Modelling Under-Five Mortality among Hospitalized Pneumonia Patients in Hawassa City, Ethiopia: A Cross-Classified Multilevel Analysis

The multilevel regression model often called hierarchical multilevel linear model (HLM) assumes that there is a hierarchical multilevel data set, with one single dependent variable that is measured at lowest level and explanatory variables at all existing levels [17].

Many kinds of data, including observational data collected in the human and biological sciences, have a hierarchical, nested, or clustered structure (Fig. 1a). Traditional linear regression techniques are not capable of taking into account data with a hierarchical multilevel structure [18]. Data with hierarchical structure violates an important assumption of linear regression model. As a result, regression coefficients are not efficient and its standard errors are understated.

Multilevel regression model can be described as the extension of regression model, which adds random effects at multiple levels. For instance, in this study the interest is to know the risk of dying from CAP if one child is in a neonatal age. In this case, both variables are patient level variables; as a result, one can use simple logistic regression. Since, patients are nested with in hospital, using multilevel models allows us to divide the variance in the outcome variable at the group and individual levels. A two-level multilevel model in health area assigns patients to level 1 and hospitals to level 2.

Consider a collection of r independent variables which will be characterized by \(\mathbf{X}= (\hbox {X}_{1}, \hbox {X2},{\ldots }\hbox {X}_{\mathrm{r}})\). Then the conditional probability of the outcome of interest in a study is present (in our case death of child due to CAP) can be denoted by \(\hbox {P}_{\mathrm{ij}}= \hbox {P}(\mathbf{Y}_{\mathrm{ij}}=1{\vert }{} \mathbf{X}=\mathbf{x}\)). The ratio of success probability to failure probability is known as odds of success.Suppose that we have a sample of n independent observations of (\(\hbox {X}_{\mathrm{ij}}, \hbox {Y}_{\mathrm{ij}})\) for r+1 variables, where \(\hbox {Y}_{\mathrm{ij}}\) denotes the value of dichotomous variable, and \(\hbox {Y}_{{ij}}\) is the value of the independent variables for the ith individual (patients). Then, the logistic model can be written using odds of success as follows.Thus, the data layout can be represented as:The relationship between success probability \(\hbox {P}_{\mathrm{i}}\) and his/her characteristics are non-linear. Hence, by applying logit transformation, in order to have a meaningful interpretation, one would have the following logistic regression model equation.where \(\beta =\left[ {\beta _0 ,\beta _1 ,\ldots ,\beta _k } \right] ^{t}\)

\(\beta _0\) is the intercept and \(\beta _1 ,..,.,.,..\beta _p\) are coefficients of the p explanatory variables (which may be continuous or dummy variables).

A two level logistic regression models allow to assess variation in a dependent variable at hospital and patient level simultaneously in which patients are nested within hospitals. The response variable is denoted by \(\hbox {Y}_{\mathrm{ij}}\):with probabilities \(\hbox {P}_{\mathrm{ij}}= \hbox {P(Y}_{\mathrm{ij}}=\hbox {1/X}_{\mathrm{lij}})\), probability of dead for patient ‘i’ in hospital ‘j’, and \(\hbox {1-P}_{\mathrm{ij}}\), probability of discharged alive for patient i and hospital j. The standard assumption is that \(\hbox {Y}_{\mathrm{ij}}\) has a Bernoulli distribution. Let the \(\hbox {P}_{\mathrm{ij}}\) be modelled using logit link function.

