Calculate Comparative Fit Measures for Regression Models

Description

Calculates goodness-of-fit metrics based on Kvalseth (1985), including Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Squared Error (MSE). This function provides a unified output for comparing different model specifications.

Usage

comp_fit(model, type = c("auto", "linear", "power"))

RMSE(model, type = c("auto", "linear", "power"))

MAE(model, type = c("auto", "linear", "power"))

MSE(model, type = c("auto", "linear", "power"))

Arguments

Details

The metrics are calculated according to the formulas in Kvalseth (1985):

• RMSE: Root Mean Squared Residual or Error

  RMSE = \sqrt{\frac{\sum (y - \hat{y})^2}{n}}

• MAE: Mean Absolute Residual or Error

  MAE = \frac{\sum |y - \hat{y}|}{n}

• MSE: Mean Squared Residual or Error (Adjusted for degrees of freedom)

  MSE = \frac{\sum (y - \hat{y})^2}{n - p}

where n is the sample size and p is the number of model parameters (including the intercept).

Note on MSE: In many modern contexts, "MSE" refers to the mean squared error without degree-of-freedom adjustment (denominator n). However, this function follows Kvalseth's definition, which uses n - p as the denominator.

Value

• For comp_fit(): An object of class comp_kvr2, which is a list containing the calculated RMSE, MAE, and MSE values.

• For individual functions (RMSE(), MAE(), MSE()): A named numeric value of the specific metric.

Note

The power regression model must be based on a logarithmic transformation.

When type = "auto", the choice between linear and power regression is determined by analyzing the model formula. It identifies a power regression if the dependent variable is a function call to log() (e.g., lm(log(y) ~ x)).

Note that simple variable names containing the string "log" (e.g., lm(log_value ~ x)) are correctly treated as linear regression. To override this automatic detection, manually specify type = "linear" or type = "power".

References

Tarald O. Kvalseth (1985) Cautionary Note about R 2 , The American Statistician, 39:4, 279-285, doi: 10.1080/00031305.1985.10479448

See Also

print.comp_kvr2()

Examples

# example data set 1. Kvålseth (1985).
df1 <- data.frame(x = c(1:6),
                 y = c(15,37,52,59,83,92))
model_intercept <- lm(y ~ x, df1)
model_without <- lm(y ~ x - 1, df1)
model_power <- lm(log(y) ~ log(x), df1)
comp_fit(model_intercept)
comp_fit(model_without)
comp_fit(model_power)

Contrast R-squared Definitions: Intercept vs. No-Intercept

Description

A specialized tool for educational and diagnostic purposes. This function automatically generates a comparison between a model with an intercept and its forced no-intercept counterpart (or vice versa), revealing how mathematical definitions of R-squared diverge under different constraints.

Usage

comp_model(model, type = c("auto", "linear", "power"), adjusted = FALSE)

Arguments

Details

This function reconstructs the alternative model using QR decomposition rather than update() to ensure robustness against environment/scoping issues.

It is particularly useful for observing how definitions like R^2_2 can exceed 1.0 or R^2_1 can become negative when an intercept is removed, illustrating the "pitfalls" discussed in Kvalseth (1985).

Value

A data frame of class comp_model containing nine R-squared definitions and three fit metrics (RMSE, MAE, MSE) for both intercept and no-intercept versions.

The original model objects are stored as attributes with_int and without_int for use by the plot method.

References

Kvalseth, T. O. (1985) Cautionary Note about R2. The American Statistician, 39(4), 279-285.

Examples

df1 <- data.frame(x = 1:6, y = c(15, 37, 52, 59, 83, 92))
model <- lm(y ~ x, data = df1)

# Compare R-squared sensitivity
comp_model(model)

# Compare adjusted R-squared
comp_model(model, adjusted = TRUE)

Get Model Information Used for Calculations

Description

Extracts the metadata and model specifications used to calculate the coefficients of determination, such as the regression type, sample size, and degrees of freedom.

Usage

model_info(x)

Arguments

Details

This function provides transparency into the calculation process of the various R-squared definitions. It is particularly useful for verifying whether a model was treated as a "power" regression (log-transformed) and how the degrees of freedom were determined for adjusted R-squared values.

Value

A list containing the following components:

• type: A string indicating the regression type ("linear" or "power").

• has_intercept: A logical value indicating if the model includes an intercept.

