How is the Std Error in nls function calculated?

Question

I'm trying to fit a theorical model to my data and the given values by fit for the parameters I want to find are good but the error (Std. Error) are very large, which causes the reliability of the adjustment to be lost

How are these Std. Error errors calculated?

My script:

y=c(133129.8,132171.4,131439,130849.8,130359.6,129942.2,129580.6,129263.1,128981.5,128729.6,128498.8,128281.9,128075.8,127878.4,127687.7,127502.7,127322.7,127146.5,126973.2,126802.1,126633.3,126467.2,126303.2,126140.8,125979.4,125810.1,125624.4,125421.6,125201.5,124964.2,124714.1,124455.8,124189.3,123914.4,123631.3,123344.3,123057.8,122772,122486.6,122201.4,121912.2,121614.8,121309,120994.6,120671.7,120342.6,120009.5,119672.4,119331.4,118986.2,118633.9,118271.9,117899.8,117517.8,117125.6,116722.6,116307.9,115881.4,115443.2,114993.2,114532.4,114061.5,113580.5,113089.6,112588.5,112077.1,111554.7,111021.5,110477.4,109922.4,109357.6,108783.8,108201.2,107609.6,107009.2,106400.9,105785.7,105163.6,104534.7,103898.9,103256.2,102606.4,101949.6,101285.7,100614.8,99936.8,99251.7,98559.5,97860.2,97153.8,96441.3,95723.7,95001,94273.3,93540.4,92804.5,92067.2,91328.6,90588.8,89847.8,89106.5,88365.8,87625.7,86886.3,86147.6,85412.2,84682.6,83958.8,83240.9,82528.8,81821,81116,80413.8,79714.3,79017.5,78324.4,77635.7,76951.3,76271.4,75596,74925.8,74261.6,73603.4,72951.2,72305.1,71665.8,71034,70409.8,69793.2,69184,68581,67983,67389.9,66801.6,66218.1,65636.9,65055.4,64473.7,63891.5,63309.1,62727.6,62148.3,61571.3,60996.6,60424,59853.1,59283.2,58714.4,58146.6,57579.9,57014.7,56451.7,55891,55332.4,54776,54222.4,53672.1,53125,52581.2,52040.7,51504,50971.7,50443.6,49919.8,49400.3,48885.7,48376.2,47872.1,47373.2,46879.6,46392.7,45914.1,45443.7,44981.5,44527.5,44081.2,43641.9,43209.7,42784.4,42366.2,41954.4,41548.4,41148.3,40754,40365.4,39982.1,39603.5,39229.4,38860,38495.1,38135.3,37780.6,37431.3,37087.3,36748.5,36415.3,36088,35766.6,35451.1,35141.4,34837,34537.3,34242.3,33952,33666.4,33385.7,33110.1,32839.8,32574.7,32314.7,32059.4,31808,31560.6,31317.2,31077.8,30843.4,30615.1,30392.8,30176.5,29966.4,29762.4,29564.9,29373.9,29189.3,29011.1,28839.1,28673.2,28513.2,28359.2,28211.2,28068.6,27930.7,27797.7,27669.3,27545.7,27425.5,27307.3,27191.1,27077,26964.8,26854.8,26747.1,26641.5,26538.2,26437.2,26339.2,26245.1,26154.8,26068.5,25985.9,25906.6,25829.9,25755.9,25684.4,25615.5,25548.9,25484.4,25421.9,25361.4,25302.9,25246.1,25190.8,25136.8,25084.3,25033.1,24982.7,24932.5,24882.3,24832.3,24782.3,24732.8,24684.1,24636.1,24588.8,24542.3,24496.1,24450.1,24404.1,24358.2,24312.3,24266.1,24219.3,24171.8,24123.6,24074.8,24025.3,23974.9,23923.8,23871.9,23819.2,23765.9,23712.3,23658.3,23603.9,23549.2,23494.1,23438.3,23382,23325.1,23267.7,23210.1,23152.9,23096,23039.5,22983.3,22926.7,22869.1,22810.3,22750.4,22689.5,22627.5,22564.7,22501,22436.4,22371,22305,22238.9,22172.7,22106.3,22039.8,21973.6,21907.8,21842.6,21777.9,21713.8,21650,21586.3,21522.6,21459.1,21395.6,21332.5,21270,21208.2,21147.1,21086.5,21026.9,20968.5,20911.2,20855,20800,20746.2,20693.4,20641.7,20591.1,20541.6,20492.8,20444.6,20396.8,20349.5,20302.7,20256.2,20210.1,20164.3,20118.8,20073.6,20028.9,19984.8,19941.3,19898.4,19856.2,19814.6,19773.9,19734.1,19695.1,19657,19619.8,19583.6,19548.5,19514.4,19481.3,19449.4,19418.6,19389,19360.6,19333.4,19307.3,19282.6,19259.1,19236.8,19215.8,19195.8,19176.7,19158.5,19141,19124.4,19108.6,19093.7,19079.5,19066.2,19053.7,19042,19031.1,19020.9,19011.4,19002.7,18994.7,18987.4,18980.7,18974.6,18969.2,18964.4,18960,18956,18952.5,18949.4,18946.7,18944.5,18942.7,18941.3,18940.3,18939.8,18939.8,18940.1,18940.9 )

