Conduct Web experiments using PHP, Part 2

By Paul Meagher
2005-03-18

Explorer output

Tables 6 and 7 and Figures 3 and 4 are screens that were generated by pointing my browser at the doe_explorer.php script. The simulated data is meant to represent realistic response data for 2,000 site visitors (or 500 simulated visitors per factor-level combination). If you refresh your browser on this script, you will observe different outcomes; it was by using this primitive method that I estimated how many subjects might be required to detect the relatively small effects (expressed as differences in the lambda parameter) one might expect to observe in a proposed Web offer study.

You could further develop this monte carlo procedure to obtain more rigorous estimates of the sample size required to reliably detect differences (in 95 percent of runs) given different estimates of the effect size (lambda value differences).

In the doe_explorer.php script, the $this->showContingencyTable() method is used to display simulated contingency table data:
Table 6. Output of the showContingencyTable() method

		text
		Short	Long	sum
image	Person	24	12	36
image	Product	6	13	19
	sum	30	25	55

The $this->showResiduals() method displays the difference between observed and expected counts (see Table 7). Note that the overall chi-square score is 6.18 and that the probability of this outcome under the independence model is 0.0130, which tells you that your test detected the interaction effect you specified. If you lower the $trials parameter value from 500 to 100, your ability to detect a specified interaction is diminished (which is one reason I chose a larger value of 500).
Table 7. Output of the showResiduals() method

Cell	O_ij	E_ij	(O_ij - E_ij)	(O_ij - E_ij)²	(O_ij - E_ij)² / E_ij
person-short	24	19.64	4.36	19.04	0.97
person-long	12	16.36	-4.36	19.04	1.16
product-short	6	10.36	-4.36	19.04	1.84
product-long	13	8.64	4.36	19.04	2.20
Sums	55	55	0.00	76.17	6.18
P(6.18, 1) = 0.0130
Critical X² for alpha=0.05 is 3.84

The $this->showBarGraph() method displays your simulated contingency table data in bar graph format:

Figure 3. Output of showBarGraph() method

The $this->showLineGraph() method displays your simulated data in line graph format:

Figure 4. Output of showLineGraph() method

Note that the line graph rendering makes it particularly easy to see that a cross-over interaction is present. You should also consider visualizing your results using grouped bar charts which is a graph type that JPGraph also supports.

Tutorial pages:

First published by IBM developerWorks

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