Missing Data Imputation and Multilevel Conditional Autoregressive Modeling of
Spatial End-Stage Renal Disease Incidence

Supraja Malladi, Indranil Sahoo, QiQi Lu

Department of Statistical Sciences and Operations Research,
Virginia Commonwealth University

###### Abstract

End-stage renal disease has many adverse complications associated with it
leading to 20-50% higher mortality rates in people than those without the
disease. This makes it one of the leading causes of death in the United
States. This article analyzes the incidence of end-stage renal disease in 2019
in Florida using a multilevel Conditional Autoregressive model under a
Bayesian framework at both the Zip Code Tabulation Area and facility levels.
The effects of some social factors and indicators of health on the
standardized hospitalization ratio of dialysis facilities are quantified.
Additionally, as kidney research studies are posed with a great burden due to
missing data, we introduce a novel method to impute missing spatial data using
spatial state space modeling. The outcomes of this study offer potentially
valuable insights for policymakers aiming to develop strategies that enhance
healthcare and service quality for disadvantaged populations.

Keywords: Bayesian hierarchical model, diabetes, dialysis facilities, federal
poverty level, Kalman smoothing, Random Forest.

##  1 Introduction

End-stage kidney/renal disease (ESKD/ESRD) is a disabling condition that often
results in premature death, causing permanent kidney failure. It is the fifth
and final stage of the chronic kidney disease (CKD) and requires dialysis or
transplantation. There has been a staggering 41.8% increase in ESRD prevalence
from 2000 to 2019 in the United States. According to the USRDS 2020 Annual
Data Report, nearly 786,000 people in the United States are living with ESRD
with 71% on dialysis and 29% with a kidney transplant. This accounted for
$37.3 billion of Medicare expenditures during 2019 (Burrows et al.,, 2022). In
2019, heart disease, chronic lower respiratory diseases, diabetes, influenza,
and pneumonia were among the 10 leading causes of death and accounted for more
than 50% of all deaths occurring in the United States (Source: National Vital
Statistics Reports: From the Centers for Disease Control and Prevention,
National Center for Health Statistics, National Vital Statistics System,
2006). Among these, diabetes and heart disease are the most common causes of
ESRD in addition to high blood pressure (check for more information on Mayo
Clinic and Penn Medicine websites).

The standardized hospitalization ratio (SHR) of admissions in dialysis
facilities is an important indicator of illness in the population. It varies
due to various demographic, socio-economic, and behavioral factors. The SHR is
calculated numerically as the ratio of the actual number of hospital
admissions for the patients in a facility over a specified time period, and
the expected number of hospital admissions for the same patients if they were
in a facility conforming to the national norm. An SHR equal to 1 indicates
that the facility’s overall hospitalization rate equals the national
hospitalization rate. The degree to which the facility’s SHR varies from 1
measures the performance of the facility in reducing overall hospitalizations.
For example, a facility’s SHR = 1.30 indicates that the facility’s covariate-
adjusted hospitalization rate exceeds the national hospitalization rate by
30%. Similarly, an SHR = 0.90 indicates that the facility’s hospitalization
rate is 10% below the national hospitalization rates (Technical Notes on the
Standardized Hospitalization Ratio (SHR), July 2021). Since 1995, the Dialysis
Facility Reports have utilized hospitalization measures. These reports are
instrumental in quality improvement, monitoring, and surveillance efforts
conducted by dialysis facilities and ESRD Networks. Hospitalization measures
play a crucial role in allocating medical costs and assisting healthcare
facilities in delivering cost-effective healthcare services. (Report for the
Standardized Hospitalization Ratio, The University of Michigan Kidney
Epidemiology and Cost Center, 2016).

In this study, we analyze the SHR of admissions in 2019 obtained at the
facility level, nested within the Zip Code Tabulation Areas (ZCTA) in the
state of Florida. We chose Florida as our study region as it had the highest
CKD prevalence rate in 2018 among Medicare beneficiaries 65 years and older
(Chronic Kidney Disease Disparities in Medicare Fee-For-Service Beneficiaries,
Data snapshot August 2020) in addition to being one of the states in the US
with the highest prevalence of diagnosed CKD among medicare beneficiaries in
2019 (Kidney Disease Surveillance System, Centers for Disease Control and
Prevention). Our goal is to investigate the incidence rate of ESRD in 2019 by
modeling associations between the SHR of hospital admissions and some select
covariates. The SHR is a good indicator of the quality and effectiveness of
care provided, the demand for services, and the ability to prevent re-
admissions. It is greatly influenced by factors such as race, gender,
excessive alcohol consumption, smoking, income, and social status among
others.

Like most healthcare data, the Medicare certified dialysis facilities data
from the Centers for Medicare & Medicaid Services (CMS) is also prone to a lot
of missing information and this poses a great burden in kidney disease
research. Due to various reasons, healthcare facilities often encounter
challenges in reporting and recording data, resulting in poor data quality and
analysis. The presence of missing data makes it challenging to draw accurate
conclusions about different patient groups. Moreover, the need for additional
data cleaning and preparation further prolongs the time required for research
development (Stiglic et al.,, 2019). In general, it is impossible to address
the mechanism that resulted in missing data since no test can prove how
missing data is generated (Marino et al.,, 2021). Ghazi et al., (2022)
examined CKD prevalence and its association with air pollution exposure in
Minnesota’s Fairview health system, adjusting for variables with more than 5%
missing data using an indicator variable under the assumption of randomness
(missing completely at random). Stiglic et al., (2019) performed a study on
type 2 diabetes mellitus data obtained from three health care centers in
Slovenia with a large amount ($\geq 73\%$) of missing data in one of the
centers. Simulating missing values from two centers, the study showed a
significant model prediction decline with incomplete record removal. Blazek et
al., (2021) implemented multivariate imputation by chained equations (MICE)
using the ‘mice’ package in R (Van Buuren and Groothuis-Oudshoorn,, 2011) on a
subset of the National Health and Nutrition Examination Survey (NHANES)
2015-2016 data set (74% complete cases). The study, focusing on hypertension-
kidney disease link in adults, imputed 26 datasets using methods like linear
regression, predictive mean matching, and logistic regression via ’mice’,
observing convergence between actual and imputed data. Shara et al., (2015)
assessed the performance of listwise deletion, mean of serial measures,
adjacent value, multiple imputation, and pattern-mixture in imputing missing
renal function data in the Strong Heart Study. The study found that there was
no method superior to the rest due to overestimating and underestimating in
different methods.

As most of the variables used in this study have missing values and are
spatially correlated, we propose a spatial state space modeling approach for
missing value imputation. We also make comparisons to imputations generated by
the Random Forest method in the ‘mice’ package. Once the missing data is
imputed, a multilevel spatial model at both the ZCTA and hospital levels is
implemented under a fully Bayesian framework to study the effect of some
social factors and indicators of health on the SHR.

