130501P - VARIABLES

Presentation at a Training Program on Biostatistics for physician managers working in Public Health Administration, Qassim Province on May 1, 2013 by Professor Omar Hasan Kasule Sr MB ChB (MUK), MPH (Harvard), DrPH (Harvard). EM: omarkasule@yahoo.com

Constants and variables

A constant has only one unvarying value under all circumstances for example p and c = speed of light. A random variable can be qualitative (descriptive with no intrinsic numerical value) or quantitative (with intrinsic numerical value). A random quantitative variable results when numerical values are assigned to results of measurement or counting. It is called a discrete random variable if the assignment is based on counting. It is called a continuous random variable if the numerical assignment is based on measurement. The numerical continuous random variable can be expressed as fractions and decimals. The numerical discrete can only be expressed as whole numbers. Choice of the technique of statistical analysis depends on the type of variable.

Qualitative random variables

Qualitative variables (nominal, ordinal, and ranked) are attribute or categorical with no intrinsic numerical value. The nominal has no ordering, the ordinal has ordering, and the ranked has observations arrayed in ascending or descending orders of magnitude.

Quantitative (numerical) discrete random variables

The discrete random variables are the Bernoulli, the binomial, the multinomial, the negative binomial, the Poisson, the geometric, the hyper geometric, and the uniform. You are not expected to remember the definitions below but reading through makes the names of various variables encountered in the medical literature seem familiar.

The Bernoulli is the number of successes in a single unrepeated trial with only 2 outcomes. The binomial is the number of successes in more than 2 consecutive trials each with a dichotomous outcome. The multinomial is the number of successes in several independent trials with each trial having more than 2 outcomes. The negative binomial is the total number of repeated trials until a given number of successes is achieved. The Poisson is the number of events for which no upper limit can be assigned a priori. The geometric is the number of trials until the first success is achieved. The hyper geometric is the number selected from a sub-sample of a larger sample for example selecting males from a sample of n persons from a population N. The uniform has the same value at repeated trials.

Quantitative (numerical) continuous random variables

The continuous random variables can be natural such as the normal, the exponential, and the uniform or artificial such as chi square, t, and F variables. The normal represents the result of a measurement on the continuous numerical scale such as height and weight. The exponential is the time until the first occurrence of the event of interest. The uniform represents results of a measurement and takes on the same value at repeated trials.

The continuous R.V can be measured on either the interval or the ratio scales. Only 2 measurements are made on the interval scale, the calendar and the thermometer. The rest of measurements are on the ratio scale. The interval scale has the following properties: the difference between 2 readings has a meaning, the magnitude of the difference between 2 readings is the same at all parts of the scale, the ratio of 2 readings has no meaning, zero is arbitrary with no biological meaning, and both negative and positive values are allowed. The ratio scale zero has the following properties: zero has a biological significance, values can only be positive; the difference between 2 readings has a meaning, the ratio of 2 readings has a meaning and can be interpreted, and intervals between 2 readings have the same meaning at different parts of the scale.

Random variables: properties and mathematical operations

A random variable has 6 properties described below. You are not expected to remember these at this stage but familiarity with the terminology will make reading scientific literature so much easier.

The expectation of a random variable is a central value around which it hovers most of the time. The variations of the random variable around the expectation are measured by its variance. Covariance measures the co-variability of the two random variables. Correlation measures the linear relation between two random variables. Skew ness measures the bias of the distribution of the random variable from the center. Kurtosis measures the peaked ness of the random variable is at the point of its expectation.

Statistical distributions are graphical representation of mathematical functions of random variables. Each random variable mentioned above has a corresponding statistical distribution that specifies all possible values of a variable with the corresponding probability. Each statistical distribution is associated with a specific statistical analytic technique.