Type:	Package
Title:	Simulate and Diagnose (Generalized) Linear Models
Version:	0.2.0
Date:	2024-08-16
Description:	Simulate samples from populations with known covariate distributions, generate response variables according to common linear and generalized linear model families, draw from sampling distributions of regression estimates, and perform visual inference on diagnostics from model fits.
URL:	https://www.refsmmat.com/regressinator/, https://github.com/capnrefsmmat/regressinator
BugReports:	https://github.com/capnrefsmmat/regressinator/issues
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.3.1
Depends:	R (≥ 4.1)
Imports:	broom, cli, DHARMa, dplyr, ggplot2, insight, nullabor, purrr, rlang, tibble, tidyr, tidyselect
Suggests:	rmarkdown, knitr, mvtnorm, palmerpenguins, patchwork, testthat (≥ 3.0.0)
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2024-08-16 15:07:24 UTC; alexreinhart
Author:	Alex Reinhart [aut, cre]
Maintainer:	Alex Reinhart <areinhar@stat.cmu.edu>
Repository:	CRAN
Date/Publication:	2024-08-16 15:40:02 UTC

regressinator: Simulate and Diagnose (Generalized) Linear Models

Description

Simulate samples from populations with known covariate distributions, generate response variables according to common linear and generalized linear model families, draw from sampling distributions of regression estimates, and perform visual inference on diagnostics from model fits.

Author(s)

Maintainer: Alex Reinhart areinhar@stat.cmu.edu (ORCID)

See Also

Useful links:

• https://www.refsmmat.com/regressinator/

• https://github.com/capnrefsmmat/regressinator

• Report bugs at https://github.com/capnrefsmmat/regressinator/issues

Augment a model fit with residuals, in "long" format

Description

Use broom::augment() to augment a model fit with residual and fit information, then reformat the resulting data frame into a "long" format with one row per predictor per observation, to facilitate plotting of the result.

Usage

augment_longer(x, ...)

Arguments

Details

The name comes by analogy to tidyr::pivot_longer(), and the concept of long versus wide data formats.

Value

A data frame (tibble) in similar form to those produced by broom::augment(), but expanded to have one row per predictor per observation. Columns .predictor_name and .predictor_value identify the predictor and its value. An additional column .obs records the original observation numbers so results can be matched to observations in the original model data.

Limitations

Factor predictors (as factors, logical, or character vectors) can't coexist with numeric variables in the .predictor_value column. If there are some numeric and some factor predictors, the factor predictors will automatically be omitted. If all predictors are factors, they will be combined into one factor with all levels. However, if a numeric variable is converted to factor in the model formula, such as with y ~ factor(x), the function cannot determine the appropriate types and will raise an error. Create factors as needed in the source data frame before fitting the model to avoid this issue.

See Also

partial_residuals(), binned_residuals()

Examples

fit <- lm(mpg ~ cyl + disp + hp, data = mtcars)

# each observation appears 3 times, once per predictor:
augment_longer(fit)

Augment data with randomized quantile residuals

Description

Generates a data frame containing a model's predictors, the residuals, and the randomized quantile residuals as additional columns.

Usage

augment_quantile(x, ...)

augment_quantile_longer(x, ...)

Arguments

Details

Randomized quantile residuals provide more interpretable residuals for generalized linear models (GLMs), such as logistic regression. See Dunn and Smyth (1996) for details, or review the examples provided in vignette("DHARMa", package="DHARMa").

Let F_Y(y; x, \beta) be the predicted cumulative distribution function for Y when X = x, using the fitted GLM. When the response is continuous, the randomized quantile residual for observation i is

r_{q,i} = F_Y(y_i; x_i, \hat \beta).

When the response is discrete, let

a_i = \lim_{y \uparrow y_i} F_Y(y; x_i, \hat \beta)

and

b_i = F_Y(y_i; x_i, \hat \beta),

then draw the randomized quantile residual as

r_{q,i} \sim \text{Uniform}(a_i, b_i).

As cumulative distributions are left-continuous, this "jitters" the values between the discrete steps, resulting in a residual that is uniformly distributed when the model is correct.

Some definitions of randomized quantile residuals transform the resulting values using the standard normal inverse cdf, so they are normally distributed. That step is omitted here, as uniform residuals are easy to work with.

Value

Data frame with one row per observation used to fit x, including a .quantile.resid column containing the quantile residuals. See broom::augment() and its methods for details of other columns.

