7 Modeling Longitudinal Responses using Generalized Least Squares

Some good general references on longitudinal data analysis are Davis (2002), Pinheiro & Bates (2000), Diggle et al. (2002), Venables & Ripley (2003), Hand & Crowder (1996), Verbeke & Molenberghs (2000), Lindsey (1997)

7.1 Notation

\(N\) subjects
Subject \(i\) (\(i=1,2,\ldots,N\)) has \(n_{i}\) responses measured at times \(t_{i1}, t_{i2}, \ldots, t_{in_{i}}\)
Response at time \(t\) for subject \(i\): \(Y_{it}\)
Subject \(i\) has baseline covariates \(X_{i}\)
Generally the response measured at time \(t_{i1}=0\) is a covariate in \(X_{i}\) instead of being the first measured response \(Y_{i0}\)
Time trend in response is modeled with \(k\) parameters so that the time “main effect” has \(k\) d.f.
Let the basis functions modeling the time effect be \(g_{1}(t), g_{2}(t), \ldots, g_{k}(t)\)

7.2 Model Specification for Effects on \(E(Y)\)

7.2.1 Common Basis Functions

\(k\) dummy variables for \(k+1\) unique times (assumes no functional form for time but may spend many d.f.)
\(k=1\) for linear time trend, \(g_{1}(t)=t\)
\(k\)–order polynomial in \(t\)
\(k+1\)–knot restricted cubic spline (one linear term, \(k-1\) nonlinear terms)

7.2.2 Model for Mean Profile

A model for mean time-response profile without interactions between time and any \(X\):
\(E[Y_{it} | X_{i}] = X_{i}\beta + \gamma_{1}g_{1}(t) + \gamma_{2}g_{2}(t) + \ldots + \gamma_{k}g_{k}(t)\)
Model with interactions between time and some \(X\)’s: add product terms for desired interaction effects
Example: To allow the mean time trend for subjects in group 1 (reference group) to be arbitrarily different from time trend for subjects in group 2, have a dummy variable for group 2, a time “main effect” curve with \(k\) d.f. and all \(k\) products of these time components with the dummy variable for group 2
Time should be modeled using indicator variables only when time is really discrete, e.g., when time is in weeks and subjects were followed at exactly the intended weeks. In general time should be modeled continuously (and nonlinearly if there are more than 2 followup times) using actual visit dates instead of intended dates (Donohue et al., n.d.).

7.2.3 Model Specification for Treatment Comparisons

In studies comparing two or more treatments, a response is often measured at baseline (pre-randomization)
Analyst has the option to use this measurement as \(Y_{i0}\) or as part of \(X_{i}\)

Comments from Jim Rochon

For RCTs, I draw a sharp line at the point when the intervention begins. The LHS [left hand side of the model equation] is reserved for something that is a response to treatment. Anything before this point can potentially be included as a covariate in the regression model. This includes the “baseline” value of the outcome variable. Indeed, the best predictor of the outcome at the end of the study is typically where the patient began at the beginning. It drinks up a lot of variability in the outcome; and, the effect of other covariates is typically mediated through this variable.

I treat anything after the intervention begins as an outcome. In the western scientific method, an “effect” must follow the “cause” even if by a split second.

Note that an RCT is different than a cohort study. In a cohort study, “Time 0” is not terribly meaningful. If we want to model, say, the trend over time, it would be legitimate, in my view, to include the “baseline” value on the LHS of that regression model.

Now, even if the intervention, e.g., surgery, has an immediate effect, I would include still reserve the LHS for anything that might legitimately be considered as the response to the intervention. So, if we cleared a blocked artery and then measured the MABP, then that would still be included on the LHS.

Now, it could well be that most of the therapeutic effect occurred by the time that the first repeated measure was taken, and then levels off. Then, a plot of the means would essentially be two parallel lines and the treatment effect is the distance between the lines, i.e., the difference in the intercepts.

If the linear trend from baseline to Time 1 continues beyond Time 1, then the lines will have a common intercept but the slopes will diverge. Then, the treatment effect will the difference in slopes.

One point to remember is that the estimated intercept is the value at time 0 that we predict from the set of repeated measures post randomization. In the first case above, the model will predict different intercepts even though randomization would suggest that they would start from the same place. This is because we were asleep at the switch and didn’t record the “action” from baseline to time 1. In the second case, the model will predict the same intercept values because the linear trend from baseline to time 1 was continued thereafter.

More importantly, there are considerable benefits to including it as a covariate on the RHS. The baseline value tends to be the best predictor of the outcome post-randomization, and this maneuver increases the precision of the estimated treatment effect. Additionally, any other prognostic factors correlated with the outcome variable will also be correlated with the baseline value of that outcome, and this has two important consequences. First, this greatly reduces the need to enter a large number of prognostic factors as covariates in the linear models. Their effect is already mediated through the baseline value of the outcome variable. Secondly, any imbalances across the treatment arms in important prognostic factors will induce an imbalance across the treatment arms in the baseline value of the outcome. Including the baseline value thereby reduces the need to enter these variables as covariates in the linear models.

Senn (2006) states that temporally and logically, a “baseline cannot be a response to treatment”, so baseline and response cannot be modeled in an integrated framework.

… one should focus clearly on ‘outcomes’ as being the only values that can be influenced by treatment and examine critically any schemes that assume that these are linked in some rigid and deterministic view to ‘baseline’ values. An alternative tradition sees a baseline as being merely one of a number of measurements capable of improving predictions of outcomes and models it in this way.

The final reason that baseline cannot be modeled as the response at time zero is that many studies have inclusion/exclusion criteria that include cutoffs on the baseline variable. In other words, the baseline measurement comes from a truncated distribution. In general it is not appropriate to model the baseline with the same distributional shape as the follow-up measurements. Thus the approaches recommended by Liang & Zeger (2000) and Liu et al. (2009) are problematic¹.

¹ In addition to this, one of the paper’s conclusions that analysis of covariance is not appropriate if the population means of the baseline variable are not identical in the treatment groups is not correct (Senn, 2006). See Kenward et al. (2010) for a rebuke of Liu et al. (2009).

7.3 Modeling Within-Subject Dependence

Random effects and mixed effects models have become very popular
Disadvantages:
- Induced correlation structure for \(Y\) may be unrealistic
- Numerically demanding
- Require complex approximations for distributions of test statistics
Conditional random effects vs. (subject-) marginal models:
- Random effects are subject-conditional
- Random effects models are needed to estimate responses for individual subjects
- Models without random effects are marginalized with respect to subject-specific effects
- They are natural when the interest is on group-level (i.e., covariate-specific but not patient-specific) parameters (e.g., overall treatment effect)
- Random effects are natural when there is clustering at more than the subject level (multi-level models)
Extended linear model (marginal; with no random effects) is a logical extension of the univariate model (e.g., few statisticians use subject random effects for univariate \(Y\))
This was known as growth curve models and generalized least squares (Goldstein, 1989; Potthoff & Roy, 1964) and was developed long before mixed effect models became popular
Pinheiro and Bates (Section~5.1.2) state that “in some applications, one may wish to avoid incorporating random effects in the model to account for dependence among observations, choosing to use the within-group component \(\Lambda_{i}\) to directly model variance-covariance structure of the response.”
We will assume that \(Y_{it} | X_{i}\) has a multivariate normal distribution with mean given above and with variance-covariance matrix \(V_{i}\), an \(n_{i}\times n_{i}\) matrix that is a function of \(t_{i1}, \ldots, t_{in_{i}}\)
We further assume that the diagonals of \(V_{i}\) are all equal
Procedure can be generalized to allow for heteroscedasticity over time or with respect to \(X\) (e.g., males may be allowed to have a different variance than females)
This extended linear model has the following assumptions:
- all the assumptions of OLS at a single time point including correct modeling of predictor effects and univariate normality of responses conditional on \(X\)
- the distribution of two responses at two different times for the same subject, conditional on \(X\), is bivariate normal with a specified correlation coefficient
- the joint distribution of all \(n_{i}\) responses for the \(i^{th}\) subject is multivariate normal with the given correlation pattern (which implies the previous two distributional assumptions)
- responses from any times for any two different subjects are uncorrelated

FGH

What Methods To Use for Repeated Measurements / Serial Data? ² ³
	Repeated Measures ANOVA	GEE	Mixed Effects Models	GLS	Markov	LOCF	Summary Statistic⁴
Assumes normality	×		×	×
Assumes independence of measurements within subject	×⁵	×⁶
Assumes a correlation structure⁷	×	×⁸	×	×	×
Requires same measurement times for all subjects	×					?
Does not allow smooth modeling of time to save d.f.	×
Does not allow adjustment for baseline covariates	×
Does not easily extend to non-continuous \(Y\)	×			×
Loses information by not using intermediate measurements						×⁹	×
Does not allow widely varying # observations per subject	×	×¹⁰				×	×¹¹
Does not allow for subjects to have distinct trajectories¹²	×	×		×	×	×
Assumes subject-specific effects are Gaussian			×
Badly biased if non-random dropouts	?	×				×
Biased in general						×
Harder to get tests & CLs			×¹³			×¹⁴
Requires large # subjects/clusters		×
SEs are wrong	×¹⁵					×
Assumptions are not verifiable in small samples	×	N/A	×	×		×
Does not extend to complex settings such as time-dependent covariates and dynamic ¹⁶ models	×		×	×		×	?

² Thanks to Charles Berry, Brian Cade, Peter Flom, Bert Gunter, and Leena Choi for valuable input.

³ GEE: generalized estimating equations; GLS: generalized least squares; LOCF: last observation carried forward.

⁴ E.g., compute within-subject slope, mean, or area under the curve over time. Assumes that the summary measure is an adequate summary of the time profile and assesses the relevant treatment effect.

⁵ Unless one uses the Huynh-Feldt or Greenhouse-Geisser correction

⁶ For full efficiency, if using the working independence model

⁷ Or requires the user to specify one

⁸ For full efficiency of regression coefficient estimates

⁹ Unless the last observation is missing

¹⁰ The cluster sandwich variance estimator used to estimate SEs in GEE does not perform well in this situation, and neither does the working independence model because it does not weight subjects properly.

