National University of Singapore
Journal of Statistics Education Volume 9, Number 2 (2001)
Copyright © 2001 by
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Key Words: Categorical variables; Data transformation; Multiple linear regression; Standardized residuals.
Many statistical problems can be satisfactorily resolved within the framework of linear regression. Business students, for example, employ linear regression to uncover interesting insights in the fields of Finance, Marketing, and Human Resources, among others. The purpose of this paper is to demonstrate how several concepts arising in a typical discussion of multiple linear regression can be motivated through the development of a pricing model for diamond stones. Specifically, we use data pertaining to 308 stones listed in an advertisement to construct a model, which educates us on the relative pricing of caratage and the different grades of clarity and colour.
Regression analysis is a most versatile tool in our students' statistical arsenal. It is perhaps the most useful statistical technique employed by them during their academic experience and later in their professional endeavours. Having gone through the complexities of independent samples t-test and ANOVA, many students are relieved when they realise that the comparison of group means can actually be conducted within the unified framework of the regression model. The latter also offers flexibility and transparency in handling exogenous factors.
In March 2000, I tasked my MBA students to develop a sensible pricing model for diamond stones using data that appeared in an advertisement in Singapore's Business Times edition of February 18, 2000. An example of such an advertisement appears in Figure 1. The analysis was to focus on data pertaining to
Figure 1. An Advertisement for Diamonds.
The website www.adiamondisforever.com educates the layperson on the factors that influence the price of a diamond stone. These are the 4 C's: Carat, Clarity, Colour and Cut.
The weight of a diamond stone is indicated in terms of carat units. One carat is equivalent to 0.2 grams. All other things being equal, larger diamond stones command higher prices in view of their rarity.
Being products of Nature, diamonds have “birthmarks” or inclusions only visible under a jeweller's magnifying glass or a microscope. Diamonds with no inclusion under a loupe with a 10 power magnification are labelled IF (“internally flawless”). Lesser diamonds are categorised in descending order as “very very slightly imperfect” VVS1 or VVS2 and “very slightly imperfect” VS1 or VS2.
The most prized diamonds display colour purity. They are not contaminated with yellow or brown tones. Top colour purity attracts a grade of D. Subsequent degrees of colour purity are rated E, F, G, … all the way down the alphabet ladder.
The cut (or faceting) of a raw diamond stone relies on the experience and the craftsmanship of the diamond cutter. The optimal cut should neither be too deep nor too shallow for it will impede the trajectory of light and thereby the brilliance or “fire” of a diamond stone.
To assist shoppers, independent certification bodies assay diamond stones and provide each of them with a certificate listing their caratage and their grades of clarity, colour and cut. The newspaper advertisement however only provided, for each stone, details on the certification body and its assessment of the caratage, clarity and colour of the stones. Three certification bodies were mentioned in the advertisement, namely New York based Gemmological Institute of America (GIA) and Antwerp based International Gemmological Institute (IGI) and Hoge Raad Voor Diamant (HRD). Their reputations could be a factor in the pricing of the diamond stones.
Given the information in the dataset, a multiple linear regression (MLR) model is a natural path to explore. Generally speaking, one would expect the price (denoted in Singapore dollars) of a stone to move in tandem with the caratage. However, the relationship may not be linear as heavier stones are more prized than the lighter ones. An examination of the scatter plot of Price against Carats would therefore be enlightening.
Figure 2. Price Against Carat.
Clearly, there is a relationship but the trend appears to fan out. This indicates higher price volatility for the heavier stones, especially those above 1 carat. Unless we transform the data, we would most likely not satisfy the homoscedascity assumption of linear regression. A transformation that is recommended in similar situations is the logarithm of prices. This is illustrated below.
Figure 3. Ln(Price) Against Carat.
The relationship between Carat and the logarithms of Price appears more homoscedastic compared to the first scatter plot. This suggests that it would be more judicious to employ ln(Price) in lieu of Price in developing a linear regression model.
Next we have to insert clarity, colour and the identity of the certification body in the regression model. Students should notice that these are all categorical in nature. Therefore the operational hurdle facing them is the following:
Discussion 1: How should the ordinal data be coded?
In the case of ordinal data like clarity (ditto for colour), some students may be tempted to employ, for example, VS2=1, VS1=2, VVS2=3, VVS1=4 and IF=5. A discussion would therefore have to be engaged on why this is not suitable.
The MINITAB® output and accompanying residual plots from the first attack on the data are reproduced below. Selecting clarity grade VS2 as my baseline category, I coded four indicator variables to help me infer on the difference between VS2 and each of VS1, VVS2, VVS1 and IF. Likewise, I defined colour I as the baseline and compared it to the other five colours using five indicator variables. Instructors may use Discussion 2 to guide their classes in assessing the results.