For augment_quantile_longer(), the output is in "long" format with one row per predictor per observation. Columns .predictor_name and .predictor_value identify the predictor and its value. An additional column .obs records the original observation numbers so results can be matched to observations in the original model data. See Limitations in augment_longer() for limitations on factor predictors.

Implementation details

Uses broom::augment() to generate the data frame, then uses the DHARMa package to generate randomized quantile residuals for the model.

References

Dunn, Peter K., and Gordon K. Smyth (1996). "Randomized Quantile Residuals." Journal of Computational and Graphical Statistics 5 (3): 236–44. doi:10.2307/1390802

See Also

vignette("logistic-regression-diagnostics") and vignette("other-glm-diagnostics") for examples of plotting and interpreting randomized quantile residuals; augment_longer(); broom::augment()

Group a data frame into bins

Description

Groups a data frame (similarly to dplyr::group_by()) based on the values of a column, either by dividing up the range into equal pieces or by quantiles.

Usage

bin_by_interval(.data, col, breaks = NULL)

bin_by_quantile(.data, col, breaks = NULL)

Arguments

Details

bin_by_interval() breaks the numerical range of that column into equal-sized intervals, or into intervals specified by breaks. bin_by_quantile() splits the range into pieces based on quantiles of the data, so each interval contains roughly an equal number of observations.

Value

Grouped data frame, similar to those returned by dplyr::group_by(). An additional column .bin indicates the bin number for each group. Use dplyr::summarize() to calculate values within each group, or other dplyr operations that work on groups.

Examples

suppressMessages(library(dplyr))
cars |>
  bin_by_interval(speed, breaks = 5) |>
  summarize(mean_speed = mean(speed),
            mean_dist = mean(dist))

cars |>
  bin_by_quantile(speed, breaks = 5) |>
  summarize(mean_speed = mean(speed),
            mean_dist = mean(dist))

Obtain binned residuals for a model

Description

Construct a data frame by binning the fitted values or predictors of a model into discrete bins of equal width, and calculating the average value of the residuals within each bin.

Usage

binned_residuals(fit, predictors = !".fitted", breaks = NULL, ...)

Arguments

Details

In many generalized linear models, the residual plots (Pearson or deviance) are not useful because the response variable takes on very few possible values, causing strange patterns in the residuals. For instance, in logistic regression, plotting the residuals versus covariates usually produces two curved lines.

If we first bin the data, i.e. divide up the observations into breaks bins based on their fitted values, we can calculate the average residual within each bin. This can be more informative: if a region has 20 observations and its average residual value is large, this suggests those observations are collectively poorly fit. We can also bin each predictor and calculate averages within those bins, allowing the detection of misspecification for specific model terms.

Value

Data frame (tibble) with one row per bin per selected predictor, and the following columns:

Limitations

Factor predictors (as factors, logical, or character vectors) are detected automatically and omitted. However, if a numeric variable is converted to factor in the model formula, such as with y ~ factor(x), the function cannot determine the appropriate type and will raise an error. Create factors as needed in the source data frame before fitting the model to avoid this issue.

References

Gelman, A., Hill, J., and Vehtari, A. (2021). Regression and Other Stories. Section 14.5. Cambridge University Press.

See Also

partial_residuals() for the related partial residuals; vignette("logistic-regression-diagnostics") and vignette("other-glm-diagnostics") for examples of use and interpretation of binned residuals in logistic regression and GLMs; bin_by_interval() and bin_by_quantile() to bin data and calculate other values in each bin

Examples

fit <- lm(mpg ~ disp + hp, data = mtcars)

# Automatically bins both predictors:
binned_residuals(fit, breaks = 5)

# Just bin one predictor, selected with tidyselect syntax. Multiple could be
# selected with c().
binned_residuals(fit, disp, breaks = 5)

# Bin the fitted values:
binned_residuals(fit, predictors = .fitted)

# Bins are made using the predictor, not regressors derived from it, so here
# disp is binned, not its polynomial
fit2 <- lm(mpg ~ poly(disp, 2), data = mtcars)
binned_residuals(fit2)

Convert factor levels to numeric values

Description

Replace each entry in a vector with its corresponding numeric value, for instance to use a factor variable to specify intercepts for different groups in a regression model.

Usage

by_level(x, ...)

Arguments

Value

Named vector of same length as x, with values replaced with those specified. Names are the original factor level name.

