Conjoint Analysis

From Wiki:

'Conjoint analysis' is a survey-based statistical technique used in market research that helps determine how people value different attributes (feature, function, benefits) that make up an individual product or service. The objective of conjoint analysis is to determine what combination of a limited number of attributes is most influential on respondent choice or decision making. A controlled set of potential products or services is shown to survey respondents and by analyzing how they make preferences between these products, the implicit valuation of the individual elements making up the product or service can be determined. These implicit valuations (utilities or part-worths) can be used to create market models that estimate market share, revenue and even profitability of new designs.

Types of Conjoint Analysis

Here are a very brief description of a few conjoint analysis methods:

1) Full-Profile Conjoint Analysis

In this type of CA, a large number of full product descriptions is displayed to the respondent yielding large amounts of data for each one of them. Different product descriptions are presented for acceptability or preference evaluations.

2) Adaptive Conjoint Analysis

This type of CA varies the choice sets presented based on the respondents’ preference. As a consequence, the features and levels shown are increasingly more competitive optimizing the data.

3) Choice-Based Conjoint

The CBC (or discrete-choice conjoint analysis) is the most common type. It requires respondents to select their preferred full-profile concept repeatedly from sets of around 3 to 5 full profile concepts. This idea is to try to simulate an actual buying scenario, and mimick shopping behavior as close as possible. From the trade-offs that are made when the respondent chooses one, or none, of the available choices, the importance of the attribute features and levels can be statistically derived. Based on the results one can estimate the value of each of the levels and also the optimal combinations that make-up products.

4) Hierarchical Bayes Conjoint Analysis (HB)

This type of CA is used to estimate utilities from respondents’ choice data. It is particularly useful when the respondent cannot provide preference evaluations for all attribute levels due to the size of the data collection.

5) Max-Diff Conjoint Analysis

This type of CA presents to the respondents an assortment of packages that must be selected under best/most preferred and worst/least preferred

We will focus on Choice-Based Conjoint Analysis in the following:

Choice-Based Conjoint Analysis

Basic Assumptions

The basic assumptions of Conjoint Analysis are:

  • Product are a bundle of attributes
  • The utility of a product is a function of the utilities of each of its attributes
  • Behavior, such as purchases, can be predicted from utilities

Steps

  • One must first choose the attributes to be included
  • The number of levels for each attribute must also be chosen
  • Definition of hypothetical products (all combinations of attribute levels would generate too many products)
  • One should make sure that:
    • All combinations of levels for pairs of attributes occur in some product
    • The subset of products should have orthogonal design i.e. the chances of finding a given level of some attribute B in a product should be the same irregardless of the level of another attribute A.
  • Estimation of utilities (usually using ordinary linear regression with dummy variables)

The linear regression model with conjoint preference data has the form:

$$R_i = u_0 + u_{j}^k X_{ij}^k$$

where, $R_i$ is the ranking/rating assigned to product $i$,

$$X_{ij}^k = \left\{ {\begin{array}{*{20}{l}} 1&{{\text{if product }}i{\text{ has level j on attribute }}k}\\ 0&{{\text{otherwise}}} \end{array}} \right.$$

and $u_j$ is the utility coefficient for level $j$ on attribute $k$.

Data

This dataset is based on [1].

Importing data

The model input data has the form below. Each row corresponds to one product profile, a combination of attributes.

