Counting Calories

If the errors in reporting the number of calories in food were random and unbiased, how would you expect the number of overreported values to compare to the number of underreported values?

• Understand that introducing randomness in the sampling scheme means that things we measure will vary from sample to sample; but this variation will have a predictable pattern if we repeated the procedure over and over.
• Understand three basic techniques fundamental to well-designed experiments: randomization, replication, and blocking. Understand that the first is used to decrease bias and that the last two help to increase precision.

If the errors were random then we would expect about an equal number of overreported values and underreported values.

This data is reported in terms of percent difference in caloric content. The data could also have been reported with two variables: labeled calories and measured calories. Which method of reporting the data would you prefer and why?

• Understand what it means for a measurement to be valid, reliable, and unbiased. Understand that the validity of a measurement involves appropriate standardization of measurements.
• Understand the difference between validity, reliability, and bias in a measurement situation.
  

Having the labeled and measured number of calories is preferable because from these the percentages can be computed. If a certain food only claimed to have 1 calorie but actually had 2 calories then the difference will appear to be 100%, but another food which claimed to have 100 calories and really had 101 calories will register only a difference of 1%. Both of these are underreported by one calorie, but in some uses, such as dieting, the actual number of calories underreported may be more meaningful than the percentage difference.

To what population are the results of this study applicable? What steps might the researchers have taken to make the study more widely applicable?

• Understand that simple random samples give an unbiased picture of the sampling frame.
• Understand the potential for bias from a convenience sample.

A population to which this study extends is that of the diet and health food products purchased for this study. To extend the conclusions to a wider geographic area and to the whole population of diet foods, some sort of random sampling design should be used.

Examine the percent difference between measured and labeled calories per item variable. Describe the shape of the distribution of this variable. What is the most appropriate measure of a typical value from this distribution?

• Be able to make (using software where appropriate) and interpret the principle methods of displaying distributions (frequency tables, pie charts, line graphs, bar graphs, and histograms).
• Be able to assess important features of the shape of a distribution by looking at a histogram (single or multiple peaks, symmetric or skewed, the existence of outliers).

Here is a histogram of the number of foods with the indicated per item percent difference between the measured and labeled calories. The distribution is highly skewed, with a long right tail. The best measure of a typical value from this data is the median, because of the skewness.

a) Does there appear to be any evidence of a systematic pattern in the difference between measured and labeled values of calories per item?

b) Can the difference between the measured and labeled values be explained by chance variation in the measuring process? Carry out an appropriate test to examine this question.

 

• Be able to calculate (using software where appropriate) and interpret the principle measures of center, relative standing, and spread (rates, percentages, mean, median, mode, percentiles, lower quartile, upper quartile, minimum, maximum, standard score, the five-number summary, variance, and standard deviation).
• Be able to carry out the nonparametric tests for comparison of two samples such as the sign test, the Wilcoxon signed-rank test, and the Wilcoxon rank-sum test. Be able to implement the Kruskal-Wallis as a nonparametric alternative to the ANOVA.

 

a) Yes, there seems to be a tendency to underrepresent the calories. The median is greater than zero, which tells us that the measured values were generally greater than the labeled values.

 

b) An appropriate test is the one-sample sign test of

H₀: Median = 0 versus H_a: Median > 0.

From the data:
Total Observations: 40.
Observations of 'Per item' > 0: 30.
Ties: 0.

Since the P-value = 0.0011, we reject H₀. This confirms the conclusion of Part a: there is a tendency to underrepresent the caloric content of the products in this study.

Use graphical or numerical summaries to examine the difference in measured and labeled calories per item according to whether the item is nationally advertised, regionally distributed, or locally prepared. Summarize your findings.

• Be able to make and interpret a boxplot. Be able to use the IQR rule for identifying outliers in a boxplot.

 

These boxplots suggest that national products are not systematically underreporting caloric content, regional products are somewhat underreporting, and local products are most prone to underreporting. There is also an increase in the variability as the classification changes from local to regional and national.