This function estimates examinees' latent ability parameters. Available scoring methods include maximum likelihood estimation (ML), maximum likelihood estimation with fences (MLF; Han, 2016), weighted likelihood estimation (WL; Warm, 1989), maximum a posteriori estimation (MAP; Hambleton et al., 1991), expected a posteriori estimation (EAP; Bock & Mislevy, 1982), EAP summed scoring (Thissen et al., 1995; Thissen & Orlando, 2001), and inverse test characteristic curve (TCC) scoring (e.g., Kolen & Brennan, 2004; Kolen & Tong, 2010; Stocking, 1996).
Usage
est_score(x, ...)
# Default S3 method
est_score(
x,
data,
D = 1,
method = "ML",
range = c(-5, 5),
norm.prior = c(0, 1),
nquad = 41,
weights = NULL,
fence.a = 3,
fence.b = NULL,
tol = 1e-04,
max.iter = 100,
se = TRUE,
stval.opt = 1,
intpol = TRUE,
range.tcc = c(-7, 7),
missing = NA,
ncore = 1,
...
)
# S3 method for class 'est_irt'
est_score(
x,
method = "ML",
range = c(-5, 5),
norm.prior = c(0, 1),
nquad = 41,
weights = NULL,
fence.a = 3,
fence.b = NULL,
tol = 1e-04,
max.iter = 100,
se = TRUE,
stval.opt = 1,
intpol = TRUE,
range.tcc = c(-7, 7),
missing = NA,
ncore = 1,
...
)
# S3 method for class 'est_item'
est_score(
x,
method = "ML",
range = c(-5, 5),
norm.prior = c(0, 1),
nquad = 41,
weights = NULL,
fence.a = 3,
fence.b = NULL,
tol = 1e-04,
max.iter = 100,
se = TRUE,
stval.opt = 1,
intpol = TRUE,
range.tcc = c(-7, 7),
missing = NA,
ncore = 1,
...
)Arguments
- x
A data frame containing item metadata (e.g., item parameters, number of categories, IRT model types, etc.), or an object of class
est_irtorest_itemobtained fromest_irt()orest_item().See
est_irt()orsimdat()for more details about the item metadata. This data frame can be easily created using theshape_df()function.- ...
Additional arguments passed to
parallel::makeCluster()whenncore > 1.- data
A matrix of examinees' item responses, with rows for examinees and columns for items in the same order as the items in
x. Responses to an item with K score categories must be integers from 0 to K - 1. Missing responses are coded asNAor, in functions that have amissingargument, as the value given inmissing.- D
A scaling constant used in IRT models to make the logistic function closely approximate the normal ogive function. A value of 1.702 is commonly used for this purpose. Default is 1.
- method
A character string indicating the scoring method to use. Available options are:
"ML": Maximum likelihood estimation"MLF": Maximum likelihood estimation with fences (Han, 2016)"WL": Weighted likelihood estimation (Warm, 1989)"MAP": Maximum a posteriori estimation (Hambleton et al., 1991)"EAP": Expected a posteriori estimation (Bock & Mislevy, 1982)"EAP.SUM": Expected a posteriori summed scoring (Thissen et al., 1995; Thissen & Orlando, 2001)"INV.TCC": Inverse test characteristic curve scoring (e.g., Kolen & Brennan, 2004; Kolen & Tong, 2010; Stocking, 1996)
Default is
"ML".- range
A numeric vector of length two specifying the lower and upper bounds of the ability scale for the
"ML","MLF","WL", and"MAP"methods. Estimates outside these bounds are set to the nearest bound. The bounds also define the search grid forstval.opt = 1and the default fence locations for"MLF". Default isc(-5, 5).- norm.prior
A numeric vector of length two specifying the mean and standard deviation of the normal prior distribution. For
"MAP", it defines the prior. For"EAP"and"EAP.SUM", it is used to generate the Gaussian quadrature points and weights whenweights = NULL. Ignored for"ML","MLF","WL", and"INV.TCC". Default isc(0, 1).- nquad
An integer indicating the number of Gaussian quadrature points to be generated from the normal prior distribution. Used only when
methodis"EAP"or"EAP.SUM"andweights = NULL. Default is41.- weights
A two-column matrix or data frame containing the quadrature points (in the first column) and their corresponding weights (in the second column) for the latent variable prior distribution. The weights and points can be conveniently generated using the function
gen.weight().If
NULLandmethodis either"EAP"or"EAP.SUM", default quadrature values are generated based on thenorm.priorandnquadarguments. Ignored ifmethodis"ML","MLF","WL","MAP", or"INV.TCC".- fence.a
A numeric value specifying the slope (a) parameter of the two fence items used in MLF. See Details below. Default is 3.0.