In [3]:
import pandas as pd
filename = 'data/mobile_services_ranking.csv'
pd.read_csv(filename)
Out[3]:
brand startup monthly service retail apple samsung google ranking
0 "AT&T" "$100" "$100" "4G NO" "Retail NO" "Apple NO" "Samsung NO" "Nexus NO" 11
1 "Verizon" "$300" "$100" "4G NO" "Retail YES" "Apple YES" "Samsung YES" "Nexus NO" 12
2 "US Cellular" "$400" "$200" "4G NO" "Retail NO" "Apple NO" "Samsung YES" "Nexus NO" 9
3 "Verizon" "$400" "$400" "4G YES" "Retail YES" "Apple NO" "Samsung NO" "Nexus NO" 2
4 "Verizon" "$200" "$300" "4G NO" "Retail NO" "Apple NO" "Samsung YES" "Nexus YES" 8
5 "Verizon" "$100" "$200" "4G YES" "Retail NO" "Apple YES" "Samsung NO" "Nexus YES" 13
6 "US Cellular" "$300" "$300" "4G YES" "Retail NO" "Apple YES" "Samsung NO" "Nexus NO" 7
7 "AT&T" "$400" "$300" "4G NO" "Retail YES" "Apple YES" "Samsung NO" "Nexus YES" 4
8 "AT&T" "$200" "$400" "4G YES" "Retail NO" "Apple YES" "Samsung YES" "Nexus NO" 5
9 "T-Mobile" "$400" "$100" "4G YES" "Retail NO" "Apple YES" "Samsung YES" "Nexus YES" 16
10 "US Cellular" "$100" "$400" "4G NO" "Retail YES" "Apple YES" "Samsung YES" "Nexus YES" 3
11 "T-Mobile" "$200" "$200" "4G NO" "Retail YES" "Apple YES" "Samsung NO" "Nexus NO" 6
12 "T-Mobile" "$100" "$300" "4G YES" "Retail YES" "Apple NO" "Samsung YES" "Nexus NO" 10
13 "US Cellular" "$200" "$100" "4G YES" "Retail YES" "Apple NO" "Samsung NO" "Nexus YES" 15
14 "T-Mobile" "$300" "$400" "4G NO" "Retail NO" "Apple NO" "Samsung NO" "Nexus YES" 1
15 "AT&T" "$300" "$200" "4G YES" "Retail YES" "Apple NO" "Samsung YES" "Nexus YES" 14

Dummy variables

We now will calculate $X_{ij}^k$ from the definition above, where we recall

$$X_{ij}^k = \left\{ {\begin{array}{*{20}{l}} 1&{{\text{if product }}i{\text{ has level j on attribute }}k}\\ 0&{{\text{otherwise}}} \end{array}} \right.$$

For example for product:

  • $X_{{\text{US Cellular}},{\text{brand}}}^{\text{0}}=1$ since the product profile on row with index $k=1$ has level $i=1={\text{Verizon}}$ on attribute $j=1={\text{brand}}$.
  • $X_{{\text{4G YES}},{\text{service}}}^{\text{8}}=1$ since the product profile on row with index $k=8$ has level $i=0={\text{4G YES}}$ on attribute $j=1={\text{service}}$.

The cell below performs the following steps:

  • The for loops over the attributes
  • The first loop fixes the attribute to brand
  • The line below counts the number of levels in the attribute brand

    nlevels = len(list(np.unique(conjoint_data_frame[brand])))
  • The aux variables is the the list of names corresponding to that attribute (brand):

    array(['"AT&T"', '"T-Mobile"', '"US Cellular"', '"Verizon"'], dtype=object)
  • The following line appends this array into an empty list of level names which becomes:

    level_name = [['"AT&T"', '"T-Mobile"', '"US Cellular"', '"Verizon"']]
  • Next the variables begin and end are calculated and a list new_part_worth is created, which contains the part worth associated with the level 'T-Mobile'. Notice that the list has three elements since the last we be obtained by imposing zero sum:

    begin = 1 
    end = 1 + 4 - 1 = 4
    
    new_part_worth = list(main_effects_model_fit.params[1:4]) = [0.0, -0.25, 0.0]
  • The command below grabs the parameters from main_effects_model_fit skipping the intercept

      main_effects_model_fit.params[1:4]) 
  • The next line calculates the next value since the utilities are zero-centered.

