A Quantitative Approach towards Developing DAPEMS
5.4. Simple and Multiple Regressions Analysis
Regression analysis is a statistical tool applied to develop DAPEMS. It helps to determine how OT changes as predictor variables change, and to predict the value of OT.
It is important to answer two questions about the contribution of predictor variables to the prediction of OT:
1. Does the entire set of predictor variables contribute significantly to the prediction of OT? (An overall test)
2. Does the addition of one predictor variable add significantly to the prediction of OT? (test for addition a single variable)
In order to answer these questions, it becomes important to determine the best-fitting model for describing the relationship between OT and other predictor variables. Best model means a reliable model that gives the best prediction of OT. There are some steps that have to be followed in selecting the best regression model (Kleinbaum et al., 2008):
1. Establishing separate simple regression models between OT and the predictor variables.
2. Establishing multiple regression models.
3. Evaluating the reliability of the model chosen.
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Many simple and multiple regression models have been performed separately. The aim is to find the best fitting models. The following procedures have been applied to get the best fitting models:
1. To find the optimal values for the coefficients in each regression function, as coefficients represent the estimated change in OT for each unit change in the predictor value. It helps also to determine how well the estimated line fits the data.
2. To identify the coefficient p-values. The coefficient value for p proves whether the association between the response and predictors is statistically significant, or not.
3. To compare the coefficient p-values to a-level. If the p-value is smaller than the a-level, the association is statistically significant. A commonly used a-level is 0.05. This value indicates that there is sufficient evidence that the coefficients are not zero.
4. The square root of the mean square error (S) and coefficient of determination (R2) are measures of how well the model fits the data. These values can help to select the model with the best fit. The S provides a measure of how spread out the data are. The R2 value is the proportion of variability in the Y variable (response) accounted for by the predictors (Taylor, 2007).
5. Adjusted square root will be observed.
6. The scatter diagrams describe the strength of the relationship between two sets of variables.
7. Pearson's r can be any value from -1.00 to +1.00. The higher the absolute value, the stronger the relationship, be it negative or positive (Taylor, 2007).
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8. A multicollinearity test was carried out to ensure that there is no exact linear relationship between the predictors. A variance Inflation Factor (VIF) test will be performed as a measure of how multicollinearity affects associated with each predictor. It is used to detect whether one predictor has a strong linear association with the remaining predictors.
It measures how much the variance of an estimated regression coefficient increases if predictors are correlated as follows (Montgomery and Peck, 1982):
VIF > 5 = normal relation; VIF between 5 and 10 = milled relation; VIF < 10 = perfect relation
9. Stepwise regressions are applied to test for errors in models that will be selected by OLS regressions. This takes place by using a sample of available data to build a model and then uses the rest of the available data to test the accuracy of the model.
10. Best subset regressions aim to test all possible sets of predictors and select the best set that provides best fitting models.
In the following part, regression analysis has been applied to calculate OT at different terminals in Damietta port including: general cargo terminal, dry bulk terminal, liquid bulk terminal and container terminal.
5.4.1 General Cargo Regression Analysis
At any terminal, there are three elements normally considered for improving terminal performance (UNCTAD, 1978). The first element is the productivity. It is normally defined as the total tonnes of general cargo handled in the port. The second element is the interruptions which tend to happen. It affects ship output, terminal productivity and the port performance. The third element is the equipment used in handling.
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5.7- General Cargo Terminal
Figure 5.7 shows the location of the general cargo terminal at Damietta port that includes four general cargo berths (berths no. 5, 6, 7, and 8) with 800 metres length and 12 metres depth. Also, the terminal is provided with a general cargo yard of 500,000 m2. It handles exports of agricultural products, fertilisers and furniture and receipt of imported goods such as petrochemicals, grains and flour. The terminal handles a total capacity of 2.1 million tonnes annually.
Many simple and multiple regression models have been performed using Ordinary Least Square (OLS)1 regressions to examine the significance of relationship between OT as a response variable and other predictor variables that are stated in Table 5.2. In all simple models, the Pearson's R is not zero. This means that there is definitely a relationship between OT and predictor variables.
1 OLS is beneficial as it minimises the sum of squared residuals. It helps to provide un-biasedness and consistency estimation because it estimates change in entirely expected ways when the units of measurement of the response and predictors change (Wooldridge, 2005, p.30).
General Cargo Terminal
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However, any relationship between OT and predictors is not necessarily statistically significant. In order to determine whether the observed relationship between the response and predictors is significant, the significance level was tested through:
1. The coefficient of p-values, as it explains whether or not the relationship between the response and predictors is significant.
2. Comparing the coefficient p-values to α- level, as if the p-value is smaller that α- level, the relationship is statistically significant. A commonly used α- level is 0.05.
Firstly, many simple regression models have been performed where OTgen is the response variable (See Appendix D). There were regression models that have been ignored from the results as the predictors have no influence on OTgen. Equation (2) shows one of the simple regression models that has not been selected, such as OTgen versus BO.
OT = 2962 + 13.3 BO (2)
Predictor Coef SE Coef T P Constant 2962 2842 1.04 0.302 BO 13.29 35.46 0.37 0.709 S = 1054.01 R-Sq = 0.2% R-Sq(adj) = 0.0%
This example shows a very weak relationship between OTgen and BO. R-sq is weak and the p-value indicates that the relationship between OTgen and BO is not significant.
S=1054.01, which is considered high as the estimated standard deviation about the regression line.
Secondly, multiple regressions have been performed to identify the best fitting model.
The purpose is to examine whether OTgen can be predicted by NCS, TTH, BO, LDR and ST variables.
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The outputs found that the best three-predictor model estimates OTgen from NCS, ST and BO. Equation (3) displays the best regression model for general cargo.
