Thanks Pedro and Zoltan, that is very much appreciated.

I agree the relationship between maximum x and latitude is the strongest. 

To put "x" into context, it is plant functional diversity. I have accounted for species richness i.e there are more plants near the equator therefore if it is not accounted for there will ultimately be more functional diversity.

So the fact, despite species number being accounted for it is interesting that this pattern still occurs.

I have two question if you don't mind?

1. My data is based on WWF Eco-regions, and this probably introduces spatial autocorrelation into the mix, can this be accounted for in quantile regression?

2. I have another variable i wish to model phylogenetic diversity - i want to see - if the patterns observed are the same and try and quantify it. I.e is there a method to describe (mathematically)  the line so i can compare?

Thank,

Peter


On 25 Jan 2011, at 19:55, Pedro Lima Pequeno wrote:

Hi, Peter.
 
X values seem to be higher near the Equator, on average. Nevertheless, the strongest relation seems to be between maximum values of X and latitude. That is, perhaps something associated with latitude limits the potential size of X. I would try a quantile regression model. Quantile regression allows you to fit regression lines through any quantile of the response variable (standard regression relates only the mean value of the response to predictors). For modeling upper values of the response, you could use upper quantiles such as 95% or 99%. You could try a non linear quantile regression model, since the relationship you are interested in is non-monotonic. Package "quantreg" has functions for fitting quantile regression models in a frequentist framework. Of course, there are several functions for modeling non-monotonic relations. Also, you need to guess starting values for model parameters, since model fitting is iterative.
 
Here is some code just for illustration; there are probably better functions for modeling this particular relation.
library(quantreg)
 
plot(latitude, x)
 
mod = nlrq(x ~ -a*latitude^2*exp(-b*latitude) + c, start = c(a=0.00001,b = 0.01,c = 0.015), tau = 0.95)
#  obs.: this function is an adaptation of Ricker's standard model for non-monotonic ecological relations

summary(mod)
newvalues = seq(-40, 80, length = 478)
lines(newvalues, predict(mod, list(latitude=newvalues)), col = 3, lw = 2)
 
Hope it helps,
Pedro A. C. Lima Pequeno
 
2011/1/25 Peter Francis <peterfrancis@me.com>

Dear List,

I am looking at the relationship between a quantity and the distance from the equator.

Asking the question does latitude effect the value of x - i.e is x higher or lower as you move away from the equator

I have plotted x against latitude(plot(latitude,x))  and there seems to be something there, however i was wanting to investigate in a more quantitative way. I had thought of models but my experience is relatively low and i am unsure what method would be best for my pattern of data. Any advice would be greatly appreciated.

