Modelling changes over time in a multivariate paired comparison: An application to window display design

Abstract

This article proposes an alternative method of making comparative judgements in multivariate paired comparisons (PCs) where judgements about change are made directly by comparing an object at two time points for each of a series of attributes. The application deals with the design of shop window displays where products should be arranged by teams of vocational students according to aesthetic principles (attributes).

The photos of the students’ window displays at time 1 (before feedback) and at time 2 (after feedback) were compared by judging each attribute as to whether it was fulfilled better at time 1 or at time 2. An advantage of this PC approach over an alternative of a scoring system is the possibility to assess even subtle changes of various aspects of attractiveness, which cannot easily be measured using a score. To analyse these data, we used earlier work which developed both a multivariate PC pattern model for multi-attribute data and a PC model over time and defined a multivariate PC model of changes (MPCC). The model can be fitted as a non-standard Poisson log-linear model and provides estimates of change for the three attributes for time 2 and we were able to check for possible interaction effects between these attributes.

Keywords

aesthetic judgements Judging change PC over time multivariate PC model of changes (MPCC)

1 Introduction

In this study, we propose a method of modelling observations of one object on a series of attributes measured at various points over time. Our application is concerned with the construction of shop window displays which are judged for aesthetic beauty using three attributes—focus, optical balance and triangle style. Students construct two window displays—one before feedback and one after feedback, and the two displays are compared using photographs, making a paired comparison (PC) for each team of students.

To measure changes one could use absolute ratings at time 1 and at time 2, but this might be problematic for various reasons. One problem when judging changes in beauty might be the fact that subtle changes cannot be assessed by absolute ratings. If we get a response of ‘fulfilled’ at both time points, we would assume there was no change. But it could have changed from minimal to maximal fulfilment. This is also true for scales with more response categories, where for each category there is a range from minimal to maximal.

In this article, we propose a comparative judgement approach which could address such problems. The method of PCs (see, e.g., Bradley and Terry (1952)) is based on comparative judgements usually made between two objects over all pairs of objects. In this study, we propose another, alternative mode of judgement. We combined the ideas of the methods of making judgements in PCs between two objects for more than one attribute at a single time point (i.e., multivariate comparisons at one time point, see Dittrich et al.(2006)Dittrich, Francis, Hatzinger, and Katzenbeisser) and the method of comparing repeatedly two objects regarding a single attribute for more than one time point (i.e., univariate comparisons over time, see, e.g., Grand et al.(2015)Grand, Dittrich, and Francis). This allowed us to construct multivariate PCs between two time points so that change can be directly judged for each attribute, which we term the multivariate PC model of changes (MPCC).

2 The measurement of beauty

Measuring beauty is not an easy task and measuring changes is even harder. In the literature, there are several studies which focus on the components of visual merchandising and the effect(s) of these in stores. Nevertheless, less attention is spent on window displays (see, e.g., Oh and Petrie (2012), Lange et al.(2016)Lange, Rosengren, and Blom, Kernsom and Sahachaisaeree (2010)) and (empirical) studies regarding the attractiveness of window display designs and the creation of an aesthetically pleasing window display are rare. We are interested in the attractiveness of window displays and there are many aspects which contribute to the aesthetics of displays. The perception of objects (e.g., things, products or a window display composition) being aesthetic is latent and subjective. Indicators for an aesthetic window display design are given in design books. (Diamond and Diamond (2007)), for example, refer to visual merchandiser and interpret good design according to a set of design principles, that is, balance, emphasis, proportion, rhythm and harmony. The same principles with the addition of lighting and colour are described in (Bastow-Shoop et al.(1991)Bastow-Shoop, Zetocha, and Passewitz). Books, which are used in commercial schools in Austria, typically refer to principles of product presentations, formal principles or guidelines for creation. The student text of (Buchegger et al.(2010)Buchegger, Reichmann, and Anderle), gives central principles for product presentations: golden section, depth effect, triangle style, focus, line management, optical balance and order. These factors should make up an aesthetically pleasing composition.

In this study we want to focus on the attributes triangle style, optical balance and focus within a window display created by a team of vocational students in Austria.

