(This post is a progression of a problem I'm working on and posted about a couple of days ago. I think that what I'm asking here is sufficiently different to warrant it's own post. I've used the information learned on answers and comments there to inform this new question)
Some data:
exdf <- structure(list(TENURE = c(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93, 94, 95, 96, 97, 98, 99, 100, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72,
73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88,
89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100), GrowthRate = c(0.522068407541269,
0.285714748576455, 0.185508515947955, 0.197014234790025, 0.0955682840302288,
0.104550241818405, 0.0729165843177029, 0.052710172057008, 0.111205299930434,
0.0842167291103291, 0.0435357167369297, 0.0342561790820088, 0.0697015114811705,
0.0567551900498326, 0.0367956747574549, 0.0300708018467208, 0.0351762377682974,
0.0355785502062602, 0.0212760525934605, 0.0363614825795171, 0.0283660359906612,
0.0276822491674977, 0.0280297636181288, 0.0176973848620943, 0.0269962225046196,
0.0232995719858735, 0.0132348057981257, 0.0231117576016402, 0.0200135248674158,
0.0138992243554679, 0.0204646137008204, 0.0147834886226441, 0.0177097949074607,
0.014717031805624, 0.0132285475095113, 0.0212633514283631, 0.00842835296173483,
0.0145954105765114, 0.0101416096952409, 0.0134118780421613, 0.00938878674984167,
0.0129743252759109, 0.0109985830231452, 0.0283401588275556, 0.0204205991521889,
0.0093976137766294, 0.00910728837933128, 0.00916694752719849,
0.00741753979164272, 0.00881249103330362, 0.0111578966749217,
0.0192836374335688, 0.0120252563211682, 0.00586430387931181,
0.0143027002879901, 0.00370704967795099, 0.00682983045111385,
0.0141476105093492, 0.00482637041824496, 0.00420690319208639,
0.00376569470107668, 0.0109937747749598, 0.0140867803731126,
0.00371637851743323, 0.00717512598946612, 0.00768842197231479,
0.00251049125312619, 0.00187703000356798, 0.0048259385225542,
0.0100417595068638, 0.00374772165540982, 0.0171614859902469,
0.0149367579256126, 0.0115188216576438, 0.00490666943969309,
0.00530930735473234, 0.00723074299963677, 0.00592049387249816,
0.00484467566849744, 0.0124063039986577, 0.00573257723329412,
0.00427949785279758, 0.0030963281113916, 0.00252914919374447,
0.00635090782429515, 0.00210452127168104, 0.00804075169058649,
0.00436670408574535, 0.0035060879778257, 0.00713604840396798,
0.0023981009089411, 0.00175704532860088, 0.000779919174233257,
0.00340671636283396, 0.00460636923636137, 0.00120733393667471,
0.00933279440430645, 0.00305741675516025, 0.00304857402624847,
0.641110630691566, 0.281083708481502, 0.21664831342118, 0.142526508746734,
0.129455004633227, 0.102003156382356, 0.0774235271374781, 0.0549670789691081,
0.0809027507369802, 0.0460263351173751, 0.0653262071979981, 0.047154349769059,
0.0385947285426713, 0.0430288757773969, 0.0406360525819185, 0.0430116335897406,
0.0588492430703393, 0.0532880434533727, 0.0428311618518791, 0.0318427099482772,
0.028791324553973, 0.023804027169529, 0.0201765096165349, 0.0333877840776626,
0.02423781438746, 0.0209281238862644, 0.0152334901166924, 0.0206101546399822,
0.0192309106927642, 0.0217159111858543, 0.0204661728941318, 0.0147045919062503,
0.00963324607321603, 0.0168095825286532, 0.00772585884807775,
0.0141943686223769, 0.0105122199425551, 0.00730692991230875,
0.01505280063156, 0.00993250009012492, 0.0190685740493759, 0.0138696114197483,
0.00991543096798608, 0.0140457461257082, 0.0238644668564838,
0.014335212464454, 0.016231440976755, 0.0136006335528513, 0.010508195169141,
0.0108223681012323, 0.00724634873393271, 0.0114878660557896,
0.010768179227167, 0.0125569796266074, 0.00746257416521345, 0.00541761544241481,
0.00915133561343495, 0.0128090265857459, 0.0128834975364693,
0.00616011529066718, 0.0124072342143222, 0.00813729534453778,
0.00918992619169501, 0.00776227156683262, 0.00770248250797501,
0.00588355687767006, 0.00690140085420765, 0.00508911144051716,
0.00911620445070582, 0.00830466702459098, 0.00520036975577831,
0.00716886970824682, 0.00303064818576892, 0.0053193981576598,
0.00327658374593653, 0.00321651271471524, 0.00513516303969652,
0.00823984725674798, 0.00417642468764257, 0.014131171763804,
0.00787422067561749, 0.00692890013640657, 0.0121753072926065,
0.0070034266847312, 0.00540091988640867, 0.00689103254072876,
0.00506455733404643, 0.00598304422775975, 0.00661625550389644,
0.00732959383078224, 0.00720982160244255, 0.00564679063948148,
0.00435863414775284, 0.00556842035294203, 0.00603506069392346,
0.00548744139395829, 0.00660639356662429, 0.0045943028618467,
0.00490921014265311, 0.713632131494036, 0.316109107238324, 0.19955189191854,
0.149362726365638, 0.140527584352695, 0.122967115856381, 0.0629710542708235,
0.0794423836726601, 0.0723283480042465, 0.048411305500041, 0.0878323397155594,
