Joint Modeling of Longitudinal Changes of Respiratory Rate, Pulse Rate and Oxygen Saturation with Time to Convalescence among Pneumonia Patients: A comparison of separate and joint models

Background: Globally, pneumonia is the first infectious disease which is the leading cause of children under age five morbidity and mortality with 98% of deaths in developing countries. Objective: The study aimed to identify the determinant factors that jointly affect the longitudinal measures of pneumonia (respiratory rate, pulse rate and oxygen saturation) and time to convalescence or recovery of under five admitted pneumonia patients at Felege Hiwot Referral Hospital, Bahir Dar, Ethiopia. Methods: A prospective cohort study design was used on 101 sampled under five admitted pneumonia patients from December 2019 to February 2020. The study was conducted using joint model of longitudinal outcomes and survival outcomes. Results: The significant values of shared parameters in the survival sub model shows that the use of joint modeling of multivariate longitudinal outcomes with the time to event outcome is the best model compared to separate models. The estimated values of the association parameters for γ_1, γ_2 and γ _3 were -0.297, -0.121 and 0.5452 respectively and indicates that; respiratory rate and pulse rate were negatively related with recovery time, whereas oxygen saturation was positively associated with recovery time. As age of patients increased by one month, the average respiratory rate and pulse rate were significantly decreased by 0.3759 bpm and 1.1012 bpm respectively keeping other variables constant, but age has no information about oxygen saturation. Conclusion: Residence, birth order, severity and visit were found as determinants of the longitudinal measures of pneumonia and time to recovery of under-five admitted pneumonia patients jointly. To improve child survival, the community should be responsible for post ponding child birth and marriage.


Background
Pneumonia is described as the inflammation of parenchymal structures of the alveoli and the bronchioles (lungs) [1]. Pneumonia is most commonly classified by where or how it was acquired.
Community-acquired pneumonia (CAP) is an infection that begins outside the hospital and/or diagnosed within 48 hours after admission to the hospital. Whereas, hospital-acquired pneumonia occurs in more than 48 hours after admission and without any antecedent signs of infection at the time of hospital admission [2,3].
Pneumonia is usually caused by infection with bacteria or viruses and bacteria are most common causes of (CAP) with Streococcus pneumonia isolated about 50% of cases. In children about 15% pneumonia cases a number of drug-resistant versions of the infections are more common, including drug resistant Streptococcus pneumonia and Methicillin-resistant Stapylococcus aurous [4,5]. The burden of medical response to pneumonia has significant challenges. Besides drug resistance to the bacteria, comorbid conditions like Malaria, TB, Sickle cell anemia, HIV/AIDS and risk factors like lack of exclusive breast feeding, alcoholism, smoking etc. commonly appear in pneumonia patients which leads to define the severity and risk scores of the disease in which used for clinicians to make care self-site decision as in-patients or out-patients [6].
Estimates from the WHO suggest that pneumonia is responsible for 20% of deaths in the underfive age group, leading to 3 million deaths per year [7]. In Africa especially in sub-Saharan Africa, by 2013 pneumonia was the second leading cause of child mortality that accounts a million child death about 15.8% of total deaths in the region [8]. A report by UNECIF indicated that 132,000 under five children killed by pneumonia in Congo which is the second cause of death next to malaria [9,10] and Kenya accounted the highest number of under-five mortality due to pneumonia which accounts about 16% of total deaths among 15 East Africa countries [11]. Pneumonia kills up to 5 million children under the age of 5 years annually in developing countries [12].
In Ethiopia, pneumonia is a leading single disease killing under five children and it contributes about 18% of all cases (370,000) of under five deaths compared to diseases like diarrhea, AIDS, malaria and measles every year [13,14]. Under five pneumonia is commonly measured through physiologic parameters (temperature, pulse rate, blood pressure, respiratory rate, and oxygen saturation) and the performance of (TCS) is decided through the longitudinal measures of those parameters [15].
Several cross-sectional studies have used scoring systems to summarize the level of symptoms within a cohort at fixed time-points following CAP [16,17]. However the understanding of which predictor affects length of hospital stay has been hampered by lack of longitudinal studies. Recent studies provide insight on the background and clinical predictors of mortality and survival of pneumonia patients among children aged under five years [18]. These studies did not consider the true and unobserved effects of longitudinal measures of physiologic parameters which correlates with recovery time to determine the length of hospital stay for under five pneumonia patients. In this study, joint model of multivariate linear mixed model and cox PH model was used to find significant factors of longitudinal measures of pneumonia (respiratory rate, pulse rate and oxygen saturation) and time to convalescence jointly.

