Function predict in unmarked

[Manuel Spínola]

Dear list members,

I want to calculate confidence interval to plot occupancy against a specific covariate included in my best model but I get the same results with the argument: interval = "confidence" and "prediction" options.

How can I get confidence intervals instead of prediction interval with function predict?

Best,

Manuel

--
Manuel Spínola, Ph.D.
Instituto Internacional en Conservación y Manejo de Vida Silvestre
Universidad Nacional
Apartado 1350-3000
Heredia
COSTA RICA
mspi...@una.ac.cr
mspin...@gmail.com
Teléfono: (506) 2277-3598
Fax: (506) 2237-7036
Personal website: Lobito de río
Institutional website: ICOMVIS

[Richard Chandler]

Hi Manuel,

This is perhaps slightly confusing, but predict returns confidence intervals, not prediction intervals. If you want to compute other types of intervals, I would recommend using parboot() or nonparboot(). 

Richard

[Manuel Spínola]

Thank you very much Richard.

Yes, is confusing for me because I am not a statistician.  I used predict (see below) and the confidence interval was to wide to plot (see the last rows), however I did also what you did in page 21 in the "Dynamic occupancy models in unmarked" vignette, and calculated the 95% CI using Predicted - 1.96*SE and Predicted + 1.96*SE and i got the "li" and "ls" values that I am showing below.  Those are completely different values, which should I use?

Aslo, could I use parboot() or nonparboot() to make a plot for a model with a covariate (continuous variable), a plot like you did in page 21?

 

> round(p, 3)

