Applied Biostatistics for the Health Sciences. Richard J. RossiЧитать онлайн книгу.

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,

z Subscript male Baseline equals StartFraction 215 minus 180 Over 30 EndFraction equals 1.17

and

z Subscript female Baseline equals StartFraction 170 minus 145 Over 15 EndFraction equals 1.67

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

upper P left-parenthesis upper A Math bar pipe bar symblom upper B right-parenthesis equals StartFraction upper P left-parenthesis upper A intersection upper B right-parenthesis Over upper P left-parenthesis upper B right-parenthesis EndFraction

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

mu equals StartFraction sum of the values of upper X Over upper N EndFraction equals StartFraction sigma-summation upper X Over upper N EndFraction

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.

upper O upper R equals StartFraction o d d s left-parenthesis DiseaseMath bar pipe bar symblomRisk Factor Present right-parenthesis Over o d d s left-parenthesis DiseaseMath bar pipe bar symblomRisk Factor Absent right-parenthesis EndFraction

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).

upper R upper R equals StartFraction upper P left-parenthesis DiseaseMath bar pipe bar symblomRisk Factor Present right-parenthesis Over upper P left-parenthesis DiseaseMath bar pipe bar symblomRisk Factor Absent right-parenthesis EndFraction

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

sigma squared equals StartFraction sum of all left-parenthesis deviations from mu right-parenthesis squared Over upper N EndFraction equals StartFraction sigma-summation Overscript zero width space Endscripts left-parenthesis upper X minus mu right-parenthesis squared Over upper N EndFraction

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.

upper Z score equals StartFraction upper X minus mu Over sigma EndFraction

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

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