1 Introduction
2 Models and Regression Trees for Ordered Evaluation Data
2.1 Binomial Trees
-
the sample size of the node is lower than a pre-specified threshold to attempt any splitting procedure;
-
the number of observations for any of the descendants of a candidate (significant) split is lower than a given threshold.
3 Residual Diagnostics for Ordinal Response Models
3.1 Model Misspecification for Missing Component
π | 0.1 | 0.25 | 0.4 | 0.6 | 0.75 | 0.9 |
---|---|---|---|---|---|---|
Binomial | 0.000 | 0.000 | 0.000 | 0.000 | 0.016 | 0.333 |
CUB | 0.433 | 0.462 | 0.455 | 0.487 | 0.472 | 0.529 |
ϕ | 0.01 | 0.05 | 0.1 | 0.15 | 0.2 | 0.3 |
---|---|---|---|---|---|---|
Binomial fit | 0.493 | 0.276 | 0.033 | 0.000 | 0.000 | 0.000 |
Beta-binomial fit | 0.505 | 0.486 | 0.481 | 0.463 | 0.453 | 0.447 |
3.2 Neglecting Sub-populations
4 Illustrative Example on Customers’ Satisfaction Survey
recom
); the extent by which the would reconsider the company for further purchases (product
); the overall satisfaction for the equipments of the purchase (equipment
); for sales and technical support (sales
, technical
); purchasing support (purchase
) and pricing
. Due to the small sample size obtained after omitting missing values list-wise (n = 212), only small trees can be grown: thus, this dataset will be used for illustration purposes only.recom
, equipment
, technical
. For equipment
, the only model that fulfils the necessary condition for correct specification is the binomial with the addition of a shelter effect (at category c = 4).
satis | recom | product | equipment | sales | technical | purchase | pricing | |
---|---|---|---|---|---|---|---|---|
Binomial | 0.392 | 0.014 | 0.106 | 0.017 | 0.255 | 0.043 | 0.333 | 0.337 |
Binomial with shelter | 0.528 | 0.353 | 0.319 | 0.497 | 0.477 | 0.242 | 0.577 | 0.457 |
CUB | 0.480 | 0.599 | 0.528 | 0.017 | 0.568 | 0.465 | 0.332 | 0.337 |
CUB with shelter | 0.480 | 0.604 | 0.569 | 0.017 | 0.581 | 0.532 | 0.331 | 0.337 |
CAUB | 0.418 | 0.033 | 0.256 | 0.005 | 0.288 | 0.014 | 0.480 | 0.520 |
recom
at the fixed significance level. This circumstance may be due to missing sub-populations: indeed, the primary split of a dissimilarity binomial tree separates recom
ratings provided by customers who are not satisfied with the sales support (sales
≤ 2) from those who are satisfied (sales
≥ 3), for which the binomial model can be safely assumed instead. Figure 5 displays diagnostics check of uniformity of residuals at the root node and at left and right descendants (top row panels), along with the barplots of the frequency distributions at the nodes, with superimposition of the fitted Binomial model.
