Lecture 4 - Biostatistics

What is biostatistics divided into?

Descriptive statistics and inferential statistics.

What are descriptive statistics?

Statistics used to summarize and describe the main features of a dataset, such as the mean, median, and standard deviation.

What are inferential statistics?

Statistics that use data from a sample to draw conclusions about a larger population, including hypothesis testing and p-values.

What is a population?

The entire group of individuals or observations of interest.

What is a sample?

A subset of the population that is included in a study.

What is a sample statistic?

A numerical summary calculated from a sample.

What is a population parameter?

A numerical summary describing an entire population.

Relationship between a population, sample, sample statistic, and population parameter

What is a variable?

An attribute or property that differs across people or objects.

What are the two major types of variables?

Quantitative and qualitative variables.

What is a quantitative variable?

A numerical variable representing an amount or quantity.

What are the two types of quantitative variables?

Continuous and discrete variables.

What is a continuous variable?

A quantitative variable that can take any value along a continuum within a defined range, such as height.

What is a discrete variable?

A quantitative variable that can take only separate, countable values, such as the number of patients seen.

What is a qualitative variable?

A categorical variable that places individuals or objects into distinct groups or categories.

What are examples of qualitative variables?

Contact lens type, glaucoma status, and blood type.

What is an independent variable?

The factor that is changed, selected, or used to explain or predict differences in an outcome.

What is a dependent variable?

The outcome or response that is measured and may be affected by the independent variable.

How can independent and dependent variables be distinguished conceptually?

The independent variable represents the cause or predictor, while the dependent variable represents the outcome or effect.

If a study examines whether the type of glaucoma medication affects intraocular pressure, what is the independent variable?

The type of glaucoma medication.

If a study examines whether the type of glaucoma medication affects intraocular pressure, what is the dependent variable?

Intraocular pressure.

What are the four scales of measurement?

Nominal, ordinal, interval, and ratio.

What four properties distinguish the scales of measurement?

Unique identity, rank ordering, equal intervals, and an absolute zero.

Four scales of measurement arranged from nominal through ratio

What is a nominal scale?

The lowest level of measurement, consisting of distinct categories with no quantitative value or inherent order.

What are examples of nominal variables?

Alive versus dead, ill versus well, vaccinated versus unvaccinated, or county of residence.

What is a dichotomous variable?

A nominal variable with two mutually exclusive categories, such as yes/no or present/absent.

What is an ordinal scale?

A scale in which categories can be ranked, but the differences between categories are not necessarily equal.

Why is cancer stage an ordinal variable?

The stages can be ranked from Stage 1 through Stage 4, but the difference between consecutive stages cannot be assumed to be equal.

What is an interval scale?

A scale with equal intervals between values but no true or absolute zero.

What are examples of interval-scale variables?

Temperature measured in Celsius or Fahrenheit and calendar years.

What is a ratio scale?

The highest level of measurement, with equal intervals and a true zero representing complete absence of the measured quantity.

What are examples of ratio-scale variables?

Height, weight, age, intraocular pressure, and number of patients.

What is the major difference between interval and ratio scales?

Both have equal intervals, but only a ratio scale has a true absolute zero.

What three aspects of data are summarized by descriptive statistics?

Distribution, central tendency, and variability.

What is a limitation of descriptive statistics?

They summarize observed data but cannot by themselves determine whether sample findings apply to the larger population.

What is a normal distribution?

A symmetric, bell-shaped distribution in which most scores are concentrated around the mean.

What is the relationship among the mean, median, and mode in a perfectly normal distribution?

Mean = median = mode.

Symmetric normal distribution centered on the mean

What is a skewed distribution?

A nonsymmetric distribution with a longer tail extending toward one side.

How is the direction of skewness determined?

Skewness is named for the direction of the distribution's long tail.

What is a positively skewed distribution?

A distribution with a long tail extending to the right.

What is a negatively skewed distribution?

A distribution with a long tail extending to the left.

Positive, normal, and negative distribution shapes distinguished by tail direction

What are the three measures of central tendency?

Mean, median, and mode.

What is the arithmetic mean?

The sum of all observed values divided by the number of observations.

What is the formula for the mean?

Mean = sum of all observations ÷ number of observations.

Population and sample mean formulas

How do you calculate the mean?

Add all observed values and divide the sum by the number of observations.

What is the mean of the values 27, 31, 15, 30, and 22?

The sum is 125, so the mean is 125 ÷ 5 = 25.

When is the mean generally the most appropriate measure of central tendency?

When data are approximately symmetric and normally distributed.

How do outliers affect the mean?

Extreme values can substantially pull the mean toward them because the mean includes every observation.

Extreme IOP values demonstrating how outliers increase the mean

Why is the mean less appropriate for severely skewed data?