###data###
x <-
c(0.015348175, 0.014102894, 0.013158871, 0.012796208, 0.012547266,
0.012499614, 0.011580142, 0.011274077, 0.011026623, 0.010934845,
0.010866659, 0.01075808, 0.010581273, 0.010548757, 0.010415088,
0.010395809, 0.010328903, 0.009754356, 0.009652272, 0.009533795,
0.009179444, 0.009141853, 0.009140609, 0.008882345, 0.008671247,
0.008426542, 0.008407421, 0.008248698, 0.008228469, 0.008169984,
0.008158929, 0.007978867, 0.007893121, 0.007886193, 0.007785512,
0.007613687, 0.007591255, 0.007542375, 0.007532009, 0.007497797,
0.007487081, 0.007469353, 0.007369905, 0.007169296, 0.007122561,
0.007070457, 0.006975594, 0.006948638, 0.006805148, 0.006791063,
0.006783499, 0.006733242, 0.006733242, 0.006701148, 0.006691116,
0.006672124, 0.006660939, 0.006571554, 0.006552405, 0.006536592,
0.006523875, 0.006480761, 0.006471293, 0.006454864, 0.006365157,
0.006356432, 0.006323978, 0.006260515, 0.006188042, 0.006187938,
0.006181114, 0.006181114, 0.006079529, 0.006017497, 0.006008061,
0.005953895, 0.005920824, 0.005899296, 0.00589274, 0.005862688,
0.005853862, 0.005811885, 0.005778694, 0.005736366, 0.005686819,
0.005686362, 0.005621206, 0.005607628, 0.005567988, 0.005532497,
0.005493249, 0.005433072, 0.005429215, 0.005420174, 0.005408397,
0.005382781, 0.005304967, 0.005298177, 0.005273268, 0.005158224,
0.005135415, 0.005092708, 0.005048508, 0.005048015, 0.005044948,
0.005028163, 0.005028163, 0.005014717, 0.004967595, 0.00489943,
0.004887419, 0.004887419, 0.004887419, 0.004861756, 0.004842995,
0.00476598, 0.004734517, 0.004689229, 0.004657052, 0.004624096,
0.004618037, 0.00460988, 0.004606092, 0.004531219, 0.004527051,
0.004506577, 0.004482963, 0.004476035, 0.004476035, 0.004476035,
0.004476035, 0.004476035, 0.004419306, 0.004413048, 0.004366397,
0.00434048, 0.00434048, 0.004302982, 0.004296803, 0.004290441,
0.004285631, 0.004275535, 0.004254518, 0.004207445, 0.004173182,
0.004155142, 0.004134336, 0.004110218, 0.004106806, 0.004105657,
0.004091356, 0.004018147, 0.004003681, 0.003994852, 0.003993325,
0.003983061, 0.003966886, 0.003918854, 0.003917177, 0.003891429,
0.003842899, 0.003842899, 0.003834539, 0.003832233, 0.003818814,
0.003789309, 0.003707218, 0.003697771, 0.003668882, 0.003668882,
0.003602582, 0.003602582, 0.003602582, 0.003591953, 0.003537737,
0.003513987, 0.003464507, 0.003459366, 0.003437983, 0.003400975,
0.003377825, 0.003366575, 0.003285648, 0.003270708, 0.003260783,
0.00318234, 0.00318234, 0.003151494, 0.0031, 0.003094114, 0.003082046,
0.003081427, 0.003069106, 0.003069106, 0.003052216, 0.003036596,
0.003035941, 0.003032508, 0.003023303, 0.003020964, 0.003019848,
0.002981245, 0.002978949, 0.002969224, 0.002958385, 0.002955199,
0.002954933, 0.002954933, 0.002950752, 0.002918346, 0.002906109,
0.002902979, 0.002901011, 0.002896504, 0.002877516, 0.002849816,
0.002843091, 0.002828998, 0.002817939, 0.002816043, 0.002803956,
0.002798723, 0.002790437, 0.002777884, 0.002777884, 0.002777884,
0.002777884, 0.002777884, 0.002777884, 0.002777884, 0.002777555,
0.002776965, 0.002771398, 0.002752354, 0.00272757, 0.002717057,
