Appendix Overview

Appendix

Flow Diagram

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.

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 normal-normal transition; thus, this is the favourable outcome and should be classified as animproving 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 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 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 = 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 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.

Justification of Age Cut-Score

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 cut-score 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-normal, the blue circles in Fig. A2 the optimal knot was 76 (75-79). The choice of cut-score 78 provides adequate sample sizes for the computation of probabilities in each cell of the transition matrices. The age 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 time-varying covariates (bmi and age as well as the initial states), Table A1BEshow the binomial 4x2 transition matrices for improving transitions. The definition of which transitions are 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 / Second State
Age<78 / 0 / 1 / 2 / 3 / margin
First State / 0 / 831 / 70 / 62 / 19 / 973
1 / 87 / 57 / 14 / 18 / 178
2 / 104 / 8 / 41 / 10 / 163
3 / 14 / 19 / 12 / 23 / 88
margin / 1038 / 154 / 129 / 61 / 1380
/ A1 B
Bene / %
Prevent / 831/973 / 85
Reverse1 / 87/178 / 49
Reverse2 / 104/163 / 64
Reverse3 / 45/88 / 88
margin / 1067/1380 / 77
/ A1 C
Harm / %
Recrudescent / 142/973 / 14
Add_to_1 / 18/178 / 10
Add_to_2 / 10/163 / 6
Persistent / 23/88 / 34
margin / 193/1380 / 14
A1 D / Second State
Age≥78 / 0 / 1 / 2 / 3 / margin
First State / 0 / 220 / 44 / 25 / 17 / 306
1 / 41 / 75 / 11 / 28 / 155
2 / 30 / 13 / 55 / 28 / 128
3 / 15 / 30 / 30 / 84 / 162
margin / 309 / 162 / 121 / 157 / 749
/ A1 E
Bene / %
Prevent / 220/306 / 72
Reverse1 / 41/155 / 28
Reverse2 / 30/126 / 24
Reverse3 / 78/162 / 48
margin / 369/749 / 49
/ A1 F
Harm / %
Recrudescent / 86/306 / 28
Add_to_1 / 28/155 / 18
Add_to_2 / 28/126 / 22
Persistent / 84/162 / 52
margin / 226/749 / 30
Transition Probability Matrices
A1 G / Second State
Age<78 / 0 / 1 / 2 / 3
First State / 0 / 0.854 / 0.072 / 0.064 / 0.010
1 / 0.494 / 0.324 / 0.080 / 0.102
2 / 0.638 / 0.049 / 0.252 / 0.061
3 / 0.206 / 0.279 / 0.179 / 0.338
/ A1 H / Second State
Age≥78 / 0 / 1 / 2 / 3
First State / 0 / 0.719 / 0.144 / 0.082 / 0.056
1 / 0.265 / 0.484 / 0.071 / 0.181
2 / 0.238 / 0.103 / 0.439 / 0.222
3 / 0.111 / 0.185 / 0.185 / 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 oldparticipants (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.

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.

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.

In384 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. Since the two curves represent paired data, the nonparametric Wilcoxon’s signed rank test (P=0.91) provides a head-to-head paired comparison.