Base de datos de variables bioquímicas

302 observaciones. Dos situaciones: * datos faltantes (missing):No se efectuó la medición * datos censurados a la izquierda: se reporta observación por debajo de un límite de detección (LD). Existen múltiples LD

Descriptiva

##        id             zona              lugar          
##  Min.   :  1.00   Length:302         Length:302        
##  1st Qu.: 76.25   Class :character   Class :character  
##  Median :151.50   Mode  :character   Mode  :character  
##  Mean   :151.50                                        
##  3rd Qu.:226.75                                        
##  Max.   :302.00                                        
##                                                        
##      fecha                          días           año      
##  Min.   :2003-01-07 00:00:00   Min.   :   1   Min.   :2003  
##  1st Qu.:2007-09-11 00:00:00   1st Qu.:1709   1st Qu.:2007  
##  Median :2010-04-01 00:00:00   Median :2642   Median :2010  
##  Mean   :2010-01-04 13:54:26   Mean   :2556   Mean   :2010  
##  3rd Qu.:2012-12-12 06:00:00   3rd Qu.:3628   3rd Qu.:2012  
##  Max.   :2015-11-30 00:00:00   Max.   :4711   Max.   :2015  
##                                                             
##        ph          dbo.orig              dbo          dbo.cen      
##  Min.   :7.200   Length:302         Min.   : 0.5   Min.   :0.0000  
##  1st Qu.:7.600   Class :character   1st Qu.: 5.0   1st Qu.:0.0000  
##  Median :7.900   Mode  :character   Median : 7.0   Median :1.0000  
##  Mean   :7.873                      Mean   :11.2   Mean   :0.5349  
##  3rd Qu.:8.100                      3rd Qu.:16.0   3rd Qu.:1.0000  
##  Max.   :9.000                      Max.   :35.0   Max.   :1.0000  
##  NA's   :1                          NA's   :1      NA's   :1       
##       odis         namon.orig            namon          namon.cen     
##  Min.   : 2.400   Length:302         Min.   : 0.100   Min.   :0.0000  
##  1st Qu.: 5.800   Class :character   1st Qu.: 0.300   1st Qu.:0.0000  
##  Median : 6.400   Mode  :character   Median : 1.000   Median :0.0000  
##  Mean   : 7.133                      Mean   : 2.433   Mean   :0.2417  
##  3rd Qu.: 8.000                      3rd Qu.: 4.875   3rd Qu.:0.0000  
##  Max.   :12.400                      Max.   :13.000   Max.   :1.0000  
##  NA's   :9                                                            
##    na.orig                na             na.cen          p.orig         
##  Length:302         Min.   : 0.100   Min.   :0.0000   Length:302        
##  Class :character   1st Qu.: 1.000   1st Qu.:0.0000   Class :character  
##  Mode  :character   Median : 5.550   Median :0.0000   Mode  :character  
##                     Mean   : 5.968   Mean   :0.2715                     
##                     3rd Qu.:10.000   3rd Qu.:1.0000                     
##                     Max.   :40.000   Max.   :1.0000                     
##                                                                         
##        p              p.cen       
##  Min.   :0.1000   Min.   :0.0000  
##  1st Qu.:0.2000   1st Qu.:0.0000  
##  Median :0.4000   Median :0.0000  
##  Mean   :0.5942   Mean   :0.1523  
##  3rd Qu.:0.9000   3rd Qu.:0.0000  
##  Max.   :3.4000   Max.   :1.0000  
##

Explorando datos faltantes (NA)

Cuántos datos faltan? De las 302 observaciones para 6 variables (ph, dbo, odis, namon,na, p) hay 11 missing, lo que consituye el 0.61% de la base

Cómo se distribuyen?

