(Non-)Linear Tests for Null Hypotheses, Joint Hypotheses, Equivalence, Non Superiority, and Non Inferiority
Description
Uncertainty estimates are calculated as first-order approximate standard errors for linear or non-linear functions of a vector of random variables with known or estimated covariance matrix. In that sense, hypotheses emulates the behavior of the excellent and well-established car::deltaMethod and car::linearHypothesis functions, but it supports more models; requires fewer dependencies; expands the range of tests to equivalence and superiority/inferiority; and offers convenience features like robust standard errors.
To learn more, read the hypothesis tests vignette, visit the package website, or scroll down this page for a full list of vignettes:
Warning #1: Tests are conducted directly on the scale defined by the type argument. For some models, it can make sense to conduct hypothesis or equivalence tests on the “link” scale instead of the “response” scale which is often the default.
Warning #2: For hypothesis tests on objects produced by the marginaleffects package, it is safer to use the hypothesis argument of the original function. Using hypotheses() may not work in certain environments, in lists, or when working programmatically with *apply style functions.
Warning #3: The tests assume that the hypothesis expression is (approximately) normally distributed, which for non-linear functions of the parameters may not be realistic. More reliable confidence intervals can be obtained using the inferences() function with method = “boot”.
specify a hypothesis test or custom contrast using a numeric value, vector, or matrix; a string equation; string; a formula, or a function.
Numeric:
Single value: the null hypothesis used in the computation of Z and p (before applying transform).
Vector: Weights to compute a linear combination of (custom contrast between) estimates. Length equal to the number of rows generated by the same function call, but without the hypothesis argument.
Matrix: Each column is a vector of weights, as describe above, used to compute a distinct linear combination of (contrast between) estimates. The column names of the matrix are used as labels in the output.
String equation to specify linear or non-linear hypothesis tests. If the term column uniquely identifies rows, terms can be used in the formula. Otherwise, use b1, b2, etc. to identify the position of each parameter. The b* wildcard can be used to test hypotheses on all estimates. If a named vector is used, the names are used as labels in the output. Examples:
hp = drat
hp + drat = 12
b1 + b2 + b3 = 0
b* / b1 = 1
String:
"pairwise": pairwise differences between estimates in each row.
"reference": differences between the estimates in each row and the estimate in the first row.
"sequential": difference between an estimate and the estimate in the next row.
"meandev": difference between an estimate and the mean of all estimates.
"meanotherdev": difference between an estimate and the mean of all other estimates, excluding the current one.
"revpairwise", "revreference", "revsequential": inverse of the corresponding hypotheses, as described above.
Formula:
comparison ~ pairs | group
Left-hand side determines the type of comparison to conduct: difference or ratio. If the left-hand side is empty, difference is chosen.
Right-hand side determines the pairs of estimates to compare: reference, sequential, or meandev
Optional: Users can supply grouping variables after a vertical bar to conduct comparisons withing subsets.
Examples:
~ reference
ratio ~ pairwise
difference ~ pairwise | groupid
Function:
Accepts an argument x: object produced by a marginaleffects function or a data frame with column rowid and estimate
Returns a data frame with columns term and estimate (mandatory) and rowid (optional).
The function can also accept optional input arguments: newdata, by, draws.
This function approach will not work for Bayesian models or with bootstrapping. In those cases, it is easy to use posterior_draws() to extract and manipulate the draws directly.
See the Examples section below and the vignette: https://marginaleffects.com/vignettes/hypothesis.html
vcov
Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values:
FALSE: Do not compute standard errors. This can speed up computation considerably.
TRUE: Unit-level standard errors using the default vcov(model) variance-covariance matrix.
String which indicates the kind of uncertainty estimates to return.
Heteroskedasticity and autocorrelation consistent: “HAC”
Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger"
Other: “NeweyWest”, “KernHAC”, “OPG”. See the sandwich package documentation.
One-sided formula which indicates the name of cluster variables (e.g., ~unit_id). This formula is passed to the cluster argument of the sandwich::vcovCL function.
Square covariance matrix
Function which returns a covariance matrix (e.g., stats::vcov(model))
conf_level
numeric value between 0 and 1. Confidence level to use to build a confidence interval.
df
Degrees of freedom used to compute p values and confidence intervals. A single numeric value between 1 and Inf. When using joint_test=“f”, the df argument should be a numeric vector of length 2.
equivalence
Numeric vector of length 2: bounds used for the two-one-sided test (TOST) of equivalence, and for the non-inferiority and non-superiority tests. See Details section below.
joint
Joint test of statistical significance. The null hypothesis value can be set using the hypothesis argument.
