In These Days of Fears of Avian Flu Pandemics We Are Given Much Advice

Experimental Question:

In these days of fears of avian flu pandemics we are given much advice. One important public health strategy is to wash our hands often. This naturally brings us to a question :

What method of washing hands is best?

The Experiment

32 people were randomly assigned to one of four hand washing groups. Each used a different method and, after washing placed one hand on a sterile media plate designed to encourage bacteria growth. After 2 days at 36º C the bacteria colonies were counted.

The Data

Hand washing method n Mean Std. Dev. Alcohol Spray (AS) 8 37.5 26.6 Antibacterial Soap (ABS) 8 92.5 42.0 Soap 8 106.0 47.0 Water 8 117.0 31.1

Preparation for analysis

Hand washing method n Mean Std. Dev. Variance SS Within Alcohol Spray (AS) 8 37.5 26.6 705 4,938 Antibacterial Soap (ABS) 8 92.5 42.0 1,761 12,324 Soap 8 106.0 47.0 2,205 15,437 Water 8 117.0 31.1 969 6784 88.0 1,410 39,483

The variance is an unbiased estimate and so to get Within SS we need to multiply by (n-1)

Between SS is easy – just square each effect, multiply them by n, and add ‘em up

Hand washing method n Mean Effects Effects ^2 x n Alcohol Spray (AS) 8 37.5 -51 2,575 20,604 Antibacterial Soap (ABS) 8 92.5 4 18 144 Soap 8 106 18 315 2,520 Water 8 117 29 827 6,612 88 29,882 ANOVA Summary Source SS DF MS F P-Value Between (Methods) 29,882 3 9,961 7.1 0.001 Within (error) 39,483 28 1,410 Total 69,365 31

So now we know that all methods of washing hands are not equal, but how big a difference between individual methods is statistically significant?

We can summarize all of the differences in a single table of differences

AS ABS Soap Water 37.5 92.5 106 117 Alcohol Spray (AS) 37.5 0 Antibacterial Soap (ABS) 92.5 -55 0 Soap 106 -68.5 -13.5 0 Water 117 -79.5 -24.5 -11 0

How big a difference is significant? Let us consider “water” as the control condition and the other three as alternative treatments.

One null hypothesis might be:

Ho: water –  soap = 0. The alternative is

H1: water –  soap ≠ 0.

A natural test statistic is ywater – y soap, and the (pooled) standard error is:

SE(water –  soap) = sp 1/nwater + 1/nsoap

1/2 sp = 37.6 and so SE = 37.6 x (1/8 + 1/8) = 37.6 x (1/4)1/2 = 37.6 x (1/2) = 18.8 Of course with equal ns all of the standard errors are the same. Now we can convert the previous table of differences into a table of t-statistics by dividing by the se of the difference.

AS ABS Soap 37.5 92.5 106 ABS 92.5 -2.93 Soap 106 -3.64 -0.72 Water 117 -4.23 -1.30 -0.59

How big is significant? These are t statistics with 28 degrees-of-freedom which requires a value greater than 2.05 to be significant at the .05 level.

But we are not doing just one test. We might be doing three tests (each treatment vs. water). Or we might be doing six tests (each treatment against all others). A way to start is the method of Least Significant Difference (or LSD for short)

½ The LSD = t x sp (1/n1+1/n2) In this case we have 2.05 x 37.6 x ½ = 38.4

So we could look at any one of the 6 comparisons and determine if it was significant.

AS ABS Soap Water 37.5 92.5 106 117 Alcohol Spray (AS) 37.5 0 Antibacterial Soap (ABS) 92.5 -55.0 0 Soap 106 -68.5 -13.5 0 Water 117 -79.5 -24.5 -11 0

But if we want to make more than a single comparison we need to control the overall Type I error rate (a). To do this we can use the Bonferroni inequality in which we set the individual test critical value (ai) based on the overall protection level (a) and the number of tests we will be doing (J).

In short the Bonferroni inequality sets

ai = a J

So if we are doing 6 tests and we want the overall a to be .05 we set ai to be .05/6 = .009 Using the t-table for a 2-tailed test at this level means that the standardized difference must be greater than 2.76 to be significant at the .05 level.

Translated back into the original score metric ½ The LSD = t x sp (1/n1+1/n2) In this case we have 2.76 x 37.6 x ½ = 51.9 We can then conclude that Alcohol Spray is better than everything else, but that none of the other differences are large enough for us to reject the hypothesis that their observed variation is anything more than chance.