## # A tibble: 6 x 3
##   variable n_miss pct_miss
##   <chr>     <int>    <dbl>
## 1 odis          9    2.98 
## 2 ph            1    0.331
## 3 dbo           1    0.331
## 4 namon         0    0    
## 5 na            0    0    
## 6 p             0    0

odis, pH y dbo presentan datos faltantes, baja proporcion (<3%)

Explorando datos censurados (por debajo de un límite de detección)

Se ve que pH y odis no presentan datos censurados. Se analizan las restantes variables

Métodos de imputación

Hay 3 métodos: KM, MLE y ROS. ROS es el más rcomendado. Usamos el paquete NADA de R

Helsel (2012) recommends the KM method for data sets with less than 50 percent censoring and multiple censoring levels. KM is nonparametric; therefore, there is no argument regarding probability distribution. Both the robust ROS and MLE rely on distribution assumptions. The MLE uses the uncensored observations, the proportion of censored observations, and a distributional assumption to compute estimates of summary statistics. A lognormal distribution is commonly assumed with water quality data; however, a variety of assumptions could be considered. Robust ROS for multiple censoring levels was introduced by Helsel and Cohn (1988),and requires an assumption that the censored data follow either a normal or lognormal distribution and there must be a minimum of three uncensored observations. The robust ROS method is based on regressing raw or transformed uncensored concentrations versus their normal score (i.e., develop a linear regression of the raw or transformed concentrations on a normal probability plot using only the detected observations). The censored observations are then imputed based on this regression. If transformations were used, the imputed values are back-transformed. Summary statistics are then computed from the uncensored data and the imputed values (in the original scale) for the censored data.

DBO

## all:
##         n     n.cen   pct.cen       min       max 
## 302.00000 162.00000  53.48837        NA        NA 
## 
## limits:
##   limit   n uncen   pexceed
## 1   0.5   1     0 0.4690908
## 2   5.0 147    15 0.4690908
## 3  10.0  13   125 0.4152824

n is the total number of observations (detects and nondetects);
n.cen is the number of nondetect/censored values;
pct.cen is the percentage of censored/censored observations; and
min and max are the minimum and the maximum values of all data.
limit is the censoring limit,
n is the number of nondetect values,
uncen is the number of detect observations, and
pexceed is the percentage of values exceeding this censoring limit.

De los 302 datos, hay 162 censurados para DBO (53%), con 3 límites de detección (LD): 0.5 5 y 10

Para decidir la distribución para la imputación, estudiamos el ajuste a normal y lognormal, comparando dist teórica y empírica

Normal parece ajustar mejor

Kaplan Meier

No funciona con datos faltantes (NA). En este caso hay 1. Genero bd sin missing e imputo. Como es un método no paramétrico, no es afectado por la distribución

##     obs n.risk n.event      prob     std.err   0.95LCL   0.95UCL
## 1   0.5      1       0 0.5309092 0.028988551 0.4740927 0.5877258
## 2   5.0    149       1 0.5309092 0.028988551 0.4740927 0.5877258
## 3   6.0    150       1 0.5344965 0.028961616 0.4777327 0.5912602
## 4   7.0    151       1 0.5380837 0.028932978 0.4813761 0.5947913
## 5   8.0    153       2 0.5416709 0.028902629 0.4850228 0.5983190
## 6   9.0    163      10 0.5488454 0.028836784 0.4923263 0.6053644
## 7  10.0    182       6 0.5847176 0.028402829 0.5290491 0.6403861
## 8  11.0    188       6 0.6046512 0.028181197 0.5494170 0.6598853
## 9  12.0    199      11 0.6245847 0.027910551 0.5698810 0.6792884
## 10 13.0    206       7 0.6611296 0.027282042 0.6076577 0.7146014
## 11 14.0    216      10 0.6843854 0.026788332 0.6318812 0.7368895
## 12 15.0    225       9 0.7176080 0.025946967 0.6667529 0.7684631
## 13 16.0    237      12 0.7475083 0.025040806 0.6984292 0.7965874
## 14 17.0    240       3 0.7873754 0.023583834 0.7411519 0.8335989
## 15 18.0    248       8 0.7973422 0.023169721 0.7519304 0.8427540
## 16 20.0    251       3 0.8239203 0.021954019 0.7808912 0.8669494
## 17 21.0    255       4 0.8338870 0.021452214 0.7918415 0.8759326
## 18 22.0    260       5 0.8471761 0.020739547 0.8065273 0.8878248
## 19 23.0    265       5 0.8637874 0.019771010 0.8250369 0.9025378
## 20 24.0    270       5 0.8803987 0.018703567 0.8437404 0.9170570
## 21 25.0    276       6 0.8970100 0.017519148 0.8626731 0.9313469
## 22 26.0    279       3 0.9169435 0.015906506 0.8857673 0.9481197
## 23 27.0    283       4 0.9269103 0.015002502 0.8975059 0.9563147
## 24 28.0    287       4 0.9401993 0.013667206 0.9134121 0.9669866
## 25 29.0    291       4 0.9534884 0.012138226 0.9296979 0.9772789
## 26 30.0    296       5 0.9667774 0.010329915 0.9465311 0.9870237
## 27 31.0    297       1 0.9833887 0.007366838 0.9689500 0.9978274
## 28 32.0    298       1 0.9867110 0.006600221 0.9737748 0.9996472
## 29 34.0    299       1 0.9900332 0.005725574 0.9788113 1.0000000
## 30 35.0    301       2 0.9933555 0.004682749 0.9841775 1.0000000

