INTRODUCTION
With the number of Pulmonary Tuberculosis (PTB) cases on the rise (Cruez, 2005; WHO, 2006), greater awareness of the disease and the need for better monitoring approach is essential. Reports by WHO (2004), the global targets for tuberculosis control are to cure 85% of the sputum smearpositive cases detected and to detect 70% of the estimated new sputum smearpositive cases.
Unnikrishnan and Jagannatha (2002) uses the definition, monitoring is a program of supervision designed to monitor the health and activities of an afflicted person. It includes ensuring compliance with an appropriate prescribed course of medication or medical therapy and with the recommendations and orders of the expert doctors. However, in this study, monitoring is defined as the investigation of a patients response to treatment by comparing a series of his chest radiographs.
The study by Rijal et al. (2006) applies the seven control point registration (SCPR) method and a resizing technique before the comparison of images is carried out. The correlation R_{F}^{2} defined from the ULFR model (Rijal et al., 2006; Dolby, 1976; Fuller, 1987) was used as a measure of quality of image registration. In this study the variables W is defined as follows;
Figure 1 shows three boxplots corresponding to three histograms
of the appropriate chest radiograph image of the PTB infected area of the same
patient taken at three consultation time points.

Fig. 1: 
Lowest boxplot corresponds to the histogram of chest radiograph of the
PTB infected area from the first consultation. Similarly topmost boxplot
is for the last consultation visit 
The leftward shift of the histogram (from bottom to top) motivates the definition
of NLSP which is the number of percentiles that shifted to the left (for example,
second image compared to the first).
After registration and resizing (Hill et al., 2001; Gonzalez et al.,
2004; Zitova and Flusser, 2003), a subset of two images, say
and
were obtained such that I(·,·)and
K(·,·) represent identical regions at two different time point, say t and t+1.
Henceforth eleven percentile of the image intensity values for each of I(·,·)and
K(·,·) were calculated and compared. For example if the median of intensity values
for K(·,·) is less than the corresponding median for I(·,·),
this suggests that the image histogram has shifted to the left. This leftward
shift suggests a decrease in intensity of snowflakes and hence in turn indicate
a positive response to treatment. The total number of percentile shifting to
the left is defined as NLSP. Estimate for NLSP was obtained from a computer
aided diagnostic for monitoring PTB (CADMPTB) system (Rijal et al.,
2006).
Selection of hospital: The data or images were collected randomly from
three types of hospitals recommended by Mahyuddin (2006). They are the District
Hospital Tanjung Karang (HTK), Kuala Selangor, Hospital Tengku Ampuan Rahimah
(HTAR), Klang and The Institute of Respiratory Medicine (IPR), Hospital Kuala
Lumpur. The HTK and HTAR were selected because
• 
They have a dedicated chest clinic. The hospitals were located in areas
of high occurrence of PTB cases and therefore a high number of PTB cases
can be obtained easily. 
• 
The medical officers are experienced in the PTB cases. 
• 
The hospitals have their own particular PTB treatment days. We may obtain
and get latest information about the progress of the PTB patients accordingly.

A total of 5 volunteer medical officers were involved in this study. The hospital
visits were limited as follows due to time consideration:

