dL or higher along with symptoms like excessive thirst, frequent urination, or unexplained weight loss, they may also be diagnosed with diabetes (Jaeger et al., 2025).

Agricultural activities, like applying chemical fertilizers and pesticides, can have environmental consequences that can indirectly impact diabetes risk factors. For instance, exposure to chemicals such as glyphosate or organophosphates used in farming has been associated with a higher likelihood of developing metabolic disorders. Additionally, environmental factors such as air pollution and climate change may exacerbate diabetes risk factors and health outcomes, potentially affecting agricultural productivity and crop yields (whiting et al., 2011). Overall, while the direct impact of type-2 diabetes on agricultural productivity and postharvest losses may be limited, the interplay between diabetes, dietary patterns, healthcare access, and environmental factors can have broader implications for agricultural communities and food systems. Addressing the complex relationship between health, agriculture, and the environment requires a holistic approach that considers socioeconomic factors, public health interventions, and sustainable agricultural practices (Whiting et al., 2011).

Overall, while the direct impact of type-2 diabetes on agricultural productivity and postharvest losses may be limited, the interplay between diabetes, dietary patterns, healthcare access, and environmental factors can have broader implications for agricultural communities and food systems. Addressing the complex relationship between health, agriculture, and the environment requires a holistic approach that considers socioeconomic factors, public health interventions, and sustainable agricultural practices (Huang et al., 2016).

This study therefore attempts to extend the existing literature and contribute to the existing body of knowledge by modeling and forecasting non insulin dependent diabetes among farmers in Benue State using autoregressive moving average (ARIMA) time series model with more recent data.

2.0       MATERIALS AND METHODS

2.1       Method of Data Collection

The data utilized in this research work are monthly secondary time series data on morbidity incidence of type-2 diabetes in Benue state for the period of January, 2005 June, 2025 making a total of 234 observations. The data was collected from Benue State Epidemiological unit, Makurdi. The data was transformed to natural logarithms using the following formula:

where is the confirmed type-2 diabetes series observation indexed by time , while is the natural logarithm. Hence forth will be regarded as a series.

2.2 Methods of Data Analysis

Find below the statistical tools employed in the analysis of data in this work.

3.2.1 Descriptive statistics and normality measures

The mean of any given set of data can be computed as follows:

The sample standard deviation of any given set of data over a given period of time is computed using the formula:

where is the sample mean, is the sample size.

Jarque-Bera test is a normality test of whether a given sample data have the skewness and kurtosis similar to that of a normal distribution. The test was proposed by Jarque and Bera (1980, 1987) and test the null hypothesis that the series is normally distributed. Given any data set, the test statistic JB is defined as:

where is the sample skewness denoted as:

and is the sample kurtosis given below:

whereT is the total number of observations. The JB normality test checks the following pair of hypothesis:

and (i.e., follows a normal distribution)

and (i.e., does not follows a normal distribution).

The test rejects the null hypothesis if the p-value of the JB test statistic is less than level of significance.

2.2.2 Augmented Dickey-Fuller (ADF) unit root test

The Augmented Dickey-Fuller (ADF) test helps to identify if a time series is stationary or has a unit root, indicating a persistent trend over time (Dickey and Fuller, 1979).

It accounts for higher-order correlations by assuming the series follows an AR(p) process and incorporates lagged differences of the series into the regression to enhance the test's precision.

.

where are optional exogenous regressors which may consist of constant, or a constant and trend, and are parameters to be estimated,β values arelagged difference terms and the are assumed to be white noise. The null and alternative hypotheses are written as:

                                                                                        (8)

and evaluated using the conventional ratio for

where is the estimate of and "the coefficient standard error is denoted as "

2.2.3 Portmanteau test

A Portmanteau test also called he Ljung-Box Q-statistic test is used to determine whether there is any remaining serial correlation or autocorrelation in the residuals of a time series. The test checks the following pairs of hypotheses:

(all lags correlations are zero)

(there is at least one lag with non-zero correlation). The test statistic is given by:

where

denotes the autocorrelation estimate of squared standardized residuals at lags. T is the sample size, Q is the sample autocorrelation at lag k. We reject if p-value is less than level of significance (Ljung and Box, 1979).

2.3 Time Series Models Specification

To specify an ARIMA model which is the model framework use in this study, we first specify autoregressive (AR) model, moving average (MA) model, autoregressive moving average (ARMA) model before specifying autoregressive integrated moving average (ARIMA) model. These models are specified as follows.

2.3.1 The autoregressive (AR) model

A stochastic time series process {} is an autoregressive process of order p, denoted AR() if it satisfied the difference equation

where is a white noise and are constants to be determined.

2.3.2 Moving average (MA) model

A time series {} which satisfies the difference equation

where are fixed constants with as white noise is called a moving average process of order q, denoted MA().

2.3.3 Autoregressive moving average (ARMA) model

A stochastic time series process {} which results from a linear combination of autoregressive and moving average processes is called an Autoregressive Moving Average (ARMA) process of order p, q, denoted ARMA () if it satisfies the following difference equation:

where are fixed constants associated with the AR terms and are fixed constants associated with the MA terms with being a white noise. The stationarity of an ARMA () process is guaranteed if the roots of the polynomial

lie outside the unit circle.

An ARMA () model is specified as:

2.3.4 Autoregressive integrated moving average (ARIMA) model

Autoregressive (AR), Moving Average (MA) or Autoregressive Moving Average (ARMA) model in which differences have been taken are collectively called Autoregressive Integrated Moving Average or ARIMA models. A time series {} is said to follow an integrated autoregressive moving average model if the ^th difference is a stationary ARMA process. If follows an ARMA(p, q) model, we say that {} is an ARIMA (p, d, q) process. For practical purposes, we can usually take or at most 2.

Consider then an ARIMA (p, 1, q) process, with , we have

In terms of the observed series,

)

2.4 Model Order Selection

We use the following information criteria for model order selection in conjunction with log likelihood function: Akaike information criterion (AIC) due to Akaike (1978), Schwarz information Criterion (SIC) due to (Schwarz, 1978) and Hannan-Quinn information Criterion (HQC) due to (Hannan, 1980). The formula for the information criteria are:



where is the number of free parameters to be estimated in the model, T is the number of observations and L is the likelihood function defined as:

Thus given a set of estimated ARMA models for a given set of data, the preferred model is the one with the minimum information criteria and maximum log likelihood.

2.5 Model Forecast Evaluation

We employed Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) accuracy measures to select an optimal model mode that is both parsimonious and accurately forecast the data based on minimum values of the accuracy measures.

2.5.1 Root Mean Square Error (RMSE)

The Root Mean Square Error is a statistical tool for measuring the accuracy of a forecast method. It is computed as:

Where is the forecast value of the series and is the actual series and is the number of forecast observations.

2.5.2 Mean Absolute Error (MAE)

The mean absolute error (MAE) is a statistical tool for measuring the average size of the errors in a collection of predictions, without taking their directions into account. It is measured as the average absolute difference between the predicted values and the actual values and is used to assess the effectiveness of a model. It is given as:

where" is the actual value of the series at time is the forecasted value of the series and is the number of observations. The lower the value of RMSE and MAE, the better the model is able to forecast future values.

3.0       RESULTS AND DISC0USSION

3.1 Summary Statistics and Normality Measures

This study seeks to provide a short-term prediction of non-insulin-dependent diabetes (Type-2 diabetes mellitus) among farmers in Benue State using the Autoregressive Moving Average (ARMA) time series model. Before model estimation, a preliminary analysis of the dataset was conducted to summarize its key characteristics and assess the normality of the distribution. Table 1 below presents the descriptive statistics and normality test results for the observed monthly diabetes cases.

Table 1: Summary Statistics and Normality Measures

Variable

Statistic

Mean

5571.321

Maximum

9661.00

Minimum

3624.000

Standard Deviation

1769.088

Skewness

0.010212

Kurtosis

1.767498

Jarque-Bera Statistic

15.57465

p-value

0.000415

Number of Observations

246

From the result of summary statistics and normality measures reported in Table 1 above, the mean value of approximately 5571 infection cases indicates the average number of recorded non-insulin-dependent diabetes cases among farmers during the study period, while the maximum and minimum values (9661 and 3624, respectively) show the range of variation in the data. The standard deviation (1769) suggests a relatively high level of fluctuation around the mean, implying moderate variability in the monthly incidence of diabetes cases.

The skewness value (0.010212), being close to zero, indicates that the distribution of the series is approximately symmetric. However, the kurtosis value (1.767498) is less than 3, signifying a platykurtic distribution, that is, the data are relatively flatter than a normal distribution with lighter tails.

The Jarque–Bera statistic (15.57465) with an associated p-value of 0.000415 is statistically significant at the 1% level, leading to the rejection of the null hypothesis of normality. This implies that the series does not follow a perfectly normal distribution, which is a common characteristic of real-world time series data.

Overall, the results suggest that while the data are fairly symmetric, they deviate slightly from normality, a factor to be considered when fitting and diagnosing the ARMA model for accurate short-term forecasting.

4.2 Graphical Examination of Diabetes Miletus Series

Examining the morbidity cases of diabetes mellitus is essential for identifying trends and patterns over time, which can provide insights into the progression and fluctuations of the disease within a population. By analyzing these visual representations, healthcare providers and policymakers can better understand peak periods, seasonal variations, and the impact of interventions. This information is crucial for planning targeted healthcare responses, optimizing resource allocation, and developing strategies to reduce disease incidence and manage complications, ultimately improving health outcomes for affected populations. The time plots of the level and log transform series of diabetes mellitus are plotted in Figures 1 and 2 respectively as shown below.

The time plots of the level series and log transformed series reported in Figures 1 and 2 below indicate that both series are covariance or weakly stationary which implies the absence of unit root in the series in level. This is indicated by the smooth trend of both series.

Figure 1: Time Series Plot of Diabetes Miletus in Benue State from 2005 to 2025

Figure 2: Time Series Plot of Natural Log of Diabetes Miletus in Benue State from 2005

            to 2025

4.3 Augmented Dickey-Fuller (ADF) Unit Root Test Result

To ensure the appropriateness of applying an Autoregressive Moving Average (ARMA) model for short-term prediction of non–insulin-dependent diabetes cases among farmers in Benue State, it is necessary to examine the time series properties of the data. A key requirement for ARMA modeling is that the underlying series must be stationary. Therefore, the Augmented Dickey–Fuller (ADF) unit root test was conducted to determine whether the series is stationary. Table 2 below presents the results of the ADF test under two specifications: with an intercept only, and with both intercept and trend.

