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220905P - FROM QUESTIONS TO VARIABLES and STATISTICAL DISTRIBUTIONS

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Presented at a Webinar on Research Methodology in Health Sciences at Northern Area Armed Forces Hospital (NAAFH) on 5th September 2022. By Professor Omar Hasan Kasule Sr. MB ChB (MUK), MPH (Harvard), DrPH (Harvard)


TYPES OF QUESTIONS

  • Substantive question
  • Statistical question
  • Statistical conclusion
  • Substantive conclusion


TYPES OF VARIABLES

  • Independent variables = determinants = causes
  • Dependent variables = outcomes
  • Confounding variables


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 random variable can only be expressed as whole numbers.


VARIABLES and STATISTICAL DISTRIBUTIONS

  • Statistical distributions are graphical representations of mathematical functions of random variables.
  • Each random variable has a corresponding statistical distribution
  • Each statistical distribution is associated with a specific statistical analytic technique.
  • The choice of the technique of statistical analysis depends on the type of statistical distribution. The statistical distribution depends on the type of random variable.


QUALITATIVE (NON-NUMERICAL) RANDOM VARIABLES

  • Qualitative variables (nominal, ordinal, and ranked) are attribute or categorical with no intrinsic numerical value.
  • The nominal has no order.
  • The ordinal has to order.
  • 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 hypergeometric, and the uniform. Each random variable is associated with a statistical distribution.
  • The Binomial is the commonest. is the number of successes in more than 2 consecutive trials each with a dichotomous outcome?


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 a continuous numerical scale such as height and weight.


QUANTITATIVE (NUMERICAL) CONTINUOUS RANDOM VARIABLES, con’t.

  • The Exponential is the time until the first occurrence of the event of interest. [Line graph of an exponential distribution]
  • The uniform represents the results of a measurement and takes on the same value at repeated trials.


SCALES OF QUANTITATIVE (NUMERICAL) CONTINUOUS RANDOM VARIABLES


  • 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 the measurements are on the ratio scale.
  • Properties of the interval scale: Zero is arbitrary with no biological meaning, and both negative and positive values are allowed.
  • Properties of the ratio scale: Zero has a biological significance, values can only be positive.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS)

  • The expectation of a random variable is a central value around which it hovers most of the time for example the mean (average).
  • The variations of the random variable around the expectation are measured by its variance.
  • Covariance measures the co-variability of the two random variables.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 1


  • Correlation measures the linear relation between two random variables.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 2


  • Skewness measures the bias of the distribution of the random variable from the center.


SIX PROPERTIES OF A RANDOM VARIABLE (DISCRETE OR CONTINUOUS), cont’. - 3


  • Kurtosis measures the peakedness of the random variable at the point of its expectation.


TRANSFORMATIONS OF RANDOM VARIABLES

  • Quantitative variables can be transformed into qualitative ones.
  • Qualitative variables can be transformed into quantitative ones but this is less desirable.
  • The continuous variable can be transformed into a discrete variable.
  • Transformation of the discrete into the continuous may be misleading.