This method provides graphical displays of the residuals between observed data and model-based predictions (Hambleton et al., 1991). For each score category of an item, it generates two types of residual plots: (a) the raw residual plot and (b) the standardized residual plot. Note that for dichotomous items, residual plots are drawn only for score category 1.
Usage
# S3 method for class 'irtfit'
plot(
x,
item.loc = NULL,
type = "both",
ci.method = c("wald", "wilson", "wilson.cr"),
show.table = TRUE,
layout.col = 2,
xlab.text,
ylab.text,
main.text,
lab.size = 15,
main.size = 15,
axis.size = 15,
line.size = 1,
point.size = 2.5,
strip.size = 12,
ylim.icc = c(0, 1),
ylim.sr.adjust = FALSE,
ylim.sr = c(-4, 4),
...
)Arguments
- x
An object of class
irtfitobtained fromirtfit().- item.loc
An integer giving the position of the item to plot in the
item_dfelement ofx. See Details below.- type
A character string indicating the type of residual plot to be displayed. Available options are:
"icc"for the raw residual plot"sr"for the standardized residual plot"both"for displaying both plots
Default is
"both".- ci.method
A character string specifying the method used to compute confidence intervals for the raw residual plot. Available options are:
"wald"for the Wald method"wilson"for the Wilson score interval"wilson.cr"for the Wilson interval with continuity correction
Default is
"wald". See Details below.- show.table
A logical value indicating whether to return the contingency table used to draw the residual plots of the item, taken from the
contingency.plotelement ofx. Default isTRUE.- layout.col
An integer specifying the number of columns in the panel layout for a polytomous item when
typeis"icc"or"sr". Withtype = "both", each plot uses one column. Default is 2.- xlab.text
A character string specifying the title for the x-axis. If omitted, a default label is used.
- ylab.text
A character string specifying the title for the y-axis. If
type = "both", a character vector of length two can be provided, corresponding to the y-axis titles for the raw residual and standardized residual plots, respectively. If omitted, default labels are used.- main.text
A character string specifying the main title for the plot. If
type = "both", a character vector of length two can be provided, corresponding to the main titles for the raw residual and standardized residual plots, respectively. If omitted, default titles are used.- lab.size
Numeric value specifying the font size of axis titles. Default is 15.
- main.size
Numeric value specifying the font size of the plot title. Default is 15.
- axis.size
Numeric value specifying the font size of axis tick labels. Default is 15.
- line.size
Numeric value specifying the thickness of plot lines. Default is 1.
- point.size
A numeric value specifying the size of points. Default is 2.5.
- strip.size
A numeric value specifying the size of facet label text. Default is 12.
- ylim.icc
A numeric vector of length two specifying the y-axis limits for the raw residual plot. Default is
c(0, 1).- ylim.sr.adjust
Logical. If
TRUE, the y-axis limits of the standardized residual plot are set to \(\pm\) the largest absolute standardized residual of the item plus 1, rounded to an integer. IfFALSE,ylim.sris used. Default isFALSE.- ylim.sr
A numeric vector of length two specifying the y-axis limits for the standardized residual plot. Default is
c(-4, 4).- ...
Additional arguments passed to
ggplot2::ggplot()from the ggplot2 package.
Value
This method displays the IRT raw residual plot, standardized
residual plot, or both for the specified item. When show.table = TRUE, a
contingency table used to generate the residual plots is also returned.
See irtfit() for more details about the contingency table.
Details
All plots are generated using the ggplot2 package.
The object of class irtfit returned by irtfit() is used to draw
raw and standardized residual plots. The plots are based mainly on the
contingency.plot element of the object.
Residual plots are drawn for one item at a time. Set item.loc to the
position of the item in the item_df element of the irtfit object; for
example, item.loc = 3 selects the third item. Items with fewer than two
responses are removed by irtfit(), so positions can differ from
those in the original item metadata.
For the raw residual plot, the ci.method argument sets the method used
to compute the confidence intervals of the observed proportions:
"wald": Wald-type interval based on the normal approximation (Laplace, 1820), \(\hat{p} \pm z_{1 - \alpha/2} \sqrt{P(1 - P)/N}\), where \(\hat{p}\) is the observed proportion, \(P\) is the model-expected probability, and \(N\) is the group size. The limits are truncated to the interval from 0 to 1."wilson": Wilson score interval (Wilson, 1927)"wilson.cr": Wilson score interval with continuity correction (Newcombe, 1998)
For more information, see
https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval.
The confidence level is \(1 - \alpha\), where \(\alpha\) is the
alpha argument of irtfit().