¹¹ Unless one knows how to properly do a weighted analysis

¹² Or users population averages

¹³ Unlike GLS, does not use standard maximum likelihood methods yielding simple likelihood ratio \(\chi^2\) statistics. Requires high-dimensional integration to marginalize random effects, using complex approximations, and if using SAS, unintuitive d.f. for the various tests.

¹⁴ Because there is no correct formula for SE of effects; ordinary SEs are not penalized for imputation and are too small

¹⁵ If correction not applied

¹⁶ E.g., a model with a predictor that is a lagged value of the response variable

Markov models use ordinary univariate software and are very flexible
They apply the same way to binary, ordinal, nominal, and continuous Y
They require post-fitting calculations to get probabilities, means, and quantiles that are not conditional on the previous Y value

Gardiner et al. (2009) compared several longitudinal data models, especially with regard to assumptions and how regression coefficients are estimated. Peters et al. (2012) have an empirical study confirming that the “use all available data” approach of likelihood–based longitudinal models makes imputation of follow-up measurements unnecessary.

7.4 Parameter Estimation Procedure

Generalized least squares
Like weighted least squares but uses a covariance matrix that is not diagonal
Each subject can have her own shape of \(V_{i}\) due to each subject being measured at a different set of times
Maximum likelihood
Newton-Raphson or other trial-and-error methods used for estimating parameters
For small number of subjects, advantages in using REML (restricted maximum likelihood) instead of ordinary MLE (Diggle et al., 2002, p. Section~5.3), (Pinheiro & Bates, 2000, p. Chapter~5), Goldstein (1989) (esp. to get more unbiased estimate of the covariance matrix)
When imbalances are not severe, OLS fitted ignoring subject identifiers may be efficient
- But OLS standard errors will be too small as they don’t take intra-cluster correlation into account
- May be rectified by substituting covariance matrix estimated from Huber-White cluster sandwich estimator or from cluster bootstrap
When imbalances are severe and intra-subject correlations are strong, OLS is not expected to be efficient because it gives equal weight to each observation
- a subject contributing two distant observations receives \(\frac{1}{5}\) the weight of a subject having 10 tightly-spaced observations

KLM

7.5 Common Correlation Structures

Usually restrict ourselves to isotropic correlation structures — correlation between responses within subject at two times depends only on a measure of distance between the two times, not the individual times
We simplify further and assume depends on \(|t_{1} - t_{2}|\)
Can speak interchangeably of correlations of residuals within subjects or correlations between responses measured at different times on the same subject, conditional on covariates \(X\)
Assume that the correlation coefficient for \(Y_{it_{1}}\) vs. \(Y_{it_{2}}\) conditional on baseline covariates \(X_{i}\) for subject \(i\) is \(h(|t_{1} - t_{2}|, \rho)\), where \(\rho\) is a vector (usually a scalar) set of fundamental correlation parameters
Some commonly used structures when times are continuous and are not equally spaced (Pinheiro & Bates, 2000, sec. 5.3.3) (nlme correlation function names are at the right if the structure is implemented in nlme):

Table 7.1: Some longitudinal data correlation structures
Structure	`nlme` Function
Compound symmetry: \(h = \rho\) if \(t_{1} \neq t_{2}\), 1 if \(t_{1}=t_{2}\) ¹⁷	`corCompSymm`
Autoregressive-moving average lag 1: \(h = \rho^{\|t_{1} - t_{2}\|} = \rho^s\) where \(s = \|t_{1}-t_{2}\|\)	`corCAR1`
Exponential: \(h = \exp(-s/\rho)\)	`corExp`
Gaussian: \(h = \exp[-(s/\rho)^2]\)	`corGaus`
Linear: \(h = (1 - s/\rho)[s < \rho]\)	`corLin`
Rational quadratic: \(h = 1 - (s/\rho)^{2}/[1+(s/\rho)^{2}]\)	`corRatio`
Spherical: \(h = [1-1.5(s/\rho)+0.5(s/\rho)^{3}][s < \rho]\)	`corSpher`
Linear exponent AR(1): \(h = \rho^{d_{min} + \delta\frac{s - d_{min}}{d_{max} - d_{min}}}\), 1 if \(t_{1}=t_{2}\)	Simpson et al. (2010)

¹⁷ Essentially what two-way ANOVA assumes

The structures 3-7 use \(\rho\) as a scaling parameter, not as something restricted to be in \([0,1]\)

7.6 Checking Model Fit

Constant variance assumption: usual residual plots
Normality assumption: usual qq residual plots
Correlation pattern: Variogram
- Estimate correlations of all possible pairs of residuals at different time points
- Pool all estimates at same absolute difference in time \(s\)
- Variogram is a plot with \(y = 1 - \hat{h}(s, \rho)\) vs. \(s\) on the \(x\)-axis
- Superimpose the theoretical variogram assumed by the model

7.7 `R` Software

Nonlinear mixed effects model package of Pinheiro & Bates
For linear models, fitting functions are
- lme for mixed effects models
- gls for generalized least squares without random effects
For this version the rms package has Gls so that many features of rms can be used:
- anova: all partial Wald tests, test of linearity, pooled tests
- summary: effect estimates (differences in \(\hat{Y}\)) and confidence limits, can be plotted
- plot, ggplot, plotp: continuous effect plots
- nomogram: nomogram
- Function: generate R function code for fitted model
- latex: representation of fitted model

In addition, Gls has a bootstrap option (hence you do not use rms’s bootcov for Gls fits).
To get regular gls functions named anova (for likelihood ratio tests, AIC, etc.) or summary use anova.gls or summary.gls * nlme package has many graphics and fit-checking functions * Several functions will be demonstrated in the case study

7.8 Case Study

Consider the dataset in Table~6.9 of Davis[davis-repmeas, pp. 161-163] from a multi-center, randomized controlled trial of botulinum toxin type B (BotB) in patients with cervical dystonia from nine U.S. sites.

Randomized to placebo (\(N=36\)), 5000 units of BotB (\(N=36\)), 10,000 units of BotB (\(N=37\))
Response variable: total score on Toronto Western Spasmodic Torticollis Rating Scale (TWSTRS), measuring severity, pain, and disability of cervical dystonia (high scores mean more impairment)
TWSTRS measured at baseline (week 0) and weeks 2, 4, 8, 12, 16 after treatment began
Dataset cdystonia from web site

7.8.1 Graphical Exploration of Data

Code

require(rms)
require(data.table)
options(prType='html')    # for model print, summary, anova, validate
getHdata(cdystonia)
setDT(cdystonia)          # convert to data.table
cdystonia[, uid := paste(site, id)]   # unique subject ID

# Tabulate patterns of subjects' time points
g <- function(w) paste(sort(unique(w)), collapse=' ')
cdystonia[, table(tapply(week, uid, g))]


            0         0 2 4   0 2 4 12 16       0 2 4 8    0 2 4 8 12 
            1             1             3             1             1 
0 2 4 8 12 16    0 2 4 8 16   0 2 8 12 16   0 4 8 12 16      0 4 8 16 
           94             1             2             4             1

Code

# Plot raw data, superposing subjects
xl <- xlab('Week'); yl <- ylab('TWSTRS-total score')
ggplot(cdystonia, aes(x=week, y=twstrs, color=factor(id))) +
       geom_line() + xl + yl + facet_grid(treat ~ site) +
       guides(color=FALSE)

Figure 7.1: Time profiles for individual subjects, stratified by study site and dose

Code

# Show quartiles
g <- function(x) {
  k <- as.list(quantile(x, (1 : 3) / 4, na.rm=TRUE))
  names(k) <- .q(Q1, Q2, Q3)
  k
}
cdys <- cdystonia[, g(twstrs), by=.(treat, week)]
ggplot(cdys, aes(x=week, y=Q2)) + xl + yl + ylim(0, 70) +
  geom_line() + facet_wrap(~ treat, nrow=2) +
  geom_ribbon(aes(ymin=Q1, ymax=Q3), alpha=0.2)

Figure 7.2: Quartiles of `TWSTRS` stratified by dose

Code

# Show means with bootstrap nonparametric CLs
cdys <-  cdystonia[, as.list(smean.cl.boot(twstrs)),
                   by = list(treat, week)]
ggplot(cdys, aes(x=week, y=Mean)) + xl + yl + ylim(0, 70) +
  geom_line() + facet_wrap(~ treat, nrow=2) +
  geom_ribbon(aes(x=week, ymin=Lower, ymax=Upper), alpha=0.2)

Figure 7.3: Mean responses and nonparametric bootstrap 0.95 confidence limits for population means, stratified by dose

Model with \(Y_{i0}\) as Baseline Covariate

Code

baseline <- cdystonia[week == 0]
baseline[, week := NULL]
setnames(baseline, 'twstrs', 'twstrs0')
followup <- cdystonia[week > 0, .(uid, week, twstrs)]
setkey(baseline, uid)
setkey(followup, uid, week)
both     <- Merge(baseline, followup, id = ~ uid)

         Vars Obs Unique IDs IDs in #1 IDs not in #1
baseline    7 109        109        NA            NA
followup    3 522        108       108             0
Merged      9 523        109       109             0

Number of unique IDs in any data frame : 109 
Number of unique IDs in all data frames: 108

Code

# Remove person with no follow-up record
both     <- both[! is.na(week)]
dd       <- datadist(both)
options(datadist='dd')

7.8.2 Using Generalized Least Squares

We stay with baseline adjustment and use a variety of correlation structures, with constant variance. Time is modeled as a restricted cubic spline with 3 knots, because there are only 3 unique interior values of week.

Code

require(nlme)
cp <- list(corCAR1,corExp,corCompSymm,corLin,corGaus,corSpher)
z  <- vector('list',length(cp))
for(k in 1:length(cp)) {
  z[[k]] <- gls(twstrs ~ treat * rcs(week, 3) +
                rcs(twstrs0, 3) + rcs(age, 4) * sex, data=both,
                correlation=cp[[k]](form = ~week | uid))
}
anova(z[[1]],z[[2]],z[[3]],z[[4]],z[[5]],z[[6]])

       Model df      AIC      BIC    logLik
z[[1]]     1 20 3553.906 3638.357 -1756.953
z[[2]]     2 20 3553.906 3638.357 -1756.953
z[[3]]     3 20 3587.974 3672.426 -1773.987
z[[4]]     4 20 3575.079 3659.531 -1767.540
z[[5]]     5 20 3621.081 3705.532 -1790.540
z[[6]]     6 20 3570.958 3655.409 -1765.479

AIC computed above is set up so that smaller values are best. From this the continuous-time AR1 and exponential structures are tied for the best. For the remainder of the analysis use corCAR1, using Gls.