The ADF statistics reported in Table 2 below for both model specifications (intercept only and intercept with trend) are -15.3344 and -15.4304, respectively. These values are far more negative than their corresponding 5% critical values (-2.8731 and -3.4283). In addition, the associated p-values are 0.0000, indicating strong statistical significance. Because the ADF test statistics are well below the critical values and the p-values are less than 0.05, the null hypothesis of a unit root is rejected under both model specifications. This confirms that the series stationary in its level form. Stationarity implies that the mean and variance of the diabetes case series remain stable over time, making it suitable for direct ARMA modeling without differencing. The strong evidence of stationarity enhances the reliability of subsequent short-term forecasts produced by the ARMA model.

Table 2: Augmented Dickey-Fuller (ADF) Unit Root Test Result

Variable

Option

ADF Test Statistic

p-value

5% Critical Value

Intercept only

-15.3344

0.0000

-2.8731

Intercept & Trend

-15.4304

0.0000

-3.4283

4.4 Autocorrelations and Partial Autocorrelations Functions of the Series

After confirming that the series of non–insulin-dependent diabetes cases among farmers in Benue State is stationary, the next step in the ARMA modeling process involves examining the autocorrelation structure of the series. The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are used to identify the dependence pattern between current and past observations, which guides the selection of appropriate autoregressive (AR) and moving-average (MA) orders.

Furthermore, the Ljung-Box Q-statistics were computed to test for the joint significance of autocorrelations up to various lags. This test determines whether the residuals are independently distributed — a key requirement for model adequacy. Table 3 below presents the ACF, PACF, and Ljung-Box Q-statistics results for the series while Figure 3 belowpresented the ACF and PACF plots of the series.

The results of ACF and PACF reported in Table 3 below and Figure 3 show that the autocorrelation (ACF) and partial autocorrelation (PACF) coefficients for all lags are small in magnitude, fluctuating around zero. This indicates the absence of significant serial correlation in the data. None of the autocorrelations exceed the approximate 95% confidence bounds (±0.1 for a large sample size of 246), suggesting that the time series behaves like a white-noise process.

The Ljung-Box Q-statistics and their corresponding p-values across all lags (p > 0.05) further confirm that there is no significant autocorrelation remaining in the residuals. This means that the null hypothesis of no autocorrelation cannot be rejected at any lag, implying that the series is adequately described by a stationary stochastic process (Ljung & Box, 1979).

Table 3: Autocorrelations and Ljung-Box Q-Statistics Test Results

Lag

ACF

PACF

Q-Statistics

p-value

1

0.014

0.014

0.0458

0.831

2

-0.019

-0.019

0.1338

0.935

3

0.004

0.005

0.1380

0.987

4

-0.049

-0.050

0.7497

0.945

5

0.022

0.024

0.8747

0.972

6

0.037

0.034

1.2165

0.976

7

0.022

0.023

1.3420

0.987

8

0.017

0.015

1.4126

0.994

9

-0.007

-0.005

1.4260

0.998

10

-0.110

-0.107

4.5659

0.918

11

-0.025

-0.022

4.7227

0.944

12

0.078

0.075

6.2944

0.901

13

-0.008

-0.012

6.3115

0.934

14

-0.017

-0.027

6.3907

0.956

15

0.052

0.055

7.0970

0.955

16

-0.035

-0.022

7.4226

0.964

17

-0.012

-0.008

7.4599

0.977

18

-0.088

-0.093

9.5213

0.946

19

-0.054

-0.050

10.302

0.945

20

-0.092

-0.114

12.567

0.895

21

-0.026

-0.032

12.750

0.917

22

-0.115

-0.115

16.369

0.797

23

0.007

0.008

16.381

0.838

24

-0.053

-0.074

17.165

0.842

25

-0.056

-0.036

18.032

0.841

26

-0.047

-0.056

18.643

0.851

27

0.055

0.057

19.482

0.852

28

-0.011

-0.032

19.514

0.882

29

0.060

0.057

20.511

0.876

30

0.056

0.042

21.381

0.876

31

0.040

0.061

21.828

0.888

32

-0.001

-0.015

21.828

0.912

33

-0.027

-0.007

22.036

0.927

34

-0.109

-0.121

25.432

0.855

35

-0.056

-0.074

26.342

0.854

36

0.066

0.025

27.604

0.841

Figure 3: Plots of ACF and PACF of Log Transformed Series

Collectively, these findings suggest that the series is not driven by persistent temporal dependence, and any ARMA model fitted to the data should yield uncorrelated and well-behaved residuals. Therefore, the dataset is suitable for ARMA model identification and estimation, and the absence of significant autocorrelation validates the appropriateness of proceeding with short-term forecasting using the ARMA framework.

4.5 Model Order Selection

Following the establishment of stationarity and the absence of significant autocorrelation in the diabetes time series, various ARMA model orders were estimated to determine the most parsimonious and best-fitting specification for short-term prediction. Model selection was based on several statistical criteria, including the Log Likelihood (LogL), Akaike Information Criterion (AIC), Schwarz Information Criterion (SIC), and Hannan–Quinn Criterion (HQC). Generally, the preferred model is the one with the highest Log Likelihood and the lowest values of AIC, SIC, and HQC. Table 4 below presents the results of the model order selection process.

Among the twenty-four ARMA model specifications estimated, the ARMA(3,3) model exhibits the highest Log Likelihood value (-24.0103) and the lowest AIC (0.2552), SIC (0.3159), and HQC (0.2958) values. These results indicate that the ARMA(3,3) model provides the best balance between goodness-of-fit and parsimony.

Table 4:Model Order Selection using Log Likelihood and Information Criteria

S/n

Model

LogL

AIC

SIC

HQC

1.

ARMA(0,1)

-34.4597

0.2964

0.3349

0.3079

2.

ARMA(1,0)

-34.8194

0.3006

0.3391

0.3121

3.

ARMA(1,1)

-32.9444

0.2934

0.3363

0.3107

4.

ARMA(0,2)

-34.4107

0.3042

0.3469

0.3214

5.

ARMA(2,0)

-35.1256

0.3125

0.3555

0.3298

6.

ARMA(1,2)

-32.9256

0.3014

0.3586

0.3245

7.

ARMA(2,1)

-33.2988

0.3057

0.3631

0.3288

8.

ARMA(2,2)

-30.3771

0.2899

0.3616

0.3188

9.

ARMA(0,3)

-34.4060

0.3122

0.3692

0.3352

10.

ARMA(3,0)

-35.4688

0.3248

0.3823

0.3480

11.

ARMA(1,3)

-28.0912

0.2701

0.3616

0.3089

12.

ARMA(3,1)

-32.9028

0.3119

0.3838

0.3409

13.

ARMA(2,3)

-30.3708

0.2981

0.3841

0.3328

14.

ARMA(3,2)

-30.5304

0.3007

0.3859

0.3354

15.

ARMA(3,3)

-24.0103

0.2552

0.3159

0.2958

16.

ARMA(0,4)

-34.1157

0.3180

0.3893

0.3467

17.

ARMA(4,0)

-35.3492

0.3335

0.4056

0.3625

18.

ARMA(1,4)

-34.4466

0.3302

0.4159

0.3647

19.

ARMA(4,1)

-35.3432

0.3417

0.4282

0.3765

20.

ARMA(2,4)

-32.0099

0.3198

0.4201

0.3602

21.

ARMA(4,2)

-26.7027

0.2785

0.3795

0.3192

22.

ARMA(3,4)

-25.4065

0.2799

0.3899

0.3213

23.

ARMA(4,3)

-33.4797

0.3428

0.4581

0.3893

24.

ARMA(4,4)

-31.4253

0.2962

0.4060

0.3285

Therefore, based on the information criteria, the ARMA(3,3) model is selected as the optimal model for forecasting short-term variations in non–insulin-dependent diabetes cases among farmers in Benue State. This suggests that both autoregressive and moving average components up to the third order significantly contribute to capturing the dynamic structure of the series.

4.6 Parameter Estimates of ARMA(3,3) Model

After selecting the ARMA(3,3) model as the optimal specification based on the information criteria, the model parameters were estimated to evaluate the dynamic relationship between past observations and random disturbances in the series of non–insulin-dependent diabetes cases among farmers in Benue State. Table 5 below presents the estimated coefficients of the ARMA(3,3) model, along with their corresponding standard errors, t-statistics, and p-values. Goodness-of-fit measures such as the R-squared, Adjusted R-squared, F-statistic, and Durbin–Watson statistic are also reported to assess the adequacy of the fitted model.

Table 5: Parameter Estimates of ARMA(3,3) Model

Variable

Coefficient

Std. Error

t-Statistic

p-value

C

8.768664

0.017218

509.2761

0.0000

AR(1)

0.366096

0.024641

14.85713

0.0000

AR(2)

0.311203

0.029382

10.59171

0.0000

AR(3)

-0.912359

0.024212

-37.68166

0.0000

MA(1)

-0.372828

0.009593

-38.86277

0.0000

MA(2)

-0.386923

0.009312

-41.55086

0.0000

MA(3)

0.982389

0.007644

128.5160

0.0000

R-squared

0.890511

AIC

0.255229

Adjusted R²

0.867389

SIC

0.315852

F-statistic

6.914400

HQC

0.295759

Prob(F-stat.)

0.000951

Durbin-Watson stat.

2.011502

The model estimation results reported in Table 5 show that all autoregressive (AR) and moving average (MA) coefficients are statistically significant at the 1% level, as indicated by their very low p-values (p < 0.01). This implies that past values and past error terms up to the third lag significantly influence the current level of non–insulin-dependent diabetes cases among farmers.

Specifically, the positive coefficients of AR(1) and AR(2) suggest a direct persistence effect, meaning that increases in diabetes cases in the immediate past periods tend to raise current cases. Conversely, the negative AR(3) coefficient indicates a corrective mechanism, implying that after about three periods, the series tends to revert toward its mean. The MA terms also show alternating positive and negative signs, suggesting that short-term shocks have both dampening and amplifying effects over time before dissipating.

