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attention to single-family residences and are equipped to make saving in this area. Someof the most important characteristics of a savings association are:1. It is generally a locally owned and privately managed home financing institution.2. It receives individuals’ savings and uses these funds to make long-term amortizedloans to home purchasers.3. It makes save for the construction, purchase, repair, or refinancing of houses.4. It is state or federally chartered.Ugandan, survey 87% of respondents said that a service delivery point in their sub-districtwould satisfy their needs as savings clients (Pelrine et al., 2005). As a result, informal andsemi-formal savings mechanisms can out-compete formal sources on proximity, helpingexplain why relatively risky informal mechanisms are still the most widely used by lowincome Ugandans. And also distance of financial sector affects saving rate of households.Return on Equity (ROE) of NBE WSACA was 37%, 32% and 30% in 2005, 2006 and2007 respectively, which was closer to ROE of commercial banks (32.7%, 40.7%and 29%in the respective years). ROE of MFI industry, on the other hand was 55%, 45% and 50%during the respective years, which was significantly higher than the later year (NBEWSACA, 2008). Similarly, in rural Ethiopia, most of the people use informal savingmechanisms because they are far away from the financial institutes or sectors.5

CHAPTER THREE: METHODOLOGY3.1 Description of the Study AreaThis study was conducted at Debre Birhan town, the capital of North Shoa or "SemenShewa" zone from February to October, 2011. According to CSA (2008) report, theestimated population size of Debre Birhan town was 65,244 out of these 31,658 weremale and 33,556 were female.3.2 Study PopulationThe target population consists of all employees at Debre Birhan town. A total of 5113employees within the age interval of 18 to 75 years were considered.3.3 Sampling Design and TechniquesThe data which is cross-sectional were collected from the target population through astructured questionnaire.3.3.1. SamplingStratified random sampling is used where the researcher draws simple random samplesfrom certain aggregation units of interest (from certain census block groups) when thestrata are heterogonous geographic units and also from successively smaller aggregationsuntil the individual subject level is reached.In this study using stratified random sampling method, government and privateemployees were selected with probability proportional to size without replacement(ppswor) as primary sample units (strata), individual employee (secondary unit samples)from each selected stratum with systematic sampling. Strata 1 : All employee that arework in the government organization. Strata 2: All employee that are work in the privateorganization. The sampling frame was prepared from government and private employeesby giving sequential order number. Finally an employee selected by systematic randomsampling procedure was contacted.3.3.2. Sample Size DeterminationThe sample size for this study was determined based on stratified sampling forproportions of 95 percent confidence level. Moreover, 3 percent of the sample size isadded to compensate for non-response rate. The sample size formula is given by(Cochran, 1977)6

. 3.1where, is the specified variance of the estimate and in case of estimatingproportion, the stratum variance is given byforcommon p. Since we use proportional allocation sample,stands for stratum weight, N N1 N2wherebe the upperpoint of standard normal distribution, wheresignificance level, which is. Suppose the relative error0.05is usuallyset by the investigator desired which is used from a similar study undertaken by the"Saving habits, needs and priorities in urban Uganda" in 2005, with a sample size of 852is taken as a reference for the purpose of fixing the standard deviation. One of theexplanatory variables used in that study is customer saving with a standard deviation of0.45. This helps us to determine the sample size to represent the population by calculatingthe acceptable absolute error, d, where. If we adopt a significance level 0.05, then the calculated margin of error from the above information is 0.03. Previousstudies indicate that the proportion of saving habits of employees varies from study tostudy. In our study, it is indicated that the proportion of saving habits of employeeswasdetermined small proportion from the results of previous work of similar population ofsaving habits of employees in Chinese and Americans 14.02% (p 0.1402) of (Feifei,2010). Thus, the sample size calculation isFinally, 3 percent of the sample size, which is 15, was added to the determined sample7

