MATH2740: Environmental Statistics
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MATH2740: Environmental Statistics Worksheet VII (Lectures 9-11, work week 8, hand in lecture 17) ALL questions count towards the continuous assessment for this module. Q1. The Office for National Statistics has divided the country into a number of “Super Output Areas” (SOAs) which contain on average 1500 people. The percentage of SOAs in 16 different wards in the central part of Leeds metropolitan district which are amongst the 20% most deprived SOAs for housing in the country are given in the table below and shown in figure 9. The regions shaded black in figure 9 are those for which the percentage of SOAs in the 20% most deprived for housing in the UK is above the median 24.5% for these 16 wards. The data are given by Leeds City Council (2011). Map Percentage of ward SOAs in code Ward name worst UK quintile for housing 1 Kirkstall 7% 2 Weetwood 31% 3 Moortown 7% 4 Roundhay 6% 5 Headingley 50% 6 Chapel Allerton 31% 7 Hyde Park and Woodhouse 62% 8 Gipton and Harehills 6% 9 Killingbeck and Seacroft 24% 10 Armley 25% 11 City and Hunslet 25% 12 Burmantofts and Richmond Hills 6% 13 Temple Newsam 38% 14 Farnley and Wortley 25% 15 Beeston and Holbeck 21% 16 Middleton Park 24% Table 3: Percentage of ward Super Output Areas in worst UK quintile for housing, taken from Leeds City Council (2011). (a) Use the map code numbering to construct the contiguity matrix W = (Wij). Note that regions 8 and 13 (Gipton & Harehills and Temple Newsam) are NOT linked, but regions 9 and 12 (Killingbeck & Seacroft and Burmantofts & Richmond Hills) ARE linked. (b) Of the 33 wards in Leeds metropolitan district, there are a proportion 16/33=0.48 with more than 24.5% SOAs in the country’s worse 20%. Examine the evidence for the presence of spatial autocorrelation in the case where the study region forms part of a larger region in which the probability of a black region is 0.48. Consider BB, BW and WW joins. (c) Examine the evidence for the presence of spatial autocorrelation when no reference is made to factors outside the study region. Consider only BW joins in this case. 308 Figure 9: 16 wards in central Leeds shaded dark if percentage of SOAs in worst UK quintile is above median 24.5%. Q2. A field used for growing barley is divided into sixteen 100 sq.m. quadrats. In each quadrat 1 the dry weight of barley xi and the dry weight of the weed Sinapis alba yi are recorded. The results are summarised below. xi yi xi yi xi yi xi yi 2.94 85.58 5.71 82.35 7.37 74.24 8.75 72.70 4.36 89.61 7.28 77.60 9.02 76.10 10.07 69.13 16.08 42.27 18.11 43.01 20.51 42.29 22.84 35.96 11.35 58.92 20.43 43.41 24.11 27.72 24.94 22.96 Table 4: Dry weights of barley xi and Sinapis alba yi in n = 16 quadrats. (a) Construct a black/white map of the data by denoting quadrats as B(lack) in which the dry weight of barley is greater than the median, and denoting all other quadrats as W(hite). (b) Assess the evidence of positive spatial autocorrelation in the dry weight of barley using BB joins. (Use “rook” joins.) (c) Suppose the dry weight of Sinapis alba is related to the dry weight of barley by the regression 2 equation yi = α + βxi, i = 1, 2,...,n. Obtain the least squares estimates for α and β. b (d) Calculate the regression residuals ri = yi − (αb + βxi) for the above data. Denote quadrats as B(lack) for which the residuals are positive, and denote all other quadrats as W(hite). Hence assess for the presence of spatial autocorrelation in the residuals using the BB joins and free sampling with πB = 0.5. Comment on the validity of the assumptions made in the test procedure. 1White mustard, sometimes grown as a “green manure”. 2You can use the R lm command! 309 References Leeds City Council (2011) Index of Deprivation 2010. Leeds Economy Briefing Note, 46, April 2011. Available online http://observatory.leeds.gov.uk/resource/view?resourceId=1943 310.