Serial data, also called longitudinal data, repeated measures, or panel data, present a special challenge in that the response variable is multivariate, and that the responses measured at different times (or other conditions such as doses) but on the same subject are correlated with each other. One expects correlations between two measurements measured closer together will exceed correlations for two distant measurements on the same subject. Analyzing the repeated measurements just as though they were independent measurements falsely inflates the sample size and results in a failure to preserve type I error and confidence interval coverage. For example, having three measurements on 10 subjects results in 30 measurements, but depending on the correlations among the three measurements within subject, the effective sample size might be 16, for example. In other words, the statistical information for collecting 3 measurements on each of 10 subjects might provide the same statistical information and power as having one measurement on each of 16 subjects. The statistical information would however be less than that from 30 subjects measured once, if the intra-subject correlation exceeds zero.

The most common problems in analyzing serial data are

B

treating repeated measurements per subject as if they were from separate subjects

using two-way ANOVA as if different measurement times corresponded to different groups of subjects

using repeated measures ANOVA which assumes that the correlation between any two points within subject is the same regardless of how far apart the measures were timed

analyzing more than 3 time points as if time is a categorical rather than a continuous variable

multiplicity problem

analyses at different times may be inconsistent since times are not connected

loss of power by not jointly analyzing the serial data

15.2 Analysis Options

There are several overall approaches to the analysis of serial measurements. Some of the older approaches such as multiple \(t\)-tests and repeated measures ANOVA are now considered obsolete because of the availability of better methods that are more flexible and robust^{1}. Separate \(t\)-tests at each time point do not make good use of available information, use an inefficient estimate of \(\sigma^2\), do not interpolate time points, and have multiple comparison problems. Since a multiplicity adjustment for multiple correlated \(t\)-tests is not model-based, the resulting confidence intervals and \(P\)-values are conservative. To preserve type I error, one always must sacrifice type II error, but in this case the sacrifice is too severe. In addition, investigators are frequently confused by the \(t\)-test being “significant” at one time point and not at another, and make the unwarranted claim that the treatment is effective only at the first time. Besides not recognizing the absence of evidence is not evidence for absence problem, the investigator can be mislead by increased variability in response at the second time point driving the \(t\) ratio towards zero. This variability may not even be real but may reflect instability in estimating \(\sigma\) separately at each time point.

C

^{1} Repeated measures ANOVA makes stringent assumptions and requires complex adjustments for within-subject correlation.

Especially when there are more than three unique measurement times, it is advisable to model time as a continuous variable. When estimation of the time-response profile is of central importance, that may be all that’s needed. When comparing time-response profiles (e.g., comparing two treatments) one needs to carefully consider the characteristics of the profiles that should be tested, i.e., where the type I and II errors should be directed, for example

D

difference in slope

difference in area under the curve

difference in mean response at the last planned measurement time

difference in mean curves at any time, i.e., whether the curves have different heights or shapes anywhere

The first 3 are 1 d.f. tests, and the last is a 2 d.f. test if linearity is assumed, and \(> 2\) d.f. if fitting a polynomial or spline function.

15.2.1 Joint Multivariate Models

E

This is the formal fully-specified statistical model approach whereby a likelihood function is formed and maximized. This handles highly imbalanced data, e.g., one subject having one measurement and another having 100. It is also the most robust approach to non-random subject dropouts. These two advantages come from the fact that full likelihood models “know” exactly how observations from the same subject are connected to each other.

Examples of full likelihood-based models include generalized least squares (GLS), mixed effects models, and Bayesian hierarchical models. GLS only handles continuous \(Y\) and assumes multivariate normality. It does not allow different subjects to have different slopes. But it is easy to specify and interpret, to have its assumptions checked, and it runs faster. Mixed effects models can handle multiple hierarchical levels (e.g., state/hospital/patient) and random slopes whereby subjects can have different trajectories. Mixed effects models can be generalized to binary and other endpoints but lose their full likelihood status somewhat when these extensions are used, unless a Bayesian approach is used.

Mixed effects model have random effects (random intercepts and possibly also random slopes or random shapes) for subjects

These allow estimation of the trajectory for an individual subject

When the interest is instead on group-level effects (e.g., average difference between treatments, over subjects), GLS squares models may be more appropriate

The generalization of GLS to non-normally distributed \(Y\) is marginalized models (marginalized over subject-level effects)

GLS, formerly called growth curve models, is the oldest approach and has excellent performance when \(Y\) is conditionally normally distributed.

15.2.2 GEE

Generalized estimating equations is usually based on a working independence model whereby ordinary univariate regressions are fitted on a combined dataset as if all observations are uncorrelated, and then an after-the-fit correction for intra-cluster correlation is done using the cluster sandwich covariance estimator or the cluster bootstrap. GEE is very non-robust to non-random subject dropout; it assumes missing response values are missing completely at random. It may also require large sample sizes for \(P\)-values and confidence intervals to be accurate. An advantage of GEE is that it extends easily to every type of response variable, including binary, ordinal, polytomous, and time-to-event responses.

