Statistics for Environmental Engineers

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Comparing Four Laboratories with a Reference Laboratory


Laboratory


Control


1


3


4


5


Average


3.97


4.30


4.46


3.12


3.34


Difference (y, — yc)



0.33


0.49


-0.85


-0.63

confidence limits are:


yy ±255( 071 l¥w


-0.81 < y — yc < 0.81


We can say with 95% confidence that any observed difference greater than 0.81 or smaller than -0.81 is unlikely to be zero. The four comparisons with laboratory 2 shown in Table 20.5 indicate that the measurements from laboratory 4 are smaller than those of the control laboratory.

Comments


Box et al. (1978) describe yet another way of making multiple comparisons. The simple idea is that if к treatment averages had the same mean, they would appear to be к observations from the same, nearly normal distribution with standard deviation a/Jn. The plausibility of this outcome is examined graphically by constructing such a normal reference distribution and superimposing upon it a dot diagram of the к average values. The reference distribution is then moved along the horizontal axis to see if there is a way to locate it so that all the observed averages appear to be typical random values selected from it. This sliding reference distribution is a .rough method for making what are called multiple comparisons.” The Tukey and Dunnett methods are more formal ways of making these comparisons.


Dunnett (1955) discussed the allocation of observations between the control group and the other p = к — 1 treatment groups. For practical purposes, if the experimenter is working with a joint confidence level in the neighborhood of 95% or greater, then the experiment should be designed so that nc In = Jp approximately, where nc is the number of observations on the control and n is the number on each of the p noncontrol treatments. Thus, for an experiment that compares four treatments to a control, p = 4 and nc is approximately 2n.

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