See Also

rfactor() to draw random factor levels, and the forcats package https://forcats.tidyverse.org/ for additional factor manipulation tools

Examples

foo <- factor(c("spam", "ham", "spam", "ducks"))

by_level(foo, spam = 4, ham = 10, ducks = 16.7)

by_level(foo, c("spam" = 4, "ham" = 10, "ducks" = 16.7))

# to define a population with a factor that affects the regression intercept
intercepts <- c("foo" = 2, "bar" = 30, "baz" = 7)
pop <- population(
  group = predictor(rfactor,
                    levels = c("foo", "bar", "baz"),
                    prob = c(0.1, 0.6, 0.3)),
  x = predictor(runif, min = 0, max = 10),
  y = response(by_level(group, intercepts) + 0.3 * x,
               error_scale = 1.5)
)
sample_x(pop, 5)

Check that the model fit uses the data argument to provide data

Description

We simulate by passing simulated data arguments to update(). If the original fit does not use the data argument, and instead refers directly to variables in the environment, the simulations will not behave as expected. This may result in the "simulated" fits all using the original data, for instance.

Usage

check_data_arg(fit)

Arguments

Details

For example, in lm(mtcars$mpg ~ mtcars$drat), simulating new data and providing it in ⁠data =⁠ will not change the data used for fitting.

Detect a missing data argument and abort. It is still possible to provide data but also refer directly to the calling environment, but this is harder to detect.

Value

No value. Raises an error if no data argument was used in fit.

Family representing a GLM with custom distribution and link function

Description

Allows specification of the random component and link function for a response variable. In principle this could be used to specify any GLM family, but it is usually easier to use the predefined families, such as gaussian() and binomial().

Usage

custom_family(distribution, inverse_link)

Arguments

Details

A GLM is specified by a combination of:

• Random component, i.e. the distribution that Y is drawn from

• Link function relating the mean of the random component to the linear predictor

• Linear predictor

Using custom_family() we can specify the random component and link function, while the linear predictor is set in population() when setting up the population relationships. A family specified this way can be used to specify a population (via population()), but can't be used to estimate a model (such as with glm()).

Value

A family object representing this family

See Also

ols_with_error() for the special case of linear regression with custom error distribution

Examples

# A zero-inflated Poisson family
rzeroinfpois <- function(ys) {
  n <- length(ys)
  rpois(n, lambda = ys * rbinom(n, 1, prob = 0.4))
}

custom_family(rzeroinfpois, exp)

Decrypt message giving the location of the true plot in a lineup

Description

Decrypts the message printed by model_lineup() indicating the location of the true diagnostics in the lineup.

Usage

decrypt(...)

Arguments

Value

The decrypted message.

Detect transmutation in formulas, such as factor(), and raise an error

Description

We rely on predictors to occur in models with only one type (such as numeric or factor), but the use of factor() would make it possible for a predictor to appear both as a factor or as numeric. The use of factor() also makes it harder to correctly detect the types of predictors, since the methods for obtaining model predictors provide them before they are converted to factor, not after. So we reject formulas that transmute types inside the formula, such as with factor().

Usage

detect_transmutation(formula, call = parent.frame())

Arguments

Details

Presently only factor() calls are rejected, but if other transmutations (such as conversions to logical or numeric) prove to be problems, they can be rejected as well.

Value

No value. Raises an error if transmutation is present.

Drop factor columns from a data frame

Description

Issues messages for the columns dropped.

Usage

drop_factors(df)

Arguments

Value

Data frame without columns that are factors

Empirically estimate response values on the link scale

Description

Calculates the average value of the response variable, and places this on the link scale. Plotting these against a predictor (by dividing the dataset into bins) can help assess the choice of link function.

Usage

empirical_link(response, family, na.rm = FALSE)

Arguments

Value

Mean response value, on the link scale.

Examples

suppressMessages(library(dplyr))
suppressMessages(library(ggplot2))

mtcars |>
  bin_by_interval(disp, breaks = 5) |>
  summarize(
    mean_disp = mean(disp),
    link = empirical_link(am, binomial())
  ) |>
  ggplot(aes(x = mean_disp, y = link)) +
  geom_point()

Check whether each column in a data frame is a factor

Description

Check whether each column in a data frame is a factor

Usage

factor_columns(df)

Arguments

Value

Logical vector with same number of entries as columns in df. Each entry is TRUE if the corresponding column is a factor.

Determine if a predictor is involved in an interaction

Description

This presently does not work for any term like poly(x, 3) or I(x^2), since the terms object contains those names, not the names of the underlying predictors.

Usage

in_interaction(formula, predictor)

Arguments

Value

TRUE if in an interaction, FALSE otherwise

Produce a lineup for a fitted model

Description

A lineup hides diagnostics among "null" diagnostics, i.e. the same diagnostics calculated using models fit to data where all model assumptions are correct. For each null diagnostic, model_lineup() simulates new responses from the model using the fitted covariate values and the model's error distribution, link function, and so on. Hence the new response values are generated under ideal conditions: the fitted model is true and all assumptions hold. decrypt() reveals which diagnostics are the true diagnostics.