OTgen = - 2054 + 47.8 NCS + 0.00468 ST + 28.5 BO (3)
Predictor Coef SE Coef T P Constant -2054.1 568.2 -3.62 0.001 NCS 47.789 2.320 20.59 0.000 ST 0.004676 0.002117 2.21 0.031 BO 28.475 7.333 3.88 0.000 S = 325.9 R-Sq = 90.8% R-Sq(adj) = 90.3%
The interpretation of the regression equation follows:
1. The slope (b1 = 47.8) is the change in OTgen when NCS increases by 1. That is, when the NCS increases by one unit, the OTgen increases by 47.8 units.
2. The slope (b2 = 0.00468) is the change in OTgen when ST increases by 1. That is, when the ST increases by one unit, the OTgen increases by 0.00468 units.
3. The slope (b3 = 28.5) is the change in OTgen when BO increases by 1. That is, when BO increases by one unit, the OTgen increases by 28.5 units.
4. The constant (intercept) value (bo = - 2054) is the predicted value of OTgen when each predictor (NCS, ST and BO) is zero. That is, when the predictors are zero the OTgen is - 2054.
The goodness-of-fit measure is not only to focus on the adjusted R squared. Adding more predictors may not enhance the prediction of OTgen, but it inflates the R square. Adding more variables to the model caused R squared to increase a little (Wooldridge, 2005(.
Hence, it was found out that TTH and LDR have no significant relationship with OTgen.
When the LDR predictor was introduced to the model, it did not help to account for a significant portion of the remaining variation in OTgen. In addition, no significant
improvement was observed when LDR was added to the model. The reason is the ST predictor plays an important role in the general cargo at Damietta port. A sufficient number of storage areas and the capability of the equipment serve to increase the handling rate (LDR predictor). Thus, LDR has no significant effect in the model as ST predictor measures the same thing.
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Also, the rate of loading and discharging differs from one type of general cargo to another, which cannot be considered as a leading factor for OTgen. In addition, LDR has
no significant effects because most general cargo discharged at Damietta port is a measurement cargo (light cargo). Therefore, the handling rate appears to be very high
as the freight tonnes are measured in cubic meters.
For the TTH predictor, the correlation matrix in Table 5.3 indicates that there is a significant relationship between OTgen and TTH, as it accounted for 95%. However, the best fitting model has excluded TTH because there is a strong relationship between NCS and TTH, as it accounted for 89%. This is called multicollinearity in the regressors, which leads to unreliable estimates of the regression coefficients if multicollinearity is present (Draper and Smith, 1998). Higher correlation is called exact dependency or exact multicollinearity. Hence, TTH predictor has been excluded to avoid multicollinearity.
Also, VIF test shows that TTH's VIF equals 5, which leads to poor estimation. It is important to highlight that the correlation matrix can be used to measure if the multicollinearity exists (Belsley, 1991).
Table 5.3- Correlations: OTgen, NCS, BO, ST, LDR, TTH
OT NCS BO ST LDR NCS 0.935
0.000
BO 0.446 0.311 0.000 0.015
ST 0.140 0.044 0.068 0.288 0.739 0.608
LDR -0.118 -0.084 -0.005 -0.100 0.371 0.521 0.971 0.446
TTH 0.953 0.891 0.300 0.064 -0.084 0.000 0.000 0.020 0.629 0.524
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The correlation coefficient calculates the relationship between each pair of predictors and the response. It measures the degree of linear relationship between variables. The correlation coefficient assumes a value between -1 and +1. The usefulness of correlation is to examine two things about the linear relationships between OTgen and the predictors:
it examines the strength of the linear relationship between the variables, and it examines the direction of the sign of the coefficient which indicates the direction of the relationship, either positive or negative. Table 5.3 shows the correlation matrix for general cargo and can be interpreted as follows:
1. It shows that NCS, ST, BO and TTH have positive effects on OTgen and on each other, except LDR where the correlation coefficient is negative.
2. This means that when NCS, ST, BO and TTH increase, OTgen also tends to increase.
3. The correlation between OTgen and TTH has the strongest relationship that accounted for 95%, followed by NCS that accounted for 93%.
4. The correlation between OTgen and BO has a moderate relationship that accounted for 44%.
5. There is a strong relationship between NCS at Damietta port and TTH that accounts for 89%, which is considered as multicollinearity (Draper and Smith, 1998).
The following figures display the residual plots that can be used to examine the goodness of fit of the model:
161 predictor. It is found that all the predictors positively affect OTgen. The figure shows that:
1. As expected, there is a strong relationship between OTgen and NCS. An increase in the number of ships calling at Damietta port will result in increased total operation time.
2. For the ST predictor, the scatter plot shows how ST meets OTgen. The problem is that the port has expanded the storage areas for general cargo.
3. For the BO predictor, there is obviously a lot of variability. This is because higher berth occupancy does not necessarily mean higher operation time. The general cargo berths may be occupied with ships whilst there are no loading and discharging operations.
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Normal - 95% C I Probability Plot of NCS
Normal - 95% C I
Probability Plot of BO Normal - 95% C I
Figure 5.9- Probability Plot of NCS, ST and BO
Figure 5.9 shows that the points generally form a straight line in both NCS and BO predictors. This means that the normality assumption is valid in these predictors. Some observations have moderate departures from normality, but it does not seriously affect the results.
For the ST probability plot, data deviate from a normal distribution, as the points of the graph do not form a straight line. In reality, almost no data are truly normal. It indicates that other variables may influence OTgen, or there are outliers. The reason is that the increase in warehouse storage areas and yards cannot be reduced once they are established, while OTgen is variable.
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Figure 5.10- Residual plots for OTgen
Figure 5.10 shows the residual plot for OTgen. It displays the following:
1. Normal probability plot- It shows that the points generally form a straight line,