F

15.2.3 Summary Measures

A simple and frequently effective approach is to summarize the serial measures from each subject using one or two measures, then to analyze these measures using traditional statistical methods that capitalize on the summary measures being independent for different subjects. This has been called

An excellent overview may be found in Matthews et al. (1990), Dupont (2008) (Chapter 11), and Senn et al. (2000).

Frequently chosen summary measures include the area under the time-response curve, slope, intercept, and consideration of multiple features simultaneously, e.g., intercept, coefficient of time, coefficient of time squared when fitting each subject’s data with a quadratic equation. This allows detailed analyses of curve shapes.

15.3 Case Study

15.3.1 Data and Summary Measures

Consider the isoproterenol dose-response analysis (Dupont, 2008) of the original data from Lang^{2}. Twenty two normotensive men were studied, 9 of them black and 13 white. Blood flow was measured before the drug was given, and at escalating doses of isoproterenol. Most subjects had 7 measurements, and these are not independent. Note: Categorization of race is arbitrary and represents a gross oversimplification of genetic differences in physiology.

# Fit subject-by-subject spline fits and either return the coefficients,# the estimated area under the curve from [0,400], or evaluate each# subject's fitted curve over a regular grid of 150 doses# Area under curve is divided by 400 to get a mean functionrequire(rms)options(prType='html')g <-function(x, y, what=c('curve', 'coef', 'area')) { what <-match.arg(what) # 'curve' is default knots <-c(20, 60, 150) f <-ols(y ~rcs(x, knots)) xs <-seq(0, 400, length=150)switch(what,coef = {k <-coef(f)list(b0 = k[1], b1=k[2], b2=k[3])},curve= {x <-seq(0, 400, length=150)list(dose=xs, fbf=predict(f, data.frame(x=xs)))},area = {antiDeriv =rcsplineFunction(knots, coef(f),type='integral')list(dose =400, fbf=y[x ==400],area =antiDeriv(400) /400,tarea =areat(x, y) /400)} )}# Function to use trapezoidal rule to compute area under the curveareat <-function(x, y) { i <-!is.na(x + y) x <- x[i]; y <- y[i] i <-order(x) x <- x[i]; y <- y[i]if(!any(x ==400)) NAelsesum(diff(x) * (y[-1] + y[-length(y)]))/2}w <- d[, j=g(dose, fbf), by =list(id, race)] # uses data.table packagea <- d[, j=g(dose, fbf, what='area'), by =list(id, race)]ggplot(d, aes(x=dose, y=fbf, color=factor(id))) +geom_line() +geom_line(data=w, alpha=0.25) +geom_text(aes(label =round(area,1)), data=a, size=2.5,position=position_dodge(width=50)) +xlab('Dose') +ylab(label(d$fbf, units=TRUE, plot=TRUE)) +facet_grid(~ race) +guides(color=FALSE)

K

Code

ggplot(a, aes(x=tarea, y=area, color=race)) +geom_point() +geom_abline(col=gray(.8)) +xlab('Area by Trapezoidal Rule / 400') +ylab('Area by Spline Fit / 400')

When a subject’s dose (or time) span includes the minimum and maximum values over all subjects, one can use the trapezoidal rule to estimate the area under the response curve empirically. When interior points are missing, linear interpolation is used. The spline fits use nonlinear interpolation, which is slightly better, as is the spline function’s assumption of continuity in the slope. @ref(fig-serial-auctwo) compares the area under the curve (divided by 400 to estimate the mean response) estimated using the the two methods. Agreement is excellent. In this example, the principle advantage of the spline approach is that slope and shape parameters are estimated in the process, and these parameters may be tested separately for association with group (here, race). For example, one may test whether slopes differ across groups, and whether the means, curvatures, or inflection points differ. One could also compare AUC from a sub-interval of \(X\).

15.3.2 Nonparametric Test of Race Differences in AUC

A minimal-assumption approach to testing for differences in isoproterenol dose-response between races is to apply the Wilcoxon test to the normalized AUCs (mean response functions).

M

Code

wilcox.test(area ~ race, data=a, conf.int=TRUE)

Wilcoxon rank sum exact test
data: area by race
W = 112, p-value = 7.639e-05
alternative hypothesis: true location shift is not equal to 0
95 percent confidence interval:
5.670113 16.348121
sample estimates:
difference in location
11.25836

There is strong evidence that the mean response is greater for whites.

15.3.3 Nonparametric Test of General Curve Differences

O’Brien (1988) proposed a method for using logistic regression to turn the Hotelling \(T^2\) test on its side. The Hotelling test is the multivariate analog of the two-sample \(t\)-test, and can be used to test simultaneously such things as whether a treatment modifies either systolic or diastolic blood pressure. O’Brien’s idea was to test whether systolic or diastolic blood pressure (or both) can predict which treatment was given. Here we use the idea to test for race differences in the shape of the dose-response curves. We do this by predicting race from a 3-predictor model—one containing the intercept, the next the coefficient of the linear dose effect and the third the coefficient of the nonlinear restricted cubic spline term (differences in cubes). These coefficients were estimated using ordinary least squares in separately predicting each subject’s relationship between dose and forearm blood flow.

