USING LOG-LINEAR MODELS AND ODD RATIOS TO DETERMINE OUTCOME OF ADMISSIONS IN A HOSPITAL

The purpose of this study is to determine the outcome of patients’ admission in the medical ward (male, female and paediatrics) in the Central Regional Hospital 
The specific objectives of the study are as follows:to determine association between Outcome of admission(alive and death), Year of admission conditioning on the type of Medical Ward, to determine association between Medical Ward and Outcome of admission(alive and death) conditioning Year of admission, to determine association between Year of admission, Medical Ward and Outcome of admission(alive and death) and determine associations among variables by means of log-linear model Statistics and likelihood ratios 
The statistical tools used in analyzing the data were SPSS Micro soft excel and SAS. The Outcome of admission variable was categories into two levels; the medical ward had three levels and year variable had five levels and is ordinal. The model approach based on log-linear models was used. In this case, the homogeneous association was tested by comparing the saturated model (SM) and a model assuming homogeneity (HAM). The conditional independence was checked by comparing the HAM model with different models assuming conditional independence. The best model was chosen to conduct further analysis on the effect of medical ward on the outcome of admission while controlling by year. 
It realized that total number of patients admitted in the hospital was twelve thousand four hundred and twenty six (12426) for the 5 year period. Ten thousand seven hundred and sixty one (10761) were discharged (alive); representing 86.6%. Also one thousand six hundred and sixty five died; representing 13.4% for the same five year period. 
We observed that the female medical ward recorded highest patients’ death in the hospital from 2006 to 2010. 
In this model, the conditional likelihood ratio between any two variable are identical. We therefore concluded that the best model is the saturated log-linear model.


Introduction
Admission to Hospital Medical Wards "The decision to admit patients to the medical wards is determined by age, co-existing illness (co morbidity), physical and laboratory findings, the ability of the patient to comply reliably with an oral medication, and the resources available to the patient outside the hospital" (Ali, Woldie and Mirkuzie, 2010). In studies done in developed countries, medical admissions accounted for 22.2%, 33.0%, and 13.0% of total hospital admissions in U.S.A, Western Australia and Hong Kong, respectively (Oregon Health &Science University Pharmacy).
In a South African study, admissions to the medical wards constituted 40% of the total hospital admissions. In developed countries non-communicable diseases namely cardiovascular diseases are the main reasons for medical admissions.(Elias Ali, 2010) For instance the Australian study, the most common reason for admissions to the medical wards (29% of patients) was cardiovascular disease. In another study, admissions to medical wards at a hospital in Hong Kong were most frequently associated with the cardiovascular system, which made up 30.3% of all medical cases. However, in cities and towns of developing countries, the increasing urbanization and westernization of the population is changing the morbidity pattern of diseases. Patrick Kwabena Amoakoh, RAJAR Volume 04 Issue 09 September 2018 It is becoming widely accepted that non-communicable or chronic diseases are also now the major causes of death and disability in low and middle-income countries, (Elias Ali, 2010). In particular, smoking is increased in underdeveloped countries. The annual cigarette consumption per adult (in cigarettes) has increased from 860 in the early 1970s 1410 in 1995. The reason was aggressive marketing of tobacco companies, delay in implementing antismoking regulations, and because of the public perception of the risk of smoking is still low. This has been supported by studies from South Africa, (Elias Ali, 2010). Circulatory disorders (22%) and infectious diseases (19%) were shown to be the main causes for admission to medical wards of GF Jooste Hospital. In another study from the same country medical admissions were mostly associated with the circulatory system (27.9%) followed by respiratory (15.9%) and infectious diseases (11.9%). Unlike the findings in developed and some Sub-Saharan African countries, the leading reasons for admission to the medical wards in Ethiopia were found to be communicable diseases. Particularly, communicable diseases such as acute febrile illness of infectious origin, pneumonia and tuberculosis were the reasons for admission to the medical wards. . All patients who were admitted to the medical wards of JUSH during the study period (January.1, 2008 to December. 31,2008) and whose case notes were available in the hospital registration room archive was included. A case note was classified according to year of admission. The recent case notes of patients admitted to the medical wards in the year 2008 was retrieved. Hence, all cases found during the study period was included in the study and no sampling technique was used. The dependent variables in the study were reasons for admission and outcome of admission, while the explanatory variables included sociodemographic characteristics of the patients in the case notes reviewed, co-morbidities, complications, duration of hospital stay and month of admission. Leng et al. (1999) stated that in recent years have seen rises in readmission both in the UK and the US. In Oxfordshire, readmission rates almost doubled between1968 and 1985, with 75% increase in emergency readmissions. A similar rise was also seen in Scotland, where the overall readmission rate rose from 7.1% in 1982 to11.4% in 1994.7 8 Unfortunately, the definition of readmission has varied considerably between many studies, including the time since discharge before readmission, and the type of readmission (elective or emergency). This focuses on unplanned, or emergency, readmissions. The rising trend in emergency readmissions is worrying partly because of implications about quality of care but also because of the burden placed on provision of hospital services. Reasons for the increase are less clear and are likely to be complex. Possible explanations include changes in the social and demographic structure of the population, falling lengths of stay, and the medical condition itself. The increasing number of elderly people in the population may be particularly important in generating the rise because admission rates increase dramatically with age, especially in those living alone. It has also been suggested that readmissions are related to recurring medical problems, indicated by a higher than expected number of admissions in the period before the readmission. Huffman (1990) stated that the medical record "must contain sufficient data to identify the patient, support the diagnosis or reason for attendance at the health care facility, justify the treatment and accurately document the results of that treatment" The main purpose of the medical record is: To record the facts about a patient's health with emphasis on events affecting the patient during the current admission or attendance at the health care facility, and for the continuing care of the patient when they require health care in the future. A patient's medical record should provide accurate information on: who the patient is and who provided health care; what, when, why and how services were provided; and the outcome of care and treatment. The medical record has four major sections: Administrative, which includes demographic and socio-economic data such as the name of the patient (identification), sex, date of birth, place of birth, patient's permanent address, and medical record number; legal data including a signed consent for treatment by appointed doctors and authorization for the release of information; financial data relating to the payment of fees for medical services and hospital accommodation; and clinical data on the patient whether admitted to the hospital or treated as an outpatient or an emergency patient.

