1. Introduction

Often in controlled comparative either prospective, retrospective or cross –sectional study involving matched samples of subjects or patients, the response of a subject or patient to a predisposing factor in a retrospective study or to a condition or treatment in a prospective study may be dichotomous but more often much finer than simply dichotomous. These responses may be of such form as yes or no, present or absent, alive or dead, positive or negative, etc. But there may also be several other possible response options. For example, in a retrospective study where the predisposing factor of a certain conditions is the respondents drinking habit, a subject may be classified as a teetotler, a light drinker or a heavy drinker etc. in a prospective study involving same conditions or test, a patient may be classified as recovered, much improved, improved, no change, worse or dead. A treatment or drug may be graded as very effective, effective or ineffective etc.

Furthermore there may not be only one pair but several pairs of case control subject combinations and other such situations. Thus specifically suppose we have a total random sample of n- matched pairs of patients or subjects matched on certain characteristics to be exposed to two treatments, drugs, tests; suppose pairs of these subject pairs are in the response category by case and response category by the control subject, for responses. In other words, suppose, one member of each pair of subjects or patients in the case control response combination is exposed to one under control of the two treatments, drugs, tests or procedures of interest and the other member of pair is exposed to the remaining (case, new drug) treatment, drug, test or procedure Furthermore, suppose that the response of members of each pair of case-control subjects or patients are more than dichotomous but numbering some c possible mutually exclusive response options or categories.

In Table 1, there is no replication, that is, there is only one pair of case-control subjects. Of the pairs of case-control subjects used in the study, and are subjects respectively in the response category for case and response category for control while pairs are in the pair of case-control subject response category for response categories.

Table 1. Format for presentation of results in diagnostic screening tests with “c” outcomes

Results from these types of study may be analyzed using several methods for example Backpar (1966); Grizzle, Starmer and Kosh (1959), Ireland, Ku and Kullback (1969); Staurt (1955) and Maxwell (1970), if or the more generalised method can be used. We present an alternative method of analysis in terms of probabilities, odds and odds-ratios of occurrence of outcomes using dummy variable regression techniques.

The proposed method assumes that there are replications; that is, there is more than one pair of case-control subjects in various case-control subject treatment combinations. It is however, for simplicity, assumed that there are no interactions between combinations or that such interaction have been removed by appropriated data transformation.

2. The Proposed Method

Suppose the pair of case-control subjects or patients is selected and members of the selected pair are each randomly assigned to treatments or and their various responses are recorded for pairs; responses and is the total number of pairs of case-control subjects studied. As already noted above, the responses by members of each matched pairs of subjects or patients are classified into c-possible mutually exclusive categories or classes. Suppose that the responses by both case and control subjects and patients are quantitative and can assume all possible values on non negative real number line.

Now. let be the pair of the scores or responses by case administered treatment responding at the response option or category and the corresponding control subject or patient administered treatment responding at the response option or category for

(1)

be the difference between the scores by case administered treatment responding at the response category and the corresponding control subject or patient administered treatment responding at the response category by the pair of case-control subjects or patients on the condition of interest for

Also, let

(2)

We propose, develop and present the use of dummy variable regression models in the analysis of diagnostic screening test results with several case-control subject –response options or categories with replications in each case-control response categories or combination and for comparing the differences between proportion of cases and controls subjects responding under various response categories or options. (Fleiss (1981), Ryan (1997)

Consistent with the use of dummy variable in regression models (Boyle 1975), we represent the c-response categories or options by case a that is, “c” levels of with dummy variables of and also the “c” response categories or options by control, “t”, that is, the “c” levels of by dummy variables of

Hence using the difference between the scores by the pair of case - control subjects or patients as the dependent or criterion variable, we may now define

Also let

Above expression is Equation (4). For and some .

Using these dummy variables, set up a dummy variable multiple regression model of regressing on the dummy variables of 1s and 0s of Eqn 3 and 4 expressing the dependence of the relative magnitude of the differences between the scores of the in the screening tests as

(5)

Equation 5 is the probability that the randomly selected case-control pair of responses on a given condition the score by case administered treatment (new drugs) is on the average higher (better, greater, less serious) than the score by the corresponding control subject administered treatment (standard drugs) or vise versa.

The odds of occurrence of this events, that is, the odds that for the case-control treatment combination, the response by case administered treatment is higher (better, greater, less serious) than the response or score by the corresponding control subject administered treatment is

(6)

are dummy variables of are partial regression coefficients and are error terms with

Now, the expected value or mean of ; the relative difference in scores or responses by the when the case responds at the response category and control subject responds at the response category or option is

(7)

Equation 5 may be alternatively expressed in matrix form as

(8)

where is an column vector of as specified in equation 1, X is an design matrix of where dummy variables of representing the response categories included in the regression model, are partial regression coefficients and is an column vector of error terms with

The expected value of is

(9)

Use of the usual methods of least squares with either Equations 2 or 4 yields unbiased estimates of the regression parameter as

(10)

where is the matrix inverse of the non singular variance – covariance matrix with full column rank

(11)

A null hypothesis that maybe of interest is that the regression model of either Equations 4 or 7 fits; that is, it is adequate representation of differences in scores or responses between pairs of case and control subjects in diagnostic screening tests. This is equivalent to the null hypothesis.