In the standardized residual plot, residuals whose absolute values exceed
the overSR argument of irtfit() are drawn as red circles, and
the others as blue crosses. Dashed lines mark \(\pm\) overSR.
References
Hambleton, R. K., Swaminathan, H., & Rogers, H. J. (1991). Fundamentals of item response theory. Newbury Park, CA: Sage.
Laplace, P. S. (1820). Theorie analytique des probabilites (3rd ed., in French). Paris: Courcier.
Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17(8), 857-872.
Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209-212.
Author
Hwanggyu Lim hglim83@gmail.com
Examples
## Import the "-prm.txt" output file from flexMIRT
flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
# Select the first two dichotomous items and the last polytomous item
x <- bring.flexmirt(file = flex_sam, "par")$Group1$full_df[c(1:2, 55), ]
# Generate examinees' abilities from N(0, 1)
set.seed(23)
score <- rnorm(1000, mean = 0, sd = 1)
# Simulate response data
data <- simdat(x = x, theta = score, D = 1)
# \donttest{
# Compute fit statistics
fit <- irtfit(
x = x, score = score, data = data, group.method = "equal.freq",
n.width = 11, loc.theta = "average", range.score = c(-4, 4), D = 1,
alpha = 0.05, overSR = 1.5
)
# Residual plots for the first item (dichotomous item)
plot(x = fit, item.loc = 1, type = "both", ci.method = "wald",
show.table = TRUE, ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.prop.0
#> 1 [-3.043234,-1.249466) -1.80368087 91 61 30 0.6703297
#> 2 [-1.249466,-0.8911049) -1.07489814 91 60 31 0.6593407
#> 3 [-0.8911049,-0.5332769) -0.70847940 91 49 42 0.5384615
#> 4 [-0.5332769,-0.3013328) -0.41173080 91 63 28 0.6923077
#> 5 [-0.3013328,-0.08359202) -0.20381460 91 49 42 0.5384615
#> 6 [-0.08359202,0.1602087) 0.02498348 90 53 37 0.5888889
#> 7 [0.1602087,0.3853747) 0.27922826 91 50 41 0.5494505
#> 8 [0.3853747,0.6428548) 0.49895029 91 47 44 0.5164835
#> 9 [0.6428548,0.9706215) 0.78936255 91 44 47 0.4835165
#> 10 [0.9706215,1.393539) 1.18366284 91 33 58 0.3626374
#> 11 [1.393539,2.903912] 1.83285727 91 22 69 0.2417582
#> obs.prop.1 exp.prob.0 exp.prob.1 raw.rsd.0 raw.rsd.1 se.0
#> 1 0.3296703 0.6820047 0.3179953 -0.01167504 0.01167504 0.04881838
#> 2 0.3406593 0.6453037 0.3546963 0.01403700 -0.01403700 0.05015217
#> 3 0.4615385 0.6200716 0.3799284 -0.08161006 0.08161006 0.05088047
#> 4 0.3076923 0.5958021 0.4041979 0.09650557 -0.09650557 0.05144312
#> 5 0.4615385 0.5766529 0.4233471 -0.03819133 0.03819133 0.05179464
#> 6 0.4111111 0.5535226 0.4464774 0.03536631 -0.03536631 0.05240180
#> 7 0.4505495 0.5253606 0.4746394 0.02408992 -0.02408992 0.05234678
#> 8 0.4835165 0.4990790 0.5009210 0.01740451 -0.01740451 0.05241415
#> 9 0.5164835 0.4619426 0.5380574 0.02157391 -0.02157391 0.05226219
#> 10 0.6373626 0.4082919 0.5917081 -0.04565452 0.04565452 0.05152505
#> 11 0.7582418 0.3173593 0.6826407 -0.07560103 0.07560103 0.04879227
#> se.1 std.rsd.0 std.rsd.1
#> 1 0.04881838 -0.2391525 0.2391525
#> 2 0.05015217 0.2798881 -0.2798881
#> 3 0.05088047 -1.6039565 1.6039565
#> 4 0.05144312 1.8759664 -1.8759664
#> 5 0.05179464 -0.7373605 0.7373605