Week 1: HTK 

Week 2: HTAR 

Week 3 and 4: IPR 
MATERIALS AND METHODS
The experiment: In this experiment, medical records (patient wallet) at each hospital are randomly selected. Then for each selected patient wallet/file, the medical officer uses the standard procedure of direct visual interpretation of xray and analyses the information (e.g., sputum test) from patient’s file, (Shaban, 2005). The same patient will then be subjected to the diagnostic procedure of the CAMPTB system where the statistic NLSP was calculated.
Only the confirmed PTB cases (37 from IPR, 29 from HTK and 16 from HTAR) are used for data analysis. The data is analyzed separately because IPR is a referral centre to which other hospital like HTAR, must report to. HTK is a smaller district hospital where respiratory specialist from HTAR will do their routine weekly visit.
Each doctor was required to make one of four responses in a survey form which
are as follow:
• 
Patient’s chest xray Worse, y = 1 
• 
Patient’s chest xray No change, y = 2 
• 
Patient’s chest xray Improvement, y = 3 
• 
Patient’s chest xray Stable, y = 4 
Increasing yvalues indicated good progress of treatment. We relabeled y =
1, y = 2 into category 0 and y = 3, y = 4 into category 1. The combination of
dependent variable y into two categories helps reduce the complication of interpreting
the four responses.
As stated in the introduction section, the NLSP values (X) ranged from 0 to 11.
To model the relationship between y and x, a simple logistic regression function
is then proposed. According to Menard (2002) and Hosmer and Lemeshow (1989),
it has become, in many fields, especially in the health sciences, the standard
method of analysis when studying the relationship between a (binary) discrete
response/ dependent variable and one or more explanatory variables.
Since the medical doctor incorporates information from the patients file before confirming his diagnosis, this study also consider the relationship of sputum test results, Z with that of Y. Finally the effect of image registration, W, is also investigated.
Brief review of logistic regression: The logistic regression model has
been used in statistical analysis for many years. It does not assume linearity
of relationship between the independent variables and the dependent, does not
require normally distributed variables, does not assume homoscedasticity and
in general has less stringent requirements (Menard, 2002). A simple logistic
regression function, Model (A), is given as follows (Hosmer and Lemeshow, 1989;
Neter et al., 1989):
where, E{Y_{i}} is viewed as a conditional mean of the doctor’s response, given the value of NLSP (X_{i}). Both parameters β_{0} and β_{1} will be estimated by using maximum likelihood estimation.
The logarithm of the likelihood functions is given as:
To maximize the likelihood function, we take partial derivatives with respect
to β_{0} and β_{1}, set these equal to zero, replace
β_{0} and β_{1} with b_{0} and b_{1},
respectively, yields the likelihood equations:
Since the Eq. 2 and 3 are nonlinear in
the parameter estimates b_{0} and b_{1}, a numerical solution
specifically designed for logistic regression in MATLAB software program is
used to estimate the parameters. Once the maximum likelihood estimates b_{0}
and b_{1} are found, the fitted response function can be obtained as
follows:
To obtain the multiple logistic regression model, we simply replace β_{0}+β_{1}X
in Eq. 1 by β_{0}+β_{1}X_{1}+β_{2}Z_{i},
where for example Z_{i} is the value of sputum test results. The multiple
logistic response function, Model (B), is then given as:
where 

The procedure for estimating the regression parameters in multiple regression is the same as explained in simple logistic regression.
Goodnessoffit test for model (A): As the case for all regression models, the aptness of the logistic regression model needs to be checked before it is accepted for use. Since our data is in the ungroup binary category, two common methods: HosmerLemeshow statistic and analogues of the R^{2}statistic are suggested by Hosmer and Lemeshow (1989) and Collet (1952) to summarize model adequacy. However, Imon (2007) commended that our sample size is small in three hospitals, respectively, thus the HosmerLemeshow statistic may not be appropriate for testing the accuracy of the fitted model. Only the analogues of the R^{2}statistic is then applied to evaluate the fit of the model.
Several analogues to the linear regression R^{2} have been proposed
for logistic regression. For general use, Nagelkerke (1991) recommended the
statistic and Menard (2002) suggested the statistic .
For statistic ,
it is defined as:
where, L_{O} is the likelihood function for the model that contains only the intercept, L_{M} is the likelihood function that contains all the predictors (NLSP or sputum results) and n is the total number of binary observations (tested PTB patients) that make up the data base.
For statistic ,
it is defined as:
where, L_{O} is the loglikelihood function for the model that contains
only the intercept, L_{F} is the loglikelihood function that contains
all the predictors (NLSP or sputum results) and L_{S} is the loglikelihood
function from the saturated model. In Eq. 8, the value of
the loglikelihood L_{S} could be easily obtained from the definition
of deviance as below:
Both and
is
then multiplied by 100 to show the percentage of accuracy the model predict
the dependent variable.
Comparing linear logistic models: In examining the effect of including terms Z in Model (B), it is important to recognize the change in deviance (Neter et al., 1989; Collet, 1952);
Let
and
where, is the maximum likelihood under Model (A), is the maximum likelihood under Model (B), given in Eq. 5 and
is the maximum likelihood under the saturated model.
Table 1: 
Analysis of LeaveOneOut method (LOU) for HTK data 

