Interaction P-values Discrepancy Linear Vs Ordinal Regression

Hey guys! Ever stumbled upon a statistical puzzle that just makes you scratch your head? Well, I recently encountered one that's been quite the brain-teaser, and I thought I'd share it with you all. It involves a fascinating divergence in interaction p-values between linear and ordinal regression models. Let's dive into the details and try to unravel this mystery together!

The Experiment Setup

So, picture this: I'm running an experiment with two independent variables (IVs) and one dependent variable (DV). It sounds straightforward enough, right? But here's where things get interesting:

• Independent Variable 1: This one's discrete, meaning it can only take on specific, separate values. Think of it like the number of items someone purchases – you can buy 2, 3, or 150 items, but not 2.5. The range for this IV is from 2 to 150.
• Independent Variable 2: Now, this one's continuous, meaning it can take on any value within a given range. Imagine it as a percentage or a proportion, like the discount rate applied to a product. In my case, this IV ranges from 0 to 1.
• Dependent Variable: This is where things get a little more nuanced. My dependent variable is ordinal, which means it has a natural order or ranking, but the intervals between the values aren't necessarily equal. Think of a customer satisfaction rating on a scale of 1 to 5 – a rating of 4 is higher than 2, but the difference between 4 and 3 might not be the same as the difference between 2 and 1.

The heart of the issue lies in understanding the interaction between these independent variables and how they influence the ordinal dependent variable. The main question is, why is there such a massive difference in interaction p-values between linear and ordinal regression models (0.991 vs. 0.001)? Let’s break this down.

The P-Value Puzzle 0.991 vs 0.001

The p-value discrepancy between the linear and ordinal regression models is striking – we're talking about 0.991 in the linear model versus 0.001 in the ordinal model! This huge difference suggests that the way these models are interpreting the interaction effect is fundamentally different. To understand this, we need to delve into the assumptions and mechanics of each type of regression.

First, let’s recap what a p-value actually tells us. In simple terms, the p-value is the probability of observing results as extreme as, or more extreme than, the results obtained, assuming that there is no true effect (the null hypothesis). A small p-value (typically ≤ 0.05) suggests strong evidence against the null hypothesis, indicating that there is a statistically significant effect. Conversely, a large p-value suggests weak evidence against the null hypothesis, implying that the observed effect could be due to random chance.

Now, let's consider the linear regression model. Linear regression assumes that the relationship between the independent and dependent variables is linear – meaning a straight line can represent the relationship. It also assumes that the residuals (the differences between the observed and predicted values) are normally distributed and have constant variance (homoscedasticity). When we apply linear regression to an ordinal dependent variable, we're essentially treating it as if it were continuous, which can be problematic because it ignores the inherent ordering and unequal intervals between the categories.

In the context of interaction effects, linear regression assesses whether the effect of one independent variable on the dependent variable changes depending on the value of the other independent variable. A p-value of 0.991 in the linear model suggests that there is virtually no statistically significant interaction effect. In other words, the model is telling us that the effect of IV1 on the DV doesn't change much as IV2 varies, and vice versa. This could be because the linear model is failing to capture the true nature of the relationship due to its assumptions being violated by the ordinal DV.

On the other hand, ordinal regression models, such as ordered logistic or ordered probit regression, are specifically designed for ordinal dependent variables. These models account for the ordered nature of the categories and do not assume equal intervals between them. They use a different approach, often involving cumulative probabilities and link functions, to model the relationship between the IVs and the DV. A p-value of 0.001 in the ordinal model strongly suggests a statistically significant interaction effect. This means that the effect of one IV on the DV does depend on the value of the other IV. The ordinal model is likely picking up a pattern that the linear model is missing.

So, why the huge difference? It boils down to the models' assumptions and how they handle the ordinal nature of the dependent variable. The linear model's assumptions are likely being violated, leading to an inaccurate assessment of the interaction effect. The ordinal model, being specifically designed for this type of data, provides a more appropriate and sensitive analysis.

Digging Deeper Why the Models Diverge

To truly grasp why these models are giving us such contrasting results, we need to understand their underlying mechanisms and assumptions. Let's break down the key differences:

Linear Regression Assumptions

Linear regression, at its core, operates under a few critical assumptions that, when violated, can lead to misleading conclusions. These assumptions include:

• Linearity: This is a big one. Linear regression assumes that the relationship between the independent variables and the dependent variable is linear. In other words, a straight line can adequately describe the relationship. When your dependent variable is ordinal, this assumption is often shaky. Ordinal data implies a ranking, but the intervals between ranks aren't necessarily equal. For example, the difference between