The two-level model is given by:where \(\beta _{0j} =\beta _0 +u_{0j} , \beta _{1j} =\beta _1 +u_{1j} ,\ldots ,\beta _{rj} =\beta _r +u_{rj}\) this becomes: \(\hbox {X}_{\mathrm{ij}}= (\hbox {X}_{\mathrm{1ij}}, \hbox {X}_{\mathrm{\mathrm{2ij}}},\ldots ,\hbox {X}_{\mathrm{rij}})\) represents the first and the second covariates, (\(\beta _{0}, \beta _{1},\cdots , \beta _{r}\)) are regression coefficients, \(\hbox {u}_{\mathrm{0j}}, \hbox {u}_{\mathrm{1j}}, \hbox {u}_{\mathrm{2j}},\cdots \hbox {u}_{\mathrm{kj}}\) are the random effect of model parameter at level two. It is assumed that these level two random effects distributed normally with mean 0 and variance \(\sigma _u^2\).

The empty two-level model for dichotomous outcome variable refers to population of groups (level-two units) and specifies the probability distribution for the group-dependent probabilities \(\hbox {P}_{ij}\), where \(\hbox {P}_{ij}\) characterised by \(\hbox {Y}_{ij}= \hbox {P}_{ij}+\hbox {e}_{ij}\), without taking further explanatory variables into account. The logit transformed models, logit(\(\hbox {P}_{ij})\), has a normal distribution. Then the empty model is expressed by formula:where \(\gamma _0\) is the population average of the transformed probabilities and \(u_{0j}\) the random deviation from this average for hospital j. The residual term associated with deviation, \(u_{0j}\) is a unique effect of hospital j on the outcome variable; it is assumed to be normally and independently distributed with mean of zero and constant variance, \(\sigma ^{2}\) that is \(u_{0j}\sim \hbox {N} (0, {\varvec{\upsigma }}^{2})\).

The level 2 residual variance captures the variation across hospital means. With this model, the amount of variance in the outcome variable that is attributable to within group characteristics (here, patients) and to between group differences (hospitals) can be identified. This model (2.5) does not include a separate parameter for the level- one variance [19]. This is because the level one residual variance of the dichotomous outcome variable follows directly from the success probability is indicated below.where \(e_{ij}\) is individual-dependent residual.

Define \(\pi _0\)- the probability corresponding to the average value by the \(\gamma _0\), as follows:For the logit function, the logistic transformation of \(\gamma _0\) is given by:Now the variance for probability of success (death of child from CAP in-hospital j) can be given by: Random Intercept Model

The extension of equation 2.5 above in which it incorporates explanatory variables is termed as random intercept model. The success probability is not necessarily the same for all individual in a given group. Therefore, the success probability now depends on the individual as well as on the group, and is denoted by \(\hbox {P}_{ij}\), [19]. Accordingly, the outcome is split into an expected value and a residual i.e.:The logistic random intercept model expresses the log-odds, i.e., the logit of \(P_{ij}\) is:As usual, the deviation \(u_{0j}\) are assumed to be independently distributed with zero mean and variance \(\sigma ^{2}\). Equation (2.10) does not include a level one residual because it is an equation for the probability rather than the outcome \(\hbox {Y}_{ij}\). Thus, the level residual has already included in the equation.

Another model which is used to assess whether the slope of any of the explanatory variable has a significant variance component between the groups is called the random slope model. Assume that effect of variable \(\hbox {X}_{\mathrm{j}}=(\hbox {X}_{\mathrm{1j}}, \hbox {X}_{\mathrm{2j}},\cdots ,\hbox {X}_{\mathrm{kj}})\) across the groups, and accordingly has a random slope. Equation (2.10) for the logit of the success probability is extended with the random effect \(u_{1j}x_{1ij,}\) (19), which leads to:The characteristic that is unique to this model, compared to others discussed here, is that the random coefficients \(\upbeta \) are themselves dependent variables in a second regression equation. Of course, in RSM we have:There are now r+1 random group effects, the random intercept \(u_{oj}\) and the random slope \(\hbox {u}_{lj}\). It is assumed that both have a zero mean. Their variances are denoted by \(\sigma _0^2 , \ldots ,\sigma _r^2\) respectively and their covariance is denoted by \(\sigma _{0r}\). This can be described by the covariance matrix \(\Omega \). 2.3.1.2 Two Level Cross-classified Multilevel Models

Unlike hierarchical multilevel data with a purely nested structure, data that are cross-classified not only may be clustered into hierarchical multilevel ordered units but also may belong to more than one unit at a given level of a hierarchy [10]. Goldstein [20] labelled multilevel model for purely nested structure as hierarchical multilevel (HM) models. However, multilevel level models for cross-classified data structure is labelled as cross-classified multilevel (CCM) models.