Keselman et al. (1998) did a simulation study to study the reliability of AIC for selecting the correct covariance structure in repeated measurement models. In choosing from among 11 structures, AIC selected the correct structure 47% of the time. Gurka et al. (2011) demonstrated that fixed effects in a mixed effects model can be biased, independent of sample size, when the specified covariate matrix is more restricted than the true one.

Code

a <- Gls(twstrs ~ treat * rcs(week, 3) + rcs(twstrs0, 3) +
         rcs(age, 4) * sex, data=both,
         correlation=corCAR1(form=~week | uid))
a

Generalized Least Squares Fit by REML

Gls(model = twstrs ~ treat * rcs(week, 3) + rcs(twstrs0, 3) + 
    rcs(age, 4) * sex, data = both, correlation = corCAR1(form = ~week | 
    uid))


Obs 522	Log-restricted-likelihood -1756.95
Clusters 108	Model d.f. 17
g 11.334	σ 8.5917
	d.f. 504

	β	S.E.	t	Pr(>\|t\|)
Intercept	-0.3093	11.8804	-0.03	0.9792
treat=5000U	0.4344	2.5962	0.17	0.8672
treat=Placebo	7.1433	2.6133	2.73	0.0065
week	0.2879	0.2973	0.97	0.3334
week'	0.7313	0.3078	2.38	0.0179
twstrs0	0.8071	0.1449	5.57	<0.0001
twstrs0'	0.2129	0.1795	1.19	0.2360
age	-0.1178	0.2346	-0.50	0.6158
age'	0.6968	0.6484	1.07	0.2830
age''	-3.4018	2.5599	-1.33	0.1845
sex=M	24.2802	18.6208	1.30	0.1929
treat=5000U × week	0.0745	0.4221	0.18	0.8599
treat=Placebo × week	-0.1256	0.4243	-0.30	0.7674
treat=5000U × week'	-0.4389	0.4363	-1.01	0.3149
treat=Placebo × week'	-0.6459	0.4381	-1.47	0.1411
age × sex=M	-0.5846	0.4447	-1.31	0.1892
age' × sex=M	1.4652	1.2388	1.18	0.2375
age'' × sex=M	-4.0338	4.8123	-0.84	0.4023

Correlation Structure: Continuous AR(1)
 Formula: ~week | uid 
 Parameter estimate(s):
      Phi 
0.8666689

\(\hat{\rho} = 0.8672\), the estimate of the correlation between two measurements taken one week apart on the same subject. The estimated correlation for measurements 10 weeks apart is \(0.8672^{10} = 0.24\).

Code

v <- Variogram(a, form=~ week | uid)
plot(v)

Figure 7.4: Variogram, with assumed correlation pattern superimposed

Check constant variance and normality assumptions:

Code

both$resid <- r <- resid(a); both$fitted <- fitted(a)
yl <- ylab('Residuals')
p1 <- ggplot(both, aes(x=fitted, y=resid)) + geom_point() +
      facet_grid(~ treat) + yl
p2 <- ggplot(both, aes(x=twstrs0, y=resid)) + geom_point()+yl
p3 <- ggplot(both, aes(x=week, y=resid)) + yl + ylim(-20,20) +
      stat_summary(fun.data="mean_sdl", geom='smooth')
p4 <- ggplot(both, aes(sample=resid)) + stat_qq() +
      geom_abline(intercept=mean(r), slope=sd(r)) + yl
gridExtra::grid.arrange(p1, p2, p3, p4, ncol=2)

Figure 7.5: Three residual plots to check for absence of trends in central tendency and in variability. Upper right panel shows the baseline score on the \(x\)-axis. Bottom left panel shows the mean \(\pm 2\times\) SD. Bottom right panel is the QQ plot for checking normality of residuals from the GLS fit.

Now get hypothesis tests, estimates, and graphically interpret the model.

Code

anova(a)

Wald Statistics for `twstrs`
	χ²	d.f.	P
treat (Factor+Higher Order Factors)	22.11	6	0.0012
All Interactions	14.94	4	0.0048
week (Factor+Higher Order Factors)	77.27	6	<0.0001
All Interactions	14.94	4	0.0048
Nonlinear (Factor+Higher Order Factors)	6.61	3	0.0852
twstrs0	233.83	2	<0.0001
Nonlinear	1.41	1	0.2354
age (Factor+Higher Order Factors)	9.68	6	0.1388
All Interactions	4.86	3	0.1826
Nonlinear (Factor+Higher Order Factors)	7.59	4	0.1077
sex (Factor+Higher Order Factors)	5.67	4	0.2252
All Interactions	4.86	3	0.1826
treat × week (Factor+Higher Order Factors)	14.94	4	0.0048
Nonlinear	2.27	2	0.3208
Nonlinear Interaction : f(A,B) vs. AB	2.27	2	0.3208
age × sex (Factor+Higher Order Factors)	4.86	3	0.1826
Nonlinear	3.76	2	0.1526
Nonlinear Interaction : f(A,B) vs. AB	3.76	2	0.1526
TOTAL NONLINEAR	15.03	8	0.0586
TOTAL INTERACTION	19.75	7	0.0061
TOTAL NONLINEAR + INTERACTION	28.54	11	0.0027
TOTAL	322.98	17	<0.0001

Code

plot(anova(a))

Figure 7.6: Results of `anova.rms` from generalized least squares fit with continuous time AR1 correlation structure

Code

ylm <- ylim(25, 60)
p1 <- ggplot(Predict(a, week, treat, conf.int=FALSE),
             adj.subtitle=FALSE, legend.position='top') + ylm
p2 <- ggplot(Predict(a, twstrs0), adj.subtitle=FALSE) + ylm
p3 <- ggplot(Predict(a, age, sex), adj.subtitle=FALSE,
             legend.position='top') + ylm
gridExtra::grid.arrange(p1, p2, p3, ncol=2)

Figure 7.7: Estimated effects of time, baseline `TWSTRS`, age, and sex

Code

summary(a)  # Shows for week 8

Effects Response: `twstrs`
	Low	High	Δ	Effect	S.E.	Lower 0.95	Upper 0.95
week	4	12	8	6.6910	1.1060	4.524	8.858
twstrs0	39	53	14	13.5500	0.8862	11.810	15.290
age	46	65	19	2.5030	2.0510	-1.518	6.523
treat --- 5000U:10000U	1	2		0.5917	1.9980	-3.325	4.508
treat --- Placebo:10000U	1	3		5.4930	2.0040	1.565	9.421
sex --- M:F	1	2		-1.0850	1.7790	-4.571	2.401

Code

# To get results for week 8 for a different reference group
# for treatment, use e.g. summary(a, week=4, treat='Placebo')

# Compare low dose with placebo, separately at each time
k1 <- contrast(a, list(week=c(2,4,8,12,16), treat='5000U'),
                  list(week=c(2,4,8,12,16), treat='Placebo'))
options(width=80)
print(k1, digits=3)

    week twstrs0 age sex Contrast S.E.  Lower  Upper     Z Pr(>|z|)
1      2      46  56   F    -6.31 2.10 -10.43 -2.186 -3.00   0.0027
2      4      46  56   F    -5.91 1.82  -9.47 -2.349 -3.25   0.0011
3      8      46  56   F    -4.90 2.01  -8.85 -0.953 -2.43   0.0150
4*    12      46  56   F    -3.07 1.75  -6.49  0.361 -1.75   0.0795
5*    16      46  56   F    -1.02 2.10  -5.14  3.092 -0.49   0.6260

Redundant contrasts are denoted by *

Confidence intervals are 0.95 individual intervals

Code

# Compare high dose with placebo
k2 <- contrast(a, list(week=c(2,4,8,12,16), treat='10000U'),
                  list(week=c(2,4,8,12,16), treat='Placebo'))
print(k2, digits=3)

    week twstrs0 age sex Contrast S.E.  Lower Upper     Z Pr(>|z|)
1      2      46  56   F    -6.89 2.07 -10.96 -2.83 -3.32   0.0009
2      4      46  56   F    -6.64 1.79 -10.15 -3.13 -3.70   0.0002
3      8      46  56   F    -5.49 2.00  -9.42 -1.56 -2.74   0.0061
4*    12      46  56   F    -1.76 1.74  -5.17  1.65 -1.01   0.3109
5*    16      46  56   F     2.62 2.09  -1.47  6.71  1.25   0.2099

Redundant contrasts are denoted by *

Confidence intervals are 0.95 individual intervals

Code

k1 <- as.data.frame(k1[c('week', 'Contrast', 'Lower', 'Upper')])
p1 <- ggplot(k1, aes(x=week, y=Contrast)) + geom_point() +
      geom_line() + ylab('Low Dose - Placebo') +
      geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0)
k2 <- as.data.frame(k2[c('week', 'Contrast', 'Lower', 'Upper')])
p2 <- ggplot(k2, aes(x=week, y=Contrast)) + geom_point() +
      geom_line() + ylab('High Dose - Placebo') +
      geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0)
gridExtra::grid.arrange(p1, p2, ncol=2)

Figure 7.8: Contrasts and 0.95 confidence limits from GLS fit

Although multiple d.f. tests such as total treatment effects or treatment \(\times\) time interaction tests are comprehensive, their increased degrees of freedom can dilute power. In a treatment comparison, treatment contrasts at the last time point (single d.f. tests) are often of major interest. Such contrasts are informed by all the measurements made by all subjects (up until dropout times) when a smooth time trend is assumed.

Code

n <- nomogram(a, age=c(seq(20, 80, by=10), 85))
plot(n, cex.axis=.55, cex.var=.8, lmgp=.25)  # Figure (*\ref{fig:longit-nomogram}*)

Figure 7.9: Nomogram from GLS fit. Second axis is the baseline score.

