Already 2010 is feeling like the year of optimization. Everywhere I look, I’m seeing conversations about A|B and MVT testing, optimizing conversion flows, and understanding statistical significance.
When I first started running A|B tests, everything I did was on faith. I had good intention, I measured all the key indicators, but I had no idea how to tackle the question of “yeah but, is it statistically significant?” Then I began to crawl as I experimented with online calculators and eventually I moved on to building out my own formulas in Excel but still there was little confidence in myself, let alone the test results.
Eventually I began to experiment with testing tools like Google Web Optimizer, Amadesa, and Omiture Test & Target. This seemed to make life so much simpler as all the questions I was being asked were answered right in the testing application. Is it significant? Amadesa says they are 98% confident in the results. What is the lift we are seeing? Google Web Optimizer says its 8.5% and as a bonus it gives the confidence interval.
While I think it is extremely valuable to have your testing and optimization platform provide the key statistical measures that relate to your test, I think it is just as important to understand the math behind the reports, after all, you can’t call yourself a “car guy” or a “car girl” if you drive on the gauges alone and you don’t understand how the underlying systems work.
Let’s walk through an example campaign to understand how Omniture Test & Target calculates the statistics behind the results.
For our campaign, lets assume the following facts:
- Our campaign has two treatments, a control and one alternative.
- The control has had 4,008 visitors
- The alternative has had 4,003 visitors
- The control has had 377 conversions
- The alternative has had 355 conversions
#1 – Conversion Rate
Conversion rate equals the number of conversions divided by the number of starts, in this example we are using visitors but this can be visits, impressions, unique starts, etc. depending on how you measure site conversion.
Conversion Rate (control) = 377 / 4,008 = 9.41%
Conversion Rate (alternative) = 355 / 4003 = 8.87%
#2 – Standard Deviation
Standard Deviation shows how much variation (measures the spread or dispersion of a set of data) there is from the “average” (mean). As conversion rate is a binomial distribution, either a visitor converts or does not convert, the binomial distribution for variance is used:
Variance (control) = .0941(1 – .0941) = 0.09
Variance (alternative) .0887(1 – .0887) = 0.08
To calculate Standard Deviation from the variance, we take the square root of the variance:
Standard Deviation (control) = SQRT(0.09) = 0.29
Standard Deviation (alternative) = SQRT(0.08) = 0.28
#3 – Standard Error
The Standard Error is the estimated Standard Deviation of the error; the “noise” in the result. The Standard Error is calculated in order to calculate to Signal-to-Noise ratio.
To calculate the Standard Error for the Control:
Standard Error (control) = SQRT(0.09 / 4008) = 0.005
To calculate the Standard Error for the alternative:
Standard Error (alternative) = SQRT((0.09 / 4008) + (0.08 / 4003)) = 0.006
# 4 – Signal-to-Noise Ratio
To calculate the Signal-to-Noise ratio:
Signal-to-Noise = (9.41 – 8.87) / 0.006 = 0.84
OK….stay with me…..we are almost there.
#5 Finally We Arrive At Confidence
We will make use of the Signal-to-Noise ratio to calculate confidence using the Student’s T-Test.
Student’s T-Test = 1 – TDIST(ABS(0.84),(4003 + 4008 -2),2) = 0.60
As reported by Test & Target, we are 60% confident in the current results.
Extra Credit: Confidence Intervals
The Confidence Interval shows how much your test results can vary and still be within a predetermined confidence level. Standard confidence levels are 90%, 95%, 99%, and 99.5%. Omniture Test & Target uses the 95% confidence level.
To calculate the Confidence Interval:
Confidence Interval = 1.96(0.28 / SQRT(4003)) = 0.008
1.96 is a constant in this formula. 1.96 is equal to z*, which is taken from a Standard Normal Critical Values table based on 95% Confidence Level. The Standard Normal Critical Values Table can be found in any introductory level statistics book.
The formulas in this post have been provided by Omniture consulting. The screenshots have been taken from Omniture Test & Target and have been modified for the purpose of this example.