N

Code

h <- d[, j=g(dose, fbf, what='coef'), by =list(id, race)]h

id race b0 b1 b2
1: 1 white -0.1264763 0.32310327 -0.34253352
2: 2 white 2.1147707 0.22798336 -0.21959542
3: 3 white 2.3378721 0.06927717 -0.03625384
4: 4 white 1.4822502 0.19837740 -0.20727584
5: 5 white 2.5366751 0.15399740 -0.14756999
6: 6 white 3.2117187 0.19910942 -0.18650561
7: 7 white 1.4264366 0.05261565 -0.03871545
8: 8 white 3.0999999 0.21833887 -0.19813457
9: 9 white 5.1507764 0.45026617 -0.48013920
10: 10 white 4.4778127 0.23853904 -0.23289815
11: 11 white 1.9052885 0.13548226 -0.13917910
12: 12 white 2.1828176 0.07558431 -0.05524955
13: 13 white 2.9318982 0.12776900 -0.10679867
14: 14 black 2.3336099 0.02679742 -0.02856275
15: 15 black 1.8356227 0.07652884 -0.07972036
16: 16 black 2.5342537 0.02290717 -0.02585081
17: 17 black 2.0254606 0.06002835 -0.06261969
18: 18 black 3.3279080 0.07620477 -0.08062536
19: 19 black 1.9308650 0.03844018 -0.04060065
20: 20 black 1.7263259 0.12358392 -0.13595538
21: 21 black 1.3215502 0.03528716 -0.03480467
22: 22 black 2.0828281 0.03143768 0.02251155
id race b0 b1 b2

lrm(formula = race ~ b0 + b1 + b2, data = h, x = TRUE, y = TRUE)

Model Likelihood Ratio Test

Discrimination Indexes

Rank Discrim. Indexes

Obs 22

LR χ^{2} 19.73

R^{2} 0.798

C 0.957

white 13

d.f. 3

R^{2}_{3,22} 0.533

D_{xy} 0.915

black 9

Pr(>χ^{2}) 0.0002

R^{2}_{3,16} 0.650

γ 0.915

max |∂log L/∂β| 8×10^{-7}

Brier 0.080

τ_{a} 0.463

β

S.E.

Wald Z

Pr(>|Z|)

Intercept

3.6618

4.1003

0.89

0.3718

b0

1.3875

2.5266

0.55

0.5829

b1

-165.3481

92.7864

-1.78

0.0747

b2

-101.1667

63.1549

-1.60

0.1092

The likelihood ratio \(\chi^{2}_{3}=19.73\) has \(P=0.0002\) indicating strong evidence that the races have different averages, slopes, or shapes of the dose-response curves. The \(c\)-index of 0.957 indicates nearly perfect ability to separate the races on the basis of three curve characteristics (although the sample size is small). We can use the bootstrap to get an overfitting-corrected index.

O

Code

set.seed(2)v <-validate(f, B=1000)

Divergence or singularity in 271 samples

Code

html(v)

Index

Original Sample

Training Sample

Test Sample

Optimism

Corrected Index

\(n\)

\(D_{xy}\)

0.9145

0.9589

0.8423

0.1166

0.798

729

\(R^{2}\)

0.7985

0.9096

0.6617

0.2479

0.5506

729

Intercept

0

0

0.115

-0.115

0.115

729

Slope

1

1

0.4259

0.5741

0.4259

729

\(E_{\max}\)

0

0

0.2056

0.2056

0.2056

729

\(D\)

0.8513

1.0728

0.6434

0.4294

0.4219

729

\(U\)

-0.0909

-0.0909

2.608

-2.6989

2.608

729

\(Q\)

0.9422

1.1637

-1.9646

3.1283

-2.1861

729

\(B\)

0.0803

0.031

0.0773

-0.0463

0.1266

729

\(g\)

6.7833

23.8206

4.4145

19.4061

-12.6228

729

\(g_{p}\)

0.4712

0.4629

0.4255

0.0373

0.4339

729

The overfitting-corrected \(c\)-index is \(c = \frac{D_{xy} + 1}{2}\) = 0.9.

15.3.4 Model-Based Analysis: Generalized Least Squares

Generalized least squares (GLS) is the first generalization of ordinary least squares (multiple linear regression). It is described in detail in Regression Modeling Strategies Chapter 7 where a comprehensive case study is presented. The assumptions of GLS are

PQ

All the usual assumptions about the right-hand-side of the model related to transformations of \(X\) and interactions

Residuals have a normal distribution

Residuals have constant variance vs. \(\hat{Y}\) or any \(X\) (but the G in GLS also refers to allowing variances to change across \(X\))

The multivariate responses have a multivariate normal distribution conditional on \(X\)

The correlation structure of the conditional multivariate distribution is correctly specified

With fully specified serial data models such as GLS, the fixed effects of time or dose are modeled just as any other predictor, with the only difference being that it is the norm to interact the time or dose effect with treatment or whatever \(X\) effect is of interest. This allows testing hypotheses such as

R

Does the treatment effect change over time? (time \(\times\) treatment interaction)

Is there a time at which there is a treatment effect? (time \(\times\) treatment interaction + treatment main effect combined into a chunk test)

Does the treatment have an effect at time \(t\)? (difference of treatments fixing time at \(t\), not assuming difference is constant across different \(t\))

In the majority of longitudinal clinical trials, the last hypothesis is the most important, taking \(t\) as the end of treatment point. This is because one is often interested in where patients ended up, not just whether the treatment provided temporary relief.