- fence.b
A numeric vector of length two specifying the difficulty (b) parameters of the lower and upper fence items used in MLF. If
NULL, the values inrangeare used. Default isNULL.- tol
A positive numeric value specifying the convergence tolerance. For
"ML","MLF","WL", and"MAP", the Fisher scoring iterations stop when the absolute update is less thantol, andtolis also the tolerance of the root search used when the iterations do not converge. For"INV.TCC", the bisection search stops when the search interval is no wider thantol. Default is 1e-4.- max.iter
A positive integer specifying the maximum number of Fisher scoring iterations for
"ML","MLF","WL", and"MAP". Default is 100. See Details for the case where the iterations do not converge.- se
Logical. If
TRUE, standard errors of ability estimates are computed. IfFALSE, the standard errors are returned asNA. For"EAP.SUM"and"INV.TCC", standard errors are always computed. Default isTRUE.- stval.opt
A positive integer specifying the starting value option for the ML, MLF, WL, and MAP scoring methods. Available options are:
1: Grid search (default)
2: Based on observed sum scores
3: Fixed at 0
See Details below for more information.
- intpol
Logical. If
TRUEandmethod = "INV.TCC", ability estimates are assigned to sum scores that cannot be mapped through the TCC (sum scores less than or equal to the sum of the guessing parameters, and the maximum possible sum score). Default isTRUE. See Details below.- range.tcc
A numeric vector of length two giving the ability estimates assigned to the lowest and the maximum possible sum scores when
method = "INV.TCC"andintpol = TRUE. Default isc(-7, 7).- missing
A value indicating missing responses in the data set. Default is
NA. See Details below.- ncore
An integer specifying the number of logical CPU cores used for parallel scoring with
"ML","MLF","WL","MAP", and"EAP". Default is 1. See Details below.
Value
For method = "ML", "MLF", "WL", "MAP", or "EAP", a data
frame with one row per examinee and two columns:
est.theta: Ability estimates.se.theta: Standard errors of the ability estimates (NAwhense = FALSE).
For "ML", "MLF", "WL", and "MAP", the standard error is set to
99.9999 when the estimate equals a bound of range. Examinees with all
responses missing receive NA in both columns.
For method = "EAP.SUM" or "INV.TCC", a list with two data frames:
est.par: One row per examinee with the columnssum.score,est.theta, andse.theta.score.table: One row per possible sum score, from 0 to the maximum possible score, with the same three columns.
For "INV.TCC", sum scores without an estimate (see Details) have
NA in est.theta and se.theta.
Details
For "MAP", the prior is the normal distribution given by
norm.prior.
Each response must be an integer from 0 to the number of categories of the
item minus 1, and data must have one column per item in x. Missing
responses must be coded as NA or declared with the missing
argument. For "ML", "MLF", "WL", "MAP", and "EAP", missing
responses are excluded from the likelihood, and examinees with all
responses missing receive NA with a warning. For "EAP.SUM" and
"INV.TCC", missing responses are recoded as 0 with a warning.
In maximum likelihood estimation with fences (MLF; Han, 2016), two
imaginary 2PL items are added to every response pattern: a lower fence
item answered correctly and an upper fence item answered incorrectly. The
lower fence should have a b-parameter below, and the upper fence a
b-parameter above, all item difficulties in the test. Both should have
steep slopes (large a-parameters). If fence.b = NULL, the fences are
placed at the bounds given in range. See Han (2016) for details.
