
line, P < 0.001).
relationship for state 3
probabilities is significantly nonlinear (non-linear regression, red line in Fig. A2, P<0.001); and is
also non-linear for state 0 probabilities (dark blue line, P < 0.001).
Multi-state Transitions
The Multi-state transition model with these 4 states has 16 possible transition probabilities for
the younger-old adults, estimated in 4x4 matrix; shown in Table A1G. Similarly, the 16 possible
transition probabilities for older-old adults are shown in Table A1H.
The Multi-state transition model with these 4 states has 16 possible transitions, so the estimated
transition frequency matrix is 4x4; shown in Table A1A for the younger old participants (age <
78). Table A1D shows the estimated 4x4 transition frequency matrix for the older old
participants (age ≥ 78). With 4 outcomes (columns), the (row) distributions are multinomial.
However, the definition of improving transitions (and worsening transitions) reduces the problem
to a binomial distribution. We use SAS’s PROC GENMOD for logistic regression. This
procedure also allows for the inclusion of covariates into the multivariate model including timevarying covariates (bmi and age as well as the initial states), Table A1B & E show the binomial
4x2 transition matrices for improving transitions. The definition of which transitions are
6
considered improving is indicated by green in table A1A; and the colour coding is carried
throughout the rest of Table A1. Table A1C & F define worsening transitions as a binary
outcome (indicated by grey).
A1 A
A1 C
Bene
%
0
0
831
1
70
2
62
3
19
margin
973
Prevent
831/973
85
1
87
57
14
18
178
Reverse1
87/178
49
2
104
8
41
10
163
Reverse2
104/163
64
3
14
19
12
23
88
Reverse3
45/88
88
margin
1038
154
129
61
1380
1067/1380
77
A1 D
A1 E
Second State
Age≥78
0
1
margin
Bene
1
44
75
2
25
11
3
17
28
margin
306
155
Prevent
220/306
72
Reverse1
41/155
28
2
30
13
55
28
128
Reverse2
30/126
24
3
15
309
30
162
30
121
84
157
162
749
Reverse3
78/162
48
margin
369/749
49
First
State
A1 G
Age<78
0
1
2
3
0
0.854
0.494
0.638
0.206
%
Recrudescent
142/973
14
Add_to_1
18/178
10
Add_to_2
10/163
6
Persistent
23/88
34
193/1380
14
margin
Transition Probability Matrices
A1 H
Second State
1
2
3
Age≥78
0
0.072 0.064 0.010
0 0.719
0.324 0.080 0.102
1 0.265
0.049 0.252 0.061
2 0.238
0.279 0.179 0.338
3 0.111
Harm
%
Recrudescent
Add_to_1
Add_to_2
86/306
28/155
28/126
28
18
22
Persistent
margin
84/162
226/749
52
30
%
0
220
41
margin
Harm
A1 F
First
State
First State
Age<78
First State
A1 B
Second State
Second State
1
2
0.144 0.082
0.484 0.071
0.103 0.439
0.185 0.185
3
0.056
0.181
0.222
0.519
Table A1. A. The Multi-state transition model estimated 4x4 transition frequency matrix for the
younger old participants of the New Mexico Aging Process Study (60 ≤ age < 78). B.&C. The
binomial 4x2 transition matrices for improving transitions (green) and worsening (grey)
transitions. D. & E. & F. Estimated 4x4 transition matrix for the older old participants (age ≥
78) and the binary classification of transitions into improving (green) and worsening (grey). G. &
H. The Multi-state transition model 4x4 transition probability matrix for younger and older old
participants are equivalent to the row % for table A1 A. & D.
7
Analysis of Improving Transitions
Figure A3. A. Interaction plot of improving transition probabilities in younger-old and older-old
participants from initial states: State 0). line 0, normal gait speed and normal cognition, State 1).
line 1, slow gait speed and normal cognition, State 2), line 2, low cognition and normal gait
speed; and State 3), line 3, slow gait speed and low cognition. The age-initial state interaction
(P = 0.02) indicates age has a differential greater, adverse effect on the improving transitions
from low cognition; however, effect of age is present for each initial state.
8
B. The adverse effect of APOE4 on the improving transition probabilities (main effect, P = 0.02)
is shown as a dashed line.
