Decrease default error in Fitting (Std Error) R

Question

I was trying to adjust a theoretical function to a point distribution through the function nls2.

However, the errors of the adjustment are very large, that is, if a parameter must be between 2 and 3, the adjustment finds 2 error of about 16, for example. I wonder if there is any way to lessen this error (Std. Error) so that it is at least in the same order as the parameter found, for example 2 with Std. Error given by the adjustment of 1.23.

Follows my code:

The data

library(minpack.lm)
library(magicaxis)

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 )

Adjustment:

mod <- nlsLM(y ~ o/(x*(x+t)^k),
             start = c(t = 0.1, o =10, k = 2),
             trace = TRUE, 
             na.action = na.exclude, 
# control= nls.lm.control(nprint=0,
#                                  ftol=.Machine$double.eps^2,
#                                  ptol=.Machine$double.eps^2,
#                                 maxfev=10000, maxiter=2024) 
 lower=c(0.1, 10, 2),
             upper=c(1,1000, 3)

)

summary(mod)

Upshot:

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

Parameters:
  Estimate Std. Error t value Pr(>|t|)
t    0.368      5.739   0.064    0.949
o   162.416    714.204   0.227   0.820
k    2.000     27.427   0.073    0.942



 Residual standard error: 56290 on 411 degrees of freedom

Number of iterations till stop: 50 
Achieved convergence tolerance: 1.49e-08
Reason stopped: Number of iterations has reached `maxiter' == 50.

Figure:

magplot(x,y,log='xy',pch=16,col="black",cex=1 )

According to the model the adjustment on these data has to give values of k between 2 and 3 and t between 0.1 and 1 what the setting can find (good!), but with Std. Error from 27,427 to k and 5,739 for t, that is, with orders of magnitude much larger than the intervals q the values should vary, I believe the adjustment loses reliability.

I wonder if there’s a way to lessen that Std. Error for the parameters without them leaving these intervals ?

Thanks

Answer 1

Standard errors are high because the adjustment of the model to the data is very bad, due to the definition of the non-linear function and the limits imposed for the values of the parameters. It is not possible to reduce standard errors without changing the definition of the non-linear model or by changing the limits for parameter values.

In the given example, the model does not even converge on the standard number of iterations (50); increasing this number allows convergence, without greatly changing the adjustment results. We can check the adjustment visually:

plot(x, y)
coefs <- coef(mod)
curve(coefs['o']/(x * (x + coefs['t'])^coefs['k']), add=T)

If you remove the limits imposed for the values of the parameters by removing the arguments upper and lower of the adjustment function, you will get results with better standard errors, although the adjustment remains questionable.

Answer 2

The response of Erikson K. is very good. He raised an excellent point: the form of the function you are trying to adjust to the data is not good. I’ll try to expand a little what he did by commenting on adjusting functions to data.

In general, when we define a parametric form for a function that fits our data, we need to have a theoretical justification for the form of this function. For example, if I am interested in determining the function that best describes the force applied when a spring is stretched, it makes sense to use a function of the type

F = k*x,

in which

F: spring force applied k: specific parameter for each spring x: displacement applied to spring

This is well defined in Hooke’s Law.

In your case, you should ask why to adjust a function like y ~ o/(x*(x+t)^k). Why would this relationship make sense in your data? Apparently, it doesn’t, because the estimated function is not adjusting well to the data.

Looking at the graph, it looks like a function like 1/y ~ x fits well to your data:

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)

dados <- data.frame(x, y)

library(ggplot2)
ggplot(dados, aes(x=x, y=y)) +
  geom_point()

fit <- lm(1/y ~ x, data=dados)
summary(fit)
Call:
lm(formula = 1/y ~ x, data = dados)

Residuals:
Min         1Q     Median         3Q        Max 
-4.399e-06 -2.534e-06  5.228e-07  1.999e-06  6.352e-06 

Coefficients:
Estimate Std. Error t value Pr(>|t|)    
(Intercept) 7.524e-07  2.714e-07   2.773  0.00581 ** 
x           1.357e-04  1.125e-06 120.596  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.736e-06 on 412 degrees of freedom
Multiple R-squared:  0.9725,    Adjusted R-squared:  0.9724 
F-statistic: 1.454e+04 on 1 and 412 DF,  p-value: < 2.2e-16

dados <- data.frame(dados, ajuste=1/fit$fitted.values)

ggplot(dados, aes(x=x, y=y)) +
  geom_point() +
  geom_line(aes(x=x, y=ajuste, col="red"))

In fact, the standard errors are better. They are at least in the same order of magnitude. However, if we evaluate the adjustment made, represented by the red curve, it cannot properly capture the behavior of the data when x < 0.05. Therefore, this adjustment, although better than the initial one, is still not good enough, even with the high R2 as it was.

So my suggestions are two:

1) If you want to infer about the parameters of the equation that best fits the data (i.e., perform the associated hypothesis tests and find the corresponding p-values), it will be necessary to define a new relationship between these data. Apparently, y ~ o/(x*(x+t)^k) and 1/y ~ x are not good. What relationship would be ideal? Good question. Consult the literature of the area where this data came from and try to locate if anyone has experienced a similar problem. From a similar problem already solved, adjust a similar parametric model.

2) If you want to make predictions (i.e., predict the particular value of y datum x), try to adjust a non-parametric model to the data. It may be something more classic, like LOESS, or something more recent in the literature, like SVM, Random Forest etc.