For "INV.TCC", the ability estimate for a sum score X is the root of
TCC(theta) = X, found by the bisection method (Howard, 2017). No root
exists for the maximum possible sum score or for sum scores less than or
equal to the sum of the guessing parameters, which is only the score 0 when
there are no guessing parameters. With
intpol = TRUE, these scores receive estimates as follows. Let
\(\theta_{min}\) and \(\theta_{max}\) be the first and second values
of range.tcc, and let \(\theta_{X}\) be the estimate for the smallest
sum score X that is greater than the sum of the guessing parameters.
Scores below X are mapped onto the line through
\((x = \theta_{min}, y = 0)\) and \((x = \theta_{X}, y = X)\), and the
maximum possible sum score receives \(\theta_{max}\). If
\(\theta_{min}\) is above \(\theta_{X}\) or \(\theta_{max}\) is
below the largest root, a warning is issued and no values are
assigned. With intpol = FALSE, these scores receive NA.
For "INV.TCC", the standard error for a sum score is the standard
deviation of the inverse TCC estimates over the conditional sum score
distribution at the estimate for that score, computed with the
Lord-Wingersky recursion (Lim et al., 2021; see lwrc()). When some
sum scores have no estimate (intpol = FALSE, or the warning described
above), the distribution is restricted to the sum scores with an estimate
and rescaled to sum to one. The implementation is based on a modified
version of SNSequate::irt.eq.tse() from the SNSequate package
(Gonzalez, 2014).
For the ML, MLF, WL, and MAP scoring methods, different strategies can be
used to determine the starting value for ability estimation based on the
stval.opt argument:
When
stval.opt = 1(default), the log-likelihood (log-posterior for"MAP") is evaluated on a grid fromrange[1]torange[2]in steps of 0.1. The grid point at the highest local maximum is the starting value. If there is no local maximum on the grid, the starting value is 0.When
stval.opt = 2, the starting value is the log-odds of the observed sum scoreobs.scorerelative to the maximum possible scoremax.score, both computed over the non-missing items:log(obs.score / (max.score - obs.score)).If
obs.score = 0, the starting value islog(1 / max.score).If
obs.score = max.score, the starting value islog(max.score).
When
stval.opt = 3, the starting value is fixed at 0.
For the ML, MLF, WL, and MAP scoring methods, the estimate is found by
Fisher scoring. This is a Newton-type iteration that divides the first
derivative of the log-likelihood (of the log-posterior for "MAP", or
Warm's weighted score function for "WL") by the expected Fisher
information instead of the observed second derivative, and a single update
is capped at 1 in absolute value. The iterations stop when the absolute
update is less than tol. If they have not converged after max.iter
iterations (for example, when the iterations alternate between two values
for a low-scoring examinee on a test with 3PLM items), the estimate is
found as follows. Among the values visited during the iterations, the
function takes the closest pair in which the score function (the
derivative above) is positive at the lower value and negative at the
upper value, so that the pair encloses a local maximum, and finds the root
of the score function between them with stats::uniroot() using the
tolerance tol. If there is no such pair, the last value of the
iterations is used. This search is not carried out when the iterations
converge. In all cases, the estimate is truncated to range, and the
standard error is computed at the final estimate.
For "ML", "MLF", "WL", "MAP", and "EAP", examinees can be scored
in parallel by setting ncore greater than 1. A warning is issued when
ncore > 1 and there are fewer than 5,000 examinees, because the parallel
overhead then exceeds the computation time.
For "ML", "MLF", "WL", and "MAP", the standard error is the
inverse square root of the expected Fisher information at the estimate.
The information includes the two fence items for "MLF" and the prior
information \(1/\sigma^2\) for "MAP". For "EAP" and "EAP.SUM",
the standard error is the posterior standard deviation. For "EAP.SUM",
the posterior for each sum score is computed with the Lord-Wingersky
recursion (see lwrc()).
