m3031

Calculation of Expected Residuals Under Normality

(Data have been sorted by increasing values of RESIDUAL)

The expected residual RESPRED for the kth sorted observation is given by

RESPRED_k = z((k - 0.5)/n)

where k is the RANK of the residual in the sorted list so that k=1 corresponds to the smallest residual and k=n the largest. (I have not multiplied by SQRT(MSE) here.) Note that the expression (k - 0.5)/n corresponds to n uniformly spaced points between 0 and 1.
EX:

by hand using a table of the standard normal distribution: for WestHam (k-0.5)/n = (RANK-0.5)/32 = (1-0.5)/32 = 0.015625; from NKNW Table B.1 z(0.015625) = -z(0.984375) = ~ -2.15; so RESPRED₁ = -2.15.
"by hand" (one case at a time) using SYSTAT's calculator: for WestHam calc zif((1-0.5)/32); the answer is -2.153875.
in practice you would let SYSTAT produce automatically a normal probability plot with the command pplot residual/normal (it does not matter whether the residuals are sorted or not).

RANK	(RANK-0.5)/32	UNION$	PAUP	OUTRATIO	ESTIMATE	RESIDUAL	RESPRED
1	0.015625	WestHam	38	49	68.593	-30.593	-2.154
2	0.046875	Wandsworth	23	18	44.866	-21.866	-1.676
3	0.078125	Fulham	31	21	47.163	-16.163	-1.418
4	...	Croydon	38	29	53.286	-15.286	-1.230
5		Hackney	33	22	47.928	-14.928	-1.078
6		Camberwell	30	14	41.805	-11.805	-0.947
7		Poplar	37	20	46.397	-9.397	-0.831
8		Lewisham	41	24	49.459	-8.459	-0.725
9		Kensington	27	5	34.916	-7.916	-0.626
10		Stepney	34	10	38.743	-4.743	-0.533
11		Greenwich	55	37	59.409	-4.409	-0.445
12		Lambeth	57	38	60.174	-3.174	-0.360
13		City	79	64	80.074	-1.074	-0.278
14		Whitechapel	35	6	35.682	-0.682	-0.197
15		StJohnHampst	61	39	60.940	0.060	-0.118
16		Westminster	52	27	51.755	0.245	-0.039
17		BethnalGreen	46	19	45.632	0.368	0.039
18		MileEnd	43	15	42.570	0.430	0.118
19		Islington	59	35	57.878	1.122	0.197
20		StGeorge	46	18	44.866	1.134	0.278
21		StGeorgeEast	37	6	35.682	1.318	0.360
22		StOlave	57	32	55.582	1.418	0.445
23		StPancras	61	35	57.878	3.122	0.533
24		StSaviour	52	22	47.928	4.072	0.626
25		Shoreditch	52	21	47.163	4.837	0.725
26		Paddington	47	12	40.274	6.726	0.831
27		Strand	64	27	51.755	12.245	0.947
28		Chelsea	64	21	47.163	16.837	1.078
29		StGiles	76	30	54.051	21.949	1.230
30		Marylebone	81	36	58.643	22.357	1.418
31		Holborn	79	33	56.347	22.653	1.676
32		Woolwich	76	20	46.397	29.603	2.154