7.8.3 Bayesian Proportional Odds Random Effects Model

Develop a \(y\)-transformation invariant longitudinal model
Proportional odds model with no grouping of TWSTRS scores
Bayesian random effects model
Random effects Gaussian with exponential prior distribution for its SD, with mean 1.0
Compound symmetry correlation structure
Demonstrates a large amount of patient-to-patient intercept variability

Code

require(rmsb)
options(mc.cores=parallel::detectCores() - 1, rmsb.backend='cmdstan')
bpo <- blrm(twstrs ~ treat * rcs(week, 3) + rcs(twstrs0, 3) +
            rcs(age, 4) * sex + cluster(uid), data=both, file='bpo.rds')
# file= means that after the first time the model is run, it will not
# be re-run unless the data, fitting options, or underlying Stan code change
stanDx(bpo)

Iterations: 2000 on each of 4 chains, with 4000 posterior distribution samples saved

For each parameter, n_eff is a crude measure of effective sample size
and Rhat is the potential scale reduction factor on split chains
(at convergence, Rhat=1)

Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

EBFMI: 0.844 0.783 0.745 0.745 

   Parameter  Rhat ESS bulk ESS tail
1   alpha[1] 1.003     2046     2023
2   alpha[2] 1.003     1529     2315
3   alpha[3] 1.005     1052     1983
4   alpha[4] 1.006      952     1882
5   alpha[5] 1.007      938     1893
6   alpha[6] 1.008      805     1789
7   alpha[7] 1.008      780     1708
8   alpha[8] 1.009      733     1563
9   alpha[9] 1.009      665     1293
10 alpha[10] 1.009      636     1296
11 alpha[11] 1.011      612     1413
12 alpha[12] 1.010      591     1211
13 alpha[13] 1.012      566     1111
14 alpha[14] 1.012      537     1048
15 alpha[15] 1.013      531     1049
16 alpha[16] 1.011      515     1099
17 alpha[17] 1.011      503      976
18 alpha[18] 1.010      489     1023
19 alpha[19] 1.010      478      975
20 alpha[20] 1.010      474     1067
21 alpha[21] 1.010      471      942
22 alpha[22] 1.010      464      906
23 alpha[23] 1.010      460      834
24 alpha[24] 1.011      457      859
25 alpha[25] 1.011      460      861
26 alpha[26] 1.010      459      912
27 alpha[27] 1.010      470      919
28 alpha[28] 1.010      475      949
29 alpha[29] 1.010      476      953
30 alpha[30] 1.010      465      794
31 alpha[31] 1.011      466      845
32 alpha[32] 1.011      473     1012
33 alpha[33] 1.010      475      826
34 alpha[34] 1.009      498      956
35 alpha[35] 1.008      500      965
36 alpha[36] 1.007      512     1081
37 alpha[37] 1.007      538     1281
38 alpha[38] 1.007      548     1323
39 alpha[39] 1.005      590     1176
40 alpha[40] 1.005      608     1300
41 alpha[41] 1.006      625     1251
42 alpha[42] 1.004      693     1447
43 alpha[43] 1.003      763     1548
44 alpha[44] 1.003      783     1716
45 alpha[45] 1.002      841     1812
46 alpha[46] 1.002      931     1733
47 alpha[47] 1.001      990     2263
48 alpha[48] 1.001     1013     2117
49 alpha[49] 1.001     1070     2086
50 alpha[50] 1.001     1121     2087
51 alpha[51] 1.001     1217     1672
52 alpha[52] 1.001     1268     1753
53 alpha[53] 1.001     1413     2342
54 alpha[54] 1.000     1437     2317
55 alpha[55] 1.000     1543     2719
56 alpha[56] 1.000     1583     2874
57 alpha[57] 1.001     1588     2611
58 alpha[58] 1.001     1722     2499
59 alpha[59] 1.001     1805     2782
60 alpha[60] 1.001     2006     3023
61 alpha[61] 1.000     2275     3210
62   beta[1] 1.004      757     1387
63   beta[2] 1.006      762     1405
64   beta[3] 1.001     2022     2740
65   beta[4] 1.001     3963     3075
66   beta[5] 1.003      826     1558
67   beta[6] 1.003      773     1543
68   beta[7] 1.006      916     1326
69   beta[8] 1.003      849     1238
70   beta[9] 1.001      879     1191
71  beta[10] 1.004      765     1320
72  beta[11] 1.001     3836     2899
73  beta[12] 1.001     2896     2707
74  beta[13] 1.000     4148     3222
75  beta[14] 1.002     3909     2921
76  beta[15] 1.008      821     1351
77  beta[16] 1.003      780     1463
78  beta[17] 1.005      889     1447
79 sigmag[1] 1.003      773     1636

Code

print(bpo, intercepts=TRUE)

Bayesian Proportional Odds Ordinal Logistic Model

Dirichlet Priors With Concentration Parameter 0.044 for Intercepts

blrm(formula = twstrs ~ treat * rcs(week, 3) + rcs(twstrs0, 3) + 
    rcs(age, 4) * sex + cluster(uid), data = both, file = "bpo.rds")

	Mixed Calibration/ Discrimination Indexes	Discrimination Indexes	Rank Discrim. Indexes
Obs 522	LOO log L -1747.2±23.67	g 3.795 [3.292, 4.281]	C 0.793 [0.785, 0.799]
Draws 4000	LOO IC 3494.4±47.33	g_p 0.434 [0.418, 0.447]	D_xy 0.585 [0.569, 0.599]
Chains 4	Effective p 178.76±7.92	EV 0.589 [0.539, 0.632]
Time 9s	B 0.149 [0.139, 0.16]	v 11.211 [8.387, 14.176]
p 17		vp 0.147 [0.135, 0.158]
Cluster on `uid`
Clusters 108
σ_γ 1.871 [1.5275, 2.2372]

	Mean β	Median β	S.E.	Lower	Upper	Pr(β>0)	Symmetry
y≥7	-1.8925	-1.9348	4.3060	-10.4593	6.4242	0.3235	1.05
y≥9	-2.9245	-2.8859	4.1909	-11.6146	4.6908	0.2448	1.02
y≥10	-4.0987	-4.1024	4.1327	-12.1431	3.8700	0.1608	1.04
y≥11	-4.5443	-4.5475	4.1279	-12.2445	3.7380	0.1320	1.03
y≥13	-4.7415	-4.7522	4.1280	-12.7626	3.1650	0.1235	1.02
y≥14	-5.0926	-5.1122	4.1261	-13.3171	2.5673	0.1045	1.03
y≥15	-5.3994	-5.4131	4.1229	-13.2727	2.6133	0.0912	1.04
y≥16	-5.7832	-5.8074	4.1193	-13.6829	2.2175	0.0808	1.04
y≥17	-6.5839	-6.5902	4.1031	-14.3847	1.2904	0.0538	1.01
y≥18	-6.8380	-6.8208	4.0984	-14.6403	1.0348	0.0490	1.01
y≥19	-7.1251	-7.1242	4.0945	-14.8641	0.7626	0.0418	1.02
y≥20	-7.3212	-7.3342	4.0950	-15.0929	0.5257	0.0372	1.01
y≥21	-7.5016	-7.5194	4.0947	-15.2810	0.2955	0.0312	1.00
y≥22	-7.9177	-7.9096	4.0906	-15.7228	-0.0901	0.0243	1.01
y≥23	-8.1852	-8.1709	4.0900	-16.0970	-0.3846	0.0215	1.00
y≥24	-8.4661	-8.4382	4.0902	-16.2124	-0.5187	0.0192	1.00
y≥25	-8.7240	-8.6958	4.0889	-16.3427	-0.6265	0.0170	1.00
y≥26	-9.1195	-9.1033	4.0900	-16.8669	-1.1156	0.0140	1.00
y≥27	-9.4088	-9.3790	4.0889	-17.1584	-1.3981	0.0115	1.01
y≥28	-9.6505	-9.6273	4.0899	-17.2485	-1.4914	0.0102	1.00
y≥29	-9.8850	-9.8647	4.0916	-17.5335	-1.8057	0.0088	1.00
y≥30	-10.1875	-10.1761	4.0931	-17.8358	-2.1341	0.0075	1.00
y≥31	-10.4768	-10.4712	4.0948	-18.1603	-2.4314	0.0073	1.00
y≥32	-10.5931	-10.5991	4.0957	-18.2305	-2.5091	0.0068	1.01
y≥33	-10.9555	-10.9465	4.0997	-18.6866	-2.8897	0.0058	1.01
y≥34	-11.2670	-11.2598	4.1016	-18.9527	-3.1342	0.0032	1.01
y≥35	-11.4901	-11.4901	4.1008	-19.2831	-3.4359	0.0027	1.00
y≥36	-11.7354	-11.7072	4.1001	-19.4760	-3.6375	0.0018	1.00
y≥37	-12.0134	-12.0031	4.1028	-19.7269	-3.9472	0.0015	0.99
y≥38	-12.2403	-12.2209	4.1053	-20.1085	-4.3351	0.0015	0.99
y≥39	-12.4861	-12.4689	4.1066	-20.3765	-4.5740	0.0015	1.00
y≥40	-12.6687	-12.6621	4.1076	-20.5231	-4.7360	0.0013	1.00
y≥41	-12.8521	-12.8450	4.1093	-20.6431	-4.8363	0.0013	1.00
y≥42	-13.1765	-13.1852	4.1114	-20.9928	-5.1592	0.0013	1.00
y≥43	-13.4033	-13.4016	4.1115	-21.1990	-5.3680	0.0013	1.00
y≥44	-13.7428	-13.7425	4.1131	-21.5027	-5.6825	0.0005	0.99
y≥45	-14.0603	-14.0603	4.1127	-21.7960	-5.8977	0.0005	1.00
y≥46	-14.3587	-14.3558	4.1139	-22.1307	-6.2589	0.0005	1.00
y≥47	-14.7783	-14.7739	4.1157	-22.6281	-6.7746	0.0005	0.99
y≥48	-15.0661	-15.0521	4.1200	-22.8265	-6.9166	0.0005	0.98
y≥49	-15.4378	-15.4149	4.1237	-23.2339	-7.2999	0.0003	0.98
y≥50	-15.7575	-15.7567	4.1266	-23.6138	-7.6148	0.0003	0.99
y≥51	-16.2863	-16.2865	4.1275	-24.1827	-8.2012	0.0003	1.00
y≥52	-16.6484	-16.6180	4.1256	-24.5473	-8.5991	0.0003	0.98
y≥53	-17.0940	-17.0594	4.1287	-25.1299	-9.1439	0.0000	0.97
y≥54	-17.5950	-17.5497	4.1311	-25.6404	-9.6396	0.0000	0.97
y≥55	-18.0042	-17.9506	4.1338	-26.0081	-10.0171	0.0000	0.98
y≥56	-18.2570	-18.2166	4.1333	-26.2906	-10.2969	0.0000	0.98
y≥57	-18.7246	-18.6750	4.1337	-26.8089	-10.8036	0.0000	0.99
y≥58	-19.2848	-19.2290	4.1365	-27.3114	-11.2621	0.0000	1.00
y≥59	-19.6375	-19.5835	4.1387	-27.7966	-11.7426	0.0000	0.99
y≥60	-19.9638	-19.9158	4.1408	-27.9487	-11.9096	0.0000	1.00
y≥61	-20.6541	-20.6027	4.1412	-28.6340	-12.7014	0.0000	0.98
y≥62	-21.0249	-20.9845	4.1434	-29.0384	-13.0626	0.0000	0.98
y≥63	-21.4276	-21.3862	4.1536	-29.2607	-13.1148	0.0000	0.98
y≥64	-21.5672	-21.5031	4.1545	-29.2026	-13.1253	0.0000	0.97
y≥65	-22.2878	-22.2548	4.1519	-29.8979	-13.6954	0.0000	0.97
y≥66	-22.6661	-22.6505	4.1554	-30.7065	-14.3955	0.0000	0.98
y≥67	-23.0760	-23.0377	4.1576	-31.5829	-15.2965	0.0000	0.97
y≥68	-23.8575	-23.8083	4.1745	-32.1561	-15.9086	0.0000	0.98
y≥71	-24.7235	-24.6678	4.1948	-32.9053	-16.6747	0.0000	0.98
treat=5000U	0.1185	0.1304	0.7254	-1.2585	1.6167	0.5702	1.05
treat=Placebo	2.3524	2.3516	0.7319	0.9497	3.8002	0.9998	1.00
week	0.1218	0.1209	0.0789	-0.0335	0.2749	0.9380	1.01
week'	0.1907	0.1902	0.0856	0.0219	0.3588	0.9862	1.00
twstrs0	0.2273	0.2275	0.0516	0.1245	0.3250	1.0000	1.01
twstrs0'	0.1285	0.1289	0.0632	0.0046	0.2492	0.9792	0.99
age	-0.0130	-0.0120	0.0795	-0.1729	0.1384	0.4335	0.93
age'	0.1921	0.1842	0.2197	-0.2576	0.6072	0.8128	1.08
age''	-1.0722	-1.0545	0.8609	-2.6773	0.6575	0.1020	0.96
sex=M	5.2646	5.2537	6.0906	-7.0597	16.7183	0.8082	1.01
treat=5000U × week	0.0504	0.0500	0.1102	-0.1673	0.2585	0.6775	0.98
treat=Placebo × week	-0.0522	-0.0528	0.1107	-0.2707	0.1658	0.3200	0.98
treat=5000U × week'	-0.1609	-0.1603	0.1205	-0.3927	0.0767	0.0922	1.00
treat=Placebo × week'	-0.1410	-0.1400	0.1210	-0.3905	0.0852	0.1195	1.00
age × sex=M	-0.1157	-0.1170	0.1457	-0.4006	0.1681	0.2120	0.99
age' × sex=M	0.1596	0.1619	0.4079	-0.6326	0.9422	0.6587	0.99
age'' × sex=M	0.0197	-0.0055	1.5849	-3.1326	2.9931	0.4993	1.05

