Testing with a nuisance parameter present only under the alternative: a score-based approach with application to segmented modelling

<p>We introduce a score-type statistic to test for a non-zero regression coefficient when the relevant term involves a nuisance parameter present only under the alternative. Despite the non-regularity and complexity of the problem and unlike the previous approaches, the proposed test statistic does not require the nuisance to be estimated. It is simple to implement by relying on the conventional distributions, such as Normal or <i>t</i>, and it justified in the setting of probabilistic coherence. We focus on testing for the existence of a breakpoint in segmented regression, and illustrate the methodology with an analysis on data of DNA copy number aberrations and gene expression profiles from 97 breast cancer patients; moreover some simulations reveal that the proposed test is more powerful than its competitors previously discussed in literature.</p

The distribution of elephant clusters in The Kruger National Park and associated private reserves.

<p>The KNP boundary is outlined in grey, with the contiguous private reserves shaded in grey. The five clusters are shown as minimum convex polygons (MCP’s), which have been clipped according to the boundaries of Kruger and other Private Reserves. These clusters are based on the distribution of collared female elephant over the entire study period. MCP’s were calculated for three collars in the Orpen-Skukuza cluster, four collars in the Satara-Nhlanguleni-Muzanduzi cluster, one collar in the Lower Sabie cluster, four collars in the Satara cluster and three collars in the Skukuza cluster. The 10 rainfall stations used are abbreviated: HOU– Houtboschrand, SAT – Satara, NWA – Nwanetsi, TSH – Tshokwane, OSA – Lower Sabie, SKZ – Skukuza, PRE – Pretoriuskop, NHL – Nlhanguleni, TAL – Talamati, KFT – Kingfisherspruit.</p

The effect of local rainfall on the average speed of elephant during the dry to wet season transition in Kruger National Park.

<p>Lower speed breakpoints are plotted against lower rainfall breakpoints for 8 collars in 2007 (blue), 14 collars in 2008 (purple) and 7 collars in 2009 (orange). Points above the line (y = x) represent speed breakpoints that occur after rainfall breakpoints; points below the line (y = x) represent speed breakpoints that occur before rainfall breakpoints. Rainfall and speed breakpoints that fall exactly on the line are equal to each other i.e. the breakpoints occur simultaneously. Vertical lines for each year represent the range in speed breakpoints (from Day 159 to 287 in 2007; Day 142 to 302 in 2008 and Day 142 to 280 in 2009). Horizontal lines represent the range in local rainfall (from Day 196 to 289 in 2007; Day 135 to 250 in 2008 and Day 158 to 279 in 2009).</p

Multiyear piecewise regression models for Collar AM93, within the period 2007–2009, in Kruger National Park.

<p>Variables modelled include: average speed in the upper row (red line), average local rainfall in the middle row (green line), and average regional rainfall in the lower row (blue line). The columns (separated by dashed-lines) represent different years: 2007, 2008 and 2009 from left to right. The X axis represents ‘day of year’. Breakpoints are given (in associated colours), together with the 95% confidence interval for each breakpoint (represented by horizontal bars).</p