3.3 Analysis tools and techniques
3.3.7 Multinomial logistic regression analysis
The multinomial logistic regression is an extension of binomial logistic regression and the chances of occurrence of a particular value of response variable are compared with the chances of occurrence of the reference value of the response variable (Iyer and Jha, 2006). This type of regression is useful in situations where one wants to be able to classify subjects based on values of a set of predictor variables. Koksal and Arditi (2004) briefly introduced multinomial logistic regression as follow.
The baseline logit simply compares each category to a baseline category where all the coefficients for the variables are 0
p ip i
i i j
i X X X
category p
category t p
logi ⎟⎟=β +β +β + +β
⎠
⎞
⎜⎜
⎝
⎛ ...
) (
) (
2 2 1 1
0 (3.5)
Where βi0 = intercept; βi1 to βip = logistic regression coefficients X1 to Xp = independent variables.
The above function is called the logit, which is the natural logarithm of the odds that the event will occur. If the baseline category is j then the function above defines the ith category of the baseline category. It is possibly to calculate the probability of a category’s occurrence by using the following equation (gi is the logit function of category i):
∑=
= j
k g g
i
k i
e category e
p
1
)
( (3.6)
The interpretation of the results is drawn mainly from “odds ratio”, “log of odds ratio”, and “the current value” of the explanatory variable which is compared with the reference value. In this thesis, the reference value is the best outcome of partnering approach in construction project, “completely successful” or “10” point. The regression procedures and results explanation in this thesis adopt the one presented in Iyer and Jha (2006).
Odds ratio (eB)
Odds ratio is the ratio of likelihood of occurrence of an event to the
event M (the current value of the response variable) is p, the chances of occurrence of performance rating not being M or other than M will be q=(1−p). Since it is a binomial case and all comparisons are made with reference value (the event N), chances of the event not being M will be reckoned with chances of the event being N. The eB value has the form:
p p q eB p
= −
= 1 (3.7)
Alternatively, the value of p and q can be determined from eB and can be written as:
B B
e p e
= +
1 and B
q e
= + 1
1 (3.8)
The event M and event N in the present study pertain to the values of response variable, i.e., event M represents the occurrence of project performance of some desired level (degree of success of partnering in construction project) called as “current value” having values as 1, 2, and so on up to 9; and event N will be the “reference level” which is taken as 10.
Log of Odds ratio
It is obvious from the name “log of odds ratio”, this quantity is denoted by B. This component is regarded more for its sign, which determines the impact of explanatory variable on the outcome of response variable. For the event M, if the analysis shows positive sign to B, it implies that any increase in the value of explanatory variable will increase the likelihood of event being M. Conversely, the negative value of B indicates that increase in the value of explanatory variable will decrease the likelihood of event being M, i.e., occurrence of the response variable being at the current level. Since the
performance level is compared with 10, decrease in the likelihood value of performance rating at the current level will indicate the increase in the likelihood value of reference performance rating and vice versa.
The magnitude of impact of explanatory variable on the current value of the response variable is determined by the magnitude of the odds ratio, eB. More precisely, one unit increase in the value of explanatory variable causes odds ratio to change by (1−eB) times. The new or changed value of odds ratio would now be:
( )
[ B ] B
B e e
e 1−1− = 2 (3.9)
Accordingly, the new value of likelihood of event M, p’ (say) and that of event N, q’ (say) after change due to one unit of explanatory variable will be e2B/(1+e2B) and 1/(1+e2B), respectively. If Δp and Δq are the changes in the values of likelihood of events M and N, they can be written as given below:
B B B B old
new e
e e
p e p p p
p − +
= +
−
=
−
=
Δ ' 1 2 1
2 (3.10)
B old B
new q q q e e
q
q − +
= +
−
=
−
=
Δ 1
1 1
' 12 (3.11)
Δp would indicate the change in likelihood of project performance being at the current level and Δq would indicate the change in likelihood of project performance of not being at the current level, i.e., being at the reference level of 10. The values of Δp and Δq are thus complementary to each other. It could be further interpreted that the negative value of Δp which indicates decreasing chances of the project performance being at the current level, is also associated with the positive value of Δq indicating increasing chances of
that a negative value of Δp indicates improvement in the performance level towards 10 from the current level. On the other hand, a positive value of Δp indicates increasing chances of performance of the project being at the same level and decreasing chances of performance being at the alternate level of 10. These lead us to conclude that with a positive value of Δp there will be diminishing chances of further improvement. This logic is used for interpretations of results of statistical analyses of responses.