Distribution-Free Location-Scale Regression

Figures & data

Fig. 1 Location-scale transformation model. The transformation (left) and cumulative distribution function (right) are shown for the baseline configuration (i.e., $\mu (\mathit{x})=0$ and $\sigma (\mathit{x})=1$) and different values of the location parameter $\mu (\mathit{x})$ and of the scale parameter $\sigma (\mathit{x})$.

Fig. 2 Stratification. Distribution (top) and density (bottom) of postpartum blood loss conditional on delivery mode estimated by the stratified transformation model (left) and location-scale transformation model (right). In addition, the empirical cumulative distribution function is shown in the top row, in-sample log-likelihoods are given in the bottom row.

Fig. 3 Crossing hazards. The survivor functions of the two groups estimated by the nonparametric Kaplan-Meier method (step function) are shown along the estimates from the location-scale Weibull model (left) and the distribution-free location-scale transformation model (right).

Fig. 4 Partial proportional hazards. Three annual quantile functions (0.25, 0.50, and 0.75th quantile) for DVCs (for a hypothetical Monday in 2002) estimated by three transformation models of increasing complexity. The in-sample log-likelihoods of the corresponding models are given in the panels.

Table 1 Partial proportional hazards.

Year	Hazard ratio	95% CI
2003–2002	0.66	0.53–0.81
2004–2003	0.74	0.60–0.91
2005–2004	0.92	0.75–1.13
2006–2005	1.12	0.91–1.38
2007–2006	0.58	0.47–0.72
2008–2007	0.88	0.72–1.09
2009–2008	0.99	0.80–1.21
2010–2009	0.99	0.80–1.21
2011–2010	1.08	0.88–1.32

NOTE: Estimates and corresponding simultaneous 95% confidence intervals (CI) of multiplicative changes in hazards by year. Hazard ratios smaller one indicate increasing DVCs when comparing two subsequent years.

Fig. 5 Location-scale transformation tree. Female orgasm frequency in heterosexual relationships as a function of questionnaire variables reported by the female respondent.

Fig. 6 Transformation additive models for location and scale (TAMLS). Conditional quantiles of head circumference along age estimated by the Box-Cox-t GAMLSS (BCT GAMLSS, top panel) and the TAMLS (bottom panel). The former model comprises four and the latter model two smooth terms.

Table 2 Model selection.

		ML		BSS
Variable	Level	$\widehat{\beta }$	$\widehat{\gamma }$	$\widehat{\beta }$	$\widehat{\gamma }$
Health	poor	0.3315	$-0.0912$	0.3997	—
	excellent	$-0.3722$	0.0417	$-0.2214$	—
Sex	male	$-0.1585$	$-0.2669$	$-0.1251$	—
Insurance	yes	0.2675	0.2541	0.2800	0.2272
Chronic		0.2542	0.1799	0.2729	0.2260
School		0.0234	$-0.0175$	0.0233	—

NOTE: Location and scale parameter estimates, $\widehat{\beta }$ and $\widehat{\gamma }$, from applying the two estimation procedures, maximum likelihood (ML) or best subset selection (BSS), to a location-scale transformation model including the following explanatory variables: Health (poor < average (baseline) < excellent), Sex (female (baseline), male), Insurance (no (baseline), yes), Chronic (number of chronic conditions) and School (number of years of education). Variables which were dropped when applying best subset selection are indicated by—.