Inferential Statistics
- Inferential Statistics – Definition, Types, Examples, Formulas
- Observational Studies and Experiments
- Sample and Population
- Sampling Bias
- Sampling Methods
- Research Study Design
- Population Distribution, Sample Distribution and Sampling Distribution
- Central Limit Theorem
- Point Estimates
- Confidence Intervals
- Introduction to Bootstrapping
- Bootstrap Confidence Interval
- Paired Samples
- Impact of Sample Size on Confidence Intervals
- Introduction to Hypothesis Testing
- Writing Hypotheses
- Hypotheses Test Examples
- Randomization Procedures
- p-values
- Type I and Type II Errors
- P-value Significance Level
- Issues with Multiple Testing
- Confidence Intervals and Hypothesis Testing
- Inference for One Sample
- Inference for Two Samples
- One-Way ANOVA
- Two-Way ANOVA
- Chi-Square Tests
Causal Conclusions
Causal Conclusions
Randomization involves randomly assigning participants to different levels of an explanatory variable, and is used to control for confounding variables. A randomized experiment allows for causal conclusions to be made, as differences in the response variable can be attributed to differences in the explanatory variable. Without randomization, only an association between variables can be noted, and causal conclusions cannot be made.
Randomization and random sampling are different concepts in statistics. Randomization involves randomly assigning experimental units to different conditions, while random sampling refers to probability-based methods for selecting a sample from a population. In other words, randomization is used to control for confounding variables in an experimental study, while random sampling is used to obtain a representative sample from a population for a study.
- Randomization
- The act of randomly assigning cases to different levels of the explanatory variable
- Causation
- Changes in one variable can be attributed to changes in a second variable
- Association
- A relationship between variables
Two teams designed studies to compare weight loss in two different fitness programs.
Example 1: Coffee Consumption
A researcher recruits a group of individuals and collects data on their coffee consumption habits and any previous or current history of heart disease. The researcher observes and records the number of cases of heart disease in both the group of individuals who consume coffee and the group of individuals who do not consume coffee. However, confounding variables such as age, smoking, diet, and exercise habits may affect the relationship between coffee consumption and heart disease. So, a causal conclusion cannot be made because of their confounding variables. The people in the two groups may be different in some key ways.
The same researcher randomly assigns a group of individuals to either a group that will consume coffee or a group that will not consume coffee. The researcher will monitor both groups over time to record the occurrence of heart disease cases. However, to eliminate confounding variables, the researcher needs to ensure that both groups are similar in terms of age, smoking, diet, and exercise habits. The researcher can do this by using stratification or randomization techniques. Because participants were randomly assigned to groups, the groups should be balanced in terms of any confounding variables and a causal conclusion may be drawn from this study.
In both cases, confounding variables such as age, smoking, diet, and exercise habits can affect the relationship between coffee consumption and heart disease risk. In an observational study, the researcher cannot control these variables, whereas in an experiment, the researcher can control them by using randomization or stratification techniques
Example 2: Fitness Programs
Two teams have designed research studies to compare the weight loss of participants in two different fitness programs. Each team used a different research study design.
The first team used an observational study design, surveying people who already participated in each program. This design only allows for association between the program and weight loss, and confounding variables may exist. A causal conclusion cannot be made because there may be confounding variables. The people in the two groups may be different in some key ways. For example, if the cost of the two programs is different, the two groups may differ in terms of their finances.
The second team used a randomized experiment, randomly assigning participants to one of the programs and measuring weight before and after. Because participants were randomly assigned to groups, the groups should be balanced in terms of any confounding variables and a causal conclusion may be drawn from this study.
Independent and Paired Samples
Independent and paired samples are two types of statistical comparisons that are commonly used in research studies. In both observational and experimental studies, we often want to compare two or more groups. When comparing two or more groups, cases may be independent or paired.
- Independent Groups
- Cases in each group are unrelated to one another.
- For example, in a study comparing the effectiveness of two different drugs, one group would receive Drug A and the other group would receive Drug B. The two groups are independent because the data for one group does not depend on the data for the other group.
- Paired Groups
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Cases in each group are meaningfully matched with one another; also known as dependent samples or matched pairs
For example, in a study comparing the effectiveness of a new therapy to an old therapy, the same group of patients would receive both treatments. The two groups are paired because the data for each patient in one group is related to the data for that same patient in the other group.
Control and Placebo Groups
Control and placebo groups are two types of groups that are commonly used in research studies, particularly in clinical trials.
The control group is a group of participants who are not exposed to the intervention being studied. The purpose of the control group is to provide a comparison against which the effects of the intervention can be evaluated. For example, in a study evaluating the effectiveness of a new drug, the control group would receive a placebo or standard treatment, while the intervention group would receive the new drug.
The placebo group is a type of control group that receives a treatment that appears identical to the intervention being studied but contains no active ingredients. The purpose of the placebo group is to control for the placebo effect, which is a phenomenon where a participant’s belief in a treatment can lead to an improvement in their symptoms, even if the treatment is ineffective. For example, in a study evaluating the effectiveness of a new pain medication, the placebo group would receive a sugar pill that looks identical to the actual medication.
- Control Group
- A level of the explanatory variable that does not receive an active treatment; they may receive no treatment or a placebo
- Placebo Group
- A group that receives what, to them, appears to be a treatment, but actually is neutral and does not contain any active treatment (e.g., a sugar pill in a medication study)