Sometimes you need an extra resource to understand a topic or concept. Below is a series of resources that provide a wide range of statistical information from understanding basic terminology, to conducting t-tests, to running regression models, and more. Our short courses dive deeper into these topics without the traditional, full workload expected while taking a for-credit course. For more in depth learning, sign up to take a for-credit course through the Department of Biostatistics and Informatics.
To begin, we must first identify the differences between what statistics defines as population data and sample data. A population is the entire set people or things in a specified group. Characteristics of a population are called parameters. A sample is a subset of a population. Characteristics of a sample are called statistics.
Biostatistics is the use of statistics for public health, biological, or medical applications, and applied to a variety of research topics and fields. The main goal is to use appropriate statistical methods to understand the factors that affect human health.
Researcher randomly assigns individuals to treatment groups.
Detailed prediction of a scientific question that can be tested.
Method to ensure there are enough observations to find a statistical difference between groups when they are, in fact, biologically different.
Significance level (α): Threshold with which null hypothesis is rejected. Standard values for α include 0.05, 0.01, 0.001
Power: Ability to detect a difference when a difference truly exists
Effect size: Clinically meaningful difference between comparisons
Any systematic error that can occur in multiple areas of a study, (e.g., study design, measurement technique, and or analyses) which will either over or under estimate a parameter and to false conclusions.
Characterizing data using graphs, tables, numerical summaries.
|Measure of Location|
|Mean: Average of the Data||Median: Middle point of the data||Mode: Most occurring data point|
|Measure of Spread|
|Standard deviation: Deviation of the data in a sample||Interquartile Range: Difference between the 75th percentile and the 25th percentile||Range: Difference between the largest and smallest values|
Outliers: Very extreme data points
Frequency: The proportions of values within a single variable
Drawing conclusions about populations based on samples
Baron, Anna. Biostatistical Methods. Lecture 1 Overview. Fall 2015.
Rosner. Fundamentals of Biostatistics. 7th ed. Brookes/Cole. 2011.
Samuels & Witmer. Statistics for the Life Sciences. 3rd ed. Pearson Education. 2003.
At the start of the study process, there are general questions, goals, and study aims that set the context for an idea. Often the questions seem clear but are not formulated in enough detail to perform analysis. Questions often range throughout research. Some examples are looking for results from groups with different profiles, assessing the effectiveness of a drug, or looking for a response pattern over time.
Supporting information is required to help move from the study aim to a hypothesis. A hypothesis is more detailed than a study aim. It is a testable, statistics-driven statement that will have evidence for or against, or no evidence at all.
It is critical to establish the null and alternative hypotheses before the study begins so that the appropriate measurements and controls can be put in place to ensure that the study is as unbiased as possible. Determining these hypotheses after the fact is a bit like ‘leading the witness’.
The primary goal of the study is to accept or reject the null hypothesis. It is through the rejection of the null hypothesis that there is enough evidence to support the possibility that the alternative hypothesis may be true.
Enrico Fermi had the following to say about hypothesis tests:
"There are two possible outcomes: if the result confirms the (null) hypothesis then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery."
A good hypothesis should have the following characteristics:
|Be measurable: Define your independent and dependent variables, how you plan to measure them, and determine ways to examine the relationship between them.||Survival rates after treatment: 1,2,3 years|
Decrease in systolic blood pressure of a minimum of 5 mm Hg after 6 months
|References a well-defined population: Describe in detail the members of the population or contrast various population groupings.||Males age 55+ with family history of heart disease|
Pregnant women with elevated hormone levels
Ethnicity - Existing conditions, comorbidity
|Proposes cause-effect or association between variables: If not true cause and effect, then make a statement about the relationship or association between variables of interest, including potential predictor variables thought to have an impact on study outcomes.||Height|
Vital measurements (potentially at different times during the study)
History of smoking/tobacco use
|Has a “biological” basis – or at least it’s plausible: Your hypothesis needs to be plausible and you need to have done your homework in the literature to propose it.||Examination of hereditary traits in an ethnic population|
Levels of serum beta-carotene distinguished in groups receiving 4 different treatments
|Is clear, focused, and in the form of a statement: Refer back to the discussion of the null and alternative hypotheses to make a clear distinction between the two.||Treatment of Stage IV pancreatic cancer with drug X will result in a reduction of tumor size in treated mice compared to mice that were not treated.|
|Can answer at least part of your research question!: Sometimes a study might not have a specific hypothesis but will have an objective, perhaps just to provide descriptive statistics or to be exploratory. The study might just provide direction for follow-on work.||Describe the characteristics of Veterans Affairs patients who participated in open heart surgery for the last 6 months at hospitals in Colorado.|
There are several different types of study designs which can be classified into two main categories: observational and experimental. Each study design has unique strengths and weaknesses which must be considered when determining the most appropriate design to test a study hypothesis.