#> 6 0.05240180 0.6749065 -0.6749065
#> 7 0.05234678 0.4601987 -0.4601987
#> 8 0.05241415 0.3320574 -0.3320574
#> 9 0.05226219 0.4128016 -0.4128016
#> 10 0.05152505 -0.8860644 0.8860644
#> 11 0.04879227 -1.5494469 1.5494469
# Residual plots for the third item (polytomous item)
plot(x = fit, item.loc = 3, type = "both", ci.method = "wald",
show.table = FALSE, ylim.sr.adjust = TRUE)
# Raw residual plot for the third item (polytomous item)
plot(x = fit, item.loc = 3, type = "icc", ci.method = "wald",
show.table = TRUE, ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.freq.2
#> 1 [-3.043234,-1.249466) -1.80368087 91 68 10 6
#> 2 [-1.249466,-0.8911049) -1.07489814 91 51 14 12
#> 3 [-0.8911049,-0.5332769) -0.70847940 91 34 18 11
#> 4 [-0.5332769,-0.3013328) -0.41173080 91 36 17 10
#> 5 [-0.3013328,-0.08359202) -0.20381460 91 38 13 12
#> 6 [-0.08359202,0.1602087) 0.02498348 90 29 14 13
#> 7 [0.1602087,0.3853747) 0.27922826 91 24 14 10
#> 8 [0.3853747,0.6428548) 0.49895029 91 25 9 13
#> 9 [0.6428548,0.9706215) 0.78936255 91 18 13 13
#> 10 [0.9706215,1.393539) 1.18366284 91 16 17 5
#> 11 [1.393539,2.903912] 1.83285727 91 10 3 6
#> obs.freq.3 obs.freq.4 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3 obs.prop.4
#> 1 2 5 0.7472527 0.10989011 0.06593407 0.02197802 0.05494505
#> 2 8 6 0.5604396 0.15384615 0.13186813 0.08791209 0.06593407
#> 3 13 15 0.3736264 0.19780220 0.12087912 0.14285714 0.16483516
#> 4 8 20 0.3956044 0.18681319 0.10989011 0.08791209 0.21978022
#> 5 12 16 0.4175824 0.14285714 0.13186813 0.13186813 0.17582418
#> 6 12 22 0.3222222 0.15555556 0.14444444 0.13333333 0.24444444
#> 7 17 26 0.2637363 0.15384615 0.10989011 0.18681319 0.28571429
#> 8 12 32 0.2747253 0.09890110 0.14285714 0.13186813 0.35164835
#> 9 10 37 0.1978022 0.14285714 0.14285714 0.10989011 0.40659341
#> 10 13 40 0.1758242 0.18681319 0.05494505 0.14285714 0.43956044
#> 11 11 61 0.1098901 0.03296703 0.06593407 0.12087912 0.67032967
#> exp.prob.0 exp.prob.1 exp.prob.2 exp.prob.3 exp.prob.4 raw.rsd.0
#> 1 0.71477689 0.11375163 0.06836062 0.03855298 0.06455787 0.032475855
#> 2 0.56957599 0.14884791 0.10278367 0.06320332 0.11558912 -0.009136425
#> 3 0.48976550 0.15944983 0.11992887 0.07818623 0.15266958 -0.116139125
#> 4 0.42532229 0.16264618 0.13183092 0.09078262 0.18941799 -0.029717899
#> 5 0.38151012 0.16172964 0.13839299 0.09938634 0.21898091 0.036072294
#> 6 0.33545340 0.15777543 0.14341041 0.10817376 0.25518700 -0.013231182
#> 7 0.28773759 0.15011991 0.14585476 0.11651096 0.29977678 -0.024001328
#> 8 0.24994169 0.14123048 0.14514018 0.12201185 0.34167580 0.024783588
#> 9 0.20531675 0.12719333 0.14027664 0.12622977 0.40098351 -0.007514555
#> 10 0.15461091 0.10608597 0.12760487 0.12566293 0.48603533 0.021213268
#> 11 0.09383143 0.07259205 0.09796229 0.11011249 0.62550173 0.016058676
#> raw.rsd.1 raw.rsd.2 raw.rsd.3 raw.rsd.4 se.0 se.1
#> 1 -0.003861524 -0.0024265574 -0.016574958 -0.009612815 0.04733222 0.03328403
#> 2 0.004998247 0.0290844659 0.024708768 -0.049655056 0.05190431 0.03731249
#> 3 0.038352370 0.0009502512 0.064670916 0.012165589 0.05240326 0.03837719
#> 4 0.024167011 -0.0219408122 -0.002870528 0.030362229 0.05182634 0.03868617