Study area and period
The data for this study was collected from FHRH, Bahir Dar, Ethiopia. Bahir Dar is the capital city of Amhara National Regional state. It is found in north western Ethiopia and is 565 kilometers from Addis Ababa. This hospital serves as referral hospital for the people who came from different surrounding areas.

Study design and sampling
The study design was prospective cohort study. Since this study is a case study conducted within three months period (12th of December 2019 to 30th February 2020), the samples were the number of children bounded in the inclusion criteria with in the study period. This study used a sample of 101 patients.

Variables in the study
Three longitudinal outcomes (Respiratory Rate in bpm, Pulse Rate in bpm and Oxygen saturation in mm Hg) and a survival outcome (time to convalescence or recovery in hour) were considered as dependent variables in this study. The time to recovery of under-five admitted pneumonia patients can be given as (pneumonia status = 0 for censored and 1 for event).

Data collection Procedure
The longitudinal and the survival data containing the socio demographic and home based information were collected using primary data collection method by face to face interview of their care givers using well-structured questionnaires. In addition, the data containing clinical information found from their charts were considered. Both primary and secondary data were collected by trained pediatrician and statistician.

Eligibility criteria
Inclusion criteria: The inclusion criteria was children 2-59 months of age with their care givers (mothers or not) and admitted at pediatric ward by community acquired pneumonia during the study period. Exclusion criteria: The exclusion criterion was children admitted at the hospital by disease other than pneumonia, pneumonia patients below two months and above 59 months and patients with incomplete medical records.

Data analysis
In this study, a longitudinal data on the three measures of pneumonia (RR, PR and Oxygen saturation), recovery time of under-five pneumonia patients for the survival data, and sociodemographic factors, home based factors, child nutritional status and child illnesses at the base line were considered. The data were coded, entered and edited using SPSS version 26 and the analysis was done using SAS 9.4 and R software and the statistical decision was made at 5% level of significance.

Survival data analysis
Survival data analysis is a class of statistical method which used to analyze data in which the time(usually measured in days, weeks, months or years) until the event (usually death, disease incidence, relapse from remission, recovery ) is of interest [19]. Cox proportional hazards model of the survival analysis was used to estimate the length of time to recover from pneumonia and to identify factors related to time to recovery [20].

Longitudinal data analysis
A longitudinal study is statistical analysis of an observational research method in which response variable is measured repeatedly over time and those measurements taken from the same subject are correlated [21]. Longitudinal response may arise in two common situations. One is when measurements taken on the same subject and the other is when measurements taken on related subjects. In both cases, the responses are likely to be correlated [22].

Linear Mixed Effects Model
The random effects contains subject specific random effect and are directly used in modeling the random variation in the dependent variable at different levels of the data. Before considering the multivariate linear mixed model, it is better to identify the covariates which have significant effect on the mean change of RR, PR and oxygen saturation measurements over time using LMM [21].
The vector (y 1ik + y 2ik +. … … . . +y nik ) T represents the observations of the response variable from the ℎ subject and vector (y 1k , y 2k , … . . y nik) T represent the observation for the ℎ response variable across all response variables and subjects, finally the vector (y 1 , y 2 , y 3 , … … … y n ) T represents the observations across all response variables and subjects. In the context of modeling the response variables, the linear mixed effect model for each response variable of subject , taken at time , can be specified by [23].
µ ( ) is the average evolution of the ℎ response over time and it is a function of fixed effect.
The subject specific random intercepts and slopes ( ) describe how the subject specific profiles deviate from the average profile for the ℎ response.

Joint modeling of multivariate longitudinal with time to event outcome
In this study three correlated and longitudinally measured response variables were considered which can be jointly modeled with time to event outcome. The separate and the joint models assume that the longitudinal sub model has the form similar to the conventional linear mixed effects model while the survival model in the joint model includes a latent association function ( ) [24]. Maximum likelihood approach was used to estimate the parameters for both longitudinal and survival sub models.

Ethical consideration
This study was carried out in the location where the approval was obtained from the ethical review committee of College of Health Sciences, Bahir Dar University, and permission for data collection was obtained from Felege Hiwot Specialized Referral Hospital Management. There were no risks due to participation in this research project, and the collected data were used only for this research purpose. The study compiled with the principles set forth in the Declaration of Helsinki (1964) and all of its subsequent amendments. The written informed consent was obtained for caregivers of each patient prior to the data collection and all information collected from each caregivers was treated with complete confidentiality.