FALSE: Hypotheses are not tested jointly.
TRUE: All parameters are tested jointly.
String: A regular expression to match parameters to be tested jointly. grep(joint, perl = TRUE)
Character vector of parameter names to be tested. Characters refer to the names of the vector returned by coef(object).
Integer vector of indices. Which parameters positions to test jointly.
joint_test
A character string specifying the type of test, either "f" or "chisq". The null hypothesis is set by the hypothesis argument, with default null equal to 0 for all parameters.
numderiv
string or list of strings indicating the method to use to for the numeric differentiation used in to compute delta method standard errors.
"fdforward": finite difference method with forward differences
"fdcenter": finite difference method with central differences (default)
"richardson": Richardson extrapolation method
Extra arguments can be specified by passing a list to the numDeriv argument, with the name of the method first and named arguments following, ex: numderiv=list(“fdcenter”, eps = 1e-5). When an unknown argument is used, marginaleffects prints the list of valid arguments for each method.
…
Additional arguments are passed to the predict() method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the marginaleffects website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the ?slopes documentation for a non-exhaustive list of available arguments.
Joint hypothesis tests
The test statistic for the joint Wald test is calculated as (R * theta_hat - r)’ * inv(R * V_hat * R’) * (R * theta_hat - r) / Q, where theta_hat is the vector of estimated parameters, V_hat is the estimated covariance matrix, R is a Q x P matrix for testing Q hypotheses on P parameters, r is a Q x 1 vector for the null hypothesis, and Q is the number of rows in R. If the test is a Chi-squared test, the test statistic is not normalized.
The p-value is then calculated based on either the F-distribution (for F-test) or the Chi-squared distribution (for Chi-squared test). For the F-test, the degrees of freedom are Q and (n - P), where n is the sample size and P is the number of parameters. For the Chi-squared test, the degrees of freedom are Q.
Equivalence, Inferiority, Superiority
\(\theta\) is an estimate, \(\sigma_\theta\) its estimated standard error, and \([a, b]\) are the bounds of the interval supplied to the equivalence argument.
Non-inferiority:
\(H_0\): \(\theta \leq a\)
\(H_1\): \(\theta > a\)
\(t=(\theta - a)/\sigma_\theta\)
p: Upper-tail probability
Non-superiority:
\(H_0\): \(\theta \geq b\)
\(H_1\): \(\theta < b\)
\(t=(\theta - b)/\sigma_\theta\)
p: Lower-tail probability
Equivalence: Two One-Sided Tests (TOST)
p: Maximum of the non-inferiority and non-superiority p values.
Thanks to Russell V. Lenth for the excellent emmeans package and documentation which inspired this feature.
# predictions(), comparisons(), and slopes()mod<-glm(am~hp+mpg, data =mtcars, family =binomial)cmp<-comparisons(mod, newdata ="mean")hypotheses(cmp, hypothesis ="b1 = b2")
# The `hypothesis` argument can be used to compute standard errors for fitted valuesmod<-glm(am~hp+mpg, data =mtcars, family =binomial)f<-function(x)predict(x, type ="link", newdata =mtcars)p<-hypotheses(mod, hypothesis =f)head(p)
# Complex aggregation# Step 1: Collapse predicted probabilities by outcome level, for each individual# Step 2: Take the mean of the collapsed probabilities by group and `cyl`library(dplyr)library(MASS)library(dplyr)dat<-transform(mtcars, gear =factor(gear))mod<-polr(gear~factor(cyl)+hp, dat)aggregation_fun<-function(x){predictions(x, vcov =FALSE)|>mutate(group =ifelse(group%in%c("3", "4"), "3 & 4", "5"))|>summarize(estimate =sum(estimate), .by =c("rowid", "cyl", "group"))|>summarize(estimate =mean(estimate), .by =c("cyl", "group"))|>rename(term =cyl)}hypotheses(mod, hypothesis =aggregation_fun)
Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 % p (NonSup) p (NonInf)
-3.94 2.05 -1.92 0.0543 4.2 -7.95 0.0727 <0.001 0.973
p (Equiv)
0.973
Term: (mean(4) - mean(3)) - (mean(5) - mean(3))
Type: response
Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv
# joint hypotheses: character vectormodel<-lm(mpg~as.factor(cyl)*hp, data =mtcars)hypotheses(model, joint =c("as.factor(cyl)6:hp", "as.factor(cyl)8:hp"))