Code

a <- anova(bpo)
a

Relative Explained Variation for `twstrs`. Approximate total model Wald χ² used in denominators of REV:262.1 [212.5, 337].
	REV	Lower	Upper	d.f.
treat (Factor+Higher Order Factors)	0.121	0.069	0.223	6
All Interactions	0.083	0.030	0.166	4
week (Factor+Higher Order Factors)	0.549	0.422	0.652	6
All Interactions	0.083	0.030	0.166	4
Nonlinear (Factor+Higher Order Factors)	0.021	0.000	0.070	3
twstrs0	0.659	0.524	0.736	2
Nonlinear	0.016	0.000	0.051	1
age (Factor+Higher Order Factors)	0.024	0.012	0.085	6
All Interactions	0.014	0.001	0.056	3
Nonlinear (Factor+Higher Order Factors)	0.021	0.003	0.067	4
sex (Factor+Higher Order Factors)	0.018	0.003	0.066	4
All Interactions	0.014	0.001	0.056	3
treat × week (Factor+Higher Order Factors)	0.083	0.030	0.166	4
Nonlinear	0.008	0.000	0.045	2
Nonlinear Interaction : f(A,B) vs. AB	0.008	0.000	0.045	2
age × sex (Factor+Higher Order Factors)	0.014	0.001	0.056	3
Nonlinear	0.014	0.000	0.051	2
Nonlinear Interaction : f(A,B) vs. AB	0.014	0.000	0.051	2
TOTAL NONLINEAR	0.053	0.022	0.127	8
TOTAL INTERACTION	0.096	0.059	0.201	7
TOTAL NONLINEAR + INTERACTION	0.128	0.093	0.238	11
TOTAL	1.000	1.000	1.000	17

Code

plot(a)

Show the final graphic (high dose:placebo contrast as function of time
Intervals are 0.95 highest posterior density intervals
\(y\)-axis: log-odds ratio

Code

wks <- c(2,4,8,12,16)
k <- contrast(bpo, list(week=wks, treat='10000U'),
                   list(week=wks, treat='Placebo'),
              cnames=paste('Week', wks))
k

           week   Contrast      S.E.     Lower      Upper Pr(Contrast>0)
1  Week 2     2 -2.2478977 0.5998165 -3.439365 -1.0775322         0.0000
2  Week 4     4 -2.1434302 0.5309202 -3.164919 -1.0687934         0.0000
3  Week 8     8 -1.7934509 0.5785167 -2.821921 -0.6136238         0.0018
4* Week 12   12 -0.8792943 0.5252163 -1.937805  0.1055448         0.0423
5* Week 16   16  0.1759067 0.6120968 -1.030554  1.3625489         0.6212

Redundant contrasts are denoted by *

Intervals are 0.95 highest posterior density intervals
Contrast is the posterior mean

Code

plot(k)

Code

k <- as.data.frame(k[c('week', 'Contrast', 'Lower', 'Upper')])
ggplot(k, aes(x=week, y=Contrast)) + geom_point() +
  geom_line() + ylab('High Dose - Placebo') +
  geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0)

For each posterior draw compute the difference in means and get an exact (to within simulation error) 0.95 highest posterior density intervals for these differences.

Code

M <- Mean(bpo)   # create R function that computes mean Y from X*beta
k <- contrast(bpo, list(week=wks, treat='10000U'),
                   list(week=wks, treat='Placebo'),
              fun=M, cnames=paste('Week', wks))
plot(k, which='diff') + theme(legend.position='bottom')

Code

f <- function(x) {
  hpd <- HPDint(x, prob=0.95)   # is in rmsb
  r <- c(mean(x), median(x), hpd)
  names(r) <- c('Mean', 'Median', 'Lower', 'Upper')
  r
}
w    <- as.data.frame(t(apply(k$esta - k$estb, 2, f)))
week <- as.numeric(sub('Week ', '', rownames(w)))
ggplot(w, aes(x=week, y=Mean)) + geom_point() +
  geom_line() + ylab('High Dose - Placebo') +
  geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0) +
  scale_y_continuous(breaks=c(-8, -4, 0, 4))

7.8.4 Bayesian Markov Semiparametric Model

First-order Markov model
Serial correlation induced by Markov model is similar to AR(1) which we already know fits these data
Markov model is more likely to fit the data than the random effects model, which induces a compound symmetry correlation structure
Models state transitions
PO model at each visit, with Y from previous visit conditioned upon just like any covariate
Need to uncondition (marginalize) on previous Y to get the time-response profile we usually need
Semiparametric model is especially attractive because one can easily “uncondition” a discrete Y model, and the distribution of Y for control subjects can be any shape
Let measurement times be \(t_{1}, t_{2}, \dots, t_{m}\), and the measurement for a subject at time \(t\) be denoted \(Y(t)\)
First-order Markov model:

\[\begin{array}{ccc} \Pr(Y(t_{i}) \geq y | X, Y(t_{i-1})) &=& \mathrm{expit}(\alpha_{y} + X\beta\\ &+& g(Y(t_{i-1}), t_{i}, t_{i} - t_{i-1})) \end{array}\]

\(g\) involves any number of regression coefficients for a main effect of \(t\), the main effect of time gap \(t_{i} - t_{i-1}\) if this is not collinear with absolute time, a main effect of the previous state, and interactions between these
Examples of how the previous state may be modeled in \(g\):
- linear in numeric codes for \(Y\)
- spline function in same
- discontinuous bi-linear relationship where there is a slope for in-hospital outcome severity, a separate slope for outpatient outcome severity, and an intercept jump at the transition from inpatient to outpatient (or vice versa)
Markov model is quite flexible in handling time trends and serial correlation patterns
Can allow for irregular measurement times:
hbiostat.org/stat/irreg.html

Fit the model and run standard Stan diagnostics.