recom
: comparison between observed and fitted distributions (top) and residuals’ QQ plots (bottom): ABC datasetrecom
: comparison between observed and fitted distributions (top) and residuals QQ plot (bottom): ABC datasetsatis
) to disentangle local association for different aspects of the customer experience: here the deviance criterion is considered.Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 8 | Node 9 | |
---|---|---|---|---|---|---|---|
Binomial | 0.392 | 0.351 | 0.707 | 0.439 | 0.356 | 0.391 | 0.684 |
Binomial with shelter | 0.528 | 0.524 | 0.709 | 0.624 | 0.579 | 0.726 | 0.673 |
CUB | 0.480 | 0.440 | 0.707 | 0.565 | 0.349 | 0.646 | 0.684 |
CUB with shelter | 0.480 | 0.440 | 0.707 | 0.654 | 0.351 | 0.729 | 0.684 |
CAUB | 0.418 | 0.562 | 0.618 | 0.604 | 0.215 | 0.410 | 0.894 |
Node | Best model | \(\hat {\pi }\) | \(1-\hat {\xi }\) | \(\hat {\delta }\) | Shelter | Split with | Diss(Bin,f) | Diss(Best,f) |
---|---|---|---|---|---|---|---|---|
1 | Binomial with shelter | 0.97 | 0.666 | 0.030 | 1 | L: recom ≤ 4; R: recom = 5 | 0.090 | 0.062 |
2 | Binomial with shelter | 0.96 | 0.650 | 0.04 | 1 | L: sales ≤ 3; R: sales ≥ 4 | 0.125 | 0.085 |
3 | Binomial | 1 | 0.711 | 0 | − | − | 0.011 | 0.011 |
4 | Binomial with shelter | 0.95 | 0.569 | 0.05 | 1 | L: sales ≤ 2; R: sales = 3 | 0.110 | 0.061 |
5 | Binomial with shelter | 0.73 | 0.713 | 0.27 | 4 | − | 0.167 | 0.048 |
8 | Binomial with shelter | 0.88 | 0.562 | 0.12 | 1 | − | 0.168 | 0.042 |
9 | Binomial | 1 | 0.656 | 0 | − | − | 0.089 | 0.089 |
-
customers’ propensity to recommend the company is the strongest indicator of overall satisfaction (indeed, node 3 refers to overall satisfaction for those who rated
recom
= 5); -
the most influential dimension of overall satisfaction is the satisfaction for the sales support: thus, overall satisfaction can be controlled by focusing primarily on the control of this aspect of the customer experience;
-
a small percentage of structurally dissatisfied customers is present (measured by \(\hat {\delta }\)), stronger for respondents who are moderately satisfied for the sales support (
sales
= 3); -
the feeling measure \((1-\hat {\xi })\), weighted for the importance \(\hat {\pi }\) of the feeling component, can be considered as an overall satisfaction indicator, and response profiles can be ranked accordingly. In the present example, customer satisfaction should be improved starting from the response profile associated with node 8 (corresponding to customers so that
recom
≤ 4 andsales
≤ 2).
5 Perceived Trust Towards Press and Television
-
Age
c: age of the respondent in ordered classes of years (1 = 18–29, 2 = 30–44, 3 = 45–59; 4 = 60–74; 5 = 75–89; 6 = more than 90); -
notwork
: a dummy indicating if the respondent is unemployed (notwork= 1
) or employed (notwork= 0
); -
Internet
: a dummy indicating if the respondent uses internet for private purposes (Internet
= 1) or not (Internet
= 0); -
(
left-right
): left-right self placement on political orientation (semantic scale with ten categories running from extreme left (left-right
= 1) to extreme right (left-right
= 10); -
univ
: a dummy variable to indicate whether the respondent has a university education (univ
= 1) or not (univ
= 0); -
west
: a dummy variable to indicate if respondent’s residence is in the Old Federal Republic (West Berlin:west
= 1) or in former German Democratic Republic (East Berlin:west
= 0).
5.1 Perceived Trust Towards Press
-
Node 4:Employed residents in former GDR (
west= 0
); -
Node 5:Unemployed residents in former GDR (
west= 0
); -
Node 6:Residents in former Federal Republic (
west= 1
) that do not use internet for private purposes; -
Node 14:Young residents (aged less than 29) in former Federal Republic (
west= 1
) that use internet for private purposes; -
Node 15:Adult and elderly respondents, resident in former Federal Republic (
west= 1
), that use internet for private purposes.
Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Node 7 | Node 14 | Node 15 | |
---|---|---|---|---|---|---|---|---|---|
Binomial | 0.07 | 0.15 | 0.16 | 0.13 | 0.33 | 0.23 | 0.25 | 0.48 | 0.25 |
Binomial + she | 0.45 | 0.52 | 0.43 | 0.43 | 0.57 | 0.50 | 0.46 | 0.56 | 0.44 |
CUB | 0.13 | 0.18 | 0.27 | 0.14 | 0.46 | 0.46 | 0.32 | 0.49 | 0.34 |
CUB + she | 0.46 | 0.52 | 0.46 | 0.43 | 0.59 | 0.56 | 0.46 | 0.56 | 0.46 |
CAUB | 0.07 | 0.16 | 0.16 | 0.15 | 0.33 | 0.23 | 0.25 | 0.48 | 0.25 |
BetaBin | 0.22 | 0.24 | 0.34 | 0.18 | 0.45 | 0.44 | 0.38 | 0.51 | 0.38 |
BetaBin + she | 0.47 | 0.52 | 0.47 | 0.43 | 0.58 | 0.55 | 0.47 | 0.56 | 0.46 |
Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Node 7 | Node 14 | Node 15 | |
---|---|---|---|---|---|---|---|---|---|
Bin - Binomial + she | 128.62 | 63.14 | 65.41 | 38.81 | 23.43 | 31.64 | 37.61 | 5.41 | 31.31 |
Bin - CUB | 57.04 | 16.17 | 37.74 | 2.94 | 12.93 | 26.98 | 15.51 | 0.40 | 16.30 |
CUB - CUB + she | 73.44 | 46.89 | 30.68 | 35.84 | 11.34 | 7.87 | 22.60 | 4.98 | 16.97 |
Bin - BetaBin | 78.24 | 27.63 | 49.03 | 12.77 | 14.23 | 25.58 | 25.43 | 2.08 | 23.18 |
Bin + she - BetaBin + she | 2.97 | 0.01 | 5.23 | 0.01 | 0.49 | 3.36 | 1.91 | 0.01 | 2.20 |
BetaBin - BetaBin + she | 53.35 | 35.44 | 21.61 | 25.88 | 9.70 | 9.42 | 14.09 | 3.33 | 10.33 |
Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Node 7 | Node 14 | Node 15 | |
---|---|---|---|---|---|---|---|---|---|
Binomial | 120.72 | 56.32 | 57.93 | 32.53 | 17.48 | 25.63 | 30.39 | 0.00 | 24.35 |
Bin + she | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.35 | 0.00 |
CUB | 71.58 | 46.97 | 27.67 | 35.87 | 10.51 | 4.66 | 22.10 | 5.35 | 15.01 |
CUB + she | 6.04 | 6.91 | 4.48 | 6.30 | 5.12 | 2.80 | 6.72 | 6.13 | 5.00 |
CAUB | 136.64 | 68.99 | 73.59 | 42.39 | 29.65 | 38.05 | 45.05 | 11.39 | 38.53 |
BetaBin | 50.38 | 35.51 | 16.38 | 26.04 | 9.20 | 6.07 | 12.18 | 3.68 | 8.13 |
BetaBin+she | 4.93 | 6.89 | 2.25 | 6.44 | 5.46 | 2.65 | 5.32 | 6.10 | 4.75 |
Node | Best model | \(\hat {\pi }\) | \(1-\hat {\xi }\) | \(\hat {\delta }\) | \(\hat {\phi }\) | Shelter | Split with | Diss(Bin,f) | Diss(Best,f) | n |
---|---|---|---|---|---|---|---|---|---|---|
1 | Bin + she | 0.95 | 0.467 | 0.05 | - | 1 | L: west= 0 ; R: west= 1 | 0.071 | 0.037 | 2692 |
2 | Bin + she | 0.94 | 0.447 | 0.06 | - | 1 | L: notwork= 0 ; R: notwork= 1 | 0.093 | 0.035 | 916 |
3 | Bin + she | 0.96 | 0.462 | 0.04 | - | 1 | L: internet= 0 ; R: internet= 1 | 0.064 | 0.038 | 1776 |
4 | Bin + she | 0.93 | 0.437 | 0.07 | - | 1 | - | 0.124 | 0.067 | 533 |
5 | Bin + she | 0.94 | 0.460 | 0.06 | - | 1 | - | 0.068 | 0.013 | 383 |
6 | Bin + she | 0.95 | 0.495 | 0.05 | - | 1 | - | 0.078 | 0.038 | 407 |
7 | Bin + she | 0.96 | 0.452 | 0.04 | - | 1 | L: Age c = 1; R: Age c ≥ 2 | 0.059 | 0.042 | 1369 |
14 | Bin + she | 0.98 | 0.475 | 0.02 | - | 1 | - | 0.055 | 0.041 | 316 |
15 | Bin + she | 0.96 | 0.444 | 0.04 | - | 1 | - | 0.065 | 0.048 | 1053 |
5.2 Perceived Trust for Television
-
Node 4:Residents in former East Germany that do not use Internet for private purposes;
-
Node 5:Residents in former West Germany that do not use Internet for private purposes;
-
Node 7:Respondents that use Internet for private purposes, aged more than 60 years;
-
Node 13:Respondents that use Internet for private purposes, aged less than 60 years, with a University education;
-
Node 24:Respondents that use Internet for private purposes, aged less than 60 years, with no University education and with left-wing political orientation (
left-right
≤ 4); -
Node 25:Respondents that use Internet for private purposes, aged less than 60 years, with no University education and with neutral or right-wing political orientation (
left-right
≥ 5).