S

Now consider the isoproterenol dataset and fit a GLS model allowing for the same nonlinear spline effect of dose as was used above, and allowing the shapes of curves to be arbitrarily different by race. We impose a continuous time AR1 correlation structure on within-subject responses. This is the most commonly used correlation structure; it assumes that the correlation between two points is exponentially declining as the difference between two times or doses increases. We fit the GLS model and examine the equal variance assumption.

The variance of the residuals is clearly increasing with increasing dose. Try log transforming both fbf and dose. The log transformation requires a very arbitrary adjustment to dose to handle zeros.

There is little evidence for a nonlinear dose effect on the log scale, implying that the underlying model is exponential on the original \(X\) and \(Y\) scales. This is consistent with Dupont (2008). Re-fit the model as linear in the logs. Before taking this as the final model, also fit the same model but using a correlation pattern based on time rather than dose. Assume equal time spacing during dose escalation.

Lower AIC is better, so it is clear that time-based correlation structure is far superior to dose-based. We will used the second model for the remainder of the analysis. But first we check some of the model assumptions.

The test for dose \(\times\) race interaction in the above ANOVA summary of Wald statistics shows strong evidence for difference in curve characteristics across races. This test agrees in magnitude with the less parametric approach using the logistic model above. But the logistic model also tests for an overall shift in distributions due to race, and the more efficient test for the combined race main effect and dose interaction effect from GLS is more significant with a Wald \(\chi^{2}_{2} = 32.45\)^{3}. The estimate of the correlation between two log blood flows measured on the same subject one time unit apart is 0.69.

W

^{3} The \(\chi^2\) for race with the dose-based correlation structure was a whopping 100 indicating that lack of fit of the correlation structure can have a significant effect on the rest of the GLS model.

The equal variance and normality assumptions appear to be well met as judged by @ref(fig-serial-glsc).

Now estimate the dose-response curves by race, with pointwise confidence intervals and simultaneous intervals that allow one to make statements about the entire curves^{4}. Anti-log the predicted values to get predictions on the original blood flow scale. Anti-logging predictions from a model that assumes a normal distribution on the logged values results in estimates of the median response.

^{4} Since the model is linear in log dose there are two parameters involving dose—the dose main effect and the race \(\times\) dose interaction effect. The simultaneous inference adjustment only needs to reflect simultaneity in two dimensions.

Finally we estimate the white:black fold change (ratio of medians) as a function of dose with simultaneous confidence bands.

Y

Code

k <-contrast(b, list(dose=dos, race='white'),list(dose=dos, race='black'), conf.type='simultaneous')k <-as.data.frame(k[c('dose', 'Contrast', 'Lower', 'Upper')])ggplot(k, aes(x=dose, y=exp(Contrast))) +geom_line() +geom_ribbon(aes(ymin=exp(Lower), ymax=exp(Upper)), alpha=0.2, linetype=0,show_guide=FALSE) +geom_hline(yintercept=1, col='red', size=.2) +ylab('White:Black Ratio of Median FBF')

By comparing the simultaneous confidence binds to the red horizontal line, one can draw the inference that the dose-response for blacks is everywhere different than that for whites when the dose exceeds zero.

15.3.5 Longitudinal Bayesian Mixed Effects Semiparametric Model

Proportional odds model with random effects

Robust with regard to distribution

Compound symmetry may not be realistic

Model estimates logits of exceedance probabilities

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

Dupont, W. D. (2008). Statistical Modeling for Biomedical Researchers (second). Cambridge University Press.

