Normalized Generalized Partial Correlation for Binomial GLMs
gcor_normalized_binom.RdComputes the normalized generalized partial correlation coefficient for binomial GLMs. The normalization scales the correlation by its maximum possible absolute value.
Arguments
- full_glm
A fitted GLM object of class `glm` with binomial family.
- terms
Character vector of variable names for which to compute normalized correlations. If `NULL` (default), computes for all non-intercept terms in the model.
- intercept_too
Logical indicating whether to include the intercept as a variable. Default is FALSE.
- algorithm.control
`list` of coltrol parameters: `n_exact` Integer specifying the sample size threshold for using exact methods (brute force). Default is 15. `thresholds` Numeric vector of threshold values for multi-start initialization. `n_random` Integer number of random starts for multi-start optimization. `max_iter` Integer maximum number of iterations per start. `topK` Integer number of top candidates to consider at each iteration. `tol` Numeric tolerance for convergence. `patience` Integer number of iterations without improvement before stopping.
- algorith
`"auto"` by default. It choose between `"intercept_only"`, `"brute_force"` and `"multi_start"`
Value
A data frame with five columns:
- terms
The variable name
- r
The generalized partial correlation coefficient
- r_n
The normalized generalized partial correlation coefficient
- null_model
The null model used to compute the generalized (partial) correlation
- algorithm
The algorithm used to compute the upper/lower bounds of the generalized partial correlation coefficient (to compute its normalized version)
Details
The normalized generalized partial correlation is computed as: $$ r_n = \begin{cases} +r / r_+ & \text{if } r > 0 \\ -r / r_- & \text{if } r < 0 \end{cases} $$ where \(r_+\) is the maximum possible correlation and \(r_-\) is the minimum.
When the (full) model has only intercept and only one predictor X, the generalized (non partial) correlation is computed and the normalization factor for X is exact. In the more general case with more predictors, for sample sizes \(n \leq n_{\text{exact}}\), brute force search is used to find the exact extrema. For larger sample sizes, a greedy multi-start algorithm is employed:
Multiple starting points are generated using thresholding and random sampling
From each start, coordinates are greedily flipped to improve the correlation
Early stopping is used when no improvements are found for several iterations
The best solution across all starts is returned
This approach provides a good trade-off between computational efficiency and solution quality for large problems where brute force is infeasible.
Examples
set.seed(123)
dt=data.frame(X=rnorm(20),
Z=factor(rep(LETTERS[1:3],length.out=20)))
dt$Y=rbinom(n=20,prob=plogis((dt$Z=="C")*2),size=1)
mod=flipscores(Y~Z+X,data=dt,family="binomial",n_flips=1000)
summary(mod)
#>
#> Call:
#> flipscores(formula = Y ~ Z + X, family = "binomial", data = dt,
#> n_flips = 1000)
#>
#> Coefficients:
#> Estimate Score Std. Error z value Part. Cor Pr(>|z|)
#> (Intercept) -0.1486 -0.2102 1.1881 -0.1770 -0.067 0.893
#> ZB -20.4539 -1.4784 0.7466 -1.9802 -0.530 0.053 .
#> ZC 20.8561 1.8043 0.8180 2.2057 0.615 0.039 *
#> X -0.4276 -0.3782 0.9574 -0.3951 -0.149 0.752
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 27.5256 on 19 degrees of freedom
#> Residual deviance: 9.4015 on 16 degrees of freedom
#> AIC: 17.401
#>
#> Number of Fisher Scoring iterations: 19
#>
(results <- gcor_normalized_binom(mod))
#> terms r r_n null_model algorithm
#> ZB ~ZB -0.5295718 -0.5304792 ~1+ZC+X multi_start
#> ZC ~ZC 0.6146304 0.6148542 ~1+ZB+X multi_start
#> X ~X -0.1493213 -0.1891066 ~1+ZB+ZC multi_start
# Compute for specific terms only
gcor_normalized_binom(mod, terms = c("X", "ZC"))
#> terms r r_n null_model algorithm
#> X ~X -0.1493213 -0.1849399 ~1+ZB+ZC multi_start
#> ZC ~ZC 0.6146304 0.6148542 ~1+ZB+X multi_start