Node 1 | Node 2 | Node 3 | Node 4 | Node 5 | Node 6 | Node 7 | Node 12 | Node 13 | Node 24 | Node 25 | |
---|---|---|---|---|---|---|---|---|---|---|---|
Binomial | 0.00 | 0.07 | 0.03 | 0.17 | 0.13 | 0.03 | 0.48 | 0.04 | 0.25 | 0.09 | 0.10 |
Binomial + she | 0.39 | 0.36 | 0.48 | 0.52 | 0.38 | 0.38 | 0.53 | 0.39 | 0.35 | 0.39 | 0.42 |
CUB | 0.05 | 0.33 | 0.10 | 0.31 | 0.48 | 0.11 | 0.48 | 0.14 | 0.31 | 0.16 | 0.26 |
CUB + she | 0.51 | 0.56 | 0.52 | 0.56 | 0.58 | 0.50 | 0.55 | 0.54 | 0.31 | 0.44 | 0.57 |
CAUB | 0.00 | 0.07 | 0.05 | 0.18 | 0.12 | 0.05 | 0.49 | 0.05 | 0.37 | 0.22 | 0.10 |
BetaBin | 0.28 | 0.38 | 0.31 | 0.38 | 0.49 | 0.29 | 0.51 | 0.32 | 0.33 | 0.34 | 0.38 |
BetaBin + she | 0.45 | 0.53 | 0.48 | 0.55 | 0.55 | 0.44 | 0.55 | 0.46 | 0.45 | 0.59 | 0.51 |
Node | Best model | \(\hat {\pi }\) | \(1-\hat {\xi }\) | \(\hat {\delta }\) | \(\hat {\phi }\) | Shelter | Split with | Diss(Bin,f) | Diss(Best,f) | n |
---|---|---|---|---|---|---|---|---|---|---|
1 | CUB + she | 0.85 | 0.373 | 0.09 | − | 1 | L: internet= 0 ; R: internet= 1 | 0.113 | 0.028 | 2692 |
2 | BetaBin + she | 0.94 | 0.450 | 0.06 | 0.05 | 1 | L: west= 0 ; R: west= 1 | 0.107 | 0.033 | 702 |
3 | CUB + she | 0.87 | 0.349 | 0.094 | − | 1 | L: Age c ≤ 3; R: Age c ≥ 4 | 0.111 | 0.029 | 1990 |
4 | Bin + she | 0.91 | 0.444 | 0.09 | − | 1 | − | 0.125 | 0.051 | 295 |
5 | BetaBin + she | 0.95 | 0.456 | 0.05 | 0.079 | 1 | − | 0.112 | 0.029 | 407 |
6 | CUB + she | 0.85 | 0.337 | 0.104 | − | 1 | L: univ= 0 ; R:univ= 1 | 0.120 | 0.030 | 1635 |
7 | Bin + she | 0.97 | 0.392 | 0.033 | − | 1 | − | 0.064 | 0.040 | 355 |
12 | CUB + she | 0.84 | 0.366 | 0.103 | − | 1 | L: leftright ≤ 4; R:leftright ≥ 5 | 0.128 | 0.032 | 996 |
13 | Bin + she | 0.90 | 0.297 | 0.103 | − | 1 | − | 0.103 | 0.034 | 639 |
24 | BetaBin + she | 0.87 | 0.283 | 0.13 | 0.116 | 4 | − | 0.167 | 0.022 | 312 |
25 | CUB+she | 0.83 | 0. 373 | 0.092 | − | 1 | − | 0.115 | 0.014 | 684 |
5.2.1 Tree Performance
Trust towards Press | Trust towards TV | |
---|---|---|
Dissimilarity with binomial | 0.078 | 0.112 |
Dissimilarity with best mixture | 0.044 | 0.042 |
Dissimilarity | Average RPS | Total RPS | ||||
---|---|---|---|---|---|---|
Binomial | Best mixture | Binomial | Best mixture | Binomial | Best mixture | |
Press | 0.114 | 0.081 | 65.38 | 64.93 | 326.85 | 324.77 |
Tv | 0.085 | 0.079 | 64.34 | 63.75 | 320.46 | 317.83 |
6 Satisfaction of Italian Ph.D. Awardees
-
gender
: a dummy indicating if the respondent is male (gender
= 0) or female (gender
= 1); -
abroad
: a dummy indicating if the respondent had any work or training experience abroad after the Ph.D. completion (abroad
= 1) or not (abroad
= 0); -
research
: a dummy indicating if the respondent currently works in the research domain (research= 1
) or in other fields (research= 0
); -
stem
, a dummy variable to indicate whether the Ph.D. program was relative to STEM disciplines (stem
= 1) or different ones (stem
= 0); -
north
, a dummy variable to indicate if respondent awarded the Ph.D. title from a University located in Northern Italy (north
= 1) or in a different geographical area (north
= 0).
