Statistical analysis is one of the principal tools employed in epidemiology, which is primarily concerned with the study of health and disease in populations and its clinical applications. Statistics is the science of collecting, analyzing, and interpreting data, and a good epidemiological study depends on statistical methods being employed correctly. At the same time, flaws in study design can affect statistics and lead to incorrect conclusions. Descriptive statistics measure, describe, and summarize features of a collection of data/sample without making inferences that go beyond the scope of that collection/sample. Common measures of descriptive statistics are those of central tendency and dispersion. Measures of central tendency describe the central distribution of data and include the mode, median, and mean. Measures of dispersion describe how data is distributed and include range, quartiles, variance, and deviation. The counterpart of descriptive statistics, inferential statistics, relies on data to make inferences that do go beyond the scope of the data collected and the sample from which it was obtained. Inferential statistics involves parameters such as sensitivity, specificity, positive/negative predictive values, confidence intervals, and hypothesis testing.
The values used to describe features of a sample or data set are called variables. Variables can be independent, in the sense that they are not dependent on other variables and can thus be manipulated by the researcher for the purpose of a study (e.g., administration of a certain drug), or dependent, in the sense that their value depends on another variable and, thus, cannot be manipulated by the researcher (e.g., a condition caused by a certain drug). Variables can furthermore be categorized qualitatively in categorical terms (e.g., eye color, sex, race) and quantitatively in numerical terms (e.g., age, weight, temperature).
Statistical analysis is used in all , including the before approval for clinical practice. These rely on inferential statistics to draw conclusions from sample groups that can be applied to the general population.
Descriptive and inferential statistics
- Descriptive statistics
- Inferential statistics: the analysis of data collected from a population sample to make inferences about the population as a whole
Sample statistics and population parameters
Measures of central tendency and outliers
Measures of central tendency
- Definition: measures to describe a common, typical value of a data set (e.g., clustering of data at a specific value)
- Approach: The type of measure used depends on the sample size.
|Measures of central tendency|
|Mean (statistics)|| || |
|Median (statistics)|| |
|Mode (statistics)|| |
- Definition: a data point/observation that is distant from other data points/observations in a data set
- Using a trimmed mean: calculate the mean by discarding extreme values in a data set and using the remaining values
- Use the median or mode: useful for asymmetrical data; these measures are not affected by extreme values because they are based on ranks of data (median) or the most commonly occurring value (mode) rather than the average score of all values
- Removing outliers can also distort the interpretation of data. It should be done with caution and with a view to reflecting the respective data set.
- Regression to mean: a phenomenon in which any measurement taken after the measurement of a random variable lying at the extreme (i.e., above or below the mean) is likely to be closer to the mean
Measures of dispersion are statistical measures that describe the variability or spread of data to determine the degree of its homogeneity or heterogeneity.
- For example, two companies with 10 employees each pay the same mean salary of $50,000, but company A has a range of $10,000–$400,000 while company B has a range of $40,000–$60,000. In company A, one employee would be earning $400,000 and another $10,000, with the remaining 8 earning a mean of $11,250. Despite paying the same mean salary, the average employee of company B has a higher salary.
- Some measures of dispersion relate to sample distribution, while others relate to sampling distribution. 
- Sample distribution
- Sampling distribution
|Measures of dispersion|
|Range (statistics)|| || |
|Interquartile range|| |
|Quartile|| || |
|Percentile|| || |
|Variance (statistics)|| || |
|Standard deviation (SD)|| |
|Standard error of the mean (SEM)|| |
- Variables: measured values of population attributes or a value subject to change
- General population: the group from which the units of observation are drawn (e.g., all the patients in a hospital)
- Unit of observation: the individual who is the subject of the study (e.g, inhabitant of a region, a patient)
- Attribute: a character of the unit of observation (e.g., gender, patient satisfaction)
- Attribute value
- Variables can be qualitative (e.g., male/female) or quantitative (e.g., temperature: 10°C, 20°C) in nature
- Quantitative variables can be discrete or nondiscrete (continuous) variables (see “Probability” below).
