### Statistics Applied to Clinical Trials

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While Averages are can be examined when comparing intervally scaled characteristics: e. They are at the core of clinically relevant indices of prevalence and incidence, and of the evaluation of the sensitivity and specificity of diagnostic findings as evidence of particular conditions. In most general terms, a form of descriptive statistical analysis which is valid for simpler mathematical scalings can be used with mathematically more complex scalings.

For example, the category containing the highest proportion of a nominal variable is termed the mode. The mode is a valid analysis of nominally scaled variables. We can count the number of patients assigned each ICD-9 coded diagnosis. We can then compare these counts to evaluate which diagnosis was most frequent that is, the mode.

### The Annals of Applied Statistics

The mode can also be used to describe variables that are ordinally scaled - e. In contrast, statistics designed specifically for more complex scalings may be invalid for measurements using mathematically simpler scalings. The situation is even more clouded with nominally scaled variables.

The numbers used as codes in the ICD-9 carry no direct implication of magnitude.

It is not meaningful to say that the diagnosis of reticulosarcoma is twice the diagnosis of leptospirosis icterohemorrhagica because reticulosarcoma's ICD-9 code of Let us now turn our attention to the associations among variables, first paying attention to how we describe that association. The strength of the association between two variables is described by correlation coefficients. The square of the correlation coefficient can be interpreted as the proportion of the variance of one variable that is predicted by the other variable.

The most frequently used correlation coefficients are phi and Cramer's V for nominal variables, Spearman's rho or rank-order correlation for ordinal variables, and Pearson's r or product-moment correlation for interval variables.

## Applied Statistics in Clinical Trials I

Kappa is also often used for binomial nominal variables. Binomial variables are nominal variables with only two values: e. Kappa adjusts in its calculation for the agreement expected by chance alone. Note that agreement does not imply accuracy. Accuracy, assessed for binary classifications by sensitivity, specificity, and receiver operating characteristic curves, will not be discussed further in this paper.

For all but nominal variables, the sign of the correlation coefficient indicates the direction of the association. Positive correlation coefficients describe situations in which increases in value of one of the variables are associated with increases in the other variable, while negative coefficients describe situations in which increases in one of the variables are associated with decreases in the other.

Correlation-based analyses using techniques such as factor analysis can be used to examine the associations among multiple measures used to investigate single events or conditions. The discussion so far has carried the implicit assumption that we are able to measure the entire course of the events we are studying. That may be true for many of the acute clinical events and processes in which cardiac anesthesiology plays a major role.

However, this is clearly true neither for all long term processes in cardiac anesthesia nor for those iatrogenic effects whose appearance is delayed, nor for cardiology, nor for clinical processes generally. Clinical and research data are often gathered within a limited time frame while the processes to which clinical attention is being given, and those which are being studied continue beyond that time frame's boundaries.

Life-table analyses typically examine median time to the target event to avoid being biased by the long times to event of those in the sample who have not experienced the event by the time the study concludes and whose experience is right-censored. Appropriate evaluation of statistical significance also uses techniques discussed below which take this right-censorship into account.

It is important that studies whose samples are right-censored use such life-table based techniques. Studies in that situation that calculate survival time by averaging time to the terminal events which have occurred will produce biased estimates unless all of those terminal events have occurred because right-censorship will be excluding those with the potentially longest survival times. Correlations measure the strength and, for all types except nominal variables, the direction of associations between variables.

Regression modeling provides the tools for making those predictions from one or more independent variables to the dependent variable.

## 1.1 - What is the role of statistics in clinical research?

If the dependent variable is a binomial, that is, a nominal variable with only two values, and it is known whether or not each member of the sample experienced that outcome, multiple logistic regression is used to model the effects of the independent variables on the odds ratio of experiencing that outcome. This model is appropriate for outcomes in, say, a study of surgical intervention in which the outcome of interest is short term and can be predicted to have occurred before discharge from hospital.

In contrast, the Cox proportional hazards model and regression is are used when the outcome data are right censored. This is likely to be the case, for example, if the study is investigating delayed effects after therapeutic interventions such as postsurgical survival in cancer patients. Cox regression models the risk of the target outcome as a hazard function which is a function of time and of the independent variables included in the model. The final principal form of regression modeling is multiple linear regression, which predicts a dependent variable measured on an interval scale based on the values of one or more predictors.

For example, linear regression can be used to model the association of the natural log of urea with age[ 30 ] taking the natural log of urea made the relationship of urea with age a straight line. Linear regressions predict straight lines or planes or their multi-dimensional analogs. There are constraints on the type of distribution and on the associations among variables suitable for linear regression analysis. Discussion of the regression modeling of these processes and of their associations is beyond the scope of this paper. Clinical decisions and research need to move beyond the initial sample of measurements of say the initial patient or group of patients to reach more generalized conclusions.

Say a change is noted in laboratory measurement following an operative procedure. How likely is it that other patients undergoing that procedure will experience the same change?

## Statistics in clinical research: Important considerations

Is that change other than the difference that would be seen in patients with the same clinical condition who are measured twice, but who do not undergo that procedure? What is the range of change in that laboratory measurement, which can be expected in future patients who do and who do not undergo that procedure? These questions explore the extent to which we can generalize from our particular clinical observations and the trustworthiness of those generalizations.

These questions are in the arena of statistical inference.

There have been many presentations of the general logic underlying statistical inference cf. The reader is referred to those sources, and to any classical statistics or biostatistics text for the logic underlying classical tests of statistical significance. We will now first discuss alternatives to the point comparison represented by classical significance testing.

Then, given the widespread use of classical significance testing, we will discuss several modifications necessary for its appropriate use in clinical studies. Classical tests of significance assess the likelihood of the study's actual results given a set of assumptions about the sources of the measures being compared.

The tests are designed to support a point judgment about the likelihood of those source groups being identical. The statistical significance test result evaluates the likelihood of the results obtained were the data drawn from identical groups, saying nothing about the magnitude or stability of any differences that were actually found. There is a long-standing argument that analyses should estimate the range of inter-group differences consistent with the collected data rather than ending with a single statement regarding statistical significance.

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These techniques initially presented by Thomas Bayes — treat probability as a statement of degree of belief in a statement rather than as an estimate of the frequency. In clinical practice, Bayesian techniques are used to calculate the predictive value positive of a diagnostic finding given prior beliefs about the finding's sensitivity and specificity and about the prevalence of the diseases being considered.

This is in contrast to classical tests of statistical significance and calculations of confidence intervals that are based only on the sets of actual measurements and assumptions about the underlying population distributions.

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The above paragraph noted that confidence intervals can be used to evaluate a likelihood of inter-group differences. These confidence intervals estimating the magnitude of the inter-group difference go beyond traditional point computations of statistical significance which only refer to the likelihood of the particular difference tested to estimate the magnitude of the inter-group difference.

They also estimate the expected stability of associations between variables. Please note that confidence intervals can also be calculated around other statistics, ranging from the proportions and means calculated as descriptive statistics through correlation coefficients to regression coefficients. In each case, the confidence interval predicts the stability of the point statistic calculated using a defined sample. While there is serious discussion about alternatives to classical tests of statistical significance as evaluations of the generalizability of findings as noted above, these classical tests continue to be widely used.

The first issue is whether or not the measures being compared are independent. Any difference or any difference in a specified direction is potentially of interest when comparing independent samples. The sets of measurements in repeat measurement of the same subjects are obviously related, with the second measurements being departures from first measurements that are already in the sample study.