Creating levels of African nations and Non-African nations for the continent variable, showed a new relationship when observing log(GDP) versus ruggedness of the terrain.

Example: Creating levels of an independent variable 

An intuitive research example of an interaction effect involves the relationship between psychotherapy and a patient's motivation. Generally, the positive impact of receiving psychotherapy is significantly stronger among individuals who are highly motivated to change compared to those who lack such motivation. This represents an interaction because the effectiveness of one independent variable (whether or not the person receives therapy) relies entirely on the specific level of a second independent variable (the individual's degree of motivation).

Interaction Between Psychotherapy and Motivation

Interaction effects can be easily understood through everyday scenarios where multiple factors combine to determine a unique outcome. For example, your decision to watch a movie might depend on both the film's genre and the specific friend accompanying you. While there might be a main effect for preferring a blockbuster movie, an interaction occurs if your choice completely changes based on who is coming along. Another practical illustration is a drug interaction, where two medications may each be independently beneficial, but taking them together produces dangerous or lethal results.

Everyday Examples of Interaction Effects

A simple effect analysis is a statistical method used to break down an interaction to understand precisely what is driving the results. Because an interaction indicates that the effect of one independent variable depends on the level of another, relying solely on main effects can be misleading. To clarify the relationship, researchers use simple effects analyses to examine the effect of each independent variable at each specific level of the other independent variable, rather than averaging across those levels.

Simple Effect

A study by Brown and colleagues illustrates an interaction effect by exploring memory recall differences based on an individual's level of hypochondriasis. The researchers hypothesized and found that participants high in hypochondriasis recalled negative health-related words more accurately than those with low hypochondriasis. However, both groups demonstrated similar recall rates for non-health-related words. This finding highlights a clear interaction, as the influence of the word type on memory recall is directly dependent on the participants' level of hypochondriasis.

Interaction Between Hypochondriasis and Word Type in Memory

In factorial research designs, the effect of one independent variable can depend on the specific level of another variable in several distinct ways. The two primary ways these complex relationships manifest are through spreading interactions, where an effect is present at one level but weak or absent at another, and cross-over interactions, where the independent variable produces opposite effects depending on the second variable's level.

Types of Interactions

Schnall and colleagues demonstrated an interaction effect by investigating the combined influence of disgust and private body consciousness on moral judgment. Rather than conducting separate studies for each independent variable, they examined them simultaneously in a single experiment. The results revealed an interaction: the impact of the disgust manipulation on moral judgment was not uniform, but instead depended fundamentally on whether participants exhibited high or low levels of private body consciousness.

Interaction Between Disgust and Private Body Consciousness

In experimental research, which of the following best describes an interaction effect?

In experimental research, an interaction effect (or simply an interaction) exists when the influence of one independent variable on a measured outcome is fundamentally altered depending on the specific level of a second independent variable. This indicates that the variables do not operate in isolation; rather, they combine to produce complex effects that cannot be fully explained by analyzing them independently.

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University of Virginia, Charlottesville

- Statistical Golems
- Plausibility
- Posterior Distribution
- Sampling the Imaginary
- Building a Bayesian Model
- Gaussian Distribution
- Correlation vs. Causation
- Linear Models
- Multiple Regression Models
- Perils of Multiple Regression
- Underfitting vs Overfitting 
- Entropy and Accuracy
- Interactions
- Markov Chain Monte Carlo
- Generalized Linear Models



Concepts in Bayesian Statistics

A factorial design is the most common experimental approach for investigating multiple independent variables, which are also known as factors. In this structure, every level of one independent variable is systematically combined with every level of the other independent variables to create all possible combinations. Each unique combination of these levels forms a distinct condition within the experiment.

Factorial Design

A main effect in a factorial design refers to the overall influence of a single independent variable on the dependent variable, determined by averaging the results across all levels of the other independent variables. Consequently, an experiment has exactly one potential main effect for each independent variable it includes.

Main Effect

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd edition)

https://kpu.pressbooks.pub/psychmethods4e/

KPU Research Methods in Psychology - 4th American Edition

- Scientific models are designed for making calculations, but they cannot determine causality on their own. 
- To determine the most appropriate statistical model to use, researchers ask questions about the type of data they have and what they wish to measure. 
- Frequentist approach is based on re-sampling methods, and it is a special case of Bayesian analysis
- We have models that are defined to do certain tests, but as researchers, you need to get behind the methods and determine if results are practically significant 

Statistical Golems 

- Bayesian Statistics considers all possible paths that yield different outcomes
- Plausibilities are probabilities – prior plausibility and posterior plausibility 

Plausibility

The posterior distribution is a combination of data, likelihood, parameters, and prior. It takes on the form of a conditional probability. 