Usage

model_lineup(fit, fn = augment, nsim = 20, ...)

Arguments

Details

To generate different kinds of diagnostics, the user can provide a custom fn. The fn should take a model fit as its argument and return a data frame. For instance, the data frame might contain one row per observation and include the residuals and fitted values for each observation; or it might be a single row containing a summary statistic or test statistic.

fn will be called on the original fit provided. Then parametric_boot_distribution() will be used to simulate data from the model fit nsim - 1 times, refit the model to each simulated dataset, and run fn on each refit model. The null distribution is conditional on X, i.e. the covariates used will be identical, and only the response values will be simulated. The data frames are concatenated with an additional .sample column identifying which fit each row came from.

When called, this function will print a message such as decrypt("sD0f gCdC En JP2EdEPn ZY"). This is how to get the location of the true diagnostics among the null diagnostics: evaluating this in the R console will produce a string such as "True data in position 5".

Value

A data frame (tibble) with columns corresponding to the columns returned by fn. The additional column .sample indicates which set of diagnostics each row is from. For instance, if the true data is in position 5, selecting rows with .sample == 5 will retrieve the diagnostics from the original model fit.

Model limitations

Because this function uses S3 generic methods such as model.frame(), simulate(), and update(), it can be used with any model fit for which methods are provided. In base R, this includes lm() and glm().

The model provided as fit must be fit using the data argument to provide a data frame. For example:

fit <- lm(dist ~ speed, data = cars)

When simulating new data, this function provides the simulated data as the data argument and re-fits the model. If you instead refer directly to local variables in the model formula, this will not work. For example, if you fit a model this way:

# will not work
fit <- lm(cars$dist ~ cars$speed)

It will not be possible to refit the model using simulated datasets, as that would require modifying your environment to edit cars.

References

Buja et al. (2009). Statistical inference for exploratory data analysis and model diagnostics. Philosophical Transactions of the Royal Society A, 367 (1906), pp. 4361-4383. doi:10.1098/rsta.2009.0120

Wickham et al. (2010). Graphical inference for infovis. IEEE Transactions on Visualization and Computer Graphics, 16 (6), pp. 973-979. doi:10.1109/TVCG.2010.161

See Also

parametric_boot_distribution() to simulate draws by using the fitted model to draw new response values; sampling_distribution() to simulate draws from the population distribution, rather than from the model

Examples

fit <- lm(dist ~ speed, data = cars)
model_lineup(fit, nsim = 5)

resids_vs_speed <- function(f) {
  data.frame(resid = residuals(f),
             speed = model.frame(f)$speed)
}
model_lineup(fit, fn = resids_vs_speed, nsim = 5)

Accept family arguments in the same way as glm(), convert to standardized form, and throw comprehensible errors.

Description

Accept family arguments in the same way as glm(), convert to standardized form, and throw comprehensible errors.

Usage

normalize_family(family, arg = caller_arg(family))

Family representing a linear relationship with non-Gaussian errors

Description

The ols_with_error() family can represent any non-Gaussian error, provided random variates can be drawn by an R function. A family specified this way can be used to specify a population (via population()), but can't be used to estimate a model (such as with glm()).

Usage

ols_with_error(error, ...)

Arguments

Value

A family object representing this family.

See Also

custom_family() for fully custom families, including for GLMs

Examples

# t-distributed errors with 3 degrees of freedom
ols_with_error(rt, df = 3)

# A linear regression with t-distributed error, using error_scale to make
# errors large
population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2,
               family = ols_with_error(rt, df = 4),
               error_scale = 2.5)
)

# Cauchy-distributed errors
ols_with_error(rcauchy, scale = 3)

# A contaminated error distribution, where
# 95% of observations are Gaussian and 5% are Cauchy
rcontaminated <- function(n) {
  contaminant <- rbinom(n, 1, prob = 0.05)

  return(ifelse(contaminant == 1,
                rcauchy(n, scale = 20),
                rnorm(n, sd = 1)))
}
ols_with_error(rcontaminated)

Simulate the distribution of estimates by parametric bootstrap

Description

Repeatedly simulates new response values by using the fitted model, holding the covariates fixed. By default, refits the same model to each simulated dataset, but an alternative model can be provided. Estimates, confidence intervals, or other quantities are extracted from each fitted model and returned as a tidy data frame.

