# Rule for Smaller factors become more important when....

• blueshifter
In summary: Let's say that you have a model that statistically predicts success based on a number of factors. How do you interpret the influence of one factor when it is highly correlated with another? One approach is to measure the effect of each factor holding the other factors constant. But the other approach, which we've been discussing, is to see how much the influence of a factor changes with the other factors. This is the heart of the question.In summary, the conversation discusses the concept of smaller factors becoming more important when two opposing individuals are closer in skill. Examples from different fields such as math, logic, and physics are given to illustrate this idea. The conversation also mentions the use of statistical methods such as step-wise linear regression
blueshifter
Hi all,

Is there a rule in math or logic that states something like this:

When Skill is the Primary factor in determining an outcome,
When two opposing people are closer in skill, then smaller factors become more important?
Lets say two warriors are battling it out and are close to being equally skilled, secondary factors or advantages become more important.

Lets say one warrior's armor protects him 40 percent more... the closer the two warriors are in skill the more important that armor might become. Now the statistical discrepancy of the armor and winning might not be that noticeable if the skill level between the two is quite different, but the armor statistically might become more and more important if the primary determination: aka "skill" is very close between the two.

Or in Physics perhaps those gravity race cars friction is the primary difference between each car.. but when friction becomes closer and closer to optimal between cars... then center of gravity becomes more and more important.

Hopefully this gives you the idea of what I am looking for... These are probably not the best examples, but I think you guys will catch on to what I am looking for.

It could be a logic, math, or statistic rule I'm thinking of... but if you know anything similar from any field of study please let me know. Thanks for your time any help would be greatly appreciated.

I think the proper terminology for what you want is not a "rule". What you want is a mathematical model for competition that represents various aspects of the combatants, some of which are "skills" and you want the formula in the model to have the property that when the combatants are nearly equal in some set of skills then small differences in those (or other) properties become important in determining the outcome. To get suggestions for a specific model, you have to be specific about what an "outcome" is. Is the "outcome" a single win-or-lose result? Is it determined by a single random draw? Or are there degrees of winning (or losing), like scores in a football match?

FactChecker
Yes. This is a concept that is very important in statistical analysis. In your first example, suppose you did a statistical analysis that predicted success based on skill. You can then remove all the influence of skill and see how strongly the remaining success was related to the quality of armor. Remember that more skilled warrior probably tends to also have better armor. This says that skill and armor quality are correlated. So a valid statistical model would only allow a warrior's armor quality to influence success to the extent that armor quality was above or below the expected armor quality for his skill level. There would be another factor which is the combination of skill and quality. The overall success is influenced by the combination of both skill and armor quality and the two can be traded off. A lesser skilled warrior might still win if his armor is much better. The subjects of step-wise linear regression and variance analysis put these ideas into formal mathematical methods.

I don't know if this is what FactChecker was referring to, but PCA (Principal Component Analysis) and Factor Analysis may be methods that interest you, which derive, as FactChecker observed from (co)variance and (auto)correlation methods.

FactChecker
WWGD said:
but PCA (Principal Component Analysis) and Factor Analysis may be methods that interest you,
Yes. Those are two more good examples. In fact, I think that the original question is fundamental to most statistical methods.

## 1. What does "Rule for Smaller factors become more important when..." mean?

The "Rule for Smaller factors become more important when..." refers to a principle in science where smaller factors, such as individual characteristics or environmental influences, play a larger role in determining an outcome or effect.

## 2. Why do smaller factors become more important?

Smaller factors become more important when they have a larger impact on the outcome compared to larger factors. This can be due to their direct influence or the cumulative effect of multiple smaller factors.

## 3. Can you provide an example of this rule in action?

Sure, an example of this rule is in genetics where smaller genetic variations can greatly influence an individual's susceptibility to a disease, even if they have a strong family history of the disease. In this case, the smaller genetic factors become more important in determining the likelihood of developing the disease.

## 4. How does this rule apply in research studies?

In research studies, this rule can help scientists understand the underlying mechanisms or causes of an effect by looking at smaller factors that may have a larger impact. This can lead to more targeted and effective interventions or treatments.

## 5. Are there any limitations to this rule?

While this rule can be useful in explaining certain phenomena, it is important to note that it is not always applicable. There are cases where larger factors may still have a significant impact on the outcome, and smaller factors may not always be the main determining factor.

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