Devadiga Raghavendra

In a clinical trial, substantial amounts of data are recorded on each subject, such as the patient's demographic characteristics, disease related risk factors, medical history, biochemical markers, medical therapies, and outcome or endpoint data at various time points. This data can be categorical or continuous. Understanding the types of data is important as they determine which method of data analysis to use and how to report the results.

For the assessment of the effectiveness and safety of an investigational pharmaceutical entity, ideally the scale for the primary clinical endpoint should be numerically continuous to provide an accurate and reliable assessment. In practice, however, it is impossible, or can be extremely expensive, to measure responses quantitatively. On the other hand, patients' responses to treatments can be easily documented according to the occurrence of some meaningful and well-defined event such as death, infection, or cure of a certain disease and any serious adverse events. In addition the intensity of these events can be graded according to some predefined categories. Therefore categorical data can be useful surrogate endpoints for some unobserved latent continuous variables in clinical trials. Sometimes, to provide an easy analysis and/or a better presentation of the results, continuous data are transformed to categorical data with respect to some predefined criteria. As a result, many efficacy and safety endpoints in clinical trials are in the form of categorical data on either a nominal or ordinal scale.

  Different statistical methods are used to analyze this data in clinical trials.  

Categorical data arises when individuals are categorized into one of two or more mutually exclusive groups.

  Data can be divided into two main types: quantitative and qualitative. Quantitative data tends to be either continuous variables that one can measure (such as height, weight, or blood pressure) or discrete variables (such as numbers of children per family or numbers of attacks of asthma per child per month). Count data are discrete and quantitative. Qualitative data tend to be categories; people are male or female, European, American, or Japanese, and they have a disease or are in good health. These are examples of categorical data. There are four types of scales that appear in social sciences: nominal, ordinal, interval, and ratio scales. They are categorized into two groups: categorical and continuous scale data. Nominal and ordinal scales are categorical data; interval and ratio scales are continuous data. When categorical data has unordered scales it is called nominal scales. "Hair color" is a good example of the nominal scale. Categorical data that has ordered scales are called ordinal scale. Rank is an example of ordinal scale. There should be distinction between them because the data analysis method is different depending on the scale of measurement.

The different statistical methods used to analyze the categorical data will be described in detail and the limitations of each method will be discussed.  

Univariate and bivariate statistical methods 

 The chi-square test of independence is used to test the association between two categorical variables. The idea behind this test is to compare the observed frequencies with the frequencies that would be expected if the null hypothesis of no association/statistical independence were true. By assuming the variables are independent, we can also predict an expected frequency for each cell in the contingency table. If the value of the test statistic for the chi-squared test of association is too large, it indicates a poor agreement between the observed and expected frequencies and the null hypothesis of independence/no association is rejected. For example in clinical trials, it will be used to test the association between adverse event and the treatment used. The chi-square test has some assumptions: independent random sampling, no more than 20% of the cells has an expected frequency less than five, and no empty cells. If the chi-square test shows significant result, then we may be interested to see the degree or strength of association among variables, but it fails to explain another situation where more than or equal to 20% of the cells have an expected frequency less than five. In this case, the usual chi-square test is not valid. Then the Fisher Exact test will be used to test the association among variables. This methods also fails to give the strength of association among variables.  

 The investigator may be interested to see the trend in response rate among the different doses of a drug. The Cochran Armitage trend test will be used for this purpose. It tests for trends in binomial proportions across levels of a single factor or covariate. This test is appropriate for a two-way table where one variable has two levels and the other variable is ordinal. The two-level variable represents the response, and the other variable represents an explanatory variable with ordered levels.  

 In clinical trials, we often test the improvement in response rate after a particular treatment. The statistical test used for this purpose is the McNemar test. It is a test on a 2x2 classification table, used to test the difference between paired proportions (e.g., studies in which patients serve as their own control, or in studies with before and after design). Assumptions: sample member must be randomly drawn from the population, within group sample scores must be independent of each other, and no expected frequencies should be less than five. The Cochran Q test is an extension to the McNemar test for related samples that provides a method for testing the differences between three or more matched sets of frequencies or proportions.  

Kappa statistics are used for measuring agreement or association among raters. In clinical measurement, comparison of a new measurement technique with an established one is often needed to check whether they agree sufficiently for the new to replace the old. Correlation is often misleading.

The kappa coefficient (k) is used to assess inter-rater agreement. One of the most important features of the kappa statistic is that it is a measure of agreement, which naturally controls for chance. If there is complete agreement, k=1. If the observed agreement is greater than or equal to chance agreement, k=>0. If the observed agreement is less than or equal to chance agreement k=<0.  The kappa coefficient (k) can be classified as follows:  

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test for the case of two related samples or repeated measurements on a single sample. It can be used as an alternative to the paired student's t-test when the population can't be assumed to be normally distributed.  

 The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a non-parametric test for assessing whether two independent samples of observations come from the same distribution. It is one of the best-known non-parametric significance tests. It was proposed initially by Wilcoxon (1945), for equal sample sizes, and extended to arbitrary sample sizes and in other ways by Mann and Whitney (1947). It can be used as an alternative to the independent student's t-test when the population can't be assumed to be normally distributed.  