The use of multilevel models to study the incidence of repeated areal Medicare
certified dialysis facilities data is notably limited within the existing
literature. Li et al., (2021) proposed a two-level spatiotemporal functional
model at the facility level and the region level, to analyze hospitalization
rates among dialysis facilities from the USRDS data set. Jamal et al., (2020)
conducted a social epidemiology study using survival data and fit four
multilevel Poisson regression models to examine the effects of facility and
regional variations in the incidence of developing hemodialysis-associated
infections (HAIs) as well as its associated factors in more than 6000 patients
with ESRD in Japan. Additionally, Sood et al., (2014) evaluated geographic and
facility-level variations using a three-level logistic model for data obtained
from the Canadian Organ Replacement Register from January 2001 to December
2010. Our dataset comprises multiple hospitals within each ZCTA, effectively
yielding replications across spatial units. Hence, we employ a Conditional
Autoregressive (CAR) covariance structure to capture the spatial correlation
in SHR at the ZCTA level, and an uncorrelated error structure at the hospital
level within each ZCTA. This approach enables us to effectively capture and
model the spatial dependencies and heterogeneity present in the dialysis
facility data.

The rest of this article is organized as follows. In Section 2, We first
present the covariates included in this study along with their missing value
information and quality restrictions imposed upon them. Section 3 introduces
the spatial local level model and evaluates its performance in spatial missing
value imputation. The multilevel spatial CAR model is developed in Section 4
with a modified neighborhood matrix. The details of the model fitting and the
results are shown in Section 5. Finally, some concluding remarks close the
article in Section 6.

##  2 Data Description

The Medicare certified dialysis facilities data furnishes comprehensive
information about clinical and patient metrics pertaining to Medicare
certified ESRD facilities. This dataset encompasses a wide array of data,
including patient attributes, treatment trends, hospitalization occurrences,
mortality rates, and patterns of transplantation. (see Medicare dialysis
facilities). For this study, we used the publicly available 2019 CMS Medicare
certified dialysis facilities data for the state of Florida. Originally, there
were 494 dialysis facilities in Florida across 334 ZCTAs consisting of 27
facilities in Miami, 18 facilities in Jacksonville, 15 in Orlando, 12 in Tampa
among others. The SHR of hospital admissions in 2019 was obtained to be
modeled as a response variable in Section 4. The following variables belonging
to the incident population at the facility level for 2019 were obtained as
covariates for the study: percentage of patients with diabetes as the primary
cause of ESRD, percentage of patients with hypertension as the primary cause
of ESRD, percentage of African Americans, number of staff members working for
the facility, percentage of septicemia/sepsis cases, and the percentage of
female patients.

Figure 1 depicts the proportion of missing values in each of the variables in
the data in the left panel and how often each combination of missing values
occurs in the right panel. After the variables were selected, the proportions
of missing covariates for each hospital were computed and hospitals that had
over 80% missing covariates were removed from the analysis. This resulted in
data for 449 facilities across 325 ZCTAs. Note that there were no missing
values for the percentage of patients with septicemia and the number of staff
working for the facility. The rest of the covariates had roughly equal
proportions of missing values, with 24.04% missing values for each. In
addition, the response variable SHR had 0.85% of missing values. To
incorporate ZCTA-level socioeconomic conditions in our study, we obtained the
Federal Poverty Level (FPL) score as an additional covariate from the Social
Deprivation Index measures developed using the American Community Survey data
at the ZCTA level. The FPL score represents the percent population less than
100% FPL and indicates the extent of economic disadvantage in a community. It
is computed using the formula:

$\mbox{FPL score}=\frac{(\mbox{Population}<0.99~{}\mbox{FPL})}{(\mbox{Total
Population})}.$

Figure 1: Proportion of missing values in the data (left) and Combinations of
missing values in the data (right)

The spatial distributions of some of the covariates and the SHR are shown in
Figure 2. Some of the ZCTAs in Spring Hill, Clermont, Tampa, Orange City,
Jacksonville, Orlando, and Gainsville appeared to have high percentages of
patients with diabetes and hypertension as the primary causes of ESRD, Federal
Poverty Level score, and septicemia. Moreover, Miami-Dade, Palm Beach, Leon,
Orange, and Duval counties indicated more than 75% of African American
patients in dialysis facilities in 2019.

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)

Figure 2: Spatial plots of a) Percentage of patients with diabetes, b)
Percentage of patients with hypertension as the primary cause of ESRD, c)
Percentage of patients with septicemia, d) Percentage of African American
patients in the facility, e) Federal poverty level score, f) log(Observed
SHR). Average values were computed for ZCTAs with multiple hospitals for each
variable.

##  3 Missing value imputation using spatial state space model

We obtained the ZCTAs for each hospital in our data using the End-Stage Renal
Disease Quality Incentive Program (ESRD QIP) data provided by the CMS. The
distances of each hospital from the centroid of Florida were calculated. Since
certain ZCTAs had multiple hospitals, all the facilities belonging to the same
ZCTA happened to have the same distance from the centroid. Therefore, to
convert the distances to unique values, hospitals were first arranged in the
ascending order of distances (including ties) and then insignificant decimal
points were added to the end of the duplicated distances.

Next, we borrowed the idea from state space models in continuous time as
described in Koopman et al., 2018b . In continuous time series, the time gaps
between consecutive observations are allowed to vary freely unlike in discrete
time series where time gaps are equal (Koopman et al., 2018a, ). The local
level model in continuous time was slightly adapted by considering the ordered
distances as continuous time. As the data has no ‘time’ variable involved, we
call our model a ‘Spatial state space’ model, which is described in detail in
Section 3.1.

###  3.1 Spatial state space model

Let $d_{i}$ denote the distance of the $i^{th}$ hospital from the centroid of
the state for $i=1,\ldots,M$. Here, $M=449$ denotes the total number of
hospitals used in the study. While $d_{i}$ can be any non-negative number, in
our case, it is strictly positive. Similar to the sequential ordering of time
in continuous time series, distances are arranged in ascending order with

$d_{1}<d_{2}<d_{3}<\ldots<d_{M}.$

Our spatial local level model for spatial data $\\{x_{i}\\}_{i=1}^{M}$ has the
form:

$\displaystyle x_{i}=\alpha_{i}+\epsilon_{i},$
$\displaystyle\quad\epsilon_{i}\sim N(0,\sigma^{2}_{\epsilon}),$ (1)
$\displaystyle\alpha_{i+1}=\alpha_{i}+\eta_{i},$
$\displaystyle\quad\eta_{i}\sim N(0,\delta_{i}\sigma_{\eta}^{2}),$ (2)

for $~{}i=1,\ldots,M$. Equation (1) is referred to as the observation or
measurement equation, while Equation (2) is called the state or transition
equation. Here, $\alpha_{i}$ is a continuous analog of Gaussian random walk.
The measurement errors $\\{\epsilon_{i}\\}$ are assumed to be independent with
a fixed variance $\sigma^{2}_{\epsilon}$. Note that the variance of
$\epsilon_{i}$ could be hospital-varying, i.e.
$\mbox{Var}(\epsilon_{i})=\sigma^{2}_{\epsilon}(i)$ for all $i$. The system
errors $\\{\eta_{i}\\}$ are also assumed to be independent. The variance of
$\\{\eta_{i}\\}$ depends on the difference in distance $\delta_{i}$ between
the $i^{th}$ hospital and the $(i-1)^{th}$ hospital after ordering, where
$\delta_{i}=d_{i}-d_{i-1}$ for $i=1,\ldots,M$ with $d_{0}=0$. In addition,
$\\{\epsilon_{i}\\}$ and $\\{\eta_{i}\\}$ are assumed to be independent of
each other for all $i$ and independent of the initial state $\alpha_{1}$.