    Predicted    SE lower upper NUMARB TRONCOS    li    ls

1       0.493 0.214 0.154 0.839  0.190    0.12 0.074 0.912

2       0.518 0.198 0.185 0.835  0.286    0.12 0.130 0.905

3       0.543 0.181 0.221 0.833  0.382    0.12 0.187 0.898

4       0.567 0.166 0.259 0.831  0.477    0.12 0.243 0.892

5       0.592 0.151 0.300 0.831  0.573    0.12 0.297 0.887

6       0.616 0.137 0.340 0.833  0.669    0.12 0.348 0.884

7       0.639 0.125 0.380 0.836  0.765    0.12 0.395 0.884

8       0.662 0.115 0.417 0.843  0.860    0.12 0.437 0.887

9       0.684 0.108 0.449 0.852  0.956    0.12 0.473 0.895

10      0.705 0.103 0.476 0.863  1.052    0.12 0.504 0.907

11      0.726 0.100 0.497 0.876  1.148    0.12 0.529 0.922

12      0.745 0.100 0.511 0.891  1.243    0.12 0.550 0.940

13      0.764 0.100 0.521 0.906  1.339    0.12 0.567 0.961

14      0.781 0.102 0.526 0.920  1.435    0.12 0.581 0.981

15      0.798 0.104 0.527 0.933  1.531    0.12 0.594 1.000

16      0.814 0.106 0.526 0.945  1.626    0.12 0.606 1.000

17      0.828 0.108 0.522 0.955  1.722    0.12 0.617 1.000

18      0.842 0.109 0.517 0.964  1.818    0.12 0.629 1.000

19      0.855 0.110 0.510 0.971  1.914    0.12 0.640 1.000

20      0.867 0.110 0.502 0.977  2.009    0.12 0.652 1.000

21      0.878 0.109 0.494 0.982  2.105    0.12 0.664 1.000

22      0.888 0.108 0.484 0.985  2.201    0.12 0.677 1.000

23      0.898 0.106 0.474 0.989  2.297    0.12 0.689 1.000

24      0.907 0.104 0.464 0.991  2.392    0.12 0.702 1.000

25      0.915 0.102 0.453 0.993  2.488    0.12 0.716 1.000

26      0.922 0.099 0.442 0.994  2.584    0.12 0.729 1.000

27      0.929 0.096 0.431 0.996  2.680    0.12 0.742 1.000

28      0.936 0.092 0.419 0.997  2.775    0.12 0.755 1.000

29      0.941 0.089 0.408 0.997  2.871    0.12 0.768 1.000

30      0.947 0.085 0.396 0.998  2.967    0.12 0.780 1.000

31      0.952 0.081 0.384 0.998  3.063    0.12 0.793 1.000

32      0.956 0.077 0.372 0.999  3.158    0.12 0.804 1.000

33      0.960 0.073 0.361 0.999  3.254    0.12 0.816 1.000

34      0.964 0.070 0.349 0.999  3.350    0.12 0.827 1.000

35      0.967 0.066 0.338 0.999  3.446    0.12 0.838 1.000

36      0.970 0.062 0.326 1.000  3.542    0.12 0.848 1.000

37      0.973 0.059 0.315 1.000  3.637    0.12 0.858 1.000

38      0.975 0.055 0.304 1.000  3.733    0.12 0.867 1.000

39      0.978 0.052 0.293 1.000  3.829    0.12 0.876 1.000

40      0.980 0.049 0.282 1.000  3.925    0.12 0.884 1.000

41      0.982 0.046 0.272 1.000  4.020    0.12 0.892 1.000

42      0.983 0.043 0.261 1.000  4.116    0.12 0.900 1.000

43      0.985 0.040 0.251 1.000  4.212    0.12 0.907 1.000

44      0.986 0.037 0.241 1.000  4.308    0.12 0.913 1.000

45      0.988 0.035 0.232 1.000  4.403    0.12 0.920 1.000

46      0.989 0.032 0.222 1.000  4.499    0.12 0.925 1.000

47      0.990 0.030 0.213 1.000  4.595    0.12 0.931 1.000

48      0.991 0.028 0.204 1.000  4.691    0.12 0.936 1.000

49      0.992 0.026 0.196 1.000  4.786    0.12 0.941 1.000

50      0.992 0.024 0.187 1.000  4.882    0.12 0.945 1.000

51      0.993 0.022 0.179 1.000  4.978    0.12 0.949 1.000

52      0.994 0.021 0.172 1.000  5.074    0.12 0.953 1.000

53      0.994 0.019 0.164 1.000  5.169    0.12 0.957 1.000

54      0.995 0.018 0.157 1.000  5.265    0.12 0.960 1.000

55      0.995 0.017 0.150 1.000  5.361    0.12 0.963 1.000

56      0.996 0.015 0.143 1.000  5.457    0.12 0.966 1.000

57      0.996 0.014 0.136 1.000  5.552    0.12 0.969 1.000

58      0.997 0.013 0.130 1.000  5.648    0.12 0.971 1.000

59      0.997 0.012 0.124 1.000  5.744    0.12 0.973 1.000

60      0.997 0.011 0.118 1.000  5.840    0.12 0.975 1.000

61      0.997 0.010 0.113 1.000  5.935    0.12 0.977 1.000

62      0.998 0.009 0.107 1.000  6.031    0.12 0.979 1.000

63      0.998 0.009 0.102 1.000  6.127    0.12 0.981 1.000

64      0.998 0.008 0.097 1.000  6.223    0.12 0.982 1.000

65      0.998 0.007 0.093 1.000  6.318    0.12 0.984 1.000

66      0.998 0.007 0.088 1.000  6.414    0.12 0.985 1.000

67      0.999 0.006 0.084 1.000  6.510    0.12 0.986 1.000

68      0.999 0.006 0.080 1.000  6.606    0.12 0.987 1.000

69      0.999 0.005 0.076 1.000  6.702    0.12 0.988 1.000

70      0.999 0.005 0.072 1.000  6.797    0.12 0.989 1.000

71      0.999 0.005 0.068 1.000  6.893    0.12 0.990 1.000

72      0.999 0.004 0.065 1.000  6.989    0.12 0.991 1.000

73      0.999 0.004 0.062 1.000  7.085    0.12 0.992 1.000

74      0.999 0.004 0.059 1.000  7.180    0.12 0.992 1.000

75      0.999 0.003 0.056 1.000  7.276    0.12 0.993 1.000

76      0.999 0.003 0.053 1.000  7.372    0.12 0.994 1.000