2.1 Attributes focus, optical balance and triangle style

In view of the literature (see, e.g., Arnheim (1982); Locher (2003); Friedenberg (2012)) and what students have learned (see, e.g., Buchegger et al.(2010)Buchegger, Reichmann, and Anderle), we characterized the three attributes focus, optical balance and triangle style as follows:

focus: the products should be arranged around the centre of the presentation.

optical balance: the window display should be created in such a way that it is visible balanced, that is, there are no large ‘empty spaces’.

triangle style: the presented products should be arranged in one huge, asymmetric (visible and $/$ or imagined) triangular form with the ground line on the base and the peak on top.

The shape of an asymmetric triangle is a relatively unusual geometric form in everyday (work)life compared to more symmetric shapes such as rectangles, cuboids or circles. It might therefore be more difficult compared to other design principles to arrange products with the aid of design elements and other tools in an asymmetric triangle style, as learned at school. (Leder et al.(2019)Leder, Tinio, Brieber, Kröner, Jacobsen, and Rosenberg), for example, found that art experts and non-art experts have another understanding regarding symmetry and complexity. The study of (Leder et al.(2019)Leder, Tinio, Brieber, Kröner, Jacobsen, and Rosenberg) showed that non-art experts (psychologists) perceive symmetric and complex stimuli as beautiful in contrast to art experts (students of art) who rated asymmetrical and simple stimuli as most beautiful. The authors (Leder et al.(2019)Leder, Tinio, Brieber, Kröner, Jacobsen, and Rosenberg) conclude that this may be due to the fact that they are educated and trained on aesthetic appreciation. Students of a vocational school are somewhere between trained artists and practitioners/experts of their retail profession. As non-art experts students might tend to overload one or several area(s) of the window display and at the same time there will be free spaces, which may result in a less balanced window display design.

We are interested if students made progress in the performance of the attributes triangle style, optical balance and focus when creating window displays after they received feedback (at time 2) from their teacher.

3 Design of the study

This study is based on frontal coloured photographs of window displays designed by students who were attending the second level of an Austrian vocational business school. All students were taught by one teacher on how to create a window display using certain design principles (attributes).

From the years 2010 till 2014, we obtained 39 photos of window displays of teams of students each at time 1 and at time 2, in total 78 from one teacher. At time 1, students were tasked to apply the theoretical design principles within two hours. They had to design an occasion related product presentation within a window display in a rectangular front opened box in a team of two. Students could select the products, design elements and other tools for the window display freely from a repertoire. After finishing the window display design a photo was taken and students got immediate feedback from the teacher about the realization of the learned design principles except for the design principle focus. After receiving feedback, at time 2, each team had to create a new window display for the same occasion with the same kind of products, arbitrary design elements and tools from a repertoire within two hours. After finishing, a second photo was taken from the window display.

3.1 How to judge change: Relative judgements versus absolute judgements

Starting with the photos of students’ window displays taken at time 1 and time 2, we questioned how a possible change in the design of a window display over time, with respect to the attributes optical balance, triangle style and focus, can be judged and measured. Between the first design at time 1 and the second design at time 2 students can vary in their performance ranging from a (bit) more beautiful, equal, to a (bit) less beautiful design.

One simple method to measure change is to look at the photos of the window displays at both time points and rate each display in two or more response categories according to the three attributes (focus, optical balance, triangle style), for example, by judging ‘fulfilled’ or ‘not fulfilled’. To obtain information about change the number of fulfilments of each attribute could be counted at both time points and compared.

On the basis of the coding of the response categories at time 1 and at time 2 interpretations about change or tests for significant differences could be made. However, in some cases essential information about change could get lost when making absolute ratings at each time point. An example: The attribute focus is named $a_{1}$ , the attribute optical balance $a_{2}$ and the attribute triangle style $a_{3}$ . The fulfilment of an attribute is coded with 1 and no fulfilment with 0. Let the response pattern at time 1 ( $t 1$ ) be $a_{t 1, 1} = 1$ , $a_{t 1, 2} = 0$ , $a_{t 1, 3} = 0$ and at time 2 ( $t 2$ ) be $a_{t 2, 1} = 0$ , $a_{t 2, 2} = 1$ , $a_{t 2, 3} = 0$ . When looking at the attribute triangle style $a_{3}$ , for example, we can see from the response pattern that it is not fulfilled at time 1 ( $a_{t 1, 3} = 0$ ) and at time 2 ( $a_{t 2, 3} = 0$ ). We can describe this as no change, but we do not know if there was a slightly better fulfilment at one of the time points. If we look at the fictitious photo of the window display at time 1 in Figure 1 (left picture), we can see that the product presentation is arranged in the shape of a rectangle, that is, the attribute triangle style is not fulfilled at time 1. At time 2 (Figure 1, right picture) the product presentation looks like an ‘incomplete’ triangle, that is, the attribute triangle style is still not fulfilled at time 2, but at least some kind of improvement can be observed, which cannot be accounted for adequately, if an absolute judgement was made.