For the implementation of the WL method, the function references the
catR::Pi(), catR::Ji(), and catR::Ii() functions from the catR
package (Magis & Barrada, 2017).
Methods (by class)
est_score(default): Default method to estimate examinees' latent ability parameters using a data framexcontaining the item metadata.est_score(est_irt): Method for an object of classest_irt. The item parameter estimates, response data, and scaling constantDare taken fromx, so supplyingdataorDstops with an error.est_score(est_item): Method for an object of classest_item. The item parameter estimates, response data, and scaling constantDare taken fromx, so supplyingdataorDstops with an error.
References
Bock, R. D., & Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6(4), 431-444. doi:10.1177/014662168200600405 .
Gonzalez, J. (2014). SNSequate: Standard and nonstandard statistical models and methods for test equating. Journal of Statistical Software, 59(7), 1-30. doi:10.18637/jss.v059.i07 .
Hambleton, R. K., Swaminathan, H., & Rogers, H. J. (1991). Fundamentals of item response theory. Newbury Park, CA: Sage.
Han, K. T. (2016). Maximum likelihood score estimation method with fences for short-length tests and computerized adaptive tests. Applied Psychological Measurement, 40(4), 289-301. doi:10.1177/0146621616631317 .
Howard, J. P. (2017). Computational methods for numerical analysis with R. New York: Chapman and Hall/CRC.
Kolen, M. J., & Brennan, R. L. (2004). Test equating, scaling, and linking (2nd ed.). Springer.
Kolen, M. J., & Tong, Y. (2010). Psychometric properties of IRT proficiency estimates. Educational Measurement: Issues and Practice, 29(3), 8-14. doi:10.1111/j.1745-3992.2010.00179.x .
Lim, H., Davey, T., & Wells, C. S. (2021). A recursion-based analytical approach to evaluate the performance of MST. Journal of Educational Measurement, 58(2), 154-178. doi:10.1111/jedm.12276 .
Magis, D., & Barrada, J. R. (2017). Computerized adaptive testing with R: Recent updates of the package catR. Journal of Statistical Software, 76(Code Snippet 1), 1-19. doi:10.18637/jss.v076.c01 .
Stocking, M. L. (1996). An alternative method for scoring adaptive tests. Journal of Educational and Behavioral Statistics, 21(4), 365-389. doi:10.3102/10769986021004365 .
Thissen, D., & Orlando, M. (2001). Item response theory for items scored in two categories. In D. Thissen & H. Wainer (Eds.), Test scoring (pp. 73-140). Mahwah, NJ: Lawrence Erlbaum.
Thissen, D., Pommerich, M., Billeaud, K., & Williams, V. S. L. (1995). Item response theory for scores on tests including polytomous items with ordered responses. Applied Psychological Measurement, 19(1), 39-49. doi:10.1177/014662169501900105 .
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427-450. doi:10.1007/BF02294627 .
