Predicting the time until employment of graduates using a prognosis index in survival analysis

Introduction

Unemployment is known to be one of the socioeconomic problems presently facing Nigeria as the labour force keep growing on a daily basis as a high number of young people graduate from either the Polytechnic or University. The rate of unemployment is measured or expressed as the percentage of the labour force that is currently actively seeking for employment but are without a job. There are several factors that contribute to the rise in the rate of unemployment in Nigeria. Some of which are labour market demands, the composition of the workforce's skills, the availability of job opportunities etc. The challenge of unemployment in Nigeria has contributed to the widespread poverty, the diminishing economic growth, increased rate of social unrest etc. Unemployment has directly and indirectly affected the standard of living, the rate of poverty, crime rate, insecurity and other socio-political problems in Nigeria which is significant among the youth population [1] and [2] . It has caused more threats to the development and peaceful coexistence of people in Nigeria.

In the light of proffering solutions to the problems of unemployment, this research aims at analysing and predicting the time until employment of graduates in Bayelsa State, Nigeria using the prognosis index in survival analysis. It will help to understand the unemployment trends and inform or enable the government or policy makers to effectively create jobs for the teeming unemployed people. The adoption of a survival analysis approach will enable the assessment and prediction of a precise model which will enable individuals to effectively predicting the time until being employed and engage in strategic planning during the period of being unemployed. Different researchers have investigated the prediction of unemployment rate using different statistical methods. For instance, [3] introduced the hybrid models for appraising Nigeria's unemployment rate using Time series and Machine learning techniques [4] . Proposed an ARIMA model in Times series for forecasting the rate of unemployment in Nigeria [5] . Also used the time series model of annual unemployment rates in Nigeria using Box-Jenkins methodology and examined the precision of forecast using this model [6] . Applied Neural Network model for forecasting unemployment while the study by [7] Karahan and Cetintas (2022) discussed the effectiveness of Artificial Neural Network model for predicting the future unemployment rate [8] . Aimed at identifying the potential determinants affecting unemployment in Nigeria using survival analysis. [9] Ugwu and Okonkwo (2025) developed an unemployment prediction system and employment tracking in Nigeria using a linear regression model and implemented the model using a trained on five years of data from the National Youth Service Corps (NYSC) covering graduates of Higher Education Institutions.

Materials and Methods

Study population and data collection

A set of data with 253 graduates from [8] was available for research. The study population was the unemployed graduates from different local government areas in Bayelsa State, Nigeria and the event of interest (survival time) was the time till first employment (duration of unemployment) after graduation from either polytechnics or universities. This is the time (approximately in months) from the date of passing out from the National Youths Service Corps (NYSC) of graduates below 30 years or date of receipt of the certificate exemption of NYSC of the graduates 30 years and above or date of receipt of OND certificate to the first day of employment. The data used for this research is right censored [10] and the censoring indicator was "1" for employed and "0" for censoring. The explanatory variables used in this research were all grouped into categories. The variables were gender ("1" for male and "2" for female), age ("1" for less than or equal to 20 years, "2" for 20 -25 years, "3" for 26 -30 years, "4" for 31 -35 years and "5" for 36 and above), marital status ("1" for single, "2" for married, "3" for divorced, "4" for widow or widower), education level ("1" for Ordinary National Diploma (OND), "2" for Higher National Diploma (HND) or Bachelors of Science (BSc.), "3" for Post Graduate Diploma (PGD) or Masters of Science (MSc.) and "4" for Doctors of Philosophy (PhD)), Cumulative Grade Point Average (CGPA) ("1" for less than 2.50, "2" for 2.50 -2.99, "3" for 3.0 -3.49, "4" for 3.5 and above), field or faculty of study ("1" for Education, "2" for Medical Science, "3" for Management Science, "4" for Arts, "5" for Sciences, "6" for Social Sciences, "7" for Environment Sciences and "8" for Others), residence lived by graduates ("1" for urban and "2" for rural), received training on job searching method ("1" for yes and "2" for no), Local Government Area ("1" for Brass, "2" for Ekeremor, "3" for Kolokuma, "4" for Nembe, "5" for Ogbia, "6" for Sagbama, "7" for Southern Ijaw and "8" for Yenagoa) and received training on field of study ("1" for yes and "2" for no).

Methodology

The Bayesian survival analysis approach will be used to analyse the data for which the event or survival time is the duration of unemployed. Bayesian inference requires the combination of prior experience (in the form of prior probability) and the observed data (in the form of a likelihood). It often involves calculations which are analytically intractable and are usually done using Markov chain Monte Carlo methods (MCMC) such as Metropolis and Metropolis -Hastings algorithm, Gibbs sampler and Metropolis within Gibbs algorithm [11] (Gilks et al., 1996). It is expected that the prior distribution should reflect the information about the parameters of the model. This is done by specifying prior information about the parameter of interest before looking at the data [12] . Bayesian inference will be applied to survival analysis using a Weibull model [13] in such a way that our prior beliefs about the unknown parameter of the survival analysis model are reflected. The proportional hazard model [14] which is mostly used by most researchers will be used for the analysis and the survival prediction by estimating the hazard ratios. The main product of the proportional hazard model is the prognostic index which is a linear predictor. This is also the logarithm of the hazard multiplier. The probability density function of the Weibull distribution is given by

where 𝜆 is the scale parameter which will be used to incorporate the explanatory variables and 𝛼 is the shape parameter. The linear predictor or prognosis index of the 𝑖 𝑡ℎ subject is given by

is the vector of the explanatory variables for the 𝑖 𝑡ℎ subject and 𝑋 𝑖 𝑇 is the transpose. and 𝛽 = (𝛽 0 , 𝛽 1 , 𝛽 2 , … . 𝛽 𝑘 ) 𝑇 is the vector of the parameters of the model. We have that