The high R-squared (0.8905) and adjusted R-squared (0.8674) values indicate that approximately 89% of the variation in diabetes cases is explained by the model, signifying a very good fit. The F-statistic (6.9144) with a significant probability value (0.000951) confirms the overall significance of the model.The Durbin–Watson statistic (2.0115) is close to 2, suggesting the absence of serial correlation in the residuals, while the information criteria (AIC = 0.2552, SIC = 0.3159, HQC = 0.2958) reaffirm that the ARMA(3,3) model remains the most parsimonious and efficient choice.

Overall, the ARMA(3,3) model adequately captures the temporal dynamics and short-term fluctuations in non–insulin-dependent diabetes cases among farmers in Benue State, making it suitable for reliable short-term forecasting.

4.7 Model Diagnostic Checks

Following the estimation of the ARMA(3,3) model for predicting non–insulin-dependent diabetes cases among farmers in Benue State, diagnostic checks such as multicolinearity test and Ljung-Box Q-statistic tests were conducted to verify the adequacy of the fitted model. This assessment ensures that the residuals behave like white noise, uncorrelated, homoscedastic, and pattern-free over time. The test are presented in the following subsections.

4.7.1 Multicolinearity test result

Multicollinearity diagnostics were performed to make sure the variables in ARMA(3,3) model weren't overlapping too much. Using the Variance Inflation Factor (VIF) for each autoregressive (AR) and moving average (MA) term, the test assessed how multicollinearity might affect the stability and reliability of parameter estimates. Generally, VIF values above 10 indicate severe multicollinearity, values between 5 and 10 suggest moderate correlation, and values below 5 imply no serious concern. The results presented in Table 6 show both uncentered and centered VIF statistics for the ARMA(3,3) model parameters.

The results of multicolinearity test reported in Table 6 below reveal that all centered VIF values are considerably low, ranging between 1.11 and 2.55, which are far below the critical threshold of 10. This indicates that there is no serious multicollinearity among the explanatory variables (AR and MA terms) in the estimated ARMA(3,3) model.

Therefore, the estimated parameters are statistically reliable, and the standard errors are not inflated by multicollinearity. This implies that the ARMA (3,3) model is well-conditioned, and the coefficients can be interpreted with confidence.

Table 6: Test for Multicolinearity (Variance Inflation Factors)

Coefficient

Uncentered

Centered

Variable

Variance

VIF

VIF

C

0.000296

1.018813

Na

AR(1)

0.000607

1.779456

1.779044

AR(2)

0.000863

2.552345

2.552344

AR(3)

0.000586

1.768375

1.768101

MA(1)

9.20E-05

1.257613

1.255458

MA(2)

8.67E-05

1.213557

1.203709

MA(3)

5.84E-05

1.121942

1.111356

4.7.2 Ljung-Box Q-statistic test result for serial correlation

The Autocorrelation Function (ACF), Partial Autocorrelation Function (PACF), and Ljung–Box Q-statistics were used to test for serial correlation. High p-values (greater than 0.05) for the Q-statistics indicate no significant autocorrelation, suggesting that the residuals are random and the model is well specified. Table 5 presents these diagnostic test results for the ARMA(3,3) model residuals.

The results of Q-statistic reported in Table 5 and the ACF as well as PACF plots reported in Figure 4 show that all residual autocorrelations (ACF and PACF) are very small and fluctuate closely around zero across all 36 lags. None of the autocorrelation coefficients appear significant, suggesting that the residuals from the ARMA(3,3) model are approximately white noise.

Furthermore, the Ljung–Box Q-statistics have p-values consistently greater than 0.05, indicating that the null hypothesis of no autocorrelation cannot be rejected at any lag. This confirms that there is no statistically significant serial correlation remaining in the residuals. In addition, the Durbin–Watson statistic from the model estimation (2.0115) supports this conclusion by indicating near-zero autocorrelation in the residuals.

Overall, these diagnostic results confirm that the ARMA(3,3) model is well specified, the residuals are independently and randomly distributed, and the model provides a statistically adequate fit to the data. Therefore, the model is suitable for reliable short-term forecasting of non–insulin-dependent diabetes cases among farmers in Benue State

Table 7: Autocorrelations and Ljung-Box Q-Statistic Test Results of Residuals

Lag

ACF

PACF

Q-Statistics

p-value

1

-0.024

-0.024

0.1415

0.707

2

-0.012

-0.012

0.1760

0.916

3

-0.069

-0.070

1.3558

0.716

4

0.007

0.003

1.3669

0.850

5

-0.126

-0.128

5.3247

0.378

6

-0.036

-0.048

5.6541

0.463

7

-0.017

-0.024

5.7294

0.572

8

0.142

0.124

10.812

0.213

9

-0.042

-0.042

11.254

0.259

10

0.046

0.032

11.802

0.299

11

-0.021

-0.015

11.918

0.370

12

0.052

0.044

12.628

0.397

13

-0.025

0.012

12.794

0.464

14

-0.009

-0.008

12.815

0.541

15

0.062

0.080

13.804

0.540

16

0.068

0.053

15.019

0.523

17

0.112

0.147

18.316

0.369

18

0.109

0.127

21.475

0.256

19

-0.008

0.027

21.493

0.310

20

-0.087

-0.066

23.529

0.264

21

-0.066

-0.032

24.707

0.260

22

-0.020

0.010

24.810

0.306

23

-0.062

-0.057

25.855

0.308

24

-0.048

-0.064

26.480

0.329

25

0.021

-0.044

26.599

0.376

26

0.020

-0.037

26.704

0.425

27

-0.033

-0.069

27.003

0.464

28

0.065

0.050

28.156

0.456

29

0.052

0.030

28.898

0.470

30

0.062

0.046

29.969

0.467

31

0.014

0.040

30.023

0.516

32

0.010

0.016

30.053

0.565

33

0.042

0.050

30.555

0.589

34

0.003

0.004

30.558

0.637

35

-0.039

-0.013

30.994

0.662

36

-0.008

-0.001

31.014

0.705

Figure 4:Plot of Correlogram of Residuals of Estimated ARMA(3,3) Model

4.8 Forecast and Forecast Evaluation

To evaluate the predictive performance of the ARMA(3,3) model in forecasting non–insulin-dependent diabetes cases among farmers in Benue State, forecast accuracy measures were computed. The Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) were used to assess both in-sample and out-of-sample forecast accuracy. Lower values of these statistics indicate better model performance and predictive reliability. The result is presented in Table 8.

The results of forecast comparison reported in Table 8below show that the out-of-sample forecast achieved slightly lower RMSE (0.2671), MAE (0.2310), and MAPE (2.6490) values compared to the in-sample forecast (RMSE = 0.2715, MAE = 0.2446, MAPE = 2.6781). This suggests that the ARMA(3,3) model demonstrates strong predictive capability, with minimal forecast error and good generalization performance. The model selected in forecast mode, as denoted by the accuracy measures, provides reliable short-term out-of-sample predictions of non–insulin-dependent diabetes cases.

Table 8: Forecast Comparison using Accuracy Measures

RMSE

MAE

MAPE

In-Sample

0.271510

0.244615

2.678116

Out-of-Sample

0.267100

0.231048

2.649005

Note: denotes forecast mode selected by accuracy measures.

4.8.1 Forecast of Diabetes Miletus in Benue State from July, 2025 to June, 2027

To evaluate the short-term predictive performance of the ARMA(3,3) model, forecasts of non–insulin-dependent diabetes (Type-2 Diabetes Mellitus) cases among farmers in Benue State were generated for the period July 2025 to June 2027. The forecasts were computed in natural logarithmic form and then converted to actual population estimates. For each forecast, the standard error, lower confidence limit (LCL), and upper confidence limit (UCL) were calculated at a 95% confidence level, using . These values provide a range within which the true number of diabetes cases is expected to fall with high probability, thereby indicating the reliability and uncertainty of the forecasts. The forecast result is reported in Table 9 below while the forecast graph is presented as Figure 5 below too.

Table 9: "Forecast of Diabetes Miletus Infection Cases in Benue State from July 2025-

            June, 2027"

Year: Month

Forecast (natural log form)

Actual Forecast (No. of Persons)

Forecast

Std. error

LCL

Forecast

UCL

2025:06

6.9967

---

---

8896

---

2025:07

8.77405

0.271243

3799

6464

11000

2025:08

8.72655

0.271669

3619

6165

10499

2025:09

8.78204

0.271670

3826

6516

11098

2025:10

8.77132

0.272065

3782

6447

10988

2025:11

8.80141

0.272672

3893

6644

11337

2025:12

8.74519

0.272672

3680

6281

10717

2026:01

8.76088

0.272790

3738

6380

10889

2026:02

8.74585

0.273455

3677

6285

10741

2026:03

8.79725

0.273466

3871

6616

11308

2026:04

8.77366

0.273476

3781

6462

11044

2026:05

8.77825

0.274040

3794

6492

11107

2026:06

8.73648

0.274110

3638

6226

10654

2026:07

8.76803

0.274114

3755

6426

10996

2026:08

8.76810

0.274473

3752

6426

11005

2026:09

8.79729

0.274652

3862

6616

11335

2026:10

8.76026

0.274669

3722

6376

10923

2026:11

8.76113

0.274824

3724

6381

10936

2026:12

8.74504

0.275111

3662

6279

10767

2027:01

8.78341

0.275121

3805

6525

11188

2027:02

8.77734

0.275152

3782

6486

11121

2027:03

8.78223

0.275481

3798

6517

11183

2027:04

8.74716

0.275481

3667

6293

10798

2027:05

8.76058

0.275481

3717

6378

10944

2027:06

8.76313

0.275759

3724

6394

10978

Total

210.40663

154075

Average

8.766942917

6419.7917

Note: For 95% confidence intervals, . LCL and UCL denote lower and upper confidence limits respectively.

Figure 5: Forecast Graph of Diabetes Miletus in Benue State from July, 2025-June, 2027

The forecast results reported in Table 9 and Figure 5 above reveals that the predicted number of non–insulin-dependent diabetes cases among farmers in Benue State is expected to fluctuate moderately over the two-year forecast horizon (July 2025–June 2027). The monthly forecasts range between approximately 3,600 and 11,300 cases, with an overall average of about 6,420 cases per month and a total forecast of 154,075 cases during the study period. The relatively narrow confidence intervals across months suggest a high level of precision in the model’s predictions.