Discussion 2: Is the regression model useful? This requires students to assess whether (a) the model has predictive power, (b) the estimates of the regression slopes are sensible, especially for the ordinal data, and (c) the standard assumptions of MLR are met. Students can also be queried on the advantage of scrutinizing standardized as opposed to raw residuals.
The regression equation is
ln_price = 6.08 + 2.86 Carat + 0.417 D + 0.387 E + 0.310 F + 0.210 G + 0.129 H + 0.299 IF + 0.298 VVS1 + 0.202 VVS2 + 0.0966 VS1 + 0.0089 GIA - 0.174 IGI Predictor Coef StDev T P Constant 6.07724 0.04809 126.37 0.000 Carat 2.85501 0.03697 77.23 0.000 D 0.41656 0.04138 10.07 0.000 E 0.38705 0.03082 12.56 0.000 F 0.31020 0.02748 11.29 0.000 G 0.21021 0.02836 7.41 0.000 H 0.12868 0.02852 4.51 0.000 IF 0.29854 0.03330 8.96 0.000 VVS1 0.29783 0.02810 10.60 0.000 VVS2 0.20192 0.02534 7.97 0.000 VS1 0.09661 0.02492 3.88 0.000 GIA 0.00886 0.02086 0.42 0.672 IGI -0.17385 0.02867 -6.06 0.000 S = 0.1382 R-Sq = 97.2% R-Sq(adj) = 97.1% Analysis of Variance Source DF SS MS F P Regression 12 197.939 16.495 863.64 0.000 Residual Error 295 5.634 0.019 Total 307 203.574
Figure 4. Residual Plot.
Figure 5. Normal Plot.
The students’ verdict should be that although the model has predictive power and the slopes adhere to the hierarchy of the grades of colour and clarity, the dome-like scatter in the residual plot is a cause for concern. The normality assumption, however, appears to be less problematic.
Discussion 3: What remedial action(s) can be undertaken?
The residual plot indicates that the regression model underestimates prices at both ends of the price range and overestimates the midrange prices.
This insight opens up several vistas for exploration. One possibility is to segregate the stones according to caratage. For instance, Figure 2 suggests that the stones may be divided into 3 clusters, say less than 0.5 carats (“small”), 0.5 to less than 1 carat (“medium”) and 1 carat and over (“large”). Separate regression models may be constructed for each cluster. The disadvantage of this approach is that results may not be consistent across the 3 clusters as these do not have an even spread of the grades of colour and clarity. This leads to the following poser,
Discussion 4: Can we construct a unified regression model that will cover all the 308 stones and will possibly deliver different pricing structures for the 3 clusters just defined?
This is where students would reckon that indicator variables coding the above three caratage ranges and their interactions with carats (to reflect different slopes) will have to be employed. This avenue has been explored in my classes. Here is the MINITAB® output where “small” was defined as the baseline caratage cluster and where the coefficient for med*carat (ditto for large*carat) is the average difference in incremental price per carat unit between “small” and “medium” stones.
The regression equation is ln_price = 5.53 + 4.26 Carat + 0.434 D + 0.349 E + 0.273 F + 0.188 G + 0.108 H + 0.311 IF + 0.213 VVS1 + 0.134 VVS2 + 0.0682 VS1 + 0.00770 GIA - 0.0167 IGI + 0.946 med + 2.38 large - 1.77 med*carat - 3.26 large*carat Predictor Coef StDev T P Constant 5.5307 0.03288 168.22 0.000 Carat 4.2572 0.08550 49.79 0.000 D 0.4336 0.01690 25.66 0.000 E 0.3487 0.01255 27.78 0.000 F 0.2728 0.01114 24.49 0.000 G 0.1879 0.01152 16.31 0.000 H 0.1079 0.01148 9.39 0.000 IF 0.3114 0.01354 22.99 0.000 VVS1 0.2133 0.01154 18.49 0.000 VVS2 0.1342 0.01035 12.96 0.000 VS1 0.0682 0.01006 6.78 0.000 GIA 0.00770 0.008473 0.91 0.364 IGI -0.0167 0.01218 -1.37 0.171 med 0.9460 0.03909 24.20 0.000 large 2.3760 0.3198 7.43 0.000 med*carat -1.7655 0.09350 -18.88 0.000 large*carat -3.2600 0.3234 -10.08 0.000 S = 0.05540 R-Sq = 99.6% R-Sq(adj) = 99.5% Analysis of Variance Source DF SS MS F P Regression 16 202.680 12.668 4126.79 0.000 Residual Error 291 0.893 0.003 Total 307 203.574
Figure 6. Residual Plot.
