Binary outcome

Marshal G, Grover FL, Henderson WG, Hammermeister KE. Assessment of predictive models for binary outcomes an empirical approach using operative death from cardiac surgery. Stat Med 1994 13 1501-11. 

The primary endpoint should not be confused with a summary measure of the benefit. For example, the primary endpoint may be a binary endpoint, survival beyond two years/death within two years, while the primary evaluation is based upon a comparison of two year survival rates between two treatments. The primary endpoint is not the proportion surviving two years, it is binary outcome survival beyond two years/death within two years, the variable measured at the patient level. 

The following data (Table 3.4) are taken from Piccart-Gebhart et al. (2005) who compared trastuzumab after adjuvant chemotherapy in HER2-positive breast cancer with observation only. The binary outcome here is one or more serious adverse events (SAEs) versus no SAEs during the one year trial. The rate in the observation only group provides the background incidence of SAEs. 

Table 4.4 Active/pLacebo comparison, binary outcome survival (hypothetical)...

Consider the data presented in table 4.4 relating to the binary outcome died/survival in a parallel group trial. 

Measures such as the difference in event rates, OR, RR, RRR and NNT do not easily translate into the categorical data context. If we want to construct such measures in these cases we would collapse the outcome categories to two, the binary case, and proceed as before. In the categorical example covered earlier this could involve collapsing categories A, B and C to produce a binary outcome death/survival. 

The odds ratio, for the binary outcome (baby admitted to the special baby unit for respiratory distress) is then 11/492 divided by 24/471, giving a value 0.439. The chi-square test comparing the treatments with regard to the rates of admission gave p = 0.02. 

Another way of avoiding adjustment is to combine the multiple measurements into a single composite variable. Examples would be disease-lfee survival in oncology, where the variable is the time to either disease recurrence or death, whichever occurs first, or a composite of death, non-fatal stroke, MI and heart failure, a binary outcome in a cardiovascular setting. This approach does not require adjustment of the significance level we are back to having a single primary endpoint. 

There are some additional requirements, however, when using composite variables and these relate to the individual components, which should all be supportive of a difference in a single direction. A large positive effect with one of the components could potentially be masking a negative effect in a different component and this would be unacceptable. One way of investigating this would be to consider treatment effects in terms of the components singly. Alternatively, clinically appropriate combinations can be considered. For example, with the binary outcome, death, non-fatal stroke, MI and heart failure, one approach could be to break down the binary outcome into three separate outcome variables at the first level ... 

Many algorithms have been used effectively with DNA microarray data for predicting of a binary outcome, e g., response versus non-response. Dudoit et al. (3) compared several algorithms using several publicly available data sets. A linear discriminant is a function... 

A commonly encountered statistical distribution is the binomial distribution. This distribution deals with the behavior of binary outcomes such as the flip of a coin (heads/tails), the gender of a child (boy/girl), or the determination if a tablet has acceptable potency (pass/fail). When dealing with a sequence of independent binary outcomes, such as multiple flips of a coin or determining whether the potencies of 20 tablets are individually acceptable, the binomial distribution can be used. The probability of observing x successes in n outcomes is C x,n) p (f Binomial expansion for X = 1 to n is C Q,n)p q + +... 

This also means that the y test for binary outcomes from Section 10.5.2 can be considered a test of the null hypothesis that the population odds ratio = 1. Values of the odds ratio appropriately < 1 or appropriately > 1 are suggestive of an association between the group and the outcome. 

Sample size for binary outcomes in superiority trials... 

We have encountered a number of statistical methods used to test the difference between two population proportions. Suppose that we are interested in estimating the sample size for a superiority trial of an investigational drug (the test treatment), which will be compared with placebo with respect to a binary outcome, for example, proportion of individuals attaining a goal SBP. The null hypothesis and its complementary alternate hypothesis typically tested in such a trial are ... 

As for continuous data, a power curve can be generated for a number of scenarios for binary outcomes. As seen in Figure 12.2, the power of a test of proportions (for a fixed value of A) is quite sensitive to the particular assumed value of the response rate in the control (for example, placebo) group. 

In a confirmatory efficacy trial the study sponsor would like to evaluate a test treatment (an antihypertensive) versus placebo with respect to a binary outcome of attaining a goal SBP < 140 mmHg. After reviewing several sources of data the sponsor estimates that the placebo response will be around 0.20 (that is, 20% of individuals will attain the goal without medical therapy). The sponsor would like to estimate the sample... 

Binary outcome data, or endpoints with exactly two possible outcomes, are commonly collected during drug development. Examples of binary (also referred to as dichotomous) endpoints include cure versus lack of cure of a disease or condition with treatment, relief versus lack of relief from symptoms, eradication versus persistence of an organism, presence versus absence of a medical outcome, and appearance versus lack of appearance of an adverse event. Binary data are a subset of what are termed discrete or categorical endpoint data. Sheiner and Beal (1) have referred to such endpoints as odd-type data, in that they are noncontinuous and therefore require the use of nonstandard methodology for proper analysis and interpretation. 

The theory and techniques described in this chapter focus on the application of logistic regression to binary outcome data and the development of models to describe the relationship between binary endpoints and one or more explanatory variables (covariates). While many software options are available for fitting fixed or mixed effects logistic regression models, this chapter endeavors to illustrate the use of nonlinear mixed effects modeling to analyze binary endpoint data as implemented in the NONMEM software. 

Percentile division is a systematic approach to finding a specific value of a covariate that can split data into subgroups to maximize the probability structure in revealing explanatory variables that can be used as predictors of the response variable in a data set. The response variable could be binary, categorical, or continuous. In a data set with a binary outcome variable, for instance, the procedure would be as follows ... 

Many different tests of validation exist, few of which have a simple pass/fail . Most validation tests are sliding subjective scales where on one end the model is valid and on the other end the model is not valid . The validity of a model under a test then lies somewhere between those two extremes. But, in order for model development to proceed iteratively, modelers need a yes/no answer to the question is the model validated Hence, the outcome from a series of subjective tests that may vary from one modeler to another is transformed into a sometimes arbitrary binary outcome, a not very scientific process. 

A further practical problem in running such designs is that the intervals at which patients arise may not permit one to determine the outcome of all previous treatments when deciding the current one. A suitable modification can deal with this. More serious is the fact that a binary outcome is required and this may force an arbitrary dichotomization. In principle this too could be overcome. 

Goetghebeur E, Molenberghs G, Katz J (1998) Estimating the causal effect of compliance on binary outcome in randomized controlled trials. Statistics in Medicine 17 341-355. 

In practice there are, of course, many different formulae for sample size determination. If the trial is not a simple parallel-group trial, if there are more than two treatments, if the outcomes are not continuous (for example, binary outcomes, or length of survival... 

Table 14.1 Summary of results for a binary outcome for a trial with two treatments run in two strata showing three possible codings. Based on an example by Chuang-Stein and Tong (1996).

In the light of the type of measurement to be taken and using the likelihood under a suitable model, a statistic Z is defined which summarizes the treatment difference at inspection i. (This statistic is related to the cumulative treatment difference.) For example, for a binary outcome, the statistic would be (S - Sq) /2 where S and Sc are the successes in experimental and treatment arms, respectively. 

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