• n: The number of observations used in the model (excluding missing values).

• k: The number of estimated parameters (including the intercept if present).

• df_res: Residual degrees of freedom (n - k).

Note

The sample size n refers to the actual number of observations used by lm(), which may be fewer than the rows in the original data frame if NA values were present.

See Also

r2(), comp_fit()

Examples

df1 <- data.frame(x = 1:6, y = c(15, 37, 52, 59, 83, 92))
model <- lm(y ~ x, data = df1)
res <- r2(model)

# Check the metadata
info <- model_info(res)
info$n
info$type

Plot Comparison of Model Specifications

Description

Generates a comprehensive 2x2 diagnostic dashboard comparing models with and without an intercept. This visualization helps identify how the absence of an intercept affects different R-squared definitions and error metrics.

Usage

## S3 method for class 'comp_model'
plot(x, ...)

Arguments

Details

The plot is organized into four panels:

• Top-Left: Grouped bar chart of the nine R-squared definitions.

• Bottom-Left: Comparison of absolute fit metrics (RMSE, MAE, MSE).

• Top-Right: Observed vs. Predicted plot for the intercept model.

• Bottom-Right: Observed vs. Predicted plot for the no-intercept model.

This layout allows for a direct "cause-and-effect" analysis: for instance, observing a data point far from the identity line in the bottom-right panel explains why certain R-squared definitions might crash or become negative in the left panels.

Value

This function is primarily called for its side effect of creating a grid-based plot. It returns the input object x invisibly.

Note

Since this plot uses the grid system to combine multiple ggplot objects, it cannot be further modified with the + operator. If you need to customize individual panels, use the internal plotting functions or extract the models from the comp_model object attributes.

See Also

comp_model(), plot_diagnostic()

Examples

df <- data.frame(x = 1:5, y = c(2, 3, 5, 4, 6))
m1 <- lm(y ~ x, data = df)
res <- comp_model(m1)
plot(res)

Plot Method for r2_kvr2 Objects

Description

Visualizes the nine definitions of R-squared to compare their values and identify potential issues (e.g., values exceeding 1 or falling below 0).

Usage

## S3 method for class 'r2_kvr2'
plot(x, ...)

Arguments

Value

A ggplot object representing the visual analysis.

Examples

df1 <- data.frame(x = 1:6, y = c(15, 37, 52, 59, 83, 92))
model <- lm(y ~ x - 1, data = df1) # No-intercept model
r2(model)

Plot Observed vs Predicted Values

Description

A diagnostic plot to visualize why R-squared might be low or negative. It compares the model predictions (identity line) against the mean (horizontal line).

Usage

plot_diagnostic(x, ...)

Arguments

Value

A ggplot object representing the visual analysis.

Examples

df1 <- data.frame(x = 1:6, y = c(15, 37, 52, 59, 83, 92))
model <- lm(y ~ x - 1, data = df1) # No-intercept model
plot_diagnostic(model)

Plot Method for Kvalseth's R-squared Objects

Description

Visualizes the different R-squared definitions or provides a diagnostic observed-vs-predicted plot to understand the model fit.

Usage

plot_kvr2(
  x,
  type = c("auto", "linear", "power"),
  plot_type = c("both", "r2", "diag"),
  ...
)

Arguments

Details

When plot_type = "r2", the function creates a bar plot comparing all nine definitions. Bars are colored based on their validity:

• Skyblue: Standard values between 0 and 1.

• Orange: Values exceeding 1.0 or falling below 0.0 (warnings).

When plot_type = "diag", the function displays a scatter plot of observed vs. predicted values. Two reference lines are added:

• Darkgreen Solid Line: The 1:1 "perfect fit" line (RSS reference).

• Red Dashed Line: The overall mean of the observed data (TSS reference).

If the data points are closer to the red dashed line than the green solid line, R^2_1 will be negative.

Combined View (plot_type = "both"): Automatically configures the plotting device to show both plots simultaneously for a comprehensive model evaluation.

Value

The return value depends on the plot_type argument:

• For "r2" and "diag": Returns a ggplot object that can be further customized.

• For "both": Generates a combined plot using the grid system and returns the input object x invisibly.