Types of variables
- Independent variable: a variable that is not dependent on other variables and can thus be manipulated by the researcher for the purpose of a study
- Dependent variable: a variable with a value that depends on another variable and therefore cannot be manipulated by the researcher
Types of quantitative variables
- Discrete variable: variables that can only assume whole number values
- Continuous variable (nondiscrete variable): variables that can assume any real number value
- Categorical variable (nominal variable): variables that have a finite number of categories that may not have an intrinsic logical order
- Definition: types of measurement scales (categorized as and )
Categorical scale (qualitative)
- The distance (interval) between two categories is undefined.
- Includes the and
Metric scale (quantitative)
- The distance between two categories is defined and the data can be ranked .
- Includes the and
|Types of scales |
|Types||Characteristics||Measure of central tendency||Measure of dispersion||Statistical analysis||Data illustration|
|Nominal scale|| || || || |
|Ordinal scale|| || || |
|Interval scale|| || |
|Ratio scale|| |
Normal distribution (Bell curve, Gaussian distribution) 
- Normal distributions differ according to their mean and variance, but share the following characteristics:
- The same basic shape
- Unimodal distribution (i.e., one peak)
- Asymptotic to the x-axis
- Symmetry (i.e., a symmetrical bell curve)
- The following assumptions about the data distribution can be made:
- 68% of the data falls within 1 SD of the mean.
- 95% of the data falls within 2 SD of the mean.
- 99.7% of the data falls within 3 SD of the mean.
- Total area under the curve = 1
- All measures of central tendency are equal (mean = median = mode)
- Standard normal distribution (Z distribution): a normal distribution with a mean of 0 and standard deviation of 1
|Types of nonnormal distributions|
|Bimodal distribution|| |
Positively skewed distribution
| || |
Negatively skewed distribution
| || |
- Enables the comparison of populations with different means and standard deviations
- Standard normal value = (value - population mean) divided by standard deviation
- A means of expressing data scores (e.g., height in centimeters or meters) in the same metric (specifically, in terms of units of standard deviation for the population)
- Determines how many standard deviations an observation is above or below the mean
|Recommended measures according to distribution|
|Distribution||Measures of central tendency||Measure of spread|
|Normal (symmetrical)|| |
|Skewed (asymmetrical)|| || |
- Presents data values for each category in a table
- Illustrates which values in a data set appear frequently
- Describes the frequency of categories in a circular graph divided into slices, with each slice representing a categorical proportion
- Useful for depicting a small number of categories and large differences between them
- Bar graph
- A histogram is similar to a bar graph but displays data on a metric scale.
- The data is grouped into intervals that are plotted on the x-axis.
- Useful for depicting continuous data
- Similar to a bar chart, but differs in the following ways:
- Used for continuous data
- The bars can be shown touching each other to illustrate continuous data.
- Bars cannot be reordered.
- Quartiles and median are used to display numerical data in the form of a box.
- Useful for depicting continuous data
- Shows the following important characteristics of data:
- Minimum and maximum values
- First and third quartiles
- Interquartile range
- Easily shows measures of central tendency, range, symmetry, and outliers at a glance
- A graph used to display values for (typically) two variables of data, plotted on the horizontal (x-axis) and vertical (y-axis) axes using cartesian coordinates, which represent individual data values
- Helps to establish correlations between dependent and independent variables
- Helps to determine whether a relationship between data sets is linear or nonlinear
Hypothesis testing and probability
Types of hypothesis
- Null hypothesis (H0): The assumption that there is no statistically significant relationship between two measured variables (e.g., the exposure and the outcome) or no significant difference between two studied populations. Statistical tests are used to either reject or accept this hypothesis.
- Alternative hypothesis (H1): : The assumption that there is a relationship between two measured variables (e.g., the exposure and the outcome) or a significant difference between two studied populations. This hypothesis is formulated as a counterpart to the null hypothesis. Statistical tests are used to either reject or accept this hypothesis.
- Correct result
Type 1 error
- The null hypothesis is rejected when it is actually true and, consequently, the alternative hypothesis is accepted, although the observed effect is actually due to chance (false positive error).