Code

# Create a new variable to hold previous value of Y for the subject
# For week 2, previous value is the baseline value
setDT(both, key=c('uid', 'week'))
both[, ptwstrs := shift(twstrs), by=uid]
both[week == 2, ptwstrs := twstrs0]
dd <- datadist(both)
bmark <- blrm(twstrs ~  treat * rcs(week, 3) + rcs(ptwstrs, 4) +
                        rcs(age, 4) * sex,
              data=both, file='bmark.rds')
# When adding partial PO terms for week and ptwstrs, z=-1.8, 5.04
stanDx(bmark)

Iterations: 2000 on each of 4 chains, with 4000 posterior distribution samples saved

For each parameter, n_eff is a crude measure of effective sample size
and Rhat is the potential scale reduction factor on split chains
(at convergence, Rhat=1)

Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

EBFMI: 0.904 0.936 1.024 1.066 

   Parameter  Rhat ESS bulk ESS tail
1   alpha[1] 1.002     4210     2266
2   alpha[2] 1.000     5448     2648
3   alpha[3] 1.001     5708     3246
4   alpha[4] 1.002     5310     3542
5   alpha[5] 1.002     5412     3452
6   alpha[6] 1.000     4983     3646
7   alpha[7] 1.000     4537     3728
8   alpha[8] 1.000     4396     3614
9   alpha[9] 1.000     4659     3673
10 alpha[10] 1.001     4294     3564
11 alpha[11] 1.000     4062     3361
12 alpha[12] 1.001     3907     3195
13 alpha[13] 1.000     3974     3305
14 alpha[14] 1.000     4083     3446
15 alpha[15] 1.000     4109     3146
16 alpha[16] 1.000     4009     3361
17 alpha[17] 1.000     4082     3272
18 alpha[18] 1.000     3929     3555
19 alpha[19] 1.000     3912     3281
20 alpha[20] 1.001     3940     3649
21 alpha[21] 1.000     3899     3651
22 alpha[22] 1.000     3811     3457
23 alpha[23] 1.000     4187     3550
24 alpha[24] 1.000     4191     3536
25 alpha[25] 1.000     4334     2950
26 alpha[26] 1.000     4628     2935
27 alpha[27] 1.000     4786     2630
28 alpha[28] 1.000     5216     2370
29 alpha[29] 1.001     5496     2361
30 alpha[30] 1.001     5947     2366
31 alpha[31] 1.000     6171     2883
32 alpha[32] 0.999     6520     2845
33 alpha[33] 0.999     7052     2853
34 alpha[34] 1.000     7180     2653
35 alpha[35] 1.002     7991     2756
36 alpha[36] 1.001     7469     2965
37 alpha[37] 1.000     7865     2831
38 alpha[38] 1.000     7703     3224
39 alpha[39] 1.000     7350     3338
40 alpha[40] 1.000     6548     3208
41 alpha[41] 1.000     5862     3407
42 alpha[42] 1.000     5915     3354
43 alpha[43] 1.000     5789     2915
44 alpha[44] 1.000     5532     3846
45 alpha[45] 1.001     5671     3483
46 alpha[46] 1.000     5546     3451
47 alpha[47] 1.001     5877     3531
48 alpha[48] 1.000     6120     3993
49 alpha[49] 1.000     5717     3726
50 alpha[50] 1.001     5591     3553
51 alpha[51] 1.000     5507     3760
52 alpha[52] 1.000     5689     3807
53 alpha[53] 1.001     5765     3798
54 alpha[54] 1.001     5980     3649
55 alpha[55] 1.000     5454     3704
56 alpha[56] 1.000     5654     3820
57 alpha[57] 1.001     5843     3575
58 alpha[58] 1.001     6060     3593
59 alpha[59] 1.000     6152     3853
60 alpha[60] 1.001     5753     3902
61 alpha[61] 1.001     5715     3199
62   beta[1] 1.001     8954     3007
63   beta[2] 1.000    10329     2842
64   beta[3] 1.000     6872     3509
65   beta[4] 1.000     8459     2891
66   beta[5] 1.000     3707     2781
67   beta[6] 1.001     6819     3336
68   beta[7] 1.001     8771     3188
69   beta[8] 1.001     9635     2451
70   beta[9] 1.001     9809     3234
71  beta[10] 1.006     9893     2584
72  beta[11] 1.002     8059     2929
73  beta[12] 1.000     8871     2859
74  beta[13] 1.001     7780     2871
75  beta[14] 1.002    11064     2953
76  beta[15] 1.000     8958     3145
77  beta[16] 1.002     9525     3036
78  beta[17] 1.003     9226     3005
79  beta[18] 1.001     9610     2941

Code

stanDxplot(bmark)

Note that posterior sampling is much more efficient without random effects.

Code

bmark

Bayesian Proportional Odds Ordinal Logistic Model

Dirichlet Priors With Concentration Parameter 0.044 for Intercepts

blrm(formula = twstrs ~ treat * rcs(week, 3) + rcs(ptwstrs, 4) + 
    rcs(age, 4) * sex, data = both, file = "bmark.rds")

Frequencies of Missing Values Due to Each Variable

 twstrs   treat    week ptwstrs     age     sex 
      0       0       0       5       0       0

	Mixed Calibration/ Discrimination Indexes	Discrimination Indexes	Rank Discrim. Indexes
Obs 517	LOO log L -1787.42±22.54	g 3.258 [3.029, 3.575]	C 0.828 [0.825, 0.83]
Draws 4000	LOO IC 3574.83±45.08	g_p 0.415 [0.403, 0.427]	D_xy 0.655 [0.65, 0.661]
Chains 4	Effective p 91.34±5.41	EV 0.531 [0.495, 0.566]
Time 4.7s	B 0.117 [0.113, 0.121]	v 8.358 [7.076, 9.88]
p 18		vp 0.133 [0.124, 0.141]

	Mode β	Mean β	Median β	S.E.	Lower	Upper	Pr(β>0)	Symmetry
treat=5000U	0.2210	0.2199	0.2249	0.5817	-0.9520	1.3083	0.6400	1.00
treat=Placebo	1.8297	1.8301	1.8332	0.5733	0.6659	2.8994	0.9998	1.01
week	0.4860	0.4856	0.4855	0.0849	0.3185	0.6467	1.0000	0.99
week'	-0.2876	-0.2859	-0.2858	0.0899	-0.4581	-0.1126	0.0008	0.98
ptwstrs	0.1996	0.2008	0.2007	0.0266	0.1473	0.2511	1.0000	0.97
ptwstrs'	-0.0621	-0.0650	-0.0643	0.0625	-0.1866	0.0594	0.1475	0.99
ptwstrs''	0.5323	0.5475	0.5453	0.2479	0.0566	1.0377	0.9848	1.02
age	-0.0295	-0.0289	-0.0290	0.0316	-0.0912	0.0346	0.1735	1.04
age'	0.1235	0.1213	0.1214	0.0881	-0.0626	0.2855	0.9115	1.01
age''	-0.5064	-0.4971	-0.4999	0.3474	-1.1322	0.2281	0.0813	1.00
sex=M	-0.4633	-0.4417	-0.4722	2.3794	-5.3183	3.7576	0.4280	1.02
treat=5000U × week	-0.0341	-0.0338	-0.0332	0.1139	-0.2564	0.1807	0.3868	0.97
treat=Placebo × week	-0.2717	-0.2714	-0.2712	0.1150	-0.4839	-0.0355	0.0098	1.01
treat=5000U × week'	-0.0339	-0.0345	-0.0357	0.1234	-0.2701	0.2009	0.3935	1.01
treat=Placebo × week'	0.1196	0.1186	0.1194	0.1244	-0.1358	0.3524	0.8288	1.01
age × sex=M	0.0112	0.0107	0.0108	0.0573	-0.0894	0.1291	0.5755	1.00
age' × sex=M	-0.0509	-0.0485	-0.0478	0.1625	-0.3634	0.2571	0.3850	0.99
age'' × sex=M	0.2614	0.2501	0.2486	0.6318	-0.9004	1.5136	0.6492	1.02

Code

a <- anova(bpo)
a

Relative Explained Variation for `twstrs`. Approximate total model Wald χ² used in denominators of REV:262.1 [215.2, 340.5].
	REV	Lower	Upper	d.f.
treat (Factor+Higher Order Factors)	0.121	0.064	0.214	6
All Interactions	0.083	0.036	0.160	4
week (Factor+Higher Order Factors)	0.549	0.424	0.656	6
All Interactions	0.083	0.036	0.160	4
Nonlinear (Factor+Higher Order Factors)	0.021	0.003	0.064	3
twstrs0	0.659	0.529	0.748	2
Nonlinear	0.016	0.000	0.048	1
age (Factor+Higher Order Factors)	0.024	0.007	0.088	6
All Interactions	0.014	0.001	0.057	3
Nonlinear (Factor+Higher Order Factors)	0.021	0.006	0.076	4
sex (Factor+Higher Order Factors)	0.018	0.005	0.072	4
All Interactions	0.014	0.001	0.057	3
treat × week (Factor+Higher Order Factors)	0.083	0.036	0.160	4
Nonlinear	0.008	0.000	0.039	2
Nonlinear Interaction : f(A,B) vs. AB	0.008	0.000	0.039	2
age × sex (Factor+Higher Order Factors)	0.014	0.001	0.057	3
Nonlinear	0.014	0.000	0.054	2
Nonlinear Interaction : f(A,B) vs. AB	0.014	0.000	0.054	2
TOTAL NONLINEAR	0.053	0.028	0.138	8
TOTAL INTERACTION	0.096	0.054	0.190	7
TOTAL NONLINEAR + INTERACTION	0.128	0.088	0.239	11
TOTAL	1.000	1.000	1.000	17

Code

plot(a)

Let’s add subject-level random effects to the model. Smallness of the standard deviation of the random effects provides support for the assumption of conditional independence that we like to make for Markov models and allows us to simplify the model by omitting random effects.