Examples

df1 <- data.frame(x = 1:6, y = c(15, 37, 52, 59, 83, 92))
model <- lm(y ~ x - 1, data = df1) # No-intercept model
plot_kvr2(model)
# Compare all definitions
plot_kvr2(model, plot_type = "r2")

# Diagnostic plot to see why some R2 might be problematic
plot_kvr2(model, plot_type = "diag")

Print Method for Model Comparison Objects

Description

A specialized print method for comp_model objects. It formats the comparison table for better readability and provides diagnostic warnings if any R-squared values fall outside the standard 0 to 1 range.

Usage

## S3 method for class 'comp_model'
print(x, ..., digits = 4)

Arguments

Details

The output is formatted using the insight package's export_table() functionality, ensuring a clean and structured display in the console.

In addition to the table, this method performs an automated check on the R-squared values (columns 2 to 10). If any value exceeds 1.0 or falls below 0.0, a warning message is displayed. This is a critical educational feature, as it flags instances where specific R^2 definitions become mathematically inappropriate due to the lack of an intercept or model misspecification.

Value

Returns the input object x invisibly.

See Also

comp_model()

Print Methods for r2 and comp_fit calculation Objects

Description

Printing objects of class "r2_kvr2" (generated by r2()) or "comp_kvr2" (generated by comp_fit()), respectively, by simple print methods.

Usage

## S3 method for class 'r2_kvr2'
print(x, ..., digits = 4, model_info = TRUE)

## S3 method for class 'comp_kvr2'
print(x, ..., digits = 4, model_info = TRUE)

Arguments

Details

These methods format the calculated statistics into a human-readable summary, displaying each index or metric with its corresponding value.

Value

The input object is returned invisibly (via invisible(x)). This function is called for its side effect of printing the results of r2() or comp_fit() calculations to the console.

See Also

r2() comp_fit() r2_adjusted()

Calculate Multiple Definitions of Coefficient of Determination (R-squared)

Description

Calculates nine types of coefficients of determination (R^2) based on the classification by Kvalseth (1985). This function is designed to demonstrate how R^2 values can vary depending on their mathematical definition, particularly in models without an intercept or in power regression models

Usage

r2(model, type = c("auto", "linear", "power"), adjusted = FALSE)

r2_1(model, type = c("auto", "linear", "power"))

r2_2(model, type = c("auto", "linear", "power"))

r2_3(model, type = c("auto", "linear", "power"))

r2_4(model, type = c("auto", "linear", "power"))

r2_5(model, type = c("auto", "linear", "power"))

r2_6(model, type = c("auto", "linear", "power"))

r2_7(model, type = c("auto", "linear", "power"))

r2_8(model, type = c("auto", "linear", "power"))

r2_9(model, type = c("auto", "linear", "power"))

Arguments

Details

The nine coefficient equations from R^2_1 to R^2_9 are based on Kvalseth (1985) and are as follows:

• R^2_1: Proportion of total variance explained.

  R_1^2 = 1 - \frac{\sum(y - \hat{y})^2}{\sum(y - \bar{y})^2}

• R^2_2: Based on the variation of predicted values.

  R^2_2 = \frac{\sum(\hat{y} - \bar{y})^2}{\sum(y - \bar{y})^2}

• R^2_3: Ratio of variation using the mean of predicted values.

  R_3^2 = \frac{\sum(\hat{y} - \bar{\hat{y}})^2}{\sum(y - \bar{y})^2}

• R^2_4: Incorporates the mean residual.

  R_4^2 = 1 - \frac{\sum(e - \bar{e})^2}{\sum(y - \bar{y})^2}, \quad e = y - \hat{y}

• R^2_5: The square of the multiple correlation coefficient between the dependent variable and the independent variable (a comprehensive indicator in linearized models).

  R_5^2 = \text{squared multiple correlation coefficient between the regressand and the regressors}

• R^2_6: Square of Pearson's correlation coefficient between observed y and predicted \hat{y}.

  R_6^2 = \frac{\left( \sum(y - \bar{y})(\hat{y} - \bar{\hat{y}}) \right)^2}{\sum(y - \bar{y})^2 \sum(\hat{y} - \bar{\hat{y}})^2}

• R^2_7: Recommended for models without an intercept.

  R_7^2 = 1 - \frac{\sum(y - \hat{y})^2}{\sum y^2}

• R^2_8: Alternative form for models without an intercept.

  R_8^2 = \frac{\sum \hat{y}^2}{\sum y^2}

• R^2_9: Robust version using medians to resist outliers.