C. The positive effect (+8%) of low baseline BMI (BMI ≤ 22.5) on the improving transition
probabilities in the younger-old is shown as a dashed line but there is a negative effect (-8%) in
the older-old. The opposite effects are found for obesity shown as a dotted line; obesity has a
negative effect (-11.5%) in the younger-old and a positive effect (+8) in the older-old. These
effects are all relative to the improving transition probabilities for normal BMI shown as a solid
red line; the interaction effects described above are significant (P =0.01).
D. The adverse effect of poorer health status on the improving transition probabilities (main
effect, P = 0.009). Good health is 10% lower than excellent health and poor health is 13% lower
than good health.
Kaplan-Meier survival curves comparing times to low gait speed and to low cognition.
In an
alternative analysis,
we excluded the
non-informative
cases where times
to low gait speed
and to low cognition
were structurally
identical because of
death or because
both times reached
the end of the study
without event
(Wilcoxon signed
rank, P=0.78).
Survival Probability to Event
In 384 participants who had a first year with normal gait speed and cognition, mean time to
develop a first occurrence of slow gait speed was 4.0 years ± 2.3 (SD) and mean time to
develop low 3MSE scores was 3.9 years ± 2.4 (SD). Since these data are time to event data,
Figure S4 below presents times to slow gait speed and to low cognition as Kaplan-Meier
survival curves; death, loss to follow, and reaching the end of the study without event are
censoring events.
1
Since the two curves
represent paired
0.9
data, the
nonparametric
0.8
Wilcoxon’s signed
rank test (P=0.91)
0.7
provides a head-tohead paired
0.6
comparison.
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
Years to Event
5
6
7
Figure A4. Kaplan-Meier 7-year survival curves to Low Cognition (blue line)
or to Low Gait Speed (red line). Circles are the corresponding censoring
times. Data are paired within participant; in a head-to-head comparison, the
paired times to event are not different (Wilcoxon's signed rank test, P=0.91).
9
Analysis of Worsening Transitions
The worsening transition probabilities are defined in Table A1 C. and F. for younger and older
old participants, respectively.
Table A2 below present results about worsening transitions in terms of odds ratios. Figure A5
below are interaction plots in terms of absolute percentages.
Factor
Younger-old participants
OR (95%CI)
Initial state
0:Normal gait speed and
normal cognitive function
1:Slow gait speed and
normal cognitive function
2:Normal gait speed and
low cognitive function
3:Slow gait speed and
low cognitive function
Sex (Female)
ApoE4 Carrier
Body mass index
Low BMI ≤ 22.5 kg/m²
Intermediate
High BMI ≥ 30 kg/m²
Hypertension
Health status
Excellent
Good
Poor
Education (College)
p
OR (95%CI)
<0.001
0.33 (0.19, 0.60)
0.36 (0.22, 0.59)
0.22 (0.11, 0.47)
0.20 (0.12, 0.36)
0.13 (0.06, 0.28)
0.27 (0.15, 0.46)
1.0 (Ref)
1.0 (Ref)
1.08 (0.76, 1.52)
1.57 (1.09, 2,25)
0.68 (0.43, 1.09)
1.0 (Ref)
1.61 (0.96, 2.72)
0.77 (0.44,1.35)
0.76 (0.47, 1.23)
0.36 (0.21, 0.61)
1.0 (Ref)
0.69 (0.49, 0.96)
0.67
0.02
0.04
1.13 (0.80, 1.58)
1.05 (0.66, 1.68)
0.33
<0.001
1.34 (0.89, 2.01)
1.0 (Ref)
0.89 (0.40, 1.98)
0.54 (0.29, 1.03)
0.69 (0.46, 1.04)
0.73 (0.47, 1.14)
1.0 (Ref)
0.94 (0.67, 1.33)
0.03
p
Interaction/
Additive*
p
<0.001
0.31/<0.001
0.49
0.83
0.35
0.91/0.43
0.21/0.05
0.03/na
0.06
0.20
0.42/0.05
0.03/na
0.72
0.25/0.07
Older-old participants
Table A2. Effect of selected factors at the time of transition on the probability of experiencing a
worsening transition
Notes for Table A2. All models for covariates above include the Initial State as a necessary part
of the definition of transitions. The binary factor (age≥78) – covariate interaction P-values are
listed in the last column; the second P-value is for the additive effect of covariate and is na (not
applicable) if the interaction is significant. Significant P-values are ≤ 0.05. APOE4 status
(additive P=0.05) and poor health status (interaction P =0.03) in addition to advancing age
(additive P < 0.001) also predicted that worsening transition probabilities are increased;
whereas sex is not a predictor (see Table S2). Again, the age-BMI interaction is significant
(P=0.03), where the worsening transition probabilities are decreased with low BMI in the
younger-old but increased in the older-old and increased with obesity in the younger-old but
decreased in the older-old. Education (college degree) P=0.03 in the younger-old but not in the
10
older-old, however the age interaction was not significant and in the additive model, education
was not significant. Thus, a Bonferroni correction (P=2x0.03=0.06) indicates our data cannot
establish that education is a significant predictor even in the younger-old.