Matthews, J. N. S., Altman, D. G., Campbell, M. J., & Royston, P. (1990). Analysis of serial measurements in medical research. BMJ, 300, 230–235. https://doi.org/10.1136/bmj.300.6719.230

```{r setup, include=FALSE}require(Hmisc)getRs('reptools.r')hookaddcap() # make knitr call a function at the end of each chunk# to try to automatically add to list of figureoptions(prType='html')getRs('qbookfun.r')knitr::set_alias(w ='fig.width', h ='fig.height',cap ='fig.cap', scap ='fig.scap')````r apacue <- 0`# Serial Data {#sec-serial}`r mrg(ddisc(19))`[For more information see [RMS Chapter 7](https://hbiostat.org/doc/rms.pdf)]{.aside}## Introduction`r mrg(sound("serial-1"))``r ipacue()`Serial data, also called longitudinal data, repeated measures, orpanel data,present a special challenge in that the response variable ismultivariate, and that the responses measured at different times (orother conditions such as doses) but on the same subject are correlatedwith each other. One expects correlations between twomeasurements measured closer together will exceed correlations for twodistant measurements on the same subject. Analyzing the repeatedmeasurements just asthough they were independent measurements falsely inflates the samplesize and results in a failure to preserve type I error and confidenceinterval coverage. For example, having three measurements on 10subjects results in 30 measurements, but depending on the correlationsamong the three measurements within subject, the effective sample sizemight be 16, for example. In other words, the statistical informationfor collecting 3 measurements on each of 10 subjects might provide thesame statistical information and power as having one measurement oneach of 16 subjects. The statistical information would however be less thanthat from 30 subjects measured once, if the intra-subject correlationexceeds zero.The most common problems in analyzing serial data are `r ipacue()`1. treating repeated measurements per subject as if they were from separate subjects1. using two-way ANOVA as if different measurement times corresponded to different groups of subjects1. using repeated measures ANOVA which assumes that the correlation between any two points within subject is the same regardless of how far apart the measures were timed1. analyzing more than 3 time points as if time is a categorical rather than a continuous variable * multiplicity problem * analyses at different times may be inconsistent since times are not connected * loss of power by not jointly analyzing the serial data## Analysis Options`r ipacue()`There are several overall approaches to the analysis of serialmeasurements. Some of the older approaches such as multiple $t$-testsand repeated measures ANOVA arenow considered obsolete because of the availability of better methods that aremore flexible and robust^[Repeated measures ANOVA makes stringent assumptions and requires complex adjustments for within-subject correlation.].Separate $t$-tests at each time point do not make good use ofavailable information, use an inefficient estimate of $\sigma^2$, donot interpolate time points, and have multiple comparison problems.Since a multiplicity adjustment for multiple correlated $t$-tests isnot model-based, the resulting confidence intervals and $P$-valuesare conservative. To preserve type I error, one always must sacrificetype II error, but in this case the sacrifice is too severe.In addition, investigators are frequently confused by the $t$-test being"significant" at one time point and not at another, and make theunwarranted claim that the treatment is effective only at the firsttime. Besides not recognizing the _absence of evidence is not evidence for absence_ problem, the investigator can be mislead byincreased variability in response at the second time point drivingthe $t$ ratio towards zero. This variability may not even be real butmay reflect instability in estimating $\sigma$ separately at each timepoint.Especially when there are more than three unique measurement times, it is advisable to model time as a continuous variable. When estimation of the time-response profile is of central importance, that may be all that's needed. When comparing time-response profiles (e.g., comparing two treatments) one needs to carefully consider the characteristics of the profiles that should be tested, i.e., where the type I and II errors should be directed, for example `r ipacue()`* difference in slope* difference in area under the curve* difference in mean response at the last planned measurement time* difference in mean curves at _any_ time, i.e., whether the curves have different heights or shapes anywhereThe first 3 are 1 d.f. tests, and the last is a 2 d.f. test if linearity is assumed, and $> 2$ d.f. if fitting a polynomial or spline function.### Joint Multivariate Models`r ipacue()`This is the formal fully-specified statistical model approach wherebya likelihood function is formed and maximized. This handles highlyimbalanced data, e.g., one subject having one measurement and anotherhaving 100. It is also the most robust approach to non-random subjectdropouts. These two advantages come from the fact that fulllikelihood models "know" exactly how observations from the samesubject are connected to each other.Examples of full likelihood-based models includegeneralized least squares (GLS), mixed effects models, and Bayesian hierarchicalmodels. GLS only handles continuous $Y$ and assumes multivariatenormality. It does not allow different subjects to have differentslopes. But it is easy to specify and interpret, to have its assumptionschecked, and it runs faster.Mixed effects models can handle multiple hierarchical levels (e.g.,state/hospital/patient) and random slopes whereby subjects canhave different trajectories. Mixed effects models can be generalizedto binary and other endpoints but lose their full likelihood statussomewhat when these extensions are used, unless a Bayesian approach is used.