The parameters in the spatial state space model are estimated via maximum
likelihood methods by Kalman filter. Once the model parameters are estimated,
the missing values are replaced by their smoothed values with routine
applications of the Kalman filter and smoother. It is known that the Kalman
filter is powerful in recursive estimation and has good compatibility in
handling missing data (Saputra et al.,, 2021). Adejumo O.A., (2021) compared
the performances of the Kalman filter algorithm to mean imputation, median
imputation, and linear interpolation with 12% of missing values in time series
and found that the Kalman filter algorithm performed better under linear trend
structure. For missing dialysis facilities data imputation, a vast amount of
dialysis facilities data (22 million data points) obtained in collaboration
with the Centers for Dialysis Care and Lytics Health AB was analyzed by
Mattias Albinsson, (2017). The study found that time series models using the
Kalman filter gave the best results when compared to traditional imputation
methods such as likelihood-based imputation, multiple imputation, and ‘mice’.
Some previous studies which compared different imputation techniques to the
Random Forest method in the ‘mice’ package found better imputations using the
latter (Waljee et al.,, 2013; Finch et al.,, 2016; Young,, 2017). In addition,
‘mice’ is found to impute better with a higher number of trees. While there
are many advantages to using ‘mice’, it can be computationally expensive when
a higher number of trees are used since they have greater run times (Umar and
Gray,, 2023).

In order to evaluate the performance of imputations using the spatial state
space model by the Kalman filter, we compared our method with the Random
Forest method in the ‘mice’ package in Section 3.2.

###  3.2 Imputing covariates

The spatial state space model was fit to each of the covariates mentioned in
Section 2 individually using the Continuous Time Structural Equation Modelling
(ctsem) package in R (Driver et al.,, 2017) to impute missing values in the
data. Since missing values in SHR (response variable) automatically get
updated under the Bayesian setting, it was not imputed here.

Additionally, we performed 1000 simulations using both the spatial state space
model and the Random Forest method in the ‘mice’ package (Van Buuren and
Groothuis-Oudshoorn,, 2011) to impute each of the covariates individually for
fair comparison although ‘mice’ can perform multivariate imputations. A cross-
validation method was used where the complete data were divided into 80%
comprising of the training data set and 20% comprising of test data. The mean
and standard deviation of the symmetric mean absolute percentage error (SMAPE)
were calculated to assess the imputations. The SMAPE is computed as follows:

$SMAPE=\frac{100}{N}\sum_{i=1}^{N}\frac{|\hat{x}_{i}-x_{i}|}{|\hat{x}_{i}|+|x_{i}|},$

where $x_{i}$ is the observed value for the hospital $i$ in the testing
dataset, $\hat{x}_{i}$ is its imputed value, and $N$ is the total number of
hospitals in the testing dataset. Table 1 provides the imputation performance
measures of both methods after simulations. The mean and standard deviation
values of SMAPE across both methods are quite close to each other.

Table 1: Performance measures of imputations using state space model and
Random Forest in ‘mice’ after 1000 simulations.

Variable | State space model | Random Forest in mice
---|---|---
Mean(SMAPE) | S.D(SMAPE) | Mean(SMAPE) | S.D(SMAPE)
% patients with diabetes as the primary cause of ESRD | 0.521 | 0.048 | 0.525 | 0.052
%patients with hypertension as the primary cause of ESRD | 0.604 | 0.052 | 0.595 | 0.050
% African American patients | 0.783 | 0.062 | 0.796 | 0.070
% Female patients in the facility | 0.250 | 0.025 | 0.241 | 0.019

To further study the efficacy of the proposed imputation technique, we
performed cross-validation on the percentage of sepsis cases and the number of
staff since they had no missing values. Data were randomly partitioned into
80%, 70%, and 60% training with 20%, 30%, and 40% testing (missing values)
respectively. Imputations using the proposed spatial state space model and
Random Forest method in the ‘mice’ package were performed. In all three cases,
the imputed distributions of the variables, as shown in Figures 3 and 4, are
closer to the true distributions when using our proposed method. In addition,
the performance measures of both methods are provided in Table 2. The SMAPE
values for the state space model are much lower than those for the Random
Forest method, showing the superior imputation performance of the proposed
spatial state space model compared to the Random Forest method in ‘mice’.

Table 2: Performance measures of imputations using state space model and
Random Forest in ‘mice’.

Variable | State space model | Random Forest in mice
---|---|---
SMAPE(60/40) | SMAPE(70/30) | SMAPE(80/20) | SMAPE(60/40) | SMAPE(70/30) | SMAPE(80/20)
% patients with septicemia | 0.162 | 0.140 | 0.093 | 0.466 | 0.476 | 0.443
Number of staff working for the facility | 0.167 | 0.125 | 0.063 | 0.485 | 0.512 | 0.493

---
(a)
(b)
(c)
Figure 3: Comparing imputations using the Random Forest method in the ‘mice’
package to those using spatial state space model for the percentage of
patients with septicemia across (a) 60/40, (b) 70/30, (c) 80/20 train/test
split and cross-validation.
---
(a)
(b)
(c)
Figure 4: Comparing imputations using Random Forest method in the ‘mice’
package to those using spatial state space model for the number of staff
across (a) 60/40, (b) 70/30, (c) 80/20 train/test split and cross-validation.

##  4 Multilevel Spatial CAR Model

Let $Y_{kj}$ denote the SHR (in the logarithmic scale) of the $j^{th}$
hospital in the $k^{th}$ ZCTA, $j=1,\ldots,m_{k},k=1,\ldots,K$, where $m_{k}$
denotes the number of hospitals within the $k^{th}$ ZCTA on which data are
available, and $K$ denotes the number of ZCTAs considered. Here, $K=325$.