Overall, the ARMA(3,3) model demonstrates strong forecasting capability, indicating that diabetes prevalence among farmers in Benue State is likely to remain fairly stable with mild month-to-month variations over the forecast period.

4.9 Implications of the Study to Farmers and Postharvest Losses in Benue State

The implications of this study for farmers and postharvest losses in Benue State are significant from both public health and socio-economic perspectives. The findings, which forecast the prevalence of non–insulin-dependent diabetes (Type-2 Diabetes Mellitus) among farmers, suggest that a substantial portion of the agricultural workforce may experience declining health and productivity over time. Poor health conditions such as diabetes can reduce farmers’ physical capacity to engage in strenuous agricultural activities, particularly during critical periods like harvesting and processing. "This in turn increases the likelihood of postharvest losses, as crops may remain un-harvested or inadequately stored due to reduced labour efficiency and absenteeism resulting from illness".

Moreover, "higher diabetes prevalence among farmers implies increased medical expenditures and a diversion of household income away from agricultural investment", further compounding the problem of low productivity and waste. The study underscores the urgent need for integrated health and agricultural policies—including improved rural healthcare services, regular medical screening, health education on diet and lifestyle, and the promotion of labour-saving technologies—to mitigate the dual burden of disease and postharvest losses. Ultimately, addressing the health challenges of farmers is crucial for achieving food security, sustaining agricultural livelihoods, and enhancing overall economic resilience in Benue State.

4.0       Conclusion

The study demonstrates that the ARMA(3,3) model effectively forecasts the incidence of non-insulin-dependent diabetes among farmers in Benue State, Nigeria, The analysis revealed that the ARMA(3,3) model provided the best fit based on information criteria and diagnostic tests, with residuals behaving like white noise, indicating a well-specified and reliable model. The forecasts from July 2025 to June 2027 suggest a steady and relatively high incidence of diabetes cases among farmers, implying that the disease poses an ongoing public health concern within the agricultural population. This condition could adversely affect farmers’ productivity, increase medical costs, and indirectly contribute to higher postharvest losses due to reduced labour availability and inefficiencies in farm management. These findings highlight the interconnectedness between health and agricultural output, emphasizing that the burden of chronic diseases like diabetes extends beyond healthcare into the realm of food security and economic stability. Therefore, proactive health interventions and policy integration between the health and agricultural sectors are vital. Ensuring farmers’ wellness through preventive care, early detection, and education can significantly reduce the impact of diabetes and its broader economic consequences. The study provides empirical evidence to guide policymakers, healthcare providers, and agricultural development agencies in formulating context-specific strategies to improve both health outcomes and agricultural sustainability in Benue State.

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John Agada¹, David Adugh Kuhe² and Ojochegbe Noah Anthony^3*

¹Department of Mathematics and Computer Science, Rev, Fr. Moses Orshio Adasu University Makurdi, Benue State, Nigeria

²Department of Statistics, Joseph Sarwuan Tarka University, Makurdi, Benue State, Nigeria

³Department of Mathematics and Computer Science, Rev, Fr. Moses Orshio Adasu University Makurdi, Benue State, Nigeria

Corresponding Author: Email: davidkuhe@gmail.com; Tel: 2348064842229

ABSTRACT

This study employs an Autoregressive Moving Average (ARMA) time series model to forecast the short-term incidence of non-insulin-dependent diabetes mellitus (Type 2 Diabetes) among farmers in Benue State, Nigeria. The data was collected from the Benue State Epidemiological Unit, Makurdi, and covered a 20-year period from January 2005 to June 2025. The study employed descriptive statistics and normality measures, Augmented Dickey-Fuller (ADF) unit root test and ARMA (p,q) model as the principal analytical techniques and procedures used to examine the data. The descriptive statistics indicated moderate variability in diabetes cases over the years, while the Augmented Dickey-Fuller (ADF) test confirmed the stationarity of the series in level. Model choice based on Akaike Information Criterion (AIC), Schwarz Information Criterion (SIC), and Hannan–Quinn Criterion (HQC) identified the ARMA(3,3) model as the best fit for forecasting diabetic cases in the study area. The model’s high coefficient of determination (R² = 0.8905) and statistically significant parameters (p < 0.05) demonstrated its robustness and predictive accuracy. Diagnostic checks using autocorrelation, partial autocorrelation, and the Ljung–Box Q-statistics showed that the residuals behaved like white noise, indicating a well-specified model. Forecast evaluations using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) confirmed that the model accurately good for predicting out-of-sample values. The forecast for July 2025 to June 2027 revealed a potential average of approximately 6,420 diabetes cases per month among farmers, with expected fluctuations over time. The study underscored the growing public health concern of diabetes among the farming population in Benue State and its implications for agricultural productivity and postharvest losses. The study concluded that predictive modeling can serve as a vital tool for health planners to design early intervention strategies, integrate health management with agricultural development, and enhance the overall well-being of rural farmers.

Keywords: Diabetes, ARIMA, Time Series Forecasting, Non-Insulin Dependent Diabetes, Farmers, Benue State, Public Health, Postharvest Losses

1.0       INTRODUCTION

Diabetes mellitus, often simply referred to as diabetes, is a group of metabolic disorders characterized by high blood sugar levels over a prolonged period. The two main types of diabetes are type-1 diabetes, which results from the body’s inability to produce insulin, and Type-2 diabetes develops when the body either becomes resistant to insulin or produces insufficient insulin to control blood sugar levels effectively. Diabetes mellitus is a multifaceted metabolic condition marked by high concentrations of glucose (sugar) in the bloodstream Glucose is a crucial source of energy for cells, and insulin, a hormone produced by the pancreas, plays a central role in regulating its uptake into cells. In diabetes mellitus, this regulation is disrupted, leading to persistent hyperglycemia (high blood sugar) (American Diabetes Association, 2022).

Diabetes mellitus is a significant public health concern worldwide, with its prevalence increasing steadily over the past few decades. According to the International Diabetes Federation (IDF, 2019), an estimated 537 million adults aged 20-79 years were living with diabetes globally in 2021 and this number is projected to rise to 783 million by 2045. The prevalence of diabetes varies by region, with higher rates observed in low- and middle-income countries, particularly in urban areas undergoing rapid socioeconomic development and lifestyle changes. (ADA, 2022).

In Nigeria, the prevalence is estimated at 7% and 11.35% in South-south zone. The Diabetes Association of Nigeria (DAN) reviewed that, mortality rate of diabetes from insufficient management far outweighs that of HIV/AIDs, Malaria and Cancer (Olamoyegun et al., 2024)

Diabetes mellitus is significantly Impacting farmers in Benue State with prevalence rate among yam farming population estimated at 24.9% and mortality rate of 8.61% and as led to reduced labor productivity, economic impact and health complications (Teran, A.D.. 2017)

Diabetes is associated with numerous complications that can affect nearly every organ system in the body. These complications includes Microvascular: Retinopathy (vision loss) neuropathy (nerve damage), nephropathy (kidney damage), and Microvascular: cardiovascular disease (such as heart attack and stroke), others are foot ulcers and amputations. The burden of diabetes-related complications is substantial, leading to increased medical costs, reduced quality of life, and higher risk of premature mortality (ADA, 2022).

Type-2 diabetes, also known as non-insulin dependent diabetes, is a long-term condition that affects how the body processes sugar (glucose), which is an important source of energy. In this condition, the body either becomes resistant to insulin, a hormone that helps move sugar into cells, or doesn’t produce enough insulin to keep blood sugar levels normal (Sun et al., 2021). Unlike type-1 diabetes, where the immune system attacks and destroys insulin-producing cells in the pancreas, type-2 diabetes usually develops slowly over time. While it was once mostly seen in adults, more children and teenagers are now being diagnosed, largely due to increasing obesity and less active lifestyles (Sun et al., 2021).

A major characteristic of type-2 diabetes is insulin resistance, which means the body's cells don't respond to insulin as they should. When this happens, the pancreas tries to make more insulin to help move sugar into the cells. However, over time, the pancreas may struggle to keep up with this increased demand. As a result, sugar starts to accumulate in the blood, causing high blood sugar levels (Cloete, 2022).

Several determinants contributes to the risk of developing type-2 diabetes, including obesity, particularly excess fat around the abdomen (central obesity), A sedentary lifestyle, unhealthy eating habits—like eating too many sugary and processed foods—having a family history of diabetes, getting older (especially after 45), and belonging to certain ethnic groups are all factors that can increase the risk of developing diabetes (ADA, 2022).

In Addition to insulin resistance, type-2 diabetes can also involve problems with the pancreas, the organ that makes insulin. Sometimes, the pancreas doesn't produce enough insulin to keep blood sugar levels in check, making high blood sugar worse (Desai & Deshmukh, 2020).

Symptoms of type-2 diabetes often develop slowly and can include increased thirst, frequent urination, fatigue, blurred vision, slow wound healing, and repeated infections. In the early stages, some people may not notice any symptoms at all, which is why regular screenings are essential (IDF, 2019).

Treatment for type-2 diabetes aims to maintain blood sugar levels within a target range to prevent serious health problems and complications. This typically involves lifestyle modifications such as regular exercise, healthy eating habits (including portion control and selecting nutrient-rich foods), weight management, and monitoring blood sugar levels. (Desai & Deshmukh, 2020).

The management and treatment of type-2 diabetes can impose financial burdens on individuals, families, and healthcare systems. In regions where healthcare costs are primarily borne by the individual or are not adequately covered by insurance, the expenses associated with diabetes care can divert resources away from agricultural investments and productivity-enhancing measures. This can directly impact agricultural communities with reduced investment into agricultural produces, reduced income and crop loss thereby affecting their livelihood (Huang et al., 2016).

Diabetes Mellitus is diagnosed when certain blood sugar levels are met or exceeded. Specifically, a person may be diagnosed if their A1C is 6.5% or higher, which reflects average blood glucose over the past few months. Alternatively, if fasting blood sugar is 126 mg/dL or higher, or if a 2-hour blood sugar reading during an oral glucose tolerance test reaches 200 mg/dL or more, a diagnosis may be made. Additionally, if an individual has a random blood sugar of 200 mg/dL or higher along with symptoms like excessive thirst, frequent urination, or unexplained weight loss, they may also be diagnosed with diabetes (Jaeger et al., 2025).