77      0.999 0.003 0.050 1.000  7.468    0.12 0.994 1.000

78      1.000 0.002 0.048 1.000  7.563    0.12 0.995 1.000

79      1.000 0.002 0.045 1.000  7.659    0.12 0.995 1.000

80      1.000 0.002 0.043 1.000  7.755    0.12 0.996 1.000

81      1.000 0.002 0.041 1.000  7.851    0.12 0.996 1.000

82      1.000 0.002 0.038 1.000  7.946    0.12 0.996 1.000

83      1.000 0.002 0.036 1.000  8.042    0.12 0.997 1.000

84      1.000 0.001 0.035 1.000  8.138    0.12 0.997 1.000

85      1.000 0.001 0.033 1.000  8.234    0.12 0.997 1.000

86      1.000 0.001 0.031 1.000  8.329    0.12 0.997 1.000

87      1.000 0.001 0.029 1.000  8.425    0.12 0.998 1.000

88      1.000 0.001 0.028 1.000  8.521    0.12 0.998 1.000

89      1.000 0.001 0.026 1.000  8.617    0.12 0.998 1.000

90      1.000 0.001 0.025 1.000  8.712    0.12 0.998 1.000

91      1.000 0.001 0.024 1.000  8.808    0.12 0.998 1.000

92      1.000 0.001 0.023 1.000  8.904    0.12 0.998 1.000

93      1.000 0.001 0.021 1.000  9.000    0.12 0.999 1.000

94      1.000 0.001 0.020 1.000  9.095    0.12 0.999 1.000

95      1.000 0.001 0.019 1.000  9.191    0.12 0.999 1.000

96      1.000 0.001 0.018 1.000  9.287    0.12 0.999 1.000

97      1.000 0.000 0.017 1.000  9.383    0.12 0.999 1.000

98      1.000 0.000 0.016 1.000  9.478    0.12 0.999 1.000

99      1.000 0.000 0.015 1.000  9.574    0.12 0.999 1.000

100     1.000 0.000 0.015 1.000  9.670    0.12 0.999 1.000

100 1.00000002012/10/25 Richard Chandler <richard....@gmail.com>

[Richard Chandler]

Hi Manuel,

I wouldn't use Pred+/-1.96*SE. I don't know why I did that in the vignette. 

Either parboot or nonparboot can be used to obtain confidence intervals for predictions. 

Richard

[Sarah Thompson]

Just to clarify...

Are you saying that the CIs generated by predict() are not appropriate? -- both the "upper" & "lower" values it generates as well as taking 1.96*SE?

predict() is considerably faster than parboot, so it sure would be nice if these intervals were reasonably accurate...

Sarah

[Richard Chandler]

No! predict works perfectly fine. I was just trying to say that one *should* use predict instead of doing the computations manually. parboot and nonparboot are alternatives, but there is no need to use them if you are content with the standard confidence intervals. Sorry for the confusion.

Richard

[Manuel Spínola]

Thank you Richard,

Is there any example on how to use parboot or nonparboot to calculate CI to make a plot with a covariate (numerical variable) vs. psi?

Best,

Manuel 

2012/10/29 Richard Chandler <richard....@gmail.com>

[Richard Chandler]

Hi Manuel,

I don't have any examples to show you. Has anyone out there done something similar? This would just require writing a function that returns predicted values (using predict), and then computing the CIs as quantiles of the bootstrap distribution. If I find time, I will add an example to the help pages for nonparboot and parboot.

Richard

[Richard Chandler]

Hi Rylee,

 

After thinking about this some more, I don't think the examples shown in the help files are the correct way of computing CIs for X="the number of sites occupied in the sample." I think the best way to do this would be to compute the posterior distribution of X using Empirical Bayes. The code below seems to do the job, but I would really appreciate it if someone could test it out, and tell me if I made any errors. If this is correct, then I will clean it up and add it to unmarked... and remove the incorrect examples. NOTE: there is nothing wrong with ranef() and bup() -- this is just a question of what is the best way to estimate X and CIs for X.

 

Richard

 

 

sitesOccupied <- function(psi.c, cuts=c(0, 1)) {
    N <- length(psi.c)          # number of sites
    n <- sum(psi.c >= cuts[2])   # number of sites known to be occupied
    n0max <- sum((cuts[1] <= psi.c) & (psi.c < cuts[2]))          # number of sites with no detections
    psi.c0 <- psi.c[psi.c < 1]  # occupancy prob for sites with no dets
    postprobs <- rep(0, N+1)    # Posterior probabilities
    names(postprobs) <- 0:N
    for(n0 in 0:n0max) {        # Loop through possible values of n0
        cat("doing", n0+1, "of", n0max+1, "\n")
        ind <- combn(n0max, n0) # all possible locations for the n0 sites
        hold <- rep(NA, ncol(ind))
        for(j in 1:ncol(ind)) {
            z <- rep(0, n0max)  # True occupancy state
            z[ind[,j]] <- 1     # Creating all possibilities
            hold[j] <- prod(dbinom(z, 1, psi.c0))
        }
        postprobs[n+n0+1] <- 1 - prod(1 - hold)
    }
    return(postprobs / sum(postprobs))
}