In a cross-classified design, patients at a given hospital might be from several different resident places/sub-cities and one sub-city might have patients who attend a number of different hospitals. In this type of scenario, hospital and sub-city are considered to be cross classified factors, and cross classified random effects modelling (CCREM) should be used to analyse these appropriately.

The consequences of ignoring an important cross-classification are similar to those of ignoring an important hierarchical multilevel classification [21]. Therefore, it is very important to consider the classification at existing level (second level in this study).

In Fig. 1 the first diagram illustrates a purely nested data structure. In this structure, there are three levels; patients, hospitals and sub-city. A typical hierarchical multilevel model has patients nested in hospitals, and hospitals nested within sub-cities. In a cross-classified multilevel structure, however, patients from a given sub-city might visited different hospital and hospital might serve patients from different sub-city. Subsequently the cross classified data structure occurs at level 2.

In this example, we can take into account influences coming from two different factors: sub-cities and hospitals. This may improve the estimation of explanatory variable effects ensuring the structure is properly specified [10].

Assume one has patients nested within a cross-classification of hospital by sub-city, which is the case illustrated in Fig. 1b and \(\hbox {P}_{\mathrm{i(hosp, subc)}}\) be the probability of child to die from CAP in hospital ‘hosp’ and came from sub-city ‘subc’. \(v_{subc(i)}\) is the sub-city effect which are mutually independent random variables and also independent to \(u_{\textit{hosp(0, j)}}\), which is hospital random effect. All \(v_{subc(i)}\) are normally distributed with mean 0 and variance \(\tau _v^2\).

Note the notation used in this paper for cross classified model is adapted from Fielding et al. [10].

Like HM model, in a CCM model, before introducing explanatory variables, the variation in outcomes could be examined by looking at the differences between hospitals, between sub-cities and between individual patients after controlling for sub-cities and hospital effects [22]. This gives a basis for extending the model to identify which sub-cities, hospitals, and patients’ characteristics might explain some part of these component variances.

After adding explanatory variables, the residual components of variances can be estimated, this will provide more information about the variation in outcomes [21].

Estimation of CCM models by existing maximum likelihood approaches run into important computational limitations, especially when large numbers of units are involved [21]. As a result, the models used in this paper are fitted using Markov Chain Monte Carlo (MCMC) based algorithms as implemented in the MLwiN package [23]. Starting values for the fixed parameters are estimated from simpler models using a maximum likelihood approach, penalized quasi- likelihood (PQL) in MLwiN [23].

Markov Chain Monte Carlo(MCMC) is a simulation techniques that generate random samples from a complex posterior distribution [22]. The simulated posterior distribution is then used to compute a point estimate and a confidence interval. The posterior distribution of \(\theta \) is defined from Bayes’ theorem asHere \(p(\theta \)) is the prior distribution for the parameter vector \(\theta \) and should represent all knowledge we have about \(\theta \) prior to obtaining the data.

In multilevel models, a non-informative prior for both fixed effect and random effect parameters are recommended [18]. The default prior distribution applied in MLwiN when MCMC is used are ‘flat’ or ‘diffuse’ priors for all parameters (23).