Author
Hwanggyu Lim hglim83@gmail.com
Examples
## Import the "-prm.txt" output file from flexMIRT
flex_prm <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
# Read item parameters and convert them into item metadata
x <- bring.flexmirt(file = flex_prm, "par")$Group1$full_df
# Generate examinee ability values
set.seed(12)
theta <- rnorm(10)
# Simulate item response data based on the item metadata and abilities
data <- simdat(x, theta, D = 1)
# \donttest{
# Estimate abilities using maximum likelihood (ML)
est_score(x, data, D = 1, method = "ML", range = c(-4, 4), se = TRUE)
#> est.theta se.theta
#> 1 -2.37597568 0.5432999
#> 2 1.54041176 0.3411440
#> 3 -0.39456473 0.2715142
#> 4 -0.80001934 0.2861069
#> 5 -2.61351340 0.6288578
#> 6 0.03097513 0.2686083
#> 7 0.07766689 0.2691152
#> 8 -0.81133014 0.2866940
#> 9 0.38832795 0.2764596
#> 10 0.42226686 0.2776366
# Estimate abilities using weighted likelihood (WL)
est_score(x, data, D = 1, method = "WL", range = c(-4, 4), se = TRUE)
#> est.theta se.theta
#> 1 -2.25452427 0.5062071
#> 2 1.50571058 0.3382932
#> 3 -0.39564529 0.2715364
#> 4 -0.79478333 0.2858408
#> 5 -2.42123985 0.5582300
#> 6 0.02247278 0.2685341
#> 7 0.06952197 0.2690142
#> 8 -0.80657712 0.2864409
#> 9 0.37324895 0.2759575
#> 10 0.40825670 0.2771420
# Estimate abilities using MLF with default fences
# based on the `range` argument
est_score(x, data,
D = 1, method = "MLF",
fence.a = 3.0, fence.b = NULL, se = TRUE
)
#> est.theta se.theta
#> 1 -2.37558947 0.5429012
#> 2 1.54040048 0.3411375
#> 3 -0.39456453 0.2715141
#> 4 -0.80001852 0.2861065
#> 5 -2.61218863 0.6274710
#> 6 0.03097511 0.2686082
#> 7 0.07766686 0.2691152
#> 8 -0.81132937 0.2866936
#> 9 0.38832773 0.2764595
#> 10 0.42226664 0.2776365
# Estimate abilities using MLF with user-specified fences
est_score(x, data,
D = 1, method = "MLF", fence.a = 3.0,
fence.b = c(-7, 7), se = TRUE
)
#> est.theta se.theta
#> 1 -2.37597472 0.5432989
#> 2 1.54041173 0.3411440
#> 3 -0.39456473 0.2715142
#> 4 -0.80001934 0.2861069
#> 5 -2.61351009 0.6288543
#> 6 0.03097513 0.2686083
#> 7 0.07766689 0.2691152
#> 8 -0.81133014 0.2866940
#> 9 0.38832795 0.2764596
#> 10 0.42226686 0.2776366
# Estimate abilities using maximum a posteriori (MAP)
est_score(x, data,
D = 1, method = "MAP", norm.prior = c(0, 1),
se = TRUE
)
#> est.theta se.theta
#> 1 -1.90935697 0.3880537
#> 2 1.37883556 0.3122734
#> 3 -0.36718683 0.2615070
#> 4 -0.73763142 0.2723683
#> 5 -2.02185605 0.4072138
#> 6 0.02869450 0.2593945
#> 7 0.07265826 0.2598135
#> 8 -0.75565388 0.2731196
#> 9 0.35887600 0.2655943
#> 10 0.39324460 0.2666123
# Estimate abilities using expected a posteriori (EAP)
est_score(x, data,
D = 1, method = "EAP", norm.prior = c(0, 1),
nquad = 30, se = TRUE
)
#> est.theta se.theta
#> 1 -1.98741065 0.4164359