- Significance level (type 1 error rate): the probability of a (denoted with “α”)
Multiple comparisons problem
When multiple hypotheses are tested simultaneously with one data set (e.g., the data of a trial is tested for different outcomes), the probability of type 1 errors increases:
- P1= 1 - P2
- P1= 1 - (1 - α)m
- Example: when testing 20 hypotheses on one set of data, each at a significance level of α = 0.05, the probability of obtaining at least one significant result by chance can be calculated as follows:
- P(at least one significant result in 20 tests at α = 0.05) = 1 - (1 - 0.05)20)
- P(at least one significant result in 20 tests at α = 0.05) ≈ 1 - 0.36 = 0.64)
- For 20 independent hypotheses that are simultaneously tested on the same data, each at α = 0.05, the probability of a type 1 error in at least 1 test is approximately 64%.
- There are methods to control the type 1 error rate for multiple comparisons; examples include:
- Results that are not adjusted for multiple comparisons should not be used to infer treatment effects or make clinical decisions to avoid interpreting random results as significant.
- When multiple hypotheses are tested simultaneously with one data set (e.g., the data of a trial is tested for different outcomes), the probability of type 1 errors increases:
Type 2 error
- The null hypothesis is accepted when it is actually false; and, consequently, the alternative hypothesis is rejected even though an observed effect did not occur due to chance (false negative error).
- Type 2 error rate: the probability of a type 2 error (denoted by “β”)
- Type 1 errors are inversely related to type 2 errors; The increase of one causes a decrease of the other.
Statistical power: (1-β)
- The probability of correctly rejecting the null hypothesis, i.e., the ability to detect a difference between two groups when there truly is a difference
- Complementary to the type 2 error rate
- Positively correlates with the sample size and the magnitude of the association of interest (e.g., increasing the sample size of a study would increase its statistical power)
- Positively correlates with measurement accuracy
- By convention, most studies aim to achieve 80% statistical power.
P-value: the probability that the result of a given statistical test will be at least as extreme as the result actually observed, assuming that the null hypothesis is correct
- Calculated using different statistical tests, depending on the type of data collected (e.g., parametric tests)
- Interpretation: The p-value is compared to the significance level (i.e., alpha or α-level), which is typically set at 0.05.
- p ≤ α-level: The association is considered statistically significant and H0 is rejected.
- The p-value is not equivalent to the probability of H0 being true, but rather to the probability of obtaining the same or more extreme results, assuming that H0 is true.
- The p-value cannot be used to prove H1 but rather to prove that observed data is inconsistent with H0.
- When multiple comparisons are performed (e.g., ANOVA), the p-value must be compared to an adjusted α-level to ensure adequate statistical significance (e.g., α-level adjusted according to the Bonferroni correction)
“Statistical significance” does not mean “clinical significance.”
Overview of errors
|Statistical test||Null hypothesis (H0) is true||Null hypothesis (H0) is false|
|Does not reject H0|| || |
|Rejects H0|| || |
- Probability of an occurring event (P)
Probability of an event not occurring (Q)
- The degree of certainty that a particular event will not take place (e.g., rolling a 6 is considered the event when tossing a die. When throwing a die, the probability of the event not occurring (rolling a “1”, “2”, “3”, “4”, or “5”) is 5/6)
- Q = number of unfavourable outcomes/total number of possible outcomes OR 1 - P
The actual probability of an event is not the same as the observed frequency of an event.
- Probability of independent events: The probability of event A is not contingent upon the probability of event B and vice versa. (e.g., eye-color and birthdays are two independent variables, with probability distributions independent of each other)
Conditional probability: the probability of event A occurring given that event B has occurred
P(A|B) = P(A and B) / P(B)
- P(B) = probability of event B
- P(A and B) = probability of events A and B occurring simultaneously
- Example: the probability of lung cancer in a smoker (A: occurrence of lung cancer; B: occurrence of smoking)
- The underlying condition is that the individual is a smoker: P(B) = probability of being a smoker = number of smokers/total population
- P(A and B) = probability of simultaneously being a smoker and having lung cancer = number of smokers with lung cancer/total population
- Therefore, P(A|B) = the probability of lung cancer arising in a smoker = P(A and B)/P(B) = number of smokers with lung cancer/number of smokers
- P(A|B) = P(A and B) / P(B)
- P(A and B): the probability of events A and B occurring simultaneously
- The multiplication rule is obtained by rearranging the formula for conditional probability.
- For dependent conditions: P(A and B) = P(B) × P(A|B)
- For independent conditions: P(A and B) = P(B) x P(A)
- The multiplication rule can be applied to a decision tree (a visual representation of all possible outcomes) in order to calculate the probability of one of the branches (a particular outcome).