size 475 to compensate for none response rate. Thus, the required sample size for thisstudy is 490 employees from 5113 who live at Debre Birhan town. Next, we carried outsample size allocation to each stratum with proportional allocation.Where,h types of employee 2Probability of stratum weight selection of hth employee,Totalnumber of employees within the hth strata, N Total number of employee in DB townTable 3.1 Number of Employee Taken from the Selected PSU at Debre BirhanSampleEmployeeGovernment (N1)35030.685336Private (N2)16100.315154Total51134903.4 The Study Variables3.4.1. Dependent VariableThe response variable in this study is the status of saving habits at Debre Birhan town.The habits of employees are identified either save out of income or no save out ofincome. The response variable is a dichotomous category and my interest of the study isno saving out of income. Thus, coded as the value 0 for 'save out of income' and 1 for 'nosave out of income'.3.4.2. Independent VariablesThe independent variables in this study are classified as occupational variables, economicvariables and personal relationship and contextual variables.3.5 Logistic Regression ModelLogistic regression analysis extends the techniques of multiple regression analysis toresearch situations in which the outcome variable is categorical.8

Generally, the response variable is binary, such as (save or no save, presence or absence,success or failure etc) in logistic regression.3.5.1. Assumptions of Logistic RegressionAssumptions were should consider for the efficient use of logistic regression as givenbelow (Hosmer and Lemeshow, 1989). Logistic regression assumes meaningful coding of the variables. Logisticcoefficients were difficult to interpret if not coded meaningfully. The conventionfor binomial logistic regression is to code the dependent class of interest as 1 andthe other class as 0. Logistic regression does not assume a linear relationship between the dependentand independent variables. The dependent variable must be categorical. The independent variables need not be interval, no normally distributed, nolinearly related and no equal variance within each group. The groups must be mutually exclusive and exhaustive; a case can only be in onegroup and every case must be a member of one of the groups. Larger samples are needed than for linear regression because maximum likelihoodcoefficients are large sample estimates. The logit regression equation should have a linear relationship with the logit formof the dependent variable. Absence of multicollinearity.3.5.2. Model DescriptionSince the response variable in logistic regression is usually dichotomous, we were definesuch a response variable as Y, and denote the even y 1, when the subject has thecharacteristic of interest and y 0, when the subject does not have that characteristic ofinterest. So an alternative form of the logistic regression equation is the logittransformation of Pi given as9

The coefficient can be interpreted as the change in the log-odds associated with a one unitchange in the corresponding independent variable or the odd increases multiplicatively byfor every one unit change increase in.3.5.3. Parameter Estimation for Logistic RegressionThe maximum likelihood and non-iterative weighted least squares are the two mostcomputing estimation methods used in fitting logistic regression model (Hosmer andLemeshow, 1989).Consider the logistic model, since observed values of Y say, Yi’s (i 1, 2 , n) are independently distributed as binomial and, the maximum likelihood function ofY is given by:3.6.1. Model SelectionIn model selection, there are two competing goals: on one hand the model should becomplex enough to fit the data well. On the other hand, it should be simple to interpret,smoothing rather than over fitting the data (Agresti, 2002).3.6.2. Goodness of Fit of the ModelAssessing goodness of fit involves investigating how close values predicted by the modelwith that of observed values (Bewick and Jonathan, 2005). The comparison of observedto predicted values using the likelihood function is based on the statistic called deviance.For purposes of assessing the significance of an independent variable, the value of D arecompared with and without the independent variable in the equation as given below: D (model without the variable)The goodness-of-fitD (model with the variable)process evaluates predictors that are eliminated from the fullmodel, or predictors (and their interactions) that are added to a smaller model.10

