r2_semipartial function

Semi-Partial (Part) Correlation Squared ($\Delta R^2$)

Compute the semi-partial (part) correlation squared (also known as $\Delta R^2$). Currently, only lm() models are supported.

r2_semipartial(
  model,
  type = c("terms", "parameters"),
  ci = 0.95,
  alternative = "greater",
  ...
)

Arguments

• model: An lm model.

• type: Type, either "terms", or "parameters".

• ci: Confidence Interval (CI) level

• alternative: a character string specifying the alternative hypothesis; Controls the type of CI returned: "greater" (default) or "less"

  (one-sided CI), or "two.sided" (two-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs .

• ...: Arguments passed to or from other methods.

Returns

A data frame with the effect size.

Details

This is similar to the last column of the "Conditional Dominance Statistics" section of the parameters::dominance_analysis() output. For each term, the model is refit without the columns on the model matrix that correspond to that term. The $R^2$ of this sub-model is then subtracted from the $R^2$ of the full model to yield the $\Delta R^2$. (For type = "parameters", this is done for each column in the model matrix.)

Note that this is unlike parameters::dominance_analysis(), where term deletion is done via the formula interface, and therefore may lead to different results.

For other, non-lm() models, as well as more verbose information and options, please see the documentation for parameters::dominance_analysis().

Confidence (Compatibility) Intervals (CIs)

Confidence intervals are based on the normal approximation as provided by Alf and Graf (1999). An adjustment to the lower bound of the CI is used, to improve the coverage properties of the CIs, according to Algina et al (2008): If the F test associated with the $sr^2$ is significant (at 1-ci

level), but the lower bound of the CI is 0, it is set to a small value (arbitrarily to a 10th of the estimated $sr^2$); if the F test is not significant, the lower bound is set to 0. (Additionally, lower and upper bound are "fixed" so that they cannot be smaller than 0 or larger than 1.)

CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - $\alpha$)% confidence interval contains all of the parameter values for which p > $\alpha$

for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen $\alpha$ level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

Plotting with see

The see package contains relevant plotting functions. See the c("plotting vignette in the ", "list("see")", " package").

Examples

data("hardlyworking")

m <- lm(salary ~ factor(n_comps) + xtra_hours * seniority, data = hardlyworking)

r2_semipartial(m)

r2_semipartial(m, type = "parameters")


# Compare to `eta_squared()`
# --------------------------
npk.aov <- lm(yield ~ N + P + K, npk)

# When predictors are orthogonal,
# eta_squared(partial = FALSE) gives the same effect size:
performance::check_collinearity(npk.aov)

eta_squared(npk.aov, partial = FALSE)

r2_semipartial(npk.aov)

# Compare to `dominance_analysis()`
# ---------------------------------
m_full <- lm(salary ~ ., data = hardlyworking)

r2_semipartial(m_full)

# Compare to last column of "Conditional Dominance Statistics":
parameters::dominance_analysis(m_full)

References

• Alf Jr, E. F., & Graf, R. G. (1999). Asymptotic confidence limits for the difference between two squared multiple correlations: A simplified approach. Psychological Methods, 4(1), 70-75. tools:::Rd_expr_doi("10.1037/1082-989X.4.1.70")
• Algina, J., Keselman, H. J., & Penfield, R. D. (2008). Confidence intervals for the squared multiple semipartial correlation coefficient. Journal of Modern Applied Statistical Methods, 7(1), 2-10. tools:::Rd_expr_doi("10.22237/jmasm/1209614460")

See Also

eta_squared(), cohens_f() for comparing two models, parameters::dominance_analysis() and parameters::standardize_parameters().