16 with hypothesis of test:
H =
β =
β
1
= β
2
= β
3
= … = β
p
= 0 H
1
= at least one βi is not the same as zero.
Furthermore, the Wald test is used to test the significance of parameter β
j
, where j = 1,2,3, .., p partially. The formula of Wald test:
where β
j
is the coefficient and SE
βj
is its standard error. Wald test denotes a random variable following the standard normal distribution Hosmer and
Lemeshow 2000. Hauck and Donner 1977 in Hosmer and Lemeshow 2000 examined the performance of the Wald test and found that it behaved in an
aberrant manner, often failing to reject the null hypothesis when the coefficient was significant. They recommended that the likelihood ratio test be used.
Table 2. Stepwise Method for Building MLR Model
Stepwise Terms Description
Forward entry This method begins with no stepwise terms in the model. At
each step, the most significant term is added to the model until none of the stepwise terms left out of the model would have a
statistically significant contribution if added to the model.
Backward elimination
This method begins by entering all terms specified on the stepwise list into the model. At each step, the least significant
stepwise term is removed from the model until all of the remaining stepwise terms have a statistically significant
contribution to the model.
Forward stepwise This method begins with the model that would be selected by
the forward entry method. From there, the algorithm alternates between backward elimination on the stepwise terms in the
model and forward entry on the terms left out of the model. This continues until no terms meet the entry or removal criteria.
Backward stepwise This method begins with the model that would be selected by
the backward elimination method. From there, the algorithm alternates between forward entry on the terms left out of the
model and backward elimination on the stepwise terms in the model. This continues until no terms meet the entry or removal
criteria.
Source: SPPS Help Topics
17
2.5 Spatial Logistic Regression
Regression can be considered as a process to extract the coefficients of the empirical relationships from observations. Commonly used regression approaches
include linear regression, log-linear regression and logistic regression. The dependent variable of logistic regression could be binary or categorical. The
independent variables of logistic regression could be a mixture of continuous and categorical variables. Normality assumption is not needed for logistic regression.
Hence, logistic regression is advantageous compared to linear regression and log- linear regression Xie et al. 2005.
Logistic regression is an approach to extract the coefficients of explanatory factors from the observation of land use conversion, since
urbanization does not usually follow normal assumption and its influential factors are usually a mixture of continuous and categorical variables. The spatial
heterogeneity of spatial data should be considered when employing logistic regression to model land conversion. Spatial statistics like spatial dependence and
spatial sampling also have to be considered in logistic regression to remove spatial auto-correlation. Otherwise, unreliable parameter estimation or inefficient
estimates and false conclusions regarding hypothesis test will result. There are two fundamental approaches to consider spatial dependence: building a more
complex model incorporating an autogressive structure and designing a spatial sampling scheme to expand the distance interval between sampled sites. Spatial
sampling leads to a smaller sample size that loses certain information and conflicts with the large-sample of asymptotic normality of maximum likelihood
method, upon which logistic regression is based. Nevertheless, it is a more sensible approach to remove spatial auto-correlation and a reasonable design of
spatial sampling scheme will make a perfect balance between the two sides Xie et al. 2005.
Data layer preparation in order to apply logistic regression into spatial manner is the most fundamental process which time-consuming trial and error
process. The dependent and independent variables of modeling land use change have different types of data binary, continuous, or categorical and spatial
resolution. Converting non-spatial data into spatial data is also another case of
18 data layer preparation in land use change modeling. In order to accommodate the
logistic regression model into spatial manner, Friesen and Lydon 1999 has proposed the utilization of ARCINFO’s GRID in term of data layer preparation.
GRID’s logistic regression command allows the generation of probability surfaces for each of the land use types.
Figure 2. The process and the elements involved in GRID’s logistic regression Friesen and Lydon 1999
In logistic regression model, the presence or absence of the outcome variable, i.e., land use type, is predicted on the basis of the explanatory variables.
Given input grids of these variables and a set of sample points indicating where a particular land use type is and is not found, an output grid can be generated in
which each cell contains the probability value that the land use type will be found at that location Friesen and Lydon 1999.
2.6 Accuracy Assessment
Accuracy assessment determines quality of the map created from remotely sensed data. Accuracy assessment can be qualitative or quantitative, expensive or
inexpensive, quick or time consuming, well designed and efficient or haphazard.