The $K\times K$ spatial neighborhood matrix is specified by
$\bm{W}=(\omega_{kk^{\prime}})_{k,k^{\prime}=1,\ldots,K}$ in our study. Since
not all ZCTAs in Florida have dialysis facilities, we end up with multiple
ZCTAs in our dataset having no neighbors. Instead of discarding these ZCTAs
with no neighbors, we find their nearest neighbors based on distances among
the ZCTAs. For ZCTAs with no neighbors, we designate the ZCTA with the
smallest distance from it as its neighbor. Thus, we create an augmented
neighborhood matrix $\bm{W}^{*}=(\omega_{kk^{\prime}}^{*})_{k,k^{\prime}}$,
where $\omega_{kk^{\prime}}^{*}=1$ if ZCTAs $k$ and $k^{\prime}$ are adjacent
$(k\sim k^{\prime})$ or if ZCTA $k^{\prime}$ is closest to ZCTA $k$ when ZCTA
$k$ has no adjacent neighbor with dialysis facilities, and is zero otherwise.

Now, we fit log(SHR) in a two-level spatial CAR model introduced in Lee,
(2013):

$\displaystyle Y_{kj}|\mu_{kj}$ $\displaystyle\sim N(\mu_{kj},\nu^{2}),$ (3)
$\displaystyle\mu_{kj}$
$\displaystyle=\bm{x}^{T}_{kj}\bm{\beta}+O_{kj}+\phi_{k}$ (4)
$\displaystyle\phi_{k}|\bm{\phi}_{-k}$ $\displaystyle\sim
N\Biggl{(}\frac{\rho\sum_{k^{\prime}\neq
k}\omega^{*}_{kk^{\prime}}\phi_{k^{\prime}}}{\rho\sum_{k^{\prime}\neq
k}\omega^{*}_{kk^{\prime}}+1-\rho},\frac{\tau^{2}}{\rho\sum_{k^{\prime}\neq
k}\omega^{*}_{kk^{\prime}}+1-\rho}\Biggl{)}$ (5)

where $\bm{\phi}=(\phi_{1},\ldots,\phi_{K})^{T}$ is the spatial random effect
at the ZCTA level and is modeled by a Conditional Autoregressive prior
proposed by Leroux et al., (2000). Also,
$\bm{\phi}_{-k}=(\phi_{1},\ldots,\phi_{k-1},\phi_{k+1},\ldots,\phi_{K})^{T}$
denotes the vector of spatial random effects after deleting the random effect
corresponding to the $k^{th}$ ZCTA. The logarithms of ZCTA population, denoted
by $O_{kj}$, were introduced as the offset for hospital $j$ within ZCTA $k$ in
Equation (4). $\bm{x}_{kj}$ in Equation (4) denotes a vector of all $p$
covariates for hospital $j$ within ZCTA $k$ and $\mu_{kj}$ is the mean
function for the hospital $j$ in the ZCTA $k$, which is a linear combination
of all the covariates (after accounting for the offset) and the spatial random
effect at the ZCTA level.

In Equation (5), $\rho$ is the spatial autocorrelation parameter. $\rho=0$
corresponds to a lack of spatial interdependence, i.e. the classical
multilevel model with 2 levels. On the other hand, $\rho=1$ leads to the
intrinsic CAR model (Congdon, (2010), p. 183 - 184). Finally, $\nu^{2}$ is the
uncorrelated error variance at the hospital level and is an additional scale
parameter required when a Gaussian family is used.

The following prior distributions were implemented on the model parameters:

$\displaystyle\bm{\beta}$ $\displaystyle\sim
N(\bm{\mu}_{\beta},\bm{\Sigma}_{\beta}),$ $\displaystyle\tau^{2}$
$\displaystyle\sim\mbox{Inverse-Gamma}(a,b),$ $\displaystyle\nu^{2}$
$\displaystyle\sim\mbox{Inverse-Gamma}(a,b),$ $\displaystyle\rho$
$\displaystyle\sim\mbox{Uniform}(0,1),$

where $\bm{\mu}_{\beta}$ and $\bm{\Sigma}_{\beta}$ are the mean and variance
of the model coefficients, respectively. For the inverse Gamma priors for the
variance components, $\tau^{2}$ and $\nu^{2}$, $a$ and $b$ were chosen to be
$a=1,b=0.01$.

##  5 Results

A fully Bayesian framework has been implemented to fit this model to the data.
After discarding 20,000 samples as burn-in, posterior inferences on model
parameters were made from 50,000 posterior samples. Posterior predictive
distributions were computed for the missing responses, and they were imputed
based on 50,000 samples from the posterior predictive distribution. Table 3
shows the posterior mean and 95% credible intervals of the model parameters.
It is evident that the ‘percentage of patients with Diabetes as the primary
cause of ESRD’, the ‘percentage of patients with Septicemia’, and the ‘ZCTA
Federal Poverty Level score’ have a significant effect on the SHR of
admissions at the facility and ZCTA levels. In addition, the variance
components in the model, $\nu^{2}$ and $\tau^{2}$, and spatial autocorrelation
$\rho$ are significant.

The Relative Squared Error (RSE) is used to compare the fitted multilevel
spatial model with a persistence (baseline) model, where averages of realized
values in ZCTAs were used as predicted values. RSE is a relative metric that
divides the squared error of a predictive model by the squared error of a
simple model. RSE is not sensitive to the mean and the scale of predictions.
It is calculated as follows:

$RSE=\frac{\sum_{k=1}^{K}\sum_{j=1}^{m_{k}}(Y_{kj}-\widehat{Y}_{kj})^{2}}{\sum_{k=1}^{K}\sum_{j=1}^{m_{k}}(Y_{kj}-\bar{Y}_{k})^{2}}$

where

$\bar{Y}_{k}=\frac{1}{m_{k}}\sum_{j=1}^{m_{k}}Y_{kj}$

for $k=1,\ldots,K$. If the RSE is lower than 1, then the predictive model
performs better than the persistence model. The relative squared error between
the observed and predicted values of log(SHR) is 0.304, indicating that our
model performs well.

Table 3: Model summary of the two-level CAR model

| Mean | 2.5% | 97.5%
---|---|---|---
Intercept | -10.3753 | -10.6752 | -10.0745
% with Diabetes as the primary cause of ESRD | -0.0036 | -0.0069 | -0.0002
% with Hypertension as the primary cause of ESRD | -0.0023 | -0.0054 | 0.0008
% African American patients in the facility | 0.0003 | -0.0021 | 0.0026
Number of Staff working for the facility | -0.0034 | -0.0088 | 0.0021
% with Septicemia in the facility | 0.0222 | 0.0144 | 0.0300
% Female patients in the facility | 0.0003 | -0.0030 | 0.0036
ZCTA Federal Poverty Level score | 0.0028 | 0.0004 | 0.0052
$\nu^{2}$ | 0.0488 | 0.0376 | 0.0630
$\tau^{2}$ | 0.2219 | 0.1816 | 0.2695
$\rho$ | 0.0018 | 0.0001 | 0.0066