  • The range of levels of the attribute brand is appended to the part_worth_range list

  • After the for loop is finished, we have a list of list, each sub-list containing the part-worths of an attribute

In [4]:
import numpy as np
import pandas as pd
import statsmodels.api as sm

def lr_params(filename):
    
    df = pd.read_csv(filename)
    
    cols = df.columns.tolist()
    
    dummies = pd.concat([pd.get_dummies(df[col], drop_first = True, prefix= col) 
                     for col in cols[0:-1]], axis=1)
    dummies.columns = [c.replace('"','').replace(" ", "_").lower() for c in dummies.columns.tolist()]

    X,y = dummies, df[cols[-1]]
    X = sm.add_constant(X)
    lr = sm.OLS(y, X).fit()
    betas = lr.params.round(3)
    v = dummies.columns.tolist()
    res = list(zip(v,betas))
    res = pd.DataFrame(res, columns=['attribute', 'beta'])
    
    attributes = ['brand', 'startup', 'monthly', 'service','retail', 'apple', 'samsung', 'google']
    levels, pw, pw_range = [],[],[]
    b = 1 
    for att in attributes:
        num_levels = len(list(np.unique(df[att])))
        levels.append(list(np.unique(df[att]))) 
        a = b 
        b = a + num_levels - 1
        pw_new = [round(i,3) for i in list(lr.params[a:b])]
        pw_new.append((-1) * sum(pw_new)) 
        pw.append(pw_new)
        pw_range.append(max(pw_new) - min(pw_new)) 
       
    
    importance = []
    for item in pw_range:
        importance.append(round(100 * (item / sum(pw_range)),2))

    name_dict = {'brand' : 'Provider', \
                 'startup' : 'Start-up Cost', 'monthly' : 'Monthly Cost', \
                 'service' : '4G Service', 'retail' : 'Nearby retail store', \
                 'apple' : 'Apple products sold', 'samsung' : 'Samsung products sold', \
                 'google' : 'Google/Nexus products sold'}  

    lst = []
    
    idx = 0 
    for att in attributes:
        print('\nAttribute and Importance:', name_dict[att],'and',importance[idx])
        print('    Level Part-Worths')
        for level in range(len(levels[idx])):
            print('       ',levels[idx][level],'-->', pw[idx][level])  
            lst.append([levels[idx][level],pw[idx][level]])
        idx = idx + 1
    
    dfnew = pd.DataFrame(list(zip(name_dict.values(),importance)), 
                         columns=['attribute', 'importance']).sort_values('importance',ascending=False)
    
    lst_new = [[lst[i][0].replace('"',''),lst[i][1]] for i in range(len(lst))]
    print(lst_new)
    tup = (lr.summary(),res,dfnew,lst_new)
    
    return tup
In [5]:
tup = lr_params(filename)
Attribute and Importance: Provider and 0.96
    Level Part-Worths
        "AT&T" --> -0.25
        "T-Mobile" --> -0.0
        "US Cellular" --> 0.25
        "Verizon" --> -0.0

Attribute and Importance: Start-up Cost and 8.61
    Level Part-Worths
        "$100" --> -0.75
        "$200" --> -0.75
        "$300" --> -1.5
        "$400" --> 3.0

Attribute and Importance: Monthly Cost and 58.85
    Level Part-Worths
        "$100" --> -3.0
        "$200" --> -6.25
        "$300" --> -10.75
        "$400" --> 20.0

Attribute and Importance: 4G Service and 13.4
    Level Part-Worths
        "4G NO" --> 3.5
        "4G YES" --> -3.5

Attribute and Importance: Nearby retail store and 1.91
    Level Part-Worths
        "Retail NO" --> -0.5
        "Retail YES" --> 0.5

Attribute and Importance: Apple products sold and 1.91
    Level Part-Worths
        "Apple NO" --> -0.5
        "Apple YES" --> 0.5

Attribute and Importance: Samsung products sold and 8.61
    Level Part-Worths
        "Samsung NO" --> 2.25
        "Samsung YES" --> -2.25