A
Harmful Transition Prob
100%
100%
Initial state x age P = .31
additive age P < 0.001
pooled age effect = + 16%
80%
60%
40%
40%
20%
20%
Young
Harmful Transition Prob
100%
80%
60%
Older
C
0%
100%
Age x BMI category P = 0.01
Average effect of low BMI = - 4%
for harmful probability in Younger
and + 7% in Older and effect of
Obesity +6% in Younger and -3%
in Older
80%
Young
Older
D
Baseline Health Status
interaction effect P = 0.03
For Poor Health vs pooled
Excellent+Good, effect = +8%
60%
40%
40%
20%
20%
0%
APOE4 additive effect P = 0.02
Average APOE4 harmful
effect = + 4%
80%
60%
0%
B
0%
Young
Young
Older
Older
Figure A5. A. Interact plots of worsening transition probabilities in younger old and older old
participants from initial states: 0) blue line, normal gait speed and normal cognition, 1) green
line, low gait speed and normal cognition, 2) purple line, low cognition and normal gait speed;
and 3)red line, low gait speed and low cognition. The age main effect (P < 0.001) indicates age
has an adverse effect (increases) on the worsening transition probabilities. Note that the
average effect of age on worsening transition probabilities is +16%.
B. The adverse effect of APOE4 on the worsening transition probabilities (main effect, P = 0.05)
is shown as the solid red line. Note that the average effect of APOE4 is +4%.
C. The effect (-4%) of low baseline BMI (BMI ≤ 22.5) on the worsening transition probabilities is
helpful in the younger-old and not (+7%) in the older-old and is shown as the dashed red line.
11
Obesity is not helpful in the younger-old (+6%) but similar in the older-old (-3%), (interaction
effect, P = 0.01)
D. The adverse effect of poorer health status on the worsening transition probabilities
(interaction effect, P = 0.03). Poor health status worsening transition probability is 8% higher
than for the pooled Good and Excellent health status.
Note. Percentages are shown in Fig. A5 as closely representing the data rather than using
predicted values obtained from the models; this simplification is done as long as it does not
distort the statistical modelling results. Where the effects of covariates in Figs. A5B-D are
independent and additive compared to the base model, we can average (pool) these effects
across the initial states. Since there is no interaction, Initial State*(Age ≥ 78), in Fig. A5A, we
also average (pool) to obtain a simple summary. There are significant age-interactions in Fig.
A5C and A5D. The interaction for BMI category is clinical important, but interaction effect for
Health status is smaller; thus we average (pool) Excellent + Good in Fig. A5D.
Similarly, since the interaction Initial State*(Age ≥ 78) in Fig. S3A above is weak
compared to the main effect, we averaged (pooled) to obtain a simple summary.
Appendix References.
1. R. A. Fisher (1950) The Significance of Deviations from Expectation in a Poisson Series.
Biometrics Vol. 6, No. 1, pp. 17-24
2. R. A. Fisher (1922) On the Interpretation of χ2 from Contingency Tables, and the Calculation
of P. Journal of the Royal Statistical Society, Vol. 85, No. 1, pp. 87-94.
3. Fallah N, Arnold Mitnitski A, Searle SD, Gahbauer EA, Gill TM, Kenneth Rockwood K. (2011)
Transitions in Frailty Status in Older Adults in Relation to Mobility: A Multi-State Modeling
Approach Employing a Deficit Count J Am Geriatr Soc. 2011 March ; 59(3): 524–529.
4. Hardy SE, Allore HG, Guo Z, Dubin JA, Gill TM. (2006) The Effect of Prior Disability History
on Subsequent Functional Transitions. J. Gerontology: MEDICAL SCIENCES Vol. 61A, No.
3, 272–277.
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