 

 

# Example

#psi.c <- bup(ranef(fm)) # conditional estimates of occurrence probability

psi.c <- c(1, 0.5, 0.1, 1, 1, 0.2, 0.4, 0.5, 0.5, 0.1, 0.001, 0.8)
nsites <- length(psi.c)
X <- sitesOccupied(psi.c)
plot(0:nsites, X, type="h",
     xlab="Sites occupied", ylab="Posterior probability")

 

q <- cumsum(X)
results <- c(Est. = as.integer(names(which.max(X))),
             lowerCI = as.integer(names(q)[max(which(q < 0.025))]),
             upperCI = as.integer(names(q)[min(which(q > 0.975))]))
results

NOTE: if n0max is much greater than 20, this will take a long time to compute, or it may not be possible. One cheap trick to avoid this is to use the "cuts" argument to indicate which values of "psi.c" are effectively 0 or 1. Hopefully the code is self-explanatory.

 

 

 

 

 

 

 

On Fri, Nov 23, 2012 at 6:00 PM, Rylee <ryl...@gmail.com> wrote:

Hi there,

I am doing a really simple single season occupancy model. I have 51 sites, 5 observations for each site (except one that only has 4 observations). I fit a bunch of models using all combinations of 10 site covariates for occupancy probability. Now I would like to calculate the overall occupancy probability for these 51 sites and calculate a 95% confidence interval. I have tried using ranef/BUP with this code using the top model from AIC selection (but not a clear winner).

re <- ranef(d23)
EBUP <- bup(re, stat="mean")
CI <- confint(re, level=0.95)
rbind(PAO = c(Estimate = sum(EBUP), colSums(CI)) / 51)

and get this output:

     Estimate      2.5%     97.5%
PAO 0.5893818 0.5882353 0.6078431

when I try the next model:

> re <- ranef(d24)
> EBUP <- bup(re, stat="mean")
> CI <- confint(re, level=0.95)
> rbind(PAO = c(Estimate = sum(EBUP), colSums(CI)) / 51)
     Estimate      2.5%     97.5%
PAO 0.5899122 0.5882353 0.6078431

The estimates are slightly different, but the CI is exactly the same.

When I try with the global model I get an estimate of 0.5896143, again, slightly different, but the confidence interval is the exact same.
Then, if I try doing all of this again but at a 90% confidence, I get whacky CI's that dont even include the estimate and again, are all the same no matter what model I use.

> re <- ranef(d23)
> EBUP <- bup(re, stat="mean")
> CI <- confint(re, level=0.9)
> rbind(PAO = c(Estimate = sum(EBUP), colSums(CI)) / 51)
     Estimate        5%       95%
PAO 0.5893818 0.5882353 0.5882353

My question is:

Am I doing something wrong first of all?
If not, and there is possibly a bug in the code, is there a way to calculate overall occupancy probability for my 51 sites using either parboot or averaging of the output from predict?

Thanks,
Rylee

[Rylee]

So, just to clarify, I can either use bup(ranef(my model)) for psi.c and that would be give me output similar to PAO examples which uses binary prediction of occupancy or I can input a vector of occupancy probabilities calculated from, for example, predict, with my model?

Rylee

[Richard Chandler]

Hi Rylee,

Yes, use psi.c <- bup(ranef(my model)). Do not use predict.

Richard

[Anna Mödrath]

Hey there,

I'm struggeling with some similar issues and just don't understand completely what I should do. I've run a single season model, survey ran over 45 days at 13 sites (so quite small dataset), with 2 categorical covariates (habitat, saltlick present/absent) and a continuous covariate (distance to watersource).

My script contains a for loop for all my files. 

I used a script I found (https://sites.google.com/site/asrworkshop/home/schedule/r-occupancy-1) and adjusted it to my data:

library(unmarked)

wd <- "D:/MA/Daten/Occupancy_2"

setwd(wd)

my.files <- list.files(wd, pattern = ".csv")

                                                                                #user defined function to create                                                                                   ordered table of aic wts and                                                                                      predictions

my_aic<-function(labels,aic,predictions,se_predictions) {

        delta<-aic-min(aic)

        wt<-exp(-delta*0.5)

        wt<-wt/sum(wt)

        wt

        output<-cbind(labels,aic,delta,wt,predictions,se_predictions)

        output<-data.frame(labels=labels,AIC=aic,Delta=delta,ModWt=wt,Prediction=predictions,SE_prediction=se_predictions)

        output[order(aic) , ]