3.6.3. Likelihood-Ratio TestThe likelihood ratio test statistic (G2) is the test statistic commonly used for assessing theoverall fit of the logistic regression model. The likelihood-ratio test uses the ratio of themaximized value of the likelihood function for the full model (L1) over the maximizedvalue of the likelihood function for the simpler model (L0). The likelihood-ratio teststatistic is given:It is compared with adistribution with 1 degree of freedom. This log transformation ofthe likelihood functions yields a chi-squared statistic.3.6.4. The Hosmer and Lemeshow Test StatisticThe final measure of model fit is the Hosmer and Lemeshow goodness-of-fit statistic,which measures the correspondence between the actual and predicted values of thedependent variable.Where g is the number of groups,is the number of covariate patterns in the kth group,is the number of responses among thecovariate patterns, andis the average estimated probability.3.6.5. The Wald StatisticThe Wald statistic is an alternative test, which is commonly used to test the significanceof individual logistic regression coefficients for each independent variable (that is to testthe null hypothesis in logistic regression model that a particular logit coefficient is zero).If the Wald test is not significant, then these explanatory variables can be omitted fromthe model. Wald 2 statistic was used to test the significance of individual coefficients inthe model and is calculated as:Each Wald statistic is compared with adistribution with 1 degree of freedom. Waldstatistic is easy to calculate but their reliability is questionable, particularly for smallsamples.11

3.6.6. R2 StatisticA number of measures have been proposed in logistic regression as an analog to R 2 inmultiple linear regressions. The Cox and Snell measure is based log-likelihoods andconsiders sample size. The maximum value that the Cox and Snell R2 attain is less than 1.The Nagelkerke R2 is an adjusted version of the Cox and Snell R2 and covers the fullrange from 0 to 1, it is often preferred.Therefore, in this study R2 statistic to indicate how useful the explanatory variables are inpredicting the response variables were used (Bewick and Jonathan, 2005).The Nagelkerke measure is as follows:Where3.7 Bayesian Analysis3.7.1. Bayesian Logistic RegressionThe foundation of Bayesian statistics is the Bayes' theorem which states that if A and B,are events and, the probability of event B, is greater than zero, then:3.7.2. The Likelihood and Prior FunctionLikelihood FunctionThe likelihood functionof the Bayesian formulation for the joint distribution of nindependent Bernoulli trials was still the product of each Bernoulli distribution, the sumof independent and identically distributed Bernoulli trials in which the sum has aBinomial distribution. Specifically, let y1, y2 yn be independent Bernoulli trials withsuccess probabilities P1, P2, P3, , Pn, that is yi 1 with probability Pi or yi 0 withprobability 1- Pi, for i 1,2, ,n. The trials are independent, the joint distribution of y1, y2,. . . yn is the product of n Bernoulli probabilities. As usual, the likelihood, function usedby Bayesians matches that from frequents inference. binomial:12

n[L( y β ) Pi i (1 Pi ) (1 yi )y]i 1Where, pi represents the probability of the event for subject i who has covariate vector Xi,yi indicates the presence, yi 1, or absence yi 0 of the event for that subject.βPi e 01 e β1β βoxi11 .xi1 β .kx βi kkxi kwhere: Pi the probability of ith employees being save, since individual subjects areassumed independent from each other likelihoods function over a data set of subjects is:nL( y β) i 1βo β1xi1 .βk xikeβ β x .β x1 e o 1 i1 k ikyiβo β1 xi1 . βk xik1 (1 yi )eβ β x .β x1 e o 1 i1 k ikPrior FunctionThe choice of an informative prior distribution typically involves a certain amount ofsubjectivity; historically, this has been a reason for disagreement between Bayesian andclassical statisticians. The assumed prior normal distribution for parameteris given by3.7.3. The Posterior DistributionBased on the prior distribution given above, the posterior distribution of the Bayesianlogistic regression contains all the available knowledge about the parameters in the modellike13