Results
The study revealed that, the median recovery time of pneumonia patients admitted at FHRH was72 hours with minimum and maximum recovery time of 18 hours and 96 hours respectively. Out of the total sampled pneumonia patients, 90 (89.1%) were recovered from pneumonia. When we fit the cox proportional hazards model using the candidate variables: residence, birth order, age of mothers, education of mothers, danger signs, cooking place, comorbidity and severity were significant factors affecting time to recovery of pneumonia patients at 5% level of significance (   Table 1).  (Table 2). Checking assumptions of the data is the first step in analyzing longitudinal data. Normal QQ plots in Fig 1 shows that, the data for the three longitudinal outcomes were approximately normally distributed and then it is better to proceed to the next steps of the analysis.  were significantly different from zero which tells the existence of a relationship between a patients baseline standing between outcomes, rate of change between outcomes as well as, between baseline standing of one outcome and rate of change of the other outcome through follow-up time.   When evaluating the overall performance of both the separate and joint models in terms of model parsimonious and goodness of fit, the joint model was preferred as it has smaller standard error than the separate model. This result also supports the study done by [25,26].
As Table 4 revealed, under MV joint model, estimate of the association parameters in the survival sub model was significantly different from zero (γ_1= -0.297, γ_2= -0.121 and γ_3=0.5452), this indicates that three longitudinal outcomes were correlated with time to recovery of under-five admitted pneumonia patients supported by [27,28,29], stats that the longitudinal and survival data are correlated. The joint model was more parsimonious fit than the separate model. Therefore, the joint model found preferable and parsimonious to fit the data better than the separate one [24] when the association parameter of the joint model is significant. Therefore, the final model for this study was joint model of MLMM and cox PH model.

Discussion
The general objective of this study was identifying the determinant factors jointly affecting Variables of urban residence, feeding exclusive breast within six months, first birth, non-danger sign and severity were significantly associated with recovery time of under-five admitted pneumonia patients. This was consistent with results of the study conducted by [31]. Increasing age of mothers increases the chance of experiencing the event of recovery (p-value=0.026). This consides with results of the study conducted by [32].The difference in the degree of significance may come from the difference in the variables as well as the model we used.
Exclusive breast feeding with in the first six months of life increases child survival by reducing the length of hospital stay. This supports results of the study done by [32]. The association parameters were significant indicates the significance of relationship between longitudinal measures of pneumonia (RR, PR and oxygen saturation) and time to recovery of under-five admitted pneumonia patients. This is in line with results of [34,35]. Higher values of average RR and PR as well as lower values of average oxygen saturation were related with longer recovery time (high risk of pneumonia).This was consistent with results of the studies done by [18,32].

Conclusion
In this study, a joint model of multivariate longitudinal changes of respiratory rate, pulse rate and oxygen saturation with time to recovery of under-five admitted pneumonia patients was discussed.
Out of the total sampled pneumonia patients 90 (89.1%) were recovered from pneumonia and the median recovery time was 72 hours. When evaluating the overall performance of both the separate (MLMM and cox PH model) and joint model in terms of model parsimonious, goodness of fit and the statistical significance of association parameters, the joint model performs better than the separate models. As a result, we concluded that the joint model was preferred for simultaneous analyses of repeated measurement and survival data. From results of the study, we can conclude that patients from urban area, borned at the first birth, having comorbid status, having lower follow-up time and having sever pneumonia have high levels of respiratory rate and pulse rate, whereas lower levels of oxygen saturation and which increases the risk of pneumonia. Patients with high levels of respiratory rate and pulse rate as well as low values of oxygen saturation requires longer recovery time. Urban residency, being first child, having sever pneumonia at the baseline, cooking food inside the living room, no-breast feeding, having dangerous signs and having comorbid status increases the recovery time of under-five admitted pneumonia patients. To improve child survival, the community should be responsible for post ponding child birth and marriage.

Ethics approval and consent to participate
This study was carried out in the location where the approval was obtained from the ethical review committee of College of Health Sciences, Bahir Dar University, and permission for data collection was obtained from Felege Hiwot Specialized Referral Hospital Management. There were no risks due to participation in this research project, and the collected data were used only for this research purpose. The study compiled with the principles set forth in the Declaration of Helsinki (1964) and all of its subsequent amendments. The written informed consent was obtained for caregivers of each patient prior to the data collection and all information collected from each caregivers was treated with complete confidentiality.

Consent for publication
Not applicable.

Availability of data and materials
The datasets analyzed during the current study are available from the corresponding author upon reasonable request.