    }

                                                                                #create a list of labels for the                                                                                   models

labels<-c("p(.)psi(.)","p(.)psi(salt)","p(.)psi(distance.water)","p(.)psi(open.shrub)",

"p(.)psi(closed.shrub)","p(.)psi(watercourse)","p(.)psi(tree.shrub)")

results.aic <- c()

results.predict <-c()

results.psi <- c()

for (i in 1:length(my.files)) {

      my.data <- read.csv(my.files[i], h=T, row.names=1) 

      

                                                                                #detection data rows are sites                                                                                     columns are detection replicates

      y<-my.data[,1:45]

      n<-nrow(my.data)

                                                                                #site level (individual)                                                                                           covariates

      Cov.site<-my.data[,46:51]

                                                                                #put everything together in                                                                                       unmarked data frame

      alles <- unmarkedFrameOccu(y = y, siteCovs = Cov.site)

                                                                                #summary of unmarked data frame

      summary(alles)

      fm1<-occu(~1 ~1,alles)

      occu(formula = ~1 ~ 1, data = alles)

                                                                                #adding covariates to modeling

                                                                                #constant detection, occupancy                                                                                     predicted by salt

      fm2<-occu(~1 ~salt,alles)

      occu(formula = ~1 ~ salt, data = alles)

                                                                                #constant detection, occupancy                                                                                     predicted by distancewater

      fm3<-occu(~1 ~distance.water,alles)

      occu(formula = ~1 ~ distance.water, data = alles)

      fm4<-occu(~1 ~open.shrub,alles)

      occu(formula = ~1 ~ open.shrub, data = alles)

      fm5<-occu(~1 ~closed.shrub,alles)

      occu(formula = ~1 ~ closed.shrub, data = alles)

      fm6<-occu(~1 ~watercourse,alles)

      occu(formula = ~1 ~ watercourse, data = alles)

      fm7<-occu(~1 ~tree.shrub,alles)

      occu(formula = ~1 ~ tree.shrub, data = alles)

                                                                                #summarize and compare the models

      fmlist<-fitList(p_dot_psi_dot=fm1,p_dot_psi_salt=fm2,p_dot_psi_distance.water=fm3,

      p_dot_psi_open.shrub=fm4,p_dot_psi_closed.shrub=fm5,p_dot_psi_watercourse=fm6,

      p_dot_psi_tree.shrub=fm7)

      

      modSel(fmlist)

      pred.i <- predict(fmlist, type="state")

      pred.i <- cbind(my.files[i], pred.i)

      results.predict <- rbind(results.predict, pred.i)

      

      psi.c.i <- ranef(fm1,fm2,fm3,fm4,fm5,fm6,fm7, type="state")

      show(psi.c.i)

      psi.c.i <- cbind(my.files[i], psi.c.i)

      results.psi <- rbind(results.psi, psi.c.i)

      

                                                                                #aic values

      aic<-c(fm1@AIC,fm2@AIC,fm3@AIC,fm4@AIC,fm5@AIC,fm6@AIC,fm7@AIC)

                                                                                #logit scale predictions for psi                                                                                  under each model

      lc1<-linearComb(fm1, c(1), type="state")

      lc2<-linearComb(fm2, c(1, 1), type="state")

      lc3<-linearComb(fm3, c(1, 1), type="state")

      lc4 <- linearComb(fm4, c(1, 1), type="state")

      lc5 <- linearComb(fm5, c(1, 1), type="state")

      lc6 <- linearComb(fm6, c(1, 1), type="state")

      lc7 <- linearComb(fm7, c(1, 1), type="state")

                                                                                #predictions and se under each                                                                                     model

      predictions<-c(lc1@estimate,lc2@estimate,lc3@estimate,lc4@estimate,lc5@estimate,

      lc6@estimate,lc7@estimate)

      se_predictions<-sqrt(c(lc1@covMat,lc2@covMat,lc3@covMat,lc4@covMat,lc5@covMat,

      lc6@covMat,lc7@covMat))

      aic_mods<-my_aic(labels=labels,aic=aic,predictions=predictions,se_predictions=se_predictions)

                                                                                #then apply linear functions to                                                                                    weighted average and se and                                                                                       backstransform

                                                                                #aic_mods

            results.aic.i <- cbind(my.files[i], aic_mods)

            results.aic <- rbind(results.aic, results.aic.i)

}     

write.csv (results.aic, "AIC.models.csv")

write.csv(results.predict, "Predictions.csv")

write.csv (results.psi, "Psi.conditional.csv")

I'll attach one of my data files as example. 