Usage

parametric_boot_distribution(
  fit,
  alternative_fit = fit,
  data = model.frame(fit),
  fn = tidy,
  nsim = 100,
  ...
)

Arguments

Details

The default behavior samples from a model and refits the same model to the sampled data; this is useful when, for example, exploring how model diagnostics look when the model is well-specified. Another common use of the parametric bootstrap is hypothesis testing, where we might simulate from a null model and fit an alternative model to the data, to obtain the null distribution of a particular estimate or statistic. Provide alternative_fit to have a specific model fit to each simulated dataset, rather than the model they are simulated from.

Only the response variable from the fit (or alternative_fit, if given) is redrawn; other response variables in the population are left unchanged from their values in data.

Value

A data frame (tibble) with columns corresponding to the columns returned by fn. The additional column .sample indicates which fit each row is from.

Model limitations

Because this function uses S3 generic methods such as model.frame(), simulate(), and update(), it can be used with any model fit for which methods are provided. In base R, this includes lm() and glm().

The model provided as fit must be fit using the data argument to provide a data frame. For example:

fit <- lm(dist ~ speed, data = cars)

When simulating new data, this function provides the simulated data as the data argument and re-fits the model. If you instead refer directly to local variables in the model formula, this will not work. For example, if you fit a model this way:

# will not work
fit <- lm(cars$dist ~ cars$speed)

It will not be possible to refit the model using simulated datasets, as that would require modifying your environment to edit cars.

See Also

model_lineup() to use resampling to aid in regression diagnostics; sampling_distribution() to simulate draws from the population distribution, rather than the null

Examples

# Bootstrap distribution of estimates:
fit <- lm(mpg ~ hp, data = mtcars)
parametric_boot_distribution(fit, nsim = 5)

# Bootstrap distribution of estimates for a quadratic model, when true
# relationship is linear:
quad_fit <- lm(mpg ~ poly(hp, 2), data = mtcars)
parametric_boot_distribution(fit, quad_fit, nsim = 5)

# Bootstrap distribution of estimates for a model with an additional
# predictor, when it's truly zero. data argument must be provided so
# alternative fit has all predictors available, not just hp:
alt_fit <- lm(mpg ~ hp + wt, data = mtcars)
parametric_boot_distribution(fit, alt_fit, data = mtcars, nsim = 5)

Augment a model fit with partial residuals for all terms

Description

Construct a data frame containing the model data, partial residuals for all quantitative predictors, and predictor effects, for use in residual diagnostic plots and other analyses. The result is in tidy form (one row per predictor per observation), allowing it to be easily manipulated for plots and simulations.

Usage

partial_residuals(fit, predictors = everything())

Arguments

Value

Data frame (tibble) containing the model data and residuals in tidy form. There is one row per selected predictor per observation. All predictors are included as columns, plus the following additional columns:

Predictors and regressors

To define partial residuals, we must distinguish between the predictors, the measured variables we are using to fit our model, and the regressors, which are calculated from them. In a simple linear model, the regressors are equal to the predictors. But in a model with polynomials, splines, or other nonlinear terms, the regressors may be functions of the predictors.

For example, in a regression with a single predictor X, the regression model Y = \beta_0 + \beta_1 X + e has one regressor, X. But if we choose a polynomial of degree 3, the model is Y = \beta_0 + \beta_1 X + \beta_2 X^2 + \beta_3 X^3, and the regressors are \{X, X^2, X^3\}.

Similarly, if we have predictors X_1 and X_2 and form a model with main effects and an interaction, the regressors are \{X_1, X_2, X_1 X_2\}.

Partial residuals are defined in terms of the predictors, not the regressors, and are intended to allow us to see the shape of the relationship between a particular predictor and the response, and to compare it to how we have chosen to model it with regressors. Partial residuals are not useful for categorical (factor) predictors, and so these are omitted.

Linear models

Consider a linear model where \mathbb{E}[Y \mid X = x] = \mu(x). The mean function \mu(x) is a linear combination of regressors. Let \hat \mu be the fitted model and \hat \beta_0 be its intercept.

Choose a predictor X_f, the focal predictor, to calculate partial residuals for. Write the mean function as \mu(X_f, X_o), where X_f is the value of the focal predictor, and X_o represents all other predictors.

If e_i is the residual for observation i, the partial residual is

r_{if} = e_i + (\hat \mu(x_{if}, 0) - \hat \beta_0).

Setting X_o = 0 means setting all other numeric predictors to 0; factor predictors are set to their first (baseline) level.