Agricultural activities, like applying chemical fertilizers and pesticides, can have environmental consequences that can indirectly impact diabetes risk factors. For instance, exposure to chemicals such as glyphosate or organophosphates used in farming has been associated with a higher likelihood of developing metabolic disorders. Additionally, environmental factors such as air pollution and climate change may exacerbate diabetes risk factors and health outcomes, potentially affecting agricultural productivity and crop yields (whiting et al., 2011). Overall, while the direct impact of type-2 diabetes on agricultural productivity and postharvest losses may be limited, the interplay between diabetes, dietary patterns, healthcare access, and environmental factors can have broader implications for agricultural communities and food systems. Addressing the complex relationship between health, agriculture, and the environment requires a holistic approach that considers socioeconomic factors, public health interventions, and sustainable agricultural practices (Whiting et al., 2011).

Overall, while the direct impact of type-2 diabetes on agricultural productivity and postharvest losses may be limited, the interplay between diabetes, dietary patterns, healthcare access, and environmental factors can have broader implications for agricultural communities and food systems. Addressing the complex relationship between health, agriculture, and the environment requires a holistic approach that considers socioeconomic factors, public health interventions, and sustainable agricultural practices (Huang et al., 2016).

This study therefore attempts to extend the existing literature and contribute to the existing body of knowledge by modeling and forecasting non insulin dependent diabetes among farmers in Benue State using autoregressive moving average (ARIMA) time series model with more recent data.

2.0       MATERIALS AND METHODS

2.1       Method of Data Collection

The data utilized in this research work are monthly secondary time series data on morbidity incidence of type-2 diabetes in Benue state for the period of January, 2005 June, 2025 making a total of 234 observations. The data was collected from Benue State Epidemiological unit, Makurdi. The data was transformed to natural logarithms using the following formula:

where is the confirmed type-2 diabetes series observation indexed by time , while is the natural logarithm. Hence forth will be regarded as a series.

2.2 Methods of Data Analysis

Find below the statistical tools employed in the analysis of data in this work.

3.2.1 Descriptive statistics and normality measures

The mean of any given set of data can be computed as follows:

The sample standard deviation of any given set of data over a given period of time is computed using the formula:

where is the sample mean, is the sample size.

Jarque-Bera test is a normality test of whether a given sample data have the skewness and kurtosis similar to that of a normal distribution. The test was proposed by Jarque and Bera (1980, 1987) and test the null hypothesis that the series is normally distributed. Given any data set, the test statistic JB is defined as:

where is the sample skewness denoted as:

and is the sample kurtosis given below:

whereT is the total number of observations. The JB normality test checks the following pair of hypothesis:

and (i.e., follows a normal distribution)

and (i.e., does not follows a normal distribution).

The test rejects the null hypothesis if the p-value of the JB test statistic is less than level of significance.

2.2.2 Augmented Dickey-Fuller (ADF) unit root test

The Augmented Dickey-Fuller (ADF) test helps to identify if a time series is stationary or has a unit root, indicating a persistent trend over time (Dickey and Fuller, 1979).

It accounts for higher-order correlations by assuming the series follows an AR(p) process and incorporates lagged differences of the series into the regression to enhance the test's precision.

.

where are optional exogenous regressors which may consist of constant, or a constant and trend, and are parameters to be estimated,β values arelagged difference terms and the are assumed to be white noise. The null and alternative hypotheses are written as:

                                                                                        (8)

and evaluated using the conventional ratio for

where is the estimate of and "the coefficient standard error is denoted as "

2.2.3 Portmanteau test

A Portmanteau test also called he Ljung-Box Q-statistic test is used to determine whether there is any remaining serial correlation or autocorrelation in the residuals of a time series. The test checks the following pairs of hypotheses:

(all lags correlations are zero)

(there is at least one lag with non-zero correlation). The test statistic is given by:

where

denotes the autocorrelation estimate of squared standardized residuals at lags. T is the sample size, Q is the sample autocorrelation at lag k. We reject if p-value is less than level of significance (Ljung and Box, 1979).

2.3 Time Series Models Specification

To specify an ARIMA model which is the model framework use in this study, we first specify autoregressive (AR) model, moving average (MA) model, autoregressive moving average (ARMA) model before specifying autoregressive integrated moving average (ARIMA) model. These models are specified as follows.

2.3.1 The autoregressive (AR) model

A stochastic time series process {} is an autoregressive process of order p, denoted AR() if it satisfied the difference equation

where is a white noise and are constants to be determined.

2.3.2 Moving average (MA) model

A time series {} which satisfies the difference equation

where are fixed constants with as white noise is called a moving average process of order q, denoted MA().

2.3.3 Autoregressive moving average (ARMA) model

A stochastic time series process {} which results from a linear combination of autoregressive and moving average processes is called an Autoregressive Moving Average (ARMA) process of order p, q, denoted ARMA () if it satisfies the following difference equation:

where are fixed constants associated with the AR terms and are fixed constants associated with the MA terms with being a white noise. The stationarity of an ARMA () process is guaranteed if the roots of the polynomial

lie outside the unit circle.

An ARMA () model is specified as:

2.3.4 Autoregressive integrated moving average (ARIMA) model

Autoregressive (AR), Moving Average (MA) or Autoregressive Moving Average (ARMA) model in which differences have been taken are collectively called Autoregressive Integrated Moving Average or ARIMA models. A time series {} is said to follow an integrated autoregressive moving average model if the ^th difference is a stationary ARMA process. If follows an ARMA(p, q) model, we say that {} is an ARIMA (p, d, q) process. For practical purposes, we can usually take or at most 2.

Consider then an ARIMA (p, 1, q) process, with , we have

In terms of the observed series,

)

2.4 Model Order Selection

We use the following information criteria for model order selection in conjunction with log likelihood function: Akaike information criterion (AIC) due to Akaike (1978), Schwarz information Criterion (SIC) due to (Schwarz, 1978) and Hannan-Quinn information Criterion (HQC) due to (Hannan, 1980). The formula for the information criteria are:



where is the number of free parameters to be estimated in the model, T is the number of observations and L is the likelihood function defined as:

Thus given a set of estimated ARMA models for a given set of data, the preferred model is the one with the minimum information criteria and maximum log likelihood.

2.5 Model Forecast Evaluation

We employed Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) accuracy measures to select an optimal model mode that is both parsimonious and accurately forecast the data based on minimum values of the accuracy measures.

2.5.1 Root Mean Square Error (RMSE)

The Root Mean Square Error is a statistical tool for measuring the accuracy of a forecast method. It is computed as:

Where is the forecast value of the series and is the actual series and is the number of forecast observations.

2.5.2 Mean Absolute Error (MAE)

The mean absolute error (MAE) is a statistical tool for measuring the average size of the errors in a collection of predictions, without taking their directions into account. It is measured as the average absolute difference between the predicted values and the actual values and is used to assess the effectiveness of a model. It is given as:

where" is the actual value of the series at time is the forecasted value of the series and is the number of observations. The lower the value of RMSE and MAE, the better the model is able to forecast future values.

3.0       RESULTS AND DISC0USSION

3.1 Summary Statistics and Normality Measures

This study seeks to provide a short-term prediction of non-insulin-dependent diabetes (Type-2 diabetes mellitus) among farmers in Benue State using the Autoregressive Moving Average (ARMA) time series model. Before model estimation, a preliminary analysis of the dataset was conducted to summarize its key characteristics and assess the normality of the distribution. Table 1 below presents the descriptive statistics and normality test results for the observed monthly diabetes cases.

Table 1: Summary Statistics and Normality Measures

Variable

Statistic

Mean

5571.321

Maximum

9661.00

Minimum

3624.000

Standard Deviation

1769.088

Skewness

0.010212

Kurtosis

1.767498

Jarque-Bera Statistic

15.57465

p-value

0.000415

Number of Observations

246

From the result of summary statistics and normality measures reported in Table 1 above, the mean value of approximately 5571 infection cases indicates the average number of recorded non-insulin-dependent diabetes cases among farmers during the study period, while the maximum and minimum values (9661 and 3624, respectively) show the range of variation in the data. The standard deviation (1769) suggests a relatively high level of fluctuation around the mean, implying moderate variability in the monthly incidence of diabetes cases.

The skewness value (0.010212), being close to zero, indicates that the distribution of the series is approximately symmetric. However, the kurtosis value (1.767498) is less than 3, signifying a platykurtic distribution, that is, the data are relatively flatter than a normal distribution with lighter tails.

The Jarque–Bera statistic (15.57465) with an associated p-value of 0.000415 is statistically significant at the 1% level, leading to the rejection of the null hypothesis of normality. This implies that the series does not follow a perfectly normal distribution, which is a common characteristic of real-world time series data.

Overall, the results suggest that while the data are fairly symmetric, they deviate slightly from normality, a factor to be considered when fitting and diagnosing the ARMA model for accurate short-term forecasting.

4.2 Graphical Examination of Diabetes Miletus Series

Examining the morbidity cases of diabetes mellitus is essential for identifying trends and patterns over time, which can provide insights into the progression and fluctuations of the disease within a population. By analyzing these visual representations, healthcare providers and policymakers can better understand peak periods, seasonal variations, and the impact of interventions. This information is crucial for planning targeted healthcare responses, optimizing resource allocation, and developing strategies to reduce disease incidence and manage complications, ultimately improving health outcomes for affected populations. The time plots of the level and log transform series of diabetes mellitus are plotted in Figures 1 and 2 respectively as shown below.

The time plots of the level series and log transformed series reported in Figures 1 and 2 below indicate that both series are covariance or weakly stationary which implies the absence of unit root in the series in level. This is indicated by the smooth trend of both series.

Figure 1: Time Series Plot of Diabetes Miletus in Benue State from 2005 to 2025

Figure 2: Time Series Plot of Natural Log of Diabetes Miletus in Benue State from 2005

            to 2025

4.3 Augmented Dickey-Fuller (ADF) Unit Root Test Result

To ensure the appropriateness of applying an Autoregressive Moving Average (ARMA) model for short-term prediction of non–insulin-dependent diabetes cases among farmers in Benue State, it is necessary to examine the time series properties of the data. A key requirement for ARMA modeling is that the underlying series must be stationary. Therefore, the Augmented Dickey–Fuller (ADF) unit root test was conducted to determine whether the series is stationary. Table 2 below presents the results of the ADF test under two specifications: with an intercept only, and with both intercept and trend.