Statement of the Problem
If critically ill patients are admitted and do not receive proper care, death may occur at any time. The medical teams must fully utilize their knowledge and skills to reduce mortality in the hospital. As the role of the hospital unit in admission outcomes at Central Regional Hospital has yet to be researched, this study explores the situation in the hospital.

Objective of the Study
The broad goal of health services delivery is to improve the health of all people living in the municipality regardless of age, ethnicity, religious conviction, political affiliation or socio-economic standing. This broad goal encompasses many specific objectives, among them, an increase in life expectancy, reduction in morbidity and fertility rates, and improvement in quality of life.

Methodology
The research method used for this work is quantitative research method. This will explain the phenomena 'outcome of admission a hospital' by collecting numerical data that are analysed using mathematically based methods. According to Aliaga and Gunderson (2000), describes what we mean by quantitative research methods very well: Quantitative research is 'Explaining phenomena by collecting numerical data that are analysed using mathematically based methods (in particular statistics)'. In order to be able to use mathematically based methods, our data have to be in numerical form. This is not the case for qualitative research. Qualitative data are not necessarily or usually numerical, and therefore cannot be analysed by using statistics. Therefore, as quantitative research is essentially about collecting numerical data to explain a particular phenomenon, particular questions seem immediately suited to being answered using quantitative methods.

Population and Sampling Technique
The population of the study consisted of the patients admitted in the medical wards (Male, Female and Pediatric wards) from 2005 to 2010 in Central Regional Hospital (Interburtin), Cape Coast -Ghana. The total number of patients admitted in the hospital was Twelve Thousand Four Hundred and Twenty Six (12,426) for the five (5) year period. The patients and medical personnel were interviewed. A purposeful sampling technique was used.

Instrumentation
The instruments used for the study were observation, interview and secondary data from biostatistics department in the hospital. Interviews were granted to the medical personnel and some patients.

Basic Strategy and Key Concepts
The basic strategy in log-linear modeling involves fitting models to the observed frequencies in the cross-tabulation of categorical variables. There can then be models represented by a set of expected frequencies that may or may not resemble the observed frequencies. Models will vary in terms of the marginal's they fit, and can be described in terms of the constraints they place on the associations or interactions that are present in the data. The pattern of association among variables can be described by a set of odds and by one or more odds ratios derived from them. Once expected frequencies are obtained, we then compare models that are hierarchical to one another and choose a preferred model, which is the most parsimonious model that fits the data. It's important to note that a model is not chosen if it bears no resemblance to the observed data. The choice of a preferred model is typically based on a formal comparison of goodness-of-fit statistics associated with models that are related hierarchically (models containing higher also implicitly include all lower order terms).
Ultimately, the preferred model should distinguish between the pattern of the variables in the data and sampling variability, thus providing a defensible interpretation.