Posterior Distribution

- Bayesian inference uses probabilities, and through Bayesian updating, one can obtain posterior plausibilities  

Building a Bayesian Model

Correlation is a relationship, yet causality indicates one event is the result of the occurence of another event.

Correlation is not equal to causality

- Comprises the likelihood in simple linear regression models
- Counts up the relative numbers of ways different combinations of means and standard deviations can produce an observation

Gaussian Distribution

- Predictor variable
- Linear model strategy
- Finding the prior 
- Finding the posterior distribution 
- Generating predictions and intervals from posterior of a fit model 

Linear Predictions

Below are some of the potential issues when adding variables to a regression model without a clear idea of the causal model - 

- Multicollinearity
- Post-Treatment Bias
- Collider Bias
- How  to recognize and prevent these confounding issues

Perils of Multiple Regression

- Sampling from a grid-approximate posterior
- Sampling to summarize
- Sampling to simulate prediction

Sampling the Imaginary

- These concepts are related to model fitting within or outside Bayesian Statistics

- Direct relation to the number of parameters used to build a model

- There is a preference to use simpler models (Ockham’s razor), then we should ask when is too simple?

- There is a measure for model fitting, R-squared, which is not always an indicator of a good fit.


Underfitting vs Overfitting

Accuracy: 

- Heuristic approach to the concept, we can have a high probability of making correct predictions and predict wrong most of the time yielding poor accuracy. 

-  Log scoring Rule for accuracy measure inherently uses the Bayes Theorem with joint probabilities (likelihood). 

Entropy: 

-  The uncertainty contained in a probability distribution. 

- Average log-prob of the event

-  Is closely related to its scientific meaning, the measure of disorder in a system, now in terms of probability. 



Entropy and Accuracy 

Thinking about how the symmetry is broken can be productive. It would help us to compare the interaction between two choices when we miss other information.

Symmetry of Interactions

If interaction exists, it is not enough to interpret the model with only posterior means and standard deviations. Thus, we need some methods to construct and interpret continuous interactions.

Continuous Interactions

- A class of techniques or methods that use a stochastic process to estimate a posterior probability – often used to generate samples/ observations from a posterior 
- STAN – Most common tool to run MCMC algorithms 
- Rstan and Rethinking package – can be used for simple models

Common MCMC Algorithms - 
- Metropolis/ Metropolis Hastings
- Gibbs Sampling 
- Hamiltonian Monte Carlo


Markov Chain Monte Carlo

Given some constraints on the parameters, the least informative distribution consistent with the constraints on the outcome variables will be nominated, according to the maximum entropy principle.

Maximum Entropy Priors 

GLM offers additional techniques for different types of data such as:
- Continuous (OLS)
- counts 
- dichotomous (two-categories) 

GLM Components:

- Link Function - serves as a model regulator
- Linear Predictor - Data fitting 
- Random Error

Exponential Family

- Every member of this family is a maximum entropy distribution,  for some set of constraints
- GLM Structure is the same for each member. 


GLM and Exponential Family 

Goal: Map linear model to [0,1];
Log-odds is natural parameter;


Rethink: Logit Link

Multiple regression is a statistical technique that extends simple regression by measuring several predictor variables ($$X_1, X_2, X_3, \dots, X_i$$) and using them to predict a single outcome variable ($$Y$$). It can also be utilized to describe the complex relationship between the outcome variable and a set of predictors. The result of this analysis is an equation that expresses the outcome variable as an additive combination of all the measured predictor variables.

Multiple Regression

Interaction Effect

Factorial designs use a specific numerical notation to describe their structure, where each number represents one independent variable and its value indicates the number of levels for that variable. By multiplying these numbers together, researchers can determine the total number of experimental conditions. For example, a $$3 	imes 2$$ factorial design indicates two independent variables—one with three levels and one with two levels—resulting in a total of six distinct conditions. While these designs can theoretically include any number of variables and levels, logistical constraints regarding the large number of conditions and required participants make designs with more than three independent variables uncommon in practice.

Factorial Design Notation

A factorial design table visually represents the different combinations of independent variables in an experiment, where the columns typically represent the levels of one variable and the rows represent the levels of another. For example, in a $$2 	imes 2$$ factorial design investigating the effects of cell phone use and time of day on driving ability, the table creates four distinct cells representing the four possible experimental conditions: using a cell phone during the day, not using a cell phone during the day, using a cell phone at night, and not using a cell phone at night.