0.002701395, 0.002687727, 0.002635401, 0.002635401, 0.002635401,
0.002635401, 0.002631182, 0.002623146, 0.002616387, 0.002584465,
0.002578125, 0.002577877, 0.002548996, 0.002488092, 0.002476084,
0.002474694, 0.002451195, 0.002443587, 0.002442069, 0.002439442,
0.002439442, 0.002425784, 0.002418847, 0.002409969, 0.002404593,
0.002401727, 0.002401727, 0.002401727, 0.002400578, 0.002364737,
0.00235511, 0.002326208, 0.002326208, 0.002323944, 0.002317546,
0.002275955, 0.002261807, 0.002219485, 0.002217373, 0.002210748,
0.002208339, 0.002199135, 0.002192113, 0.002171856, 0.002171856,
0.002156217, 0.002126684, 0.002090031, 0.002086738, 0.002086209,
0.002078779, 0.002047235, 0.002040628, 0.002036937, 0.002036773,
0.002036773, 0.002017393, 0.002009864, 0.001999733, 0.001998787,
0.001990045, 0.001986944, 0.001975691, 0.001971625, 0.001945183,
0.001936149, 0.001926163, 0.001919142, 0.001897503, 0.001897503,
0.001887366, 0.001866615, 0.001865528, 0.001860441, 0.001851106,
0.001850471, 0.001807395, 0.001807281, 0.001794796, 0.001790554,
0.001778587, 0.001773777, 0.001770118, 0.001726876, 0.001716987,
0.001705133, 0.001692117, 0.001675794, 0.001605834, 0.001605667,
0.001598937, 0.001575879, 0.001565384, 0.001546179, 0.001534643,
0.001525447, 0.001522089, 0.001502833, 0.001481944, 0.001473202,
0.001452109, 0.001449217, 0.001446246, 0.001446246, 0.001422652,
0.001422652, 0.001375286, 0.001361211, 0.001356151, 0.001347116,
0.001339627, 0.001339627, 0.00133279, 0.001331517, 0.001331517,
0.001331517, 0.001331517, 0.001322368, 0.001306458, 0.001296254,
0.001257304, 0.001238302, 0.001238302, 0.001238302, 0.001223937,
0.001191425, 0.001189038, 0.001180128, 0.001176136, 0.001144119,
0.001122304, 0.001109047, 0.001072805, 0.001072805, 0.001072805,
0.001072805, 0.001072805, 0.001072805, 0.001072805, 0.001072805,
0.001072805, 0.001072805, 0.001072805, 0.001072805, 0.001072805,
0.001072805, 0.001072805, 0.001072805, 0.001072805, 0.001072805,
0.001072805, 0.001072805, 0.001060498, 0.001051146, 0.001051146,
0.001015594, 0.000997827, 0.000979584, 0.000979125, 0.000959362,
0.000932058, 0.000927945, 0.000927867, 0.000909772, 0.000898857,
0.000891068, 0.000816096, 0.000804403, 0.000783013, 0.000781831,
0.000750918, 0.000744673, 0.000739314, 0.000738508, 0.000735344,
0.000712073, 0.000679757, 0.000679757, 0.000674523, 0.000636956,
0.000629264, 0.000623775, 0.000621972, 0.000620247, 0.000617503,
0.000615432, 0.000580207, 0.000548295, 0.000540084, 0.000527764,
0.000518084, 0.000510753, 0.000507758, 0.000493544, 0.000490135,
0.000468122, 0.000468122, 0.000466777, 0.000440226, 0.000420408,
0.000409094, 0.000393234, 0.000373562, 0.000373562, 0.000373562,
0.000373562, 0.000373562, 0.000373562, 0.000373562, 0.000373562,
0.000347605, 0.000344458, 0.000343825, 0.000335533, 0.000315615,
0.000313374, 0.000313148, 0.000246113, 0.000239747, 0.000227964,
0.000224992, 0.000215867, 0.000197565, 0.000196167, 0.000171385,
0.000170994, 0.000156109, 0.000129517, 0.000126808, 0.000126808,
9.35e-05, 6.5e-05, 6.5e-05, 5.95e-05, 5.59e-05, 4.36e-05, 4.36e-05,
4.36e-05, 3.74e-05, 3.34e-05, 2.14e-05, 2.02e-05)