* Mixed effects model have random effects (random intercepts and possibly also random slopes or random shapes) for subjects* These allow estimation of the trajectory for an individual subject* When the interest is instead on group-level effects (e.g., average difference between treatments, over subjects), GLS squares models may be more appropriate* The generalization of GLS to non-normally distributed $Y$ is _marginalized models_ (marginalized over subject-level effects)GLS, formerly called growth curve models, is theoldest approach and has excellent performance when $Y$ isconditionally normally distributed.### GEE`r ipacue()`_Generalized estimating equations_ is usually based on a workingindependence model whereby ordinary univariate regressions are fitted on acombined dataset as if all observations are uncorrelated, and then anafter-the-fit correction for intra-clustercorrelation is done using the cluster sandwich covariance estimator or thecluster bootstrap. GEE is very non-robust to non-random subject dropout; itassumes missing response values are missing completely at random. Itmay also require large sample sizes for$P$-values and confidence intervals to be accurate. An advantage of GEE isthat it extends easily to every type of response variable, including binary,ordinal, polytomous, and time-to-event responses.### Summary Measures`r mrg(sound("serial-2"))``r ipacue()`A simple and frequently effective approach is to summarize the serialmeasures from each subject using one or two measures, then to analyzethese measures using traditional statistical methods that capitalizeon the summary measures being independent for different subjects.This has been called `r ipacue()`1. Two-stage derived variable analysis [@diggle-longit]1. Response feature analysis [@dupmod]1. Longitudinal analysis through summary measures [@mat90ana]An excellent overview may be found in @mat90ana,@dupmod (Chapter 11), and @sen00rep.Frequently chosen summary measures include the area under thetime-response curve, slope, intercept, and consideration of multiplefeatures simultaneously, e.g., intercept, coefficient of time,coefficient of time squared when fitting each subject's data with aquadratic equation. This allows detailed analyses of curve shapes.## Case Study### Data and Summary Measures`r mrg(sound("serial-3"))``r ipacue()`Consider the isoproterenol dose-response analysis [@dupmod] ofthe original data from Lang^[CC Lang etal NEJM 333:155-60, 1995]. Twenty two normotensive men were studied, 9 ofthem black and 13 white. Blood flow was measured before the drug wasgiven, and at escalating doses of isoproterenol. Most subjects had 7measurements, and these are not independent. **Note**: Categorization of race is arbitrary and represents a gross oversimplification of genetic differences in physiology.```{r download}require(Hmisc)require(data.table) # elegant handling of aggregationrequire(ggplot2)d <-csv.get('https://hbiostat.org/data/repo/11.2.Long.Isoproterenol.csv')d <-upData(d, keep=c('id', 'dose', 'race', 'fbf'),race =factor(race, 1:2, c('white', 'black')),labels=c(fbf='Forearm Blood Flow'),units=c(fbf='ml/min/dl'))setDT(d, key=c('id', 'race'))````r ipacue()````{r spag,w=7,h=4,cap="Spaghetti plots for isoproterenol data showing raw data stratified by race. Next to each curve is the area under the curve divided by 400 to estimate the mean response function. The area is computed analytically from a restricted cubic spline function fitted separately to each subject's dose-response curve. Shadowing the raw data are faint lines depicting spline fits for each subject",scap='Spaghetti plots for isoproterenol data'}#| label: fig-serial-spag# Fit subject-by-subject spline fits and either return the coefficients,# the estimated area under the curve from [0,400], or evaluate each# subject's fitted curve over a regular grid of 150 doses# Area under curve is divided by 400 to get a mean functionrequire(rms)options(prType='html')g <-function(x, y, what=c('curve', 'coef', 'area')) { what <-match.arg(what) # 'curve' is default knots <-c(20, 60, 150) f <-ols(y ~rcs(x, knots)) xs <-seq(0, 400, length=150)switch(what,coef = {k <-coef(f)list(b0 = k[1], b1=k[2], b2=k[3])},curve= {x <-seq(0, 400, length=150)list(dose=xs, fbf=predict(f, data.frame(x=xs)))},area = {antiDeriv =rcsplineFunction(knots, coef(f),type='integral')list(dose =400, fbf=y[x ==400],area =antiDeriv(400) /400,tarea =areat(x, y) /400)} )}# Function to use trapezoidal rule to compute area under the curveareat <-function(x, y) { i <-!is.na(x + y) x <- x[i]; y <- y[i] i <-order(x) x <- x[i]; y <- y[i]if(!any(x ==400)) NAelsesum(diff(x) * (y[-1] + y[-length(y)]))/2}w <- d[, j=g(dose, fbf), by =list(id, race)] # uses data.table packagea <- d[, j=g(dose, fbf, what='area'), by =list(id, race)]ggplot(d, aes(x=dose, y=fbf, color=factor(id))) +geom_line() +geom_line(data=w, alpha=0.25) +geom_text(aes(label =round(area,1)), data=a, size=2.5,position=position_dodge(width=50)) +xlab('Dose') +ylab(label(d$fbf, units=TRUE, plot=TRUE)) +facet_grid(~ race) +guides(color=FALSE)````r ipacue()````{r auctwo,cap='AUC by curve fitting and by trapezoidal rule'}#| label: fig-serial-auctwoggplot(a, aes(x=tarea, y=area, color=race)) +geom_point() +geom_abline(col=gray(.8)) +xlab('Area by Trapezoidal Rule / 400') +ylab('Area by Spline Fit / 400')```When a subject's dose (or time) span includes the minimum and maximum valuesover all subjects, one can use the trapezoidal rule to estimate the area underthe response curve empirically. When interior points are missing, linearinterpolation is used. The spline fits use nonlinear interpolation, which isslightly better, as is the spline function's assumption of continuity in theslope. \@ref(fig-serial-auctwo) compares the area under the curve (dividedby 400 to estimate the mean response) estimated using the the two methods.Agreement is excellent. In this example, the principle advantage of the splineapproach is that slope and shape parameters are estimated in the process, andthese parameters may be tested separately for association with group (here,race). For example, one may test whether slopes differ across groups, andwhether the means, curvatures, or inflection points differ. One couldalso compare AUC from a sub-interval of $X$.