#> 5 -0.018872495 -0.0065248628 0.032481794 -0.043156729 0.05092120 0.03859812
#> 6 -0.002219870 0.0010340388 0.025159570 -0.010742557 0.04976885 0.03842487
#> 7 0.003726247 -0.0359646468 0.070302223 -0.014062496 0.04745671 0.03744357
#> 8 -0.042329386 -0.0022830374 0.009856283 0.009972552 0.04538854 0.03650748
#> 9 0.015663809 0.0025805066 -0.016339656 0.005609896 0.04234367 0.03492774
#> 10 0.080727222 -0.0726598139 0.017194216 -0.046474892 0.03789899 0.03228168
#> 11 -0.039625020 -0.0320282211 0.010766629 0.044827936 0.03056736 0.02719940
#> se.2 se.3 se.4 std.rsd.0 std.rsd.1 std.rsd.2
#> 1 0.02645491 0.02018231 0.02576098 0.6861257 -0.11601733 -0.09172429
#> 2 0.03183391 0.02550774 0.03351698 -0.1760244 0.13395641 0.91363168
#> 3 0.03405650 0.02814272 0.03770352 -2.2162576 0.99935330 0.02790220
#> 4 0.03546419 0.03011719 0.04107602 -0.5734130 0.62469381 -0.61867504
#> 5 0.03619853 0.03136259 0.04335241 0.7083944 -0.48894855 -0.18025213
#> 6 0.03694498 0.03274009 0.04595488 -0.2658527 -0.05777171 0.02798861
#> 7 0.03700032 0.03363282 0.04802823 -0.5057521 0.09951635 -0.97200911
#> 8 0.03692501 0.03431031 0.04971716 0.5460319 -1.15947164 -0.06182903
#> 9 0.03640419 0.03481439 0.05137620 -0.1774658 0.44846325 0.07088489
#> 10 0.03497595 0.03474740 0.05239380 0.5597317 2.50071304 -2.07742206
#> 11 0.03116170 0.03281445 0.05073626 0.5253537 -1.45683459 -1.02780733
#> std.rsd.3 std.rsd.4
#> 1 -0.82126153 -0.3731541
#> 2 0.96867740 -1.4814898
#> 3 2.29796274 0.3226645
#> 4 -0.09531193 0.7391716
#> 5 1.03568597 -0.9954863
#> 6 0.76846374 -0.2337631
#> 7 2.09028632 -0.2927965
#> 8 0.28726885 0.2005857
#> 9 -0.46933624 0.1091925
#> 10 0.49483460 -0.8870305
#> 11 0.32810632 0.8835483
# Standardized residual plot for the third item (polytomous item)
plot(x = fit, item.loc = 3, type = "sr", ci.method = "wald",
show.table = TRUE, ylim.sr.adjust = TRUE)
#> interval point total obs.freq.0 obs.freq.1 obs.freq.2
#> 1 [-3.043234,-1.249466) -1.80368087 91 68 10 6
#> 2 [-1.249466,-0.8911049) -1.07489814 91 51 14 12
#> 3 [-0.8911049,-0.5332769) -0.70847940 91 34 18 11
#> 4 [-0.5332769,-0.3013328) -0.41173080 91 36 17 10
#> 5 [-0.3013328,-0.08359202) -0.20381460 91 38 13 12
#> 6 [-0.08359202,0.1602087) 0.02498348 90 29 14 13
#> 7 [0.1602087,0.3853747) 0.27922826 91 24 14 10
#> 8 [0.3853747,0.6428548) 0.49895029 91 25 9 13
#> 9 [0.6428548,0.9706215) 0.78936255 91 18 13 13
#> 10 [0.9706215,1.393539) 1.18366284 91 16 17 5
#> 11 [1.393539,2.903912] 1.83285727 91 10 3 6
#> obs.freq.3 obs.freq.4 obs.prop.0 obs.prop.1 obs.prop.2 obs.prop.3 obs.prop.4
#> 1 2 5 0.7472527 0.10989011 0.06593407 0.02197802 0.05494505
#> 2 8 6 0.5604396 0.15384615 0.13186813 0.08791209 0.06593407
#> 3 13 15 0.3736264 0.19780220 0.12087912 0.14285714 0.16483516
#> 4 8 20 0.3956044 0.18681319 0.10989011 0.08791209 0.21978022
#> 5 12 16 0.4175824 0.14285714 0.13186813 0.13186813 0.17582418
#> 6 12 22 0.3222222 0.15555556 0.14444444 0.13333333 0.24444444
#> 7 17 26 0.2637363 0.15384615 0.10989011 0.18681319 0.28571429
#> 8 12 32 0.2747253 0.09890110 0.14285714 0.13186813 0.35164835