is the posterior distribution which is the product of the logistic regressionWherelikelihood and the normal prior distributions for theparameters.Note that the posterior distribution is a conditional distribution, conditional observing thesample. The posterior distribution is now used to make statements about, which is stilla random quantity. The most popular method of simulation technique is Markov ChainMonte Carlo (MCMC) methods.3.7.4. Markov Chain Monte Carlo (MCMC) MethodsThe use of Markov chain Monte Carlo (MCMC) methods to evaluate integral quantitieshas exploded over the last fifteen years. The primary distinction made here is betweenstandard Monte Carlo simulation and the Markov chain type of Monte Carlo methods.The initial definition required is that of a more primitive concept that underlies for thesecond MC which is called Markov chains.3.7.5. The Gibbs Sampler AlgorithmThe Gibbs sampler (David, 2006) is the most widely used MCMC technique. It is atransition kernel created by a series of full conditional distributions that is a Markovianupdating scheme based on conditional probability statements. If the limiting distributionof interest iswhereis an k length vector of coefficients to be estimated, then theobjective is to produce a Markov chain that cycles through these conditional statementsmoving toward and then around this distribution. The set of full conditional distributionsforare denotednotationand defined by( ) indicates a specific parametric form fromfor i 1, 2 k, where thewithout thecoefficient. Theserequirement facilities the iterative nature of the Gibbs sampling algorithm described as:1. Start with an initial value:2. Sample for each i 0, 1, 2. , n-1:14

Generate β 0(i 1)(f (βfrom f β 0 β 1Generate β1 (i 1) from1(Generate β 2 (i 1) from f βGenerate β k( i 1)2 β0 β(i),β2(i 1), β1(i 1)0(i),β3(i),β1(i),β2(i 1),., β k(i),β,., β k(i)2)(i)(i ),., β()(i)k))from f β k β 0 ( i 1 ) , β 1 ( i 1 ) , β 2 ( i 1 ) ,., β k 1 ( i 1 ) . .( 3 . 16 )3. Return β (1),. ., β (n )Once convergence is reached, all simulation values are from the target posteriordistribution and a sufficient number should then be drawn so that all areas of the posteriorare explored.3.7.6. Assessment of ConvergenceThere are basically three approaches to determining convergence for Markov chains:assessing the theoretical and mathematical properties of particular Markov chain,diagnosing summary statistics from in-progress models, and avoiding the issue altogetherwith perfect sampling, which uses the idea of “coupling from the past” to produce asample from the exact stationary distribution (Congdon, 2005). The second convergenceassessment methodology involves monitoring the performance of the chain as part of theestimation process and making an often subjective determination about when to stop thechain. Among several ways the most popular and straight forward convergenceassessment methods are:1. Autocorrelation: High correlation between the parameters of a chain tends to giveslow convergence, whereas high autocorrelation within a single parameter chain leadsto slow mixing and possibly individual non convergence to the limiting distributionbecause the chain will tend to explore less space in finite time. In analyzing Markovchain autocorrelation, it is helpful to identify lags in the series in order to calculate thelonger-run trends in correlation, and in particular whether they decrease withincreasing lags. These can be accessed in WinBUGS using the “autcor” button on theSample Monitor Tool.2. Time series plots: Iteration numbers on x-axis and parameter value on y-axis arecommonly used to assess convergence. If the plot looks like a horizontal band, with nolong upward or downward trends, then we have evidence that the chain has converged.These can be accessed in WinBUGS by using the “history” button.15

3. Gelman-Rubin statistic: for a given parameter, this statistic assesses the variabilitywithin parallel chains as compared to variability between parallel chains. The model isjudged to have converged if the ratio of between to within variability is close to 1.Plots of this statistic can be obtained in WinBUGS by using the “bgr diag” button. Thegreen line represents between variability, the blue line represents within variability, andthe red line represents the ratio. Evidence for convergence comes from the red linebeing close to 1 on the y-axis and from the blue and green lines being stable(horizontal) across the width of the plot.4. Density plot: a classic sign of non convergence is multimodality of the density estimate.3.7.7. The Burn-in Peri

Debre Birhan University, Alpha University College/Addis Ababa University, Hawassa University 15 November 2011 Online at https://mpra.ub.uni-muenchen.de/42301/ . 2Alpha University College/Addis Ababa University, Tel: 251911405942, buttegotu@gmail.com 3Hawassa University, Po. Box 05, Tel: 251911380766 yegna2009@gmail.com