`r ipacue()````{r auc,w=5,h=1.5,cap='Mean blood flow computed from the areas under the spline curves, stratified by race, along with box plots',scap='Mean blood flow by race'}#| label: fig-serial-aucggplot(a, aes(x=race, y=area)) +geom_boxplot(alpha=.5, width=.25) +geom_point() +coord_flip() +ylab(expression(paste('Mean Forearm Blood Flow, ', scriptstyle(ml/min/dl))))```### Nonparametric Test of Race Differences in AUC`r mrg(sound("serial-4"))`<br>`r ipacue()`A minimal-assumption approach to testing for differences inisoproterenol dose-response between races is to apply the Wilcoxontest to the normalized AUCs (mean response functions).```{r wil}wilcox.test(area ~ race, data=a, conf.int=TRUE)```There is strong evidence that the mean response is greater for whites.### Nonparametric Test of General Curve Differences`r ipacue()`@obr88com proposed a method for using logistic regression toturn the Hotelling $T^2$ test on its side. The Hotelling test is themultivariate analog of the two-sample $t$-test, and can be used totest simultaneously such things as whether a treatment modifies eithersystolic or diastolic blood pressure. O'Brien's idea was to testwhether systolic or diastolic blood pressure (or both) can predictwhich treatment was given. Here we use the idea to test for racedifferences in the shape of the dose-response curves. We do this bypredicting race from a 3-predictor model---one containing the intercept, the next the coefficient of the linear dose effect and the third the coefficient of the nonlinear restricted cubic spline term (differences in cubes).These coefficients were estimatedusing ordinary least squares in separately predicting each subject's relationshipbetween dose and forearm blood flow.```{r coefs}h <- d[, j=g(dose, fbf, what='coef'), by =list(id, race)]h``````{r lrm,results='asis'}f <-lrm(race ~ b0 + b1 + b2, data=h, x=TRUE, y=TRUE)f````r ipacue()`The likelihood ratio $\chi^{2}_{3}=19.73$ has $P=0.0002$ indicating strong evidence that the races have different averages,slopes, or shapes of the dose-response curves. The $c$-index of 0.957indicates nearly perfect ability to separate the races on the basis of threecurve characteristics (although the sample size is small). We can usethe bootstrap to get an overfitting-corrected index.```{r boot,results='asis'}set.seed(2)v <-validate(f, B=1000)html(v)```The overfitting-corrected $c$-index is $c = \frac{D_{xy} + 1}{2}$ = `r round((v['Dxy','index.corrected'] + 1) / 2, 2)`.### Model-Based Analysis: Generalized Least Squares`r mrg(sound("serial-5"))``r ipacue()`Generalized least squares (GLS) is the first generalization of ordinaryleast squares (multiple linear regression). It is described in detailin _Regression Modeling Strategies_ Chapter 7 where acomprehensive case study is presented. The assumptions of GLS are `r ipacue()`* All the usual assumptions about the right-hand-side of the model related to transformations of $X$ and interactions* Residuals have a normal distribution* Residuals have constant variance vs. $\hat{Y}$ or any $X$ (but the G in GLS also refers to allowing variances to change across $X$)* The multivariate responses have a multivariate normal distribution conditional on $X$* The correlation structure of the conditional multivariate distribution is correctly specifiedWith fully specified serial data models such as GLS, the fixed effectsof time or dose are modeled just as any other predictor, with the onlydifference being that it is the norm to interact the time or doseeffect with treatment or whatever $X$ effect is of interest. Thisallows testing hypotheses such as `r ipacue()`* Does the treatment effect change over time? (time $\times$ treatment interaction)* Is there a time at which there is a treatment effect? (time $\times$ treatment interaction + treatment main effect combined into a chunk test)* Does the treatment have an effect at time $t$? (difference of treatments fixing time at $t$, not assuming difference is constant across different $t$)In the majority of longitudinal clinical trials, the last hypothesis is the most important, taking $t$ as the end of treatment point. Thisis because one is often interested in where patients ended up, notjust whether the treatment provided temporary relief. `r ipacue()`Now consider the isoproterenol dataset and fit a GLS model allowing forthe same nonlinear spline effect of dose as was used above, andallowing the shapes of curves to be arbitrarily different by race. Weimpose a continuous time AR1 correlation structure on within-subjectresponses. This is the most commonly used correlation structure; itassumes that the correlation between two points is exponentiallydeclining as the difference between two times or dosesincreases.We fit the GLS model and examine the equal variance assumption.`r ipacue()````{r glsa,cap='Residual plot for generalized least squares fit on untransformed `fbf`'}#| label: fig-serial-glsarequire(nlme)dd <- datadist(d); options(datadist='dd')a <- Gls(fbf ~ race * rcs(dose, c(20,60,150)), data=d, correlation=corCAR1(form = ~ dose | id))plot(fitted(a), resid(a))```The variance of the residuals is clearly increasing with increasingdose. Try log transforming both `fbf` and dose. The logtransformation requires a very arbitrary adjustment to dose to handle zeros.