    x=c(0.003,0.004,0.005,0.006,0.007,0.008,0.009,0.01,0.011,0.012,0.013,0.014,0.015,0.016,0.017,0.018,0.019,0.02,0.021,0.022,0.023,0.024,0.025,0.026,0.027,0.028,0.029,0.03,0.031,0.032,0.033,0.034,0.035,0.036,0.037,0.038,0.039,0.04,0.041,0.042,0.043,0.044,0.045,0.046,0.047,0.048,0.049,0.05,0.051,0.052,0.053,0.054,0.055,0.056,0.057,0.058,0.059,0.06,0.061,0.062,0.063,0.064,0.065,0.066,0.067,0.068,0.069,0.07,0.071,0.072,0.073,0.074,0.075,0.076,0.077,0.078,0.079,0.08,0.081,0.082,0.083,0.084,0.085,0.086,0.087,0.088,0.089,0.09,0.091,0.092,0.093,0.094,0.095,0.096,0.097,0.098,0.099,0.1,0.101,0.102,0.103,0.104,0.105,0.106,0.107,0.108,0.109,0.11,0.111,0.112,0.113,0.114,0.115,0.116,0.117,0.118,0.119,0.12,0.121,0.122,0.123,0.124,0.125,0.126,0.127,0.128,0.129,0.13,0.131,0.132,0.133,0.134,0.135,0.136,0.137,0.138,0.139,0.14,0.141,0.142,0.143,0.144,0.145,0.146,0.147,0.148,0.149,0.15,0.151,0.152,0.153,0.154,0.155,0.156,0.157,0.158,0.159,0.16,0.161,0.162,0.163,0.164,0.165,0.166,0.167,0.168,0.169,0.17,0.171,0.172,0.173,0.174,0.175,0.176,0.177,0.178,0.179,0.18,0.181,0.182,0.183,0.184,0.185,0.186,0.187,0.188,0.189,0.19,0.191,0.192,0.193,0.194,0.195,0.196,0.197,0.198,0.199,0.2,0.201,0.202,0.203,0.204,0.205,0.206,0.207,0.208,0.209,0.21,0.211,0.212,0.213,0.214,0.215,0.216,0.217,0.218,0.219,0.22,0.221,0.222,0.223,0.224,0.225,0.226,0.227,0.228,0.229,0.23,0.231,0.232,0.233,0.234,0.235,0.236,0.237,0.238,0.239,0.24,0.241,0.242,0.243,0.244,0.245,0.246,0.247,0.248,0.249,0.25,0.251,0.252,0.253,0.254,0.255,0.256,0.257,0.258,0.259,0.26,0.261,0.262,0.263,0.264,0.265,0.266,0.267,0.268,0.269,0.27,0.271,0.272,0.273,0.274,0.275,0.276,0.277,0.278,0.279,0.28,0.281,0.282,0.283,0.284,0.285,0.286,0.287,0.288,0.289,0.29,0.291,0.292,0.293,0.294,0.295,0.296,0.297,0.298,0.299,0.3,0.301,0.302,0.303,0.304,0.305,0.306,0.307,0.308,0.309,0.31,0.311,0.312,0.313,0.314,0.315,0.316,0.317,0.318,0.319,0.32,0.321,0.322,0.323,0.324,0.325,0.326,0.327,0.328,0.329,0.33,0.331,0.332,0.333,0.334,0.335,0.336,0.337,0.338,0.339,0.34,0.341,0.342,0.343,0.344,0.345,0.346,0.347,0.348,0.349,0.35,0.351,0.352,0.353,0.354,0.355,0.356,0.357,0.358,0.359,0.36,0.361,0.362,0.363,0.364,0.365,0.366,0.367,0.368,0.369,0.37,0.371,0.372,0.373,0.374,0.375,0.376,0.377,0.378,0.379,0.38,0.381,0.382,0.383,0.384,0.385,0.386,0.387,0.388,0.389,0.39,0.391,0.392,0.393,0.394,0.395,0.396,0.397,0.398,0.399,0.4,0.401,0.402,0.403,0.404,0.405,0.406,0.407,0.408,0.409,0.41,0.411,0.412,0.413,0.414,0.415,0.416 )

Fitting:

start_ini=data.frame(t=c(0.1,1),o=c(10,1000),k=c(2,3))

py=nls2(y ~ o/(x*(x+t)^k), start=start_ini, algorithm="brute-force")

summary(py)

Results:

> Formula: y ~ o/(x * (x + t)^k)

Parameters:
  Estimate Std. Error t value Pr(>|t|)
t     0.70      17.90   0.039    0.969
o   670.00   23162.61   0.029    0.977
k     2.00      46.48   0.043    0.966

Residual standard error: 55100 on 411 degrees of freedom

Number of iterations to convergence: 64 
Achieved convergence tolerance: NA

Answer 1

The standard errors are large because your model does not fit at all. Take a look at the fitted values (on a log-log plot):

plot(x,y,log="xy",ylim=c(1000,1000000))
points(x,predict(py),col="red")
legend("bottomleft",legend = c("Data","Model"),col=c("black","red"),pch=1)

Either your model is not appropriate at all, or you're unable to find good parameter values because the grid in your grid search is too restrictive.