The mix of outcomes over time, with simultaneous intervals and trend tests
Source:R/justice.R
dispmix.RdFor a table of counts – periods in rows, outcomes in columns, such as cases withdrawn, pleaded guilty, or tried – gives each outcome's share of its period with Goodman's simultaneous confidence intervals, tests whether the mix is the same in every period, and tests each outcome for a linear trend in its share.
Arguments
- cnts
A matrix or data frame of non-negative counts, one row per period and one column per outcome. Columns that are not numeric are refused; pass only the outcome columns.
- prd
Optional period labels (default: the row names, or
1:nrow(counts)).- scores
Numeric scores for the periods in the trend tests. Default:
1:nrow(counts), i.e. equally spaced.- conflvl
Confidence level. Default 0.95.
Value
A list of class rmbl_disposition_mix: shares, a long
data frame (period, outcome, count, total,
share, lower, upper); homogeneity, the
Pearson chi-square test that every period has the same mix (with
Cramer's V); trend, one row per outcome with the
Cochran-Armitage trend statistic, its p-value, the Holm-adjusted
p-value, and the change in share from the first to the last period.
Details
Goodman's intervals hold for all \(k\) shares of a period at once: with \(A\) the upper \(\alpha/k\) point of the chi-square distribution on one degree of freedom, the limits for a share \(n_i / N\) are $$\frac{A + 2 n_i \pm \sqrt{A (A + 4 n_i (N - n_i) / N)}}{2 (N + A)}.$$
The trend test for each outcome is the Cochran-Armitage test of its
count against the period totals (stats::prop.trend.test()); the
Holm adjustment covers the \(k\) tests. Counts of court cases are
large, so these tests detect small changes; the shares and their
intervals show whether a change is large enough to matter.
Errors: cnts not a numeric matrix or data frame, with a
non-numeric column, fewer than one period or two outcomes, or a negative,
missing or infinite count; a period with no cases; scores not one
finite number per period, or all equal; prd of the wrong length
or with repeats; conflvl outside (0, 1).
References
Goodman, L. A. (1965). On simultaneous confidence intervals for multinomial proportions. Technometrics, 7(2), 247-254.
Armitage, P. (1955). Tests for linear trends in proportions and frequencies. Biometrics, 11(3), 375-386.
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(2), 65-70.
See also
shrs for one period's shares with Wilson
intervals.
Examples
d <- read.csv(system.file("extdata", "ocj_dispositions.csv",
package = "rmoriebricklayer"))
mix <- dispmix(d[, -(1:2)], prd = d$year)
mix
#> ╭── Outcome shares by period ─────────────────────────────────────────────────╮
#> │ Periods 12 │
#> │ Outcomes 7 │
#> │ Cases 2,651,849 │
#> │ Intervals 95% Goodman simultaneous (in $shares) │
#> │ Same mix in every period chi-square 43792.1 on 66 df, p < 0.001, │
#> │ Cramer's V 0.052 │
#> │ │
#> │ Share of each period's cases │
#> │ outcome 2012 2013 2014 2015 │
#> │ ----------------------------- ------- ------- ------- ------- │
#> │ withdrawn_stayed_before_trial 0.37958 0.39219 0.39803 0.40161 │
#> │ guilty_plea_before_trial 0.42359 0.41458 0.40938 0.41909 │
#> │ other_before_trial 0.03608 0.03489 0.03582 0.03635 │
#> │ withdrawn_at_trial 0.05538 0.05490 0.05471 0.04824 │
#> │ guilty_plea_at_trial 0.05356 0.05213 0.04988 0.04619 │
#> │ other_at_trial 0.00118 0.00108 0.00120 0.00097 │
#> │ following_trial 0.05063 0.05022 0.05098 0.04755 │
#> │ │
#> │ outcome 2016 2017 2018 2019 │
#> │ ----------------------------- ------- ------- ------- ------- │
#> │ withdrawn_stayed_before_trial 0.40928 0.42702 0.44114 0.43649 │
#> │ guilty_plea_before_trial 0.41992 0.41070 0.39989 0.41120 │
#> │ other_before_trial 0.03710 0.03769 0.04347 0.04176 │