Figure 5 shows the fitted log(SHR) and the observed log(SHR) for comparison
across all ZCTAs in Florida containing dialysis facilities. Average values of
the log(SHR) were computed for ZCTAs with multiple hospitals before plotting.
Some of the ZCTAs in Miami, Oviedo, West Palm-Beach, Orlando, and Bartow
indicated high SHR ($>2$) among facilities. While some of the ZCTAs belonging
to Naples, Pensacola, Fort Lauderdale, Titusville, and Jacksonville indicated
values of the SHR $<1$. Facilities in Walton, Tallahassee, Gainsville, and
Collier have higher predicted values than the observed, while facilities in
Lafayette and Hendry have lower values than predicted. Overall, the predicted
values fall very close to the observed values of SHR. It is evident that
facilities in ZCTAs with high poverty levels, high African American population
and higher percentage of patients with septicemia are more susceptible to
higher SHR values when compared to the SHR of admissions for the state of
Florida. This is consistent with findings from previous studies on nephrology
care. The lowest rates of pre-ESRD care among dialysis facilities were
discovered in counties with high African American populations and low levls of
educational attainment (Hao et al.,, 2015; Prakash et al.,, 2010).
Moreover,Chang et al., (2020) and A et al., (2005) indicated an association
between septicemia and increased risks of adverse outcomes and
hospitalization.

|
---|---

Figure 5: a) Observed log(SHR) and b) fitted log(SHR) across dialysis
facilities in Florida. Average values of the SHR were computed for ZCTAs with
multiple hospitals before plotting.

##  6 Discussion

In this study, we implemented a multilevel spatial model to study the effect
of some comorbidities, social, and economic factors on the SHR of dialysis
facilities in 2019, obtained from the Centers for Medicare and Medicaid
Services for the state of Florida. A major challenge in working with such data
is that they usually have a lot of missing values. To overcome this challenge,
we have proposed a novel spatial state space model for missing data imputation
in the Medicare certified dialysis facilities data. We also established that
the proposed model is very efficient in imputing missing values based on
simulations and comparisons to the Random Forest method in the ‘mice’ package.
This method can therefore be used to impute spatial data since it accounts for
spatial autocorrelation in the data. Moreover, the multilevel CAR model
efficiently incorporated spatial correlation and hierarchy inherent in the
data. Thus, studies like ours can be very useful to ESRD networks and dialysis
facilities in targeting patient inclusion, improving facility level care, and
reducing the number of re-admissions.

In our study, it was shown that covariates such as “percentage of patients
with diabetes as the primary cause of ESRD”, “percentage of patients with
septicemia”, and the “Federal Poverty Level score” had a significant impact on
the SHR of admissions when compared to other variables. However, we found
facilities in Florida with a higher percentage of patients with diabetes as
the primary cause of ESRD to have significantly lower SHR in 2019, which is
contrary to what we anticipated. In the past, ESRD burden has been found to be
greatly associated with comorbid conditions that include diabetes and
hypertension (Qi et al.,, 2019; Hsu et al.,, 2009; McCullough et al.,, 2019;
Go et al.,, 2018), poverty at the county level (Crews et al.,, 2014), being
African American (Bock et al.,, 2019), and deprivation (Occelli et al.,,
2014). Bilgel, (2019) studied county level ESRD prevalence averaged over the
2006-2015 period across the US. The study found the strongest ESRD increasing-
effects of poverty, high levels of income inequality, and a high percentage of
uninsured people in most counties of Florida, Georgia, Carolinas, and
Mississippi. The study also found rising hypertension prevalence in Florida in
contrast to the results in our study. Facility and ZCTA level ESRD incidence
rates are much lower compared to prevalence rates since ESRD is a long-term
condition that requires monitoring over a period and incidence only includes
one year of data. The risk factors for both prevalence and incidence may also
vary due to this. Similar to hypertension, “the percentage of African
Americans” were not significant for Florida, although these factors are well
established as important contributors to ESRD in literature. Since we also
focus on missing spatial data imputation for a single year, we did not use
prevalence rates for our study. Although age was not included as a covariate
in our study, the state of Florida has been one of the states with the highest
percentage of the older population (65 years and above) over the past few
years. Therefore, a spatial causal inferential framework is needed to
establish the relationship between SHR and the demographic, socio-economic,
behavioral, and environmental covariates while accounting for potential
confounding variables like age, race, comorbidities, and access to healthcare.
Spatial causal inference can help identify spatially varying risk factors,
such as environmental exposures, socioeconomic disparities, and healthcare
accessibility, that contribute to ESRD prevalence and incidence. Moreover, it
enables the evaluation of the effectiveness of interventions and policies
targeted at reducing the burden of ESRD within specific geographic areas. By
understanding spatial causality, policymakers can allocate resources more
efficiently, implement targeted prevention strategies, and tailor healthcare
interventions to address the specific needs of high-risk populations in
particular regions. Reich et al., (2021) reviews computational methods for
causal inference under the spatial setting, which can be easily inherited into
kidney research studies.

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# Classical and quantum Cosmology of the Sáez-Ballester theory

J. Socorro<EMAIL_ADDRESS>M. Sabido<EMAIL_ADDRESS>L. Arturo
Ureña-López<EMAIL_ADDRESS>Departamento de Física, DCI, Campus León,
Universidad de Guanajuato, C.P. 37150, Guanajuato, México

###### Abstract

We study the generalization of the Sáez-Ballester theory applied to a flat FRW
cosmological model. Classical exact solutions up to quadratures are easily
obtained using the Hamilton-Jacobi approach. Contrary to claims in the
specialized literature, it is shown that the Sáez-Ballester theory cannot
provide a realistic solution to the dark matter problem of Cosmology.
Furthermore the quantization procedure of the theory can be simplified by
reinterpreting the theory in the Einstein frame, where the scalar field can be
interpreted as part of the matter content of the theory, in this approach,
exact solutions are also found for the Wheeler-DeWitt equation in the quantum
regime.

###### pacs:

04.20.Fy; 04.20.Jb; 04.60.Kz; 98.80.Qc.

##  I Introduction

The inclusion of scalar fields into homogeneous cosmologies is a typical
practice to study different scenarios, such as inflation, dark matter, and
dark energyCopeland:2006wr . However, since the early seventies, the problem
exists of finding the appropriate sources of matter and its corresponding
Lagrangian to solve an specific scenarioryan1 ; ryan .

In this respect, Saez and Ballester (SB)s-b  formulated a scalar-tensor theory
of gravitation in which the metric is coupled to a dimensionless scalar field
in order to solve the so-called missing matter problem in Cosmology. Some
works about the classical regime are already present in the literaturesingh ;
shri ; mohanty ; singh-shri . In particular, in ref. singh-shri  the authors
consider the coupling parameter time-dependent and take a particular ansatz
for mathematical convenience for solving the field equations.

In spite of a the dimensionless character of the scalar field, an antigravity
regime appears, and this fact has been used to suggest a new possible way to
solve the missing matter problem in non-flat FRW cosmologies. On the other
hand, the quantization program of the theory has yet to be made.