The ADF statistics reported in Table 2 below for both model specifications (intercept only and intercept with trend) are -15.3344 and -15.4304, respectively. These values are far more negative than their corresponding 5% critical values (-2.8731 and -3.4283). In addition, the associated p-values are 0.0000, indicating strong statistical significance. Because the ADF test statistics are well below the critical values and the p-values are less than 0.05, the null hypothesis of a unit root is rejected under both model specifications. This confirms that the series stationary in its level form. Stationarity implies that the mean and variance of the diabetes case series remain stable over time, making it suitable for direct ARMA modeling without differencing. The strong evidence of stationarity enhances the reliability of subsequent short-term forecasts produced by the ARMA model.

Table 2: Augmented Dickey-Fuller (ADF) Unit Root Test Result

Variable

Option

ADF Test Statistic

p-value

5% Critical Value

Intercept only

-15.3344

0.0000

-2.8731

Intercept & Trend

-15.4304

0.0000

-3.4283

4.4 Autocorrelations and Partial Autocorrelations Functions of the Series

After confirming that the series of non–insulin-dependent diabetes cases among farmers in Benue State is stationary, the next step in the ARMA modeling process involves examining the autocorrelation structure of the series. The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are used to identify the dependence pattern between current and past observations, which guides the selection of appropriate autoregressive (AR) and moving-average (MA) orders.

Furthermore, the Ljung-Box Q-statistics were computed to test for the joint significance of autocorrelations up to various lags. This test determines whether the residuals are independently distributed — a key requirement for model adequacy. Table 3 below presents the ACF, PACF, and Ljung-Box Q-statistics results for the series while Figure 3 belowpresented the ACF and PACF plots of the series.

The results of ACF and PACF reported in Table 3 below and Figure 3 show that the autocorrelation (ACF) and partial autocorrelation (PACF) coefficients for all lags are small in magnitude, fluctuating around zero. This indicates the absence of significant serial correlation in the data. None of the autocorrelations exceed the approximate 95% confidence bounds (±0.1 for a large sample size of 246), suggesting that the time series behaves like a white-noise process.

The Ljung-Box Q-statistics and their corresponding p-values across all lags (p > 0.05) further confirm that there is no significant autocorrelation remaining in the residuals. This means that the null hypothesis of no autocorrelation cannot be rejected at any lag, implying that the series is adequately described by a stationary stochastic process (Ljung & Box, 1979).

Table 3: Autocorrelations and Ljung-Box Q-Statistics Test Results

Lag

ACF

PACF

Q-Statistics

p-value

1

0.014

0.014

0.0458

0.831

2

-0.019

-0.019

0.1338

0.935

3

0.004

0.005

0.1380

0.987

4

-0.049

-0.050

0.7497

0.945

5

0.022

0.024

0.8747

0.972

6

0.037

0.034

1.2165

0.976

7

0.022

0.023

1.3420

0.987

8

0.017

0.015

1.4126

0.994

9

-0.007

-0.005

1.4260

0.998

10

-0.110

-0.107

4.5659

0.918

11

-0.025

-0.022

4.7227

0.944

12

0.078

0.075

6.2944

0.901

13

-0.008

-0.012

6.3115

0.934

14

-0.017

-0.027

6.3907

0.956

15

0.052

0.055

7.0970

0.955

16

-0.035

-0.022

7.4226

0.964

17

-0.012

-0.008

7.4599

0.977

18

-0.088

-0.093

9.5213

0.946

19

-0.054

-0.050

10.302

0.945

20

-0.092

-0.114

12.567

0.895

21

-0.026

-0.032

12.750

0.917

22

-0.115

-0.115

16.369

0.797

23

0.007

0.008

16.381

0.838

24

-0.053

-0.074

17.165

0.842

25

-0.056

-0.036

18.032

0.841

26

-0.047

-0.056

18.643

0.851

27

0.055

0.057

19.482

0.852

28

-0.011

-0.032

19.514

0.882

29

0.060

0.057

20.511

0.876

30

0.056

0.042

21.381

0.876

31

0.040

0.061

21.828

0.888

32

-0.001

-0.015

21.828

0.912

33

-0.027

-0.007

22.036

0.927

34

-0.109

-0.121

25.432

0.855

35

-0.056

-0.074

26.342

0.854

36

0.066

0.025

27.604

0.841

Figure 3: Plots of ACF and PACF of Log Transformed Series

Collectively, these findings suggest that the series is not driven by persistent temporal dependence, and any ARMA model fitted to the data should yield uncorrelated and well-behaved residuals. Therefore, the dataset is suitable for ARMA model identification and estimation, and the absence of significant autocorrelation validates the appropriateness of proceeding with short-term forecasting using the ARMA framework.

4.5 Model Order Selection

Following the establishment of stationarity and the absence of significant autocorrelation in the diabetes time series, various ARMA model orders were estimated to determine the most parsimonious and best-fitting specification for short-term prediction. Model selection was based on several statistical criteria, including the Log Likelihood (LogL), Akaike Information Criterion (AIC), Schwarz Information Criterion (SIC), and Hannan–Quinn Criterion (HQC). Generally, the preferred model is the one with the highest Log Likelihood and the lowest values of AIC, SIC, and HQC. Table 4 below presents the results of the model order selection process.

Among the twenty-four ARMA model specifications estimated, the ARMA(3,3) model exhibits the highest Log Likelihood value (-24.0103) and the lowest AIC (0.2552), SIC (0.3159), and HQC (0.2958) values. These results indicate that the ARMA(3,3) model provides the best balance between goodness-of-fit and parsimony.

Table 4:Model Order Selection using Log Likelihood and Information Criteria

S/n

Model

LogL

AIC

SIC

HQC

1.

ARMA(0,1)

-34.4597

0.2964

0.3349

0.3079

2.

ARMA(1,0)

-34.8194

0.3006

0.3391

0.3121

3.

ARMA(1,1)

-32.9444

0.2934

0.3363

0.3107

4.

ARMA(0,2)

-34.4107

0.3042

0.3469

0.3214

5.

ARMA(2,0)

-35.1256

0.3125

0.3555

0.3298

6.

ARMA(1,2)

-32.9256

0.3014

0.3586

0.3245

7.

ARMA(2,1)

-33.2988

0.3057

0.3631

0.3288

8.

ARMA(2,2)

-30.3771

0.2899

0.3616

0.3188

9.

ARMA(0,3)

-34.4060

0.3122

0.3692

0.3352

10.

ARMA(3,0)

-35.4688

0.3248

0.3823

0.3480

11.

ARMA(1,3)

-28.0912

0.2701

0.3616

0.3089

12.

ARMA(3,1)

-32.9028

0.3119

0.3838

0.3409

13.

ARMA(2,3)

-30.3708

0.2981

0.3841

0.3328

14.

ARMA(3,2)

-30.5304

0.3007

0.3859

0.3354

15.

ARMA(3,3)

-24.0103

0.2552

0.3159

0.2958

16.

ARMA(0,4)

-34.1157

0.3180

0.3893

0.3467

17.

ARMA(4,0)

-35.3492

0.3335

0.4056

0.3625

18.

ARMA(1,4)

-34.4466

0.3302

0.4159

0.3647

19.

ARMA(4,1)

-35.3432

0.3417

0.4282

0.3765

20.

ARMA(2,4)

-32.0099

0.3198

0.4201

0.3602

21.

ARMA(4,2)

-26.7027

0.2785

0.3795

0.3192

22.

ARMA(3,4)

-25.4065

0.2799

0.3899

0.3213

23.

ARMA(4,3)

-33.4797

0.3428

0.4581

0.3893

24.

ARMA(4,4)

-31.4253

0.2962

0.4060

0.3285

Therefore, based on the information criteria, the ARMA(3,3) model is selected as the optimal model for forecasting short-term variations in non–insulin-dependent diabetes cases among farmers in Benue State. This suggests that both autoregressive and moving average components up to the third order significantly contribute to capturing the dynamic structure of the series.

4.6 Parameter Estimates of ARMA(3,3) Model

After selecting the ARMA(3,3) model as the optimal specification based on the information criteria, the model parameters were estimated to evaluate the dynamic relationship between past observations and random disturbances in the series of non–insulin-dependent diabetes cases among farmers in Benue State. Table 5 below presents the estimated coefficients of the ARMA(3,3) model, along with their corresponding standard errors, t-statistics, and p-values. Goodness-of-fit measures such as the R-squared, Adjusted R-squared, F-statistic, and Durbin–Watson statistic are also reported to assess the adequacy of the fitted model.

Table 5: Parameter Estimates of ARMA(3,3) Model

Variable

Coefficient

Std. Error

t-Statistic

p-value

C

8.768664

0.017218

509.2761

0.0000

AR(1)

0.366096

0.024641

14.85713

0.0000

AR(2)

0.311203

0.029382

10.59171

0.0000

AR(3)

-0.912359

0.024212

-37.68166

0.0000

MA(1)

-0.372828

0.009593

-38.86277

0.0000

MA(2)

-0.386923

0.009312

-41.55086

0.0000

MA(3)

0.982389

0.007644

128.5160

0.0000

R-squared

0.890511

AIC

0.255229

Adjusted R²

0.867389

SIC

0.315852

F-statistic

6.914400

HQC

0.295759

Prob(F-stat.)

0.000951

Durbin-Watson stat.

2.011502

The model estimation results reported in Table 5 show that all autoregressive (AR) and moving average (MA) coefficients are statistically significant at the 1% level, as indicated by their very low p-values (p < 0.01). This implies that past values and past error terms up to the third lag significantly influence the current level of non–insulin-dependent diabetes cases among farmers.

Specifically, the positive coefficients of AR(1) and AR(2) suggest a direct persistence effect, meaning that increases in diabetes cases in the immediate past periods tend to raise current cases. Conversely, the negative AR(3) coefficient indicates a corrective mechanism, implying that after about three periods, the series tends to revert toward its mean. The MA terms also show alternating positive and negative signs, suggesting that short-term shocks have both dampening and amplifying effects over time before dissipating.

The high R-squared (0.8905) and adjusted R-squared (0.8674) values indicate that approximately 89% of the variation in diabetes cases is explained by the model, signifying a very good fit. The F-statistic (6.9144) with a significant probability value (0.000951) confirms the overall significance of the model.The Durbin–Watson statistic (2.0115) is close to 2, suggesting the absence of serial correlation in the residuals, while the information criteria (AIC = 0.2552, SIC = 0.3159, HQC = 0.2958) reaffirm that the ARMA(3,3) model remains the most parsimonious and efficient choice.