#> │ withdrawn_at_trial 0.04520 0.04207 0.03858 0.03785 │
#> │ guilty_plea_at_trial 0.04263 0.03878 0.03660 0.03421 │
#> │ other_at_trial 0.00085 0.00074 0.00061 0.00063 │
#> │ following_trial 0.04501 0.04301 0.03971 0.03787 │
#> │ │
#> │ outcome 2021 2022 2024 2025 │
#> │ ----------------------------- ------- ------- ------- ------- │
#> │ withdrawn_stayed_before_trial 0.53260 0.54286 0.51803 0.50879 │
#> │ guilty_plea_before_trial 0.34898 0.33376 0.34176 0.34300 │
#> │ other_before_trial 0.03676 0.03798 0.03623 0.03581 │
#> │ withdrawn_at_trial 0.02858 0.02960 0.03809 0.04260 │
#> │ guilty_plea_at_trial 0.02103 0.02090 0.02766 0.03166 │
#> │ other_at_trial 0.00055 0.00054 0.00052 0.00058 │
#> │ following_trial 0.03149 0.03435 0.03771 0.03756 │
#> │ │
#> │ Cochran-Armitage trend in each share (Holm-adjusted) │
#> │ outcome statistic p_value pholm change │
#> │ ----------------------------- --------- ------- ------- -------- │
#> │ withdrawn_stayed_before_trial 29,543.56 < 0.001 < 0.001 0.12920 │
#> │ guilty_plea_before_trial 10,090.69 < 0.001 < 0.001 -0.08060 │
#> │ other_before_trial 20.29 < 0.001 < 0.001 -0.00026 │
#> │ withdrawn_at_trial 3,208.77 < 0.001 < 0.001 -0.01278 │
#> │ guilty_plea_at_trial 7,080.55 < 0.001 < 0.001 -0.02190 │
#> │ other_at_trial 187.10 < 0.001 < 0.001 -0.00061 │
#> │ following_trial 2,172.85 < 0.001 < 0.001 -0.01306 │
#> │ │
#> │ From 2012 to 2025: │
#> │ withdrawn_stayed_before_trial rose most, +12.9 points of the share; │
#> │ guilty_plea_before_trial fell most, -8.1 points. │
#> │ 7 of 7 shares trend at the Holm-adjusted 0.05 level: 1 up │
#> │ (withdrawn_stayed_before_trial), 6 down (guilty_plea_before_trial, │
#> │ other_before_trial, withdrawn_at_trial and 3 more). │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
plot(mix)
# the parts of the result
head(mix$shares)
#> period outcome count total share lower
#> 1 2012 withdrawn_stayed_before_trial 93775 247047 0.3795836 0.3769607
#> 2 2013 withdrawn_stayed_before_trial 90864 231681 0.3921944 0.3894691
#> 3 2014 withdrawn_stayed_before_trial 83626 210098 0.3980333 0.3951641
#> 4 2015 withdrawn_stayed_before_trial 82430 205249 0.4016098 0.3987024
#> 5 2016 withdrawn_stayed_before_trial 84787 207160 0.4092827 0.4063797
#> 6 2017 withdrawn_stayed_before_trial 91440 214137 0.4270163 0.4241433
#> upper
#> 1 0.3822136
#> 2 0.3949265
#> 3 0.4009096
#> 4 0.4045240
#> 5 0.4121920
#> 6 0.4298943
mix$trend
#> outcome statistic p_value pholm
#> 1 withdrawn_stayed_before_trial 29543.56265 0.000000e+00 0.000000e+00
#> 2 guilty_plea_before_trial 10090.69293 0.000000e+00 0.000000e+00
#> 3 other_before_trial 20.29329 6.643309e-06 6.643309e-06
#> 4 withdrawn_at_trial 3208.77296 0.000000e+00 0.000000e+00
#> 5 guilty_plea_at_trial 7080.54897 0.000000e+00 0.000000e+00
#> 6 other_at_trial 187.10154 1.364671e-42 2.729342e-42
#> 7 following_trial 2172.84508 0.000000e+00 0.000000e+00
#> change
#> 1 0.1292036827
#> 2 -0.0805960172
#> 3 -0.0002635828
#> 4 -0.0127761314
#> 5 -0.0218969033
#> 6 -0.0006064793
#> 7 -0.0130645687
# years as trend scores, so the gap where no row exists is respected
dispmix(d[, -(1:2)], prd = d$year, scores = d$year)$trend[, 1:3]
#> outcome statistic p_value
#> 1 withdrawn_stayed_before_trial 30150.915355 0.000000e+00
#> 2 guilty_plea_before_trial 10825.914695 0.000000e+00
#> 3 other_before_trial 7.296441 6.909133e-03
#> 4 withdrawn_at_trial 2890.841438 0.000000e+00
#> 5 guilty_plea_at_trial 6782.412494 0.000000e+00
#> 6 other_at_trial 175.420561 4.845824e-40
#> 7 following_trial 2044.569303 0.000000e+00
# outcome columns only: a text column is refused
try(dispmix(data.frame(year = c("a", "b"), x = 1:2, y = 3:4)))
#> Error : `cnts` has a column that is not numeric (year); pass only the outcome columns