Generalized linear models

Consider a generalized linear model where g(\mathbb{E}[Y \mid X = x]) = \mu(x), where g is a link function. Let \hat \mu be the fitted model and \hat \beta_0 be its intercept.

Let e_i be the working residual for observation i, defined to be

e_i = (y_i - g^{-1}(x_i)) g'(x_i).

Choose a predictor X_f, the focal predictor, to calculate partial residuals for. Write \mu as \mu(X_f, X_o), where X_f is the value of the focal predictor, and X_o represents all other predictors. Hence \mu(X_f, X_o) gives the model's prediction on the link scale.

The partial residual is again

r_{if} = e_i + (\hat \mu(x_{if}, 0) - \hat \beta_0).

Interpretation

In linear regression, because the residuals e_i should have mean zero in a well-specified model, plotting the partial residuals against x_f should produce a shape matching the modeled relationship \mu. If the model is wrong, the partial residuals will appear to deviate from the fitted relationship. Provided the regressors are uncorrelated or approximately linearly related to each other, the plotted trend should approximate the true relationship between x_f and the response.

In generalized linear models, this is approximately true if the link function g is approximately linear over the range of observed x values.

Additionally, the function \mu(X_f, 0) can be used to show the relationship between the focal predictor and the response. In a linear model, the function is linear; with polynomial or spline regressors, it is nonlinear. This function is the predictor effect function, and the estimated predictor effects \hat \mu(X_{if}, 0) are included in this function's output.

Limitations

Factor predictors (as factors, logical, or character vectors) are detected automatically and omitted. However, if a numeric variable is converted to factor in the model formula, such as with y ~ factor(x), the function cannot determine the appropriate type and will raise an error. Create factors as needed in the source data frame before fitting the model to avoid this issue.

References

R. Dennis Cook (1993). "Exploring Partial Residual Plots", Technometrics, 35:4, 351-362. doi:10.1080/00401706.1993.10485350

Cook, R. Dennis, and Croos-Dabrera, R. (1998). "Partial Residual Plots in Generalized Linear Models." Journal of the American Statistical Association 93, no. 442: 730–39. doi:10.2307/2670123

Fox, J., & Weisberg, S. (2018). "Visualizing Fit and Lack of Fit in Complex Regression Models with Predictor Effect Plots and Partial Residuals." Journal of Statistical Software, 87(9). doi:10.18637/jss.v087.i09

See Also

binned_residuals() for the related binned residuals; augment_longer() for a similarly formatted data frame of ordinary residuals; vignette("linear-regression-diagnostics"), vignette("logistic-regression-diagnostics"), and vignette("other-glm-diagnostics") for examples of plotting and interpreting partial residuals

Examples

fit <- lm(mpg ~ cyl + disp + hp, data = mtcars)
partial_residuals(fit)

# You can select predictors with tidyselect syntax:
partial_residuals(fit, c(disp, hp))

# Predictors with multiple regressors are supported:
fit2 <- lm(mpg ~ poly(disp, 2), data = mtcars)
partial_residuals(fit2)

# Allowing an interaction by number of cylinders is fine, but partial
# residuals are not generated for the factor. Notice the factor must be
# created first, not in the model formula:
mtcars$cylinders <- factor(mtcars$cyl)
fit3 <- lm(mpg ~ cylinders * disp + hp, data = mtcars)
partial_residuals(fit3)

Define the population generalized regression relationship

Description

Specifies a hypothetical infinite population of cases. Each case has some predictor variables and one or more response variables. The relationship between the variables and response variables are defined, as well as the population marginal distribution of each predictor variable.

Usage

population(...)

Arguments

Value

A population object.

See Also

predictor() and response() to define the population; sample_x() and sample_y() to draw samples from it

Examples

# A population with a simple linear relationship
linear_pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2, error_scale = 1.0)
)

# A population whose response depends on local variables
slope <- 2.2
intercept <- 0.7
sigma <- 2.5
variable_pop <- population(
  x = predictor(rnorm),
  y = response(intercept + slope * x, error_scale = sigma)
)

# Response error scale is heteroskedastic and depends on predictors
heteroskedastic_pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2,
               error_scale = 1 + x2 / 10)
)

# A binary outcome Y, using a binomial family with logistic link
binary_pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2,
               family = binomial(link = "logit"))
)

# A binomial outcome Y, with 10 trials per observation, using a logistic link
# to determine the probability of success for each trial
binomial_pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2,
               family = binomial(link = "logit"),
               size = 10)
)

