Preventive Medicine: Biostatistics 2

Sources of Variation in Medicine

• Variation in clinical medicine can arise from:
  • Biologic differences (e.g., genetic differences, nutrition, environmental exposures, age, gender, race)
  • Presence or absence of disease (e.g., cancer, comorbidity)
  • Techniques and conditions of measurement (e.g., time of day, ambient temperature, fatigue or anxiety in the patient)
  • Errors in measurement (e.g., defective blood pressure cuffs, differences in laboratory instruments)
  • Random variation (e.g., slight inaccuracies in obtaining measurements)

Types of Variables

• Nominal variables:
  • Categorical variables with no measurement scale and no rank order (e.g., blood groups, occupations, food groups, skin color)
  • Examples: blood groups (O, A, B, AB), occupations
• Dichotomous (Binary) variables:
  • Variables with only two levels (e.g., normal/abnormal skin color, well/sick, living/dead)
  • Often inadequate by themselves to fully describe a phenomenon
• Ordinal (Ranked) variables:
  • Variables with three or more qualitative values that have a clearly implied direction from better to worse (e.g., satisfaction with care, amount of swelling in a patient's legs)
  • Examples: respiratory distress (absent, mild, moderate, or severe), pain intensity (0-10 scale)
• Continuous (Dimensional) variables:
  • Variables measured on a continuous scale (e.g., patients' heights, weights, systolic and diastolic blood pressures, serum glucose levels)
  • Examples: birth weight and probability of survival of newborns
• Ratio variables:
  • Continuous scales with a true 0 point (e.g., Kelvin temperature scale)
• Risks and proportions as variables:
  • Conditional probability of an event (e.g., death or disease) in a defined population in a defined period
  • Share characteristics of both discrete and continuous variables

Statistics and Variables

• Statistics help clinicians and investigators understand and explain variation in medical data
• The first step in understanding variation is to describe it
• Types of variables are important to understand in order to choose the correct statistical tools for analysis

Properties of the Mean

• The mean of a random sample is an unbiased estimator of the population mean.
• The mean is the mathematical expectation and is different from the mode, which is the value observed most often.
• The sum of squared deviations from the mean is smaller than the sum of squared deviations from any other number.
• The sum of squared deviations from the mean is fixed for a given set of observations.

Measures of Dispersion Based on Percentiles

• Percentiles are the percentage of observations below a certain point in a ranked dataset.
• The median is the 50th percentile, the 75th percentile is the point below which 75% of observations lie, and the 25th percentile is the point below which 25% of observations lie.
• The interquartile range (IQR) is the distance between the 75th and 25th percentiles.

Measures of Dispersion Based on the Mean

• Mean deviation is the average of the absolute values of the deviations from the mean, but it's not widely used.
• Variance is the sum of squared deviations from the mean divided by the number of observations minus one.
• Standard deviation is the square root of variance and is a measure of spread.

Properties of Variance

• The variance of a random sample is an unbiased estimator of the population variance.
• The variance of the sum of two independently sampled variables is equal to the sum of their variances.
• The variance of the difference between two independently sampled variables is equal to the sum of their variances.

Standard Deviation

• The standard deviation is a measure of spread that is easy to interpret.
• One standard deviation above and below the mean includes 68% of the data, and two standard deviations above and below the mean include 95.4% of the data.

Problems in Analyzing a Frequency Distribution

• Skewness occurs when the data is not symmetrical and the mean is not equal to the median.
• Kurtosis occurs when the data is more or less peaked than a normal distribution.
• Extreme values (outliers) can affect the mean and should be checked for plausibility.

Methods of Depicting a Frequency Distribution

• Histograms and line graphs are commonly used to display frequency distributions.
• Stem and leaf diagrams, quantiles, and boxplots are other methods of visualizing data.

Unit-Free (Normalized) Data

• Data can be normalized by subtracting the mean and dividing by the standard deviation to create z-values.
• Z-values have a mean of 0 and a standard deviation of 1.
• Normalizing data eliminates the effect of units of measurement.

Frequency Distributions of Dichotomous Data and Proportions

• Dichotomous data can be analyzed using the binomial distribution.
• The binomial distribution approaches the normal distribution as the number of trials increases.

Frequency Distributions of Other Types of Data

• Nominal and ordinal data require nonparametric tests that do not assume a specific distribution.

• The chi-square distribution is used to analyze counts in frequency tables.

• Ordinal data can be analyzed using nonparametric tests such as the Wilcoxon test and the Mann-Whitney U test.### Understanding Variation in Clinical Medicine

• Variation in clinical medicine can be caused by biologic differences, presence or absence of disease, differences in measurement techniques, errors in measurement, and random variation.

• Statistics helps in describing and understanding variation, but it cannot correct for measurement errors or bias.

Types of Variables or Data

• Nominal data: categorical data with no inherent order or ranking
• Dichotomous data: binary data with two distinct categories
• Ordinal data: categorical data with inherent order or ranking
• Continuous data: measured data with a range of values
• Ratio data: continuous data with a true zero point
• Risks, rates, and proportions: special types of data used to describe events or outcomes

Continuous Data and Frequency Distribution

• Frequency distribution can be described by two parameters: measure of central tendency and measure of dispersion
• Measure of central tendency: median and mean are the most important
• Measure of dispersion: variance and standard deviation are the most important
• Normal (Gaussian) distribution: a common distribution type where mean and median coincide, and 95% of observations are within 1.96 standard deviations above and below the mean

Skewed Distribution and Normalization

• Skewed distribution: a distribution where the mean is farther in the direction of the long tail than the median
• Normalization: making data unit-free by creating z-values, which express the value of each observation as the number of standard deviations above or below the mean

Probability Distributions

• Binomial distribution: describes probability of dichotomous data, where the probability of success and failure are 0.5 each, and the number of trials is large
• Poisson distribution: used to describe rare events, and has the convenient property that the mean equals the variance

Non-Parametric Statistics

• Used when the normal distribution cannot be assumed, to study differences and associations among variables