latitude <- c(4.845, -21.9, 6.395, 19.79, 5.385, -25.565, -27.83, 19.93,
-10.995, 6.255, 7.035, 2.51, 15.16, 20.985, -29.59, -10.12, 27.655,
14.405, -0.165, 24.04, 20.765, -12.385, -12.03, 9.955, 11.46,
3.425, 1.01, -4.22, 12.52, 36.325, 8.46, 22.655, 15.11, 41.925,
34.395, 12.12, -30.375, -0.535, 17.47, -11.415, 30.11, 4.84,
-32.045, -18.435, 36.37, 20.55, 2.395, 1.895, -1.17, 35.7, 24.72,
-1.365, -1.035, 37.045, 17.745, 0.955, -15.34, 4.65, -22.675,
-34.015, -23.56, 39.615, -4.415, 28.98, 36.96, 16.39, 41.26,
-0.75, 18.58, 4.395, -32.605, -0.365, 4.975, -17, 40.67, 7.43,
5.07, 2.125, 1.25, 21.28, 3.695, -6.16, 11.13, -35.655, 21.905,
10.54, 5.145, 1.615, 10.29, 13.345, 42.71, 10.405, 39.17, -6.625,
7.845, 19.115, 23.295, 18.065, 7.31, -5.53, 56.1, 10.23, -25.705,
44.31, 17.135, -20.11, -6.87, -5.21, 15.635, -11.675, -46.71,
39, 42.845, -11.505, 18.855, -6.895, 29.55, 11.865, 16.875, -32.09,
-14.44, 32.46, 33.49, -11.655, 7.275, -12.305, -3.35, -34.16,
-3.855, 33.455, 38.615, 54.455, 10.63, 30.775, 37.525, 25.325,
30.935, 43.085, 7.69, -2.235, 11.225, 10.51, 22.26, 9.005, 40.255,
-11.045, 6.33, -1.46, 33.385, -26.69, 9, 29.405, -29.125, 6.57,
34.56, -4.67, -17.94, 28.615, -28.205, 10.95, -2.545, 30.49,
26.935, -4.45, -7.855, -14.835, 29.88, 7.95, 34.84, 39.8, -28.745,
38.275, 43.15, -19.85, 10.695, 0.43, -0.78, -5.69, -5.285, 38.085,
0.685, -3.385, -2.515, 15.44, -20.16, 40.185, 50.705, -6.71,
-37.8, 28.555, -1.58, -0.065, -10.97, 36.505, 48.14, -19.835,
-5.755, 18.3, 14.4, 7.28, 6.965, 36.95, 24.62, -29.885, 16.33,
10.795, -12.5, -0.58, -7.085, 14.885, -14.515, 0.53, 5.84, -32.88,
29.945, -13.44, 23.045, 65.91, 31.525, 20.04, 17.94, -7.855,
-27.86, -8.355, 15.41, 18.165, 18.425, 47.655, 51.59, 56.745,
-2.39, -42.475, 18.05, -14.01, 23.055, -42.19, -24.14, 26.935,
66.305, 68.565, 71.13, 74.72, 38.305, -6.09, -14.18, 31.73, 5.025,
-4.035, -22.18, -16.525, -15.675, 37.645, 23.95, 36.48, -12.39,
47.24, 48.425, -37.45, 18.13, 6.78, 22.71, -40.75, -17.15, 30.16,
41.445, -1.52, 8.795, -2.91, 27.955, 22.075, -12.75, -19.575,
-20.53, 12.665, -7.415, -12.54, 16.46, 23.89, 41.805, -13.615,
-1.475, 38.37, 33.15, 46.965, 33.04, 18.575, -6.08, 10.935, 19.14,
17.165, 38.645, 40.49, 25.725, 25.785, 16.285, 46.095, 8.95,
15.15, -18.045, 45.095, 26.115, 37.54, -0.295, -2.77, -3.445,
46.19, -10.32, 11.33, -6.09, -5.435, 17.76, -26.505, -24.855,
-2.89, 31.965, 42.825, -19.13, 30.43, 29.525, -13.855, 22.8,
0.29, 33.635, 19.055, 37.14, 4.755, -5.4, -1.7, 26.87, -7.1,
19.81, 33.005, 15.805, 1.15, 42.47, 10.73, -15.695, 32.38, 27.93,
45.155, -26.425, -21.545, 36.165, -21.265, 5.775, 20.55, -0.885,
2.82, 8.16, -7.37, 19.11, 39.72, 40.5, -7.91, 19.43, 23.605,
16.935, -11.65, 8.99, 9, 18.845, 41.575, 33.51, -29.935, 37.055,
-28.54, -31.89, 18.34, -32.745, -15.29, -0.96, 0.41, 3.235, 4.12,
12.505, 22.675, 26.05, 35.625, 38.825, 39.015, 40.825, 42.075,
43.48, 44.33, 46.735, 53.115, 57.18, 60.945, 42.555, 21.825,
30.85, 37.205, -26, -28.705, -25.1, 13.195, 22.515, 17.925, -24.31,
-3.495, 3.08, 20.91, -34.72, 18.98, 22.035, 41.88, -1.83, 7.37,
17.175, 39.735, -21.705, 4.835, -28.215, -21.96, 29.755, -5.74,
32.945, 35.14, 21.065, 20.885, -19.72, -5.18, 8.745, 36.335,
17.01, 46.885, 16.285, 27.935, -36.145, -0.85, 12.935, 18.34,
26.99, 26.335, 14.32, 32.01, 5.675, -25.82, -32.665, 4.035, 14.055,
16.65, 20.585, 30.365, 38.725, 46.025, -42.595, 81.835, -32.405,
24.455, 12.87, 30.595, 20.505, -29.405, 19.99, 42.565, 6.78,
12.84, 39.43, 20.065, 35.13, 35.91, 18.39, -34.33, -5.805, 20.715,
18.18, 19.575, 20.535, 44.73, 42.605, -13.615, 7.745, 11.37,
9.455, 42.595, 31.305, 9.835)

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