ROS Asumiendo distribución lognormal:

## 
## Call:
## lm(formula = obs.transformed ~ pp.nq, na.action = na.action)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.67544 -0.03352  0.03809  0.07830  0.16467 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.22640    0.02075  107.31   <2e-16 ***
## pp.nq        0.67988    0.02030   33.49   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.137 on 138 degrees of freedom
## Multiple R-squared:  0.8905, Adjusted R-squared:  0.8897 
## F-statistic:  1122 on 1 and 138 DF,  p-value: < 2.2e-16

Asumiendo distribución normal (sin transformación):

## 
## Call:
## lm(formula = obs.transformed ~ pp.nq, na.action = na.action)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7916 -0.5052 -0.0510  0.7726  2.0192 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.4993     0.1777   42.19   <2e-16 ***
## pp.nq        12.2542     0.1739   70.47   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.174 on 138 degrees of freedom
## Multiple R-squared:  0.973,  Adjusted R-squared:  0.9728 
## F-statistic:  4966 on 1 and 138 DF,  p-value: < 2.2e-16

MLE Asumiendo distribución lognormal:

##             Value Std. Error     z        p
## (Intercept) 1.653     0.0918 18.02 1.35e-72
## Log(scale)  0.218     0.0686  3.18 1.46e-03
## 
## Scale = 1.24 
## 
## Log Normal distribution
## Loglik(model)= -733.3   Loglik(intercept only)= -733.3 
## Loglik-r:  0 
## 
## Number of Newton-Raphson Iterations: 3 
## n = 301

Asumiendo distribución normal:

##             Value Std. Error    z        p
## (Intercept)  9.87     0.5231 18.9 1.87e-79
## Log(scale)   2.20     0.0412 53.4 0.00e+00
## 
## Scale = 9 
## 
## Gaussian distribution
## Loglik(model)= -824   Loglik(intercept only)= -824 
## Loglik-r:  0 
## 
## Number of Newton-Raphson Iterations: 5 
## n = 301

Resumen DBO

Elegimos ROS como método de imputación y comparamos

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -25.4478  -0.7234   7.1265   7.5247  16.0000  35.0000

ROS con normal como método de imputación genera valores negativos! Además se altera la forma de la distribución. ROS, por su parte, imputa valores que exceden el LD

Son 147 observaciones con LD<5. Sin embargo, enlength(p1$modeled[p1$modeled >5]) casos (length(p1$modeled[p1$modeled >5])/147*100) imputa valores >5!

es más razonable la imputación asumiendo lognormal?

El promedio sin imputar es 11.1976744 Los estadísticos descriptivos imputando asumiendo lognormal son:

##         n     n.cen   pct.cen 
## 301.00000 161.00000  53.48837

##       median     mean        sd
## K-M       NA 11.00966  8.069292
## ROS 8.966334 11.55900  7.798353
## MLE 5.225092 11.32816 21.791219

##        5%       10%       25%       50%       75%       90%       95% 
##  3.103336  3.937272  5.865563  8.966334 16.000000 25.000000 28.000000

Atención: De 302 datos originales: 1 missing - 162 son censurados - Maximo de no censurados es 35. Pero solo imputa 140? Dropped censored values that exceed max of uncensored values.