0.0552740672330323, 0.0540408087274855, 0.0711265884999275, 0.0461176344041601,
0.0421688829740532, 0.0408229960327535, 0.0337191263188075, 0.0365627089380531,
0.0343486264915729, 0.0369672518324808, 0.0324027127903612, 0.0335280505737625,
0.0241231062680551, 0.0315868501489174, 0.0260876281368159, 0.0358190565406051,
0.031770274327048, 0.0307189115217597, 0.0204031908865456, 0.0304622366438121,
0.0488170927658, 0.0395661396435187, 0.0234030909209384, 0.0225458543480528,
0.0312754025113176, 0.0135110740643416, 0.0187184090181205, 0.013568316123143,
0.0170345821576348, 0.0184100113763801, 0.019219623166352, 0.0137349197436514,
0.0131633552715194, 0.0157758351970561, 0.0117732814253362, 0.0111181879788198,
0.0193989282426799, 0.0121149862839278, 0.0136437991248339, 0.0111608014682734,
0.011828405955594, 0.00944009280512503, 0.0123120446979428, 0.0175349768061963,
0.0243703580512893, 0.00775361404112473, 0.0115755699642719,
0.00869789602322868, 0.0230748185953917, 0.011388761136665, 0.0144514590777103,
0.0114791528596783, 0.0121005997846328, 0.00959154057558642,
0.0125793490404558, 0.0106768149670486, 0.00824512497201191,
0.0076573705531775, 0.0104520640098951, 0.0115961878675463, 0.0132315972263175,
0.00453620536761079, 0.00996315028677053, 0.00699906839894915,
0.0100602309904119, 0.00881275005589721, 0.00840590467502622,
0.00410243169790725, 0.00393864651260856, 0.0027632056756115,
0.00604623344076316, 0.00644190206596029, 0.0073296018951492,
0.0062144137386646, 0.00779570731803325, 0.00465780201362875,
0.0141815826452891, 0.0121929551749531, 0.00573235434597308,
0.00512844095153575, 0.0122186637666211, 0.00354975243105926,
0.0158151971904505, 0.006405119131486, 0.00504244314036839, 0.00623419419033411,
0.00664127009174464, 0.00428700468190435)), row.names = c(NA,
-297L), class = c("tbl_df", "tbl", "data.frame"))
I am modeling GrowthRate ~ TENURE and have 4 approaches:
# lm models
mod.lm.hasintercept <- lm(log(GrowthRate) ~ log(TENURE), data = exdf)
mod.lm.nointercept <- lm(log(GrowthRate) ~ 0 + log(TENURE), data = exdf)
# nls models
mod.nls.hasintercept <- nls(GrowthRate ~ i + I(TENURE^power), start = list(power = 1, i = 1), trace = T, data = exdf)
mod.nls.nointercept <- nls(GrowthRate ~ I(TENURE^power), start = list(power = 1), trace = T, data = exdf)
From my previous post linked at the top, I understand that the 1st and 3rd models are equivalent, as are the 2nd and 4th ones.
Here's the summaries:
mod.lm.hasintercept |> summary()
mod.lm.nointercept |> summary()
mod.nls.hasintercept |> summary()
mod.nls.nointercept |> summary()
mod.lm.hasintercept |> summary()
Call:
lm(formula = log(GrowthRate) ~ log(TENURE), data = exdf)
Residuals:
Min 1Q Median 3Q Max
-1.90334 -0.22186 0.01676 0.24309 1.11874
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.13426 0.10617 1.265 0.207
log(TENURE) -1.18576 0.02815 -42.125 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4134 on 295 degrees of freedom
Multiple R-squared: 0.8575, Adjusted R-squared: 0.857
F-statistic: 1775 on 1 and 295 DF, p-value: < 2.2e-16
> mod.lm.nointercept |> summary()
Call:
lm(formula = log(GrowthRate) ~ 0 + log(TENURE), data = exdf)
Residuals:
Min 1Q Median 3Q Max
-1.92663 -0.21544 0.03445 0.25076 1.09509
Coefficients:
Estimate Std. Error t value Pr(>|t|)
log(TENURE) -1.151079 0.006367 -180.8 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4138 on 296 degrees of freedom
Multiple R-squared: 0.991, Adjusted R-squared: 0.991
F-statistic: 3.269e+04 on 1 and 296 DF, p-value: < 2.2e-16
> mod.nls.hasintercept |> summary()
Formula: GrowthRate ~ i + I(TENURE^power)
Parameters:
Estimate Std. Error t value Pr(>|t|)
power -0.090765 0.005964 -15.22 <2e-16 ***
i -0.686270 0.015701 -43.71 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.04828 on 295 degrees of freedom
Number of iterations to convergence: 13
Achieved convergence tolerance: 1.975e-06
> mod.nls.nointercept |> summary()
Formula: GrowthRate ~ I(TENURE^power)
Parameters:
Estimate Std. Error t value Pr(>|t|)
power -1.08564 0.01093 -99.28 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.02001 on 296 degrees of freedom
Number of iterations to convergence: 13
Achieved convergence tolerance: 1.417e-06
I am trying to understand how to pick the 'best' model by fit.
Of the 2 lm() models, the no intercept model has a higher R^2. However, RSE is smaller. Of the 2 nls() models, there's no R^2, just RSE and the no intercept version has lower RSE.
Suppose that based on R^2 I decide the no intercept linear model is best of those two while based on RSE I decide that the no intercept model is also superior of the nls two, then how can I compare fit between these two winners? i.e. is there a sound way to evaluate and compare mod.lm.nointercept Vs. mod.nls.nointercept ? (without testing on out of sample new data, am interested in fit here)
Which of the 4 models are 'the best' fit to my data?