Theory and Calculations The Log-linear Model
The following model refers to the traditional chi-square test where two variables, each with two levels (2 x 2 tables), are evaluated to see if an association exists between the variables: Log(m ij ) = is the log of the expected cell frequency of the cases for cell ij in the Contingency table. µ = is the overall mean of the natural log of the expected frequencies λ = terms each represent "effects" which the variables have on the cell frequencies X and Y = are treated as outcome variables i and j refer to the categories or level within the variables X and Y Therefore: The motivation for the use of log-linear models is that statistical independence can be express in terms of a linear combination of the logarithms of the cell probabilities. In particular if the variables X and Y in a 2 × 2 table are statistically independent, then the probability of individuals being in the first row (level 1 of X) among those in the first column (level 1 of Y) would be the same as the probability for the first row among those in the second column (level 2 of Y).
The likelihood ratio test of independence can be derived directly from the multinomial likelihood.
The unrestricted maximum likelihood estimates (MLEs) of the π ij values are given By P ij = . the maximized likelihood is then: Under the independence hypothesis H 0 :π ij = π i + π + j , the likelihood is: The MLEs for the π ij under this model are P ij = n i + n + j / n and the maximized log-likelihood is The likelihood ratio is: Where m ij = n i+ n + n and the likelihood ratio statistics i The statistic G 2 has an asymptotic chi-square distribution with 1 df ifH 0 is true, and it is asymptotically equivalent to the Pearson chi-square statistic. Where m ij= nπ ij is the expected frequency in the (I, j) cell. The parameters µ is fixed by the sample size n and the model has s + r + s r parameters λ j X , λ j Y , and λ i j

XY
. The sum-to-zero constraints

Fitting Log-linear Models
Once a model has been chosen for investigation the expected frequencies need to be tabulated. For two variable models, the following formula can be used to compute the direct estimates for non-saturated models. (Column total) * (row total)/grand total for larger tables, an iterative proportional fitting algorithm (Deming-Stephan algorithm) is used to generate expected frequencies. This procedure uses marginal tables fitted by the model to insure that the expected frequencies sum across the other variables to equals the corresponding observed marginal tables

Testing for goodness of fit
Once the model has been fitted, it is necessary to decide which model provides the best fit. The overall goodness-offit of a model is assessed by comparing the expected frequencies to the observed cell frequencies for each model. The Pearson Chi-square statistic or the likelihood ratio (G 2 ) can be used to test a models fit. However, the (G 2 ) is more commonly used because it is the statistic that is minimized in maximum likelihood estimation and can be partitioned uniquely for more powerful test of conditional independence in multi way tables:  n! n 11 ! n 12 ! n 21 ! n 22 ! π 1+ π 2+ π + π +2 n 1 + n 2 + n +1 n +2 …….. (14) n 11 ! n 12 ! n 21  G 2 follows a chi-square distribution with the degrees of freedom (df) equal to the number of lambda terms set equal to zero. Therefore, the G 2 statistic tests the residual frequency that is not accounted for by the effects in the model (the λ parameters set equal to zero). The larger the G 2 relative to the available degrees of freedom, the more the expected frequencies depart from the actual cell entries. Therefore, the larger G 2 values indicate that the model does not fit the data well and thus, the model should be rejected. It is often found that more than one model provides an adequate fit to the data as indicated by the non-significance of the likelihood ratio. At this point, the likelihood ratio can be used to compare an overall model within a smaller, nested model (i.e. comparing a saturated model with one interaction or main effect dropped to assess the importance of that term). The equation is as follows: Model 1 is the model nested within model 2. The degrees of freedom (df) are calculated by subtracting the df of model 2 from the df of model 1. If the L 2 comparison statistic is not significant, then the nested model (1) is not significantly worse than the saturated model (2). Therefore, choose the more parsimonious (nested) model.

Log-linear Residuals
In order to further investigate the quality of fit of a model, one could evaluate the Individual cell residuals. Residual frequencies can show why a model fits poorly or can point out the cells that display a lack of fit in a generally goodfitting model The process involves standardizing the residuals for each cell by dividing the difference between frequencies observed and frequencies expected by the square root of the frequencies expected (F obs -F exp /F exp ).
The cells with the largest residuals show where the model is least appropriate. Therefore, if the model is appropriate for the data, the residual frequencies should consist of both negative and positive values of approximately the same magnitude that are distributed evenly across the cells of the table.