Code

bmarkre <- blrm(twstrs ~  treat * rcs(week, 3) + rcs(ptwstrs, 4) +
                          rcs(age, 4) * sex + cluster(uid),
                data=both, file='bmarkre.rds')
stanDx(bmarkre)

Iterations: 2000 on each of 4 chains, with 4000 posterior distribution samples saved

For each parameter, n_eff is a crude measure of effective sample size
and Rhat is the potential scale reduction factor on split chains
(at convergence, Rhat=1)

Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
2 of 4000 (0.05%) transitions ended with a divergence.
These divergent transitions indicate that HMC is not fully able to explore the posterior distribution.
Try increasing adapt delta closer to 1.
If this doesn't remove all divergences, try to reparameterize the model.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete.
Divergent samples: 1 0 0 1 

EBFMI: 1.042 0.887 0.937 0.989 

   Parameter  Rhat ESS bulk ESS tail
1   alpha[1] 1.002     3945     2235
2   alpha[2] 1.001     4008     2617
3   alpha[3] 1.001     3186     3143
4   alpha[4] 1.000     2813     2535
5   alpha[5] 1.001     2693     2591
6   alpha[6] 1.000     2618     2471
7   alpha[7] 1.000     2511     2560
8   alpha[8] 1.000     2360     2421
9   alpha[9] 1.000     2347     2700
10 alpha[10] 1.000     2294     2807
11 alpha[11] 1.000     2137     2154
12 alpha[12] 1.000     2196     2518
13 alpha[13] 1.000     2048     2404
14 alpha[14] 1.000     2055     2446
15 alpha[15] 1.000     2060     2622
16 alpha[16] 1.000     2066     2785
17 alpha[17] 1.000     2041     2487
18 alpha[18] 1.000     1936     2547
19 alpha[19] 1.000     1929     2202
20 alpha[20] 1.001     1978     2372
21 alpha[21] 1.000     1895     2345
22 alpha[22] 1.000     1802     1946
23 alpha[23] 1.000     1792     1769
24 alpha[24] 1.000     1834     2317
25 alpha[25] 1.000     1999     2355
26 alpha[26] 1.000     2201     2248
27 alpha[27] 1.000     2459     2700
28 alpha[28] 1.000     2509     2849
29 alpha[29] 1.000     2615     2714
30 alpha[30] 1.001     2738     2644
31 alpha[31] 1.000     3214     2496
32 alpha[32] 1.001     3315     2250
33 alpha[33] 1.001     3627     2668
34 alpha[34] 1.002     3842     2425
35 alpha[35] 1.001     4230     2976
36 alpha[36] 1.002     4757     2939
37 alpha[37] 1.003     4895     3213
38 alpha[38] 1.000     4842     3148
39 alpha[39] 1.001     4736     3356
40 alpha[40] 1.001     4539     3264
41 alpha[41] 1.002     4304     3365
42 alpha[42] 1.002     4238     3251
43 alpha[43] 1.001     3945     3324
44 alpha[44] 1.003     3744     3227
45 alpha[45] 1.002     3404     3017
46 alpha[46] 1.001     3436     3195
47 alpha[47] 1.002     3200     3122
48 alpha[48] 1.002     3174     3111
49 alpha[49] 1.003     3181     3028
50 alpha[50] 1.000     3382     2772
51 alpha[51] 1.001     3543     2832
52 alpha[52] 1.002     3512     2833
53 alpha[53] 1.001     3425     2904
54 alpha[54] 1.001     3415     2931
55 alpha[55] 1.001     3376     3152
56 alpha[56] 1.001     3427     3079
57 alpha[57] 1.001     3138     3054
58 alpha[58] 1.001     3315     3150
59 alpha[59] 1.002     2907     3059
60 alpha[60] 1.000     3553     3026
61 alpha[61] 1.001     4031     2978
62   beta[1] 1.001     5933     2651
63   beta[2] 1.000     4441     2850
64   beta[3] 1.002     4251     2765
65   beta[4] 1.001     5745     2538
66   beta[5] 1.001     1745     2440
67   beta[6] 1.000     3954     3004
68   beta[7] 1.001     4701     2865
69   beta[8] 1.001     5532     2605
70   beta[9] 1.004     6115     2597
71  beta[10] 1.001     5363     2381
72  beta[11] 1.002     5335     2899
73  beta[12] 1.000     6075     2504
74  beta[13] 1.002     5991     2838
75  beta[14] 1.001     5661     2772
76  beta[15] 1.004     6543     2720
77  beta[16] 1.004     5364     2611
78  beta[17] 1.001     5527     2517
79  beta[18] 1.001     6023     2615
80 sigmag[1] 1.003     1264     1702

Code

bmarkre

Bayesian Proportional Odds Ordinal Logistic Model

Dirichlet Priors With Concentration Parameter 0.044 for Intercepts

blrm(formula = twstrs ~ treat * rcs(week, 3) + rcs(ptwstrs, 4) + 
    rcs(age, 4) * sex + cluster(uid), data = both, file = "bmarkre.rds")

Frequencies of Missing Values Due to Each Variable

      twstrs        treat         week      ptwstrs          age          sex 
           0            0            0            5            0            0 
cluster(uid) 
           0

	Mixed Calibration/ Discrimination Indexes	Discrimination Indexes	Rank Discrim. Indexes
Obs 517	LOO log L -1786.26±22.28	g 3.258 [2.942, 3.537]	C 0.828 [0.825, 0.83]
Draws 4000	LOO IC 3572.52±44.55	g_p 0.415 [0.401, 0.427]	D_xy 0.655 [0.65, 0.66]
Chains 4	Effective p 93.09±4.62	EV 0.531 [0.492, 0.569]
Time 6.4s	B 0.117 [0.113, 0.121]	v 8.359 [6.825, 9.832]
p 18		vp 0.133 [0.123, 0.142]
Cluster on `uid`
Clusters 108
σ_γ 0.1165 [0, 0.3384]

	Mean β	Median β	S.E.	Lower	Upper	Pr(β>0)	Symmetry
treat=5000U	0.2155	0.2174	0.5864	-0.9299	1.3469	0.6518	0.99
treat=Placebo	1.8441	1.8421	0.5814	0.7542	3.0127	0.9992	1.02
week	0.4865	0.4873	0.0801	0.3304	0.6428	1.0000	0.99
week'	-0.2862	-0.2856	0.0836	-0.4493	-0.1219	0.0005	0.99
ptwstrs	0.1999	0.2003	0.0269	0.1482	0.2532	1.0000	0.99
ptwstrs'	-0.0649	-0.0637	0.0622	-0.1856	0.0537	0.1532	1.02
ptwstrs''	0.5474	0.5462	0.2479	0.0659	1.0186	0.9875	1.04
age	-0.0294	-0.0298	0.0316	-0.0908	0.0305	0.1767	0.98
age'	0.1240	0.1249	0.0882	-0.0487	0.2966	0.9202	1.00
age''	-0.5099	-0.5108	0.3477	-1.1647	0.2046	0.0722	1.01
sex=M	-0.4235	-0.4078	2.4502	-5.1794	4.2852	0.4328	0.97
treat=5000U × week	-0.0323	-0.0297	0.1120	-0.2718	0.1710	0.3918	0.99
treat=Placebo × week	-0.2731	-0.2715	0.1117	-0.5055	-0.0666	0.0070	0.97
treat=5000U × week'	-0.0360	-0.0372	0.1196	-0.2593	0.2073	0.3812	1.03
treat=Placebo × week'	0.1197	0.1187	0.1217	-0.1148	0.3585	0.8383	1.00
age × sex=M	0.0100	0.0104	0.0591	-0.1034	0.1242	0.5635	1.02
age' × sex=M	-0.0477	-0.0480	0.1682	-0.3958	0.2537	0.3938	0.98
age'' × sex=M	0.2522	0.2564	0.6546	-0.9502	1.5686	0.6450	0.99

The random effects SD is only 0.11 on the logit scale. Also, the standard deviations of all the regression parameter posterior distributions are virtually unchanged with the addition of random effects:

Code

plot(sqrt(diag(vcov(bmark))), sqrt(diag(vcov(bmarkre))),
     xlab='Posterior SDs in Conditional Independence Markov Model',
     ylab='Posterior SDs in Random Effects Markov Model')
abline(a=0, b=1, col=gray(0.85))

So we will use the model omitting random effects.

Show the partial effects of all the predictors, including the effect of the previous measurement of TWSTRS. Also compute high dose:placebo treatment contrasts on these conditional estimates.

Code

ggplot(Predict(bmark))

Code

ggplot(Predict(bmark, week, treat))

Code

k <- contrast(bmark, list(week=wks, treat='10000U'),
                     list(week=wks, treat='Placebo'),
              cnames=paste('Week', wks))
k

           week   Contrast      S.E.      Lower      Upper Pr(Contrast>0)
1  Week 2     2 -1.2872979 0.3788043 -2.0676854 -0.5851706         0.0003
2  Week 4     4 -0.7444594 0.2532338 -1.2155523 -0.2251157         0.0005
3  Week 8     8  0.2226112 0.3626920 -0.4921401  0.9308765         0.7312
4* Week 12   12  0.7152567 0.2657876  0.1879102  1.2379469         0.9958
5* Week 16   16  1.0892960 0.3934977  0.3237274  1.8605355         0.9970

Redundant contrasts are denoted by *

Intervals are 0.95 highest posterior density intervals
Contrast is the posterior mean

Code

plot(k)

Code

k <- as.data.frame(k[c('week', 'Contrast', 'Lower', 'Upper')])
ggplot(k, aes(x=week, y=Contrast)) + geom_point() +
  geom_line() + ylab('High Dose - Placebo') +
  geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0)

Using posterior means for parameter values, compute the probability that at a given week twstrs will be \(\geq 40\) when at the previous visit it was 40. Also show the conditional mean twstrs when it was 40 at the previous visit.