```{r glsb,results='asis'}a <- Gls(log(fbf) ~ race * rcs(log(dose + 1), log(c(20,60,150)+1)), data=d, correlation=corCAR1(form = ~ dose | id))anova(a)```There is little evidence for a nonlinear dose effect on the log scale, `r mrg(sound("serial-6"))`implying that the underlying model is exponential on the original $X$and $Y$ scales. This is consistent with @dupmod. Re-fit themodel as linear in the logs. Before taking this as the final model,also fit the same model but using a correlation pattern based on timerather than dose. Assume equal time spacing during dose escalation. `r ipacue()````{r glstwocorr,results='asis'}a <-Gls(log(fbf) ~ race *log(dose +1), data=d,correlation=corCAR1(form =~ dose | id))d$time <-match(d$dose, c(0, 10, 20, 60, 150, 300, 400)) -1b <-Gls(log(fbf) ~ race *log(dose +1), data=d,correlation=corCAR1(form =~ time | id))AIC(a);AIC(b)banova(b)```Lower AIC is better, so it is clear that time-based correlationstructure is far superior to dose-based. We will used the secondmodel for the remainder of the analysis. But first we check some ofthe model assumptions.`r ipacue()````{r glsc,results='asis',w=6,h=4,cap='Checking assumptions of the GLS model that is linear after logging dose and blood flow. The graph on the right is a QQ-plot to check normality of the residuals from the model, where linearity implies normality.',scap='Checking assumptions of GLS model'}#| label: fig-serial-glscw <-data.frame(residual=resid(b), fitted=fitted(b))p1 <-ggplot(w, aes(x=fitted, y=residual)) +geom_point()p2 <-ggplot(w, aes(sample=residual)) +stat_qq()gridExtra::grid.arrange(p1, p2, ncol=2)````r ipacue()`The test for dose $\times$ race interaction in the above ANOVA summary of Wald statistics shows strong evidence for difference in curvecharacteristics across races. This test agrees in magnitudewith the less parametric approach using the logistic model above. Butthe logistic model also tests for an overall shift in distributionsdue to race, and the more efficient test for the combined race maineffect and dose interaction effect from GLS is more significantwith a Wald $\chi^{2}_{2} = 32.45$^[The $\chi^2$ for race with the dose-based correlation structure was a whopping 100 indicating that lack of fit of the correlation structure can have a significant effect on the rest of the GLS model.]. Theestimate of the correlation between two log blood flows measured onthe same subject one time unit apart is 0.69.The equal variance and normality assumptions appear to be well met asjudged by \@ref(fig-serial-glsc).`r mrg(sound("serial-7"))`Now estimate the dose-response curves by race, with pointwise confidence intervals and simultaneous intervals that allow one to makestatements about the entire curves^[Since the model is linear in log dose there are two parameters involving dose---the dose main effect and the race $\times$ dose interaction effect. The simultaneous inference adjustment only needs to reflect simultaneity in two dimensions.]. Anti-log the predicted values toget predictions on the original blood flow scale. Anti-loggingpredictions from a model that assumes a normal distribution on thelogged values results in estimates of the median response. `r ipacue()````{r glsd,w=6,h=3.75,cap='Pointwise and simultaneous confidence bands for median dose-response curves by race',scap='Confidence bands for median dose-response curves',cache=TRUE}#| label: fig-serial-glsddos <-seq(0, 400, length=150)p <-Predict(b, dose=dos, race, fun=exp)s <-Predict(b, dose=dos, race, fun=exp, conf.type='simultaneous')ps <-rbind(Pointwise=p, Simultaneous=s)ggplot(ps, ylab=expression(paste('Median Forearm Blood Flow, ',scriptstyle(ml/min/dl))))```Finally we estimate the white:black fold change (ratio of medians) asa function of dose with simultaneous confidence bands. `r ipacue()````{r glse,cap='White:black fold change for median response as a function of dose, with simultaneous confidence band',cache=TRUE}#| label: fig-serial-glsek <-contrast(b, list(dose=dos, race='white'),list(dose=dos, race='black'), conf.type='simultaneous')k <-as.data.frame(k[c('dose', 'Contrast', 'Lower', 'Upper')])ggplot(k, aes(x=dose, y=exp(Contrast))) +geom_line() +geom_ribbon(aes(ymin=exp(Lower), ymax=exp(Upper)), alpha=0.2, linetype=0,show_guide=FALSE) +geom_hline(yintercept=1, col='red', size=.2) +ylab('White:Black Ratio of Median FBF')```By comparing the simultaneous confidence binds to the red horizontalline, one can draw the inference that the dose-response for blacks iseverywhere different than that for whites when the dose exceeds zero.### Longitudinal Bayesian Mixed Effects Semiparametric Model* Proportional odds model with random effects* Robust with regard to distribution* Compound symmetry may not be realistic* Model estimates logits of exceedance probabilities* Convert these to means on the original scale```{r blrm,results='asis',cap='Stan Hamiltonion MCMC convergence diagnostics for non-intercepts',cap='MCMC diagnostics'}#| label: fig-serial-blrmrequire(rmsb)options(mc.cores = parallel::detectCores() -1)b <-blrm(fbf ~ race *rcs(dose, 5) +cluster(id), data=d)bstanDxplot(b)``````{r blrm2,cap='Posterior mean logit of exceedance probability', scap='Posterior mean logit'}#| label: fig-serial-blrm2doses <-seq(0, 400, by=5)ggplot(Predict(b, dose=doses, race))``````{r blrm3,cap='Posterior mean of true mean forearm blood flow',scap='Posterior mean for mean blood flow'}#| label: fig-serial-blrm3M <-Mean(b)ggplot(Predict(b, dose=doses, race, fun=M))k <-contrast(b, list(race='white', dose=seq(0, 400, by=100)),list(race='black', dose=seq(0, 400, by=100)))k``````{r echo=FALSE}saveCap('15')```