# Another binomial outcome, but the number of trials depends on another
# predictor
binom_size_pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  trials = predictor(rpois, lambda = 20),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2,
               family = binomial(link = "logit"),
               size = trials)
)

# A population with a simple linear relationship and collinearity. Because X
# is bivariate, there will be two predictors, named x1 and x2.
library(mvtnorm)
collinear_pop <- population(
  x = predictor(rmvnorm, mean = c(0, 1),
                sigma = matrix(c(1, 0.8, 0.8, 1), nrow = 2)),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2, error_scale = 1.0)
)

Get the predictors of a population

Description

Get the predictors of a population

Usage

population_predictors(population)

Arguments

Value

Named list of predictors

Get the response variables of a population

Description

Get the response variables of a population

Usage

population_response(population)

Arguments

Value

Named list of response variables

Specify the distribution of a predictor variable

Description

Predictor variables can have any marginal distribution as long as a function is provided to sample from the distribution. Multivariate distributions are also supported: if the random generation function returns multiple columns, multiple random variables will be created. If the columns are named, the random variables will be named accordingly; otherwise, they will be successively numbered.

Usage

predictor(dist, ...)

Arguments

Details

The random generation function must take an argument named n specifying the number of draws. For univariate distributions, it should return a vector of length n; for multivariate distributions, it should return an array or matrix with n rows and a column per variable.

Multivariate predictors are successively numbered. For instance, if predictor X is specified with

library(mvtnorm)
predictor(dist = rmvnorm, mean = c(0, 1),
          sigma = matrix(c(1, 0.5, 0.5, 1), nrow = 2))

then the population predictors will be named X1 and X2, and will have covariance 0.5.

If the multivariate predictor has named columns, the names will be used instead. For instance, if predictor X generates a matrix with columns A and B, the population predictors will be named XA and XB.

Value

A predictor_dist object, to be used in population() to specify a population distribution

Examples

# Univariate normal distribution
predictor(dist = rnorm, mean = 10, sd = 2.5)

# Multivariate normal distribution
library(mvtnorm)
predictor(dist = rmvnorm, mean = c(0, 1, 7))

# Multivariate with named columns
rmulti <- function(n) {
  cbind(treatment = rbinom(n, size = 1, prob = 0.5),
        confounder = rnorm(n)
  )
}
predictor(dist = rmulti)

Get a prototype data frame for partial residuals

Description

All predictors, except the one we are calculating partial residuals for, are set to 0 (or their baseline level, for factors).

Usage

prototype_for(df, predictor)

Arguments

Value

prototype data frame

Specify a response variable in terms of predictors

Description

Response variables are related to predictors (and other response variables) through a link function and response distribution. First the expression provided is evaluated using the predictors, to give this response variable's value on the link scale; then the inverse link function and response distribution are used to get the response value. See Details for more information.

Usage

response(expr, family = gaussian(), error_scale = NULL, size = 1L)

Arguments

Details

Response variables are drawn based on a typical generalized linear model setup. Let Y represent the response variable and X represent the predictor variables. We specify that

Y \mid X \sim \text{SomeDistribution},

where

\mathbb{E}[Y \mid X = x] = g^{-1}(\mu(x)).

Here \mu(X) is the expression expr, and both the distribution and link function g are specified by the family provided. For instance, if the family is gaussian(), the distribution is Normal and the link is the identity function; if the family is binomial(), the distribution is binomial and the link is (by default) the logistic link.

Response families

The following response families are supported.

gaussian()

The default family is gaussian() with the identity link function, specifying the relationship

Y \mid X \sim \text{Normal}(\mu(X), \sigma^2),

where \sigma^2 is given by error_scale.

ols_with_error()

Allows specification of custom non-Normal error distributions, specifying the relationship

Y = \mu(X) + e,

where e is drawn from an arbitrary distribution, specified by the error argument to ols_with_error().

binomial()

Binomial responses include binary responses (as in logistic regression) and responses giving a total number of successes out of a number of trials. The response has distribution

Y \mid X \sim \text{Binomial}(N, g^{-1}(\mu(X))),

where N is set by the size argument and g is the link function. The default link is the logistic link, and others can be chosen with the link argument to binomial(). The default N is 1, representing a binary outcome.

poisson()

Poisson-distributed responses with distribution

Y \mid X \sim \text{Poisson}(g^{-1}(\mu(X))),

where g is the link function. The default link is the log link, and others can be chosen with the link argument to poisson().

custom_family()

Responses drawn from an arbitrary distribution with arbitrary link function, i.e.

Y \mid X \sim \text{SomeDistribution}(g^{-1}(\mu(X))),

where both g and SomeDistribution are specified by arguments to custom_family().