Results and Discussion
Preliminary Analysis  Table 4 deals with Cross tabulation Outcome of Admission and Medical Ward. It was observed that more patients were discharged / alive in the paediatric ward Representing 4893 (91.19%) than those discharge in the male and female medical ward representing 3389 (87, 77%) and 2279 (75.58%) respectively.
On the other hand, it was also noted that female medical ward recorded highest number of death representing 736 (24,71%) than paediatrics and female medical wards representing 429 (8.06) and 500 (12.23) respectively.   We could evaluate the difference in the Chi-square statistics, based on the difference in the degrees of freedom; if the differential Chi-square statistic is significant, then we would conclude that the three-way interaction model provides a significantly better fit to the observed table than the model without this interaction. Therefore, the three-way interaction is statistically significant.

Year (Z), Medical ward (X), Outcome(Y) Variables
YZ ………………………………………… (4.6) Homogeneous model Log (m ijk ) = µ+ λ i X + λ j Y + λ K Z …………………..(4.5) Mutually independent model The log-linear models were used to check for associations. In this approach, based on fitted models, it was checked whether the homogeneity assumption holds and thus, whether conditional independence models, joint independence models or the mutually independent model gave a good fit to the data. First of all, the goodness of fit was tested for each model. The table 7 summarizes the results. The likelihood ratio statistic (G 2 ) was used when testing goodness of fit. The choice of G 2 was suggested mainly by the fact that it can be partitioned. This property allows the comparison of nested models and the saturated model. The values of G 2 in 14 are differences of likelihood ratio for a given model and the likelihood ratio for the saturated model. A large P-value indicates that there is no significant difference of the related model and the saturated model. It can be seen that the homogeneity assumption holds since the model ((ZX, ZY, XY) is significantly different of the saturated model (P-value < 0.0001), thus there is a homogeneous association between the three variables. The homogeneous association model can therefore be considered as baseline to check conditional independence among the variables. Table 8 gives results on comparison of nested models to the HAM. The model (ZY, XY) (P-value< 0.0001) is significantly different from the HAM. Thus, year and outcome are conditionally independent given the medical ward and outcome of Patrick Kwabena Amoakoh, RAJAR Volume 04 Issue 09 September 2018 admission. The model (ZX, XY), assuming conditional independence of Medical ward and the outcome of admission given year, showed a p-value=0.0001. This means that interaction ZY cannot be removed from the HAM. It is deduced that year and the outcome of admission are not independent. Thus, the three variables are not mutually independent. The last association checked is the joint independence of medical ward to year and outcome of admission. By taking the conditional independence model (ZY, ZX) as baseline, it is shown in table 15 that joint independence model (X, ZX) is not significantly different from the model (ZY,ZX) (P-value=0.0001) which means that medical ward is jointly independent to therapy and response-to-chemotherapy. Also the models (X, ZY), (Y, ZX) and (Z, XY) are jointly independent.  The model with no threefactor interaction provides a good fit to the observed data. Thus, no pair of variables is conditionally independent. In this model, the conditional likelihood ratio between any two variable are identical. We therefore concluded that the model is three-factor interaction model.

Conclusion
Based on the results of the research, we observed that the use of log-linear models is successfully employed to determine the associations between Year of admission (Z), Medical Ward(Y) and Outcome of admission(X). From the results, we can infer that the female medical ward recorded more death than the male and the paediatrics ward. Again the results reveal that more patients died in 2006 and year 2009 recorded least number of patients death. We father conclude that female medical ward recorded the highest number of death and paediatric ward recorded least number of death from 2006 to 2010 We can then conclude that there is association between medical ward and the outcome of admission, medical ward and the year, and year and outcome of admission. The three-way association tested was between category of medical ward, outcome of admission and year of admission (model 2d), and it was found to be significant. After that, a two-way association between medical ward, outcome of admission and year (models 1a, and 2a) were tested and all of them were found to be significant. This shows that there is an association between the three variables. Therefore, the best model is the saturated model.

Recommendations
Based on the findings from the log-linear model for outcome on admissions in Hospital from 2006 -2010, the following recommendations have been made:  The Hospital Authorities could use the log-linear modeling and likelihood ratio methods to analyze data periodically.  Medical personnel should make maximum use of available facilities to help reduced mortality rate in the hospital. This is because current rate is not very good.  Doctor to patient ratio must be improved for store further death in the hospital.  Further studies should be conducted in other hospitals to see whether or not same problem exist. Patrick Kwabena Amoakoh, RAJAR Volume 04 Issue 09 September 2018