Package 'ipr'

Iterative Proportional Repartition (IPR) algorithm

Description

Estimating the health cost repartition among diseases in the presence of multimorbidity, i.e. when some patients have multiple diseases. Using the Iterative Proportional Repartition algorithm (see reference below), the goal is to estimate the average cost for each disease, starting from the global health costs available for each patient.

Usage

ipr(X, y, print.it=FALSE, start=rep(1,dim(X)[2]), cutup=Inf, cutlow=cutup,
epsrel=0.001, epsabs=0.1, maxiter=1000, det=FALSE)

Arguments

X	Matrix with $x_{ij}=1$ if patient $i$ suffers from disease $j$ and $x_{ij}=0$ otherwise. Each row thus refers to one patient and each column to one disease. The number of columns of X corresponds to the number of diseases considered.
y	Vector where $y_i$ is the global health cost of patient $i$. The length of y must be equal to the number of rows of X.
print.it	Logical. If TRUE, the number of the current iteration and the current estimates are printed.
start	Vector of initial estimates of the average cost for each disease to start IPR algorithm. Default is an initial average cost of 1 for all diseases. The length of start must be equal to the number of columns of X.
cutup, cutlow	Options which can be used to get a robust version of IPR. If the current allocated cost of disease $j$ for patient $i$ is more than cutup times more expansive (or less then cutlow times less expansive) than the current average cost estimate of that disease $j$, then this outlying allocated cost is not taken into account in the next iteration to compute the average cost of disease $j$. By default, cutup and cutlow are set to Inf.
epsrel	Stopping criterion such that the IPR algorithm stops if for all diseases, the current estimated average cost differs by less than 100*epsrel percent from what it was at the previous iteration. The default value is 0.001. Should be set to 0 to ignore that criterion.
epsabs	Stopping criterion such that the IPR algorithm stops if for all diseases, the current estimated average cost differs (in absolute value) by less than epsabs percent from what it was at the previous iteration. The default value is 0.1. Should be set to 0 to ignore that criterion.
maxiter	Maximal number of iterations of IPR algorithm. The default value is 1000.
det	Logical. If TRUE, the allocated costs of each disease for each patient are given, by returning a matrix $Y$ where $y_{ij}$ is the estimated cost of disease $j$ for patient $i$.

Details

Let us consider $n$ patients and $p$ diseases. We are given a matrix $X$ such that $x_{ij}=1$ if the patient $i$ suffers from disease $j$ and $x_{ij}=0$ otherwise. We are also given a vector $y$, where $y_i$ is the global health cost of patient $i$. In order to estimate the average cost of each disease, the IPR algorithm works as follows:

1. Start with some initial estimates $\mu_j$, e.g. $\mu_j=1$ for all $j=1,\dots,p$. Those initial estimates are stored in the vector start.

2. Allocate the cost $y_i$ among the diseases diagnosed for patient $i$, proportionally to the current estimates $\mu_j$.

3. Update the current estimate of $\mu_j$ by averaging the specific costs obtained in step 2 for the disease $j$ over the patients having that disease.

4. Repeat steps 2 and 3 until a stopping criterion, based on relative or absolute distance between two consecutive iterations. The stopping criterion can be defined with epsabs or epsrel.

By construction, the IPR algorithm satisfies two properties. First, it allows to obtain positive estimates for each average disease cost. Secondly, it allows to retrieve the total health costs. In other words, the sum of the estimates $\mu_j$ multiplied by the number of patients suffering from $j$ is equal to the sum of the costs $y_i$.

The estimate of total cost $\tau_j$ spent for disease $j$ as well as the estimated proportion $\pi_j$ of the total costs which is allocated to disease $j$ are also returned by our function. Mathematically, $\tau_j = \sum_{i=1}^n X_{ij}\cdot\mu_j$, while $\pi_j$ is defined by $\pi_j = \tau_j/\sum_{k=1}^p \tau_k$.

Confidence intervals for each $\mu_j$ may be obtained with the ipr_boot function, which is based on bootstrapping technique.

Value

coef	A vector with the estimated average cost $\mu_j$ of each disease.
total	A vector with the estimated total cost $\tau_j$ spent for each disease.
proportions	A vector with the estimated proportion $\pi_j$ of total cost spent for each disease.
niter	The number of iterations of IPR algorithm until the stopping criterion is achieved.
esprel	The stopping criterion based on a relative distance between two consecutive iterations which has been used.
epsabs	The stopping criterion based on an absolute distance between two consecutive iterations which has been used.
detail	A matrix with the allocated costs of each disease for each patient, if det is set to TRUE.

Author(s)

Jean-Benoit Rossel ([email protected]), Valentin Rousson, and Yves Eggli.

References

Rousson, V., Rossel, J.-B. & Eggli, Y. (2019). Estimating Health Cost Repartition Among Diseases in the Presence of Multimorbidity. Health Services Research and Managerial Epidemiology, 6.

Rossel, J.-B., Rousson, V. & Eggli, Y. (2021). A comparison of statistical methods for allocating disease costs in the presence of interactions. Statistics in Medicine, 40(14).

See Also

ipr_boot.