Evaluation and scoping

The expr, error_scale, and size arguments are evaluated only when simulating data for this response variable. They are evaluated in an environment with access to the predictor variables and the preceding response variables, which they can refer to by name. Additionally, these arguments can refer to variables in scope when the enclosing population() was defined. See the Examples below.

Value

A response_dist object, to be used in population() to specify a population distribution

See Also

predictor() and population() to define populations; ols_with_error() and custom_family() for custom response distributions

Examples

# Defining a binomial response. The expressions can refer to other predictors
# and to the environment where the `population()` is defined:
slope1 <- 2.5
slope2 <- -3
intercept <- -4.6
size <- 10
population(
  x1 = predictor(rnorm),
  x2 = predictor(rnorm),
  y = response(intercept + slope1 * x1 + slope2 * x2,
               family = binomial(), size = size)
)

Draw random values from a factor variable

Description

To specify the population distribution of a factor variable, specify the probability for each of its factor levels. When drawn from the population, factor levels are drawn with replacement according to their probability.

Usage

rfactor(n, levels, prob = rep_len(1/length(levels), length(levels)))

Arguments

Value

Sample of n values from levels, drawn in proportion to their probabilities. By default, levels are equally likely.

See Also

by_level() to assign numeric values based on factor levels, such as to set population regression coefficients by factor level

Examples

rfactor(5, c("foo", "bar", "baz"), c(0.4, 0.3, 0.3))

Draw a data frame from the specified population.

Description

Sampling is split into two steps, for predictors and for response variables, to allow users to choose which to simulate. sample_x() will only sample predictor variables, and sample_y() will augment a data frame of predictors with columns for response variables, overwriting any already present. Hence one can use sample_y() as part of a simulation with fixed predictors, for instance.

Usage

sample_x(population, n)

sample_y(xs)

Arguments

Value

Data frame (tibble) of n rows, with columns matching the variables specified in the population.

Examples

# A population with a simple linear relationship
pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2, error_scale = 1.0)
)

xs <- pop |>
  sample_x(5)

xs

xs |>
  sample_y()

Simulate the sampling distribution of estimates from a population

Description

Repeatedly refits the model to new samples from the population, calculates estimates for each fit, and compiles a data frame of the results.

Usage

sampling_distribution(fit, data, fn = tidy, nsim = 100, fixed_x = TRUE, ...)

Arguments

Details

To generate sampling distributions of different quantities, the user can provide a custom fn. The fn should take a model fit as its argument and return a data frame. For instance, the data frame might contain one row per estimated coefficient and include the coefficient and its standard error; or it might contain only one row of model summary statistics.

Value

Data frame (tibble) of nsim + 1 simulation results, formed by concatenating together the data frames returned by fn. The .sample column identifies which simulated sample each row came from. Rows with .sample == 0 come from the original fit.

Model limitations

Because this function uses S3 generic methods such as model.frame(), simulate(), and update(), it can be used with any model fit for which methods are provided. In base R, this includes lm() and glm().

The model provided as fit must be fit using the data argument to provide a data frame. For example:

fit <- lm(dist ~ speed, data = cars)

When simulating new data, this function provides the simulated data as the data argument and re-fits the model. If you instead refer directly to local variables in the model formula, this will not work. For example, if you fit a model this way:

# will not work
fit <- lm(cars$dist ~ cars$speed)

It will not be possible to refit the model using simulated datasets, as that would require modifying your environment to edit cars.

See Also

parametric_boot_distribution() to simulate draws from a fitted model, rather than from the population

Examples

pop <- population(
  x1 = predictor(rnorm, mean = 4, sd = 10),
  x2 = predictor(runif, min = 0, max = 10),
  y = response(0.7 + 2.2 * x1 - 0.2 * x2, error_scale = 4.0)
)

d <- sample_x(pop, n = 20) |>
  sample_y()

fit <- lm(y ~ x1 + x2, data = d)
# using the default fn = broom::tidy(). conf.int argument is passed to
# broom::tidy()
samples <- sampling_distribution(fit, d, conf.int = TRUE)
samples

suppressMessages(library(dplyr))
# the model is correctly specified, so the estimates are unbiased:
samples |>
  group_by(term) |>
  summarize(mean = mean(estimate),
            sd = sd(estimate))

# instead of coefficients, get the sampling distribution of R^2
rsquared <- function(fit) {
  data.frame(r2 = summary(fit)$r.squared)
}
sampling_distribution(fit, d, rsquared, nsim = 10)