Categorical, or discrete, data are characterized by having a finite number of categories or values, whereas continuous data can take on any real value within a given range. For a categorical variable with more than two categories, we distinguish between nominal and ordered variables. Ordered variables have a natural ordering to the categories, for example, degree of pain classified as none, mild, moderate, or severe.

Sometimes, we have data that are essentially categorical, but with numerical properties, or numerical data that can take on only a small number of values. We shall refer to such data as discrete numerical [[1], p.11]. In medical research, discrete numerical data arise mostly in situations where we count the number of events per individual, such as the number of clinical visits, the number of adverse events, or the number of units of blood transfused. As a preliminary assessment of the prevalence of variables reporting the number of events, we considered all randomized, controlled trials (RCTs) published in January and February 2010 in the New England Journal of Medicine, Lancet, Journal of the American Medical Association, and BMJ. Out of a total of 52 papers, 24 (46%) papers reported at least one variable describing the number of events; 16 (31%) papers reported baseline variables, and 15 (29%) papers reported outcome variables.

Discrete numerical data are a blend between categorical and continuous data, and it is not obvious how to analyze such data. Of particular interest is how to compare two independent discrete numerical variables, a common problem in comparisons of two treatment or exposure groups. Should we analyze discrete numerical variables using methods for continuous or for ordered categorical data?

The main problem with using methods for ordered categorical data is information loss. Statistical methods for ordered categorical data do not treat the distance between values or categories as constant. Thus, we may not be getting the most out of the data. Even worse in that regard is to combine the outcomes into two categories and use methods for binary data. By doing so, we may throw away a lot of information. Poor power is often the result, and estimates may be inaccurate [2, 3].

If we intend to analyze discrete numerical data without discarding relevant information, we should consider treating the variables as if they were continuous. Continuous variables with an approximately normal distribution are best analyzed using parametric methods for confidence intervals and hypothesis tests [4, 5]. The usual alternative is a non-parametric test and a non-parametric or bootstrap confidence interval, or a transformation, for example, the logarithmic, prior to parametric methods. To decide if parametric methods are appropriate, the shapes of the underlying distributions are estimated by inspecting histograms, QQ-plots, and sample moments, or by using prior knowledge about the variable of interest. For discrete numerical variables, however, such tools may not be relevant because of the discrete nature of the underlying distributions.

In the survey of 52 RCTs published in four leading medical journals, 12 (23%) papers used statistical methods to compare discrete numerical variables between groups. All these 12 papers reported p-values, but only two papers reported effect measures and confidence intervals. Seven papers used non-parametric methods, three papers used parametric methods, one paper used negative binomial regression, one paper stated that the Cochran-Mantel-Haenszel test was used, and one paper used either the two-sample T test or the Wilcoxon-Mann-Whitney test.

The literature on statistical methods for analyzing discrete numerical variables is sparse. Newcombe [6] compares eight confidence interval methods for the mean of a single variable on the scale {0, 1, 2}, but does not consider comparisons of two independent variables. For comparing two continuous variables, on the other hand, a large body of literature exists. A relevant study for the hypothesis tests under investigation in this paper is Fagerland and Sandvik [7]. Confidence intervals for the difference in means of two independent continuous variables are considered in Zhou and Dinh [5] and Wilcox [[8], chapter 5]. Ordered categorical data is the topic of many papers, see for example Ryu and Agresti [9].

Returning to our survey, we counted nine different methods of presenting discrete numerical variables. The most common methods were to tabulate the data using categories such as {0, 1-3, 4+}, present the group means and standard deviations, or present medians and interquartile ranges (IQRs). Other methods included various combinations of means, medians, ranges, IQRs, and confidence intervals. Only two (8%) of 24 papers reported complete non-categorized data.

There is thus lack of a consensus on how to compare and present discrete numerical variables. In this paper, we shall investigate the performance of standard methods for continuous data applied to discrete numerical variables with outcomes such as {0, 1, 2, 3}. We assume that we are faced with samples from two independent random variables of equal type but with possibly different distributions. We shall further assume that we do not have a composite upper (or lower) limit, such as {0, 1, 2, 3, 4+}, where 4+ indicates outcomes with four or more events. The aim of this paper is to establish strong empirical evidence for recommending a suitable effect measure, methods for hypothesis testing and confidence intervals, and overall manner of presentation.