The absolute rating process itself seems to be tricky as it might be difficult to interpret and rate the degree of the fulfilment of aesthetic design principles in given response categories at time 1 and time 2 directly. Therefore, we asked the judge(s) to make relative judgements by considering each change directly on each attribute.

In this study, 39 pairs of window displays were judged by the same teacher. The teacher judged for each team whether the window display at time 1 or the window display at time 2 was better or if there was no change with respect to the defined attributes focus, optical balance and triangle style. In this study, no ties occurred. A fictitious example is illustrated in Figure 1 and the pairwise judgement of these two fictitious window displays is shown in Figure 2. In this example, it was judged, for example, that the presented products were arranged more in the shape of a triangle at time 2. But it was not judged if this attribute was fulfilled or not at time 1 and at time 2. Only changes were judged. One team of students did not change their window display at time 2 at all, so this team was excluded from the analysis. We therefore analysed 38 pairs of window displays (N = 38).

Figure 1:

Simple fictitious window display of one fictitious team of students at two time points. Left picture: Window display at time 1 (students’ first design). Right picture: Window display at time 2 (students’ second design). See also Figure 2.

Figure 2:

Pairwise judgement of the fictitious window displays of the fictitious team #1 at time 1 (students’ first design) and time 2 (students’ second design) with respect to the attributes focus, optical balance and triangle style. See also Figure 1.

4 Method

We were interested in the comparison of one object at two time points with respect to three attributes. In this study, the number of time points defines the number of PCs. For two time points we can only compare time 1 with time 2, that is, giving one PC $(12)$ , $J choose 2$ = $(2 * 1) / 2 = 1$ (where $J$ denotes here the number of time points). A modification of the multivariate PC pattern model of (Dittrich et al.(2006)Dittrich, Francis, Hatzinger, and Katzenbeisser) was considered by modelling a response pattern over the three attributes within one PC. The proposed approach was based on the Bradley–Terry (BT) model (Bradley and Terry (1952)). We defined the probability $(P_{(1 2)})$ that the object at time 1 (named 1) is preferred ( ) over the object at time 2 (named 2) with respect to an attribute in the PC $(12)$ , $P_{(1 2)} = π_{1} / (π_{1} + π_{2})$ . The parameters $π$ are the (positive) worth parameters of the object at time 1 (1) and the object at time 2 (2) which are constrained to sum to one. As a starting point to fit a Bradley–Terry PC model in the framework of generalised linear models (GLMs, see Aitkin et al.(2009)Aitkin, Francis, Hinde, and Darnell), the probability, $P_{(1 2)}$ was reformulated according to (Sinclair (1982)),

P_{(1 2)} = \frac{π_{1}}{π_{1} + π_{2}} = \frac{(π_{1} / π_{2})^{\frac{1}{2}}}{(π_{1} / π_{2})^{\frac{1}{2}} + (π_{2} / π_{1})^{\frac{1}{2}}} = \frac{\sqrt{π_{1} / π_{2}}}{\sqrt{π_{1} / π_{2}} + \sqrt{π_{2} / π_{1}}} = c (\frac{\sqrt{π_{1}}}{\sqrt{π_{2}}}),

(4.1)

where $c$ $^{- 1}$ $= \sqrt{π_{1} / π_{2}} + \sqrt{π_{2} / π_{1}}$ .