NAMON

Sin missing

## all:
##         n     n.cen   pct.cen       min       max 
## 302.00000  73.00000  24.17219   0.10000  13.00000 
## 
## limits:
##   limit  n uncen   pexceed
## 1   0.1 38    88 0.8392200
## 2   1.0 35   141 0.4668874

Ajusta mejor lognormal. Imputamos

Kaplan Meier

##      obs n.risk n.event      prob     std.err   0.95LCL   0.95UCL
## 1   0.10     41       3 0.1607800 0.023453608 0.1148118 0.2067482
## 2   0.20     63      22 0.1734731 0.024133137 0.1261731 0.2207732
## 3   0.30     78      15 0.2665563 0.027748023 0.2121712 0.3209414
## 4   0.35     81       3 0.3300221 0.029116714 0.2729544 0.3870898
## 5   0.40    104      23 0.3427152 0.029299833 0.2852886 0.4001418
## 6   0.50    112       8 0.4400294 0.029775392 0.3816707 0.4983881
## 7   0.60    118       6 0.4738779 0.029563239 0.4159350 0.5318207
## 8   0.70    123       5 0.4992642 0.029274043 0.4418881 0.5566402
## 9   0.80    124       1 0.5204194 0.028945311 0.4636877 0.5771512
## 10  0.90    126       2 0.5246505 0.028869748 0.4680668 0.5812341
## 11  1.00    165       4 0.5331126 0.028708604 0.4768448 0.5893804
## 12  1.10    168       3 0.5463576 0.028647837 0.4902089 0.6025063
## 13  1.20    170       2 0.5562914 0.028588846 0.5002583 0.6123245
## 14  1.30    176       6 0.5629139 0.028543092 0.5069705 0.6188573
## 15  1.40    185       9 0.5827815 0.028374694 0.5271681 0.6383948
## 16  1.50    191       6 0.6125828 0.028032923 0.5576393 0.6675263
## 17  1.60    196       5 0.6324503 0.027743916 0.5780733 0.6868274
## 18  1.70    200       4 0.6490066 0.027464428 0.5951773 0.7028359
## 19  1.80    201       1 0.6622517 0.027214770 0.6089117 0.7155916
## 20  1.90    204       3 0.6655629 0.027148654 0.6123525 0.7187733
## 21  2.00    208       4 0.6754967 0.026941248 0.6226928 0.7283006
## 22  2.10    209       1 0.6887417 0.026643126 0.6365222 0.7409613
## 23  2.20    212       3 0.6920530 0.026564656 0.6399872 0.7441187
## 24  2.30    215       3 0.7019868 0.026319569 0.6504013 0.7535722
## 25  2.50    216       1 0.7119205 0.026059642 0.6608446 0.7629965
## 26  3.10    217       1 0.7152318 0.025969625 0.6643323 0.7661313
## 27  3.40    218       1 0.7185430 0.025877893 0.6678233 0.7692628
## 28  3.90    219       1 0.7218543 0.025784426 0.6713178 0.7723909
## 29  4.10    220       1 0.7251656 0.025689206 0.6748156 0.7755155
## 30  4.50    221       1 0.7284768 0.025592214 0.6783170 0.7786366
## 31  4.60    223       2 0.7317881 0.025493428 0.6818219 0.7817543
## 32  4.70    225       2 0.7384106 0.025290392 0.6888423 0.7879789
## 33  4.80    226       1 0.7450331 0.025079923 0.6958774 0.7941889
## 34  4.90    228       2 0.7483444 0.024971843 0.6994005 0.7972883
## 35  5.00    229       1 0.7549669 0.024749867 0.7064580 0.8034757
## 36  5.10    230       1 0.7582781 0.024635918 0.7099926 0.8065637
## 37  5.20    232       2 0.7615894 0.024519960 0.7135312 0.8096476
## 38  5.30    234       2 0.7682119 0.024281896 0.7206203 0.8158036
## 39  5.40    235       1 0.7748344 0.024035434 0.7277259 0.8219430
## 40  5.50    239       4 0.7781457 0.023908973 0.7312850 0.8250064
## 41  5.80    242       3 0.7913907 0.023380770 0.7455653 0.8372162
## 42  5.90    257      15 0.8013245 0.022960053 0.7563236 0.8463254
## 43  6.00    259       2 0.8509934 0.020490965 0.8108318 0.8911549
## 44  6.10    262       3 0.8576159 0.020108223 0.8182045 0.8970273
## 45  6.20    264       2 0.8675497 0.019506090 0.8293184 0.9057809
## 46  6.30    267       3 0.8741722 0.019084613 0.8367670 0.9115773
## 47  6.40    269       2 0.8841060 0.018419559 0.8480043 0.9202076
## 48  6.60    277       8 0.8907285 0.017952407 0.8555424 0.9259145
## 49  6.70    279       2 0.9172185 0.015856213 0.8861409 0.9482961
## 50  6.80    280       1 0.9238411 0.015263552 0.8939250 0.9537571
## 51  7.20    281       1 0.9271523 0.014954777 0.8978415 0.9564631
## 52  7.70    283       2 0.9304636 0.014637011 0.9017756 0.9591516
## 53  7.80    284       1 0.9370861 0.013972033 0.9097014 0.9644708
## 54  8.10    285       1 0.9403974 0.013623384 0.9136960 0.9670987
## 55  8.40    286       1 0.9437086 0.013262838 0.9177139 0.9697033
## 56  8.80    288       2 0.9470199 0.012889397 0.9217571 0.9722826
## 57  9.00    291       3 0.9536424 0.012099010 0.9299288 0.9773560
## 58  9.90    294       3 0.9635762 0.010780339 0.9424471 0.9847052
## 59 10.00    296       2 0.9735099 0.009240772 0.9553984 0.9916215
## 60 11.00    297       1 0.9801325 0.008029917 0.9643941 0.9958708
## 61 11.10    299       2 0.9834437 0.007342650 0.9690524 0.9978350
## 62 12.00    301       2 0.9900662 0.005706710 0.9788813 1.0000000
## 63 13.00    302       1 0.9966887 0.003305772 0.9902095 1.0000000