  R_9^2 = 1 - \left( \frac{M\{|y_i - \hat{y}_i|\}}{M\{|y_i - \bar{y}|\}} \right)^2

where M represents the median of the sample.

For degree of freedom adjustment adjusted = TRUE, refer to r2_adjusted.

Value

• For r2(): An object of class r2_kvr2, which is a list containing calculated values for all R^2 formulas.

• For individual functions (r2_1() to r2_9()): A named numeric value of the specific R^2 definition. calculated values for each R^2 formula.

Note

The power regression model must be based on a logarithmic transformation.

When type = "auto", the choice between linear and power regression is determined by analyzing the model formula. It identifies a power regression if the dependent variable is a function call to log() (e.g., lm(log(y) ~ x)).

Note that simple variable names containing the string "log" (e.g., lm(log_value ~ x)) are correctly treated as linear regression. To override this automatic detection, manually specify type = "linear" or type = "power".

References

Tarald O. Kvalseth (1985) Cautionary Note about R 2 , The American Statistician, 39:4, 279-285, doi:10.1080/00031305.1985.10479448

Box, George E. P., Hunter, William G., Hunter, J. Stuart. (1978) Statistics for experimenters: an introduction to design, data analysis, and model building. New York, United States, J. Wiley, p. 462-473, ISBN:9780471093152.

See Also

print.r2_kvr2() r2_adjusted()

Examples

# Example data set 1. Kvalseth (1985).
df1 <- data.frame(x = c(1:6),
                 y = c(15,37,52,59,83,92))
# Linear regression model with intercept
model_intercept1 <- lm(y ~ x, df1)
# Linear regression model without intercept
model_without1 <- lm(y ~ x - 1, df1)
# Power regression model
model_power1 <- lm(log(y) ~ log(x), df1)
r2(model_intercept1)
r2(model_without1)
r2(model_power1)
# Example data set 2. Kvalseth (1985).
df2 <- data.frame(x = 6:13,
                  y = c(3882, 1266, 733, 450, 410, 305, 185, 112))
power_model2 <- lm(log((y/7343)) ~ log(x), data = df2)
r2(power_model2)
# Example of a Multiple Regression Analysis Model.
# The data for two independent variables given by Box et al. (1978, p. 462)
# as used in Kvalseth (1985).
df3 <- data.frame(x1 = c(0.34, 0.34, 0.58, 1.26, 1.26, 1.82),
                  x2 = c(0.73, 0.73, 0.69, 0.97, 0.97, 0.46),
                  y = c(5.75, 4.79, 5.44, 9.09, 8.59, 5.09))
# Multiple regression analysis model with intercept
model_intercept3 <- lm(y ~ x1 + x2, df3)
# Multiple regression analysis model without intercept
model_without3 <- lm(y ~ x1 + x2 - 1, df3)
# Multiple power regression analysis model
model_power3 <- lm(log(y) ~ log(x1) + log(x2), df3)
r2(model_intercept3)
r2(model_without3)
r2(model_power3)

Calculate the Adjusted Determination Coefficient

Description

Calculate the adjusted coefficient of determination by entering the regression model and coefficient of determination. See details.

Usage

r2_adjusted(model, r2)

Arguments

Details

The adjustment factor a is calculated using the following formula.

a = (n - 1) / (n - k - 1)

n is the sample size, and k is the number of parameters in the regression model.

R^2_a (R^2 adjusted) is calculated using the following formula.

R^2_a = 1 - a (1 - R^2)

This function performs freedom-of-degrees adjustment for all coefficients based on the above formula. However, Kvalseth (1985) recommends applying freedom-of-degrees adjustment only to R^2_1 and R^2_9, based on the principle of consistency in coefficients. Furthermore, there is no basis for applying the same type of adjustment to R^2_6 (the square of the correlation coefficient) or to R^2_7 and R^2_8, which depend on specific model forms.

For details on each coefficient of determination, refer to r2().

Value

A numeric vector or a list of class r2_kvr2 containing the adjusted R^2 values. Each element represents the adjusted version of the corresponding R^2 definition, accounting for the degrees of freedom.

References

Tarald O. Kvalseth (1985) Cautionary Note about R 2 , The American Statistician, 39:4, 279-285, doi:10.1080/00031305.1985.10479448

See Also

r2()