The log-linear PC model of changes that the object at time 1 is preferred over the object at time 2 (see also Dittrich et al.(1998)Dittrich, Hatzinger, and Katzenbeisser) is defined by:

ln m_{(1 2)} = ln N + ln P_{(1 2)} = μ + λ_{1} - λ_{2},

(4.2)

where $m_{(1 2)}$ is the expected number of preferences for the object at time 1 and $N$ is the total number of observed preferences in comparison (12), which is the sum of the observed number of preferences for the object at time 1 ( $n_{(1 2)}$ ) and the observed number of preferences for the object at time 2 ( $n_{(1 2)}$ ), $N = n_{(1 2)} + n_{(1 2)}$ . The parameter $μ$ is a nuisance parameter for the PC $(12)$ and defined by $μ = ln N - ln (\sqrt{π_{1} / π_{2}} + \sqrt{π_{2} / π_{1}})$ . The parameter $λ_{1}$ is the parameter for the object at time 1 and $λ_{2}$ for the object at time 2, where $λ_{1} = \frac{1}{2} ln π_{1}$ and $λ_{2} = \frac{1}{2} ln π_{2}$ . Note that the model (4.2) is the simplest and special case of a more general model. In this study, we compared the window display at time 1 with the window display at time 2 with respect to three attributes ( $a$ ), that is, focus ( $a$ 1), optical balance ( $a 2$ ) and triangle style ( $a 3$ ). In one PC $Y_{a}$ of two time points $(12)$ with respect to attribute $a$ there are two possible responses. Either the window display at time 1 is fulfilled better with respect to attribute $a$ (i.e., $Y_{a} = 1$ ) or the window display at time 2 is fulfilled better (i.e., $Y_{a} = - 1$ ). For three attributes ( $a 1, a 2, a 3$ ), we obtain a sequence of three responses. For example, the object at time 1 is fulfilled better ( $Y_{a 1} = 1$ ) with regard to attribute $a 1$ , the object at time 1 is fulfilled better ( $Y_{a 2} = 1$ ) with regard to attribute $a 2$ and the object at time 2 is fulfilled better ( $Y_{a 3} = - 1$ ) regarding attribute $a 3$ . These decisions resulted in the response pattern ( $Y_{a 1} = 1, Y_{a 2} = 1, Y_{a 3} = - 1$ ) or in shortened form ( $1, 1, - 1$ ). In total, there are $2^{3} = 8$ different response patterns for three attributes ( $A = 3$ ) which were modelled through a log-linear MPCC model. The 8 possible response patterns $ℓ$ within one comparison ( $12$ ) are (see also, e.g., Grand et al.(2015)Grand, Dittrich, and Francis):

\begin{matrix} ℓ_{1} & = & [y_{a 1} = 1, y_{a 2} = 1, y_{a 3} = 1] & = & (1, 1, 1); & n_{ℓ_{1}} = 1 \\ ℓ_{2} & = & [y_{a 1} = - 1, y_{a 2} = 1, y_{a 3} = 1] & = & (- 1, 1, 1); & n_{ℓ_{2}} = 0 \\ ℓ_{3} & = & [y_{a 1} = 1, y_{a 2} = - 1, y_{a 3} = 1] & = & (1, - 1, 1); & n_{ℓ_{3}} = 5 \\ ℓ_{4} & = & [y_{a 1} = - 1, y_{a 2} = - 1, y_{a 3} = 1] & = & (- 1, - 1, 1); & n_{ℓ_{4}} = 7 \\ ℓ_{5} & = & [y_{a 1} = 1, y_{a 2} = 1, y_{a 3} = - 1] & = & (1, 1, - 1); & n_{ℓ_{5}} = 3 \\ ℓ_{6} & = & [y_{a 1} = - 1, y_{a 2} = 1, y_{a 3} = - 1] & = & (- 1, 1, - 1); & n_{ℓ_{6}} = 2 \\ ℓ_{7} & = & [y_{a 1} = 1, y_{a 2} = - 1, y_{a 3} = - 1] & = & (1, - 1, - 1); & n_{ℓ_{7}} = 10 \\ ℓ_{8} & = & [y_{a 1} = - 1, y_{a 2} = - 1, y_{a 3} = - 1] & = & (- 1, - 1, - 1); & n_{ℓ_{8}} = 10 \end{matrix}

Let $n_{ℓ_{i}}$ be the number of times where the response pattern $ℓ_{i}$ occurs over the three attributes. The $n_{ℓ}$ s are the random variables which are multinomially distributed. The expected number of $n_{ℓ}$ is $m_{ℓ}$ given by $m_{ℓ} = N P_{ℓ}$ , with probability $P_{ℓ}$ for the response pattern $ℓ$ . The total number of times $N$ comparing the window displays between time 1 and time 2 over three attributes is, $N = \sum_{i = 1}^{8} n_{ℓ_{i}}$ . Conditioned on fixed $N$ , a Poisson log-linear model can be fitted (see for the multinomial-Poisson relationship Aitkin et al.(2009)Aitkin, Francis, Hinde, and Darnell, p. 287f).