ROS

## 
## Call:
## lm(formula = obs.transformed ~ pp.nq, na.action = na.action)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.68646 -0.22724  0.01378  0.16049  0.71024 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.20812    0.02727  -7.632 6.36e-13 ***
## pp.nq        1.64099    0.03170  51.763  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3742 on 227 degrees of freedom
## Multiple R-squared:  0.9219, Adjusted R-squared:  0.9216 
## F-statistic:  2679 on 1 and 227 DF,  p-value: < 2.2e-16

MLE

##              Value Std. Error     z        p
## (Intercept) -0.284     0.1071 -2.65 7.94e-03
## Log(scale)   0.573     0.0487 11.76 6.13e-32
## 
## Scale = 1.77 
## 
## Log Normal distribution
## Loglik(model)= -609.6   Loglik(intercept only)= -609.6 
## Loglik-r:  0 
## 
## Number of Newton-Raphson Iterations: 6 
## n = 302

Resumen NAMON

En resumen, estadísticos descriptivos según método de imputación:

##         n     n.cen   pct.cen 
## 302.00000  73.00000  24.17219

##        median     mean        sd
## K-M 0.7000000 2.351370  2.993459
## ROS 0.7000000 2.355245  2.988941
## MLE 0.7524449 3.628063 17.113066

El promedio sin imputar es

## [1] 2.433278

obs<-namon.ros$obs 
cens<-namon.ros$censored 
modeled<-namon.ros$modeled 
pp<-namon.ros $pp 
n<-data.frame(obs, cens, modeled, pp)
mean(n$modeled)

## [1] 2.355245

NA

Sin missing

## all:
##         n     n.cen   pct.cen       min       max 
## 302.00000  82.00000  27.15232   0.10000  40.00000 
## 
## limits:
##   limit  n uncen   pexceed
## 1   0.1  4     0 0.7284768
## 2   1.0 78   220 0.7284768