In this paper, we shall study a generalization of the SB theory and transform
it into a conventional tensor theory, where the dimensionless scalar field is
interpreted as an exotic matter. We found the general behaviour for the
kinetic scalar field dependent to the scale factor of the universe, but the
behaviour corresponds to stiff matter and not for a dust universe, then the
missing matter problem is not solved.

With respect to the quantization program, in this approach we can construct
the quantization program of the theory using the usual ADM formalismryan1 .
Also, we can in principle quantize the theory following the Loop Quantum
Cosmoloy program.

In this work, we shall use this formulation to obtain classical and quantum
solutions in quadratures, for the flat barotropic FRW cosmology, including a
cosmological term $\lambda$.

The paper is arranged as follows, In section II we write the generalization
Sáez-Ballester formalism in the usual manner, that is, we calculate the
corresponding energy-momentum tensor to the scalar field and give the
equivalent lagrangian density. Next, we proceed to obtain the corresponding
canonical lagrangian ${\cal L}_{can}$ to a flat FRW universe through the
lagrange transformation, we calculate the classical hamiltonian, we also
present solutions to some models. In section III, using the transformation and
the Hamiltonian constraint ${\cal H}$, f we find the Wheeler-DeWitt (WDW)
equation of the corresponding cosmological model under study. Section IV is
devoted to conclusions and outlook.

##  II Generalized Saez-Ballester theory

The simplest generalization of the Sáez-Ballester theorys-b  with a
cosmological term is

${\cal
L}_{geo}=\left(R-2\lambda-F(\phi)\phi_{,\gamma}\phi^{,\gamma}\right)\,,$ (1)

where $R$ the scalar curvature,
$\phi^{,\gamma}=g^{\gamma\alpha}\phi_{,\alpha}$, and $F(\phi)$ is a
dimensionless and arbitrary functional of the scalar field. According to
common wisdom, the Lagrangian (1) would correspond to a scalar field theory
without scalar potential but with an exotic kinetic term.

The complete action is then

$I=\int_{\Sigma}\sqrt{-g}({\cal L}_{geo}+{\cal L}_{mat})\,d^{4}x\,,$ (2)

where we have included a matter Lagrangian ${\cal L}_{mat}$, and $g$ is the
determinant of metric tensor. The field equations derived from the above
action are

$\displaystyle
G_{\alpha\beta}+g_{\alpha\beta}\lambda-F(\phi)\left(\phi_{,\alpha}\phi_{,\beta}-\frac{1}{2}g_{\alpha\beta}\phi_{,\gamma}\phi^{,\gamma}\right)$
$\displaystyle=$ $\displaystyle 8\pi GT_{\alpha\beta}\,,$ (3a) $\displaystyle
2F(\phi)\phi^{,\alpha}_{\,\,;\alpha}+\frac{dF}{d\phi}\phi_{,\gamma}\phi^{,\gamma}$
$\displaystyle=$ $\displaystyle 0\,,$ (3b)

in which $G$ is the gravitational constant, and a semicolon means covariant
derivative.

The same set of equations(3a,3b) is obtained if we consider the scalar field
$\phi$ as part of the matter budget, i.e. say $\rm{\cal L}_{\phi}=\rm
F(\phi)g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}$. In this new line of
reasoning, action (2) can be rewritten as a geometrical part (Hilbert-Einstein
with $\Lambda$) and matter content (usual matter plus a term that corresponds
to the scalar field component of Sáez-Ballester theory),

$I=\int_{\Sigma}\sqrt{-g}\left(R-2\lambda+{\cal L}_{mat}+{\cal
L}_{\phi}\right)\,d^{4}x\,.$ (4)

Even though the philosophy is different to that of the original SB theory, the
similarity of the latter to a standard scalar field theory at the classical
level will help us to infer the correspondence quantum formulation. We expect
the quantum picture will also be the correct one for the SB theory, as all the
formulation is based upon the same (classical) Hamiltonian constraint.

Using this action we obtain the classical Hamiltonian of the generalized SB
theory for a Friedmann-Robertson-Walker background. Let us start with the line
element for a homogeneous and isotropic universe,

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-\kappa
r^{2}}+r^{2}d\Omega^{2}\right]\,,$ (5)

where $a(t)$ is the scale factor, $N(t)$ is the lapse function, and $\kappa$
is the curvature constant that can to take the values $0$, $1$ and $-1$, for
flat, closed and open universe, respectively. The total Lagrangian density
then reads

${\cal L}=\frac{6\dot{a}^{2}a}{N}-6\kappa
Na+\frac{F(\phi)a^{3}}{N}\dot{\phi}^{2}+16\pi GNa^{3}\rho-2Na^{3}\lambda\,,$
(6)

where $\rho$ is the matter energy density; we will assume that it complies
with a barotropic equation of state of the form $p=\gamma\rho$, where $\gamma$
is a constant. The conjugate momenta are obtained from

$\displaystyle\Pi_{a}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}}{\partial\dot{a}}=\frac{12a\dot{a}}{N},\qquad\rightarrow\qquad\dot{a}=\frac{N\Pi_{a}}{12a}\,,$
$\displaystyle\Pi_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}}{\partial\dot{\phi}}=\frac{2Fa^{3}\dot{\phi}}{N}\,,\qquad\rightarrow\qquad\dot{\phi}=\frac{N\Pi_{\phi}}{2Fa^{3}}\,.$
(7)

From the canonical form of the Lagrangian density (6) and the solution for the
barotropic fluid equation of motion we find the Hamiltonian density for this
theory

${\cal
H}=\frac{a^{-3}}{24}\left[a^{2}\Pi_{a}^{2}+\frac{6}{F(\phi)}\Pi_{\phi}^{2}+144\kappa
a^{4}+48a^{6}\lambda-384\pi G\rho_{\gamma}a^{3(1-\gamma)}\right],$ (8)

where $\rho_{\gamma}$ is an integration constant.

