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Test Hypotheses Concerning Individual Parameters

Source: R/comparison.R
hypothesis.Rd

Returns posterior probabilities and Bayes Factors for flexible hypotheses involving model parameters. The documentation for the argument hypotheses below shows examples of how to specify hypotheses, and read worked examples on the mcp website. For directional hypotheses, hypothesis executes the hypothesis string in a data-frame environment and summarises the proportion of posterior and prior samples where the expression evaluates to TRUE. The Bayes factor is the posterior odds divided by the prior odds. For equals-hypotheses, a Savage-Dickey ratio is computed. Both kinds of Bayes factor require prior samples, so remember mcp(..., sample = "both"). This function is heavily inspired by the hypothesis function from the brms package.

Usage

hypothesis(fit, hypotheses, width = 0.95, digits = 3, prior = FALSE)

Arguments

fit

An mcpfit object.

hypotheses

String representation of a logical test involving model parameters. Takes R code that evaluates to TRUE or FALSE in a vectorized way.

Directional hypotheses are specified using <, >, <=, or >=. hypothesis returns the posterior probability and the Bayes factor in favor of the stated hypothesis. The Bayes factor requires both prior and posterior samples from mcp(sample = "both"). For example:

  • "cp_1 > 30": the first change point is above 30.

  • "Intercept_1 > Intercept_2": the intercept is greater in segment 1 than 2.

  • "x_2 - x_1 <= 3": the difference between slope 1 and 2 is less than or equal to 3.

  • "Intercept_1 > -2 & Intercept_1 < 2": Intercept_1 is between -2 and 2 (an interval hypothesis). This can be useful as a Region Of Practical Equivalence test (ROPE).

  • "cp_1^2 < 30 | (log(x_1) + log(x_2)) > 5": be creative.

  • "`cp_1_id[1]` > `cp_1_id[2]`": id1 is greater than id2, as estimated through the varying-by-"id" change point in segment 1. Note that `` required for varying effects.

Hypotheses can also test equality using the equal sign (=). This runs a Savage-Dickey test, i.e., the proportion by which the probability density has increased from the prior to the posterior at a given value. Therefore, it requires mcp(sample = "both"). There are two requirements: First, there can only be one equal sign, so don't use and (&) or or (|). Second, the point to test has to be on the right, and the variables on the left.

  • "cp_1 = 30": is the first change point at 30? Or to be more precise: by what factor has the credence in cp_1 = 30 risen/fallen when conditioning on the data, relative to the prior credence?

  • "Intercept_1 + Intercept_2 = 0": Is the sum of two intercepts zero?

  • "`cp_1_id[John]`/`cp_1_id[Erin]` = 2": is the varying change point for John (which is relative to `cp_1“) double that of Erin?

width

Float. The width of the central posterior interval (between 0 and 1).

digits

a non-null value for digits specifies the minimum number of significant digits to be printed in values. The default, NULL, uses getOption("digits"). (For the interpretation for complex numbers see signif.) Non-integer values will be rounded down, and only values greater than or equal to 1 and no greater than 22 are accepted.

prior

TRUE/FALSE. Summarise prior instead of posterior?

Value

A data.frame with a row per hypothesis and the following columns:

  • hypothesis is the hypothesis; often re-arranged to test against zero.

  • mean is the posterior mean of the left-hand side of the hypothesis.

  • lower is the lower bound of the central posterior interval of width width.

  • upper is the upper bound of ditto.

  • p Posterior probability. For "=" (Savage-Dickey), it is the BF converted to p. For directional hypotheses, it is the proportion of samples that returns TRUE.

  • BF Bayes Factor in favor of the hypothesis. For "=" it is the Savage-Dickey density ratio. For directional hypotheses, it is the posterior odds divided by the prior odds.

Author

Jonas Kristoffer Lindeløv jonas@lindeloev.dk