In some recent improvement project work that was sent to me to review, I noticed the project leader had undertaken a data analysis and had referenced the Coefficient R² to validate correlation between certain variables.

Because this was not the correct way to do it, and having seen this on a number of occasions, I thought it timely write about this topic and share with our network of business improvement practitioners.

Please note – if you are not a user of quantitative analysis tools as applied in Six Sigma methods, this will probably not be of interest to you.

# Limitations of R²

**First let’s look at the limitations of using the R² value to draw conclusions about correlation.**

The R-squared (R²) value is a commonly used metric in regression analysis to assess the goodness-of-fit of a model. It measures the proportion of the variance in the dependent variable that is predictable from the independent variables.

While R² can be useful in determining the strength of the relationship between variables, it has several limitations when it comes to drawing conclusions about correlation:

### 1. Linearity assumption:

R² is most appropriate for linear regression models, meaning it assumes a linear relationship between the independent and dependent variables. If the relationship is non-linear, R² may not be an accurate representation of the strength of the relationship.

### 2. Sensitivity to outliers:

R² can be heavily influenced by outliers in the data. A few extreme observations can disproportionately affect the R² value, making it difficult to draw accurate conclusions about the correlation.

### 3. Correlation ≠ causation:

A high R² value indicates a strong relationship between the independent and dependent variables, but it does not imply causation. It is crucial to remember that correlation does not equal causation, and further investigation is needed to determine if there is a causal relationship between the variables.

### 4. Multiple independent variables:

In multiple regression, the R² value measures the proportion of variance explained by all the independent variables together. It does not indicate the individual contributions of each independent variable to the dependent variable, making it difficult to determine which variables have the strongest correlations.

### 5. Overfitting:

A high R² value might result from overfitting the model to the data, particularly when there are many independent variables or when the sample size is small.

In such cases, the model may capture noise in the data rather than the true underlying relationship, leading to misleading conclusions.

### 6. R² increases with more independent variables:

R² value will never decrease when adding more independent variables to the model, even if they have little to no correlation with the dependent variable.

This can give a false impression of improved model performance when, in reality, the added variables may not be meaningful.

### 7. Scale dependency:

R² is sensitive to the scale of the variables, which may affect its interpretation. When comparing models with different units or scales, R² values may not be directly comparable.

To address some of these limitations, other metrics like Adjusted R² (Adj-R²), which takes into account the number of independent variables and sample size, or examining residuals and their patterns can be helpful.

It is essential to use multiple evaluation metrics and consider the context of the data and research question when drawing conclusions about correlation.

# Choosing Between R² and R

**So what is the best statistic to use when studying correlation between variables – is it R² or R?**

The best statistic to use when studying the correlation between variables depends on the context and the specific research question.

Here’s a brief comparison between R² and R (correlation coefficient) to help you decide which one is more appropriate for your study:

### 1. R (Correlation Coefficient):

The Pearson correlation coefficient (denoted as ‘r’) measures the strength and direction of a linear relationship between two continuous variables.

R ranges from -1 to 1, where -1 indicates a strong negative linear relationship, 1 indicates a strong positive linear relationship, and 0 indicates no linear relationship.

R is more appropriate when you want to assess the bivariate relationship between two variables without considering other factors.

ADVANTAGES:

– Easy to interpret and understand

– Takes into account both the strength and direction of the relationship

LIMITATIONS:

– Assumes a linear relationship between variables

– Sensitive to outliers

– Cannot be used to infer causality

### 2. R² (Coefficient of Determination):

R² is the square of the correlation coefficient (R), used primarily in regression analysis.

It represents the proportion of variance in the dependent variable that can be explained by the independent variables in the model.

R² is more suitable when you are interested in understanding how well a set of independent variables explains the variance in the dependent variable.

ADVANTAGES:

– Indicates the proportion of variance explained by the model

– Useful for comparing the goodness-of-fit of different models

LIMITATIONS:

– Assumes a linear relationship between variables

– Sensitive to outliers

– Cannot be used to infer causality

– Can be misleading in multiple regression (as adding more variables always increases R²)

In summary, if you want to assess the strength and direction of the linear relationship between two variables, use the correlation coefficient (R).

If you want to understand how well a set of independent variables explains the variance in the dependent variable, use R².

In both cases, remember that correlation does not imply causation, and additional analyses may be necessary to establish causal relationships.

# More Information

This article was written by George Lee Sye, author of Process Mastery with Lean Six Sigma. For more information CLICK HERE where you’ll discover why this is one of the most important text books in the business improvement world today.