Bootstrap,Jackknife and other resampling methods Part V:Permutation tests Rozenn Dahyot Room 128,Department of Statistics Trinity College Dublin,Ireland dahyot mee.tcd.ie 2005 R.Dahyot (TCD) 453 Modern statistical methods 20051/22
Bootstrap, Jackknife and other resampling methods Part V: Permutation tests Rozenn Dahyot Room 128, Department of Statistics Trinity College Dublin, Ireland dahyot@mee.tcd.ie 2005 R. Dahyot (TCD) 453 Modern statistical methods 2005 1 / 22
So far The resampling methods are: o Bootstrap resampling:generate samples with the same size n as x with replacement. Jackknife subsampling generate samples with a smaller size than x without replacement. Used for: Compute accuracy measures (standard error,bias,etc.)of a statistic 0 from one set x (x1,...,xn). oCompare two sets of observations:the example of the mouse data R.Dahyot (TCD) 453 Modern statistical methods 20052/22
So far The resampling methods are: Bootstrap resampling: generate samples with the same size n as x with replacement. Jackknife subsampling : generate samples with a smaller size than x without replacement. Used for: Compute accuracy measures (standard error, bias, etc.) of a statistic θˆ from one set x = (x1, · · · , xn). Compare two sets of observations: the example of the mouse data R. Dahyot (TCD) 453 Modern statistical methods 2005 2 / 22
Example on the mouse data Data (Treatment group) 94;197:16;38;99:141;23 Data(Control group) 52;104;146;10:51;30:40;27:46 Table:The mouse data [Efron].16 mice divided assigned to a treatment group (7)or a control group(9).Survival in days following a test surgery.Did the treatment prolong survival R.Dahyot (TCD) 453 Modem statistical methods 20053/22
Example on the mouse data Data (Treatment group) 94; 197; 16; 38; 99; 141; 23 Data (Control group) 52; 104; 146; 10; 51; 30; 40; 27; 46 Table: The mouse data [Efron]. 16 mice divided assigned to a treatment group (7) or a control group (9). Survival in days following a test surgery. Did the treatment prolong survival ? R. Dahyot (TCD) 453 Modern statistical methods 2005 3 / 22
Example on the mouse data Compute B bootstrap samples for each group ·x8=(x81…,x8) *(b) x=(…,温g) B bootstrap replications are computed:(= you can approximate the p.d.f.of the replications by a histogram. R.Dahyot (TCD) 453 Modern statistical methods 20054/22
Example on the mouse data 1 Compute B bootstrap samples for each group I x ∗(b) Treat = (x ∗(b) Treat 1 , · · · , x ∗(b) Treat 7 ) I x ∗(b) Cont = (x ∗(b) Cont 1 , · · · , x ∗(b) Cont 9 ) 2 B bootstrap replications are computed: θˆ∗ (b) = x ∗(b) Treat − x ∗(b) Cont 3 you can approximate the p.d.f. of the replications by a histogram. R. Dahyot (TCD) 453 Modern statistical methods 2005 4 / 22
Example on the mouse data 003 0.05 P(0*) 0.04 0.03 0.02 0,01 -100 00 150 200 Figure:P.d.f.P()(histogram)of the replication (=30.63 and 5seB=26.85). R.Dahyot (TCD) 453 Modem statistical methods 2005 5/22
Example on the mouse data Figure: P.d.f. P(θˆ∗ ) (histogram) of the replication θˆ∗ ( θˆ = 30.63 and seˆ B = 26.85). R. Dahyot (TCD) 453 Modern statistical methods 2005 5 / 22