- P(A or B): the probability of either event A or B occurring
P(A or B) = P(A)+ P(B) − P(A and B): the probability of either event A or B occurring equals the sum of probabilities of events A and B minus the probability that they will occur simultaneously (nonmutually exclusive probability)
- Example: the probability that an individual has a history of either myocardial infarction (event A) or stroke (event B) equals the probability of a history of myocardial infarction P(A) plus the probability of a history of stroke P(B) minus the probability of a history of both myocardial infarction and stroke P(A and B)
- Example: meeting someone who has coronary artery disease (CAD) OR is obese : 0.3 + 0.4 - (0.3 × 0.4) = 0.58
- If the events are mutually exclusive (mutually exclusive probability), P(A and B) = 0 and P(A or B) = P(A) + P(B)
- Bayes theorem
- Confidence intervals (CIs) are calculated using data from a sample and give an estimate of the true population value, which can never be determined by an experiment.
- CIs can be used to infer the statistical significance of sample values. CIs and p-values are interrelated and either can be used to determine whether a result is statistically significant.
- Definition: the range of values estimated to contain the true population value with a certain level of confidence
Formula: The formula depends on the kind of data for which the confidence interval is calculated (e.g., means, proportions).
- For confidence intervals for the mean: mean +/- Z score x (standard error of the mean)
- For confidence intervals for the proportion: p +/- Z score x (√p x (1 - p)/n)
- Requires the following values:
- What kind of value does the confidence interval describe? Examples include:
- Mean value (e.g., the height of students in a class)
- Difference between the means of two values (e.g., the difference between the mean height of students in class A and the mean height of students in class B)
- Relative risk (e.g., the relative risk of lung cancer in smokers vs. nonsmokers)
- Odds ratio (e.g., odds of melanoma in residents of Hawaii compared to residents of Massachusetts)
- What is the ?
- Are several confidence intervals compared?
- What is the null value of the effect tested? 1 for ratios (e.g., relative risk, odds ratio) and 0 for differences (e.g., attributable risk, absolute risk reduction)
- Does the CI include the null value?
- If the CI includes the null value (i.e., 0 for differences or 1 for ratios), the result is not statistically significant, and the null hypothesis cannot be rejected.
- If the CI does not include the null value, the result is statistically significant, and the null hypothesis can be rejected. Therefore, if the results of a study are statistically significant, i.e., p-value < α-level, the associated CI does not include the null value.
- How wide is the confidence interval?
- What kind of value does the confidence interval describe? Examples include:
Statistical significance vs. clinical significance 
Significance (epidemiology): the statistical probability that a result did not occur by chance alone
- Statistical significance
Clinical significance (epidemiology)
- Describes an important change in a patient's clinical condition, which may or may not be due to an intervention introduced during a clinical study
- If the clinical significance is high, the intervention is likely to have had a great impact on patient outcome or measures.
- Statistical and clinical significance do not necessarily correlate. A study might have a high statistical significance but the tested intervention did not have any clinical impact on patient outcome.
Correlation and regression
- Example: how does y change if x is changed?
Interpretation: A correlation coefficient measures the strength (i.e., the degree) and direction (i.e., a positive or negative relationship) of a linear relationship (does not require causality).
- Direction or relationship: can be positive or negative (which are identified by a plus or minus, respectively)
- Strength of relationship
- See “.““ and “
- Definition: the process of developing a mathematical relationship between the dependent variable (the outcome; y) and one or more independent variables (the exposure; x)
Linear regression: a type of regression in which the dependent variable is continuous
- Simple linear regression
- 1 independent variable is analyzed
- If y has a linear relationship with an independent variable x, a graph plotting this relationship takes the form of a straight line (called regression line).
- In the case of simple linear regression, the equation of the regression line is: y = mx + b, with m representing the slope of the regression line, y the dependent variable, x the independent variable, and b the y-intercept (the value of y where the line crosses the y-axis)
- Multiple linear regression: > 1 independent variable is analyzed
- Simple linear regression
- Logistic regression: a type of regression in which the dependent variable is categorical
Pearson correlation coefficient (r)
- Compares interval level variables
- Calculates the estimated strength and direction of a relationship between two variables
- r is always a value between -1 and 1.
