1 Wasserstein Regression
wass_regress is the estimation function which works similar to lm. To compute the fitted values, it requires a formula, response and predictor data. We give explanation of other arguments below.
Ytype: whether the response matrixYmatcontains'quantile'or'density'functions.Sup: the common grid for density/quantile functions inYmat.
1.0.1 The effect of Sup grid vector when Ytype == 'quantile'
Since most derivation in WRI works in the space of quantile functions and its derivatives, the probability density functions are converted into quantile functions. However, the transformation will result in certain deviation between the original density function and \(1/q(t)\), where \(q(t) = Q'(t), t = F(x)\). Note that it is \(q(t)\)’s that are directly used in the WRI functions.
Below we set t as equally spaced grid vector and nonequally spaced vector, which is denser near the boundary to compare the resulting \(1/q(t)\).
equal_t = den2Q_qd(densityCurves, dSup, t_vec = seq(0, 1, length.out = 120))
nonequal_t = den2Q_qd(densityCurves, dSup, t_vec = unique(c(seq(0, 0.05, 0.001), seq(0.05, 0.95, 0.05), seq(0.95, 1, 0.001))))
When the quantile support vector is finer near the boundary, \(1/q(t)\) is closer to original density function \(f(x)\). Thus, when user inputs density functions as response curves, i.e. Ytype == 'density', the support for quantile functions is set as nonequal_t.
1.0.2 Usage of wass_regress function
The density curves and predictor variables are input into wass_regress separately, as illustrated below.
res = wass_regress(rightside_formula = ~., Xfit_df = predictor, Ytype = 'density', Ymat = densityCurves, Sup = dSup)The wass_regress function returns a WRI object. This object can be used with the other functions in this package to run hypothesis tests, calculate Wasserstein \(R^2\), and compute confidence bands.
1.1 Summary
The summary method for WRI objects combines the global F-test, Wassertstein \(R^2\), and partial F-tests for individual effects into one easily-readable output.
summary(res)
#> Call:
#> wass_regress(rightside_formula = ~., Xfit_df = predictor, Ytype = "density",
#> Ymat = densityCurves, Sup = dSup)
#>
#> Partial F test for individual effects:
#>
#> F-stat p-value(truncated) p-value(satterthwaite)
#> log_b_vol 0.256 0.002 0.000
#> b_shapInd 0.062 0.002 0.000
#> midline_shift 0.035 0.002 0.000
#> weight 0.028 0.002 0.000
#> DM 0.012 0.022 0.022
#> AntiPt 0.007 0.138 0.116
#> age 0.000 0.940 0.885
#> B_TimeCT 0.000 0.920 0.857
#> Warfarin 0.003 0.341 0.372
#>
#> Wasserstein R-squared: 0.224
#> F-statistic (by Satterthwaite method): 141.008 on 11.218 DF, p-value: 1.356e-241.1.1 Wasserstein Coefficient of Determination, \(R^2\)
The Wasserstein coefficient of determination, \(R^2\) can be calculated with wass_R2(res). The formula for the Wasserstein \(R^2\) is as follows:
\[R^2=1-\frac{\sum^n_{i=1} d_W^2(f_{i}, \hat{f}_i)}{\sum^n_{i=1} d_W^2(f_{i}, \overline{f_{i}})},\] Where \(\overline{f_{i}}\) is the unconditional Wasserstein mean estimate and \(\hat{f}_i\) is the conditional mean estimate.
This value represents the fraction of Wasserstein variability explained by the model, and can therefore be used to assess the goodness of fit for a model.
1.2 Hypothesis Testing
1.2.1 Global F-test
globalFtest function performs the global F-tests. It provides four methods of computing the p-value, two (truncated and satterthwaite) through asymptotic analysis and two resampling techniques (permutation and bootstrap). Please note that the resampling methods can be slow.
- Setting
permutation = TRUEwill also compute permutation p-value. The number of permutation samples can be controlled with thenumPermuargument. - Setting
bootstrap = TRUEwill also compute bootstrap p-value. The number of bootstrap samples can be controlled with thenumBootargument.