Overall, the ARMA(3,3) model adequately captures the temporal dynamics and short-term fluctuations in non–insulin-dependent diabetes cases among farmers in Benue State, making it suitable for reliable short-term forecasting.

4.7 Model Diagnostic Checks

Following the estimation of the ARMA(3,3) model for predicting non–insulin-dependent diabetes cases among farmers in Benue State, diagnostic checks such as multicolinearity test and Ljung-Box Q-statistic tests were conducted to verify the adequacy of the fitted model. This assessment ensures that the residuals behave like white noise, uncorrelated, homoscedastic, and pattern-free over time. The test are presented in the following subsections.

4.7.1 Multicolinearity test result

Multicollinearity diagnostics were performed to make sure the variables in ARMA(3,3) model weren't overlapping too much. Using the Variance Inflation Factor (VIF) for each autoregressive (AR) and moving average (MA) term, the test assessed how multicollinearity might affect the stability and reliability of parameter estimates. Generally, VIF values above 10 indicate severe multicollinearity, values between 5 and 10 suggest moderate correlation, and values below 5 imply no serious concern. The results presented in Table 6 show both uncentered and centered VIF statistics for the ARMA(3,3) model parameters.

The results of multicolinearity test reported in Table 6 below reveal that all centered VIF values are considerably low, ranging between 1.11 and 2.55, which are far below the critical threshold of 10. This indicates that there is no serious multicollinearity among the explanatory variables (AR and MA terms) in the estimated ARMA(3,3) model.

Therefore, the estimated parameters are statistically reliable, and the standard errors are not inflated by multicollinearity. This implies that the ARMA (3,3) model is well-conditioned, and the coefficients can be interpreted with confidence.

Table 6: Test for Multicolinearity (Variance Inflation Factors)

Coefficient

Uncentered

Centered

Variable

Variance

VIF

VIF

C

0.000296

1.018813

Na

AR(1)

0.000607

1.779456

1.779044

AR(2)

0.000863

2.552345

2.552344

AR(3)

0.000586

1.768375

1.768101

MA(1)

9.20E-05

1.257613

1.255458

MA(2)

8.67E-05

1.213557

1.203709

MA(3)

5.84E-05

1.121942

1.111356

4.7.2 Ljung-Box Q-statistic test result for serial correlation

The Autocorrelation Function (ACF), Partial Autocorrelation Function (PACF), and Ljung–Box Q-statistics were used to test for serial correlation. High p-values (greater than 0.05) for the Q-statistics indicate no significant autocorrelation, suggesting that the residuals are random and the model is well specified. Table 5 presents these diagnostic test results for the ARMA(3,3) model residuals.

The results of Q-statistic reported in Table 5 and the ACF as well as PACF plots reported in Figure 4 show that all residual autocorrelations (ACF and PACF) are very small and fluctuate closely around zero across all 36 lags. None of the autocorrelation coefficients appear significant, suggesting that the residuals from the ARMA(3,3) model are approximately white noise.

Furthermore, the Ljung–Box Q-statistics have p-values consistently greater than 0.05, indicating that the null hypothesis of no autocorrelation cannot be rejected at any lag. This confirms that there is no statistically significant serial correlation remaining in the residuals. In addition, the Durbin–Watson statistic from the model estimation (2.0115) supports this conclusion by indicating near-zero autocorrelation in the residuals.

Overall, these diagnostic results confirm that the ARMA(3,3) model is well specified, the residuals are independently and randomly distributed, and the model provides a statistically adequate fit to the data. Therefore, the model is suitable for reliable short-term forecasting of non–insulin-dependent diabetes cases among farmers in Benue State

Table 7: Autocorrelations and Ljung-Box Q-Statistic Test Results of Residuals

Lag

ACF

PACF

Q-Statistics

p-value

1

-0.024

-0.024

0.1415

0.707

2

-0.012

-0.012

0.1760

0.916

3

-0.069

-0.070

1.3558

0.716

4

0.007

0.003

1.3669

0.850

5

-0.126

-0.128

5.3247

0.378

6

-0.036

-0.048

5.6541

0.463

7

-0.017

-0.024

5.7294

0.572

8

0.142

0.124

10.812

0.213

9

-0.042

-0.042

11.254

0.259

10

0.046

0.032

11.802

0.299

11

-0.021

-0.015

11.918

0.370

12

0.052

0.044

12.628

0.397

13

-0.025

0.012

12.794

0.464

14

-0.009

-0.008

12.815

0.541

15

0.062

0.080

13.804

0.540

16

0.068

0.053

15.019

0.523

17

0.112

0.147

18.316

0.369

18

0.109

0.127

21.475

0.256

19

-0.008

0.027

21.493

0.310

20

-0.087

-0.066

23.529

0.264

21

-0.066

-0.032

24.707

0.260

22

-0.020

0.010

24.810

0.306

23

-0.062

-0.057

25.855

0.308

24

-0.048

-0.064

26.480

0.329

25

0.021

-0.044

26.599

0.376

26

0.020

-0.037

26.704

0.425

27

-0.033

-0.069

27.003

0.464

28

0.065

0.050

28.156

0.456

29

0.052

0.030

28.898

0.470

30

0.062

0.046

29.969

0.467

31

0.014

0.040

30.023

0.516

32

0.010

0.016

30.053

0.565

33

0.042

0.050

30.555

0.589

34

0.003

0.004

30.558

0.637

35

-0.039

-0.013

30.994

0.662

36

-0.008

-0.001

31.014

0.705

Figure 4:Plot of Correlogram of Residuals of Estimated ARMA(3,3) Model

4.8 Forecast and Forecast Evaluation

To evaluate the predictive performance of the ARMA(3,3) model in forecasting non–insulin-dependent diabetes cases among farmers in Benue State, forecast accuracy measures were computed. The Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) were used to assess both in-sample and out-of-sample forecast accuracy. Lower values of these statistics indicate better model performance and predictive reliability. The result is presented in Table 8.

The results of forecast comparison reported in Table 8below show that the out-of-sample forecast achieved slightly lower RMSE (0.2671), MAE (0.2310), and MAPE (2.6490) values compared to the in-sample forecast (RMSE = 0.2715, MAE = 0.2446, MAPE = 2.6781). This suggests that the ARMA(3,3) model demonstrates strong predictive capability, with minimal forecast error and good generalization performance. The model selected in forecast mode, as denoted by the accuracy measures, provides reliable short-term out-of-sample predictions of non–insulin-dependent diabetes cases.

Table 8: Forecast Comparison using Accuracy Measures

RMSE

MAE

MAPE

In-Sample

0.271510

0.244615

2.678116

Out-of-Sample

0.267100

0.231048

2.649005

Note: denotes forecast mode selected by accuracy measures.

4.8.1 Forecast of Diabetes Miletus in Benue State from July, 2025 to June, 2027

To evaluate the short-term predictive performance of the ARMA(3,3) model, forecasts of non–insulin-dependent diabetes (Type-2 Diabetes Mellitus) cases among farmers in Benue State were generated for the period July 2025 to June 2027. The forecasts were computed in natural logarithmic form and then converted to actual population estimates. For each forecast, the standard error, lower confidence limit (LCL), and upper confidence limit (UCL) were calculated at a 95% confidence level, using . These values provide a range within which the true number of diabetes cases is expected to fall with high probability, thereby indicating the reliability and uncertainty of the forecasts. The forecast result is reported in Table 9 below while the forecast graph is presented as Figure 5 below too.

Table 9: "Forecast of Diabetes Miletus Infection Cases in Benue State from July 2025-

            June, 2027"

Year: Month

Forecast (natural log form)

Actual Forecast (No. of Persons)

Forecast

Std. error

LCL

Forecast

UCL

2025:06

6.9967

---

---

8896

---

2025:07

8.77405

0.271243

3799

6464

11000

2025:08

8.72655

0.271669

3619

6165

10499

2025:09

8.78204

0.271670

3826

6516

11098

2025:10

8.77132

0.272065

3782

6447

10988

2025:11

8.80141

0.272672

3893

6644

11337

2025:12

8.74519

0.272672

3680

6281

10717

2026:01

8.76088

0.272790

3738

6380

10889

2026:02

8.74585

0.273455

3677

6285

10741

2026:03

8.79725

0.273466

3871

6616

11308

2026:04

8.77366

0.273476

3781

6462

11044

2026:05

8.77825

0.274040

3794

6492

11107

2026:06

8.73648

0.274110

3638

6226

10654

2026:07

8.76803

0.274114

3755

6426

10996

2026:08

8.76810

0.274473

3752

6426

11005

2026:09

8.79729

0.274652

3862

6616

11335

2026:10

8.76026

0.274669

3722

6376

10923

2026:11

8.76113

0.274824

3724

6381

10936

2026:12

8.74504

0.275111

3662

6279

10767

2027:01

8.78341

0.275121

3805

6525

11188

2027:02

8.77734

0.275152

3782

6486

11121

2027:03

8.78223

0.275481

3798

6517

11183

2027:04

8.74716

0.275481

3667

6293

10798

2027:05

8.76058

0.275481

3717

6378

10944

2027:06

8.76313

0.275759

3724

6394

10978

Total

210.40663

154075

Average

8.766942917

6419.7917

Note: For 95% confidence intervals, . LCL and UCL denote lower and upper confidence limits respectively.

Figure 5: Forecast Graph of Diabetes Miletus in Benue State from July, 2025-June, 2027

The forecast results reported in Table 9 and Figure 5 above reveals that the predicted number of non–insulin-dependent diabetes cases among farmers in Benue State is expected to fluctuate moderately over the two-year forecast horizon (July 2025–June 2027). The monthly forecasts range between approximately 3,600 and 11,300 cases, with an overall average of about 6,420 cases per month and a total forecast of 154,075 cases during the study period. The relatively narrow confidence intervals across months suggest a high level of precision in the model’s predictions.

Overall, the ARMA(3,3) model demonstrates strong forecasting capability, indicating that diabetes prevalence among farmers in Benue State is likely to remain fairly stable with mild month-to-month variations over the forecast period.