Extreme values in the long tail influence the mean and can make it less representative of the typical observation.

What is the median?

The middle value of a dataset after the observations are placed in order.

How is the median determined when there is an odd number of observations?

The median is the single middle observation.

How is the median determined when there is an even number of observations?

The median is the average of the two middle observations.

What is a major advantage of the median over the mean?

The median is less affected by extreme values and outliers.

Median identified from the middle observations of an ordered IOP dataset

When is the median generally more appropriate than the mean?

When data are skewed or contain extreme outliers.

Why does an outlier affect the mean more than the median?

The mean incorporates the magnitude of every value, while the median depends primarily on the position of values after they are ordered.

What is the mode?

The value that occurs most frequently in a dataset.

Can the mode be used with categorical data?

Yes. The mode can be used with categorical or numerical data.

If the values are 0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, what is the mode?

3, because it occurs more frequently than any other value.

Which measure of central tendency is generally preferred for normally distributed data?

The mean.

Which measure of central tendency is generally preferred for skewed data or data with substantial outliers?

The median.

Which measure of central tendency identifies the most frequently occurring value?

The mode.

What are measures of variability?

Measures that describe the spread or dispersion of scores within a dataset.

What are the major measures of variability?

Range, variance, and standard deviation.

What is the range?

The difference between the highest and lowest values in a dataset.

How is range calculated?

Range = highest value − lowest value.

What does a larger range indicate?

Greater spread or variability.

What does a smaller range indicate?

Less spread or variability.

Can two datasets have the same mean but different amounts of variability?

Yes. Two groups can have identical means while their observations have very different spreads.

Two groups with equal means but different variability

What is variance?

Variance measures the average squared deviation of observations from the mean.

What is a deviation score?

The difference between an individual observation and the mean of the dataset.

Why is variance expressed in squared units?

Variance uses squared deviations from the mean.

What is standard deviation?

Standard deviation is a measure of dataset variability equal to the square root of the variance.

What does a small standard deviation indicate?

Values are clustered relatively close to the mean.

What does a large standard deviation indicate?

Values are more widely spread around the mean.

What is standard error?

Standard error describes how much a sample statistic, such as a sample mean, is expected to vary from sample to sample.

How does sample size affect standard error?

Larger samples generally produce smaller standard errors.

What is the difference between standard deviation and standard error?

Standard deviation describes the spread of observations within a dataset, while standard error describes how much a sample statistic is expected to vary across samples.

What is a z-score?

A standardized score indicating how many standard deviations a value is above or below the mean.

What does a positive z-score indicate?

The value is above the mean.

What does a negative z-score indicate?

The value is below the mean.

What does a z-score of 0 indicate?

The value is exactly equal to the mean.

What does a larger absolute z-score indicate?

The value is farther from the mean.

What is the formula for calculating a z-score?

Z = (observed value − mean) ÷ standard deviation.

If a student's score is 80, the mean is 90, and the standard deviation is 10, what is the z-score?

Z = (80 − 90) ÷ 10 = −1.

How should a z-score of −1 be interpreted?

The value is 1 standard deviation below the mean.

Worked z-score example yielding a z-score of negative one

What is hypothesis testing?

A statistical method used to determine whether evidence from a sample is sufficient to support a claim about a population.

What types of research questions can hypothesis testing evaluate?

Differences, associations, and effects.

What are the two major hypotheses used in hypothesis testing?

The null hypothesis (H₀) and the research hypothesis (H₁).

Null hypothesis H₀ and research hypothesis H₁

What is the null hypothesis (H₀)?

The hypothesis that there is no difference, effect, or relationship between the groups or variables being studied.

What is the research hypothesis (H₁)?

A specific, testable statement predicting an expected relationship, difference, or effect between variables.

What is the basic distinction between H₀ and H₁?

H₀ states that there is no difference or effect, while H₁ states that a difference or effect exists.

If the research question asks whether a new glaucoma medication lowers IOP, what is the null hypothesis?

The new glaucoma medication does not lower IOP.

If the research question asks whether a new glaucoma medication lowers IOP, what is the research hypothesis?

The new glaucoma medication lowers IOP.

Which hypothesis is initially assumed to be true during hypothesis testing?

The null hypothesis.

What are researchers fundamentally asking during hypothesis testing?

How surprising the observed results would be if the null hypothesis were actually true.

What is a p-value?

The probability of observing a result at least as extreme as the result obtained, assuming the null hypothesis is true.

What does a small p-value indicate?

Stronger evidence against the null hypothesis.

Does the p-value represent the probability that the null hypothesis is true?

No.

What is alpha (α)?

The predetermined significance level used as the cutoff for determining statistical significance.

What is a commonly used alpha level?

α = 0.05.

When is a result statistically significant?

When the p-value is less than the predetermined alpha level.