Note that the likelihood estimating equation 2. Thus, the estimating equation 2.
Thus, we first use '. By similar calculations as in 2.
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We then write a vector of distinct parameters, say , collecting them from all levels and append the estimating equation 2. These coefficient matrices are given by. Illustration 2. Note that Table 2.
Because the individual status was recorded with regard to both snoring and heart disease problems, it is reasonable to consider the snoring status and heart disease status as two response variables. One would then analyze this data set by using a bivariate multinomial model to be constructed by accommodating the correlation between two multinomial response variables.
This will be discussed in Chap. If one is, however, interested to examine the effect of snoring levels on the heart disease status, then the same data set may be analyzed by conditioning on the snoring levels and fitting a binary distribution at a given snoring level.
This leads to a product binomial model that we use in this section to fit this snoring and heart disease data. To be specific, following the notations from Sect. The responses of these individuals are distributed into two categories with regard to the heart disease problem. This product binomial 2. Next in this special case with.
Now to see that this estimating equation is the same as 2. As the algebra below shows, this equality holds. Hence, as expected, all three estimating equations are same. Note that the estimating equation 2. However, unlike in 2.
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Thus, it is up to the users to choose between 2. In this book we will mostly follow the global parameters based estimating equation 2. These estimates were then used in 2. The estimates and their corresponding estimated standard errors are given in Table 2. Note that as the snoring status is considered to be a fixed covariate as opposed to a response variable with four levels, the heart disease status of an individual follow a binary distribution at a given level. For example, it is clear from Sect. Table 2. Regression parameters Quantity 10 20 30 40 Estimate 4. Heart disease Yes Observed Estimated 0.
These probabilities at all four levels may now be estimated by using the parameter estimates from Table 2. These estimated probabilities along with their respective observed probabilities are shown in Table 2.
Notice from the results in Table 2. This result is expected because of the fact that the product binomial model is constructed with four independent regression parameters to fit data in four independent cells. This type of models are known as saturated models. In summary, the product binomial model and the estimation of its parameters by using the likelihood approach appear to be perfect for both fitting and interpretation of the data. Note that the observed and estimated probabilities appear to support that as the snoring level increases the probability for an individual to have a heart disease gets larger.
Remark that when the same snoring data is analyzed by using the snoring as a covariate with arbitrary codes, as it was done in Sect. Agresti , Table 4. In any case, these estimated probabilities, as opposed to the estimated probabilities shown in Table 2.
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Thus, it is recommended not to use any modeling approach based on arbitrary coding for the fixed categorical covariates. To illustrate the application of product multinomial model 2. We reproduce the data set below in Table 2. We describe and analyze this data using product multinomial approach. This data set was originally recorded from a report on the relationship between aspirin use and heart attacks by the Physicians Health Study Research Group at.
Harvard Medical School. The Physicians Health Study was a 5 year randomized study of whether regular aspirin intake reduces mortality from cardiovascular disease. A physician participating in the study took either one aspirin tablet or a placebo, every other day, over the 5 year study period. This was a blind study for the participants as they did not know whether they were taking aspirin or a placebo for all these 5 years. By considering the heart attack status as one multinomial response variable with three categories fatal, non-fatal, and no attacks and the treatment as another multinomial response variable with two categories placebo and aspirin use , Agresti used a full multinomial approach and described the association correlation equivalent between the two variables through computing certain odds ratios.
Then many existing approaches write the joint probabilities, for example, for the aspirin use and heart attack data, as r. These parameters are restricted by the dependence of the last category of each variable on their remaining independent categories. Thus, in this example, one may use.
This full multinomial approach, that is, considering the treatment as a response variable, lacks justification. This can be understood simply by considering a question that, under the study condition, can the response of one randomly chosen individual out of 22, participants belong to one of the six cells in the Table 2. This is not possible, because, even though, the placebo pill or aspirin was chosen by some one for a participant with a prior probability, the treatment was made fixed for an individual participant for the whole study period.
Thus, treatment variable here must be considered as a fixed regression covariate with two levels. This prompted one to reanalyze this data set by using the product multinomial model 2. By this token, for both crosssectional and longitudinal analysis, this book emphasizes on appropriate modeling for the categorical data by distinguishing categorical covariates from categorical responses. Product multinomial global regression approach: Turning back to the analysis of the categorical data in Table 2.
Note that in notation of the model 2. When the model 2. Also, even though. This is because, 11 and 21 in 2. But, there is a definition problem with these odds ratio parameters in this situation, because treatment here cannot represent a response variable. Now for the product multinomial model 2.
Longitudinal Data Analysis, Including Categorical Outcomes
Regression parameters Quantity 10 11 20 21 Estimate 6. Now following 2. These estimates and their corresponding standard errors computed by using 2. Other estimates can be interpreted similarly. Now by using these estimates from Table 2. These estimated probabilities along.