Binary, categorical and ordinal data

Again let z = 0 for patients in the control group and z = 1 for patients in the test treatment group and assume that we have several covariates, say Xj, and X3. The main effects model looks at the dependence of pr(y= 1) on treatment and the covariates  

The coefficient c measures the impact that treatment has on pr(y= 1). If c = 0 then pr(y = I) is unaffected by which treatment group the patients are in there is no treatment effect. Having fitted this model to the data and in particular obtained an estimate of c and its standard error then we can test the hypothesis Hq c = 0 in the usual way through the signal-to-noise ratio. 

The quantity c is very closely related to the odds ratio in fact c is the log of the OR, adjusted for the covariates. The anti-log of c (given by e ) gives the adjusted OR. Confidence intervals in relation to this OR can be constructed initially by obtaining a confidence interval for c itself and then taking the anti-log of the lower and upper confidence limits for c. 

Example 6.1 Effect of betamethasone on incidence of neonatal respiratory distress 

This randomised trial (Stutchfield et al. (2005)) investigated the effect of betamethasone on the incidence of neonatal respiratory distress after elective caesarean section. Of the 503 women randomised to the active treatment, 11 babies were subsequently admitted to the special baby unit with respiratory distress compared to 24 babies out of 495 women randomised to the control group. 

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As we shall see later the data type to a large extent determines the class of statistical tests that we undertake. Commonly for continuous data we use the t-tests and their extensions analysis of variance and analysis of covariance. For binary, categorical and ordinal data we use the class of chi-square tests (Pearson chi-square for categorical data and the Mantel-Haenszel chi-square for ordinal data) and their extension, logistic regression. 

In this section we will discuss the extension of the t-tests for continuous data and the chi-square tests for binary, categorical and ordinal data to deal with more than two treatment arms. 

For binary, categorical and ordinal data there is also an approach which is a further form of the Mantel-Haenszel chi-square test. You will recall that the MH test is used for ordinal responses comparing two treatments. Well, this procedure generalises to allow ordering across the treatment groups in addition, for each of... 

We saw in the previous chapter how to account for centre in treatment comparisons using two-way ANOVA for continuous data and the CMH test for binary, categorical and ordinal data. These are examples of so-called adjusted analyses, we have adjusted for centre differences in the analysis. 

For binary data in multi-centre trials we will have a series of 2 x 2 tables, one for each of the centres. For categorical and ordinal data with c categories, we will have a series of 2 x c tables. The CMH test in the first instance provides a single p-value for the main effect of treatment. 

The simplest form of qualitative data is binary data in which there are only two possible values, for example, death/survival or success/failure each of which needs to be defined within a specified time interval has pain relief been achieved within two hours of treatment, success - or not, failure. This form of data is extremely common in medical research and yet it ignores the possibility of gradation, success may not be total but only partial and yet not be total failure. These considerations lead naturally to the concept of ordered categorical or ordinal data. 

As with binary and categorical data, is there an issue with small sample sizes Well, in fact, no, there is not. The MH test is a different kind of chi-square test and is not built around expected frequencies. As a consequence it is not affected by small expected frequencies and can be used in all cases for ordinal data. There are some pathological cases where it will break down but these should not concern us in practical settings. 

Before delving into the specific similarity calculation, we start our discussion with the characteristics of attributes in multidimensional data objects. The attributes can be quantitative or qualitative, continuous or binary, nominal or ordinal, which determines the corresponding similarity calculation (Xu and Wunsch, 2005). Typically, distance-based similarity measures are used to measure continuous features, while matching-based similarity measures are more suitable for categorical variables. 

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