Non hay mejor ajuste con alguna. Imputamos con ambas

LOGNORMAL

Kaplan Meier

##     obs n.risk n.event      prob     std.err   0.95LCL   0.95UCL
## 1   0.1      4       0 0.2715232 0.025592214 0.2213634 0.3216830
## 2   1.0     82       0 0.2715232 0.025592214 0.2213634 0.3216830
## 3   1.1     83       1 0.2715232 0.025592214 0.2213634 0.3216830
## 4   1.9     85       2 0.2748344 0.025689206 0.2244845 0.3251844
## 5   2.0     93       8 0.2814570 0.025877893 0.2307372 0.3321767
## 6   2.3     94       1 0.3079470 0.026564656 0.2558813 0.3600128
## 7   2.6     96       2 0.3112583 0.026643126 0.2590387 0.3634778
## 8   2.8     98       2 0.3178808 0.026795312 0.2653629 0.3703986
## 9   3.0    107       9 0.3245033 0.026941248 0.2716994 0.3773072
## 10  3.5    108       1 0.3543046 0.027523191 0.3003602 0.4082491
## 11  3.6    109       1 0.3576159 0.027580512 0.3035591 0.4116727
## 12  3.8    110       1 0.3609272 0.027636401 0.3067608 0.4150935
## 13  4.0    117       7 0.3642384 0.027690866 0.3099653 0.4185115
## 14  4.1    118       1 0.3874172 0.028032923 0.3324737 0.4423607
## 15  4.5    120       2 0.3907285 0.028076276 0.3357000 0.4457570
## 16  4.8    121       1 0.3973510 0.028158914 0.3421605 0.4525415
## 17  5.0    146      25 0.4006623 0.028198211 0.3453948 0.4559297
## 18  5.2    149       3 0.4834437 0.028755989 0.4270830 0.5398044
## 19  5.3    151       2 0.4933775 0.028769243 0.4369908 0.5497642
## 20  5.8    152       1 0.5000000 0.028771767 0.4436084 0.5563916
## 21  6.0    175      23 0.5033113 0.028771136 0.4469209 0.5597016
## 22  7.0    182       7 0.5794702 0.028406025 0.5237954 0.6351450
## 23  8.0    192      10 0.6026490 0.028158914 0.5474585 0.6578395
## 24  9.0    224      32 0.6357616 0.027690866 0.5814885 0.6900347
## 25 10.0    251      27 0.7417219 0.025186098 0.6923580 0.7910857
## 26 11.0    279      28 0.8311258 0.021558152 0.7888726 0.8733790
## 27 12.0    293      14 0.9238411 0.015263552 0.8939250 0.9537571
## 28 13.0    298       5 0.9701987 0.009784635 0.9510211 0.9893762
## 29 14.0    300       2 0.9867550 0.006578513 0.9738613 0.9996486
## 30 15.0    301       1 0.9933775 0.004667295 0.9842298 1.0000000
## 31 40.0    302       1 0.9966887 0.003305772 0.9902095 1.0000000

ROS

## 
## Call:
## lm(formula = obs.transformed ~ pp.nq, na.action = na.action)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.14444 -0.09623  0.06548  0.15445  0.32705 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.63685    0.01919   85.31   <2e-16 ***
## pp.nq        0.66372    0.02291   28.97   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2397 on 218 degrees of freedom
## Multiple R-squared:  0.7938, Adjusted R-squared:  0.7928 
## F-statistic: 839.1 on 1 and 218 DF,  p-value: < 2.2e-16

MLE

##             Value Std. Error     z        p
## (Intercept) 1.175     0.0834 14.10 3.94e-45
## Log(scale)  0.325     0.0518  6.27 3.70e-10
## 
## Scale = 1.38 
## 
## Log Normal distribution
## Loglik(model)= -895.6   Loglik(intercept only)= -895.6 
## Loglik-r:  0 
## 
## Number of Newton-Raphson Iterations: 5 
## n = 302

Resumen NA LOGNORMAL