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The Millennium Development Goal (MDG) 5 advocated the reduction of maternal mortality rates significantly by 2015, however, maternal mortality rates continue to rise. Here, we modelled maternal mortality data for the years 2000 to 2013 obtained from a public hospital in Kumasi, Ghana. We applied the Box-Jenkins approach of univariate form of time series autoregressive integrated moving average (ARIMA). The output revealed that the ARIMA (1, 1, 1) model was most appropriate to model and predict monthly maternal cases with Akaike information criterion (AIC) value of 117.02 and Bayesian information criterion (BIC) value of 125.91. The Shapiro-Wilk normality test confirmed normality of the residuals. The Ljung-Box test on the residuals showed no serial correlation. The model was then validated based on the measures of accuracy. The results showed that the maternal mortality cases for the years 2000 to 2011 are high: minimum 3, median 11, mean 12 and maximum cases of 26 per month. The predicted mortality cases were 10 to 11 monthly for years 2012 to 2013, indicating that the target of MDG 5 could not be achieved by 2015. Fresh and perceptive strategies are urgently needed to arrest the unacceptably high death rates.

Issues of maternal health have continuously received attention globally and nationally since the 1980s. The UN Millennium Summit (2000) involving the UN member states, including Ghana, adopted Millennium Development Goals (MDGs) 8 which were geared towards the improvement of life of all people across the globe [

The deaths that occur in women during pregnancy or within 42 days after pregnancy termination are referred to as maternal mortality [

Maternal mortality reduction is one of the MDG that Ghana seeks to achieve since it affects the development of the nation [

Despite interventions and several efforts by governments and other development partners towards achieving the goals of MDG 5, the MMR for developing countries still remains high [

Using the Box-Jenkins methodology [

In this report, we applied the principles of Box-Jenkins methodology to maternal mortality cases recorded at the Konfo Anokye Teaching Hospital (KATH), Kumasi, Ashanti Region, Ghana. The study seeks to model, validate and forecast the monthly maternal mortality at the hospital and highlight the trend of maternal mortality in the presence of programmes implemented to achieve the MDG 5. The findings of the study could serve as a guide for a review of MDG 5 with the passage of time and help assess the current interventions to curb the high numbers of maternal mortality.

Box-Jenkins methodology is a statistical procedure that is used to model time series data by using autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) models [

A dataset Y_{t} follows ARIMA model if the d^{th} differences ∇^{d}Y_{t} follows a stationary ARMA model. The parameters that help build the ARIMA model are three; p, which determines the AR order; d, denotes the number of differencing required before stationarity, and MA order is given by q [

Hence, ARIMA (p, d, q) is represented in a general form according to Tebbs [

∅ ( B ) ( 1 − B ) d Y t = θ ( B ) e t (1)

where, the AR and MA characteristic operators are

∅ ( B ) = 1 − ∅ 1 B − ∅ 2 B 2 − ⋯ − ∅ p B d (2)

θ ( B ) = 1 − θ 1 B − θ 2 B 2 − ⋯ − θ q B q (3)

And,

( 1 − B ) d Y t = ∇ d Y t (4)

where,

f is the autoregressive parameter to be estimated; θ is the moving average parameter to be estimated; ∇, the difference operator; B, the backward shift operator; e_{t}, random process having zero mean and variance not depending on time ( σ e 2 ). Box and Jenkins [

In order to make inferences in time series analysis, it is necessary to determine whether the time series is stationary or not. Prior studies have relied on Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test [_{t} follows a randomness in the time series data:

y t = ρ y t − 1 + e t (5)

where, ρ, the characteristic root of an AR polynomial and e_{t}, white noise with mean zero and variance σ^{2} [

H_{0}: ρ = 1 (non-stationary) versus H_{1}: ρ < 1 (stationary).

Phillips and Perron [

There are techniques under ARIMA model identification which estimate the p, q and d values. The autocorrelation function (ACF) and partial autocorrelation function (PACF) help to determine the p, q and d values. The theoretical PACF of ARIMA (p, q, d) process usually show non-zero PACF at first p lags, with remaining lags having zero PACF. The first q lags also report non-zero ACF and the remaining lags having zero ACF for the theoretical ACF. We determine q and p by the total frequency of the significant lags which are not zero for ACF and PACF respectively. If the values of p, d, and q are inaccurately selected, models derived can be inadequate, hence cannot be used for predictions [

If the ARIMA model is identified, then the maximum likelihood approach to estimating the parameters is used. In estimating the parameters, the log likelihood of a given p, d, q is maximized so that the probability of obtaining the observed data is maximized. Model estimation is followed by model selection, and it is done by considering minimum values of Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) [

AIC = − 2 ln ( L ¯ ) + 2 h (6)

and BIC = − 2 ln ( L ¯ ) + ln ( n ) h , (7)

where, L ¯ is the likelihood value of the likelihood function, h and n are number of parameters to be estimated and number of residuals respectively. For any two competing models, the model with the minimum AIC or BIC will be selected as a better one.

Ljung-Box Q test is used to assess serial correlation of the residuals and it helps to determine the randomness of the residuals and model adequacy. Therefore,

Q m = n ( n + 2 ) ∑ k = 1 n ( n − k ) − 1 r k 2 ≈ χ m − r 2 . (8)

where, r k 2 = the residuals autocorrelation at lag k, n= the number of residual, and m= the number of time lags included in the test.

