Applied Biostatistics for the Health Sciences. Richard J. Rossi
scores are unitless measures of relative standing and provide a meaningful measure of relative standing only for mound-shaped distributions. Furthermore, Z scores can be used to compare the relative standing of individuals in two mound-shaped distributions.
Example 2.41
The weights of men and women both follow mound-shaped distributions with different means and standard deviations. In fact, the weight of a male adult in the United States is approximately normal with mean µ = 180 and standard deviation σ = 30, and the weight of a female adult in the United States is approximately normal with mean µ = 145 and standard deviation σ = 15. Given a male weighing 215 lb and a female weighing 170 lb, which individual weighs more relative to their respective population?
The answer to this question can be found by computing the Z scores associated with each of these weights to measure their relative standing. In this case,
and
Since the female’s weight is 1.67 standard deviations from the mean weight of a female and the male’s weight is 1.17 standard deviations from the mean weight of a male, relative to their respective populations a female weighing 170 lb is heavier than a male weighing 215 lb.
Glossary
Absolute RiskThe absolute risk of a condition or disease is the probability that an individual develops the condition or disease.Binomial Probability ModelThe binomial probability model is a probability model for a discrete random variable that counts the number of successes in n independent trials of a chance experiment having only two possible outcomes.Chance ExperimentA task where the outcome cannot be predetermined is called a random experiment or a chance experiment.Conditional ProbabilityThe conditional probability of the event A given that the event B has occurred is denoted by P(A|B) and is defined as
Continuous VariableA quantitative variable is a continuous variable when the variable can take on any value in one or more intervals.Discrete VariableA quantitative variable is a discrete variable when there are either a finite or a countable number of possible values for the variable.DistributionThe distribution of a variable explicitly describes how the values of the variable are distributed in terms of percentages.EventAn event is a subcollection of the outcomes in the sample space is associated with a chance experiment.Explanatory VariableAn explanatory variable is a variable that is believed to cause changes in the response variable.Independent EventsTwo events A and B are independent when P(A|B)=P(A) or P(B|A)=P(B).Interquartile RangeThe Interquartile range of a population is the distance between the 25th and 75th percentiles and will be denoted by IQR.MeanThe mean of a variable X measured on a population consisting of N units is
MedianThe median of a population is the 50th percentile of the possible values of the variable X and will be denoted by μ~.ModeThe mode of a population is the most frequent value of the variable X in the population and will be denoted by M.Multivariate VariableA collection of variables that will be measured on each unit is called a multivariate variable.Negative Predictive ValueIn a diagnostic test, the negative predictive value (NPV) is the probability of a correct negative test result, P(−|not D).Nominal VariableA qualitative variable is called a nominal variable when the values of the variable have no intrinsic ordering.Non-standard NormalA non-standard normal is any normal distribution that does not have a standard normal distribution (i.e., either μ≠ or σ≠1).OddsThe odds of an event A is odds(A)=P(A)1−P(A).Odds RatioThe odds ratio for a disease is the ratio of the odds of the disease when the risk factor is present to the odds when the risk factor is absent.
Ordinal VariableA qualitative variable is called an ordinal variable when there is a natural ordering of the possible values of the variable.ParameterA parameter is a numerical value that summarizes a particular characteristic of the population.PercentileThe pth percentile of a quantitative variable is the value in the population where p percent of the population falls below this value. The pth percentile is denoted by xp for values of p between 0 and 100.Positive Predictive ValueIn a diagnostic test, the positive predictive value (PPV) is the probability of a correct positive test result, P(+|D).ProbabilityA probability is a number between 0 and 1 that measures how likely it is for an event to occur.Qualitative VariableA variable that takes on non-numeric values is called a qualitative variable.Quantitative VariableA variable that takes on only numeric values is called a quantitative variable.Relative RiskThe relative risk is of a condition or disease for a dichotomous risk factor is the ratio of the probabilities P(Disease|Risk Factor Present) and P(Disease|Risk Factor Absent).
Response VariableThe response variable is the outcome variable of primary interest to a researcher.Sample SpaceThe set of all possible outcomes of a chance experiment is called the sample space and is denoted by S.SensitivityThe sensitivity is the conditional probability of a positive test for the subpopulation of individuals having the disease (i.e., P(+|D)).SpecificityThe specificity is the conditional probability of a negative test for the subpopulation of individuals who do not have the disease (i.e., P(−|not D)).Standard DeviationThe standard deviation of a population is defined to be the square root of the variance and will be denoted by σ.Standard NormalThe standard normal is a normal distribution with mean 0 and standard deviation 1.VarianceThe variance of a variable X measured on a population consisting of N units is
Z ScoreA Z score is a measure of relative position within a distribution and measures how many standard deviations a point is above or below the mean.
Exercises
1 2.1 What is the difference between a qualitative and a quantitative variable?
2 2.2 What is the difference between a discrete and a continuous variable?
3 2.3 What is the difference between a nominal and an ordinal variable?
4 2.4 Determine whether each of the following variables is a qualitative or quantitative variable.AgeSystolic blood pressureRaceGenderPain