#> 9 10 37 0.1978022 0.14285714 0.14285714 0.10989011 0.40659341
#> 10 13 40 0.1758242 0.18681319 0.05494505 0.14285714 0.43956044
#> 11 11 61 0.1098901 0.03296703 0.06593407 0.12087912 0.67032967
#> exp.prob.0 exp.prob.1 exp.prob.2 exp.prob.3 exp.prob.4 raw.rsd.0
#> 1 0.71477689 0.11375163 0.06836062 0.03855298 0.06455787 0.032475855
#> 2 0.56957599 0.14884791 0.10278367 0.06320332 0.11558912 -0.009136425
#> 3 0.48976550 0.15944983 0.11992887 0.07818623 0.15266958 -0.116139125
#> 4 0.42532229 0.16264618 0.13183092 0.09078262 0.18941799 -0.029717899
#> 5 0.38151012 0.16172964 0.13839299 0.09938634 0.21898091 0.036072294
#> 6 0.33545340 0.15777543 0.14341041 0.10817376 0.25518700 -0.013231182
#> 7 0.28773759 0.15011991 0.14585476 0.11651096 0.29977678 -0.024001328
#> 8 0.24994169 0.14123048 0.14514018 0.12201185 0.34167580 0.024783588
#> 9 0.20531675 0.12719333 0.14027664 0.12622977 0.40098351 -0.007514555
#> 10 0.15461091 0.10608597 0.12760487 0.12566293 0.48603533 0.021213268
#> 11 0.09383143 0.07259205 0.09796229 0.11011249 0.62550173 0.016058676
#> raw.rsd.1 raw.rsd.2 raw.rsd.3 raw.rsd.4 se.0 se.1
#> 1 -0.003861524 -0.0024265574 -0.016574958 -0.009612815 0.04733222 0.03328403
#> 2 0.004998247 0.0290844659 0.024708768 -0.049655056 0.05190431 0.03731249
#> 3 0.038352370 0.0009502512 0.064670916 0.012165589 0.05240326 0.03837719
#> 4 0.024167011 -0.0219408122 -0.002870528 0.030362229 0.05182634 0.03868617
#> 5 -0.018872495 -0.0065248628 0.032481794 -0.043156729 0.05092120 0.03859812
#> 6 -0.002219870 0.0010340388 0.025159570 -0.010742557 0.04976885 0.03842487
#> 7 0.003726247 -0.0359646468 0.070302223 -0.014062496 0.04745671 0.03744357
#> 8 -0.042329386 -0.0022830374 0.009856283 0.009972552 0.04538854 0.03650748
#> 9 0.015663809 0.0025805066 -0.016339656 0.005609896 0.04234367 0.03492774
#> 10 0.080727222 -0.0726598139 0.017194216 -0.046474892 0.03789899 0.03228168
#> 11 -0.039625020 -0.0320282211 0.010766629 0.044827936 0.03056736 0.02719940
#> se.2 se.3 se.4 std.rsd.0 std.rsd.1 std.rsd.2
#> 1 0.02645491 0.02018231 0.02576098 0.6861257 -0.11601733 -0.09172429
#> 2 0.03183391 0.02550774 0.03351698 -0.1760244 0.13395641 0.91363168
#> 3 0.03405650 0.02814272 0.03770352 -2.2162576 0.99935330 0.02790220
#> 4 0.03546419 0.03011719 0.04107602 -0.5734130 0.62469381 -0.61867504
#> 5 0.03619853 0.03136259 0.04335241 0.7083944 -0.48894855 -0.18025213
#> 6 0.03694498 0.03274009 0.04595488 -0.2658527 -0.05777171 0.02798861
#> 7 0.03700032 0.03363282 0.04802823 -0.5057521 0.09951635 -0.97200911
#> 8 0.03692501 0.03431031 0.04971716 0.5460319 -1.15947164 -0.06182903
#> 9 0.03640419 0.03481439 0.05137620 -0.1774658 0.44846325 0.07088489
#> 10 0.03497595 0.03474740 0.05239380 0.5597317 2.50071304 -2.07742206
#> 11 0.03116170 0.03281445 0.05073626 0.5253537 -1.45683459 -1.02780733
#> std.rsd.3 std.rsd.4
#> 1 -0.82126153 -0.3731541
#> 2 0.96867740 -1.4814898
#> 3 2.29796274 0.3226645
#> 4 -0.09531193 0.7391716
#> 5 1.03568597 -0.9954863
#> 6 0.76846374 -0.2337631
#> 7 2.09028632 -0.2927965
#> 8 0.28726885 0.2005857
#> 9 -0.46933624 0.1091925
#> 10 0.49483460 -0.8870305
#> 11 0.32810632 0.8835483
# }