In this study, the level of significance is set to 5% and a model attains adequacy when the Q test statistics report p-value > 0.05. Absence of this renders the model inadequate and a better model should be identified and assessed. In addition, in order to achieve homoscedasticity, ACF and PACF will be plotted of the squared residuals. The Shapiro-Wilk test of normality and histogram will be used to assess the normality of the residuals. To validate the model selected, the dataset was modelled using a training set which comprised of data from 2000 to 2011 and validated using a testing sample from 2012 to 2013. The validation measures included root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE).

RMSE = 1 n ∑ i = 1 n r i (9)

MAE = 1 n ∑ i = 1 n | r i | (10)

MAPE = 100 n ∑ i = 1 n | r i o i | (11)

where r i , n and o i are the residuals, number of observations and the observed values respectively. The closer the validation measures for the errors from both models the better the training model.

Log transformation of data (y_{t}) is among the family of methods called Box-Cox approach which is usually applied when there is high volatility in the data in order to stabilize the variance over time [_{0} was rejected. The R statistical software was used for the analysis.

The data for this study is a time series (2000-2013) data from KATH, Kumasi on maternal mortality. The time plot in

The trend of the data over the years was assessed using the time plot. Examination of the dataset, revealed an existence of unstable trend. This was confirmed by computing the mean and variance of the dataset which revealed that the value for the variance was greater than that of the mean hence the instability of the data. This, consequently, led to the natural log-transformation of the data to stability.

The ADF, PP and KPSS tests were used to test for further stationarity. The results of the no differenced log data revealed that the ADF and PP test confirmed stationarity (

The output in

ARIMA (1, 1, 1* (Row 2)) in

Different stationarity tests | The order of differencing | The test statistic values | p-value |
---|---|---|---|

ADF | 0 | −4.2998 | 0.01 |

PP | 0 | −11.270 | 0.01 |

KPSS | 0 | 0.64742 | 0.02 |

ADF | 1 | −7.305 | 0.01 |

PP | 1 | −28.292 | 0.01 |

KPSS | 1 | 0.012779 | 0.10 |

Model | BIC | AIC | |
---|---|---|---|

1 | ARIMA (1, 1, 0) | 166.13 | 160.2 |

2* | ARIMA (1, 1, 1) | 125.91* | 117.02* |

3 | ARIMA (2, 1, 1) | 130.79 | 118.94 |

4 | ARIMA (3, 1, 1) | 133.51 | 118.69 |

5 | ARIMA (2, 1, 0) | 147.63 | 138.75 |

6 | ARIMA (3, 1, 0) | 143.74 | 131.89 |

Component | Coefficient | S.E | P-Value |
---|---|---|---|

AR(1) | 0.0496 | 0.0894 | 0.5791 |

MA(1) | −0.9504 | 0.0346 | <0.0001 |

to forecast two years ahead in order to validate the model.

Therefore,

Y t = ( 1 − 0.9504 B ) e t = − 0.9504 e t − 1 + e t (12)

From

0, which may occur as a result of randomness. The Ljung-Box test also showed that the ARIMA (1, 1, 1) was adequate with p-value > 5% and could be used to forecast maternal mortality cases at the hospital.

The histogram (

The dataset was partitioned as training and testing sample. The training sample contains about 85.7% (2000 to 2011) portion of the dataset for modeling the data. The sample for testing the validity of the model (test sample) contains the remaining portion, 14.3% (2012 to 2013) of the dataset. Based on the estimates of RMSE, MAE and MAPE for both training model and testing model in

Our study confirms that, instead of maternal mortality cases declining as being sought by Target 6 of the MDG 5, there were increases in death rates at the KATH, Kumasi, Ghana over the years to 2013. This study has revealed that maternal mortality rates are expected to be on a constant trajectory over the years 2012 to 2013 even into 2014 if the prevailing conditions remain from the previous years. These observations have also been made in other studies [

Model Fit Indexes | Training Model | Testing Model |
---|---|---|

RMSE | 0.3527 | 0.2583 |

MAE | 0.2706 | 0.2016 |

MAPE | 13.5079 | 9.5522 |

Our study supports the report by Commonwealth Health Online [

According to the UN Agencies report [

This study applied the Box-Jenkins methodology to model the maternal mortality cases recorded at KATH, Kumasi, Ghana, using data from 2000 to 2013. The time series modeling was employed by first assessing the time plot, ACF and the PACF of the series. The time plot showed fluctuations in mortality from 2000 to 2011, with 2011 recording the highest mean maternal mortality of 12 cases. The dataset was natural log transformed because it was volatile. Finally, the appropriate model ARIMA (1, 1, 1) was used to forecast two years (24 months) for the maternal mortality cases at KATH, Kumasi. The model adequacy and validation have also shown to be appropriate in predicting the maternal mortality cases, and was used to forecast data from 2012 to 2013. The forecast values fell within the required 95% confidence interval highlighting the adequacy of the fitted model. The results of the forecasting showed that from 2012 to 2013, the maternal mortality rates were stable, and were estimated to be 10 to 11 cases monthly. These predicted monthly maternal mortality cases are unacceptably high and this is not in favour of the target of MDG 5. These findings could serve as a guide for a review of MDG 5 and help scrutinise the on-going interventions to curb maternal mortality. The MDG 5 was not achieved by the set time of 2015.

The authors would like thank the KATH, Kumasi, Ghana, for allowing access to the secondary data.

We declare no conflicts of interests.

Adedia, D., Nanga, S., Appiah, S.K., Lotsi, A. and Abaye, D.A. (2018) Box-Jenkins’ Methodology in Predicting Maternal Mortality Records from a Public Health Facility in Ghana. Open Journal of Applied Sciences, 8, 189-202. https://doi.org/10.4236/ojapps.2018.86016