The survival function of the Weibull distribution [15] is given by

and the hazard function of the Weibull distribution is given by

where ℎ 0 (𝑡) is the baseline hazard function. The likelihood contribution [16] from the data can be simplified as

where C is the set of subjects that were censored, E is the set of subjects that had the event (employed) and 𝑛 𝐷 is the number of subjects that had the event. We will also note that the shape parameter accounts for additional possible hazard shapes.

We follow [12] to incorporate priors. We construct the prior distributions of the vector of parameters of explanatory We follow [12] to incorporate priors. We construct the prior distributions of the vector of parameters of explanatory variables 𝛽 = (𝛽 0 , 𝛽 1 , 𝛽 2 , … . 𝛽 𝑘 ) 𝑇 by giving them multivariate normal distribution with vector of means of parameters µ = (µ 0 , µ 1 , µ 2 , … … µ 𝑘 ) 𝑇 and the vector of covariance matrix V. The multivariate normal distribution for the prior distribution is given by

We will also give the shape parameter 𝛼, a gamma prior distribution given by

The posterior distribution is given by 𝑓(𝛽, 𝛼|𝐷) ∝ 𝑝𝑟𝑖𝑜𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 The joint posterior density will be simulated using Markov chain Monte Carlo (McMC) techniques via the JAGS program. The JAGS program which stands for Just another Gibbs sampler [17] . It is a program designed to work closely with the R language for the analysis of Bayesian models using McMC techniques. The RJAGS package can be used from within R to run JAGS [18] .

Results and Discussions

We will apply the procedure discussed in Section 2.2 to the data set. The explanatory variables, categories and notations are given in Table 3.1. The prior values of the parameters are elicited following discussions in [12] . The construction of prior distributions of the parameters might involve elicitation of prior beliefs from either a person or analyst in the field of study. The elicitation also depends on the type of variable. A Weibull distribution is used for the survival time for the 𝑖 𝑡ℎ individual which is given as Weibull(𝜆 𝑖 , 𝛼) where 𝜆 𝑖 is the hazard multiplier which depends on the linear predictor ɳ 𝑖 . The linear predictor is given by where 𝛿 𝑖,𝑗,𝑑 = 1 if 𝑥 𝑖,1 = 𝑑 and 𝛿 𝑖,𝑗,𝑑 = 0 otherwise for 𝑗 = 2, 3, 4, 5, 6, 9.

The prior means and standard variances of the parameters of the model are given in Table 3 chains starting with different values. Visual inspection of the trace plots of the parameters of the model showed that the mixing was appeared very satisfactory as shown in figure 1.

Figure 1: Trace plot of the Baseline parameter 𝛽 0 , 𝛽 2,1 (𝐴𝑔𝑒 1), 𝛽 5,1 (𝐶𝐺𝑃𝐴 1) and 𝛽 6,1 (𝐹𝑖𝑒𝑙𝑑 1).

The posterior means and standard deviations of the parameters of the model are given in Table 3.3.

Prognostic index

The prognostic index is calculated by substituting the posterior means of the parameters using Equation 3.1. The prognostic index will be used for predicting the duration of unemployment of graduates on the basis of their explanatory variables. It helps in planning and making decisions on the need of creation of jobs for the unemployed. It is known that high values of the prognostic index will indicate the best prognosis outcome for the event of interest (employment). It might also be preferable to index the prognostic index in a range of 0 to 100 by finding the value 100ɸ -1 (

where ɸ -1 is the standard normal distribution function.

The values of m and s are the sample mean and standard deviation of the values of the linear predictor of all subjects in the data set using the posterior means in Table 3

Conclusion and Recommendations 4.1 Conclusion

This study presents the survival analysis approach to predicting the time until employment of a graduate in Bayelsa State, Nigeria. This research will be helpful for making inference, monitoring and controlling the rate of unemployment in Nigeria using the prognosis index in survival analysis. The prognosis index provides a reliable platform for predicting the time until employment of a graduate. This will help the government to strategize ways to tackle unemployment at all levels.

Recommendations This research recommends the following

The government and private sector should make adequate solutions to increasing employment and reducing poverty by investment in agriculture, industrialization, diversification in the economy, modification in education curriculum among others have been identified in this research work. It is necessary that there should be significant policy interventions or changes in the economic structure to regulate the rate of unemployment in Bayelsa State and Nigeria at large. Hence, the Nigerian government should reduce reliance on a very few sectors such as oil and promote the growth or make adequate solutions such as investments in other sectors like agriculture, industrialization, technology and diversification in the economy.

The government should also modify the education curriculum to the enhancement of the quality of education and the alignment with the needs of the market to reduce the skills gap.

That the government should create an enabling environment for the private sector or businesses to thrive and so that new jobs employment can be created.

The government of Nigeria should also introduce skills development and entrepreneurship programs. The government of Nigeria can intelligently detect the fraud in ministries by using an employee tracking system and also make the correct estimate of the rate of employment. This will enable them make effective plan to create jobs for the unemployed graduates.

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