(12)

The null hypothesis of Equation 11 is tested using the usual F test presented in the traditional analysis of variance Table.

The null hypothesis of Equation 11 is rejected at the if the calculated F ratio is greater than the tabulated or critical F ratio with “p’ and degrees of freedom; that is, if

Otherwise is accepted.

If the null hypothesis of Equation 11 is rejected, in which case not all the are equal to zero, then one may proceed further to determine which of these regression coefficients or their combinations are statistically different from zero and hence, may have been responsible for the rejection of . In particular, one may wish to compare the responses of cases and controls in the paired case-control subject responses or scores.

Note that the expected difference in scores or responses between case and control subjects when the responses by the control subjects in the paired case-control subjects responses are held at constant level, that is when the mean response or score by case in all the pairs is of interest, is obtained by setting in Equation 5 yielding

(13)

Similarly, the expected difference in scores or responses between case and control subjects when the response by case in the paired case-control responses or scores are held at constant levels, that is when the mean response or score by the control subjects in all the pairs of interest is obtained by setting in Equation 5 yielding

(14)

(15)

(16)

Having rejected the initial null hypothesis of Equation 11, one may then wish to test the null hypothesis that is equal to some specified value say that is

(17)

The null hypothesis of Equation 16 is tested using the “t” test statistic

(18)

where is a row vector of with occurring at and occurring at , and MSE is the error mean sum of squares obtained from the corresponding analysis of variance Table.

The null hypothesis of Equation 16 is rejected at the level of significance if otherwise is accepted.

Further research interest may also be in performing pairwise comparison of the effects scores by case and control subject at the response category for case and response category for control subjects have on the variation in the differences between scores or responses by the pair of case-control subjects in the two response categories. This is equivalent to testing the null hypothesis

(19)

The null hypothesis of Equation 18 is tested using the student “t” test statistic;

(20)

is the element in the row and column of for

The null hypothesis of Equation 18 is rejected at the level of significance if is satisfied otherwise is accepted.

If in particular, the “c” response categories are ordered from the least serious (most serious) or the most serious (least serious) or if any of the response categories is treated or known to be in the best (least serious) condition or the worst (most serious) condition, then, the null hypothesis of Equation 18 would be equivalent to testing the null hypothesis that pairs of case-control subjects are equivalent in the same level of seriousness or condition in the test being conducted.

The test statistic of Equation 19 is used to test this null hypothesis; that is, the null hypothesis that case and control subjects are in the same condition, that is, do not differ in their responses or scores in the screening test.

Note that in practice, the data of interest are not usually in their summary form as frequencies as in Table 1 but as raw scores, that is, pairs of case-control subject responses or scores in these cases, the frequencies if required may nevertheless be obtained from the design matrix X used for the analysis. In other words, the cell frequencies are obtained from the columns X as;

(21)

Also, and are obtained as the scores in the omitted categories for case and control response categories respectively.

(22)

The number of pairs of case-control responses or scores in the case –control response category, that is the number of scores common to the “omitted categories” for case and control subjects in the design matrix X of the regression model is obtained as the number of zeros common to all the “included categories” for case and control subjects respectively, or equivalently as

(23)

with these frequencies, one may now apply an alternative generalized method based on only frequencies (Oyeka et al, 2013) to the data for comparative purposes.

2.1. Illustrative Example

We, now use data on matched subjects of matched patients from a controlled comparative clinical trial who manifest four possible responses to illustrate the proposed method. Suppose the data in Table 2 are obtained by assigning a standard treatment regarded as control drug and a new treatment regarded as case at random to members of each pair of a random sample of 54 pairs of case and control malaria patients matched on age, gender and body weight used in controlled clinical trials to compare the effectiveness of two malaria drugs.

Table 2. Sample Data of Response Scores by Malaria patients in controlled Clinical Trials with Replications

Using these values of Table 2 with Equations 1-4, we obtain the design matrix X shown in Table 3 for use in the regression model of Equation 9.

Table 3. Design Matrix X for the data of Table 2

Using the design matrix X of Table 3 in Equation 6, we obtain the fitted dummy variable regression model of the difference in case-control subject response scores as function of the seriousness of it or levels of these responses as

(23)

The fitted regression model of Equation 23 with an independent variables can explain only about 5 percent of the total variation in case and control subjects in their response scores, suggesting that its regression coefficients are all probably not different from zero.

With these conclusions, further analysis would ordinarily become unnecessary, however, for illustrative purposes only, one could assume that not all the regression coefficients are zero thereby enabling some comparisons of these parameters if so desire.

3. Summary and Conclusions

From the result of the analysis, implies that independent variables can only explain 5 percent of the total variation in case and control subjects in their response scores, suggesting that its regression coefficients are all probably not different from zero (insignificant). Also, the p-value of 0.869 indicates that the model is insignificant. The illustrative example shows that the proposed method is adequate for modeling and determination of significance/insignificance models.