Code

ex <- ExProb(bmark)
ex40 <- function(lp, ...) ex(lp, y=40, ...)
ggplot(Predict(bmark, week, treat, ptwstrs=40, fun=ex40))

Code

ggplot(Predict(bmark, week, treat, ptwstrs=40, fun=Mean(bmark)))

Semiparametric models provide not only estimates of tendencies of Y but also estimate the whole distribution of Y
Estimate the entire conditional distribution of Y at week 12 for high-dose patients having TWSTRS=42 at week 8
Other covariates set to median/mode
Use posterior mean of all the cell probabilities
Also show pointwise 0.95 highest posterior density intervals
To roughly approximate simultaneous confidence bands make the pointwise limits sum to 1 like the posterior means do

Code

# Get median/mode for covariates including ptwstrs (TWSTRS in previous visit)
d <- gendata(bmark)
d

   treat week ptwstrs age sex
1 10000U    8      42  56   F

Code

d$week <- 12
p <- predict(bmark, d, type='fitted.ind')   # defaults to posterior means
yvals <- as.numeric(sub('twstrs=', '', p$y))
lo <- p$Lower / sum(p$Lower)
hi <- p$Upper / sum(p$Upper)
plot(yvals, p$Mean, type='l', xlab='TWSTRS', ylab='',
     ylim=range(c(lo, hi)))
lines(yvals, lo, col=gray(0.8))
lines(yvals, hi, col=gray(0.8))

Repeat this showing the variation over 5 posterior draws

Code

p <- predict(bmark, d, type='fitted.ind', posterior.summary='all')
cols <- adjustcolor(1 : 10, 0.7)
for(i in 1 : 5) {
  if(i == 1) plot(yvals, p[i, 1, ], type='l', col=cols[1], xlab='TWSTRS', ylab='')
  else lines(yvals, p[i, 1, ], col=cols[i])
}

Turn to marginalized (unconditional on previous twstrs) quantities
Capitalize on PO model being a multinomial model, just with PO restrictions
Manipulations of conditional probabilities to get the unconditional probability that twstrs=y doesn’t need to know about PO
Compute all cell probabilities and use the law of total probability recursively \[\Pr(Y_{t} = y | X) = \sum_{j=1}^{k} \Pr(Y_{t} = y | X, Y_{t-1} = j) \Pr(Y_{t-1} = j | X)\]
predict.blrm method with type='fitted.ind' computes the needed conditional cell probabilities, optionally for all posterior draws at once
Easy to get highest posterior density intervals for derived parameters such as unconditional probabilities or unconditional means
Hmisc package soprobMarkovOrdm function (in version 4.6) computes an array of all the state occupancy probabilities for all the posterior draws

Code

# Baseline twstrs to 42 in d
# For each dose, get all the posterior draws for all state occupancy
# probabilities for all visit
ylev <- sort(unique(both$twstrs))
tlev <- c('Placebo', '10000U')
R <- list()
for(trt in tlev) {   # separately by treatment
  d$treat <- trt
  u <- soprobMarkovOrdm(bmark, d, wks, ylev,
                        tvarname='week', pvarname='ptwstrs')
  R[[trt]] <- u
}
dim(R[[1]])    # posterior draws x times x distinct twstrs values

[1] 4000    5   62

Code

# For each posterior draw, treatment, and week compute the mean TWSTRS
# Then compute posterior mean of means, and HPD interval
Rmean <- Rmeans <- list()
for(trt in tlev) {
  r <- R[[trt]]
  # Mean Y at each week and posterior draw (mean from a discrete distribution)
  m <- apply(r, 1:2, function(x) sum(ylev * x))
  Rmeans[[trt]] <- m
  # Posterior mean and median and HPD interval over draws
  u <- apply(m, 2, f)   # f defined above
  u <- rbind(week=as.numeric(colnames(u)), u)
  Rmean[[trt]] <- u
}
r <- lapply(Rmean, function(x) as.data.frame(t(x)))
for(trt in tlev) r[[trt]]$treat <- trt
r <- do.call(rbind, r)
ggplot(r, aes(x=week, y=Mean, color=treat)) + geom_line() +
  geom_ribbon(aes(ymin=Lower, ymax=Upper), alpha=0.2, linetype=0)

Use the same posterior draws of unconditional probabilities of all values of TWSTRS to get the posterior distribution of differences in mean TWSTRS between high and low dose

Code

Dif <- Rmeans$`10000U` - Rmeans$Placebo
dif <- as.data.frame(t(apply(Dif, 2, f)))
dif$week <- as.numeric(rownames(dif))
ggplot(dif, aes(x=week, y=Mean)) + geom_line() +
  geom_ribbon(aes(ymin=Lower, ymax=Upper), alpha=0.2, linetype=0) +
  ylab('High Dose - Placebo TWSTRS')

Get posterior mean of all cell probabilities estimates at week 12
Distribution of TWSTRS conditional high dose, median age, mode sex
Not conditional on week 8 value

Code

p <- R$`10000U`[, '12', ]   # 4000 x 62
pmean <- apply(p, 2, mean)
yvals <- as.numeric(names(pmean))
plot(yvals, pmean, type='l', xlab='TWSTRS', ylab='')

7.9 Study Questions

Section 7.2

When should one model the time-response profile using discrete time?

Section 7.3

What makes generalized least squares and mixed effect models relatively robust to non-completely-random dropouts?
What does the last observation carried forward method always violate?

Section 7.4

Which correlation structure do you expect to fit the data when there are rapid repetitions over a short time span? When the follow-up time span is very long?

Section 7.8

What can go wrong if many correlation structures are tested in one dataset?
In a longitudinal intervention study, what is the most typical comparison of interest? Is it best to borrow information in estimating this contrast?

Davis, C. S. (2002). Statistical Methods for the Analysis of Repeated Measurements. Springer.

Diggle, P. J., Heagerty, P., Liang, K.-Y., & Zeger, S. L. (2002). Analysis of Longitudinal Data (second). Oxford University Press.

Donohue, M. C., Langford, O., Insel, P. S., van Dyck, C. H., Petersen, R. C., Craft, S., Sethuraman, G., Raman, R., Aisen, P. S., & Initiative, F. the A. D. N. (n.d.). Natural cubic splines for the analysis of Alzheimer’s clinical trials. Pharmaceutical Statistics, n/a(n/a). https://doi.org/10.1002/pst.2285

Gardiner, J. C., Luo, Z., & Roman, L. A. (2009). Fixed effects, random effects and GEE: What are the differences? Stat Med, 28, 221–239.

nice comparison of models; econometrics; different use of the term "fixed effects model"

Goldstein, H. (1989). Restricted unbiased iterative generalized least-squares estimation. Biometrika, 76(3), 622–623.

derivation of REML

Gurka, M. J., Edwards, L. J., & Muller, K. E. (2011). Avoiding bias in mixed model inference for fixed effects. Stat Med, 30(22), 2696–2707. https://doi.org/10.1002/sim.4293

Hand, D., & Crowder, M. (1996). Practical Longitudinal Data Analysis. Chapman & Hall.

Kenward, M. G., White, I. R., & Carpener, J. R. (2010). Should baseline be a covariate or dependent variable in analyses of change from baseline in clinical trials? (Letter to the editor). Stat Med, 29, 1455–1456.

sharp rebuke of liu09sho

Keselman, H. J., Algina, J., Kowalchuk, R. K., & Wolfinger, R. D. (1998). A comparison of two approaches for selecting covariance structures in the analysis of repeated measurements. Comm Stat - Sim Comp, 27, 591–604.

use of AIC and BIC for selecting the covariance structure in repeated measurements;serial data;longitudinal data;when chosing from 11 covariance patterns, AIC selected the correct structure 0.47 of the time; BIC was correct in 0.35

Liang, K.-Y., & Zeger, S. L. (2000). Longitudinal data analysis of continuous and discrete responses for pre-post designs. Sankhyā, 62, 134–148.

makes an error in assuming the baseline variable will have the same univariate distribution as the response except for a shift;baseline may have for example a truncated distribution based on a trial’s inclusion criteria;if correlation between baseline and response is zero, ANCOVA will be twice as efficient as simple analysis of change scores;if correlation is one they may be equally efficient

Lindsey, J. K. (1997). Models for Repeated Measurements. Clarendon Press.

Liu, G. F., Lu, K., Mogg, R., Mallick, M., & Mehrotra, D. V. (2009). Should baseline be a covariate or dependent variable in analyses of change from baseline in clinical trials? Stat Med, 28, 2509–2530.

seems to miss several important points, such as the fact that the baseline variable is often part of the inclusion/exclusion criteria and so has a truncated distribution that is different from that of the follow-up measurements;sharp rebuke in ken10sho

Peters, S. A., Bots, M. L., den Ruijter, H. M., Palmer, M. K., Grobbee, D. E., Crouse, J. R., O’Leary, D. H., Evans, G. W., Raichlen, J. S., Moons, K. G., Koffijberg, H., & METEOR study group. (2012). Multiple imputation of missing repeated outcome measurements did not add to linear mixed-effects models. J Clin Epi, 65(6), 686–695. https://doi.org/10.1016/j.jclinepi.2011.11.012

Pinheiro, J. C., & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer.

Potthoff, R. F., & Roy, S. N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika, 51, 313–326.

included an AR1 example

Senn, S. (2006). Change from baseline and analysis of covariance revisited. Stat Med, 25, 4334–4344.

shows that claims that in a 2-arm study it is not true that ANCOVA requires the population means at baseline to be identical;refutes some claims of lia00lon;problems with counterfactuals;temporal additivity ("amounts to supposing that despite the fact that groups are difference at baseline they would show the same evolution over time");causal additivity;is difficult to design trials for which simple analysis of change scores is unbiased, ANCOVA is biased, and a causal interpretation can be given;temporally and logically, a "baseline cannot be a \(<\)i\(>\)response\(<\)/i\(>\) to treatment", so baseline and response cannot be modeled in an integrated framework as Laird and Ware’s model has been used;"one should focus clearly on “outcomes” as being the only values that can be influenced by treatment and examine critically any schemes that assume that these are linked in some rigid and deterministic view to “baseline” values. An alternative tradition sees a baseline as being merely one of a number of measurements capable of improving predictions of outcomes and models it in this way.";"You cannot establish necessary conditions for an estimator to be valid by nominating a model and seeing what the model implies unless the model is universally agreed to be impeccable. On the contrary it is appropriate to start with the estimator and see what assumptions are implied by valid conclusions.";this is in distinction to lia00lon

Simpson, S. L., Edwards, L. J., Muller, K. E., Sen, P. K., & Styner, M. A. (2010). A linear exponent AR(1) family of correlation structures. Stat Med, 29, 1825–1838.

Venables, W. N., & Ripley, B. D. (2003). Modern Applied Statistics with S (Fourth). Springer-Verlag.

Verbeke, G., & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer.

7.1 Notation

7.2 Model Specification for Effects on \(E(Y)\)

7.2.1 Common Basis Functions

7.2.2 Model for Mean Profile

7.2.3 Model Specification for Treatment Comparisons

7.3 Modeling Within-Subject Dependence

7.4 Parameter Estimation Procedure

7.5 Common Correlation Structures

7.6 Checking Model Fit

7.7 R Software

7.8 Case Study

7.8.1 Graphical Exploration of Data

Model with \(Y_{i0}\) as Baseline Covariate

7.8.2 Using Generalized Least Squares

7.8.3 Bayesian Proportional Odds Random Effects Model

7.8.4 Bayesian Markov Semiparametric Model

7.9 Study Questions

7.7 `R` Software