Introduction

This R package proposes different functions to make inference on a random changepoint framework for longitudinal data. The work is still in progress and the package is not yet fully functional. Already implemented or being implemented are :

• testRCPMM: a test for the existence of a random changepoint for longitudinal data
• rcpme: an estimation algorithm for random changepoint mixed models
• bircpme: an estimation algorithm for bivariate random changepoint mixed models taking into account an eventual correlation between two markers

In the following, we will see how to use these functions and how to manipulate them with a toy dataset.

The dataset: PAQUID cohort

The PAQUID cohort is an epidemiologic study on cogntive ageing launched in South-West France in 1988. In the rcpm package, a sample of one hundred randomly selected demented subjects has been extracted and provided as a toy dataset in the package. Let us import the dataset into the working environment and let us look at the data !

library(rcpm)
data(paquid)
head(paquid)

##   id ist      delay el
## 1  1  25 -1.9700001  1
## 2  1  25 -1.8621301  1
## 3  1  28 -1.6521401  1
## 4  1  23 -1.1809501  1
## 5  1  16 -0.1474333  1
## 6  1  15  0.1474333  1

The dataset contains one line per visit. On each line, we can read id the patient identifier , ist its current Isaacs Set Test value which measures memory impairment, delay the current delay to patient diagnostic of dementia divided by ten and el a binary variable for educational level (0 for non primary school certificate patients, 1 for other).

It is interesting to look at the all longitudinal trajectory according to educational level

Testing the existence of the random changepoint

A first step is to assess if there is a random changepoint in these trajectories for both groups. This will be done with the rcpm function testRCPMM. But first let us create two datasets for each educational level.

paquid_el0 <- paquid[paquid$el == 0, ]
paquid_el1 <- paquid[paquid$el == 1, ]

To apply the test, we need to specify a formula indicating the score variable, the time variable and the grouping variable considered. This formula must be written this way score ~ time | group so that in our example, it should look like ist ~ delay | id. This first two arguments are essential for the function to run. Next argument is covariate but the underlying functionnality has not been developed yet so I will not give details about it. The gamma argument is used to smooth the trajectory at the date of the changepoint, it has to be fixed at a small value regarding the timescale, here delay. Be careful, by default, its value is 0.1 so if your timescale is very reduced, let us say from 0 to 0.1, you need to chose a smaller value for gamma. The argument nbnodes indicates the number of nodes used for the pseudo-adaptive gaussian quadrature which is used tu numerically compute the integrals. I would advise not to touch this value. Finally, the nbpert is important because it fixed the size of the empirical null test statistics which will be used to compute the empirical pvalue. Default is 500 which might induced quite a long computing time.

test_el0 <- testRCPMM(longdata = paquid_el0, formu = ist ~ delay | id)
test_el0

## $`empirical p-value`
## [1] 0.092
## 
## $`obs. stat`
## [1] -5.513509

test_el1 <- testRCPMM(longdata = paquid_el1, formu = ist ~ delay | id)
test_el1

## $`empirical p-value`
## [1] 0
## 
## $`obs. stat`
## [1] -23.24749

Estimating the random changepoint model

Now that we tested for the existence of a random changepoint and did not rejected its absence for high educational level, we can estimate the random changepoint model. This is done with the rcpme function.

esti <- rcpme(paquid_el1, ist ~ delay | id, nbnodes = 10, model = "bw", link = "linear")
esti