Figure 7. Normal Plot.
Discussion 5: Is this regression model satisfactory? Are the standard assumptions of linear regression validated? Are the numerical estimates sensible? Interpret the interaction parameter med*carat. Which is more highly valued: colour or clarity? What can we infer on the incremental pricing of caratage in the 3 clusters? All other things being equal, what is the average price difference between a grade D diamond and another one graded (a) I (b) E? etc. All other things being equal, are there price differences amongst the stones appraised by the GIA, IGI and HRD?
Another remedial option, which avoids the subjectivity of cluster definitions, is to employ the square of carat, as suggested by the curvature in Figure 3. The statistical output and diagnostic plots are shown below:
The regression equation is ln_price = 5.31 + 5.67 Carat + 0.443 D + 0.363 E + 0.287 F + 0.198 G + 0.104 H + 0.177 IF + 0.226 VVS1 + 0.143 VVS2 + 0.0757 VS1 + 0.00622 GIA - 0.0192 IGI - 2.10 Caratsq Predictor Coef StDev T P Constant 5.30634 0.02961 179.20 0.000 Carat 5.67062 0.07928 71.52 0.000 D 0.44261 0.01774 24.95 0.000 E 0.36336 0.01322 27.48 0.000 F 0.28662 0.01179 24.31 0.000 G 0.19757 0.01215 16.26 0.000 H 0.10351 0.01224 8.46 0.000 IF 0.17670 0.01259 14.03 0.000 VVS1 0.22617 0.01220 18.54 0.000 VVS2 0.14348 0.01098 13.07 0.000 VS1 0.07571 0.01069 7.08 0.000 GIA 0.006223 0.008938 0.70 0.487 IGI -0.01919 0.01300 -1.48 0.141 Caratsq -2.10292 0.05802 -36.24 0.000 S = 0.05920 R-Sq = 99.5% R-Sq(adj) = 99.5% Analysis of Variance Source DF SS MS F P Regression 13 202.543 15.580 4445.36 0.000 Residual Error 294 1.030 0.004 Total 307 203.574
Figure 8. Residual Plot.
Figure 9. Normal Plot.
Discussion 6: Which remedial option is preferable? Students here would scrutinize the adjusted R-squares, the standard deviation of the residuals, the residual plots and the sensibilities of the regression estimates. The issue of interpretability may also be raised. Specifically, do we learn more about pricing using the variables medium, large, med*carat and large*carat as opposed to caratsq?
In many textbook exercises, students are provided with neat datasets where often “everything works out” at first attempt. In real life, this is rarely the case. Students should be exposed to real-life datasets where they would have to exercise judgment before arriving at practical results.
In this regression application, students get to infer the pricing of the caratage and the grades of the colour and clarity of diamond stones. Unlike the hard sciences where physical laws exist to guide knowledge, statistics is about the only tool that students in business or the social sciences can use to get a grip on phenomena arising in their disciplines.
Instructors only interested in a simple linear regression application linking caratage to price may refer to an earlier publication (jse.amstat.org/v4n3/datasets.chu.html)
The basic data are collated in the file 4Cdata.txt. The dataset with the indicator or "dummy" codes and transformed variables, as employed in the above analyses, is in 4C1data.txt. A synopsis of the application and a description of the variables are provided in the 4C.txt file.
Columns 1 - 4 Carat - Weight of diamond stones in carat units 6 Colour - D, E, F, G, H or I 8 - 11 Clarity - IF, VVS1, VVS2, VS1 or VS2 13 - 15 Certification Body - GIA, IGI or HRD 18 - 21 Price (Singapore $)
Columns 1 - 4 Carat - Weight of diamond stones in carat units 6 Indicator for colour D 8 Indicator for colour E 10 Indicator for colour F 12 Indicator for colour G 14 Indicator for colour H 16 Indicator for clarity IF 18 Indicator for clarity VVS1 20 Indicator for clarity VVS2 22 Indicator for clarity VS1 24 Indicator for certification body GIA 26 Indicator for certification body IGI 28 Indicator for medium stones between 0.5 to less than 1 carat 30 Indicator for large stones weighing 1 carat or more 32 - 35 Interaction variable med*carat 37 - 40 Interaction variable large*carat 42 - 48 Carat squared 50 - 53 Price (Singapore $) 55 - 65 Ln(Price)
Faculty of Business Administration
National University of Singapore
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