Examples

# Here is a first example with 10 patients and 4 diseases:
X <- matrix(c(1,0,0,0,
0,1,1,0,
0,1,0,1,
1,0,0,1,
1,1,1,0,
0,0,1,1,
0,1,0,0,
1,1,0,0,
0,1,1,1,
0,0,0,1),ncol=4,byrow=TRUE)

y <- c(500,200,100,400,1000,500,100,300,800,2000)

# If we would use a linear model without intercept to estimate the average
# disease costs, we would obtain a negative value for disease 2.
lm(y~X-1)

# The IPR algorithm provides only positive estimates
ipr(X,y)


# Here is a second example:
X <- matrix(c(1,0,0,1,1,1),nrow=3,byrow=TRUE)
y <- c(5000,500,6600)

# We have three patients. The first one has only disease 1 with a cost of 5000.
# The second one has only disease 2 with a cost of 500 (i.e. ten times less
# expansive than disease 1). The third patient has both diseases with
# a cost of 6600 (i.e. 5000 + 500 + an extra cost of 1100).

# Using a linear model, one would allocate the extra cost equally between
# the three patients. The estimated average cost would thus be 5000+(1100/3)
# for disease 1 and 500+(1100/3) for disease 2.
lm(y~X-1)

# Using IPR algorithm, one allocates the extra cost taking into account that
# disease 1 is ten times more expansive than disease 2 when occuring alone.
# One thus gets an estimated average cost of 5500 for disease 1 and
# of 550 for disease 2.
ipr(X,y)

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Bootstrap Confidence Intervals for Iterative Proportional Repartition (IPR) algorithm

Description

Obtaining confidence intervals for IPR estimates via bootstrapping. Three types of bootstrap methods are implemented.

Usage

ipr_boot(X, y, method = "normal", conf.level = 0.95, B = 100, ...)

Arguments

X	Matrix with $x_{ij}=1$ if patient $i$ suffers from disease $j$ and $x_{ij}=0$ otherwise. Each row thus refers to one patient and each column to one disease. The number of columns of X corresponds to the number of diseases considered.
y	Vector where $y_i$ is the global health cost of patient $i$. The length of y must be equal to the number of rows of X.
method	A string representing the type of intervals to be computed. The value can be normal (default), basic, or perc, in order to get the normal, basic, or percentile bootrstrap intervals, respectively. See details below. A value of all can also be used, to get the three intervals.
conf.level	A scalar containing the confidence level. Default value is 0.95.
B	Number of resamplings to get the intervals. Default value is 100.
...	Other arguments of the ipr function.

Details

Let $\hat{\mu}$ be a $p$-dimensional vector with the estimated costs of the diseases, obtained via the IPR algorithm. The function ipr_boot computes confidence intervals for these costs, via a bootstrapping technique. We draw B bootstrap samples and apply the IPR algorithm on each of them. Three possibilities are then implemented to get confidence intervals.

The normal bootstrap is based on a normal distribution of the estimator. We compute $\hat{\sigma}$, a vector with the estimated standard errors of the costs of each disease, using our B samples. Confidence intervals are then obtained by computing $\hat{\mu} \pm z_{1-\alpha/2}\cdot\hat{\sigma}$.

The two other methods are computed with the $\alpha/2$- and $1-\alpha/2$-quantiles of the bootstrap distribution. Let us notice them by $q_{\alpha/2}$ and $q_{1-\alpha/2}$. The percentile bootstrap is given by these two quantiles (see Davison and Hinkley 1997, Equation (5.6) on page 194). The basic bootstrap is given by the equations $2\hat{\mu}-q_{1-\alpha/2}$ for the lower bound, and $2\hat{\mu}-q_{\alpha/2}$ for the upper bound (see Davison and Hinkley 1997, Equation (5.18) on page 203).

Considering B = 100 bootstrap samples may be enough with the normal bootstrap. However, higher values should be used with percentile or basic bootstrap.

Since the bootstrap samples are randomly defined, the user should define a random seed (e.g., set.seed(1234)) before using ipr_boot.

Value

estimate	A vector with the estimated average cost of each disease.
lower	A vector with the lower bounds of the confidence intervals.
upper	A vector with the upper bounds of the confidence intervals.
lower_normal	If the method is all, lower bounds obtained via normal bootstrap.
upper_normal	If the method is all, upper bounds obtained via normal bootstrap.
lower_perc	If the method is all, lower bounds obtained via percentile bootstrap.
upper_perc	If the method is all, upper bounds obtained via percentile bootstrap.
lower_basic	If the method is all, lower bounds obtained via basic bootstrap.
upper_basic	If the method is all, upper bounds obtained via basic bootstrap.

Author(s)

Jean-Benoit Rossel ([email protected]), Valentin Rousson, and Yves Eggli.

References

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge university press.

Rousson, V., Rossel, J.-B. & Eggli, Y. (2019). Estimating Health Cost Repartition Among Diseases in the Presence of Multimorbidity. Health Services Research and Managerial Epidemiology, 6.

Rossel, J.-B., Rousson, V. & Eggli, Y. (2021). A comparison of statistical methods for allocating disease costs in the presence of interactions. Statistics in Medicine, 40(14).

See Also

ipr

Examples

# Simulation of data with 30 patients and 7 diseases.
  set.seed(123456)
  n <- 30
  p <- 7
  X <- matrix(0,n,p,byrow=TRUE)
  for(i in 1:n) X[i,] <- rbinom(p, size=1, prob=0.5)

  # Simulation of costs. We consider lognormally distributed prices.
  mean <- 1000
  sdlog <- 0.2
  cost <- matrix(rlnorm(n*p,log(mean)-sdlog^2/2,sdlog),n,p,byrow=TRUE)
  y <- apply(X*cost,1,sum)

  # Here are the estimated costs provided by IPR algorithm,
  # and confidence intervals based on bootstrap techniques:
  ipr_boot(X,y,method="all",B=1000)

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