"Page 151, third paragraph. The argument given in this paragraph still does not explain why the zone of equivalence is not symmetrical around 100%. It is true that the zones of equivalence are the same for the test/reference ratio and for the reference/test ratio (with inverted limits), but this fact (although convenient) does not explain why each of these two zones is asymmetrical. The zone is asymmetrical, because it is supposed to contain an asymmetrical confidence interval. The 90% CI is asymmetrical, because this is the CI of the geometric mean ratio (or ratio of geometric means) of peak drug concentrations (Cmax). (On page 95 you state: “Note that the CI of the geometric mean is not symmetrical around the geometric mean”.) The mean is geometric, rather than arithmetic, because the pharmacokinetic measures AUC and Cmax are statistically analyzed after log transformation, as recommended by the FDA. The primary outcome of the analysis is thus the 90% CI for the difference in the arithmetic means of the log transformed data, which is then backconverted to the 90% CI for the ratio of the geometric means. The 90% CI for the difference in the means of log transformed data is perfectly symmetrical (around the mean difference), so is the zone of equivalence if expressed as logarithms: log100+/ log1.25, if the logarithms to the base 10 are used and the ratio is expressed as percentage. The zone of equivalence for the ratio is also perfectly symmetrical but in the multiplicative (not only practical) sense: 100*1.25=125 for the upper limit, and 100/1.25=80 for the lower limit.
For this reason, Figures 21.2 & 21.3 are incorrect. Since the 80125% zone of equivalence applies to geometric means (see the previous comment), the CIs should be asymmetrical around the means. The figures, as printed, show symmetrical confidence intervals."
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They used stepwise logistic regression to develop a model to predict divorce in one group of subjects. They had lots of variables (and interactions) so were able to find a model that worked quite well. When predicting a couple would divorce, the equation was correct 65% of the time (positive predictive value). When predicting that a couple would not divorce, the equation was correct 98% of the time (negative predicted value). Sounds useful.
They then applied that same model to a new group of subjects. Not surprisingly, the predictions don't work very well when used with new data. When predicting a couple would divorce, the model was correct only 29% of the time. Not very impressive, considering that 33% of their subjects had divorced.
This makes an interesting data showing the dangers of overfitting, and the need to test models on new data sets.
Heyman, R.E, and SmithSlep, A.M. The Hazards of Predicting Divorce Without Crossvalidation. J Marriage Fam (2001) vol. 63 (2) pp. 473479
This same point is made, in plainer language, in an article in Slate.
]]>One consequence of heart failure is that the heart gets larger (cardiomegaly). This is a physiological adaptation to allow the heart to pump enough blood to perfuse the organs. Eventually the heart grows so large it is not able to pump blood efficiently. Therefore it makes sense to seek a drug that would reduce cardiomegaly in heart failure.
This study tested such a drug in an animal model of heart failure. Half the rats were given the surgery to create heart failure, and half were given sham surgery. In each of those groups, half of the rats were injected with an experimental drug and half were injected with vehicle as a control. (The data are real, but the investigators prefer to remain anonymous).

Sham surgery 
CHF 


Mean 
SD 
n 
Mean 
SD 
n 
Vehicle 
0.3124 
0.0211 
10 
0.5481 
0.0723 
9 
Drug 
0.3518 
0.0251 
10 
0.4669 
0.0768 
10 
Is the drug effective in blunting the increase in heart weight?
Think about how you would analyze the data before reading the casestudy. Better, download the raw data and do your own analyses.
When you have thought about how you would analyze the data, read this 15 page casestudy. Here is the corresponding Prism file. Please email me with your suggestions about this case study, or ideas for future case studies.
Here is my attempt at explaining the twelve key concepts in statistics.
]]>Reference: American Scientist, 97:310316, 2009