Note on Degrees of Freedom : The degrees of freedom are approximated by a chi-square distribution, so there is only 1 degree of freedom for our F-statistic. This is done because the F-statistic is asymptotically equivalent to a chi-squared distribution.
globalF_res = globalFtest(res, alpha = 0.05, permutation = TRUE, numPermu = 200)
kable(globalF_res$summary_df, digits = 3)| method | statistic | critical_value | p_value |
|---|---|---|---|
| truncated | 0.338 | 0.047 | 0.002 |
| satterthwaite | 0.338 | 0.048 | 0.000 |
| permutation | 0.338 | 0.062 | 0.005 |
1.2.2 Partial F-test
partialFtest can be used to test individual effects or submodel fits. Using the stroke data as an example, we test whether the clinical variables are significant for head CT hematoma densities when radiological variables are in the model.
# the reduced model only has four radiological variables
reduced_res = wass_regress(~ log_b_vol + b_shapInd + midline_shift + B_TimeCT, Xfit_df = predictor, Ymat = densityCurves, Ytype = 'density', Sup = dSup)
full_res = wass_regress(rightside_formula = ~., Xfit_df = predictor, Ymat = densityCurves, Ytype = 'density', Sup = dSup)
partialFtable = partialFtest(reduced_res, full_res, alpha = 0.05)
kable(partialFtable, digits = 3)| method | statistic | critical_value | p_value | |
|---|---|---|---|---|
| 95% | truncated | 0.056 | 0.100 | 0.828 |
| satterthwaite | 0.056 | 0.099 | 0.839 |
With p-value greater than 0.05, we are confident to conclude that when radiological variables are in the model, clinical variables are not significant for explaining the variance in head CT hematoma densities.
1.3 Confidence Bands
1.3.1 Usage of confidenceBands function
The confidenceBands function computes the intrinsic Wasserstein\(-\infty\) bands and Wasserstein density bands. In the function, these refer to quantile band and density band respectively, which are controlled by type argument (options are ‘quantile’, ‘density’ or ‘both’). By default, the function visualizes confidence bands for one object. But it allows to compute \(k\) confidence bands simultaneously if a \(k \times p\) dataframe Xpred_df is provided. All the results, including upper and lower bounds, predicted density function etc, are returned in a list.
1.3.2 Investigate the effect of hematoma volume
We set log(hematoma volume) equal to the first quartile (Q1) or third quartile (Q3) of the observed values, with all other predictors set at their mean (for continuous variables) or mode (for binary variables). Then compare the CT hematoma densities in these two cases.
mean_Mode <- function(vec) {
return(ifelse(length(unique(vec)) < 3, modeest::mfv(vec), mean(vec)))
}
mean_mode_vec = apply(predictor, 2, mean_Mode)
predictorDF = rbind(mean_mode_vec, mean_mode_vec)
predictorDF[ , 1] = quantile(predictor$log_b_vol, probs = c(1/4, 3/4))res_cb = confidenceBands(res, predictorDF, level = 0.95, delta = 0.01, type = 'both', figure = F)
m = ncol(res_cb$quan_list$Q_lx)
na.mat = matrix(NA, nrow = 2, ncol = m - ncol(res_cb$den_list$f_lx))
cb_plot_df = with(res_cb, data.frame(
fun = rep(c('quantile function', 'density function'), each = 2*m),
Q1Q3 = rep(rep(c('Q1', 'Q3'), each = m), 2),
value_m = c(as.vector(t(quan_list$Qpred)), as.vector(t(cbind(cdf_list$fpred)))),
value_u = c(as.vector(t(quan_list$Q_ux)), as.vector(t(cbind(den_list$f_ux, na.mat)))),
value_l = c(as.vector(t(quan_list$Q_lx)), as.vector(t(cbind(den_list$f_lx, na.mat)))),
support_full = c(rep(quan_list$t, 2), as.vector(t(cbind(cdf_list$Fsup)))),
support_short = c(rep(quan_list$t, 2), as.vector(t(cbind(den_list$Qpred, na.mat))))
))
ggplot(data = cb_plot_df, aes(color = Q1Q3)) +
theme_linedraw()+
geom_line(aes(x = support_full, y = value_m)) +
geom_ribbon(aes(x = support_short, ymin = value_l, ymax = value_u, fill = Q1Q3), alpha = 0.25) +
facet_wrap( ~ fun, scales = "free_y") +
ylab('Confidence band') +
xlab('Support')