4.9 Implications of the Study to Farmers and Postharvest Losses in Benue State

The implications of this study for farmers and postharvest losses in Benue State are significant from both public health and socio-economic perspectives. The findings, which forecast the prevalence of non–insulin-dependent diabetes (Type-2 Diabetes Mellitus) among farmers, suggest that a substantial portion of the agricultural workforce may experience declining health and productivity over time. Poor health conditions such as diabetes can reduce farmers’ physical capacity to engage in strenuous agricultural activities, particularly during critical periods like harvesting and processing. "This in turn increases the likelihood of postharvest losses, as crops may remain un-harvested or inadequately stored due to reduced labour efficiency and absenteeism resulting from illness".

Moreover, "higher diabetes prevalence among farmers implies increased medical expenditures and a diversion of household income away from agricultural investment", further compounding the problem of low productivity and waste. The study underscores the urgent need for integrated health and agricultural policies—including improved rural healthcare services, regular medical screening, health education on diet and lifestyle, and the promotion of labour-saving technologies—to mitigate the dual burden of disease and postharvest losses. Ultimately, addressing the health challenges of farmers is crucial for achieving food security, sustaining agricultural livelihoods, and enhancing overall economic resilience in Benue State.

4.0       Conclusion

The study demonstrates that the ARMA(3,3) model effectively forecasts the incidence of non-insulin-dependent diabetes among farmers in Benue State, Nigeria, The analysis revealed that the ARMA(3,3) model provided the best fit based on information criteria and diagnostic tests, with residuals behaving like white noise, indicating a well-specified and reliable model. The forecasts from July 2025 to June 2027 suggest a steady and relatively high incidence of diabetes cases among farmers, implying that the disease poses an ongoing public health concern within the agricultural population. This condition could adversely affect farmers’ productivity, increase medical costs, and indirectly contribute to higher postharvest losses due to reduced labour availability and inefficiencies in farm management. These findings highlight the interconnectedness between health and agricultural output, emphasizing that the burden of chronic diseases like diabetes extends beyond healthcare into the realm of food security and economic stability. Therefore, proactive health interventions and policy integration between the health and agricultural sectors are vital. Ensuring farmers’ wellness through preventive care, early detection, and education can significantly reduce the impact of diabetes and its broader economic consequences. The study provides empirical evidence to guide policymakers, healthcare providers, and agricultural development agencies in formulating context-specific strategies to improve both health outcomes and agricultural sustainability in Benue State.

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	RMSE	MAE	MAPE
In-Sample	0.271510	0.244615	2.678116
Out-of-Sample**	0.267100	0.231048	2.649005

	RMSE	MAE	MAPE
In-Sample	0.271510	0.244615	2.678116
Out-of-Sample**	0.267100	0.231048	2.649005

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An Autoregressive Moving Average Model for Short Term Prediction of Non-Insulin Dependent Diabetes Among Farmers in Benue State

2.3.4 Autoregressive integrated moving average (ARIMA) model

2.3.4 Autoregressive integrated moving average (ARIMA) model

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Variable	Statistic
Mean	5571.321
Maximum	9661.00
Minimum	3624.000
Standard Deviation	1769.088
Skewness	0.010212
Kurtosis	1.767498
Jarque-Bera Statistic	15.57465
p-value	0.000415
Number of Observations	246

Variable	Option	ADF Test Statistic	p-value	5% Critical Value
	Intercept only	-15.3344	0.0000	-2.8731
	Intercept & Trend	-15.4304	0.0000	-3.4283

Lag	ACF	PACF	Q-Statistics	p-value
1	0.014	0.014	0.0458	0.831
2	-0.019	-0.019	0.1338	0.935
3	0.004	0.005	0.1380	0.987
4	-0.049	-0.050	0.7497	0.945
5	0.022	0.024	0.8747	0.972
6	0.037	0.034	1.2165	0.976
7	0.022	0.023	1.3420	0.987
8	0.017	0.015	1.4126	0.994
9	-0.007	-0.005	1.4260	0.998
10	-0.110	-0.107	4.5659	0.918
11	-0.025	-0.022	4.7227	0.944
12	0.078	0.075	6.2944	0.901
13	-0.008	-0.012	6.3115	0.934
14	-0.017	-0.027	6.3907	0.956
15	0.052	0.055	7.0970	0.955
16	-0.035	-0.022	7.4226	0.964
17	-0.012	-0.008	7.4599	0.977
18	-0.088	-0.093	9.5213	0.946
19	-0.054	-0.050	10.302	0.945
20	-0.092	-0.114	12.567	0.895
21	-0.026	-0.032	12.750	0.917
22	-0.115	-0.115	16.369	0.797
23	0.007	0.008	16.381	0.838
24	-0.053	-0.074	17.165	0.842
25	-0.056	-0.036	18.032	0.841
26	-0.047	-0.056	18.643	0.851
27	0.055	0.057	19.482	0.852
28	-0.011	-0.032	19.514	0.882
29	0.060	0.057	20.511	0.876
30	0.056	0.042	21.381	0.876
31	0.040	0.061	21.828	0.888
32	-0.001	-0.015	21.828	0.912
33	-0.027	-0.007	22.036	0.927
34	-0.109	-0.121	25.432	0.855
35	-0.056	-0.074	26.342	0.854
36	0.066	0.025	27.604	0.841

S/n	Model	LogL	AIC	SIC	HQC
1.	ARMA(0,1)	-34.4597	0.2964	0.3349	0.3079
2.	ARMA(1,0)	-34.8194	0.3006	0.3391	0.3121
3.	ARMA(1,1)	-32.9444	0.2934	0.3363	0.3107
4.	ARMA(0,2)	-34.4107	0.3042	0.3469	0.3214
5.	ARMA(2,0)	-35.1256	0.3125	0.3555	0.3298
6.	ARMA(1,2)	-32.9256	0.3014	0.3586	0.3245
7.	ARMA(2,1)	-33.2988	0.3057	0.3631	0.3288
8.	ARMA(2,2)	-30.3771	0.2899	0.3616	0.3188
9.	ARMA(0,3)	-34.4060	0.3122	0.3692	0.3352
10.	ARMA(3,0)	-35.4688	0.3248	0.3823	0.3480
11.	ARMA(1,3)	-28.0912	0.2701	0.3616	0.3089
12.	ARMA(3,1)	-32.9028	0.3119	0.3838	0.3409
13.	ARMA(2,3)	-30.3708	0.2981	0.3841	0.3328
14.	ARMA(3,2)	-30.5304	0.3007	0.3859	0.3354
15.	ARMA(3,3)**	-24.0103	0.2552	0.3159	0.2958
16.	ARMA(0,4)	-34.1157	0.3180	0.3893	0.3467
17.	ARMA(4,0)	-35.3492	0.3335	0.4056	0.3625
18.	ARMA(1,4)	-34.4466	0.3302	0.4159	0.3647
19.	ARMA(4,1)	-35.3432	0.3417	0.4282	0.3765
20.	ARMA(2,4)	-32.0099	0.3198	0.4201	0.3602
21.	ARMA(4,2)	-26.7027	0.2785	0.3795	0.3192
22.	ARMA(3,4)	-25.4065	0.2799	0.3899	0.3213
23.	ARMA(4,3)	-33.4797	0.3428	0.4581	0.3893
24.	ARMA(4,4)	-31.4253	0.2962	0.4060	0.3285

Variable	Coefficient	Std. Error	t-Statistic	p-value
C	8.768664	0.017218	509.2761	0.0000
AR(1)	0.366096	0.024641	14.85713	0.0000
AR(2)	0.311203	0.029382	10.59171	0.0000
AR(3)	-0.912359	0.024212	-37.68166	0.0000
MA(1)	-0.372828	0.009593	-38.86277	0.0000
MA(2)	-0.386923	0.009312	-41.55086	0.0000
MA(3)	0.982389	0.007644	128.5160	0.0000
R-squared	0.890511		AIC	0.255229
Adjusted R²	0.867389		SIC	0.315852
F-statistic	6.914400		HQC	0.295759
Prob(F-stat.)	0.000951		Durbin-Watson stat.	2.011502

	Coefficient	Uncentered	Centered
Variable	Variance	VIF	VIF
C	0.000296	1.018813	Na
AR(1)	0.000607	1.779456	1.779044
AR(2)	0.000863	2.552345	2.552344
AR(3)	0.000586	1.768375	1.768101
MA(1)	9.20E-05	1.257613	1.255458
MA(2)	8.67E-05	1.213557	1.203709
MA(3)	5.84E-05	1.121942	1.111356

Year: Month	Forecast (natural log form)		Actual Forecast (No. of Persons)
Year: Month	Forecast	Std. error	LCL	Forecast	UCL
2025:06	6.9967	---	---	8896	---
2025:07	8.77405	0.271243	3799	6464	11000
2025:08	8.72655	0.271669	3619	6165	10499
2025:09	8.78204	0.271670	3826	6516	11098
2025:10	8.77132	0.272065	3782	6447	10988
2025:11	8.80141	0.272672	3893	6644	11337
2025:12	8.74519	0.272672	3680	6281	10717
2026:01	8.76088	0.272790	3738	6380	10889
2026:02	8.74585	0.273455	3677	6285	10741
2026:03	8.79725	0.273466	3871	6616	11308
2026:04	8.77366	0.273476	3781	6462	11044
2026:05	8.77825	0.274040	3794	6492	11107
2026:06	8.73648	0.274110	3638	6226	10654
2026:07	8.76803	0.274114	3755	6426	10996
2026:08	8.76810	0.274473	3752	6426	11005
2026:09	8.79729	0.274652	3862	6616	11335
2026:10	8.76026	0.274669	3722	6376	10923
2026:11	8.76113	0.274824	3724	6381	10936
2026:12	8.74504	0.275111	3662	6279	10767
2027:01	8.78341	0.275121	3805	6525	11188
2027:02	8.77734	0.275152	3782	6486	11121
2027:03	8.78223	0.275481	3798	6517	11183
2027:04	8.74716	0.275481	3667	6293	10798
2027:05	8.76058	0.275481	3717	6378	10944
2027:06	8.76313	0.275759	3724	6394	10978
Total	210.40663			154075
Average	8.766942917			6419.7917

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An Autoregressive Moving Average Model for Short Term Prediction of Non-Insulin Dependent Diabetes Among Farmers in Benue State

2.3.4 Autoregressive integrated moving average (ARIMA) model

2.3.4 Autoregressive integrated moving average (ARIMA) model

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