The PC model in this study, that is, the Poisson log-linear MPCC model over two time points according to three attributes $(a = 1, 2, 3)$ , is defined by,

ln m_{(y_{a 1}, y_{a 2}, y_{a 3})} = μ + \sum_{a = 1}^{A} y_{a} (λ_{1 a} - λ_{2 a}),

(4.3)

where $m_{(y_{a 1}, y_{a 2}, y_{a 3})}$ is the expected number of the response pattern $(y_{a 1}, y_{a 2}, y_{a 3})$ in PC $(12)$ and the parameter $μ$ for PC $(12)$ is a normalizing constant. Although the $λ$ s are unknown and have to be estimated, the $μ$ is the same for all 8 patterns and can therefore be treated as a constant. The parameter $λ_{1 a}$ is the object parameter of the object at time 1 with respect to attribute $a$ and $λ_{2 a}$ the object parameter for the object at time 2 with respect to attribute $a$ . For model estimation and as we are interested in the relative change at time 2 compared to time 1, we set all $λ_{1 a}$ object parameters at time 1 to be zero (i.e., $λ_{1 a 1} = 0, λ_{1 a 2} = 0, λ_{1 a 3} = 0$ ). From model estimation, we then got estimated object parameters (and change parameters) for each of the three attributes of the window display at time 2. Note that the presented model (4.3) would result in the same estimated object parameters as if fitted separately for each attribute. One advantage of modelling a response sequence for each PC is that all possible two-way interaction parameters (see Dittrich et al.(2006)Dittrich, Francis, Hatzinger, and Katzenbeisser) between the attributes can be incorporated into the model (4.3):

\begin{matrix} ln m_{(y_{a 1}, y_{a 2}, y_{a 3})} = μ + \sum_{a = 1}^{A} y_{a} (λ_{1 a} - λ_{2 a}) + y_{a 1} y_{a 2} θ_{a 1 a 2} + y_{a 1} y_{a 3} θ_{a 1 a 3} + y_{a 2} y_{a 3} θ_{a 2 a 3}, \end{matrix}

(4.4)

where $θ_{a 1 a 2}$ , for example, represents the interaction between the judgements of attributes $a 1$ and $a 2$ in the comparison of time 1 and time 2 $(12)$ . For three attributes we obtain three possible two-way interaction parameters $θ_{a 1 a 2}, θ_{a 1 a 3}, θ_{a 2 a 3}$ .

The interaction parameter of model (4.4) could be interpreted in the way of conditional log-odds ratio for consistently designing window displays (i.e., ( $1, 1$ ), ( $- 1, - 1$ )) compared to inconsistent designs (i.e., ( $1, - 1$ ), ( $- 1, 1$ )) over the attributes optical balance (a = 2) and triangle style (a $=$ 3), given $y_{a 1} = 1$ :

\begin{matrix} ln \frac{P_{ℓ_{1} (1, 1 | y_{a 1} = 1)} \cdot P_{ℓ_{7} (- 1, - 1 | y_{a 1} = 1)}}{P_{ℓ_{3} (- 1, 1 | y_{a 1} = 1)} \cdot P_{ℓ_{5} (1, - 1 | y_{a 1} = 1)}} = \\ ln m_{ℓ_{1}} + ln m_{ℓ_{7}} - ln m_{ℓ_{3}} - ln m_{ℓ_{5}} = 4 θ_{a 2 a 3} . \end{matrix}

5 Results

The MPCC model was fitted as a conditional Poisson log-linear model (see Aitkin et al.(2009)Aitkin, Francis, Hinde, and Darnell) in R (R Development Core Team, R Development Core Team (2019)), with the package gnm (Turner and Firth (2018)) and parts of the package prefmod (Hatzinger and Maier (2017)). The latter assisted us to generate all 8 possible response patterns for three attributes, which are the basis for the design structure. The matrix notation of the MPCC model (4.3) is shown in Table 1.