Attribute and Importance: Google/Nexus products sold and 5.74
    Level Part-Worths
        "Nexus NO" --> 1.5
        "Nexus YES" --> -1.5
[['AT&T', -0.25], ['T-Mobile', -0.0], ['US Cellular', 0.25], ['Verizon', -0.0], ['$100', -0.75], ['$200', -0.75], ['$300', -1.5], ['$400', 3.0], ['$100', -3.0], ['$200', -6.25], ['$300', -10.75], ['$400', 20.0], ['4G NO', 3.5], ['4G YES', -3.5], ['Retail NO', -0.5], ['Retail YES', 0.5], ['Apple NO', -0.5], ['Apple YES', 0.5], ['Samsung NO', 2.25], ['Samsung YES', -2.25], ['Nexus NO', 1.5], ['Nexus YES', -1.5]]
/Users/marcotavora/miniconda3/lib/python3.6/site-packages/scipy/stats/stats.py:1394: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
  "anyway, n=%i" % int(n))

Results:

In [11]:
print('Summary of statistics:')
tup[0]
print('Utilities:')
tup[1]
print('Importances:')
tup[2]
Summary of statistics:
Out[11]:
OLS Regression Results
Dep. Variable: ranking R-squared: 0.999
Model: OLS Adj. R-squared: 0.989
Method: Least Squares F-statistic: 97.07
Date: Sun, 15 Jul 2018 Prob (F-statistic): 0.0794
Time: 00:45:34 Log-Likelihood: 10.568
No. Observations: 16 AIC: 8.864
Df Residuals: 1 BIC: 20.45
Df Model: 14
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 11.1250 0.484 22.980 0.028 4.974 17.276
brand_t-mobile -0.2500 0.354 -0.707 0.608 -4.742 4.242
brand_us_cellular -1.865e-14 0.354 -5.28e-14 1.000 -4.492 4.492
brand_verizon 0.2500 0.354 0.707 0.608 -4.242 4.742
startup_$200 -0.7500 0.354 -2.121 0.280 -5.242 3.742
startup_$300 -0.7500 0.354 -2.121 0.280 -5.242 3.742
startup_$400 -1.5000 0.354 -4.243 0.147 -5.992 2.992
monthly_$200 -3.0000 0.354 -8.485 0.075 -7.492 1.492
monthly_$300 -6.2500 0.354 -17.678 0.036 -10.742 -1.758
monthly_$400 -10.7500 0.354 -30.406 0.021 -15.242 -6.258
service_4g_yes 3.5000 0.250 14.000 0.045 0.323 6.677
retail_retail_yes -0.5000 0.250 -2.000 0.295 -3.677 2.677
apple_apple_yes -0.5000 0.250 -2.000 0.295 -3.677 2.677
samsung_samsung_yes 2.2500 0.250 9.000 0.070 -0.927 5.427
google_nexus_yes 1.5000 0.250 6.000 0.105 -1.677 4.677
Omnibus: 29.718 Durbin-Watson: 2.000
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2.667
Skew: -0.000 Prob(JB): 0.264
Kurtosis: 1.000 Cond. No. 9.06


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Utilities:
Out[11]:
attribute beta
0 brand_t-mobile 11.125
1 brand_us_cellular -0.250
2 brand_verizon -0.000
3 startup_$200 0.250
4 startup_$300 -0.750
5 startup_$400 -0.750
6 monthly_$200 -1.500
7 monthly_$300 -3.000
8 monthly_$400 -6.250
9 service_4g_yes -10.750
10 retail_retail_yes 3.500
11 apple_apple_yes -0.500
12 samsung_samsung_yes -0.500
13 google_nexus_yes 2.250
Importances:
Out[11]:
attribute importance
2 Monthly Cost 58.85
3 4G Service 13.40
1 Start-up Cost 8.61
6 Samsung products sold 8.61
7 Google/Nexus products sold 5.74
4 Nearby retail store 1.91
5 Apple products sold 1.91
0 Provider 0.96