Thus, according to Browne et al. [23] the prior for a fixed parameter, \(\beta \) in MLwiN is an improper uniform distribution, \(p(\beta )\propto 1\), which is equivalent to normal prior with large variance. The prior for scalar variance, \(\sigma ^{2}\)is specific as inverse gamma, \(p(\sigma ^{2})\sim invGamma(\alpha ,\beta )\). In MLwiN, the prior applied for covariance matrices,\(\Omega \) is an inverse Wishart distribution, \(p(\Omega ^{-1})\sim Wishart_p (p,p,\overline{\Omega })\), where p is the number of rows in the variance matrix and \(\overline{\Omega }\) is the estimate for the true value of \(\Omega \).

Consider a two-level random slope binary logistic regression in equation (2.12),with \(Y_{ij}= P_{ij}+ R_{ij}\) is the outcome variables where \({{\varvec{\Sigma }}}_\varepsilon =\sigma ^{2}{\hbox {I}}_{nj}\) is level one residual is i.i.d. \({{\varvec{\Omega }}}_u \) is a level two covariance matrix. In this study, the prior distribution used for all fixed parameters is non-informative normal distribution with variance \(10^{4}\) and different values of \(\alpha \) and \(\beta \) in gamma distribution for scalar variance prior was used.

Consider Eq. (2.12) above, random slope model, in which each observation is considered as an outcome of a Bernoulli trial (\(y_{ij} \sim { Bernoulli}(p_{ij} ))\), and hence for the \(\hbox {i}{\mathrm{th}}\) patient in \(\hbox {j}{\mathrm{th}}\) hospital:Assuming the n observations are independent, the likelihood function is given by:where \({{\varvec{\uptheta }}}\) stands for the joint parameters, \(\left( {u_j ,\varSigma _\varepsilon ,\varOmega _u } \right) \).

The posterior distribution is obtained by multiplying the prior distribution over all parameters by the full likelihood functions. The posterior distribution is:

In this study, non-informative priors have been used which express the complete ignorance of the value of the parameter. Accordingly, as we mentioned on chapter three Sect. 3.5.3, we have used normal distribution with mean 0 and variance \(10^{4}\)(huge variance) for fixed parameters.

However inverse gamma distribution with parameter \(\alpha \) and \(\beta \) for single variance was used. The value of parameters for this distribution was chosen based on Bayesian deviance information criteria (BDIC). Table 6 shows the proposed parameter value for prior distribution of single variance. On Table 6, BDIC is smaller when we use \(\alpha =100\) and \(\beta =100\) in each model except on cross classified empty model. Since, there was no significant change on the values of BDIC, the same prior parameter for these models were used too.

The number of iteration used in the analysis for all of the parameters (fixed and random effect) was set up to be 50,000 with ‘thining’ number of 1; the first 5000 was discarded(burn-in) and the rest of the chains was used to summarize the posterior distribution.

Now like the case of HM model, an empty model can be extended to random intercept CCM model using explanatory variables those were significantly associated with discharge status of patients. In CCM random intercept model we vary the intercept across both hospitals and sub-city.

In this analysis, the coefficients of Intercept (−10.323), number of patients attended (3.024), spring (2.021), AGE (1.893), AFI (2.017), Malaria (1.967), postnatal age group (−2.383), Child age group (−2.466), duration of onset to diagnosis from 4-7days (1.556), duration of diagnosis more than a week (2.165), Patient to Physician ratio (0.014) coefficient of Patient to Nurse ratio (0.330) were statistically significant at 5% level of significance. In the model, more variation (19.3%) of discharge status of the patients (dead or alive) was due to hospital as compared to variation attributable to residents/sub-city (18.9%).

The proportion of variation of discharge status explained by explanatory variables is 78.95% in random intercept of hierarchical multilevel logistic regression. In other words, season of diagnosed, CAP complicated disease, duration till diagnosed, age, patient refer status, patient to nurse ratio and patient to physician ration explains about 78.95% of variation in whether patients were dead or alive due to community acquired pneumonia (Tables 7, 8).

In MLwiN statistical Packages, MCMC estimation methods can be easily run by taking either PQL result or MQL results as an initial point for simulation. The packages finally display MCMC summery statistics and MCMC plots which is important to test the diagnostic convergence of MCMC chain. Under MCMC summery statistics the package computes chain means (posterior means), the chain standard deviation, etc.