Studies of risk factors on health or other outcomes based on population or group (aggregate) data and not individual level data.
Identify a group of individuals with an outcome of particular interest and describe the characteristics of the group. Studies with this type of design are based on prevalent cases since they represent a snapshot in time.
Identify subjects before they have the outcome of interest, but they may be designed prospectively or based on retrospective data.
Study that compares patients who have the disease of interest (cases) with patients who do not have the disease of interest (controls - who are from the same source population as the cases) and then looks back retrospectively to compare how frequently the exposure to a risk factor is present in each group to determine the relationship between the risk factor and disease. Studies with this type of design should use odds ratios to summarize the association between exposure and disease.
Randomized controlled trial
Study participants are randomly allocated to receive one or more treatment assignments that may include novel clinical interventions/treatments, placebo treatments, or existing interventions that serve as the standard of care. In randomized controlled trials, the investigator controls the exposure and then the study participants are followed-up for outcomes of interest. This type of study is generally considered the gold standard and is often used to test efficacy or effectiveness of various types of medical interventions.
A longitudinal study in which study participants receive a sequence of different treatments (or exposures) of interest during different time periods, i.e. the patients cross over from one treatment to another during the course of the trial with a predetermined "wash-out" period between treatments. This type of study design can be experimental or observational in nature.
Manipulate intervention but do not randomize subjects. Known as the "natural experiment" and exposure is often dictated by policy or legislation (e.g., seat belt laws).
Errors arising in various study designs
Random error: Natural variation in the underlying data the will be different for each sample.
Non-random error (systematic)
Confounding: Distortion of the exposure-outcome association due to their mutual association with another factor. In order for a variable to be considered a confounder, it has to be associated with the exposure of interest and cause (or precede) the disease/outcome of interest.
Mediation: A mediator is present when the relationship between your exposure of interest (x) and your outcome (y) is mediated by a third variable (z). In other words, your mediation variable z is on the causal pathway between x and y.
To clearly identify the measures and variables being used in a study and determine levels of confidence in reporting results.
To begin, let us look at an example of a Table 1. Typically, studies begin with a summary of the patient characteristics to show the properties of the sample being studied. This table is often referred to as "Table 1" and shows characteristics associated with the group of participants. Continuous and categorical variables are depicted in the table.
Suppose there is a drug treatment (Drug X) designed to reduce the risk of stroke among people aged 60 years or older with isolated systolic hypertension.
Table 1: Characteristics of Hypertension Participants
|Baseline||After 6 Months|
|Characteristic||Active (N=2365)||Placebo (N=2371)||Total (N=4736)||Active (N=2330)||Placebo (N=23500||Total (N=4680)|
|Age, mean (SD), y||71.6 (6.7)||71.5 (6.7)||71.6 (6.7)||72 (6.7)||72 (6.7)||72 (6.7)|
|Systolic Blood Pressure, mean (SD), mmHg||170.5 (9.5)||170.1 (9.2)||170.3 (9.4)||160.5 (11)||170.1 (9.2)||165.3 (10.1)|
|Current Smokers (%)||12.6||12.9||12.7||12.5||12.2||12.3|
|Past Smokers (%)||36.6||37.6||37.1||36.2||36.3||36.2|
|Never Smokers (%)||50.8||49.6||50.2||51.3||51.5||51.5|
The use of simple graphs/visuals is a great way to get started with a set of data. For example, in the set of histograms above in which the Distribution of SBP is shown by gender, it spurs the question of why the two groups look a little
different. By drawing out the linear relationship, one can start to see patterns of a positive or negative correlation between these two variables to start teasing out additional thoughts on why this would be true.
Confidence intervals are a range of values in which we feel confident that the true parameter is contained. It is important to distinguish confidence intervals from probabilities as we cannot attach a probability to the true value of
the statistic based on a single sample of data. Confidence intervals are calculated in different ways depending on the type of statistic we are evaluating. In the example above, Systolic Blood Pressure (SBP) is a continuous variable for which a mean
and standard deviation are given for the total participants in the study.
To calculate a confidence interval: Unknown mean (μ) and known standard deviation (σ)
Note: x is the sample mean and the critical value (Z) for a 95% Confidence Interval is 1.96.
To calculate a confidence interval: Unknown mean (μ) and unknown standard deviation (σ)
Note: When a proportion is the statistic being examined, confidence intervals are generated in a different way.