###  II.1 Classical solutions for flat FRW

Using the transformation $\Pi_{q}=\frac{dS_{q}}{dq}$, the Einstein-Hamilton-
Jacobi corresponding to Eq. (8) is

$a^{2}\left(\frac{dS_{a}}{da}\right)^{2}+\frac{6}{F(\phi)}\left(\frac{dS_{\phi}}{d\phi}\right)^{2}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}=0\,,.$ (9)

The EHJ equation can be further separated in the equations

$\displaystyle\frac{6}{F(\phi)}\left(\frac{dS_{\phi}}{d\phi}\right)^{2}$
$\displaystyle=$ $\displaystyle\mu^{2}\,,$ (10) $\displaystyle
a^{2}\left(\frac{dS_{a}}{da}\right)^{2}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}$ $\displaystyle=$ $\displaystyle-\mu^{2}\,,$
(11)

where $\mu$ is a separation constant. With the help of Eqs. (7), we can obtain
the solution up to quadratures of Eqs. (10) and (11),

$\displaystyle\int\sqrt{F(\phi)}\,d\phi$ $\displaystyle=$
$\displaystyle\frac{\mu}{2\sqrt{6}}\int a^{-3}(\tau)\,d\tau\,,$ (12a)
$\displaystyle\Delta\tau$ $\displaystyle=$
$\displaystyle\int\frac{a^{2}da}{\sqrt{\frac{8}{3}\pi
G\rho_{\gamma}a^{3(1-\gamma)}-\frac{\lambda}{3}a^{6}-\nu^{2}}}\,,$ (12b)

with $\nu=\frac{\mu}{12}$

Eq. (12a) readily indicates that

$F(\phi)\dot{\phi}^{2}=6\nu^{2}a^{-6}(\tau)\,,$ (13)

despite of the particular form of the functional $F(\phi)$. Also, this
structure is directly obtained for this model solving the equation (3b).
Moreover, the matter contribution of the SB scalar field to the rhs of the
Einstein equations would be

$\rho_{\phi}=\frac{1}{2}F(\phi)\dot{\phi}^{2}\propto a^{-6}\,.$ (14)

That is, the contribution of the scalar field is the same as that of stiff
matter with a barotropic equation of state $\gamma=1$.

This is an interesting result, since the original SB theory was thought of as
a form to solve the missing matter problem of Cosmology, now generically
called the dark matter problem; to solve the latter, one needs a fluid
behaving as dust with $\gamma=0$. It is surprising that such a general result
remain unnoticed until now in the literature about SB.

Also, that we have identified the general evolution of the scalar field with
that of a stiff fluid means that the Eq. (12b) can be integrated separately
without a complete solution for the scalar field. For completeness, we give
below a compilation of exact solutions in the case of the original SB theory.

If $F(\phi)=\omega\phi^{m}$, then we have two cases that correspond to $m=-2$
and $m\neq-2$; the general solution for the scalar field is

$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{6\nu}{\sqrt{6\omega}}\int
a^{-3}(\tau)d\tau\right]$&\qquad m = -2\\\
$\left[\frac{2\nu(m+2)}{\sqrt{6\omega}}\int
a^{-3}(\tau)d\tau\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$\\\
\end{tabular}\right.$ (15)

which can be completely integrated once the time dependence of the scale
factor $a$ has been resolved.

  * •

Stiff plus a cosmological constant, $\gamma=-1$. The master equation become

$\Delta\tau=\int\frac{a^{2}da}{\sqrt{b_{-1}a^{6}-\nu^{2}}}\,,$ (16)

where $b_{-1}=\frac{8}{3}\pi G\rho_{-1}-\frac{\lambda}{3}$, whose solution is

$\Delta\tau=\frac{1}{3\sqrt{b_{-1}}}\,Ln\left[b_{-1}a^{3}+\sqrt{b_{-1}}\sqrt{b_{-1}a^{6}-\nu^{2}}\right]\,.$
(17)

The volume function is then

$a^{3}=\frac{1}{2b_{-1}}\left(e^{3\sqrt{b_{-1}}\,\Delta\tau}+b_{-1}\nu^{2}e^{-3\sqrt{b_{-1}}\,\Delta\tau}\right)\,,$
(18)

whereas that of the scalar field is

$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{4}{\sqrt{6\omega}}\arctan\left(\frac{Exp[3\sqrt{b_{-1}}\Delta\tau]}{\nu\sqrt{b_{-1}}}\right)\right]$&\qquad$m=-2$
\, ;\\\
$\left[\frac{2(m+2)}{\sqrt{6\omega}}\arctan\left(\frac{Exp[3\sqrt{b_{-1}}\Delta\tau]}{\nu\sqrt{b_{-1}}}\right)\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$
\, .\\\ \end{tabular}\right.$ (19)

For the case $\gamma=1$ the same solutions are found and only a redefinition
of the constants is needed.

  * •

Stiff plus a cosmological constant plus dust, $\gamma=0$. In this case the
master equation becomes

$\Delta\tau=\int\frac{a^{2}da}{\sqrt{\frac{8}{3}\pi
G\rho_{0}a^{3}-\frac{\lambda}{3}a^{6}-\nu^{2}}}$ (20)

whose solution is

$\Delta\tau=\frac{1}{\sqrt{3|\lambda|}}\,Ln\left[\frac{b_{0}+\frac{2|\lambda|}{3}a^{3}}{\sqrt{\frac{|\lambda|}{3}}}+2\sqrt{b_{0}a^{3}+\frac{|\lambda|}{3}a^{6}-\nu^{2}}\right]\,,$
(21)

with $|\lambda|>0$ and $\rm b_{0}=\frac{8}{3}\pi G\rho_{0}$. The volume
function is now

$a^{3}=\frac{3}{4\sqrt{3|\lambda|}}e^{-\sqrt{3|\lambda|}\tau}\left[4\nu^{2}+\left(e^{\sqrt{3|\lambda|}\tau}-\frac{3b_{0}}{\sqrt{3|\lambda|}}\right)^{2}\right]\,.$
(22)

In this way, the solution for the field $\phi$ is

$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{4}{\sqrt{6\,\omega}}\arctan\left(\frac{\sqrt{3|\lambda|}Exp[\sqrt{3|\lambda|}\Delta\tau]-3b_{0}}{2\nu\sqrt{3|\lambda|}}\right)\right]$&\qquad$m=-2$
\, ;\\\
$\left[\frac{4(m+2)}{\sqrt{6\omega}}\arctan\left(\frac{\sqrt{3|\lambda|}Exp[\sqrt{3|\lambda|}\Delta\tau]-3b_{0}}{2\nu\sqrt{3|\lambda|}}\right)\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$
\, .\\\ \end{tabular}\right.$ (23)

The classical solution when $\rm F(\phi)=we^{m\phi}$ have the following
structure

$\rm\phi(\tau)=\frac{2}{m}Ln\left[\frac{m}{2}\sqrt{\frac{6\nu^{2}}{w}}\int
a^{-3}(\tau)d\tau+e^{\frac{m}{2}\phi_{0}}\right],$ (24)

where the integration value must be consider the last calculations over the
scale factor.

The solutions above were checked to comply with the Einstein field equations
encoded in equations (3b), using the REDUCE 3.8 package.