It is known that D_{e}~ χ^{2} (d_{D1}–d_{D2})
where d_{D1} and d_{D2}is the degree of freedom, respectively
for model (A) and model (B). If D_{e} < χ^{2} (d_{D1}–d_{D2})
for a given level of significance (α = 0.05), the Model (A) and Model (B)
are considered similar, in other words choose the simpler model.
If the reduction in deviance is highly significant, the term Z should be added in model (A). Otherwise, we may ignore the variable.
The leaveoneout method (LOU): Since the sample size is important in
deciding the choice of model, the robustness of the model is investigated by
using the LOU method. Suppose there are n observations (x_{1}, y_{1}),
(x_{2}, y_{2}),..., (x_{n}, y_{n}). The LOU
method is as follows;
• 
Omit point (x_{1}, y_{1}), estimate β_{0},
β_{1} and E{Y}. 
• 
Omit point (x_{2}, y_{2}), replace with (x_{1},
y_{1}) and again estimate β_{0}, β_{1}
and E{Y}. 
• 
Repeat the process until (x_{n}, y_{n}) is omitted. 
Some of the results are shown in Table 1 for HTK. For large
values of X, E{Y} is close to 1 (as expected). However for X values less than
seven, E{Y} takes values around 0.5. This suggests that Model (A) should be
recommend for use for values of X greater or equal to seven.
RESULTS AND DISCUSSION
Imon (2007) noted that in practice the coefficient of determination, and
with values approximately 0.5 is considered reasonably high. Thus, the goodnessoffit
test showed the simple logistic model fits the data in HTK and HTAR. But, the
model is not fitted well in IPR data. The reason might be the biased data since
IPR is a referral centre and the chances that a patient referred to this hospital
is very likely a positive PTB case.
In Table 2, is
compared to .
The former is suitable as a measure of goodnessoffit because Eq.
7 does not critically depend on n, whilst the latter is suitable as a measure
of goodnessoffit because Menard (2002) noted that it is a common measure used
in the current versions of SPSS and SAS statistical package.
Table 2: 
Results of and
for
three hospitals 

Table 3: 
Analysis of deviance table and test statistic for IPR, HTK and HTAR 

D: Deviance 
Table 4: 
Prediction interval for E(Y) from several selected x_{0} values
for HTK 

Nevertheless, the general conclusion is the same, namely the simple logistic
model fits the data ‘better’ for HTK and HTAR.
Table 3 shows that the change of deviance in the three hospitals,
respectively is not significant when the variable Z is added to Model (A). This
result suggests that the visual interpretation of the chest Xray film may be
carried out independently of the outcome of the sputum test. In view of the
possible errors when conducting sputum test, for example, a patient not being
able to give a good sputum sample, omission of variable Z would mean simplifying
the diagnosis process. The results of Table 3 also suggest
that it is possible to omit the variable W. The omission of W implies that two
Xray films need not be perfectly aligned before any comparison can be carried
out.
Based on a sample of 29 patients for HTK and a sample of 16 patients for HTAR, the following models were obtained:
The LOU method for HTK (Table 1) shows that the β_{1}
parameter is not very different from the value 0.5223, except when patient number
eight is omitted (this is a patient where his chest radiograph shows lung scarring).
Hence Model (A) gives robust estimates of β_{1}. However for β_{0},
estimates are reliable for X values larger than seven. Similar remarks may be
made for the HTAR model.
The estimate for β_{0} and β_{1} for HTK and HTAR
suggest that this parameter values are geographically dependent. Comparison
of these models is not trivial. Instead of performing any hypothesis test on
the existing data, prediction interval for E{Y} was derived. For values of X≥7
narrow prediction intervals were obtained as shown in Table 4.
CONCLUSION
The results from the experiment suggested that the simple logistic regression Model (A) is appropriate to model the relationship between the doctors diagnosis (or response) (Y) and NLSP (X) for hospital HTK and HTAR. Further the LOU method suggests that the logistic regression model may predict Yvalues (doctors’ response) accurately for Xvalues greater than seven, but should be treated with caution otherwise.
Since films scanners are affordable by most hospitals, the CADMPTB system used may be easily applied to estimate the statistics NLSP which in turn can give an objective assessment of the patient’s progress to treatment. This facility would be most suitable when doctors involved are not in the category of specialist or consultant.
ACKNOWLEDGMENTS
We would like to acknowledge the contribution from Datin Dr. Hjh Aziah Ahmad Mahyuddin (Director) and Dr. Azwayati Abas, The Respiratory Unit, Kuala Lumpur General Hospital and Dr. Hamidah Shaban, Selangor Medical Centre.
This research was funded under an IRPA grant from the Ministry of Science, Technology and the Environment and Universiti Teknologi Malaysia.