Table 1:
Matrix notation of the MPCC model

$(\begin{matrix} ln m_{ℓ_{1}} \\ ln m_{ℓ_{2}} \\ ln m_{ℓ_{3}} \\ ln m_{ℓ_{4}} \\ ln m_{ℓ_{5}} \\ ln m_{ℓ_{6}} \\ ln m_{ℓ_{7}} \\ ln m_{ℓ_{8}} \end{matrix})$ $=$ $(\begin{matrix} 1 & 1 & - 1 & 1 & - 1 & 1 & - 1 \\ 1 & - 1 & 1 & 1 & - 1 & 1 & - 1 \\ 1 & 1 & - 1 & - 1 & 1 & 1 & - 1 \\ 1 & - 1 & 1 & - 1 & 1 & 1 & - 1 \\ 1 & 1 & - 1 & 1 & - 1 & - 1 & 1 \\ 1 & - 1 & 1 & 1 & - 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 & 1 & - 1 & 1 \\ 1 & - 1 & 1 & - 1 & 1 & - 1 & 1 \end{matrix})$ $(\begin{matrix} μ \\ λ_{1 a 1} \\ λ_{2 a 1} \\ λ_{1 a 2} \\ λ_{2 a 2} \\ λ_{1 a 3} \\ λ_{2 a 3} \end{matrix})$

Table 2:

Parameter estimates of model C

,model C	estimates	std. error	p-value
time1: focus, balance, triangle	0	NA
time2.focus	0.000	0.162	1.000
time2.balance	0.837	0.222	0.000*
time2.triangle	0.327	0.171	0.056

Table 3:

Model selection

,	deviance	df	dev. change, df change, p-value
model A	1.076	1
model B	1.534	2	0.458, 1, p-value = 0.499
model C	3.000	4	1.466, 2, p-value = 0.480
model D	13.473	6	10.473, 2, p-value = 0.005

We started with a MPCC model by estimating all possible change parameters and included all possible two-way interaction parameters between the three attributes focus $(a 1)$ , optical balance $(a 2)$ and triangle style $(a 3)$ into the model (named model A, see Formula (4.4)). We obtained the following interaction parameters ${\hat{θ}}_{a 1 a 2} = 0.197$ (s.e. $0.237$ ), ${\hat{θ}}_{a 1 a 3} = - 0.036$ (s.e. $0.175$ ) and ${\hat{θ}}_{a 2 a 3} = - 0.268$ (s.e. $0.291$ ). We tried to reduce these parameters by setting two of them to be equal ( ${\hat{θ}}_{a 1 a 3} = {\hat{θ}}_{a 2 a 3}$ ), in model B. Neither this summarized interaction parameter ${\hat{θ}}_{a 1 a 3, a 2 a 3} = - 0.108$ (s.e. $0.135$ ) nor the interaction parameter ${\hat{θ}}_{a 1 a 2}$ was significantly different from zero. Therefore we eliminated them in model C (see Formula (4.3) and estimates in Table 2). In model D we set the change parameters for all three attributes to be equal, ${\hat{λ}}_{2 a 1} = {\hat{λ}}_{2 a 2} = {\hat{λ}}_{2 a 3}$ (see Aitkin et al.(2009)Aitkin, Francis, Hinde, and Darnell, p. 248ff) and obtained ${\hat{λ}}_{2 a 1, 2, 3} = 0.347$ (s.e. $0.099$ ). We checked the deviance differences and the differences of the degrees of freedom of the computed models (see Table 3). For model selection, we defined a significance criterion of $0.05$ . We found that model C cannot be simplified to model D with only one common parameter ( $p$ -value = 0.005).

The estimated object parameters of the MPCC model (model C) with all three change parameters are shown in Table 2. From Table 2 we can see that the estimated object parameters of the attributes optical balance (time2.balance) and triangle style (time2.triangle) are positive at time 2 (after feedback was given) compared to time 1, which is set to be zero (i.e., reference level, $λ_{1 a 1} = λ_{1 a 2} = λ_{1 a 3} = 0$ ). The positive object parameter estimates indicate a positive change (i.e., improvement) from time 1 to time 2 (i.e., ${\hat{λ}}_{2 a 2} = 0.837$ , ${\hat{λ}}_{2 a 3} = 0.327$ ). Whereas the attribute optical balance (time2.balance) is significantly more often better fulfilled within the window display at time 2 (after feedback was given) compared to time 1. But there is no change at all for the attribute focus ( $a 1$ ), ${\hat{λ}}_{2 a 1} = 0$ , over time. The method is telling us that in the case of no feedback there is no change at all.

