Ratio metrics and the delta method

A/B Testing in Python

Moe Lotfy, PhD

Principal Data Science Manager

Ratio metrics A/B testing

Mean metrics

Example of mean metrics: mean order value and mean time on page

Unit of analysis:
- The entity being analyzed in an A/B test
- Denominator in ratio metrics
Randomization unit:
- The subject randomly allocated to each variant

Ratio metrics A/B testing

Per-user Ratio metrics

Example of ratio metrics: click-through-rate and revenue per session

Delta method motivation

print(checkout.groupby('checkout_page')[['order_value','purchased']].agg({'sum','count','mean'}))

              order_value                     purchased              
                     mean           sum count      mean     sum count
checkout_page                                                        
A               24.956437  61417.791564  2461  0.820333  2461.0  3000
B               29.876202  75915.430125  2541  0.847000  2541.0  3000
C               34.917589  90890.484142  2603  0.867667  2603.0  3000

checkout.groupby('checkout_page')['order_value'].sum()/
checkout.groupby('checkout_page')['purchased'].count()

checkout_page
A    20.472597
B    25.305143
C    30.296828
dtype: float64

Delta method variance

Delta method ratio metrics variance estimation:{

Delta method variance formula

# Delta method variance of ratio metric
def var_delta(x,y): 
    x_bar = np.mean(x)
    y_bar = np.mean(y)
    x_var = np.var(x,ddof=1)
    y_var = np.var(y,ddof=1)
    cov_xy = np.cov(x,y,ddof=1)[0][1]
    # Note that we divide by len(x) here because the denominator of the test statistic is standard error (=sqrt(var/n))
    var_ratio = (x_var/y_bar**2 + y_var*(x_bar**2/y_bar**4) - 2*cov_xy*(x_bar/y_bar**3))/len(x)
    return var_ratio

¹ Budylin, Roman & Drutsa, Alexey & Katsev, Ilya & Tsoy, Valeriya. (2018). Consistent Transformation of Ratio Metrics for Efficient Online Controlled Experiments. 55-63. 10.1145/3159652.3159699.

Delta method z-test

# Delta method ztest calculation 
ztest_delta(x_control,y_control,x_treatment,y_treatment, alpha = 0.05)

Input arguments:

x_control: control variant user-level ratio numerator column
y_control: control variant user-level ratio denominator column
x_treatment: treatment variant user-level ratio numerator column
y_treatment: treatment variant user-level ratio denominator column
alpha: significance level.

Output:

mean_control: control group ratio metric mean
mean_treatment: treatment group ratio metric mean
difference: difference between treatment and control means
diff_CI: confidence interval of the difference in means
p-value: the two-tailed z-test p-value

¹ https://medium.com/@ahmadnuraziz3/applying-delta-method-for-a-b-tests-analysis-8b1d13411c22

Python example

# Create DataFrames for per user metrics for variants A and B
A_per_user = pd.DataFrame({'order_value':checkout[checkout['checkout_page']=='A'].groupby('user_id')['order_value'].sum()
                            ,'page_view':checkout[checkout['checkout_page']=='A'].groupby('user_id')['user_id'].count()})
B_per_user = pd.DataFrame({'order_value':checkout[checkout['checkout_page']=='B'].groupby('user_id')['order_value'].sum()
                           ,'page_view':checkout[checkout['checkout_page']=='B'].groupby('user_id')['user_id'].count()})

# Assign the control and treatment ratio columns
x_control = A_per_user['order_value']
y_control = A_per_user['page_view']
x_treatment = B_per_user['order_value']
y_treatment = B_per_user['page_view']

# Run a z-test for ratio metrics
ztest_delta(x_control,y_control,x_treatment,y_treatment)

{'mean_control': 20.472597188012,
 'mean_treatment': 25.30514337484097,
 'difference': 4.833,
 'diff_CI': '[4.257, 5.408]',
 'p-value': 5.954978880467735e-61}

Let's practice!

A/B Testing in Python