#> 2 1.39979503 0.3163898
#> 3 -0.36445308 0.2470032
#> 4 -0.76884012 0.2757507
#> 5 -2.12589479 0.4317501
#> 6 0.03088081 0.2998967
#> 7 0.09087299 0.2819628
#> 8 -0.79743301 0.2484775
#> 9 0.35318919 0.2606670
#> 10 0.37559150 0.2512978
# Estimate abilities using EAP summed scoring
est_score(x, data,
D = 1, method = "EAP.SUM", norm.prior = c(0, 1),
nquad = 30
)
#> $est.par
#> sum.score est.theta se.theta
#> 1 15 -1.84945881 0.4193692
#> 2 55 1.01201284 0.3327285
#> 3 34 -0.41587865 0.2986418
#> 4 31 -0.62087399 0.3218871
#> 5 13 -2.04576501 0.4430296
#> 6 40 -0.04315730 0.3144901
#> 7 41 0.02805944 0.3150827
#> 8 29 -0.76191091 0.3134347
#> 9 46 0.35095185 0.2972605
#> 10 50 0.63635923 0.3261820
#>
#> $score.table
#> sum.score est.theta se.theta
#> 1 0 -3.12665588 0.4947348
#> 2 1 -3.06696536 0.4959440
#> 3 2 -3.00363668 0.4968896
#> 4 3 -2.93631019 0.4975675
#> 5 4 -2.86462002 0.4978945
#> 6 5 -2.78826682 0.4976399
#> 7 6 -2.70712182 0.4963949
#> 8 7 -2.62133160 0.4936339
#> 9 8 -2.53136890 0.4888928
#> 10 9 -2.43797666 0.4820164
#> 11 10 -2.34200326 0.4733294
#> 12 11 -2.24421762 0.4635516
#> 13 12 -2.14525247 0.4533911
#> 14 13 -2.04576501 0.4430296
#> 15 14 -1.94671250 0.4319439
#> 16 15 -1.84945881 0.4193692
#> 17 16 -1.75547421 0.4051694
#> 18 17 -1.66571260 0.3904342
#> 19 18 -1.58007691 0.3772410
#> 20 19 -1.49739146 0.3675672
#> 21 20 -1.41598904 0.3619472
#> 22 21 -1.33464611 0.3588298
#> 23 22 -1.25335609 0.3552883
#> 24 23 -1.17343058 0.3487230
#> 25 24 -1.09676958 0.3384327
#> 26 25 -1.02470912 0.3261176
#> 27 26 -0.95715839 0.3151243
#> 28 27 -0.89249371 0.3088107
#> 29 28 -0.82819457 0.3087049
#> 30 29 -0.76191091 0.3134347
#> 31 30 -0.69254754 0.3192458
#> 32 31 -0.62087399 0.3218871
#> 33 32 -0.54922749 0.3187256
#> 34 33 -0.48033020 0.3099879
#> 35 34 -0.41587865 0.2986418
#> 36 35 -0.35572290 0.2891027
#> 37 36 -0.29796720 0.2852870
#> 38 37 -0.23977641 0.2887030
#> 39 38 -0.17850221 0.2975782
#> 40 39 -0.11277486 0.3077038
#> 41 40 -0.04315730 0.3144901
#> 42 41 0.02805944 0.3150827
#> 43 42 0.09798419 0.3096104
#> 44 43 0.16459743 0.3010856
#> 45 44 0.22770650 0.2940215
#> 46 45 0.28894048 0.2923086
#> 47 46 0.35095185 0.2972605
#> 48 47 0.41626072 0.3069569
#> 49 48 0.48617270 0.3173587
#> 50 49 0.56020071 0.3244417
#> 51 50 0.63635923 0.3261820
#> 52 51 0.71230767 0.3235277
#> 53 52 0.78672325 0.3198855
#> 54 53 0.86006860 0.3192397
#> 55 54 0.93434155 0.3238112
#> 56 55 1.01201284 0.3327285
#> 57 56 1.09473828 0.3427152
#> 58 57 1.18255707 0.3503152
#> 59 58 1.27411421 0.3541637
#> 60 59 1.36784810 0.3559114
#> 61 60 1.46342571 0.3591339
#> 62 61 1.56252098 0.3666446
#> 63 62 1.66819684 0.3779267
#> 64 63 1.78281178 0.3892442
#> 65 64 1.90635687 0.3975784
#> 66 65 2.03833819 0.4047557
#> 67 66 2.18202764 0.4156153