6 Discussion

This article addresses the problem of making absolute ratings for objects at each of two or more time points to obtain measures of change. We suggest an alternative, PC, method to simply judge if one attribute of an object is fulfilled better at time 1 or at time 2. In this article an attempt is made by modelling response patterns resulting from the pairwise judgement of an object between time 1 and time 2 with respect to three attributes.

The study gives insights as to whether students have improved their ability to design window displays with respect to aesthetic attributes such as focus, optical balance and triangle style between time 1 and time 2. We questioned if feedback could have an effect on a possible change in performance. We assumed that the aesthetic attributes focus, optical balance and triangle style were influenced by the same possible factors between time 1 and time 2. To control possible confounding effects (e.g., learning effect, resulting from the fact that students design twice a window display with the same kind of products for the same occasion) we looked at the attribute focus which was excluded from feedback, compared to the other two attributes optical balance and triangle style to which feedback was given. The estimated parameters for the attribute focus did not change over time. We found that students improved in designing a triangle style and significantly in optical balance at time 2 after feedback was given. We carefully conclude that these findings of improvement are due to the feedback given between the two time points. Refinements of this study could be that multiple experts $/$ non-experts are asked for judgements to check the degree of agreement and that the judges are presented with the photos of the window designs at time 1 and time 2 in a randomized order. Further studies have to be done with a larger sample size.

The model presented here could be applied to studies from different disciplines, where it might be easier for judges to state a change. For example, users could judge a website of an online service with regards to different aspects and decide pairwise whether an aspect is fulfilled better before or after relaunch or in sports where based on video tapes the technique of athletes is compared between pairs of times.

The MPCC model could be extended by allowing a third possible response category no change (see, e.g., Dittrich et al.(1998)Dittrich, Hatzinger, and Katzenbeisser) or by including subject covariates (see Dittrich et al.(1998)Dittrich, Hatzinger, and Katzenbeisser) into the model. Furthermore it could be applied to more than two time points. For $J$ time points $J choose 2$ PCs can be obtained. In this case all $J choose 2$ PCs and A different attributes could be regarded within one long response pattern by applying a PC pattern model (see Dittrich et al.(2006)Dittrich, Francis, Hatzinger, and Katzenbeisser). One advantage of this method is that dependencies between object (time) pairs can be incorporated into the model (see Dittrich et al.(2006)Dittrich, Francis, Hatzinger, and Katzenbeisser). For many time points and attributes we would suggest modelling short response patterns, containing the responses regarding A attributes, for each PC separately (see Grand et al.(2015)Grand, Dittrich, and Francis), as otherwise the response pattern would become very large. Within the MPCC model short response patterns for each pair of time can be modelled separately. For example, for three time points (time 1, time 2, time 3) with three pairs of time (12), (13), (23) and A attributes the MPCC model is defined by three equations,

\begin{matrix} ln m_{12 (y_{12 a 1}, y_{12 a 2}, y_{12 a 3})} = μ_{12} + \sum_{a = 1}^{A} y_{12 a} (λ_{1 a} - λ_{2 a}), \\ ln m_{13 (y_{13 a 1}, y_{13 a 2}, y_{13 a 3})} = μ_{13} + \sum_{a = 1}^{A} y_{13 a} (λ_{1 a} - λ_{3 a}), \\ ln m_{23 (y_{23 a 1}, y_{23 a 2}, y_{23 a 3})} = μ_{23} + \sum_{a = 1}^{A} y_{23 a} (λ_{2 a} - λ_{3 a}) . \end{matrix}

Thanks to Murray Aitkin and his work on generalised linear models (GLMs) which has inspired the Paired Comparison Project at WU Vienna. This project aimed to develop an integrated statistical modelling approach to the analysis of PC data using underlying Poisson and multinomial generalised linear models and allowing standard software to be used where possible. The team at WU Vienna (including the authors of this article) were motivated by Murray's early work on categorical data analysis, including his articles on log-linear model selection (Aitkin (1980)) and the fitting of covariate models for the multinomial distribution (Aitkin and Francis (1992)), his books on statistical modelling—first in GLIM (Aitkin et al.(1989)Aitkin, Anderson, Francis, and Hinde) and then in R (Aitkin et al.(2009)Aitkin, Francis, Hinde, and Darnell), and his evangelical short courses on generalized linear models which he and his team gave throughout Europe. The excitement which he and his body of work generated has motivated an entire generation of researchers.

Footnotes

Declaration of conflicting interests

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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