Example of a Factorial Design Table

A between-subjects factorial design is an experimental structure in which all of the independent variables are manipulated between subjects. This means that each participant is exposed to exactly one specific combination of the variables' levels, resulting in them being tested in one and only one experimental condition. For instance, in a $$2 	imes 2$$ study examining cell phone use and time of day, a participant would be assigned to drive either with a cell phone or without, and either during the day or at night, but never across multiple conditions.

Between-Subjects Factorial Design

A within-subjects factorial design is an experimental structure in which all of the independent variables are manipulated within subjects. In this approach, each participant is exposed to every possible combination of the independent variables' levels, requiring them to be tested across all conditions of the experiment. For example, in a $$2 	imes 2$$ study investigating cell phone use and time of day, a single participant would perform driving tests under all four distinct conditions: with and without a cell phone, both during the day and at night.

Within-Subjects Factorial Design

A mixed factorial design is an experimental structure that incorporates both between-subjects and within-subjects manipulations. Because factorial experiments involve multiple independent variables, researchers can opt to manipulate at least one variable between subjects (testing participants in only one of its levels) and another variable within subjects (testing the same participants across all of its levels). For example, a researcher might test all participants' driving ability both with and without a cell phone (a within-subjects factor) while assigning each participant to drive exclusively during the day or exclusively at night (a between-subjects factor).

Mixed Factorial Design

A non-experimental factorial design is a study structure that includes multiple independent variables, but unlike a true experiment, all of these variables are merely measured rather than actively manipulated. Because every factor in this design—such as pre-existing participant variables—is solely measured, the study is non-experimental and correlational in nature. Consequently, while researchers can examine the relationships between these non-manipulated variables, they cannot use this design to establish definitive cause-and-effect relationships.

Non-Experimental Factorial Design

In a factorial experiment with two independent variables, researchers often graph the results by plotting one independent variable along the $$x$$-axis and the second independent variable using differently colored bars or distinct line formats. The $$y$$-axis is always used to plot the dependent variable.

Graphing Factorial Experiments

The factorial ANOVA is a statistical test utilized when an experiment includes more than one independent variable, such as in a factorial design. While its foundational logic is identical to that of the one-way and repeated-measures ANOVAs, a factorial ANOVA is distinct because it computes a separate $$F$$ ratio and $$p$$-value for every main effect and for every interaction effect present in the study. The specific mathematical calculations must be modified depending on whether the study utilizes a between-subjects, within-subjects, or mixed design.

Factorial ANOVA

In a factorial design study by Schnall and colleagues, private body consciousness was utilized as a non-manipulated independent variable alongside a manipulated variable. The researchers actively manipulated disgust by placing participants in either a clean or messy room, but they simply measured participants' pre-existing levels of private body consciousness. This illustrates how experimental studies often combine manipulated factors with measured participant variables to examine their joint effects.

Example of a Non-Manipulated Independent Variable: Private Body Consciousness

An experiment by Schnall and colleagues on disgust and moral judgments serves as a key example of a factorial design because it includes more than one independent variable. To examine a complex psychological phenomenon, the researchers investigated the impact of two distinct independent variables—the manipulated physical cleanliness of a testing room (clean versus messy) and the measured participant variable of private body consciousness—on the dependent variable of how harshly participants judged moral transgressions. Because the study goes beyond a simple one-variable question and incorporates multiple independent variables, it typifies a factorial design.

Example of a Factorial Design

A $$2 \times 2 \times 2$$ factorial design contains three independent variables with two levels each, producing eight total conditions. For example, psychotherapy type, psychotherapy length, and therapist sex could each have two levels, creating all combinations of those factors.

Example of a $$2 \times 2 \times 2$$ Factorial Design

What is the defining characteristic of a factorial design?

In a factorial experiment, the main effects of different independent variables operate entirely independently of each other. Knowing that a main effect exists for one independent variable provides no predictive information regarding the presence, absence, or magnitude of a main effect for another independent variable in the same study.

Independence of Main Effects

Consider a $$2 	imes 2$$ factorial study on driving performance involving the independent variables of cell phone use and time of day. A main effect of cell phone use is evident if, when averaging across both day and night conditions, participants without cell phones demonstrate better driving performance than those using them. Likewise, a main effect of time of day occurs if daytime driving is superior to nighttime driving, regardless of cell phone use. As another example, in a study evaluating psychotherapy, a main effect of psychotherapy length is demonstrated if longer therapy consistently yields better outcomes when averaging across all types of therapy.

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