##  III Quantum FRW cosmological model

One of the open problem of SB is the lack of a quantum model, in this section
using the generalization of the ideas presented in the previos sections we use
canonical quantuization. By the usual representation for the momenta operators
$\rm\Pi_{q}=-i\frac{\partial}{\partial q}$, $(\hbar=1),$ including the factor
ordering problem in the $a$ and $\phi$ variables, we obtain the Wheeler-DeWitt
equation

$\rm\left[-a^{2}\frac{\partial^{2}}{\partial a^{2}}-qa\frac{\partial}{\partial
a}-\frac{6}{F(\phi)}\frac{\partial^{2}}{\partial\phi^{2}}-\frac{6s}{F(\phi)}\phi^{-1}\frac{\partial}{\partial\phi}+144\kappa
a^{4}+48a^{6}\lambda-384\pi G\rho_{\gamma}a^{3(1-\gamma)}\right]\Psi=0,$ (25)

where q and s are real constants that measures the ambiguity in the factor
ordering in the operators $\Pi_{a}$ and $\Pi_{\phi}$, $\Psi$ is the wave
function for this cosmological model. Employing the variables separation
method, $\Psi(a,\phi)={\cal A}(a){\cal B}(\phi)$, (25) gives the set of
equations

$\displaystyle\rm-a^{2}\frac{d^{2}{\cal A}}{da^{2}}-qa\frac{\partial{\cal
A}}{\partial a}+\left(144\kappa a^{4}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}-\mu^{2}\right){\cal A}$ $\displaystyle=$
$\displaystyle 0,$ (26) $\displaystyle\rm\phi\frac{d^{2}{\cal
B}}{d\phi^{2}}+s\frac{d{\cal B}}{d\phi}-\frac{\mu^{2}}{6}\phi F(\phi){\cal B}$
$\displaystyle=$ $\displaystyle 0.$ (27)

The equation (26) does have not a general solution for any $\kappa$, then we
solve for flat case and the particular values in the $\gamma$ parameter. When
$\gamma=-1$, the exact solution is

$\rm{\cal
A}(a)=a^{\frac{1-q}{2}}\,Z_{\nu}\left(\frac{\sqrt{b}}{3}a^{3}\right),$ (28)

where $\nu=\frac{1}{6}\sqrt{(1-q)^{2}-4\mu^{2}}$ and $b=384\pi
G\rho_{-1}-48\lambda$. We can see that when $b>0$, the generic Bessel function
$Z_{\nu}\to J_{\nu}$, and when $b<0$, $Z_{\nu}\to(K_{\nu},I_{\nu})$

Other soluble case is when $\gamma=1$, the solution is the same, and the
changes appear in the constants $\mu^{2}\to 384\pi G\rho_{1}+\mu^{2}$ and
$b=-48\lambda$. In this form, we obtain the exact solution to the wave
function $\Psi(a,\phi)$ in this theory.

For solve the equation (27), we apply this approach at Sáez-Ballester theory.
The case when $m\not=-2$ polyanin  is written in term of generic Bessel
function $\rm Z_{\eta}$ as

$\rm
B(\phi)=c\phi^{\frac{1-s}{2}}Z_{\eta}\left(\frac{2\sqrt{-\xi}}{m+2}\phi^{\frac{m+2}{2}}\right),$
(29)

where c is a integration constants, and $\eta=\frac{1-s}{m+2}$,
$\xi=\frac{\mu^{2}\omega}{6}$. Also, we can see that the generic Bessel
function $\rm Z_{\eta}\to J_{\eta}$ when $\omega<0$, or
$\rm(K_{\eta},I_{\eta})$ when $\omega>0$.

We can build the wave packet, introducing the continuum parameters $\eta$ and
$\nu$ as

$\rm\Psi_{\eta\nu}=\int_{\eta}\int_{\nu}{\cal F}(\eta){\cal
G}(\nu)\phi^{\frac{1-s}{2}}Z_{\eta}\left(\frac{2\sqrt{-\xi}}{n+2}\phi^{\frac{n+2}{2}}\right)a^{\frac{1-q}{2}}\,Z_{\nu}\left(\frac{\sqrt{b}}{3}a^{3}\right)d\eta
d\nu$ (30)

For particular values in the constant $m$, the exact solutions are very
simple. For instant when $m=-2$, we have the Euler equation who solution is

$\rm
B(\phi)=\phi^{\frac{1-s}{2}}\left\\{\begin{tabular}[]{lr}$\rm\left[c_{1}\phi^{\alpha}+c_{2}\phi^{-\alpha}\right]$&\qquad$(1-s)^{2}>4b$\\\
$\rm\left[c_{1}+c_{2}Ln\phi\right]$&\qquad$(1-s)^{2}=4b$\\\
$\rm\left[c_{1}sin(\alpha Ln\phi)+c_{2}cos(\alpha
Ln(\phi))\right]$&\qquad$(1-s)^{2}<4b$\end{tabular}\right.$ (31)

with $\alpha=\frac{1}{2}\sqrt{(1-s)^{2}-4b}$ and $b=-\frac{\omega\mu^{2}}{6}$.

When $m=-6$ and $s=-1$, making the transformations $z=\phi^{-2}$ and
$B=\frac{u}{z}$, leads to a constant coefficient linear equation, (27) is
transformed to $4\frac{d^{2}u}{dz^{2}}-\frac{\mu^{2}\omega}{6}u=0$ who exact
solutions becomes

$u(z)=\left\\{\begin{tabular}[]{lr}$\rm
c_{1}\,sinh\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)+c_{2}\,cosh\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)$&\qquad$\omega>0$\\\
$\rm
c_{1}\,sin\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)+c_{2}\,cos\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)$&\qquad$\omega<0$\\\
\end{tabular}\right.$ (32)

in the original variables

${\cal B}(\phi)=\phi^{2}\left\\{\begin{tabular}[]{lr}$\rm
c_{1}\,sinh\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)+c_{2}\,cosh\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)$&\qquad$\omega>0$\\\
$\rm
c_{1}\,sin\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)+c_{2}\,cos\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)$&\qquad$\omega<0$\\\
\end{tabular}\right.$ (33)

##  IV conclusions

We studied the generalization of the Sáez-Ballester theory by including a
dimensionless functional of the scalar field $F(\phi)$. The classical dynamics
of the theory were obtained from the corresponding classical Lagragian and
Hamiltonian densities; the solutions were in turn given up to quadratures.

One general result here is that the evolution of the scale factor of the
Universe does not depend upon the particular form of the functional $F(\phi)$;
actually, the contribution of the scalar field in the SB theory is that of
perfect fluid with a stiff (barotropic) equation of state. If any, its
contribution to the matter budget of the Universe is only relevant at early
times.

A separate conclusion is that the SB, whether in its original form as given in
Ref.s-b  or in the generalized case studied here, cannot be an answer to the
dark matter riddle of Cosmology.

In the quantum regime was necessary to build one equivalent density lagrangian
in order to apply this, and does not possible to write this solution in closed
form. In this sense, we check this approach using the original Sáez-Ballester
formalism, obtaining the exact solutions in both regimes, classical and
quantum for particular values in the $\gamma$ parameter. This formalism will
be used with anisotropic cosmological models, which will be reported in other
work.

###### Acknowledgements.

This work was partially supported by CONACYT grants 47641, 56946 and 62253,
DINPO 38.07 and PROMEP UGTO-CA-3. This work is part of the collaboration
within the Instituto Avanzado de Cosmología.

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