#> 68 67 2.34491493 0.4325989
#> 69 68 2.53427044 0.4563203
#> 70 69 2.75296022 0.4866583
#> 71 70 3.00707957 0.5256893
#>
# Estimate abilities using inverse TCC scoring
est_score(x, data,
D = 1, method = "INV.TCC", intpol = TRUE,
range.tcc = c(-7, 7)
)
#> $est.par
#> sum.score est.theta se.theta
#> 1 15 -2.18410156 0.8408969
#> 2 55 1.12566406 0.3712260
#> 3 34 -0.45707031 0.3196167
#> 4 31 -0.66792969 0.3287452
#> 5 13 -2.55285156 1.0468414
#> 6 40 -0.04410156 0.3146460
#> 7 41 0.02519531 0.3152754
#> 8 29 -0.81285156 0.3382760
#> 9 46 0.38222656 0.3243666
#> 10 50 0.69035156 0.3392010
#>
#> $score.table
#> sum.score est.theta se.theta
#> 1 0 -7.00000000 1.1103831
#> 2 1 -6.74910156 1.1119776
#> 3 2 -6.49820313 1.1139931
#> 4 3 -6.24730469 1.1165384
#> 5 4 -5.99640625 1.1197472
#> 6 5 -5.74550781 1.1237801
#> 7 6 -5.49460938 1.1288239
#> 8 7 -5.24371094 1.1350844
#> 9 8 -4.99281250 1.1427654
#> 10 9 -4.74191406 1.1520241
#> 11 10 -3.62566406 1.2024653
#> 12 11 -3.12660156 1.1913471
#> 13 12 -2.79917969 1.1339512
#> 14 13 -2.55285156 1.0468414
#> 15 14 -2.35332031 0.9450795
#> 16 15 -2.18410156 0.8408969
#> 17 16 -2.03613281 0.7431657
#> 18 17 -1.90363281 0.6571415
#> 19 18 -1.78292969 0.5851965
#> 20 19 -1.67152344 0.5273994
#> 21 20 -1.56753906 0.4822437
#> 22 21 -1.46964844 0.4475187
#> 23 22 -1.37675781 0.4208710
#> 24 23 -1.28800781 0.4002422
#> 25 24 -1.20269531 0.3840154
#> 26 25 -1.12042969 0.3710377
#> 27 26 -1.04066406 0.3604623
#> 28 27 -0.96300781 0.3517186
#> 29 28 -0.88714844 0.3444143
#> 30 29 -0.81285156 0.3382760
#> 31 30 -0.73980469 0.3330989
#> 32 31 -0.66792969 0.3287452
#> 33 32 -0.59691406 0.3250969
#> 34 33 -0.52667969 0.3220755
#> 35 34 -0.45707031 0.3196167
#> 36 35 -0.38785156 0.3176693
#> 37 36 -0.31902344 0.3162006
#> 38 37 -0.25035156 0.3151809
#> 39 38 -0.18175781 0.3145908
#> 40 39 -0.11300781 0.3144157
#> 41 40 -0.04410156 0.3146460
#> 42 41 0.02519531 0.3152754
#> 43 42 0.09503906 0.3163010
#> 44 43 0.16542969 0.3177190
#> 45 44 0.23667969 0.3195335
#> 46 45 0.30894531 0.3217490
#> 47 46 0.38222656 0.3243666
#> 48 47 0.45683594 0.3274011
#> 49 48 0.53292969 0.3308677
#> 50 49 0.61066406 0.3347878
#> 51 50 0.69035156 0.3392010
#> 52 51 0.77207031 0.3441483
#> 53 52 0.85613281 0.3497026
#> 54 53 0.94285156 0.3559647
#> 55 54 1.03261719 0.3630787
#> 56 55 1.12566406 0.3712260
#> 57 56 1.22261719 0.3806898
#> 58 57 1.32394531 0.3918367
#> 59 58 1.43042969 0.4052066
#> 60 59 1.54308594 0.4215904
#> 61 60 1.66324219 0.4421928
#> 62 61 1.79253906 0.4689061
#> 63 62 1.93371094 0.5049851
#> 64 63 2.09019531 0.5558304
#> 65 64 2.26738281 0.6304313
#> 66 65 2.47363281 0.7424616
#> 67 66 2.72269531 0.9093035
#> 68 67 3.04050781 1.1450429
#> 69 68 3.48441406 1.4331872
#> 70 69 4.23816406 1.6207472
#> 71 70 7.00000000 0.7318590
#>
# }