Benefit of confidence intervals
Categorical vs. continuous data
Let’s go back to our ASA example which can take on the values 1-5. You want to look at the ASA values of patients at Children’s Hospital compared to those at University of Colorado Hospital. Your table would look something like this:
|ASA||Children's Hospital||University of Colorado Hospital|
This table describes how many patients fall into each category based on ASA value (the outcome) and hospital (the exposure). Now, you want to analyze your data—there are many ways to do this based on your research question.
The chi-square test is used for categorical variables and tests whether ASA level is associated with hospital. In our example, we would be testing if ASA level differed between the two hospitals. It is important to pay attention to the
cell counts, as the test expects at least 5 values in each cell. Since the cells coinciding with ASA values of 5 are 0 and 2, this test would not work. When the chi-square test is not an option, one may use Fisher’s exact test.
Risk ratio (RR)
Describes the risk of a certain event happening in one group compared to another.
The risk of disease for a person exposed to the causal factor is (RR) times greater than for a person who was not exposed.
Interpretation: the odds of cancer are OR times lower in the group who got the treatment compared to the group that didn’t get the treatment.
Once you calculate an estimated mean or measure of association for categorical data, you can also calculate a confidence interval around that mean. This is important because it gives a measure of uncertainty about where the true mean lies. Once a confidence interval is calculated, it is read as "We have a certain level (i.e., 95%) of confidence that the true population parameter is within this interval."
Know your assumptions. In general, we assume that the data were collected without bias, it is normally distributed for continuous variables and there are no unmeasured variables that actually explain the difference between the two means.
Roberts, Donna. "Qualitative vs Quantitative Data." Qualitative vs Quantitative Data. 2012. Web. 13 Oct. 2015.
"Risk Differences and Rate Differences." Risk Difference. Boston University School of Public Health, 16 Sept. 2015. Web. 13 Oct. 2015.
Szumilas, Magdalena. “Explaining Odds Ratios." Journal of the Canadian Academy of Child and Adolescent Psychiatry 19.3 (2010): 227–229. Print.
"Understanding Data Concepts." Understanding Data Concepts. Canada.ca, 9 Dec. 2013. Web. 13 Oct. 2015.
Aligning power or sample size analysis with planned data analysis helps to avoid the problems of 1) sample size too small to detect important alternative hypotheses and 2) sample size so large that the design squanders precious resources.
What do you need for sample size justification?
Discuss your science and study design before calculating your sample size. This information will be used to create a table of sample size choices and a written paragraph justifying the sample size.
Data you need to provide from historical literature, pilot data, or other clinical information:
Sample size is a function of:
Based on the data, we make a decision to reject the null hypothesis (H0) or fail to reject H0. We quantify the evidence against the H0 in the form of a p-value. Remember, we want α and β small. Note: β increases as α decreases.
Evaluating the performance of a hypothesis test
There are 4 important quantities that vary together.
Derivation of power for two-sided, one-sample Z-test
As an example, we illustrate derivations for a Z-test. We usually set α (typically at 0.05), and then by fixing two of the other parameters we can calculate the last:
Detectable difference (difference in the means that can be detected)
Note: For a one sided test, replace α/2 with α.
How power, detectable difference, and sample size relate to each other:
As sample size n increases:
As the difference to be detected | μ0 - μ1 |, increases:
As desired power increases:
"Reproducible Research (RR) is the practice of distributing, along with a research publication, all data, software source code, and tools required to reproduce the results discussed in the publication. As such the RR package not only describes the research and its results, but becomes a complete laboratory in which the research can be reproduced and extended." (Source: CTSPedia)
Strive for your analysis to be reproducible and document your code.
Prepare your data for efficient analysis by creating a data dictionary prior to sharing your data. Data dictionaries give a list of variable names, type of variable (categorical, continuous, text), and interpretation of codes, e.g. 1="Female", 2="Male".
Use REDCap to efficiently set up your database to make it easily accessible to your research team and usable for data analysis at the end of the study. REDCap is a secure web application for building and managing online surveys and databases.
If you want one of our biostatisticians to develop your database and forms for you, please submit our Request Biostatistics Consulting form and we will work with you to develop a scope of work and timeline for your project.
The Department of Biostatistics and Informatics offers an introductory applied statistics sequence designed for those without a calculus background and requires minimal mathematical derivations.
More in-depth training is also taught in the MS biostatistics graduate courses. They require calculus and cover the theory of the methods in addition to the application.
Other applied courses are offered in longitudinal analysis, genetics/genomics analysis, etc.
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