Raftry and Lewis diagnostic [24] is a diagnostic based on a particular quantile of the distribution. The diagnostic that is used to estimate the length of Markov chain required to estimate a particular quintile to a given accuracy. In MLwiN the diagnostic is calculated for the two quintiles (the defaults are the 2.5 and 97.5% quintiles) that will form a central interval estimate.

The dependency among the Markov chain samples are measured by the autocorrelation which is one method of model diagnostic of convergence. MCMC algorithms do not produce independent samples from the distribution. Rather, it generates samples that are auto correlated. As in time series analysis, we can compute the autocorrelation function at each lag (1+) and include the number of iterations we need to skip in order to have a non-significant autocorrelation. High correlations between long lags indicate poor mixing. Thus, low autocorrelation indicates good convergence.

All the coefficients were tested using kernel density plot, Lewis and Reftry test PACF,and ACF. Thus, the results were obtained after all were satisfied which can be attained with increasing number of iteration.

Model was selected based on deviance information criteria (DIC) found for each model, where model with small deviance information criteria was selected for interpretation of the results. We have used deviance information criteria to choose better model for this analysis. The DIC for cross-classified random intercept model was small as compared to the other models.

All of the variables significantly predicted discharge status of patients in both random intercept models except referred status of patients. This variable was significant in CCM random intercept model. However, it was insignificant in HM random intercept model. In addition, the explained proportion of variation \((\hbox {R}^{2})\) in outcome variable is large for CCM random intercept model.

The model comparison result suggests CCM logistic model as the best fit for discharge status of hospitalized pneumonia patients. Thus, the effects of each covariate can be interpreted using the estimated odds observed under cross-classified multilevel logistic. The odds of dying due to community acquired pneumonia are the probability that the patient died divided by the probability that the patient discharged alive.

Most of the patient level variables included in the model significantly affect the discharge status of hospitalized pneumonia patients. Season diagnosed, CAP complication, duration of onset to diagnosis and age are patient level variable that significantly affect the discharge status of hospitalized pneumonia patients. Whereas, patient to physician ratio and patient to nurse ratio are hospital level variable that significantly affect the outcome of response variable.

The value of Exp(\(\beta \)) for spring season is 7.54 which implies that the odds of being at risk to death during spring season were 7.54 times more likely than patients diagnosed in winter season. In CAP complication variable, there was high probability of death on patients with AGE, AFI, and malaria than patients only with CAP. The odds of these coefficients are 6.64, 7.52 and 7.15, respectively. There was high probability of dying from CAP if the patients are complicated with these diseases. There is less likely to die from CAP, if patients are postnatal and child age than those patients in neonatal age group. Accordingly, the risk of dying from CAP for postnatal age group and child age group patients was less by 91 and 92% respectively. On the result, it was found that patients who late diagnosis has high probability of being at risk than those admitted before 3 days. The odds of being at risk for patients who diagnosed after a week is 8.71 times than those patients diagnosed before three days. The odds of being at risk for patients who diagnosed with 4 to 7 days was also 4.74, which means the risk of dying from pneumonia if patients diagnosed between these days was 4.74 times than those diagnosed before three days keeping another risk factors constant.

Patient to nurse ratio, patient to physician ratio were variables significantly predicts the discharge status of patients in the chosen CCM logistic model among hospital level variables. The odds of being at risk increases by 1.39 for a unit increases of patient to nurse ratio keeping the others risk factors constant. In the same manner, the odds of being at risk increases by 1.01 for a unit increase of patient to physician ratio keeping the risk factors constant. These means number of patients increased by the number physician/nurses keeping number of physician/nurse constant. Generally, patients diagnosed during patient to nurse ratio and/or patient to physician ratio was large had high probability of dying from CAP.