1. Introduction

Categorical data are obtained when the variables which are discrete in nature are cross-classified and subjects having the same levels of the cross-classification are aggregated to form counts. Clearly such variables are at most ordinal in nature. Variables that are purely metric are reduced appropriately for categorical data analysis to be effected. In a follow-up (longitudinal) study the progression of positive outcome is critical and should be examined.

Cross-classified data can have any of full-multinomial, hypergeometric, independent Poisson or product multinomial distributions, Bishop, Feinberg and Holland[1], Agresti[2], Sanni and Jolayemi[3], Adejumo[4] among many authors. All these distributions have fixed, but unknown, parameters. Each underlying distribution is dictated by the sampling scheme, even though the parameter estimates within each are identical as demonstrated by Birch[5], see also Jolayemi et al[6]. It is possible, however, that the parameters involved in the categorical data, have a specific pattern, especially when one or more of the categorical variables are metric but of constant interval. A statistical analysis approach for such data may be appropriate to use some models for probability outcomes. The model used, if appropriate can then be used to determine termination of management. This approach is in focus in this work.

In this research, the main objective is to examine a model fitting-algorithm for a longitudinal categorical data.

The follow-up data of this form becomes a panel data if the period for reassessment is constant.

2. Methodology

Consider an r x c contingency table. The row (r) categories are the sup-populations to be compared and column (c) equals the number of possible follow-ups. Let the matrix of observation be represented by

,

where

(1)

Within the foregoing, assume the product multinomial distribution for. Thus

(2)

where is as represented in 1.1,

and

such that

and

(3)

Furthermore, assume that for each i, the vector P_i has a known or suspected pattern . The mixture model is with a compelling assumption if each is unique, see for example Brooks et al.[7], when the variable characterizing the column is ordinal.

The main aim of this study is to test some hypothesis regarding . In particular, we assume that is exponential in this research paper with parameters

In this formulation,

(4)

where j=1,2, …, c; indicating the outcome of the column variable. If β_i < 0, the probability reduces over j (usually indexing time) or over jth follow-up time of constant period. What may be of interest here are various hypotheses regarding. Some of these include.

(i) which represents all r rows are identical before follow-up

(ii) which can be interpreted to be identical reactions of the r subpopulation for the intervention of the follow-up.

(iii) is the combination of (i) and (ii) above.

Note that other forms of are possible. Such other forms includes which is essentially used when the response is quadratic. It is also used for studying medical intervention.

Let be the likelihood function for.

Then,

so that the log likelihood L under the constraint in equation (1) is given by

(5)

where λ is the Langrange multiplier (indicating the boundary limit). Clearly the log likelihood of equation (5) does not give normal equations which are linear in the parameters, see McCulagh and Nelder[8], Jolayemi and Okoro[9] for example.

Let be the likelihood function using estimation under the null hypothesis and be the similar likelihood under the parameter space.

The Likelihood ratio test statistic can be obtained from

Under some regularity conditions, see Bickel et al.[10] and Adejumo[4],

has the chi-square distribution with (k-m) degrees of freedom, where k and m are the number of parameters estimated under Ω and Ho respectively.

2.1. Estimation of Parameters

First consider the log likelihood function of equation (5) and let the null hypothesis H_o be given by

This is equivalent to

which represents gender insensitivity. Other forms of H_o can be used.

The likelihood function L_Ho is given by

(6)

The normal equations from equation (6) are obtained as follows:

(7)

(8)

(9)

From equation (7) it is clear that λ = -n_..

Thus equations (7) and (8) become respectively

(10)

(11)

Let S_2x1 represent the vector of normal equations.

Then S is given by

So that the Hessian matrix, Morisson[11] is given by

(12)

Let , then by mid-value theorem: Mood et al.[12]

(13)

so that the can be obtained as

An iterative procedure is then used to obtain an estimate using an initial vector and tolerance

(14)

It is easy to note that the vector is given by

And the cell values of the Hessian matrix is given by

Under Ω, the above procedure is obtained for each i.

Thus

Finally, the estimate is obtained as explained earlier. The test statistic in this case which is the likelihood ratio test statistic is given by

So that where d = (k - m) degrees of freedom. (k - m) = (2r - 2)=2(r - 1).

If under H_o is given as

And under Ω is given as

Then

And -2log∆ is given as

The above is a demonstration of how to produce software to perform the process for execution.

3. Empirical Results, Discussions and Conclusions

The method of application of mixture models for the 2-dimensional categorical data is demonstrated using a data set from a disease management from a hospital, the University of Ilorin Teaching Hospital (UITH), Nigeria, spanning the period between 1996 and 1998 on the management of Tuberculosis patients. The data excluded those who were lost to follow-up, so that, those who were successfully discharged were considered in Table 1 using approximated periods.

Table 1. 109 Tuberculosis patients classified by length of treatment and gender using one month follow up interval Duration (in month) 1 2 3 4 5 6 Total No. of Male 44 17 6 3 3 1 74 No. of Female 22 8 4 1 0 0 35 Total 66 25 10 4 3 1 109

The analysis of the data followed equation (2) and the imposed models in equation (4). Using the tolerance limit δ = 0.001 for maximum difference in the parameter estimates as dictated by

of equation (13), the following estimates were obtained:

The likelihood ratio test statistic for of G²=-2log∆=15.24 with d=2 degrees of freedom with p-value of 0.001 provided a bad fit

This implies that a uniform distribution cannot be used for both males and females. Consequently, different models existed for males and females which were θ₁ and θ₂. This showed that the period of treatment was gender sensitive. While males would be treated for seven months the female counterpart would be treated for 4 months.