Densities For Two Proportions: Statistics: A Bayesian Perspective: Chapter 9

The best way to explain this chapter is to outline one of the examples.

In this example, a researcher tests the effectiveness of semi-personal letters against form letters. The results from a test were:

-          Sent 1,018 semi-personal letters, 325 responses.

-          Sent 1,022 form letters, 225 responses

How much of an improvement in response is gained by using semi-personal letters?

Consider two population models: semi-personal (1) and form letters (2)

Prior density for each population is beta (1,1).

Therefore, the posterior density is:

Semi-personal: beta (326, 694)

Form letter: beta (226, 798)

We can calculate the following:

	Semi-Person	Form
r	0.3196	0.2207
r+	0.3203	0.2215
t  (std deviation)	0.01459	0.01295

We then calculate the z-score to obtain the probability that difference d = p1-p2 is less than zero:

Z = ( 0 – (r1 – r2)) / √ (t2 + t2)       = -5.07

The probability for this z-score is p < 0.0001, therefore the probability of that difference being at least zero = 1

Similiarly, PdAL (probabilities that d is at least as big as) 0.05:

Z = ( 0.05  – (r1 – r2)) / √ (t2 + t2)       = -2.506  à therefore PdAL0.05 = 0.994

Z = ( 0.10  – (r1 – r2)) / √ (t2 + t2)       = 0.056  à therefore PdAL0.10 = 0.52

In other words, the probability of additional response rate of 5% using semi-personal letters as opposed to form letters is 99%.

The z-score can also be used to obtain probability intervals for the difference :

r1 – r2 + / -  z *  √ (t2 + t2).

If the probability interval for d contains zero, then the null hypothesis is supported.

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I’m still thinking about the differences between Frequentist and Bayesian statistics:

Bayesian statistics probably provides more information than Frequentist statistics:-

Bayesian

-          Probability / credible interval

-          Null hypothesis testing (using probability interval)

-          Probability that difference between treatment & control success proportions is at least x

-          Probability of next success observation

Frequentist

-          Significant / p-value / null hypothesis testing

-          Effect size