Data set: Skinfold7
Celiac disease results in an inability to absorb carbohydrates and fats. Crohn’s disease is another chronic intestinal disease in which the body’s immune system attacks the intestines. Both Crohn’s disease and celiac disease often result in malnutrition or impaired growth in children. A skinfold thickness measurement is a simple technique assessing body fat percentages by pinching the skin near the biceps and then using a calipers to measure the skin thickness.
a. Transform the original data into √Thickness, ln(Thickness), log10(Thickness), and the reciprocal (1/Thickness). For each transformation, conduct two-sample t-tests for differences between the mean of the Crohn’s disease and celiac disease groups. Create a table with p-values and confidence intervals for the difference between the two means.
b. Back transform the confidence intervals in Part A. For example, square the upper and lower bounds of the confidence interval created from the √Thickness, data. Conduct the appropriate back transformation on the other three confidence intervals as well. Create a table of the four back-transformed confidence intervals. What do these back-transformed confidence intervals tell you?
Confidence limits for the difference between means often cannot be transformed back to the original scale. When reciprocal transformations have very small bounds, the back transformation provides unreasonably large bounds. For example, a skinfold transformation of 1 / 0.022 = 45.5 mm is not realistic. In addition, if the lower bound of a confidence interval is negative when the square-root transformation is used, back transforming the results (by squaring the bounds) will result in a confidence interval that does not contain zero. Thus, there are not reasonable practical interpretations of the back-transformed scales.
The log (this includes the natural log and log10) is often preferable over other transformations because the back transformation has a practical interpretation. The log10 back-transformed confidence interval (0.89, 2.03) provides results that can be interpreted, but not in the original units (millimeters). Notice that the confidence interval does not contain zero, but the results are not significant. Recall from your introductory statistics class that if a two-sided confidence interval contains zero, we fail to reject the null hypothesis for the corresponding two-sided hypothesis test. This is a 95% confidence interval for the ratio of the means. Thus, a value of one represents no difference between group means.