My aim is to make a plot of the probability of occupancy for every species at every site. I'm confused about the differences with predict and ranef. Can anybody help me with that? Which values can I use to plot? The deviations are really high in predict, why are they like that?

I'm sorry, if I'm asking silly questions, but I'm just a total beginner with R.

Kind regards 

Anna

[Kery Marc]

Dear Anna,

you say this: "My aim is to make a plot of the probability of occupancy for every species at every site." --- I can see how in a more management-oriented study, there will often be an interest that is focused on the exact list of sites studied. However, is there any lasting scientific interest in the particular sites in your study or why would you want to predict the occurrence state of each ? Wouldn't you rather want to use this sample of sites to learn something general about how the use of a site depends on site characteristics such as those embodied in your covariates ?

If the latter is what you want, then you should use predict(). However, if you really, really want to get the best possible estimate of the occurrence state (i.e., whether a species is present or absent) at each of your study sites, then you should use ranef().

A little algebra is always very useful. Here is a sketch of the model with a single covariate in occupancy probability X:

(1) The true presence/absence state (z_i) of site i is a Bernoulli (coin-flip) random variable governed by parameter psi_i, which may be modelled as a function of covariate X as in a logistic regression:

z_i ~ Bern(psi_i)
logit(psi_i) = a + b *X_i

(2) What you observe, the "presence/absence measurement" y_ij at site i during replicate j, is linked to the true presence/absence state via another logistic regression:
y_ij ~ Bern(z_i * p_ij)

Whenever p<1 (detection is imperfect), presence/absence (z) is not observed without error and you may (sometimes) want to estimate it. Technically, z is binary random effect in what can be called a Bernoulli-Bernoulli random effects (or mixed) model, so to estimate it you can use the ranef() function. However, although z is a binary thing, its estimate will be any number between 0 and 1, see page 97-98 in the MacKenzie et al. (2006, occupancy bible, section "Probability of Occupancy Given Species Not Detected at a Site").

To study the tendency for some values of the covariate X to be associated with more, or less, presences of a species, you want to look at the relationship between the parameter that governs the coin-flip-like process that produces the presences and the absences and that covariate. That parameter is occupancy probability, psi, and this always a value between 0 and 1 (i.e., psi is not binary). To characterise the system underlying the observed data you want to plot the logistic-linear relationship between psi and the covariate X.

Hope this clarifies things a little.

Finally, I permit myself to make the following comment:
Basically, your question is one about logistic regression, so you should make sure that you really understand it by reading up in books about generalized linear models. Or even more fundamental, about the distinction between the outcome of a random variable (z) and the parameter(s) governing it (psi). I think that really, really good intro books for ecologists about these fundamentals of applied stats have yet to be written. However, in the meantime, there is a lot of reading that can be done to better understand this, e.g., see books by Crawley, Zuur, McCarthy, Royle & Dorazio, Kery, MacKenzie et al., and couple of others.

I am convinced that the majority of questions on this list (and no doubt on many other, similar ones, e.g., for programs MARK or PRESENCE) are not really about the specific types of models (e.g., occupancy model, N-mixture model, CJS models ....), but rather about the fundamentals of applied statistics: things such as random variables, linear models, and generalized linear models in particular. We have a very far way to go until these concepts really sink in among ecologists and similar scientists. (And I claim that I know exactly what I write about here, since I am an ecologist !)

Kind regards  ---  Marc

 I'm confused about the differences with predict and ranef. Can anybody help me with that? Which values can I use to plot? The deviations are really high in predict, why are they like that?

I'm sorry, if I'm asking silly questions, but I'm just a total beginner with R.

"

---

From: unma...@googlegroups.com [unma...@googlegroups.com] on behalf of Anna Mödrath [choria...@gmail.com]
Sent: 05 December 2014 09:28
To: unma...@googlegroups.com
Subject: Re: [unmarked] Function predict in unmarked

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[Anna Mödrath]

Dear Marc, 

I'm sorry for the late respond, but I'm just super busy with my thesis.

Your answer really helped me out and I finally got some definitions right and understood the differences between those outputs. 

Thank you very much for your fast and very useful answer! 

You're very right that I need some deeper knowledge on the general algebra and I'm working on that. 

Kind regards 

Anna