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 0.028 0.022
#> AntiPt 0.007 0.114 0.116
#> age 0.000 0.914 0.885
#> B_TimeCT 0.000 0.920 0.857
#> Warfarin 0.003 0.365 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$summary_df, digits = 3)| method | statistic | critical_value | p_value |
|---|---|---|---|
| truncated | 0.338 | 0.049 | 0.002 |
| satterthwaite | 0.338 | 0.048 | 0.000 |
| permutation | 0.338 | 0.062 | 0.005 |
sprintf('The wasserstein F-statistic is %.3f on %.3f degrees of freedom', globalF_res$wasserstein.F_stat, globalF_res$chisq_df)
#> [1] "The wasserstein F-statistic is 141.008 on 11.218 degrees of freedom"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.
xpred = colMeans(predictor)
confidence_Band = confidenceBands(res, Xpred_df = xpred, type = 'both')
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')