A laboratory assessment of the effect of varying roughness on dissolved oxygen using error correction method

Figures & data

Table 1. Aggregate sizes, combinations and their respective roughness coefficients.

Sizes (mm)	Control	6.3	9.5	12.5	13.2	13.2, 12.5, 9.5	12.5, 13.2, 9.5	9.5, 12.5, 6.3	6.3, 9.5, 12.5
K	0.9312	1.0768	1.1064	1.1233	1.1167	1.1685	1.2024	1.1873	1.182

Table 2. Descriptive statistics of the explanatory variables

	Velocity (V)	Roughness (K)	Froude no	DO	Dispersion (d)
Mean	0.466400	1.147218	0.025090	0.240800	0.107000
Median	0.460000	1.123343	0.025566	0.210000	0.116900
Maximum	0.550000	1.378147	0.041540	0.850000	0.227800
Minimum	0.390000	0.931157	0.016556	−0.12	0.007300
Std. dev.	0.065503	0.119296	0.005427	0.210771	0.051552
Skewness	0.167092	0.139313	1.007095	0.809878	−0.065782
Kurtosis	1.563884	2.004969	4.505250	3.996328	2.769926
Jarque-Bera	2.264697	1.112207	6.586184	3.766960	0.073170
Probability*	0.322275	0.573439	0.037139	0.152060	0.964076
Sum	11.66000	28.68045	0.627260	6.020000	2.675000
Sum sq. dev.	0.102976	0.341558	0.000707	1.066184	0.063783

* Null hypothesis: reject when p < 0.05 that the variable is normally distributed.

Table 3. Correlation statistics of all variables used for ECM modelling

Variable	Velocity	K	Froude nos	DO	Dispersion (d)
Velocity	1.000000	0.875588	−0.686848	−0.09334	0.247989
K	0.875588	1.000000	−0.94233	0.113135	0.169903
Froude	−0.686848	−0.94233	1.000000	−0.25079	−0.09159
DO	−0.09334	0.113135	−0.250787	1.000000	−0.366679
Dispersion (d)	0.247989	0.169903	−0.09159	−0.36668	1.000000

Table 4. Augmented Dickey Fuller test for stationarity of the variables after first difference

Variables	ADF test statistics*	ADF critical values*		Remark	Order of integration
		1% level	5% level
DDO	−13.11472	−3.769597	−3.004861	Stationary	I(1)
Dvelocity	−7.745967	−3.769597	−3.004861	Stationary	I(1)
Ddispersion	−10.49917	−3.769597	−3.004861	Stationary	I(1)
Dfroude	−7.572152	−3.769597	−3.004861	Stationary	I(1)
DK	−7.840730	−3.769597	−3.004861	Stationary	I(1)

* Null hypothesis: reject when ADF critical values are greater than ADF test statistics.

Table 5a. Cointegration test for all the variables considered for ECM showing no deterministic trend (Trace)

Hypothesized no. of CE(s)	Eigenvalue	Trace statistic	Critical value	Prob.** (0.05)
None*	0.8442	111.1685	60.0614	0.0000
At most 1*	0.7677	70.2631	40.1749	0.0000
At most 2*	0.5650	38.1472	24.2760	0.0005
At most 3*	0.3854	19.8323	12.3209	0.0023
At most 4*	0.3394	9.1227	4.1299	0.0030

Note: Trace test indicates 5 cointegrating eqn(s) at the 0.05 level with Unrestricted Cointegration Rank Test (Trace).

* Denotes rejection of the hypothesis at the 0.05 level.

** MacKinnon-Haug-Michelis (1999) p-values.

Table 5b. Cointegration test for all the variables considered for ECM showing no deterministic trend (Max-Eigen)

Hypothesized no. of CE(s)	Eigenvalue	Max-eigen statistic	Critical value	Prob.** (0.05)
None*	0.8442	40.90547	30.43961	0.0018
At most 1*	0.7677	32.11586	24.15921	0.0034
At most 2*	0.5650	18.31486	17.79730	0.0418
At most 3	0.3854	10.70960	11.22480	0.0615
At most 4*	0.3394	9.1227	4.1299	0.0030

Note: Max-eigenvalue test indicates 3 cointegrating eqn(s) at the 0.05 level with Unrestricted Cointegration Rank Test (Trace).

* Denotes rejection of the hypothesis at the 0.05 level.

** MacKinnon-Haug-Michelis (1999) p-values.

Table 6. Error correction estimates for dissolved oxygen modelling

Dependent variable: DDO
Variable	Coefficients	Standard error (S.E)	t-ratio	Probability	R ^2	Adj. R ^2	Durbin-Watson stat (DW)
C	−0.0236	0.0543	−0.4344	0.6702
Dvelocity	−0.6533	1.2292	−0.5315	0.6028	0.839	0.764	2.059
Dfroude	−77.7071	28.2076	−2.7548	0.0147
DDispersion (d)	−2.0388	0.4994	−4.0828	0.0010
DK	−0.7962	1.2954	−0.6147	0.5480
ECM (−1)	−0.8177	0.1540	−5.3088	0.0001
AR (1)	0.0405	0.3080	0.1317	0.8970
MA (3)	0.8751	0.0581	15.0722	0.0000

Source: Author’s computation achieved with e-Views 8.0 statistical software.

Table 7. Heteroscedasticity test using the Breusch-Pagan-Godfrey test

F-statistics	0.729329	Prob. F (5,17)*	0.6111
Observed R ^2	4.062299	Prob. χ (5)	0.5405
Scaled explained SS	1.669750	Prob. χ (5)	0.8927

* Null hypothesis: reject when p < 0.05 that heteroscedasticity does not exist in the model.

Table 8. Breusch-Godfrey serial correlation LM test

F-statistics	0.497489	Prob. F (1,14)*	0.4922
Observed R ^2	0.787539	Prob. χ (1)	0.3748

* Null hypothesis: reject when p < 0.05 that autocorrelation does not exist in the model.

Figure 1. Parameter stability test.