## $call
## $call[[1]]
## rcpme
## 
## $call$longdata
## paquid_el1
## 
## $call$formu
## ist ~ delay | id
## 
## $call$nbnodes
## [1] 10
## 
## $call$model
## [1] "bw"
## 
## $call$link
## [1] "linear"
## 
## 
## $Loglik
## [1] -1291.964
## 
## $formula
## ist ~ delay | id
## 
## $fixed
##           par se(par)   ICinf  ICsup Wald stat. pvalue
## beta0  27.817   0.907  26.040 29.594     30.679      0
## beta1 -14.158   2.281 -18.628 -9.688     -6.208      0
## beta2 -12.216   2.306 -16.737 -7.696         NA     NA
## mutau  -0.122   0.084  -0.287  0.042         NA     NA
## 
## $sdres
## [1] 3.232696
## 
## $VarEA
##           [,1]      [,2]      [,3]       [,4]
## [1,]  28.35515 -29.39257 -34.36392 0.00000000
## [2,] -29.39257  50.63468  59.61890 0.00000000
## [3,] -34.36392  59.61890  70.30699 0.00000000
## [4,]   0.00000   0.00000   0.00000 0.01688214
## 
## $optpar
##  [1]  27.8172111 -14.1581333 -12.2161527  -0.1221993   3.2326965
##  [6]   5.3249554  -5.5197786  -6.4533720   4.4907376   5.3438246
## [11]  -0.3232900  -0.1299313
## 
## $covariate
## [1] "NULL"
## 
## $REadjust
## [1] "no"
## 
## $invhessian
##                [,1]         [,2]         [,3]          [,4]          [,5]
##  [1,]  0.8221283864  0.003256801 -0.322263001 -0.0302992253  0.0007620257
##  [2,]  0.0032568005  5.201618190  5.136104983 -0.1591676721  0.0338709627
##  [3,] -0.3222630014  5.136104983  5.319437089 -0.1443233842  0.0354469470
##  [4,] -0.0302992253 -0.159167672 -0.144323384  0.0070579483 -0.0011077503
##  [5,]  0.0007620257  0.033870963  0.035446947 -0.0011077503  0.0174172400
##  [6,] -0.0503389725 -0.079384693 -0.037034571  0.0028950866 -0.0060285502
##  [7,]  0.3835600575  2.355777041  2.152674755 -0.0921726414  0.0224263041
##  [8,]  0.4000618250  2.293349515  2.060616694 -0.0887801542  0.0279814187
##  [9,] -0.2373158487 -1.449131269 -1.329475971  0.0523857537 -0.0194222738
## [10,] -0.2142962986 -1.437028186 -1.326793812  0.0513514716 -0.0142777716
## [11,] -0.0325631282 -0.003939966  0.062965561  0.0017983674  0.0748164005
## [12,]  0.0028119960  0.011199215  0.009922797 -0.0008555675  0.0002318826
##               [,6]        [,7]        [,8]        [,9]       [,10]
##  [1,] -0.050338972  0.38356006  0.40006183 -0.23731585 -0.21429630
##  [2,] -0.079384693  2.35577704  2.29334952 -1.44913127 -1.43702819
##  [3,] -0.037034571  2.15267476  2.06061669 -1.32947597 -1.32679381
##  [4,]  0.002895087 -0.09217264 -0.08878015  0.05238575  0.05135147
##  [5,] -0.006028550  0.02242630  0.02798142 -0.01942227 -0.01427777
##  [6,]  0.485304032 -0.49814411 -0.73220980  0.06348452  0.03471641
##  [7,] -0.498144113  2.97514176  3.17448992 -0.42944152 -0.72438996
##  [8,] -0.732209800  3.17448992  3.60379824 -0.44737608 -0.71780847
##  [9,]  0.063484522 -0.42944152 -0.44737608  1.84574621  1.53164933
## [10,]  0.034716409 -0.72438996 -0.71780847  1.53164933  1.55655114
## [11,] -0.078875922 -0.80408190 -0.54589031 -0.90384496 -0.08585452
## [12,]  0.001585263  0.02100994  0.01887481  0.03190308  0.02096557
##              [,11]         [,12]
##  [1,] -0.032563128  0.0028119960
##  [2,] -0.003939966  0.0111992146
##  [3,]  0.062965561  0.0099227969
##  [4,]  0.001798367 -0.0008555675
##  [5,]  0.074816400  0.0002318826
##  [6,] -0.078875922  0.0015852634
##  [7,] -0.804081899  0.0210099438
##  [8,] -0.545890314  0.0188748082
##  [9,] -0.903844956  0.0319030773
## [10,] -0.085854519  0.0209655732
## [11,]  5.098681491 -0.0275491168
## [12,] -0.027549117  0.0023434409
## 
## $conv
## [1] 1
## 
## $init
##  [1] 21.7614447 -6.6189204 -0.5000000 -0.6194516  4.1875530  3.9549840
##  [7]  0.0000000  0.0000000  1.0000000  0.0000000  1.0000000  1.0000000
## 
## $model
## [1] "bw"
## 
## $gamma
## [1] 0.1
## 
## $link
## [1] "linear"

To assess the quality of the obtained estimation, we can use the function IndPres that estimates individual prediction based on the output of the rcpme function. We plotted for twelve randomly selected subjects their individual prediction (plain line) vs. their observed measures (dots).

pred <- unlist(IndPred(esti))