Conjoint analysis is a multivariate technique developed specifically to understand how respondents develop preferences for any type of object. This analysis were developed with the assumption that each consumer evaluate the value of the object feature based on its utility. Utility is a subjective judgement of preference unique to each individual.

Utility represents the total worth or overall preference of an object, can be thought of as the sum of part-worth.

\[Utility = part-worth\ of\ X_1 + part-worth\ of\ X_2\] Define product attributes that can be quantified and corresponding levels. A level is the specific value or realization of the attribute. Part-worths are generated by OLS Regression. OLS is the method of calculation traditionally used in most conjoint studies. However, OLS is not appropriate for conjoint data consisting of rank orders.

For example we used study on preferences of tea consumers carried out in 2004 on a sample group of students of Wroc law University of Economics3. The main aim of the study was to identify determines of consumers’ choice of specific brands and types of tea. This data collected using rating method. Source

In the conjoint analysis first we must construct the profile or stimuli combination of the observed factor. In this example the factor and the level are:

First, lets evalute single respondent and see how his attitude toward choice of specific brands and types of tea.

library(conjoint)
data(tea)
caModel(y = tprefm[1,], x = tprof)
## 
## Call:
## lm(formula = frml)
## 
## Residuals:
##       1       2       3       4       5       6       7       8       9 
##  1.1345 -1.4897  0.3103 -0.2655  0.3103  0.1931  1.5931 -1.4310 -1.4310 
##      10      11      12      13 
##  1.1207  0.3690  1.1931 -1.6069 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)   
## (Intercept)          3.3937     0.5439   6.240  0.00155 **
## factor(x$price)1    -1.5172     0.7944  -1.910  0.11440   
## factor(x$price)2    -1.1414     0.6889  -1.657  0.15844   
## factor(x$variety)1  -0.4747     0.6889  -0.689  0.52141   
## factor(x$variety)2  -0.6747     0.6889  -0.979  0.37234   
## factor(x$kind)1      0.6586     0.6889   0.956  0.38293   
## factor(x$kind)2     -1.5172     0.7944  -1.910  0.11440   
## factor(x$aroma)1     0.6293     0.5093   1.236  0.27150   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.78 on 5 degrees of freedom
## Multiple R-squared:  0.8184, Adjusted R-squared:  0.5642 
## F-statistic:  3.22 on 7 and 5 DF,  p-value: 0.1082

the first respondent showed that he is preferred toward high price, because in \(price1\) (low) and \(price2\) (medium) the estimated part-worth are negative, \(-1.5172\) and \(-1.1414\). This respondent preferred red tea among other because the estimated part-worth of black \(variety1\) and green \(variety2\) is negative.

Conjoint(y = tprefm, x = tprof, z = tlevn)
## 
## Call:
## lm(formula = frml)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5,1888 -2,3761 -0,7512  2,2128  7,5134 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         3,55336    0,09068  39,184  < 2e-16 ***
## factor(x$price)1    0,24023    0,13245   1,814    0,070 .  
## factor(x$price)2   -0,14311    0,11485  -1,246    0,213    
## factor(x$variety)1  0,61489    0,11485   5,354 1,02e-07 ***
## factor(x$variety)2  0,03489    0,11485   0,304    0,761    
## factor(x$kind)1     0,13689    0,11485   1,192    0,234    
## factor(x$kind)2    -0,88977    0,13245  -6,718 2,76e-11 ***
## factor(x$aroma)1    0,41078    0,08492   4,837 1,48e-06 ***
## ---
## Signif. codes:  0 '***' 0,001 '**' 0,01 '*' 0,05 '.' 0,1 ' ' 1
## 
## Residual standard error: 2,967 on 1292 degrees of freedom
## Multiple R-squared:  0,09003,    Adjusted R-squared:  0,0851 
## F-statistic: 18,26 on 7 and 1292 DF,  p-value: < 2,2e-16
## [1] "Part worths (utilities) of levels (model parameters for whole sample):"
##        levnms    utls
## 1   intercept  3,5534
## 2         low  0,2402
## 3      medium -0,1431
## 4        high -0,0971
## 5       black  0,6149
## 6       green  0,0349
## 7         red -0,6498
## 8        bags  0,1369
## 9  granulated -0,8898
## 10      leafy  0,7529
## 11        yes  0,4108
## 12         no -0,4108
## [1] "Average importance of factors (attributes):"
## [1] 24,76 32,22 27,15 15,88
## [1] Sum of average importance:  100,01
## [1] "Chart of average factors importance"

Based on the part-worth, most respondent attracted to low price, black, leafy and aroma tea.

These results mean that the most attractive is a black, leaf tea with low price and aroma. Besides that, it is also possible to calculate average importance of attributes. In this example most important are by turns: color of tea (32,22%), its type (27,15%) and price of tea (24,76%). The minimum importance has aroma (15,88%).