- A positive r-value = a positive correlation
- A negative r-value = negative correlation
- The closer the r-value is to 1, the stronger the correlation between the compared variables.
- The coefficient of determination = r2 (the coefficient may be affected by extreme values and indicates the proportion of a variable's variance that can be predicted by the variance of another variable)
- Calculates the difference between the means of two samples or between a sample and population or a value subject to change; especially when samples are small and/or the population or a value subject to change distribution is not known
- Used to determine the confidence intervals of a t-distribution (a collection of distributions in which the standard deviation is unknown and/or the sample size is small)
One sample t-test
- The t-value can be classified according a table that lists t-values and their corresponding quantiles based on the number of degrees of freedom (df) and the significance level (α value).
- Alternatively, one may calculate the confidence intervals of the sample observations and check if the population mean (μ0) falls within the range given by the confidence intervals.
- Formula: t-value = (sample mean - population mean)/standard deviation) * √(n)
- Prerequisite: normal distribution (the variance is known and depends on the degrees of freedom.)
- Calculates whether a sample mean differs from the population mean (μ0)
Two sample t-test
- Calculates whether the means of two groups differ from one another
- Formula: t-value = (mean difference between the two samples/standard deviation) * √(n)
- Interpretation: The t-value is compared with a table of t-values in order to determine whether the difference is statistically significant (similar to the one sample t-test described above).
- Unpaired t-test (independent samples t-test)
- Paired t-test (dependent samples t-test)
Analysis of variance (ANOVA)
- Calculates if there is a statistically significant difference between ≥ 2 groups by comparing their means
- The aim is to determine whether there is a statistically significant effect of one or more independent variable(s) on a dependent variable (the mean).
- Can be seen as an extension of the t-test (which can only be used for the analysis of two groups)
- One-way ANOVA: assesses if there is a statistically significant difference in the means of 1 independent variable (e.g., the mean height of women in clinics A, B, and C; the independent variable is the clinic, the dependent variable is the height)
- Two-way ANOVA: assesses if there is a statistically significant difference in the means of 2 independent variables (e.g., the mean height of women and men in clinics A, B, and C at a point in time; the independent variables are sex category and clinic, the dependent variable is the height)
Spearman correlation coefficient
- Calculates the relationship between two variables according to their rank
- Compares ordinal level variables
- Extreme values have a minimal effect on Spearman coefficient.
- Not precise because not all information from the data set is used.
- See “ .”
Mann-Whitney U test
- Compares ordinal, interval, or ratio scales
- Calculates whether two independently chosen samples originate from the same population and have identical distributions and/or medians
- Example: comparing two groups of high school students – one with an average GPA of 4.2 and the other an average GPA of 3.2 – to determine if both came from the same larger group.
Wilcoxon test (rank sum and signed rank)
- Rank sum test: compares the means between groups of different sizes
- Signed rank test: compares the means between pairs of scores that can be matched; substitute for the one-sample t-test when a pre-intervention measure is compared with a post-treatment measure and the null hypothesis is that the treatment has no effect
Kruskal-Wallis H test
- Extension of the
- Compares multiple groups by testing the null hypothesis (that there is no median difference between at least two groups)
- Examines whether the observed frequency of an event with binary outcomes (e.g., heads/tails, dead/alive) is statistically probable or not
- Example: if a coin is tossed 20 times, it is likely to land on heads approx. 10 times. If only 9 heads come up, the result is still acceptable.
- If only 6 heads come up (p = 0.04), one is left wondering whether the coin is biased
- From a statistical perspective, the result of a coin toss should be questioned if fewer than 25% of the coin tosses result in heads because the probability of such an event is < 2.5%
Chi-square test (X2 test)
- Used to compare the distributions of two categorical variables.
- A chi-square test compares proportions in two or more sets of categorical data to determine whether there is a statistically significant difference in the distribution (e.g., the proportion of patients with lung disease in clinics A, B, and C at a certain point in time or the proportion of individuals with diabetes in four different ethnic groups)
Fisher exact test
- Also calculates the difference between the frequencies in a sample but, unlike a Chi-square test, is used when the study sample is small
- Also aims to determine how likely it was the outcomes occurred due to chance
“Chi-tegorical:” Chi-square test is used for categorical variables.