10.3: General Linear Model

These patterns or these results are generated by the statistical tests we will address in this book. Statistical tests are derived from statistical models. A model is how we describe the pattern among the variables. Think of a model as an equation of variables and the process of assembling the variables together to produce an outcome. Models are good for prediction. If we include these variables and describe the relationships between them, we can predict an outcome.

You can think of a statistical model like a recipe. A recipe has your ingredients; these are your independent variables. How you put the variables together is similar to how you describe the relationships among the variables. In baking, you put the dry ingredients together and the wet ingredients together. A recipe has an outcome, your finished product. In baking, you make cookies. The relationships among the ingredients, or your independent variables, will be associated with the outcome. One association is that the more sugar you put into your cookie, the sweeter the cookie. The more baking powder you put into your cookie, the more the cookie will rise. The more baking soda you put into your cookie, the more the cookie will spread (I learned this baking association – powder puffs, soda spreads). Another association is that if you use white sugar, your cookie will rise more and if you use brown sugar, your cookie will be denser. If you use regular flour, your cookie will be chewy and dense, and if you use cake flour, your cookie will be softer and lighter. We select our ingredients, put them together, and create an outcome. That is one metaphor you can use to learn how statistical models work and how statistical tests are used to predict an outcome.

These tests are based on the general linear model, which is based on frequency statistics, based on the central limit theorem. There are many ways to describe the general linear model, but for a starting point, think of the general linear model describing relationships between variables using straight lines. The linear model is one of the following scenarios: a) as variable X increases, variable Y increases, or b) as variable X increases, Variable Y decreases, or c) group A is higher than group B, or d) group A is lower than group B.

In psychology, here is an example of scenario

A – as your client attends more counseling sessions, your client’s positive well-being increases; scenario

B – as your client attends more counseling sessions, your client’s symptom severity decreases; scenario

C – mindfulness treatment has more positive well-being compared to medication only; scenario

D – mindfulness treatment has less symptom distress compared to medication only.

Of note, when describing linear relationships, it is best to always think of the variable X as increasing. This practice is good for describing relationships between several variables. When describing relationships, you want to keep these descriptions consistent, so it is easy to read. By stating that variable X is increasing across all the relationships, the reader does not have to do engage in cognitive mind flips to understand which variables are increasing or decreasing.

You are right to wonder if there are any situations where you can state, “as variable X decreases, variable Y decreases?” Technically speaking, that description is the same as this statement – “as variable X increases, variable Y increases.” Low scores on variable X correspond to low scores on variable Y. There are situations where that description fits the relationship between the variables. The less minutes you spend exercising, your health decreases. The less minutes you spend on homework, your test scores decrease. If your intent is to provide results to support those statements, then phrasing the relationship in terms of both variable X and Y decreasing could make sense. There is nothing wrong with those statements. Conventionally, we are used to reading relationships between variables as, when variable X increases, variable Y increases, or decreases, because it is easier to have the first variable increase, and then determine if variable Y increases or decreases. Phrasing the relationships between the variables takes practice and knowledge of the concepts you are trying to describe. While technically you can phrase the relationship between variables X and Y anyway you want, for ease of reading and keeping the relationships consistent, it is simpler to describe your relationships by starting off with stating that variable X is increasing.

The general linear model means that you want to think about your variables and the linear relationships they have with each other as you embark on your statistical analyses. These linear relationships are what you are looking for in terms of patterns or results.

There are other statistical models, and there are other ways of describing relationships between variables. The general linear model uses frequency statistics. It means that we count the number of times we see an observation and determine its significance based on how rare or infrequent it is. Other models, such as Bayesian models, use probability models. Briefly, these models determine the likelihood that something is going to happen. There are other statistical patterns, such as latent variable models, where we determine if there is a latent or hidden trait that is responsible for the variations in observed variables. The point is that there are many models that we use to describe the variations in the observed variables, and how these variables relate to each other.

The implication is that if the general linear model does not seem like a good fit for describing the variations in your variables and the relationships between the variables, then there are other models that suit your purposes. The rat’s ass part of this discussion is this – you want to get comfortable with a statistical model, in this case the general linear model, so you can organize your understanding of how results from statistical tests can help you understand the phenomenon you are trying to study.

You might think that these patterns are too simple. Psychology is much more complicated than the general linear model and its two patterns – as X increases, Y increases or decreases, or Group A is higher or lower than Group B. In one sense, you are right. Psychology is complex and intricate, and relationships are complicated. Advanced thinking, advanced models, advanced equations, advanced theories, and advanced discussions are needed to understand these relationships. The general linear model may not be well suited to understand these complex situations. The model does have limitations, such as a) if the data do not conform to a normal distribution, we cannot use statistics based on the general linear model, and b) if the patterns between variables do not follow a linear model, then the results obtained from statistical models do not apply. In these cases, we cannot use the linear model.

However, to learn this craft of statistics, we want to start at a simple place. We need to understand the basics of statistics and start with the general linear model before advancing. Advanced statistics do use statistics based on the general linear model. For example, the concept of correlation permeates the process of understanding a latent variable model, such as factor analysis. It is not like you will leave basic statistics behind once you venture into advanced statistics.

Before writing off basic statistics and the general linear model as too simplistic for the issues you want to investigate, consider Albert Einstein’s statement: “If you can't explain it simply, you don't understand it well enough.” The art of statistics and research is to understand complicated phenomena in simple terms. The K.I.S.S method applies here (keep it simple, stupid). Racism, the LGBTQ+ coming out process, PTSD, autism, treatment resistance, and mental health stigma in the Muslim religion —all are complicated issues that deserve simple explanations. Mind you, simple models might not fit these issues, but with simple analyses, we can begin the process of understanding these complicated issues and then move on to complex models.

Beware of people who try to sound, talk, and act in a complex and complicated way. Intelligence is not about going off on possibilities, or about spiraling and going down rabbit holes, by getting off target (“what if…”), or thinking about different contexts (“it depends on….”). Intelligence is not about criticizing (“your sample size is too small…”; “you did not include this variable…”; “your sample is not representative and generalizable”). Intelligence is not about intimidating others based on what you think you know. The sign of intelligence is to be simple. We want to have good bedside manners when discussing statistics and research. We want others to understand us. The art of intelligence is to be succinct. We want economical, elegant explanations for complex phenomena. We also don’t want to make things too complicated for ourselves. Recovery in mental health is a complicated process. But in some ways, the recovery process is simple – pay attention to your mental health, value your mental health, build your physical and mental and emotional strength to do things that are in your best interest, and avoid the things you know are not in your best interests.

Keep it simple, in life, and in statistics.

Helpful Websites

https://conjointly.com/kb/general-linear-model/

https://statsthinking21.github.io/statsthinking21-core-site/the-general-linear-model.html