Node | Best model | \(\hat {\pi }\) | \(1-\hat {\xi }\) | \(\hat {\delta }\) | \(\hat {\phi }\) | Shelter | Split with | Diss(CUB,f) | Diss(Best,f) |
---|---|---|---|---|---|---|---|---|---|
1 | CUB+she | 0.52 | 0.664 | 0.037 | − | 6 | L: research = 0, R: research = 1 | 0.038 | 0.027 |
2 | CUB+she | 0.39 | 0.604 | 0.040 | − | 6 | L: stem = 0; R: stem = 1 | 0.039 | 0.025 |
3 | CUB+she | 0.67 | 0.683 | 0.016 | − | 2 | L: stem = 0; R: stem = 1 | 0.045 | 0.036 |
4 | CUB+she | 0.39 | 0.564 | 0.030 | − | 2 | L: north = 0; R: north = 1 | 0.037 | 0.022 |
5 | CUB+she | 0.45 | 0.631 | 0.082 | − | 6 | - | 0.059 | 0.014 |
6 | CUB+she | 0.52 | 0.662 | 0.020 | − | 8 | L: abroad = 0; R: abroad = 1 | 0.053 | 0.044 |
7 | CUB+she | 0.79 | 0.695 | 0.016 | − | 2 | L: gender = 0; R: gender = 1 | 0.046 | 0.036 |
8 | CUB | 0.34 | 0.610 | 0.000 | − | − | − | 0.027 | 0.027 |
9 | CUB+she | 0.44 | 0.516 | 0.034 | − | 2 | − | 0.071 | 0.055 |
12 | CUB+she | 0.63 | 0.669 | 0.033 | − | 8 | L: gender = 0; R: gender = 1 | 0.043 | 0.028 |
13 | CUB+she | 0.41 | 0.643 | 0.048 | − | 2 | − | 0.076 | 0.052 |
14 | CUB+she | 0.77 | 0.718 | 0.016 | − | 2 | L: north = 0; R: north = 1 | 0.042 | 0.034 |
15 | CUB+she | 0.73 | 0.649 | 0.079 | − | 6 | − | 0.065 | 0.028 |
24 | CUB+she | 0.62 | 0.708 | 0.072 | − | 8 | − | 0.088 | 0.051 |
25 | CUB+she | 0.60 | 0.638 | 0.025 | − | 5 | − | 0.031 | 0.025 |
28 | CUB+she | 0.63 | 0.739 | 0.026 | − | 4 | − | 0.033 | 0.021 |
29 | CUB+she | 0.75 | 0.706 | 0.058 | − | 6 | − | 0.064 | 0.044 |
-
Ph. Doctors with studies in disciplines different from STEM experience lower feeling than Ph. Doctors with studies in STEM disciplines, especially if the current job does not involve research. For Ph. Doctors in disciplines different from STEM, feeling of evaluation decreases if the respondent had any work or training experience abroad after the Ph. doctors;
-
Among the Ph. Doctors working in research, women are less satisfied than men, especially if the Ph.D. program concerned disciplines different from STEM and if no abroad experience has occurred after the Ph.D. completion. In the latter case, the lower feeling towards satisfaction experienced by women is revealed also in terms of the location of the shelter effect at a lower category than it is found for men (even if in both cases concentration is found at the center of the scale). Thus, even for neutral evaluation, women tend to assess their evaluations with lower scores.