The Kruskal-Wallis test is a non-parametric method for testing equality of population medians among groups. Intuitively, it is identical to a one-way analysis of variance with the data replaced by their ranks. It is an extension of the Mann-Whitney U test to three or more groups.  

 A non-parametric test (distribution-free) used to compare observations repeated on the same subjects. This test is an alternative to the repeated measures ANOVA, when the assumption of normality or equality of variance is not met. This, like many non-parametric tests, uses the ranks of the data rather than their raw values to calculate the statistic. If there are only two measures for this test, it is equivalent to the sign test.   

. The odds ratio is the ratio of the odds of an event occurring in one group to the odds of it occurring in another group. It is used to assess the risk of a particular outcome (or disease) if a certain factor (or exposure) is present. The odds ratio is a relative measure of risk, telling us how much more likely it is that someone who is exposed to the factor under study will develop the outcome as compared to someone who is not exposed. For a 2x2 table:  

OR=1 means there is an equal chance or likelihood of getting the disease among exposed group compared to unexposed group. 

OR>1 means there is a more chance or likelihood of getting the disease among exposed group compared to unexposed group. 

OR<1 means there is a less chance or likelihood of getting the disease among exposed group compared to unexposed group. Odds ratio can be used in both retrospective and prospective studies. 

Relative risk is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group. In clinical trials, it is used to compare the risk of developing a disease in people not receiving the treatment (or receiving a placebo) versus people who are receiving the treatment. Alternatively, it is used to compare the risk of developing a side effect in people receiving a drug as compared to the people who are not receiving the treatment. For a 2x2 table:  

RR=1 means there is no difference in the risk of getting disease between two groups.

RR>1 means there is a more risk of getting disease among exposed group compared to unexposed group. RR<1 means there is a less risk of getting disease among exposed group compared to unexposed group. It can be used only in prospective studies. For the rare disease, the RR will be approximately equal to OR. 

 The probability of the test finding disease among those who have the disease or the proportion of people with disease that are positive with the test. 

The probability of the test finding no disease among those who do not have the disease or the proportion of people free of a disease who have a negative test.  

. The percentage of people with a positive test result who actually have the disease.  

The percentage of people with a negative test who do not have the disease.  

Regression methods used in clinical trials 

A form of regression, which is used when the dependent variable is dichotomous that is, the dependent variable can take the value 1 with a probability of success θ, or the value 0 with probability of failure 1-θ. 

The independent or predictor variables in logistic regression can take any form. That is, logistic regression makes no assumption about the distribution of the independent variables. They do not have to be normally distributed, linearly related, or of equal variance within each group. The relationship between the predictor and response variables is not a linear function in logistic regression; instead, the logistic regression function is used, which is the logit transformation of q.  

Where α = the constant of the equation and,  β = the coefficient of the predictor variables.  

Hosmer and Lemeshow chi-square test of goodness of fit. If the p value of H-L goodness of fit test is greater than 0.05, we fail to reject the null hypothesis that there is no difference between observed and model predicted values, implying that the model's estimates fit the data at an acceptable level.

Omnibus tests of model coefficients. It tests if the model with the predictors is significantly different from the model with only the intercept. The omnibus test may be interpreted as a test of the capability of all predictors in the model jointly to predict the response variable. A finding of significance corresponds to a research conclusion that there is adequate fit of the data to the model, meaning that at least one of the predictors is significantly related to the response variable.

Lowest AIC (Akaike Information Criterion). AIC is commonly used to compare models. The lower AIC is the better model.

Lowest BIC (Bayesian Information Criterion). BIC is also used to compare models; lower BIC is the better model.

. It is used to investigate the relationship between an outcome and a set of prognostic factors in matched case-control studies. The outcome is whether the subject is a case or a control. If there is only one case and one control, the matching is 1:1.  

An extension of the binary logistic regression, it is used when the dependent variable has more than two nominal (unordered) categories. In multinomial logistic regression the dependent variable is dummy coded into multiple 1/0 variables. There is a variable for all categories but one, so if there are M categories, there will be M-1 dummy variables. All but one category has its own dummy variable. Each category's dummy variable has a value of 1 for its category and a 0 for all others. One category, the reference category, doesn't need its own dummy variable, as it is uniquely identified by all the other variables being 0. The multinomial logistic regression then estimates a separate binary logistic regression model for each of those dummy variables. The result is M-1 binary logistic regression models. Each one tells the effect of the predictors on the probability of success in that category, in comparison to the reference category. Each model has its own intercept and regression coefficients—the predictors can affect each category differently.   

It is a form of regression analysis used to model the count data. It assumes the response variable y has a poisson distribution and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. The assumptions include:

  logarithm of the disease rate changes linearly with equal increment increases in the exposure variable; changes in the rate from combined effects of different exposures or risk factors are multiplicative; at each level of the covariates the number of cases has variance equal to the mean; observations are independent. 

A plot of observed versus predicted values.

Deviance residuals versus the predicted values.

Global goodness of fit statistics of the null hypothesis, model fits; alternative hypothesis, doesn't fit can be found by using the Pearson chi-squared and deviance test statistics given in the SAS Proc GENMOD output. Large values of these statistics, and small p-values imply evidence that the model does not fit the observed data. 

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